international journal of acoustics and vibration...editor’s space the twenty-first international...

EDITORIAL OFFICE

EDITOR-IN-CHIEFMalcolm J. Crocker

MANAGING EDITORMarek Pawelczyk

ASSOCIATE EDITORSDariusz BismorNickolay IvanovZhuang Li

ASSISTANT EDITORSTeresa GlowkaSebastian Kurczyk

EDITORIAL ASSISTANTAubrey Wood

EDITORIAL BOARD

Jorge P. ArenasValdivia, Chile

Jonathan D. BlotterProvo, USA

Leonid GelmanCranfield, UK

Samir GergesFlorianopolis, Brazil

Victor T. GrinchenkoKiev, Ukraine

Colin H. HansenAdelaide, Australia

Hanno HellerBraunschweig, Germany

Hugh HuntCambridge, England

Dan MarghituAuburn, USA

Manohar Lal MunjalBangalore, India

David E. NewlandCambridge, England

Kazuhide OhtaFukuoka, Japan

Goran PavicVilleurbanne, France

Subhash SinhaAuburn, USA

International Journal ofAcoustics and Vibration

A quarterly publication of the International Institute of Acoustics and Vibration

Volume 19, Number 2, June 2014

EDITORIAL

The Twenty-First International Congress on Sound and VibrationJing Tian and Malcolm J. Crocker . . . . . . . . . . . . . . . . . 70

ARTICLES

Elastodynamics for Non-linear Seismic Wave Motion in Real-TimeExpert Seismology

Evangelos G. Ladopoulos . . . . . . . . . . . . . . . . . . . . 71

3D Analysis of the Sound Reduction Provided by ProtectiveSurfaces Around a Noise Source

Luıs Godinho, Edmundo G. A. Costa, Jose A. F. Santiago,Andreia Pereira and Paulo Amado-Mendes . . . . . . . . . . . 78

Study on Linear Vibration Model of Shield TBM CutterheadDriving System

Xianhong Li, Haibin Yu, Peng Zeng, Mingzhe Yuan, JiandaHan and Lanxiang Sun . . . . . . . . . . . . . . . . . . . . . 89

An Algorithm for Solving Torsional Vibration Problems Based onthe Invariant Imbedding Method

Antonio Lopes Gama and Rafael Soares de Oliveira . . . . . . . 107

Free Flexural Vibration Response of Integrally-Stiffened and/orStepped-Thickness Composite Plates or Panels

Jaber Javanshir, Touraj Farsadi and Umur Yuceoglu . . . . . . 114

Vibrational Power Flow Analysis of a Cylindrical Shell Using aFour-Point Technique

H. Salimi-Mofrad, S. Ziaei-Rad and M. Moradi . . . . . . . . . 127

Non-Linear Thickness Variation on the Thermally-InducedVibration of a Rectangular Plate: A Spline Technique

Arun Kumar Gupta and Mamta . . . . . . . . . . . . . . . . . 131

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . 137

INFORMATION

Book Reviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Editor’s Space

The Twenty-First International Congress

on Sound and Vibration

On behalf of the Acoustical Society of China (ASC) andthe Institute of Acoustics, Chinese Academy of Sciences (IA-CAS), it is our great pleasure to invite you to participate in the21st International Congress on Sound and Vibration (ICSV21)held from 13 to 17 July 2014 at the China National ConventionCenter (CNCC), Beijing, China. The Congress is sponsored bythe International Institute of Acoustics and Vibration (IIAV).ICSV21 will be a great event in the annual series of congresseson acoustics and vibration, and a fruitful opportunity to pro-mote scientific research and technological development in var-ious respects. The theme of ICSV21 is “In-Depth Sound andVibration Research,” by which we want to stress the physicalaspects of the mechanisms of sound and vibration.

The ICSV21 Scientific Programme consists of six plenarykeynote lectures, 58 structured sessions chaired by well knownscientists in special areas of interest in acoustics and vibra-tion, and 57 regular sessions. More than 950 abstracts in thefields of acoustics and vibration from 51 countries and regionshave been accepted for presentation. About 400 submissionsare from China, nearly 300 from Europe and the others arefrom Japan, South Korea, United States, Australia, Singapore,Malaysia, Brazil, Mexico, etc. All abstracts were reviewed bythe organizers of structured sessions and by the members of thescientific committee, respectively. A total of 126 refereed fullpapers have been peer reviewed by more than two scientists foreach paper.

Distinguished plenary lectures will be given by six out-standing scientists, including “Acoustics of Ancient ChineseChimes”, by Jie Pan from Perth, Australia; “Aircraft Noise:an Industry Perspective at COMAC”, by Cyrille Breard fromShanghai, China; “Railway Noise and Vibration: the Use ofAppropriate, Models to Solve Practical Problems”, by DavidThompson from Southampton, UK; “Parametric Acoustic Ar-rays in Air: from Theory to Applications”, by Jun Yang fromBeijing, China; “Understanding Their Interaction and Contri-bution to Vehicle Noise and Fuel Consumption”, by Ines LopezArteaga from Stockholm, Sweden; and “Vehicle NVH Designand Technologies”, by Shaobo Young from Chongqing, China.

There are 33 exhibitors and sponsors which have taken 37booths, including Sound PLAN, Norsonic, G.R.A.S., RION,

Acustekpro, LMS, BSW, Changrong, ESI, SINUS, AN-TYSOUND, Cirrus Research, Vibration Research, HangzhouAihua Instruments Co.,Ltd, Beijing AcousticSensing Co.,Ltd,PCB Piezotronics, Inc., m+p international, B&K, MSC Soft-ware Corporation, BSWA, Polytec GmbH, Nti China Co. Ltd,WILSON, ODEON, DataKustik, Microflown, Beijing Dean-well Technology Co., Crystal Instruments (CI), COMSOL,ACOEM, and One Measurement Group Limited.

Beijing is a historic city, and also a modern metropolis. It isthe cultural centre and the administrative capital of China. Itshistory dates back three millennia. As the last of the four greatancient capitals of China, Beijing has been the political centreof the country in much of the past seven centuries. The cityis renowned for its opulent palaces, temples, gardens, tombs,great walls and gates, and also the hundreds of famous uni-versities, research institutes, and professional organizations.In the city, you will find all the contrasts of modern China.Teeming neighborhoods and birch trees are woven together byelevated highways and modern skyscrapers. At ICSV21 youcan take part in well-organized social and technical activities,and pre- and post-congress tours. You will be able to observenot only traditional Chinese culture, but also the ever-changingworld of modern China.

Jing Tian Malcolm J. CrockerGeneral Chair of ICSV21 Executive Director, IIAV

70 International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014

Elastodynamics for Non-linear Seismic WaveMotion in Real-Time Expert SeismologyEvangelos G. LadopoulosInterpaper Research Organization, 8, Dimaki Str., Athens, GR - 106 72, Greece

(Received 15 May 2012; provisionally accepted 9 February 2013; accepted 19 May 2013)

Based on the new theory of Real-Time Expert Seismology, a non-linear 3-D elastic wave real-time expert systemis used for the exploration globa on-shore and off-shore petroleum and gas reserves. This highly innovative andground-breaking technology uses elastic (seismic) waves moving in an unbounded subsurface medium to searchingthe on-shore and off-shore hydrocarbon reserves developed on the continental crust and in deeper water rangingfrom 300 to 3000+ m. This modern technology can be used in any depth of sea, in oceans all over the world, andfor any depth in the subsurface of existing oil and gas reserves.

Furthermore, the various mechanical properties of the rock regulating the wave propagation phenomenon ap-pear as spatially-varying coefficients in a system of time-dependent hyperbolic partial differential equations. Thepropagation of the seismic waves through the earth subsurface is described by the wave equation, which is finallyreduced to a Helmholz differential equation. Then the Helmholtz differential equation is numerically evaluated byusing the Singular Integral Operators Method (SIOM). Also, several properties are analysed and investigated forthe wave equation.

Finally, an application is proposed for the determination of the seismic field radiated from a pulsating sphere intoan infinite homogeneous medium. The acoustic pressure radiated from the above pulsating sphere is determinedby the SIOM.

1. INTRODUCTION

The new technology Non-Linear Real-Time Expert Seismol-ogy is the main and best tool which can be used by the oil andgas industry to map petroleum deposits in the Earth’s uppercrust. Environmental and civil engineers can also use variantsof the above modern technique to locate bedrock, aquifers, andother near-surface features. Academic geophysicists can ex-tend it into a tool for imaging the lower crust and mantle. Thismethod was proposed and investigated by Ladopoulos26–31 asan extension of his methods on non-linear singular integralequations in fluid mechanics, potential flows, structural analy-sis, solid mechanics, hydraulics, and aerodynamics.16–25

Seismic wave propagation is the physical phenomenon un-derlying the Non-Linear Real-Time Expert Seismology, aswell as other types of seismology. It is modelled with rea-sonable accuracy as small-amplitude displacement of a con-tinuum, using various specializations and generalizations oflinear elastodynamics. In these models, the various mechan-ical properties of rock regulating the wave propagation phe-nomenon appear as spatially-varying coefficients in a systemof time-dependent, hyperbolic, partial differential equations.

The Non-Linear Real-Time Expert Seismology seeks to ex-tract maps of the Earth’s sedimentary crust from transient,near-surface recordings of echoes stimulated by explosions orother controlled sound sources positioned near the surface.Reasonably accurate models of seismic energy propagationtake the form of hyperbolic systems of partial differential equa-tions, in which the coefficients represent the spatial distribu-tion of various mechanical characteristics of rock, like den-sity, stiffness, etc. The exploration geophysics community hasdeveloped various methods for estimating the Earth’s struc-ture from seismic data; however the modern technology Non-Linear Real-Time Expert Seismology seems to be the best toolfor on-shore and off-shore petroleum and gas exploration forvery deep drillings, ranging up to 20,000 or 30,000 m.

Over the past years, several variants of the integral equa-tions methods were used for the solution of elastodynamic andacoustic problems. It was already at the end of the 1960swhen H.A. Shenk1 stated that the integral equation for po-tential mathematically failed to yield unique solutions to theexterior acoustic problem. A method was proposed in whichan over-determined system of equations at some characteristicfrequencies was formed by combining the surface Helmholtzequation with the corresponding interior Helmholtz equation.It was analytically proved that the system of equations wouldprovide a unique solution at the same characteristic frequen-cies, to some extent. However, the above method might fail toproduce unique solutions, when the interior points used in thecollocation of the boundary integral equations are located on anodal surface of an interior standing wave.

Furthermore, at the beginning of the 1970s Burton andMiller2 proposed a combination of the surface Helmholtz in-tegral equation for potential and the integral equation for thenormal derivative of potential at the surface, in order to circum-vent the problem of nonuniqueness at characteristic frequen-cies. Their method was named the Composite Helmholtz Inte-gral Equation Method. Some years later, Meyer, Bell, Zinn andStallybrass,3 as well as Terai,4 developed regularization tech-niques for planar elements for the calculation of sound fieldsaround three dimensional objects by integral equation meth-ods.

On the other hand, Reut5 further investigated the Compos-ite Helmholtz Integral Equation Method by introducing thehypersingular integrals. Furthermore, in the above numeri-cal method, the accuracy of the integrations affects the re-sults and the conventional Gauss quadrature cannot be useddirectly. Okada, Rajiyah, and Alturi,6 as well as Okada andAlturi9 introduced the basic idea of using the gradients of thefundamental solution to the Helmholtz differential equationfor velocity potential as vector test functions. This could becompleted to write the weak form of the original Helmholtz

International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014 (pp. 71–77) 71

E. G. Ladopoulos: ELASTODYNAMICS FOR NON-LINEAR SEISMIC WAVE MOTION IN REAL-TIME EXPERT SEISMOLOGY

differential equation for potential and so directly derive non-hypersingular boundary integral equations for velocity poten-tial gradients. They used the displacement and velocity gra-dients to directly establish the numerically-tractable displace-ment and displacement gradient boundary integral equationsin elasto-plastic solid problems and traction boundary integralequations. Chien, Rajiyah and Atluri7 employed some knownidentities of the fundamental solution from the associated inte-rior Laplace problem, to regularize the hypersingular integrals.

Beyond the above, Wu, Seybert and Wan8 proposed theregularized normal derivative equation, to be converged inthe Cauchy principal value sense. The computation of tan-gential derivatives was required everywhere on the boundary.Hwang10 then reduced the singularity of the Helmholtz integralequation by using some identities from the associated Laplaceequation. On the other hand, the value of the equipotentialinside the domain must be computed, because the source dis-tribution for the equipotential surface from the potential theorywas used to regularize the weak singularities.

The identities of the fundamental solution of the Laplaceproblem were used further by Yang11 to efficiently solve theproblem of acoustic scattering from a rigid body. Furthermore,Yan, Hung and Zheng,12 in order to solve the intensive com-putation of double surface integral, employed the concept of adiscretized operator matrix to replace the evaluation of doublesurface integral with the evaluation of two discretized operatormatrices.

Han and Atluri13 used further traction boundary integralequations for the solution of the Helmholtz equation. Atluri,Han and S. Shen14 recently used the meshless method as an al-ternative numerical method to eliminate the drawbacks in theFinite Element Method and the Boundary Element Method.

Griffa, et al.15 studied the deformation characteristicsof a sheared granular layer during stick-slip from a meso-mechanical viewpoint, both in the absence and in the presenceof externally-applied vibration. They performed a 2D DiscreteElement Method (DEM) simulation of stick-slip dynamics fora granular layer confined and sheared by thick, elastically-deformable blocks.

In the present research, the Singular Integral OperatorsMethod (SIOM) will be used for the solution of elastodynamicproblems using the Helmholtz differential equation. In thisderivation, the gradients of the fundamental solution to theHelmholtz differential equation for the velocity potential willbe used. Also, several basic identities governing the funda-mental solution to the Helmholtz differential equation for thevelocity potential are analysed and investigated.

Thus, by using the SIOM, the acoustic velocity potential willbe computed. Moreover, some properties of the wave equation,which is a Helmholtz differential equation, are proposed andinvestigated. Also, some basic properties of the fundamentalsolution will be derived.

Finally, an application is proposed for the determination ofthe seismic field radiated from a pulsating sphere into an infi-nite homogeneous medium. By using the SIOM, the acousticpressure radiated from the above pulsating sphere will be com-puted. This is very important in petroleum reservoir engineer-ing in order to evaluate the size of the reservoir.

Consequently, the SIOM which was used with great suc-cess for the solution of several engineering problems of fluidmechanics, hydraulics, aerodynamics, solid mechanics, poten-tial flows, and structural analysis, are further extended in thepresent investigation for the solution of oil reservoir engineer-ing problems in elastodynamics.

2. NON-LINEAR SEISMIC WAVE MOTION INELASTODYNAMICS

Generally, seismic wavelengths run in the tens of meters,so it is reasonable to presume that the mechanical propertiesof rocks responsible for seismic wave motion might be locallyhomogeneous on length scales of millimetres or less, whichmeans that the Earth might be modelled as a mechanical con-tinuum. Except possibly for a few metres around the sourcelocation, the wave field produced in seismic reflection experi-ments does not appear to result in extended damage or defor-mation, so the waves are entirely transient. These considera-tions suggest a non-linear wave motion as a mechanical modelin elastodynamics.

The equations of elastodynamics in homogeneous media aregiven by:

ρ∂v

∂t= ∇·σ + b; (1)

and∂σ

∂t=

1

2C(∇v +∇vT

)+ γ; (2)

where v denotes the particle velocity field, σ the stress tensor,b a body force density, γ a defect in the elastic constitutive law,ρ the mass density, t the time and C the Hooke’s tensor.Furthermore, the right hand sides b and γ provide a variety ofrepresentations for external energy input to the system.

The new technique for on-shore and off-shore petroleumand gas reserves exploration. Non-linear Real-Time ExpertSeismology, uses transient energy sources to produce transientwave fields. Therefore, the appropriate initial conditions forthe system of Eqs. (1) and (2) are

v = 0 and, σ = 0, for t 0. (3)

For isotropic elasticity, the Hooke’s tensor has only twoindependent parameters, the compressional and shear wavespeeds cp and cs. It is instructive to examine direct measure-ments of these quantities, made in a borehole. Therefore, thereare two types of elastic waves produced: 1.) P-waves, whichare primary or compressional waves, and 2.) S-waves, or shearwaves.

In the current research, the seismic problem will be not de-veloped in the generalized context of the elastodynamic systemseen in Eqs. (1) and (2). Instead, our research will be limited toa special case of seismology. Thus, in this present model, it issupposed that the material does not support shear stress. Thestress tensor becomes scalar, σ = −pI with, p representingpressure, and only one significant component, the bulk modu-lus κ, is left in the Hooke tensor.

Then the system (1) and (2) reduces to:

ρ∂v

∂t= −∇p+ b; (4)

and1

κ

∂p

∂κ= −∇ · v + h; (5)

where the energy source is represented as a constitutive lawdefect h.

The proposed model predicts wave motion c with spatially-varying wave speed as

c =

√κ

ρ; (6)



with ρ the mass density and κ the bulk modulus.Furthermore, it is very convenient to represent the elastody-

namics in terms of the acoustic velocity potential: u(x, t) =t∫−∞

p(x, s)ds, which results in

p =∂u

∂t; (7)

andv =

1

ρ∇u. (8)

By using Eqs. (6) and (8), the acoustic system representedby Eqs. (4) and (5) reduces to the wave equation, because ofthe propagation of seismic waves through an unbounded ho-mogeneous solid; it is represented as

1

ρc2∂2u

∂t2−∇ · 1

ρ∇u = h. (9)

Furthermore, by assuming that density ρ is constant and thatthe source (transient constitutive law defect h) is an isotropicpoint radiator located at the source point, then the wave Eq. (9)reduces to the following Helmholtz differential equation as

1

c2∂2u

∂t2−∇2u = 0. (10)

For time harmonic waves with a time factor e−iωt, then thewave Eq. (10) reduces to:

∇2u+ k2u = 0; (11)

where the wave number k is equal to:

k =ω

c; (12)

with ω representing the angular frequency and c representingthe speed of sound in the medium at the equilibrium state.

The fundamental solution of the wave Eq. (1) at any fieldpoint y due to a point sound source x, for the two dimensionsis given by the formula

u∗(x,y) =i

4H

(1)0 (kr); (13)

and∂u∗

∂r(x,y) = − i

4kH

(1)1 (kr); (14)

where i =√−1, H(0)

1 (kr) denotes the Hankel function of thefirst kind and r is the distance between the field point y andthe source point x (r = |x− y|).

Furthermore, the fundamental solution of the wave Eq. (1)for the three dimensions is given as

u∗(x,y) =1

4πre−ikr (15)

and∂u∗

∂r(x,y) =

e−ikr

4πr2(−ikr − 1) (16)

The fundamental solution u∗(x,y) is further governed bythe wave equation:

∇2u∗(x,y) + k2u∗(x,y) + ∆(x,y) = 0. (17)

Thus, Eq. (17) is referred as the Helmholtz potential equa-tion governing the fundamental solution.

Beyond the above, consider the weak form of the Helmholtzequation to be given by∫

Ω

(∇2u+ k2u

)u∗dΩ = 0; (18)

in the solution domain Ω.By applying further the divergence theorem once in Eq. (18),

we obtain a symmetric weak form:∫∂Ω

niu,iu∗dS −

∫Ω

u,iu∗,idΩ−

∫Ω

k2uu∗dΩ = 0; (19)

where n denotes the outward normal vector of the surface S.Consequently, in the symmetric weak form the function u

and the fundamental solution u∗ are only required to be first-order differentiable. Furthermore, by applying the divergencetheorem twice in Eq. (18) we then have∫∂Ω

niu,iu∗dS −

∫∂Ω

niuu∗,idS +

∫Ω

u(u∗,ii + k2u∗

)dΩ = 0;

(20)Therefore, Eq. (20) is the asymmetric weak form and the

fundamental solution u∗ is required to be second-order differ-entiable. Furthermore, u is not required to be differentiable inthe domain Ω.

By combining further Eqs. (17) and (20), we then have

u(x) =

∫∂Ω

ni(y)u,i(y)u∗(x,y)dS

−∫∂Ω

ni(y)u(y)u∗,i(x,y)dS; (21)

which can be further written as

u(x) =

∫∂Ω

q(y)u∗(x,y)dS −∫∂Ω

u(y)R∗(x,y)dS; (22)

where q(y) denotes the potential gradient along the outwardnormal direction of the boundary surface as

q(y) =∂u(y)

∂ny= nk(y)u,k(y), y ∈ ∂Ω; (23)

and the kernel function is

R∗(x,y) =∂u∗(x,y)

∂ny= nk(y)u∗,k(x,y), y ∈ ∂Ω. (24)

By differentiating Eq. (22) with respect to xk, one can obtainthe integral equation for potential gradients u,k(x) by

∂u(x)

∂xk=

∫∂Ω

q(y)∂u∗(x,y)

∂xkdS −

∫∂Ω

u(y)∂R∗(x,y)

∂xkdS.

(25)

International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014 73


3. MATHEMATICAL PROPERTIES OF THEFUNDAMENTAL SOLUTION

The weak form of Eq. (6) governing the fundamental solu-tion, can be rewritten as∫

Ω

[∇2u∗(x,y) + k2u∗(x,y)

]cdΩ + c = 0, x ∈ Ω; (26)

where c denotes a constant, considering as the test function.Eq. (26) can be further written as∫

Ω

[u∗,ii(x,y) + k2u∗(x,y)

]dΩ + 1 = 0, x ∈ Ω. (27)

Beyond the above, Eq. (27) takes the following form as∫∂ω

ni(y)u∗,i(x,y)dS +

∫Ω

k2u∗(x,y)dΩ + 1 = 0, x ∈ Ω.

(28)By considering further an arbitrary function u(x) in Ω as the

test function, then the weak form of Eq. (6) may be written as∫Ω

[∇2u∗(x,y) + k2u∗(x,y) + ∆(x,y)

]u(x)dΩ =

= 0, x ∈ Ω; (29)

and further as:∫Ω

[u∗,ii(x,y) + k2u∗(x,y)

]u(x)dΩ + u(x) = 0, x ∈ Ω;

(30)Finally, Eq. (30) can be written as∫∂Ω

Φ∗(x,y)u(x)dS

+

∫Ω

k2u∗(x,y)u(x)dΩ + u(x) = 0, x ∈ Ω; (31)

Beyond the above, if x approaches the smooth boundary(x ∈ ∂Ω), then the first term in Eq. (31) may be written as

limx→∂Ω

∫∂Ω

Φ∗(x,y)u(x)dS =

CPV∫∂Ω

Φ∗(x,y)u(x)dS − 1

2u(x);

(32)in the sense of a Cauchy Principal Value (CPV) integral.

For the understanding of the physical meaning of Eq. (32),Eqs. (28) and (31) can be written as

CPV∫∂Ω

Φ∗(x,y)dS+

∫Ω

k2u∗(x,y)dΩ+1

2= 0, x ∈ ∂Ω; (33)

and

CPV∫∂Ω

Φ∗(x,y)u(x)dS +

∫Ω

k2u∗(x,y)u(x)dΩ+

+1

2u(x) = 0, x ∈ ∂Ω; (34)

From Eq. (33) it can be seen that only a half of the sourcefunction at point x is applied to the domain Ω, when the pointx approaches a smooth boundary, x ∈ ∂Ω.

Consider further another weak form of Eq. (31) by suppos-ing the vector functions to be the gradients of an arbitrary func-tion u(y) in Ω, chosen in such a way that they have constantvalues:

u,k(y) = u,k(x), for k = 1, 2, 3. (35)

Consequently, the weak form of Eq. (31) may be written as∫Ω

[u∗,ii(x,y) + k2u∗(x,y)

]u,k(y)dΩ + u,k(x) = 0. (36)

By applying further the divergence theorem, then Eq. (36)takes the form:∫∂Ω

Φ∗(x,y)u,k(x)dS+

∫Ω

k2u∗(x,y)u,k(x)dΩ+u,k(x) = 0.

(37)Furthermore, the following property exists:∫∂Ω

ni(y)u,i(x)u∗,k(x,y)dS−∫∂Ω

nk(y)u,i(x)u∗,i(x,y)dS =

=

∫Ω

ui(x)u∗,ki(x,y)dS −∫∂Ω

u,i(x)u∗,ik(x,y)dS = 0.

(38)

By adding Eqs. (37) and (38), one then has∫∂Ω

ni(y)u,i(x)u∗,k(x,y)dS

−∫∂Ω

nk(y)u,i(x)u∗,i(x,y)dS

+

∫∂Ω

Φ∗(x,y)u,k(x)dS

+

∫Ω

k2u∗(x,y)u,k(x)dΩ + u,k(x) = 0; (39)

which finally takes the form of∫∂Ω

ni(y)u,i(x)u∗,k(x,y)dS

+

∫∂Ω

eiktRiu(x)u∗,i(x,y)dS

+

∫Ω

k2u∗(x,y)u,k(x)dΩ + u,k(x) = 0. (40)

4. REGULARIZATION OF THE SINGULARINTEGRAL OPERATORS METHOD

In the present section, the regularization of the SingularIntegral Operators Method will be considered together withthe possibility of satisfying the SIOM in a weak form at ∂Ω,through a generalized Petrov-Galerkin formula.



By subtracting Eq. (31) from Eq. (22), we then have

∫∂Ω

q(y)u∗(x,y)dS

−∫∂Ω

[u(y)− u(x)]R∗(x,y)dS

+

∫Ω

k2u∗(x,y)u(x)dΩ = 0. (41)

Consequently, by using Eq. (34), then Eq. (41) can be ap-plied at point x on the boundary ∂Ω, as follows

∫∂Ω

q(y)u∗(x,y)dS −∫∂Ω

[u(y)− u(x)]R∗(x,y)dS =

=

CPV∫∂Ω

R∗(x,y)u(x)dS +1

2u(x), x ∈ ∂Ω. (42)

Furthermore, the Petrov-Galerkin scheme can be used in or-der for the weak form of Eq. (42) to be written as

∫∂Ω

f(x)dSx

∫∂Ω

q(y)u∗(x,y)dSy

−∫∂Ω

f(x)dSx

∫∂Ω

[u(y)− u(x)]R∗(x,y)dSy =

=

∫∂Ω

f(x)dSx

CPV∫∂Ω

R∗(x,y)u(x)dSy

+1

2

∫∂Ω

f(x)u(x)dSx; (43)

where u(x) denotes a test function on the boundary ∂Ω.By using further Eq. (34), then from Eq. (43) it follows

1

2

∫∂Ω

f(x)u(x)dSx =

∫∂Ω

f(x)dSx

∫∂Ω

q(y)u∗(x,y)dSy

−∫∂Ω

f(x)dSx

CPV∫∂Ω

R∗(x,y)u(y)dSy. (44)

Finally, if we choose the test function f(x) in such way tobe identical to a function which is energy-conjugate to u(x),then the following Galerkin SIOM is obtained:

1

2

∫∂Ω

q(x)u(x)dSx =

∫∂Ω

q(x)dSx

∫∂Ω

q(y)u∗(x,y)dSy

−∫∂Ω

q(x)dSx

CPV∫∂Ω

R∗(x,y)u(y)dSy. (45)

Thus, Eq. (45) is referred to a symmetric Galerkin SIOM.

Figure 1. Field Radiated by a Pulsating Sphere into an Infinite HomogeneousMedium.

Figure 2. Real Part of Dimensionless Surface Acoustic Pressure of a PulsatingSphere.

5. APPLICATION OF NON-LINEARSEISMIC WAVE MOTION BYA PULSATING SPHERE

The previously mentioned theory will be further applied tothe determination of the seismic field radiated from a pulsatingsphere into an infinite homogeneous medium (Fig.1).

Consequently, by using the Singular Integral OperatorsMethod (SIOM) described in the previous paragraphs, then thecomputation of the acoustic pressure radiated from the abovepulsating sphere is determined.

Furthermore, the analytical solution of the acoustic pressurefor a sphere of radius a, pulsating with uniform radial velocityva, is given as7

p(r)

z0va=a

r

ika

(1 + ika)e−ik(r−a); (46)

where p(r) denotes the acoustic pressure at distance r, z0 is thecharacteristic impedance and k represents the wave number.

In Table 1 and Table 2, the real and imaginary parts of di-mensionless surface acoustic pressures are shown with respectto the reduced frequency ka. So, the computational results byusing the SIOM were compared to the analytical solutions ofthe same problem. From the tables it can be seen that thereis very small difference between the computational results andthe analytical solutions. Finally, the same results are plotted, inFigs. 2 and 3, and in three-dimensional form in Figs 2a and 3a.

6. CONCLUSIONS

The new technology of Real-time Expert Seismology as wasintroduced and investigated by Ladopoulos26–31 is used for theexploration of on-shore and off-shore oil and gas reserves. Thismodern theory can be used at any depth of seas and oceans allover the world ranging from 300–3000 m, or even deeper andfor any depth such as 20,000 m or 30,000 m in the subsurfaceof existing oil and gas reserves.



Figure 3. 3-D Distribution of Real Part of Dimensionless Surface AcousticPressure of a Pulsating Sphere.

Figure 4. Imaginary Part of Dimensionless Surface Acoustic Pressure of aPulsating Sphere.

The benefits of the new theory of Real-Time Expert Seismol-ogy in comparison to the old theory of Reflection Seismologyare as follows:

1. The new theory uses the special form of the crests of thegeological anticlines of the bottom of the sea, in order todecide which areas of the bottom have the most possibili-ties to include petroleum. On the other hand, the existingtheory is only based to the best chance and do not includeany theoretical or sophisticated model.

2. The new theory of elastic (sound) waves is based on thedifference of the speed of the sound waves which are trav-elling through solid, liquid, or gas. In a solid the elasticwaves are moving faster than in a liquid and the air, and ina liquid faster than in the air. The existing theory is basedon the application of Snell’s Law and Zoeppritz equa-tions, which do not produce as good of results as thosethat we are expecting with the new method.

3. The new theory is based on a Real-Time Expert Systemworking under Real Time Logic, which gives results in

Table 1. Real Part of Dimensionless Surface Acoustic Pressure of a PulsatingSphere.

ka Re(p(a)/z0va) Re(p(a)/z0va)Analytical SIOM

0.00 0.00 0.000.40 0.10 0.110.60 0.30 0.320.75 0.40 0.411.00 0.50 0.521.25 0.60 0.601.50 0.70 0.712.00 0.80 0.802.50 0.86 0.873.00 0.90 0.913.50 0.92 0.934.00 0.94 0.944.50 0.96 0.965.00 0.97 0.976.00 0.98 0.987.00 0.99 0.997.50 0.99 0.99

Figure 5. 3-D Distribution of Imaginary Part of Dimensionless SurfaceAcoustic Pressure of a Pulsating Sphere.

Table 2. Imaginary Part of Dimensionless Surface Acoustic Pressure of aPulsating Sphere.

ka Im(p(a)/z0va) Im(p(a)/z0va)Analytical SIOM

0.00 0.00 0,000.20 0.22 0.230.40 0.30 0.310.50 0.40 0.410.60 0.45 0.450.80 0.48 0.481.00 0.50 0.491.50 0.45 0.452.00 0.40 0.402.50 0.35 0.363.00 0.30 0.313.50 0.26 0.264.00 0.24 0.244.50 0.21 0.215.00 0.19 0.205.50 0.17 0.176.00 0.15 0.156.50 0.14 0.147.00 0.12 0.127.50 0.11 0.11

real time, i.e., every second. Existing theory does not in-clude real time logic.

From the above three points the evidence of the applicabilityof the new method of Real-Time Expert Seismology is evident.The method is also novel as it is based mostly on a theoreti-cal and very sophisticated Real-Time Expert Model and not onpractical tools like the existing method.

In the present investigation, the Singular Integral OperatorsMethod (SIOM) has been used for the solution of the elasto-dynamic problems used in Non-Linear Real-Time Expert Seis-mology by applying the Helmholtz differential equation. Insuch a derivation the gradients of the fundamental solution tothe Helmholtz differential equation for the velocity potentialhave been used. Beyond the above, several basic identitiesgoverning the fundamental solution to the Helmholtz differ-ential equation for the velocity potential were analysed and in-vestigated.

Consequently, by using the SIOM, the acoustic velocity po-tential has to be computed. Beyond the above, several prop-erties of the wave equation, which is a Helmholtz differentialequation, were proposed and investigated. Furthermore, somebasic properties of the fundamental solution have been derived.

Finally, an application was proposed and investigated forthe determination of the seismic field radiated from a pulsat-ing sphere into an infinite homogeneous medium. Thus, byusing the SIOM, then the acoustic pressure radiated from theabove pulsating sphere has be computed. This is very impor-tant in hydrocarbon reservoir engineering in order for the sizeof the reservoir to be evaluated.



REFERENCES1 Schenk, H.A. Improved integral formulation for acoustic

radiation problems, Journal of the Acoustical Society ofAmerica, 44, 41–58, (1968).

2 Burton, A.J. and Miller, G.F. The application of the integralequation method to the numerical solution of some exteriorboundary value problems, Proceedings of the Royal Societyof London, Ser A, 323, 201–210, (1971).

3 Meyer, W.L., Bell, W.A., Zinn, B.T. and Stallybrass M.P.Boundary integral solutions of three dimensional acousticradiation problems, Journal of Sound and Vibration, 59,245–262, (1978).

4 Terai, T. On calculation of sound fields around three di-mensional objects by integral equation methods, Journal ofSound and Vibration, 69, 71–100, (1980).

5 Reut, Z. On the boundary integral methods for the exte-rior acoustic problem, Journal of Sound and Vibration, 103,297–298,(1985).

6 Okada, H., Rajiyah, H. and Atluri, S.N. Non-hypersingularintegral representations for velocity (displacement) gradi-ents in elastic/plastic solids (small or finite deformations),Computional Mechanics, 4, 165–175, (1989).

7 Chien, C.C., Rajiyah, H. and Atluri, S.N. An effectivemethod for solving the hypersingular integral equations in3-D acoustics, Journal of the Acoustical Society of America,88, 918–937, (1990).

8 Wu, T.W., Seybert, A.F. and Wan, G.C. On the numericalimplementation of a Cauchy principal value integral to in-sure a unique solution for acoustic radiation and scattering,Journal of the Acoustical Society of America, 90, 554–560,(1991).

9 Okada, H. and Atluri, S.N. Recent developments in the fieldboundary element method for finite / small strain elastoplas-ticity, International Journal of Solids and Structures, 31,1737–1775, (1994).

10 Hwang, W.S. Hyper-singular boundary integral equationsfor exterior acoustic problems, Journal of the AcousticalSociety of America, 101, 3336–3342, (1997).

11 Yang, S.A. An investigation into integral equation meth-ods involving nearly singular kernels for acoustic scatter-ing, Journal of Sound and Vibration, 234, 225–239, (2000).

12 Yan, Z.Y., Hung, K.C. and Zheng, H. Solving the hyper-singular boundary integral equation in three-dimensionalacoustics using regularization relationship, Journal of theAcoustical Society of America, 113, 2674–2683, (2003).

13 Han, Z.D. and Atluri, S.N. On simple formulations ofweakly-singular tBIE & dBIE, and Petrov-Galerkin ap-proaches, Computer Modeling in Engineering Sciences, 4,5–20, (2003).

14 Atluri, S.N., Han, Z.D. and Shen, S. Meshless local Patrov-Galerkin (MLPG) approaches for weakly singular traction& displacement boundary integral equations, ComputerModeling in Engineering Sciences, 4, 507–516, (2003).

15 Griffa, M., Ferdowsi, B., Daub, E.G., Guyer, R.A., John-son, P.A., Marone C. and Carmeliet J. Meso-mechanicalanalysis of deformation characteristics for dynamically trig-gered slip in a granular medium, Philosophical Magazine,92, 3520–3539, (2012).

16 Ladopoulos, E.G. Non-linear integro-differential equationsused in orthotropic spherical shell analysis, Mechanics Re-search Comunications, 18, 111–119, (1991).

17 Ladopoulos, E.G. Non-linear integro-differential equationsin sandwich plates stress analysis, Mechanics Research Co-munications, 21, 95–102, (1994).

18 Ladopoulos, E.G. Non-linear singular integral representa-tion for unsteady inviscid flowfields of 2-D airfoils, Me-chanics Research Comunications, 22, 25–34, (1995).

19 Ladopoulos, E.G. Non-linear singular integral computa-tional analysis for unsteady flow problems, Renewable En-ergy, 6, 901–906, (1995).

20 Ladopoulos, E.G. Non-linear singular integral representa-tion analysis for inviscid flowfields of unsteady airfoils,International Journal of Non-Linear Mechanics, 32, 377–384, (1997).

21 Ladopoulos, E.G. Non-linear multidimensional singular in-tegral equations in 2-dimensional fluid mechanics analysis,International Journal of Non-Linear Mechanics, 35, 701–708, (2000).

22 Ladopoulos, E.G. Singular Integral Equations, Linear andNon-Linear Theory and its Applications in Science and En-gineering, Springer, New York, Berlin, 2000.

23 Ladopoulos, E.G. Non-linear two-dimensional aerodynam-ics by multidimensional singular integral computationalanalysis, Forsch. Ingen., 68, 105–110, (2003).

24 Ladopoulos, E.G. Non-linear singular integral equations inelastodynamics, by using Hilbert transformations, Nonlin-ear Analysis, Real World Applications, 6, 531–536, (2005).

25 Ladopoulos, E.G. Unsteady inviscid flowfields of 2-D air-foils by non-linear singular integral computational analysis,Internationa Journal of Nonlinear Mechanics, 46, 1022–1026, (2011).

26 Ladopoulos, E.G. Non-linear singular integral representa-tion for petroleum reservoir engineering, Acta Mechanica,220, 247–253, (2011).

27 Ladopoulos, E.G. Petroleum reservoir engineering by non-linear singular integral equations, Journal of MechanicalEngineering Research, 1, 2–11, (2011).

28 Ladopoulos, E.G. Oil reserves exploration by non-linearreal-time expert seismology, Oil Asia Journal, 32, 30–35,(2012).

29 Ladopoulos, E.G. Hydrocarbon reserves exploration byreal-time expert seismology and non-linear singular inte-gral equations, Internationa Journal of Oil, Gas and CoalTechnology, 5, 299–315, (2012).

30 Ladopoulos, E.G. Non-linear singular integral equations formultiphase flows in petroleum reservoir engineering, Jour-nal of Petroleum Engineering Technology, 2, 29–39, (2012).

31 Ladopoulos, E.G. New aspects for petroleum reservoir ex-ploration by real-time expert seismology, Oil Gas BusinessJournal, 2012, 314–329, (2012).


3D Analysis of the Sound Reduction Provided byProtective Surfaces Around a Noise SourceLuıs GodinhoCICC, Department of Civil Engineering, University of Coimbra, Coimbra, Portugal

Edmundo G. A. Costa and Jose A. F. SantiagoCOPPE/Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

Andreia Pereira and Paulo Amado-MendesCICC, Department of Civil Engineering, University of Coimbra, Coimbra, Portugal

(Received 12 July 2012; provisionally accepted 23 June 2013; accepted 20 August 2013)

In the present paper, the authors present a numerical analysis of the sound reduction provided by simple acousticprotective measures to attenuate the noise emitted by equipment placed near a facade. The noise source is assumedto be surrounded by an enclosed space, defined as a parallelepiped chamber with a rectangular opening. To performthe numerical analysis, a 3D Boundary Element formulation is implemented. This formulation makes use of thedomain decomposition, together with Green’s functions, specifically defined to reduce the size of the involvedsystem matrices and to allow consideration for surface absorption. Indeed, these Green’s functions, defined usingthe image-source technique, allow modelling of the building’s facade and the ground as infinite surfaces with agiven absorption coefficient. The numerical model is verified against an analytical solution known for the case ofa point load acting within a parallelepiped space; additionally, it is validated by comparing its results with thoseobtained experimentally for a simple case. The implemented model is then used to perform a number of numericalsimulations, illustrating the effect of different configurations of the protective surfaces in the sound reduction.

1. INTRODUCTION

Many numerical methods have been used to model soundpropagation in both 2D and 3D environments. During the lastfour decades, the Boundary Element Method (BEM) has es-tablished itself as one of the preferred methods to be used inacoustics and vibro-acoustics engineering analysis. In fact, theBEM has a number of advantages over other numerical meth-ods that contribute to its success, as noted by Brebbia.1 First,it only requires the discretization of the problem boundaries,and thus only involves a more compact description of the envi-ronment. Second, it has a very good accuracy, since it is basedon the use of Greens functions, which are, themselves, a solu-tion of the governing equation. Finally, it is very well suitedto the analysis of infinite or semi-infinite domains, as the far-field radiation conditions are automatically satisfied. Some ofthese advantages are even more pronounced when a 3D analy-sis in an infinite/semi-infinite domain is considered, for whichalternative methods frequently require millions of degrees offreedom, together with a truncation of the propagation domainand the use of approximate absorbing boundary conditions.

Through the years, many researchers have used the methodin acoustic analysis of different systems. Many resources forthe BEM can be found, such as the excellent books by Wu2 andVon Estorff,3 which describe the fundamentals of the boundaryelement based acoustic analysis. Interesting developments canbe found in many scientific papers, such as the early worksof Lacerda et al.,4 in which a dual BEM formulation is usedto analyse the 2D sound propagation around acoustic barri-ers, over an infinite plane, considering both the ground and the

barrier to be absorptive. Later, the 3D propagation of soundaround an absorptive barrier has been studied by Lacerda etal.,5 introducing a dual boundary element formulation that al-lowed the barrier to be modelled as a simple surface. Severalstudies were also published concerning the convergence andthe discretization requirements of the BEM, such as the worksof Tadeu et al.,6 Tadeu and Antonio,7 or Marburg.8 Many re-cent works have also focused on the development of the BEMand its variants for complex acoustic problems, including moreintricate acoustic models. For example, these have been ap-plied to silencer or baffle analysis9–11 and have been used topropose highly efficient numerical approaches to tackle large-scale problems.12, 13

Perhaps the strongest drawbacks of the BEM are its com-plex mathematical formulation and the fact that it requires theprior knowledge of fundamental solutions, which are availableonly for some specific types of differential equations with spe-cific boundary conditions. However, for acoustic spaces, withtypically homogeneous propagation media, those solutions arewell known, and the method has been applied with success.2, 3

Another point that should not be neglected is that the accuracyof a BEM model depends on the special treatment of analyti-cal and numerical integration of the singular and hypersingularintegrals. This treatment is not trivial in the general case, andspecific numerical integration strategies need to be devised insome cases to allow for accurately performing those integra-tions (as it is the case of the strategy devised in Telles14 andTelles and Oliveira15). Recently, a new strategy based on aclosed form integration of singular and hypersingular integrals

78 (pp. 78–88) International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014

L. Godinho, et al.: 3D ANALYSIS OF THE SOUND REDUCTION PROVIDED BY PROTECTIVE SURFACES AROUND A NOISE SOURCE

was published by Tadeu and Antonio.16

An interesting feature of the BEM is that it can be adaptedto include more complex Greens functions, accounting for thepresence of specific features of the propagation medium. God-inho et al.17 and Tadeu et al.,18 analysed the specific case of2D configurations subject to the effect of a 3D pressure source,using the BEM to study the effect of acoustic barriers and ofthin screens coupled to a building facade in the reduction oftraffic noise. In those studies, both the rigid ground and therigid facade were taken into account by using the image-sourcemethod, thus avoiding their discretization. Additionally, thoseauthors synthesized the 3D sound field as a summation of sim-pler 2D problems (also known as a 2.5D formulation), withmuch lower computational cost.

Recently, an interesting work by Jean19 investigated thesound propagation through an opening. The author usedtwo different models to compute sound transmission throughopened windows; one was a hybrid model that combinedmodal, geometrical, wave, and integral approaches, while theother was a multi-domain BEM approach.

One relevant issue in acoustics is related to the propagationof noise produced by equipment, which is a major concernfor building designers. In fact, nowadays it is common for abuilding to have a large quantity of equipment, such as liftsor pumps, which can greatly affect the comfort of their inhab-itants. HVAC units are also included in that group, and it iscommon to adopt specific noise-control measures to minimizetheir influences. In many cases, those devices are installedoutside the buildings, either on the roof or at specific spacesthat may be acoustically treated to minimize their negative in-fluence. For these units, noise control procedures usually in-volve partially enclosing the equipment. However, due to theirnature, these machines require large ventilation openings thatconstitute acoustical weak points in the protective structure.

In the present work, the authors make use of a 3D BEMmodel to study the propagation of sound generated by an en-closed equipment, which propagates to the external space bya ventilation opening. The specific case of a parallelepipedspace with a rectangular opening, placed next to a facade, isanalysed. For the application of the BEM model, the authorsmake use of domain decomposition together with Greens func-tions, designed to reduce the size of the involved system ma-trices, and to allow consideration of surface absorption. Thosefunctions are derived using the image-source technique, andautomatically account for a rigid floor or absorptive floor (bothinside and outside) and for other internal and external verti-cal surfaces, such as walls. The numerical model is verifiedagainst an analytical solution known for the case of a pointload acting within a parallelepipedic space, and is validated bycomparing its results with those obtained experimentally. Anapplication example is presented, illustrating for several fre-quencies the pressure field generated around the enclosure.

2. MATHEMATICAL FORMULATION

2.1. Governing EquationsThe propagation of sound within a three-dimensional space

can be mathematically represented in the frequency domain by

the Helmholtz partial differential equation,

∇2p+ k2p =NS∑k=1

Qkδ(ξfk , ξ

); (1)

where ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 , p is the acoustic pres-sure, k = ω

c , ω = 2πf , f is the frequency, c is thesound propagation velocity within the acoustic medium, NSis the number of sources in the domain, Qk is the magni-tude of the existing sources ξfk located at

(xξfk

, yξfk, zξfk

),

ξ is a domain point located at (xξ, yξ, zξ) and δ(ξfk , ξ

)is the Dirac delta generalized function. In the above de-fined Helmholtz equation, the Sommerfeld radiation condition(

limx→∞

[∂p(x)∂n − ikp(x)

]= 0)

is automatically satisfied at in-

finity, where x is the field point located at (x, y, z), n is theunit outward normal vector and i =

√−1. Considering that a

point source is placed within this propagation domain, at x0, itis possible to establish the fundamental solution for the soundpressure at a point x, which can be written as

G(ξ,x) =e−ikr

4πr, with

r =√

(x− x0)2 + (y − y0)2 + (z − z0)2. (2)

2.2. Definition of Greens Functions UsingImage-Sources

In an acoustic analysis, the presence of perfectly reflectingplane surfaces can be taken into account using the well-knownimage-source method. In this technique, the effect of a pointsource in the presence of a given plane surface can be sim-ulated by considering an additional virtual source, positionedin a symmetrical position with respect to the reflecting plane.Thus, following Godinho et al.,17 if such plane is defined byz = 0 m (see Fig. 1a), the corresponding Greens function canbe written as

Ghalf(ξ,x) =e−ikr

4πr+e−ikr1

4πr1, with

r1 =√

(x− x0)2 + (y − y0)2 + (z + z0)2. (3)

The above expression is only valid if the plane is perfectlyrigid, with a reflection coefficient of one. However, it may bepossible to account for a partially-reflecting plane by follow-ing the strategy defined by Antonio et al.,20 and used togetherwith the Method of Fundamental Solutions by the same au-thors,21 multiplying the effect of the virtual source by a genericreflection coefficient R (one should note that this parametermay be related to the standard sound absorption coefficient byR =

√1− α). In this case, Eq. (3) becomes

Ghalf(ξ,x) =e−ikr

4πr+R

e−ikr1

4πr1. (4)

This approach can be further extended to incorporate moresurfaces. For example, for a ”quarter-space,” defined by twoorthogonal planes–one located at z = 0 m and at x = 0 m (see



Figure 1. Domain of the Greens functions: a) Half-space, b) Quarter-space,c) 3 surfaces, and d) 6 surfaces.

Fig. 1b)–the corresponding Greens function could be writtenas:

Gquarter(ξ,x) =e−ikr

4πr+R1

e−ikr1

4πr1

+R2e−ikr2

4πr2+R1R2

e−ikr3

4πr3; (5)

with r2 =√

(x+ x0)2 + (y − y0)2 + (z − z0)2 and r3 =√(x+ x0)2 + (y − y0)2 + (z + z0)2 . Following the de-

scribed procedure, if we consider a space defined by two verti-cal orthogonal planes and a horizontal plane placed at x = 0 m,y = 0 m and z = 0 m, respectively, as displayed in Fig. 1c, thecorresponding Greens function can be expressed as:

G3S(ξ,x) =

e−ikr

4πr+R1

e−ikr1

4πr1+R2

e−ikr2

4πr2+R1R2

e−ikr3

4πr3

+R1R3e−ikr4

4πr4+R3

e−ikr5

4πr5+R2R3

e−ikr6

4πr6

+R1R2R3e−ikr7

4πr7; (6)

with r4 =√

(x− x0)2 + (y + y0)2 + (z + z0)2,r5 =

√(x− x0)2 + (y + y0)2 + (z − z0)2,

r6 =√

(x+ x0)2 + (y + y0)2 + (z − z0)2,andr7 =

√(x+ x0)2 + (y + y0)2 + (z + z0)2.

Finally, considering a space defined by four vertical orthog-onal planes (x = 0, x = Lx, y = 0, y = Ly) and two horizon-tal planes, placed at z = 0, z = Lz , as defined in Fig. 1d, thecorresponding Greens function can be expressed as in Eq. (7)(see top of page 81), where NSX , NSY and NSZ repre-sent the number of sources used in the x, y, and z directionsrespectively, for the correct definition of the signal. In thiscase, rijk =

√(xi)

2 + (yj)2 + (zk)2, with x0 = (x − x0),

y0

= (y − y0); z0 = (z − z0); x1 = (x + x0 + 2Lxm);y

1= (y + y0 + 2Lyn); z1 = (z + z0 + 2Lzl); x2 =

(x − 2Lx − x0 − 2Lxm); y2

= (y − 2Ly − y0 − 2Lyn);

z2 = (z − 2Lz − z0 − 2Lzl); x3 = (x+ 2Lx − x0 + 2Lxm);y

3= (y + 2Ly − y0 + 2Lyn); z3 = (z + 2Lz − z0 + 2Lzl);

x4 = (x− 2Lx +x0− 2Lxm); y4

= (y− 2Ly + y0− 2Lyn);and z4 = (z − 2Lz + z0 − 2Lzl).

In this expression Rijk = Ri00R0j0R00k, with Eq. (8)shown on page 81, where Rce, Rfl, Rre, Rfr, Rri and Rlecorrespond to the reflection coefficients of the ceiling, floor,rear wall, front wall, right wall and left wall, respectively.

The above defined functions can be incorporated within nu-merical methods to avoid the full discretization of the simu-lated surfaces, and thus considerably reduce the computationalcost of such methods. This approach will be used to developthe numerical models described in the next section. Eq. (7)will be used in subsequent sections as a reference to performverifications of the numerical models.

2.3. Boundary Element FormulationAccording to Greens Second Identity, Eq. (1) can be trans-

formed into the following boundary integral equation:

C(ξ)p(ξ) = −iρω∫

Γ

G(ξ,x)vn(x)dΓ

−∫

Γ

G(ξ,x)

∂np(x)dΓ +

NS∑k=1

QkG(ξfk , ξ); (9)

where Γ is the boundary surface, ρ is the density, ω is theangular frequency, G(ξ,x) is the fundamental solution andG(ξfk , ξ) is the incident field regarding the acoustic pressuregenerated by the real source placed at position ξfk ; p(x) andvn(x) represent the acoustic pressure and the normal compo-nent of the particle velocity, respectively. The coefficient C(ξ)depends on the boundary geometry at the source point.

In order to solve Eq. (9), the Boundary Element Methodmay be used, requiring discretization of all surfaces. Afterdiscretization, if NEB constant elements with linear geome-try are assumed and the collocation method is applied to theintegral equation in terms of intrinsic coordinates η1 and η2,the following equation can be obtained:

C(ξs)p(ξs) =

−iρωNEB∑q=1

∫ 1

−1

∫ 1

−1

G(ξs,xq)vn(xq)|J |dη1dη2

−NEB∑q=1

∫ 1

−1

∫ 1

−1

G(ξs,xq)

∂np(xq)|J |dη1dη2

+Ns∑k=1

QkG(ξfk , ξq); (10)

where ξs refers to the functional node s with s ranging from1 to NEB , and |J | is the Jacobian. Different boundary condi-tions may be prescribed at each boundary element, such as theDirichlet (prescribing p(x) = p(x)), the Neumann (prescrib-ing v(x) = v(x)), or the Robin ( p(x)

v(x) = Z(x)) boundary con-ditions. After prescribing such conditions, a system withNEBequations on NEB unknowns can be assembled. Its solutionmakes it possible to obtain the acoustic pressure and the nor-mal velocity at each boundary element, and consequently the



G6S(ξ,x) =e−irω/c

4πr+NSY∑n=0

4∑j=1

e−ir0j0ω/cR0j0

4πr0j0

+NSX∑m=0

4∑i=1

e−iri00ω/cRi00

4πri00+NSY∑n=0

4∑j=1

e−irij0ω/cRij04πrij0

+NSZ∑l=0

4∑k=1

e−ir00kω/cR00k

4πr00k+NSY∑n=0

4∑j=1

e−ir0jkω/cR0jk

4πr0jk

+NSX∑m=0

4∑i=1

e−iri0kω/cRi0k4πri0k

+NSY∑n=0

4∑j=1

e−irijkω/cRijk4πrijk

; (7)

R000 = 1.0;

R100 = Rre(RreRfr)m; R010 = Rle(RleRri)

n; R001 = Rfl(RflRce)l;

R200 = (RreRfr)m+1; R020 = (RleRri)

n+1; R002 = (RflRce)l+1;

R300 = (RreRfr)m+1; R030 = (RleRri)

n+1; R003 = (RflRce)l+1;

R400 = Rfr(RreRfr)m; R040 = Rri(RleRri)

n; R004 = Rce(RflRce)l. (8)

pressure at any point of the domain can be computed by apply-ing the boundary integral, as shown in Eq. (10). If the probleminvolves more than one subdomain, then one boundary inte-gral equation (Eq. (10)) must be written for each subdomain,and continuity conditions must be enforced at the interface be-tween subdomains. These continuity conditions can be writtenas

p+ = p− in Γc;

v+n = −v−n in Γc; (11)

where Γc is the common interface to the two subdomains, pis the pressure, and vn is the particle velocity along the nor-mal direction pointing outwards of each subdomain (i.e., withopposing directions for the two subdomains).

It should be mentioned that under specific circumstances itmay be possible to avoid the discretization of all surfaces of themodel, incorporating them a-priori into the Greens functionsused within the boundary element model. For the problem tobe solved in this paper, the authors make use of the Greensfunctions defined in the previous section, which may incor-porate the effect of the ground and of one or two orthogonalwalls.

Considering that the problem to be solved corresponds to anenclose parallelepiped space containing a noise source, placednext to the facade of a building, and connected to an externaldomain by an opening, the system is subdivided in two subdo-mains: Ωin, corresponding to the interior of the enclosure, andΩext, corresponding to the external, semi-infinite space. Theinner surface of the enclosure is here designated as Γin, whilethe outer surface is designated as Γext. The common interface(where continuity of pressure and velocity is imposed) is des-ignated as Γc. For this configuration, it is possible to use theGreens function of Eqs. (4) or (5) for the outer problem (eitherconsidering just the ground or the ground and a vertical wall)and that of Eq. (6) for the inner problem. Thus, for this spe-cific problem, considering that the enclosure is made of rigid

surfaces, both inside and outside, the boundary conditions toconsider can be written as:

pin = pext in Γc

vinn = −vextn in Γc

continuity of pressure

and normal velocity

vinn = 0 in Γin

vextn = 0 in Γext

rigid boundary,

both inside and outside. (12)

Considering that NEB1 elements are used to discretize the in-ternal surface, NEB2 elements are used to discretize the ex-ternal surface, and NEBC elements are used to discretize theconnecting opening, a system of NEB1 + NEB2 + 2NEBCequations on NEB1 +NEB2 + 2NEBC unknowns may thenbe written. In contrast with Dual-BEM formulation, this equa-tion system is not fully populated, being basically made oftwo blocks, coupled by 2NEBC equations; this fact leads tosome savings in terms of computation effort (only a part ofthe coefficients of the matrix need to be calculated). Addition-ally, the direct BEM formulation makes use of two subdomainsand therefore avoids the calculation of hypersingular integrals,which are required in the Dual-BEM approach, which becomevery complex for 3D problems.

2.4. Model Verification Against a BenchmarkSolution

In order to verify the proposed BEM model, consider a par-allelepiped space with dimensions of 1.3 m × 1.4 m × 1.5 m,filled with air, with a density of 1.22 kg

m3 and sound propagationvelocity of 340 m

s . Consider that all the surfaces of this spaceare rigid, with null normal particle velocities, and that withinthe space a point source is located at (0.9 m, 1.0 m, 0.5 m),oscillating with an angular frequency of ω = 2πf .

For the above defined configuration, it becomes possible toestablish an analytical solution in the form of an infinite se-ries of image sources, reproducing the reflections at the variouswalls of the space, as seen in Eq. (7). A detailed description



(a)

(b)

Figure 2. a) Real and imaginary parts and absolute value of the response forfrequencies between 100 Hz and 300 Hz; b) Convergence of the BEM resultsfor the frequencies of 100 Hz and 300 Hz.

of this solution can be found in Antonio et al.20 It is impor-tant to note that, in theory, the number of sources required toreproduce the sound field in such a closed space tends to bevery high, and thus the series converges slowly to the correctsolution. However, the convergence of the series is greatly im-proved if a damped system is considered, making use of com-plex frequencies of the form ωc = ω − iζ∆ω, where ζ is adamping coefficient and ∆ω = 2π∆f , with ∆f being the fre-quency step. With the aim of verifying the proposed method,this frequency increment and the damping factor were fixed at4 Hz and 0.7, respectively. Those values allow just consider-ing image sources within a distance of 1

4 Hz × 340 ms = 85 m

from the analysed domain, since sources placed further awayare greatly damped and will not influence the response signifi-cantly.

The same configuration was modelled using the 3D BEMmodel, making use of Greens function defined before for asemi-infinite space limited by one horizontal and two verti-cal orthogonal planes. In this case, the boundary conditionsof three faces of the closed space are automatically satisfied,and only the remaining three must be discretized. For this pur-pose, 675 boundary elements were used. It should be notedthat, if a full discretization of the chamber was performed, atotal of 1350 boundary elements would be required, leading toa system of equations four times larger (1350×1350 for the fulldiscretization vs. 675×675 when using the proposed strategy).In terms of performance, the expected computational cost re-

Figure 3. a) Photography of the test chamber and b) image of the numericalmodel.

quired to assemble the equation system would then be fourtimes bigger if the traditional approach was used; addition-ally, the solution time of the equation system is known to beof O(n3), which means that the relation between the compu-tational times required by both models for this purpose wouldbe around 13503

6753 = 8. These are very relevant improvementsin terms of computational time and memory requirements thatreveal the benefits of using the proposed strategy.

Figure 2a illustrates the real, imaginary, and absolute valuesof the computed pressure at a receiver located at (0.4 m, 1.0 m,0.5 m), for frequencies between 100 Hz and 300 Hz. In thisplot, it is clear that the agreement between the analytical andthe BEM solutions is excellent, and that the method is correctlyimplemented. In Fig. 2b, two curves representing the conver-gence behaviour of the proposed boundary element model forfrequencies of 100 Hz and 300 Hz are shown. The presentedcurves clearly reveal a steady and regular convergence of thismodel to the correct solution, with the response improving asfurther refined meshes are used. In addition, the two plots re-veal that the relative error calculated for the higher frequency issomewhat higher than that computed for the lower frequency.This is a typical behaviour of most numerical methods, whichusually require finer meshes to correctly reproduce the acousticresponse at higher frequencies.

3. EXPERIMENTAL VALIDATION

Besides the verification illustrated in the previous section, anexperimental validation has been performed to ensure that thenumerical results can be representative of a realistic situation.For this purpose, a laboratorial test apparatus has been devel-oped, consisting of a concrete chamber illustrated in Fig. 3a.This parallelepipedic chamber is made of concrete and has in-ternal dimensions of 1.3×1.4×1.5 m3. In one of its faces, a0.5×0.5 m2 opening is inserted, allowing access to its inte-rior. Inside this chamber, a dodecahedrical sound source (B&KOmni Power 4292) has been placed, 0.4 m from its front wall,0.4 m from its right side wall, and with its center 0.2 m abovethe floor. Sound levels were registered at a microphone (type40AF from Gras Sound & Vibration), located 0.4 m from thefront wall, 0.4 m from the left side wall, and 0.6 m above thefloor. Time responses were acquired using a dBBati32 systemfrom 01 dB, and then transformed to the frequency domain bymeans of FFT.

The same system has also been modeled numerically, mak-ing use of the above described BEM model. For this purpose,



(a)

(b)

Figure 4. Sound pressure levels obtained experimentally and numerically in-side the concrete test chamber, when the window was a) closed and b) open.

square elements with 0.05× 0.05 m2 were used along the nec-essary surfaces, while the Greens function defined in Eq. (6),that accounts for the floor, the left wall, and the back wall, isused to minimize the discretization requirements (see Fig. 3b).It should be noted that, on the front wall of the chamber, theexisting opening is also adequately discretized, accounting forthe correct thickness of the wall (0.10 m). For this model, atotal of 2428 elements were used.

Comparing the presented curves in Fig. 4a, it is clear thatthey follow very similar trends, despite visible amplitude dif-ferences. It is important to note that the numerical result hasbeen computed considering rigid surfaces, while in the exper-imental model a small amount of acoustic absorption exists,influencing the results. More importantly, the peak positionsof the two curves match, in general, very well, and also corre-spond to the analytical normal modes prediction. However, thepeak occurring at 220 Hz in the experimental result is not sowell reproduced by the numerical results, and appears slightlyshifted to the right. It is important to note that the difference iseven larger for the analytical solution, indicating that the pres-ence of the irregularity and of a plate in the front wall may beintroducing some modifications to the dynamic behaviour.

For the second configuration, the chamber window was con-sidered to be open. Observation of the corresponding experi-mental results in Fig. 4b concluded that they exhibit a similartrend to those in Fig. 4a. However, it is important to note thatsome shifts occur in the peaks, revealing a modified dynamicbehaviour. This change can be identified, for example, in the

Figure 5. Geometry of the problem.

peak occurring around 135 Hz, which is shifted to the right inthose plots.

For this case, the BEM model was also applied, making useof Greens functions that account for a possible inner absorbingfloor, rear and left walls, as well as for the outer floor surfaceand for a vertical wall behind the chamber. Two sub-regionswere defined, one for the interior and the other for the exteriorsub-domains, allowing to account for the coupling of the inter-nal space with a much larger external domain, and giving someinsight about the influence of this coupling (performed usingEq. (12)). In this case, two different sets of Greens functionsare adopted: Eq. (6) to model the inner problem, and Eq. (5)to account for the outer space. It should be noted that someuncertainty will always exist in the computed results, sincepart of the input data may not exactly correspond to that ofthe real model, which may even present some spatial variabil-ity. Indeed, to accurately reproduce the acoustic behavior ofthe experimental enclosure, it would be necessary to know indetail the properties of its surfaces, namely in what concernstheir impedance, elastic properties, etc.. In addition, since aconcrete enclosure is here considered, slight changes in thedimensions may occur with respect to the ones considered inthe numerical model. All these uncertainties will lead to dis-crepancies between the numerical model and the experimentalmeasurements, which are quite difficult to estimate. Thus, inFig. 4b, two numerical results are included, one correspondingto a completely rigid model, and the other to a small degree ofsound absorption in the internal rear wall and floor (α =0.04).By considering these two cases, it becomes possible to assessthe effect of a small change in the sound absorption coefficientof the internal surfaces. In both cases, the shift in the normalmodes is correctly reproduced, and the numerical results fol-low approximately the experimental curve. It is interesting tonote that the absorption effect of the inner surfaces contributesto a better match between experimental and numerical results,particularly in what concerns the amplitude of the sound fieldat the normal modes. In the remaining part of the curve, verylittle changes occur, and essentially the same result is obtained.

4. APPLICATION EXAMPLES

The above defined numerical formulation was used to studythe propagation of noise generated by an equipment near abuilding facade, considering different scenarios. For this pur-pose, a point sound source, simulating a possible equipment,was assumed to be placed at (0.9 m, 1.0 m, 0.2 m), radiat-ing energy in all directions. The propagation medium was as-



No protection Top and side barriers

a1) a2)

b1) b2)

c1) c2)

Figure 6. SPL maps near the facade of the building for different frequencies: a) f = 63 Hz, b) f = 125 Hz, and c) f = 250 Hz.

sumed to be air, with a density of 1.22 kgm3 , which allows sound

waves to travel at 340 ms . The building facade and the external

ground were assumed to be perfectly rigid, and their presencewas simulated using the image-source method.

4.1. Responses Provided by the Presence ofTop and Side Barriers

In a first set of simulations, the sound field generated bythe point source without the implementation of any noise con-trol measures was determined and compared with the field ob-tained when three rigid barriers were placed around the equip-ment (one on top and the remaining two on either side). Thedimensions of these barriers are 1.3 m along the x direction,perpendicular to the building facade, 1.4 m along the y di-rection, parallel to the building plane, and 1.5 m along the zdirection. To model those barriers, the propagation domainwas separated in two sub-regions, defining a parallelepipedic,

closed sub-region in which the equipment was located, and anexternal, semi-infinite sub-region. The interface between bothdomains was modelled using a total of 832 boundary elements,approximately 0.125 × 0.125 m2 each. The geometry of theproblem is represented in Fig. 5.

Sound pressure level (SPL) maps were computed at a verti-cal grid of receivers 0.50 m away from the facade. In all cases,a small damping effect was considered, in the form of a verysmall imaginary part of the frequency, to account for the ab-sorbing effect of the air. The first set of results is presentedin Fig. 6, with the left column corresponding to the situation inwhich no protective devices are used, and the right column cor-responding to the case of a protective device made with top andside barriers. Results are displayed for frequencies of 63 Hz,125 Hz, and 250 Hz. In the absence of any protection, thesound levels registered along the grid reach relatively high val-ues near the source, and show a progressive decrease with thedistance to the floor, while energy is spreading away from the



Figure 7. SPL registered at a point 9 m above the ground, centred with theprotective lateral barriers and 0.5 m from the facade.

point source. The results for all frequencies were very similar,although for 250 Hz some traces of interaction between the in-cident wave and reflections at the facade and ground can befound (see Fig. 6c1).

When the protective surfaces are introduced, some changesbecome visible in the SPL results, due to multiple interactionswith those rigid surfaces, the facade and the ground. For thelower frequency (see Fig. 6a2), a general decrease is visible,with reductions around 5 to 10 dB. It is clear that the SPL mapis not symmetrical, since the source is not centred with the lat-eral barriers. For the frequency of 125 Hz, the obtained resultsindicate that there is an amplification of the SPL, probably dueto multiple interactions between the floor and the top barrier(see Fig. 6b2). For the higher frequency, the SPL decreasessignificantly over the grid of receivers, indicating effective pro-tection provided by the sound barriers.

To help understand the behaviour observed in Fig. 6, in par-ticular for the frequency of 125 Hz, the SPL has been calcu-lated at a single point, 9 m above the ground, centred with theprotective lateral barriers and 0.5 m from the facade for fre-quencies between 100 Hz and 250 Hz. In Fig. 7, results aredisplayed with and without the protective devices. In the ab-sence of those devices, the SPL is almost constant along theanalysed frequency range, and only reveals a smooth and slightdecrease as frequency increases. When the barriers are consid-ered, it can be seen that the SPL is, in general, lower, but twopeaks occur along the analysed frequency range. The first oneis registered at 125 Hz, and corroborates the effect observed inFig. 6, when constructive wave interferences lead to increasedSPL at this plane near the facade.

4.2. Responses Provided by an EnclosingSpace with Sound Absorption and anOpening

A second configuration of the protective device was anal-ysed, corresponding to an enclosure with a small opening,0.5×0.5 m2, centred in the front wall (seen in Fig. 8). Forthis geometric configuration, two different cases are studied:in the first, all the internal surfaces are assumed to be rigid; inthe second, a reflection coefficient of 0.8 (representing a soundabsorption of α =0.36) is included in the Greens function, andascribed to the ground and rear wall. The results were, onceagain, computed over the same grid of receivers, at a distance

Figure 8. Geometry of the enclosure, with a small opening.

of 0.5 m from the facade. It should be noted that in many realapplications the construction of such an enclosure around thesound source would additionally require associated mechani-cal ventilation devices, which are here not taken into account.

Figure 9 illustrates the results computed for 63 Hz, 125 Hz,and 250 Hz, in the presence of the above defined enclosure.For this case, the attenuation of sound levels is much morepronounced in relation to the unprotected case, with importantreductions of near 15 to 20 dB being observed. For higherfrequencies, interaction between the different surfaces becamemore evident, with very low sound levels occurring at specificregions of the vertical grid. The presence of absorbing interiorsurfaces (results on the right column) also seems to be influ-encing the final SPLs, with slightly lower levels being regis-tered for this situation, as some energy was dissipated alongthe internal absorptive surfaces.

Additional calculations were performed at a horizontal gridof receivers, 1.5 m above the floor, and the corresponding SPLsare presented in Fig. 10. In this figure, three situations are il-lustrated, corresponding, respectively, to the top and side pro-tective barriers and to the rigid and absorbing enclosures withan opening opposite from the facade. Results are illustratedfor 125 Hz and 250 Hz. It is clear in these figures, that thereis a very significant difference between the performance ofthe first configuration (see Fig. 10a1 and 10a2) and of the re-maining two, with much larger sound reductions being reachedwhen the protection corresponds to an enclosure. This effect isobservable for both analysed frequencies along the horizontalgrids of receivers.

To obtain further insight on these behaviours, results werealso calculated over a range of frequencies from 100 Hz to250 Hz, at a specific receiver 9 m above the ground and 0.5 mfrom the facade, centred with the protective lateral devices. Inthese results, presented in Fig. 11, it is interesting to note theeffect in the computed SPLs of introducing absorptive surfacesinside the enclosure, with the sound pressure curve becomingsmoother, and with a significant reduction of the amplificationeffect registered at normal modes. In fact, for this case, theSPL curve is always below the reference curve, and the soundreduction is always at least 10 dB. In the absence of these ab-sorptive surfaces, localized peaks are observed at the normalmode frequencies, which generate amplifications, leading toSPLs that may even surpass those of the reference situation.



α =0.0 α =0.36

a1) a2)

b1) b2)

c1) c2)

Figure 9. SPL maps near the facade of the building, in the presence of an enclosure with a small opening.

5. CONCLUSIONS

In this paper, a 3D numerical model based on the Bound-ary Element Method was presented and used to simulate thesound field produced by an equipment (noise source) near abuilding facade. The proposed model makes use of Greensfunctions defined using the image-source method and account-ing for absorption, which allows significant reduction of therequired discretization. The method was implemented andverified against an analytical solution known for the case ofa closed parallelepipedic space containing a point source, re-vealing good accuracy. A laboratorial apparatus, consisting ofa concrete chamber with a small opening excited by an om-nidirectional source, was also used to experimentally validatethe model and verify if realistic results could be obtained. Acomparison between the experimental and numerical resultsrevealed that the dynamic behaviour of the acoustic space wascorrectly reproduced by the BEM. Additionally, it was possible

to observe that the coupling between an internal and an exter-nal acoustic media introduced small changes in the sound fieldthat were also visible in the BEM results. By these procedures,the proposed numerical model demonstrated its adequacy forcorrectly and efficiently simulating the acoustic behaviour ofthe described systems.

The model was then used to simulate several applicationcases. Sound pressure levels were computed for different sce-narios in which protective measures were used to attenuate thenoise produced by a point source, simulating the presence ofa loud equipment in the vicinity of a building facade. Both anenclosure with a small opening and a simpler set of three rigidbarriers around the equipment, defining an open protection de-vice, were tested with the BEM methodology being a valuableanalysis tool. Several sets of results were computed for dif-ferent frequencies in SPL maps along plane grids of receivers(parallel to the facade and to the ground), and also analysedby plotting the responses at a specific receiver for a range of



125 Hz 250 Hz

a1) a2)

b1) b2)

c1) c2)

Figure 10. SPL maps at a horizontal grid of receivers 1.5 m above the ground: a) top and side barriers, b) rigid enclosure with opening, and c) absorbingenclosure with opening.

frequencies. Results revealed that different efficiencies can beobtained, and that the presence of absorbing surfaces within aprotective enclosure can aid in the minimization of the noisethat reaches the building facade. The acoustic behaviour ofthis kind of protective devices is shown to be quite dependentof the interaction of the acoustic waves with the rigid and/orabsorptive surfaces defining the geometry of the modelled sys-tems.

ACKNOWLEDGMENTS

The second author would like to thank CNPq for providingfinancial support to this research.

REFERENCES1 Brebbia, C. A. The boundary element method for engineers,

Pentech Press, London, (1984).

2 Wu, T. Boundary element acoustics, WIT Press, Southamp-ton, (2000).

3 Von Estorff, O. Boundary element in acoustics advancesand applications, WIT Press, Southampton, (2000).

4 Lacerda, L.A., Wrobel, L.C., and Mansur, W.J. Adual boundary element formulation for sound propagationaround barriers over an infinite plane, Journal of Sound andVibration, 202, 235–347, (1997).

5 Lacerda, L.A., Wrobel, L.C., Power, H., and Mansur,W.J. A novel boundary integral formulation for three-dimensional analysis of thin acoustic barriers over animpedance plane, Journal of the Acoustical Society ofAmerica, 104(2), 671–8, (1999).

6 Tadeu, A., Godinho, L., and Santos, P. Performance of the



Figure 11. SPL registered at a point 9 m above the ground, centred with theenclosure and 0.5 m from the facade.

BEM solution in 3D acoustic wave scattering, Advances inEngineering Software, 32, 629–639, (2001).

7 Tadeu, A. and Antonio, J. Use of constant, linear, andquadratic boundary elements in 3D wave diffraction anal-ysis, Engineering Analysis with Boundary Elements, 24,131–144, (2000).

8 Marburg, S. Six boundary elements per wavelength: Is thatenough?, Journal of Computational Acoustics, 11, 25–51,(2002).

9 Ji, Z.L. Boundary element acoustic analysis of hybrid ex-pansion chamber silencers with perforated facing, Engi-neering Analysis with Boundary Elements, 34(7), 690–696,(2010).

10 Jiang, C., Wu, T.W., and Cheng, C.Y.R. A single-domainboundary element method for packed silencers with mul-tiple bulk-reacting sound absorbing materials, Engineer-ing Analysis with Boundary Elements, 34(11), 971–976,(2010).

11 Chen, Z.S., Kreuzer, W., Waubke, H., and Svobodnik,A.J. A Burton-Miller formulation of the boundary elementmethod for baffle problems in acoustics and the BEM/FEMcoupling, Engineering Analysis with Boundary Elements,35(3), 279–288, (2011).

12 Li, S. and Huang, Q. A fast multipole boundary elementmethod based on the improved Burton-Miller formula-tion for three-dimensional acoustic problems, EngineeringAnalysis with Boundary Elements, 35(5), 719–728, (2011).

13 Wang, X., Zhang, J, Zhou, F., and Zheng, X. An adap-tive fast multipole boundary face method with higher orderelements for acoustic problems in three-dimension, Engi-neering Analysis with Boundary Elements, 37(1), 144–152,(2013).

14 Telles, J.C.F. A self-adaptive co-ordinate transformation forefficient numerical evaluation of general boundary elementintegrals, International Journal for Numerical Methods inEngineering, 24(5), 959–973, (1987).

15 Telles, J.C.F. and Oliveira, R.F. Third degree polynomialtransformation for boundary element integrals, EngineeringAnalysis with Boundary Elements, 13, 135–141, (1994).

16 Tadeu, A. and Antonio, J. 3D acoustic wave simulation us-ing BEM formulations: closed form integration of singu-lar and hypersingular integrals, Engineering Analysis withBoundary Elements, 36, 1389–1396, (2012).

17 Godinho, L., Antonio, J., and Tadeu, A. 3D sound scatteringby rigid barriers in the vicinity of tall buildings, AppliedAcoustics, 62(11), 1229–1248, 2001.

18 Tadeu, A., Antonio, J., Amado Mendes, P., and Godinho,L. Sound pressure level attenuation provided by thin rigidscreens coupled to tall buildings, Journal of Sound and Vi-bration, 304(3-5), 479–496, (2007).

19 Jean, P. Sound transmission through opened windows, Ap-plied Acoustics, 70(1), 41–49, (2009).

20 Antonio, J., Godinho, L., and Tadeu, A. Reverberationtimes obtained using a numerical model versus those givenby simplified formulas and measurements, Journal Acus-tica - Acta Acustica, 88, 252–261, (2002).

21 Antonio, J., Tadeu, A., and Godinho, L. A three-dimensional acoustics model using the method of funda-mental solutions, Engineering Analysis with Boundary Ele-ments, 32(6), 525–531, (2008).


Study on Linear Vibration Model of Shield TBMCutterhead Driving SystemXianhong Li, Haibin Yu, Peng Zeng and Mingzhe YuanKey Laboratory of Industrial Control Network and System, Shenyang Institute of Automation Chinese Academy ofSciences, Shenyang, Liaoning province, China, 110016

Jianda Han and Lanxiang SunState Key Laboratory of Robotics, Shenyang Institute of Automation Chinese Academy of Sciences, Shenyang,Liaoning province, China, 110016

(Received 22 August 2012; provisionally accepted 23 October 2012; accepted 18 February 2013)

In this paper, a general linear time-varying multiple-axis (LTVMA) vibration model of shield tunnel boring ma-chine (TBM) cutterhead driving system is established. The corresponding multiple inputs and multiple outputs(MIMO) state-space model is also presented. The linear vibration model is analysed, and the vibration-torquetransfer function matrix and the vibration-torque static gain matrix are obtained. The linear vibration model is sim-ulated, and the physical parameters’ effects on the vibration response are investigated. A preliminary approach isproposed to reduce vibration by increasing motor rotor inertia and viscous damped. The LTVMA vibration modelprovides a solid foundation for fault detection and diagnosis (FDD), as long as the health monitoring of cutterheaddriving system.

NOMENCLATURE

n The number of cutterhead driving motors;q the reducer speed reduction ratioJd,i i-th induction motor rotor inertia after equiva-

lent coupling-ibd,i i-th induction motor rotor viscous damped af-

ter equivalent coupling-iJz,i Inertia of the i-th coupling between motor-i

and reducer-ibz,i Viscous damped of the i-th coupling between

motor-i and reducer-iJr,i i-th induction motor rotor inertiabr,i i-th induction motor rotor viscous dampedJm Large gear inertiabm large gear viscous dampedrm large gear radiusθp,i Angular displacement of i-th active pinionθi angular displacement of i-th motor rotorθm Large gear angular displacementωm large gear angular speedJc,i i-th pinion inertia after equivlalent coupling-ibc,i i-th pinion viscous damped after equivalent

coupling-iJw,i Inertia of the i-th coupling between reduce-i

and pinion-ibw,i Viscous damped of the i-th coupling between

reducer-i and pinion-iJp,i i-th pinion inertia;bp,i i-th pinion viscous dampedTe,i i-th induction motor electrical magnetic torqueωp,i Angular speed of pinion-iωi angular speed of induction motor-i

Fi Elastic mesh force of the pinion-i and largegear

ri radius of pinion-iMc,i Elastic mesh torque of the pinion-i and large

gearpi Relative position of the pinion-i and large gearki Mesh stiffness of the pinion-i and large gearci Mesh damped of the pinion-i and large gearmp,i Mass of the pinion-iky,i support stiffness of pinion-icy,i support damped of pinion-img Mass of the large gearky,m support stiffness of large gearcy,m support damped of large gearyi The contact direction (line-of-action direction)

vibration displacement of i-th pinionyi

m The large gear’s contact vibration displace-ment along i-th pinion’s line-of-action direc-tion

TL Shield TBM cutterhead’s load torquekf ,i, cf ,i elastic mesh force coefficient of pinion-ikt,i, ct,i elastic mesh torque coefficient of pinion-iim,i gear transmission ratioT1 The resistant torque of soil and rocksT2 The friction torque of soil and rocks which

chafe with the front of the cutterheadT3 The friction torque of soil and rocks which

chafe with the back of the cutterheadT4 The friction torque of soil and rocks which

chafe with bulkheadT5 The stirring torque of soil and rocks by cutter-

head stirring rodT6 The friction torque of cutterhead bearings and

sealed chamber


X. Li, et al.: STUDY ON LINEAR VIBRATION MODEL OF SHIELD TBM CUTTERHEAD DRIVING SYSTEM

Figure 1. (a) Shield head; (b) Panel cutterhead; (c) Shield TBM profile; (d)Thrust system.

1. INTRODUCTION

The shield TBM is a large-scale, underground machine usedto excavate tunnels or subways. Modern shield TBM integratesthe information and communication, network control, intelli-gent sensing, laser measuring guide, infrared detection, andother advanced science and technology with cutting geotech-nical, transporting rock and soil, support excavation section,orientation correction, tunnel segments assembly, and otherfunctions. Therefore, the shield TBM excavates tunnels withhigh safety and reliability, low manpower, minor environmen-tal damages, and rapid speed. The cutterhead driving system isa core system of shield TBM, and it drives the cutterhead to cutrock and soil (the cutterhead has a thrust system). The compo-nents of a shield TBM are shown in Fig. 1. To adapt to geologi-cal conditions, the shield TBM has different cutterhead forms:spokes form, panel form, and hybrid form. The shield TBMcutterhead includes two driven modes: hydraulic-driven andmotor-driven. In this paper, the motor-driven mode’s shieldTBM cutterhead is studied. The profile and transmission struc-ture of the shield TBM cutterhead is shown in Fig. 2.20–22 Mul-tiple gear structure synthesizes the motors’ electrical magnetictorque (EMT). Overall, the principle structure of the cutter-head driving system is shown in Fig. 3.20–22 The cutterheadand the central large gear have identical bearing, central axis,and rotation speed.

The development of the shield TBM cutterhead drivingsystem mainly goes through the human-driven, mechanical -driven, hydraulic-driven, and motor-driven stages. For themotor-driven mode’s cutterhead driving system, it needs tosolve the cutterhead speed tracking (CST) problems, multiplemotors torque synchronization (MMTS) problems, and FDDproblems of the cutterhead driving system. The LTVMA vi-bration model provides a theoretical basis for fault detectionand diagnosis of the cutterhead driving system. Therefore, theLTVMA vibration model’s establishment and analysis is re-searched to address the FDD problems of the cutterhead driv-ing system.

The literature7–14 introduces the composition, load torque,thrust system, and driving structure of shield TBM. Later,15–18

the load torque of the shield TBM cutterhead was studied. Avertical vibration model was then established for the scrap-

Figure 2. Shield TBM cutterhead profile and transmission structure.

ing the cutter of the shield TBM,19 and a nonlinear and linearkinematics model was established for the cutterhead drivingsystem.20–22 A two-dimensional, nonlinear vibration model ofcutterhead driving system has been presented,23 and the systemperformances of a hydraulic-driven cutterhead driving systemwas also studied.24 This paper presents the LTVMA vibra-tion model of cutterhead driving system. The vibration-torquetransfer function matrix and its static gain matrix are obtained.LTVMA vibration model is simulated to analyse physical pa-rameters’ effects on dynamic vibration response, and a prelim-inary approach is concluded to reduce vibration. For a cutter-head driving system, the LTVMA vibration model provides atheoretical foundation for FDD and health monitoring.

This paper is arranged as follows: in Section 2, a generalLTVMA vibration model of the cutterhead driving system ispresented, and MIMO state-space is also presented. In Sec-tion 3, the linear vibration model is analysed, and the vibration-torque transfer function matrix and vibration-torque static gainmatrix are obtained. In Section 4, the LTVMA vibration modelis simulated, and a preliminary approach is presented to re-duce the cutterhead driving system’s vibration. In Section 5,the study contents and results are briefly reviewed, and someconclusions are made.

2. GENERAL LTVMA VIBRATION MODEL OFCUTTERHEAD DRIVING SYSTEM

2.1. Preliminary and PreparationTo establish the linear vibration model of cutterhead driving

system, the gear backlash and transmission error is ignored,and the spring-damped is usually used to depict the gear meshprocess. Figure 4 shows the linear vibration dynamic modelof the gear mesh process, where k and c are the mesh stiff-ness and damped, respectively, ky,1 and ky,m are the pinion-1support stiffness and large gear support stiffness, respectively,and cy,1 and cy,m are the pinion-1 support damped and largegear support damped respectively. Mesh stiffness k and meshdamped c are the physical parameters that are affected by meshpoint. In addition, the gear mesh point will change along withgear mesh action line. Therefore, mesh stiffness k and meshdamped c are time-varying physical parameters. The linear vi-bration model of the shield TBM cutterhead driving system isshown in Fig. 5.

Based on the gear mesh dynamic,1–6 the relative positiondeviation function, elastic mesh force, and elastic mesh torque



Figure 3. Overall principle structure of shield TBM cutterhead driving system (motor-driven mode).

Figure 4. Linear vibration dynamic model.

Figure 5. Linear vibration dynamic model of cutterhead driving system.

can be obtained where yi(t) is the contact direction (line-of-action) vibration displacement of i-th pinion, and yim(t) is thelarge gear’s contact vibration displacement along contact di-rection of i-th pinion.

pi = ri ∗ θp,i − rm ∗ θm +(yi(t)− yim(t)

); (1)

Fi = kipi + cipi; (2)

Mc,i = Fi ∗ ri = kipiri + cipiri. (3)

Then, the elastic mesh force Fi and mesh torque Mc,i could beexpressed as

Fi = ki(ri ∗ θp,i − rm ∗ θm +

(yi(t)− yim(t)

))+ci

(ri ∗ θp,i − rm ∗ θm +

(yi(t)− yim(t)

)); (4)

pi = ri ∗ θp,i − rm ∗ θm +(yi(t)− yim(t)

)⇔

piri

= θp,i − im,i ∗ θm +

(yi(t)− yim(t)

)ri

; (5)

Mc,i = kiri(ri ∗ θp,i − rm ∗ θm +

(yi(t)− yim(t)

))+ciri

(ri ∗ θp,i − rm ∗ θm +

(yi(t)− yim(t)

)). (6)

Therefore, the elastic mesh force and torque of i-th pinion canbe equivalently expressed as

Fi = kf,i

(θp,i − im,i ∗ θm +

yi(t)− yim(t)

ri

)+cf,i


yi(t)− yim(t)

ri

); (7)

and

Mc,i = kt,i


yi(t)− yim(t)

ri

)+ct,i


yi(t)− yim(t)

ri

). (8)

The physical parameters in Eqs. (7) and (8) are defined as

kf,i = kiri , (i = 1, 2, ..., n); (9)

kt,i = kf,iri = kir2i , (i = 1, 2, ..., n); (10)

cf,i = ciri , (i = 1, 2, ..., n); (11)

ct,i = cf,iri = cir2i , (i = 1, 2, ..., n); (12)

im,i =rmri

, (i = 1, 2, ..., n). (13)

where kf ,i and cf ,i are elastic mesh force coefficients, kt,i

and ct,i are the mesh torque coefficients, and im,i is the gear



transmission ratio. The mesh stiffness ki and mesh dampedci are time-varying parameters, which can be expressed as theFourier series

ki(t) = ki,0 +∞∑j=1

ki,j cos(j2πfz,it+ φki,j

); (14)

ci(t) = ci,0 +∞∑j=1

ci,j cos(j2πfz,it+ φci,j

); (15)

where fz,i is the mesh frequency, ki,0 and ci,0 are the meanmesh stiffness and mean mesh damped respectively, and φk

i,j

and φci,j are the phase angle of mesh stiffness and mesh

damped, respectively.

2.2. Linear Dynamic Vibration ModelWithout losing generality, it was assumed that the cutterhead

is driven by n motors when establishing the general LTVMAvibration model. If the coupling mass is far less than the motorrotor mass, then the coupling inertia can be ignored. In fact,motor rotor mass cannot be far greater than coupling mass, andtherefore coupling inertia cannot be ignored. According to thetorque balance principle, torque balance equations of the firstinduction motor will yield

Te,1 = Jr,1θ1 + br,1θ1 + To,1; (16)

To,1 = Jz,1θ1 + bz,1θ1 +M1,1. (17)

For the second induction motor, the corresponding torque bal-ance equations are obtained as

Te,2 = Jr,2θ2 + br,2θ2 + To,2; (18)

To,2 = Jz,2θ2 + bz,2θ2 +M1,2. (19)

Likewise, the torque balance equations of the n-th inductionmotor will yield

Te,n = Jr,nθn + br,nθn + To,n; (20)

To,n = Jz,nθn + bz,nθn +M1,n; (21)

where To,i is the output torque of the i-th motor and M1,i isthe input torque of the i-th reducer (i = 1, 2, ..., n). The Equa-tions (17), (19), and (21) are then substituted into Eqs. (16),(18), and (20), respectively. The following equations will beobtained:

Te,1 = (Jr,1 + Jz,1) θ1 + (br,1 + bz,1) θ1 +M1,1; (22)

Te,2 = (Jr,2 + Jz,2) θ2 + (br,2 + bz,2) θ2 +M1,2; (23)

Te,n = (Jr,n + Jz,n) θn + (br,n + bz,n) θn +M1,n; (24)

Then, Eqs. (22)–(24) can be rewritten

Te,i = Jd,iθi + bd,iθi +M1,i , (i = 1, 2, ..., n); (25)

where the physical parameters are defined as Jd,i = Jr,i +Jz,i,bd,i = br,i + bz,i(i = 1, 2, ..., n). The input and out-put relationships of the i-th reducer can be described by theequation

θ = qθp,i,M2,i = qM1,i , (i = 1, 2, ..., n); (26)

where M2,i is the output torque of the i-th reducer. The torquebalance equation of the reduce-i or pinion-i will yield

M2,i = Jw,iθp,i + bw,iθp,i + Tp,i , (i = 1, 2, ..., n); (27)

Tp,i = Jp,iθp,i + bp,iθp,i +Mc,i , (i = 1, 2, ..., n); (28)

where Tp,i is the input torque of pinion-i, and Mc,i is the elas-tic mesh torque of pinion-i or gear pair-i. Substituting Eq. (28)into the Eq. (27), then the torque balance equation can be writ-ten as

M2,i = (Jp,i + Jw,i) θp,i + (bp,i + bw,i) θp,i

+Mc,i , (i = 1, 2, ..., n). (29)

Then, Eq. (29) can be rewritten as

M2,i = Jc,iθp,i + bc,iθp,i +Mc,i , (i = 1, 2, ..., n); (30)

where the system parameters are defined as Jc,i = Jp,i +Jw,i,and bc,i = bp,i +bw,i(i = 1, 2, ..., n). The elastic mesh forcewill excite contact direction (line-of-action) vibration displace-ment, because the mesh contact point is along with contactline direction. Thus, the pinions’ contact direction (line-of-action) vibration displacement in their own coordinates will beobtained as

mp,1y1(t) + cy,1y1(t) + ky,1y1(t) + F1 = 0

mp,2y2(t) + cy,2y2(t) + ky,2y2(t) + F2 = 0

, ...,

mp,n−1yn−1(t) + cy,n−1yn−1(t)

+ky,n−1yn−1(t) + Fn−1 = 0

mp,nyn(t) + cy,nyn(t) + ky,nyn(t) + Fn = 0

⇔ (31)

mp,iyi(t) + cy,iyi(t) + ky,iyi(t) + Fi = 0 , (i = 1, 2, ..., n)(32)

where mp,i is the mass of pinion-i, ky,i and cy,i is the sup-port stiffness and viscous damped of pinion-i (i = 1, 2, ..., n).Likewise, the large gear’s contact vibration displacement alongeach pinion’s contact direction will be obtained as

mg y1m(t) + cy,my

1m(t) + ky,my

1m(t) = F1

mg y2m(t) + cy,my

2m(t) + ky,my

2m(t) = F2

, ...,

mg yn−1m (t) + cy,my

n−1m (t) + ky,my

n−1m (t) = Fn−1

mg ynm(t) + cy,my

nm(t) + ky,my

nm(t) = Fn

⇔

(33)

mg yim(t) + cy,my

im(t) + ky,my

im(t) = Fi , (i = 1, 2, ..., n)

(34)



where mg is the mass of large gear, and ky,m and cy,m are thelarge gear support stiffness and viscous damped. The elasticmesh torque of pinion-i or gear pair-i has been presented inSection 2.1, and the elastic mesh force Fi and mesh torqueMc,i of pinion-i will be obtained

Fi = kf,i


yi(t)−yim(t)ri

)+cf,i


yi(t)−yim(t)ri

)Mc,i = kt,i


yi(t)−yim(t)ri

)+ct,i


yi(t)−yim(t)ri

) . (35)

Then, the torque balance equations of the large gear will beobtained as

Jmθm + bmθm + TL = Mm; (36)Mm = im,1Mc,1 + im,2Mc,2 + · · ·+ im,nMc,n

=n∑k=1

im,kMc,k; (37)

where TL is total load torque of the shield TBM cutterhead.Cutterhead design parameters and geology conditions will di-rectly affect the load torque, for instance, the cutterhead cuttingdepth, opening ratio, excavation diameter, cutterhead forms,cutter amount and distribution, rocks and soil properties, boul-ders, and so on. The shield TBM cutterhead’s load torque is es-timated and calculated,15–18 and the total load torque includessix parts of torque:17

TL = T1 + T2 + T3 + T4 + T5 + T6 =6∑j=1

Tj . (38)

Detailed derivations and descriptions of load torque compo-nents and parameters can be found in previous work.17 There-fore, calculation formulas of load torque are directly presentedhere.

T1 =D2vsqu

8Nc; (39)

T2 =πγH0µD

3

12; (40)

T3 =πγH0µ1

(D3 −D3

1

)12

; (41)

T4 =πγH0µ1B (D1 +D2)

2

8 cosα; (42)

tanα =D1 −D2

2B; (43)

T5 = γH0µ1DbLbRb, T6 =(1.414G0 +KHH)µ2Dg0

2.

(44)

The general LTVMA dynamic vibration model of the shieldTBM cutterhead driving system is obtained, and it can be de-picted as in Eq. (45). The LTVMA dynamic vibration modelcan be combined and simplified, and LTVMA vibration modelis equivalently expressed as in Eq. (46). Hence, the linear dy-namic vibration model of shield TBM cutterhead driving sys-tem is finally expressed as in Eq. (47). The LTVMA vibra-tion model can be transformed into the state-space dynamic

vibration model via selecting suitable state variables, outputvariables, and control variables. It could select suitable statevariables for motion equations (Eq. (48)—see the next page).The motion Eq. (48) is transformed into the MIMO state-spaceform via selecting the the state vector, output vector, and con-trol vector as in Eqs. (49)–(51)—see the next page.

xρ = Aρxρ +Bρuρ

yρ = Cρxρ; (53)

Aρ =

(Aρ11 Aρ12

Aρ21 Aρ22

); (54)

Bρ =(B11ρ B12

ρ

); (55)

Cρ =(Cρ11 Cρ12

). (56)

where xρ is a state vector, yρ is an output vector, and uρ isa control vector, Aρ is the state matrix, Bρ is a control ma-trix, and Cρ is an output matrix. The matrices Aρ, Bρ, andCρ are presented in Appendix 1. Likewise, it could select suit-able state variables for vibration equations of linear dynamicvibration model, Eq. (47):

Fi = kf,i


yi(t)− yim(t)

ri

)+cf,i


yi(t)− yim(t)

ri

)mp,iyi(t) + cy,iyi(t) + ky,iyi(t) + Fi = 0;

(57)

mg yim(t) + cy,my

im(t) + ky,my

im(t) = Fi , (i = 1, 2, ..., n).

(58)

The vibration Eq. (57) is transformed into the MIMO state-space form by selecting the following state vector, output vec-tor, and control vector. See Eq. (50), where xδ is a state vector,yδ is an output vector, and uδ is a control vector; Aδ is the statematrix, Bδ is a control matrix, and Cδ is an output matrix. Thematrices Aδ , Bδ , and Cδ are presented in Appendix 1. Then,the state Eq. (53) can be equivalently expressed as

xρ = Aρxρ +Bρuρ ⇔ xρ = Aρxρ +B11ρ uρ1 +B12

ρ uρ2.

(59)

It can be seen that the control variables uρ2 and uδ have rela-tionships with the state variables xδ and xρ respectively; there-fore, the state space equations are expressed as

xρ = Aρxρ +B11ρ uρ1 +B12

ρ uρ2

= Aρxρ +B11ρ uρ1 +B12

ρ A0xδ

xδ = Aδxδ +Bδuδ = Aδxδ +BδA1uρ

⇔ (60)

uρ2 = A0xδ

uδ = A1xρ; (61)(

xσxρ

)=

(Aδ BδA1

B12ρ A0 Aρ

)(xδxρ

)+

(0B11ρ

)uρ1. (62)

where A0, A1 is the transformation matrix, and 0 is zero ma-trix or vector. Therefore, the MIMO state space model of the



Te,i = Jd,iθi + bd,iθi +M1,i, θi = qθp,i,M2,i = qM1,i,

M2,i = Jc,iθp,i + bc,iθp,i +Mc,i,mp,iyi(t) + cy,iyi(t) + ky,iyi(t) + Fi = 0,

mg yim(t) + cy,my

im(t) + ky,my

im(t) = Fi , (i = 1, 2, ..., n)

Jmθm + bmθm + TL = Mm,Mm =n∑k=1

im,kMc,k, TL =6∑j=1

Tj ,

Fi = kf,i


yi(t)−yim(t)ri

)+ cf,i


yi(t)−yim(t)ri

),

Mc,i = kt,i


yi(t)−yim(t)ri

)+ ct,i


yi(t)−yim(t)ri

).

(45)

mp,iyi(t) + cy,iyi(t) + ky,iyi(t) + Fi = 0,mg yim(t) + cy,my

im(t) + ky,my

im(t) = Fi, , (i = 1, 2, ..., n)

qTe,i = Jiθp,i + biθp,i +Mc,i, Jmθm + bmθm + TL =n∑k=1

im,kMc,k, TL =6∑j=1

Tj , Ji =(q2Jd,i + Jc,i

),

bi =(q2bd,i + bc,i

), Fi = kf,i


yi(t)−yim(t)ri

)+ cf,i


yi(t)−yim(t)ri

)Mc,i = kt,i


yi(t)−yim(t)ri

)+ ct,i


yi(t)−yim(t)ri

).

(46)

qTe,i = Jiθp,i + biθp,i + kt,i


yi(t)−yim(t)ri

)+ ct,i


yi(t)−yim(t)ri

),

Ji =(q2Jd,i +Kc,i

), bi =

(q2bd,i + bc,i

)Jmθm + bmθm + TL =

n∑k=1

im,k

kt,k

(θp,k − im,k ∗ θm +

yk(t)−ykm(t)rk

)+ct,k


yk(t)−ykm(t)rk

), (i = 1, 2, ..., n)

Fi = kf,i


yi(t)−yim(t)ri

)+ cf,i


yi(t)−yim(t)ri

)mp,iyi(t) + cy,iyi(t) + ky,iyi(t) + Fi = 0,mg y

im(t) + cy,my

im(t) + ky,my

im(t) = Fi

(47)

qTe,i =(q2Jd,i + Jc,i

)θp,i +

(q2bd,i + bc,i

)θp,i

kt,i

(θp,i

−im,i ∗ θm +yi(t)−yim(t)

ri

)+ ct,i


yi(t)−yim(t)ri

),

Jmθm + bmθm + TL =n∑k=1

im,k

kt,k


yk(t)−ykm(t)rk

)+ct,k


yk(t)−ykm(t)rk

), (i = 1, 2, ..., n)

(48)

xρ = (θp,1θp,2 · · · θp,nθmωp,1ωp,2 · · ·ωp,nωm)

T ∈ R2n+2

θp,1 = ωp,1, · · · , θp,n = ωp,n, θm = ωm, yρ = (ωp,1ωp,2 · · ·ωp,nωm)T ∈ Rn+1

uρ =(uTρ1, u

Tρ2

)T ∈ R5n+1, uρ1 = (Te,1Te,2 · · ·Te,nTL)T ∈ Rn+1

uρ2 =(y1y2, · · · , yny1

my2m, · · · , ynmy1y2, · · · , yny1

my2m, · · · , ynm

)T ∈ R4n

(49)

xδ =

(y1y2 · · · yny1

my2m · · · ynmy1y2 · · · yny1

my2m · · · ynm

)T ∈ R4n

yδ =(y1y2 · · · yny1

my2m · · · ynm

)T ∈ R2n

uρ = (θp,1θp,2 · · · θp,nθmωp,1ωp,2 · · ·ωp,nωm)T ∈ R2n+2

(50)

xδ = Aδxδ +Bδuδ

yδ = Cδxδ; (51)

Aδ =

(Aδ11 Aδ12

Aδ21 Aδ22

), Bδ =

(B11δ B12

δ

), Cδ =

(Cδ11 Cδ12

). (52)



general LTV dynamic vibration model for shield the TBM cut-terhead driving system is finally obtained as(

xδxρ

)=

(Aδ BδA1

B12ρ A0 Aρ

)(0B11ρ

)uρ1; (63)(

yδyρ

)=

(Cδ 00 Cρ

)(xδxρ

). (64)

3. THEORETICAL ANALYSIS OFTHE LINEAR VIBRATION MODEL

The general LTVMA vibration model of the cutterhead driv-ing system has been established in the previous section. Atpresent, there is no good theoretical analysis method for theLTVMA vibration model seen in Eq. (47). In order to anal-yse the LTVMA dynamic vibration model in theory, the time-varying system parameter’s mesh stiffness, mesh damped, andload torque are replaced by the mean mesh stiffness, meanmesh damped, and mean load torque. Therefore, the vibra-tion model of the cutterhead driving system can be seen asthe linear time-invariant dynamic vibration system, and the dy-namic vibration model can be analysed by a transfer functionanalysis method or frequency domain analysis method. Then,the Laplace transform of the dynamic vibration model seen inEq. (47) will be obtained.

qTe,i(s) =(Jis

2 + bis)θp,i(s)

+ (kt,i + ct,is)

(θp,i(s)− im,i ∗ θm(s)

+yi(s)− yim(s)

ri

); (65)

(Jms

2 + bms)θm(s) + TL(s) =

n∑k=1

im,k

(kt,k + ct,ks)

(θp,k(s)

− im,k ∗ θm(s) +yk(s)− ykm(s)

rk

); (66)

Fi(s) = (kf,i + cf,is)

(θp,i(s)

− im,i ∗ θm(s) +yi(s)− yim(s)

ri

); (67)

mp,is2yi(s) + cy,isyi(s) + ky,iyi(s) + Fi(s) = 0; (68)

mgs2yim(s) + cy,msy

im(s) + ky,my

im(s) = Fi(s). (69)

Substituting Eq. (67) into vibration Eq. (68) and Eq. (69), thevibration Eq. (68) and Eq. (69) can be expressed as

mp,is2yi(s) + cy,isyi(s) + ky,iyi(s)+

(kf,i + cf,is)


+yi(s)− yim(s)

ri

)= 0; (70)

mgs2yim(s) + cy,msy

im(s) + ky,my

im(s) =

(kf,i + cf,is)


+yi(s)− yim(s)

ri

). (71)

Then, the vibration equation (Eqs. (70)–(71)) can be furtherexpressed as

ri(mp,is

2 + cy,is+ ky,i)yi(s)

+ (kf,i + cf,is)(yi(s)− yim(s)

)=

−ri (kf,i + cf,is) (θp,i(s)− im,i ∗ θm(s)) ; (72)

ri(mgs

2 + cy,ms+ ky,m)yim(s)

− (kf,i + cf,is)(yi(s)− yim(s)

)= ri (kf,i + cf,is) (θp,i(s)− im,i ∗ θm(s)) . (73)

Similarly, the motion (Eqs. (65)–(66)) can be equivalently ex-pressed as

qTe,i(s) =(Jis

2 + bis+ kt,i + ct,is)θp,i(s)

−im,i (kt,i + ct,is) θm(s)

+ (kt,i + ct,is)yi(s)− yim(s)

ri⇔ (74)

qriTe,i(s) = ri(Jis

2 + bis+ kt,i + ct,is)θp,i(s)

−riim,i (kt,i + ct,is) θm(s)

+ (kt,i + ct,is)(yi(s)− yim(s)

); (75)

and(Jms

2 + bms)θm(s) + TL(s)

=n∑k=1

im,k (kt,k + ct,ks) θp,k(s)

−n∑k=1

i2m,k (kt,k + ct,ks) θm(s)

+n∑k=1

im,k (kt,k + ct,ks)yk(s)− ykm(s)

rk⇔ (76)

TL(s) +(Jms

2 + bms)θm(s)

+n∑k=1

i2m,k (kt,k + ct,ks) θm(s)

=n∑k=1

i2m,k (kt,k + ct,ks) θp,k(s)

+n∑k=1


rk. (77)

Then, the motion equation could be further expressed as

Di(s)θp,i(s) = qriTe,i(s) + riim,i (kt,i + ct,is) θm(s)

− (kt,i + ct,is)(yi(s)− yim(s)

); (78)



TL(s) +Dm(s)θm(s)

=n∑k=1

im,k (kt,k + ct,ks) θp,k(s)

+n∑k=1


rk(79)

Di(s) = ri(Jis

2 + bis+ kt,i + ct,is)

; (80)

Dm(s) =(Jms

2 + bms)

+n∑k=1

i2m,k (kt,k + ct,ks) (81)

In order to simplify the analysis, it is assumed that constitutedequipments of the cutterhead driving system have identicalphysical parameters with the formula Jd,i = Jd, Jc,i = Jc,bd,i = bd, bc,i = bc, ri = r, im,i = im, kt,i = kt, ct,i = ct,kf ,i = kf , and cf ,i = cf , mp,i = mp, ky,i = ky, cy,i = cy

(Ji = q2Jd,i + Jc,i = J = q2Jd + Jc,bi = q2bd,i + bc,i =b = q2bd + bc), then Eqs. (78)–(80) can be rewritten as

D(s)θp,i(s) = qriTe,i(s) + rim (kt + cts) θm(s)

− (kt + cts)(yi(s)− yim(s)

); (82)

TL(s) +Dm(s)θm(s) =n∑k=1

im (kt + cts) θp,k(s)

+

n∑k=1

im (kt + cts)yk(s)− ykm(s)

r; (83)

D(s) = r(Js2 + bs+ cts+ kt

); (84)

Dm(s) =(Jms

2 + bms)

+ ni2m (kt + cts) . (85)

Likewise, the vibration Eqs. (72)–(73)) can be rewritten as

r(mps

2 + cys+ ky)yi(s)

+ (kf + cfs)(yi(s)− yim(s)

)= −r (kf + cfs) (θp,i(s)− im ∗ θm(s)) ; (86)

r(mgs

2 + cy,ms+ ky,m)yim(s)

− (kf + cfs)(yi(s)− yim(s)

)= r (kf + cfs) (θp,i(s)− im ∗ θm(s)) . (87)

Substituting Eq. (82) into Eq. (83), Eq. (83) can be equiva-lently expressed in Eq. (88). Substituting Eqs. (82)–Eq. (88)into vibration Eq. (86), the vibration Eq. (86) can be equiv-alently expressed as in Eq. (89). Likewise, substitutingEq. (82) and Eq. (88) into Eq. (87), the vibration Eq. (87)can be equivalently expressed as in Eq. (93), where the Dy(s),Dy,m(s), Dλ(s), DL(s), DΣ(s), DΩ(s), and DΛ(s) are definedas in Eq. (97). Then, Eqs. (91) and Eq. (94) can be equivalentlytransformed into the transfer function matrix (see Eq. (100)).

Finally, the vibration transfer function matrix is acquired as:

Y (s) =Gvib(s)−1GT (s)

U(s). (108)

Thus, the vibration-torque static gain matrix of the shield TBMcutterhead driving system could be obtained as

KV = Gvib(s)−1GT (s)|s=0,

Dy(s)|s=0 = − kykf , Dy,m(s)|s=0 = 0, Dλ(s)|s=0 = qkt,

DL(s)|s=0 = 1−nktim , DΣ(s)|s=0 = q

nkt,

DΩ(s)|s=0 = 0, DΛ(s)|s=0 =ky,m

kf

;

(109)

G11vib(s)|s=0 = diag

(−kykf− kykf· · · − ky

kf

); (110)

−G12vib(s)|s=0 = G21

vib(s)|s=0 = 0n×n (111)

G22vib(s)|s=0 = diag

(ky,mkf

ky,mkf· · · − ky,m

kf

),

G11T (s)|s=0 = G21

T (s)|s=0

=

qkt

+ qnkt

qnkt

· · · qnkt

1−nktim

qnkt

qkt

+ qnkt

· · · qnkt

1−nktim

......

. . ....

...qnkt

qnkt

· · · qkt

+ qnkt

1−nktim

.

(112)

KV = Gvib(s)−1GT (s)|s=0

= Gvib(s)|s=0−1GT (s)|s=0

=

kv11 kv12 · · · kv1,n kv1,n+1

kv21 kv22 · · · kv2,n kv2,n+1...

.... . .

......

kvn,1 kvn,2 · · · kvn,n kvn,n+1

kvn+1,1 kvn+1,2 · · · kvn+1,n kvn+1,n+1

kvn+2,1 kvn+2,2 · · · kvn+2,n kvn+2,n+1...

.... . .

......

kv2n,1 kv2n,2 · · · kv2n,n kv2n,n+1

. (113)

4. SIMULATION RESULTS OF THE LTVMAVIBRATION MODEL

In order to analyse the LTVMA vibration model, time-varying physical parameters such as mesh stiffness, meshdamped, and load torque are replaced by mean mesh stiff-ness, mean mesh damped, and mean load torque in the simu-lation. The corresponding simulation parameters are presentedin Appendix 2, and the simulation results of the linear vibrationmodel are shown in Figs. 6–7 (in the unload case), respectively.The vibration response results of the LTVMA dynamic modelin Figs. 6–7 obviously demonstrate that the vibration speeds ofpinions and the large gear appear as oscillation behaviours inthe initial stage. The vibration response results also show thatthe vibration speeds of pinions and the large gear is not stan-dard simple harmonic vibration. In addition, Figs. 6–7 showthat the vibration speeds of pinions and the large gear appearas complex harmonic components. The maximum vibrationspeed appears in the initial stage and the maximum vibrationspeed attenuates slowly. The maximum vibration speed ampli-tude (MVSA) of the pinions and large gear is about 1.297 mm

sand 0.1082 mm

s , respectively. Pinions and large gear’s atten-uation time of maximum vibration speed (ATMVS) is about80.00 s, but the vibration setting time (VST) of pinions andlarge gear is much bigger than their ATMVS. The pinions and



(rD(s)Dm(s)− nr2i2m (kt + cts)

2)θm(s) = −rD(s)TL(s) +

n∑k=1

qr2im (kt + cts)Te,k(s)−n∑k=1

imr (kt + cts)2 (yk(s)− ykm(s)

)+D(s)×

n∑k=1

im (kt + cts)(yk(s)− ykm(s)

)⇔

θm(s) = 1∆(s) ×

−rD(s)TL(s) + qr2im (kt + cts)

n∑k=1

Te,k(s)− Ω(s)n∑k=1

(yk(s)− ykm(s)

)∆(s) =

(rD(s)Dm(s)− nr2i2m (kt + cts)

2),Ω(s) =

(r (kt + cts)

2 −D(s) (kt + cts))im.

(88)

(mps

2 + cys+ ky)

− (kf + cfs)yi(s)−

1

r

(yi(s)− yim(s)

)=

1

D(s)

qrTe,i(s) + rim (kt + cts) θm(s)− (kt + cts)

(yi(s)− yim(s)

)− im ∗ θm(s)

; (89)

⇔

−(mps

2 + cys+ ky)

(kf + cfs)− 1

r+

(kt + cts)

D(s)

yi(s) +

1

r− (kt + cts)

D(s)

yim(s) =

qr

D(s)Te,i(s) +

rim (kt + cts)− imD(s)

D(s)θm(s)

; (90)

⇔

−(mps

2 + cys+ ky)

(kf + cfs)− 1

r+

(kt + cts)

D(s)

yi(s) +

1

r− (kt + cts)

D(s)

yim(s) =

qr

D(s)Te,i(s) +


D(s)

× 1

∆(s)

−rD(s)TL(s) + qr2im(kt + cts)

n∑k=1

Te,k(s)− Ω(s)n∑k=1

(yk(s)− ykm(s)

); (91)

⇔ Dy(s)yi(s) +Dy,m(s)yim(s) = Dλ(s)Te,i(s) +DL(s)TL(s) +DΣ(s)n∑k=1

Te,k(s) +DΩ(s)n∑k=1

(yk(s)− ykm(s)

). (92)

(mgs

2 + cy,ms+ ky,m)

(kf + cfs)yim(s)− 1

r

(yi(s)− yim(s)

)=

1

D(s)

qrTe,i(s) + rim (kt + cts) θm(s)− (kt + cts)

(yi(s)− yim(s)

)− im ∗ θm(s)

; (93)

⇔

(mgs

2 + cy,ms+ ky,m)

(kf + cfs)+

1

r− (kt + cts)

D(s)

yim(s) +

−1

r+

(kt + cts)

D(s)

yi(s)

=

qr

d(s)Te,i(s) +


D(s)θm(s)

; (94)

⇔

(mgs

2 + cy,ms+ ky,m)

(kf + cfs)+

1

r− (kt + cts)

D(s)

yim(s) +

−1

r+

(kt + cts)

D(s)

yi(s)

=

qr

D(s)Te,i(s) +


D(s)

× 1

∆(s)

−rD(s)TL(s) + qr2im(kt + cts)

n∑k=1

Te,k(s)− Ω(s)

n∑k=1

(yk(s)− ykm(s)

); (95)



⇔ DΛ(s)yim(s)−Dy,m(s)yi(s) = Dλ(s)Te,i(s) +DL(s)TL(s) +DΣ(s)n∑k=1

Te,k(s) +DΩ

n∑k=1

(yk(s)− ykm(s)

); (96)

Dy(s) = −(mps

2 + cys+ ky)

kf + cfs− 1

r+

(kt + cts)

D(s), Dy,m(s) =

1

r− (kt + cts)

D(s), Dλ(s) =

qr

D(s); (97)

DL(s) =r2im (kt + cfs)− rimD(s)

−∆(s), DΣ(s) =


D(s)× qr2im (kt + cts)

∆(s); (98)

DΩ(s) =rim (kt + cts)− imD(s)

D(s)× −Ω(s)

∆(s), DΛ(s) =

(mgs

2 + cy,ms+ ky,m)

(kf + cfs)+

1

r− (kt + cts)

D(s); (99)

Dy(s)yi(s) +Dy,m(s)yim(s) = Dλ(s)Te,i(s) +DL(s)TL(s) +DΣ(s)

n∑k=1


(yk(s)− ykm(s)

)DΛ(s)yim(s)−Dy,m(s)yi(s) = Dλ(s)Te,i(s) +DL(s)TL(s) +DΣ(s)

n∑k=1


(yk(s)− ykm(s)

)(100)

⇔ Gvib(s)Y (s) = GT (s)U(s), Y (s) =(y1(s)y2(s) · · · yn(s)y1

m(s)y2m(s) · · · ynm(s)

)T ∈ R2n (101)

U(s) = (Te,1(s)Te,2(s) · · ·Te,n(s)Te,L(s))T ∈ Rn+1, Gvib(s) =

(G11

vib(s) G12vib(s)

G21vib(s) G22

vib(s)

), GT (s) =

(G11T (s)

G21T (s)

); (102)

G11vib(s) =

Dy(s)−DΩ(s) −DΩ(s) · · · −DΩ(s)−DΩ(s) Dy(s)−DΩ(s) · · · −DΩ(s)

......

. . ....

−DΩ(s) −DΩ(s) · · · Dy(s)−DΩ(s)

∈ Rn×n (103)

G12vib(s) =

Dy,m(s) +DΩ(s) DΩ(s) · · · DΩ(s)

DΩ(s) Dy,m(s) +DΩ(s) · · · DΩ(s)...

.... . .

...DΩ(s) DΩ(s) · · · Dy,m(s) +DΩ(s)

∈ Rn×n (104)

G21vib(s) =

−Dy,m(s)−DΩ(s) −DΩ(s) · · · −DΩ(s)

−DΩ(s) −Dy,m(s)−DΩ(s) · · · −DΩ(s)...

.... . .

...−DΩ(s) −DΩ(s) · · · −Dy,m(s)−DΩ(s)

∈ Rn×n (105)

G22vib(s) =

DΛ(s) +DΩ(s) DΩ(s) · · · DΩ(s)

DΩ(s) DΛ(s) +DΩ(s) · · · DΩ(s)...

.... . .

...DΩ(s) DΩ(s) · · · DΛ(s) +DΩ(s)

∈ Rn×n (106)

G11T (s) = G21

T (s) =

Dλ(s) +DΣ(s) DΣ(s) · · · DΣ(s) DL(s)

DΣ(s) Dλ(s) +DΣ(s) · · · DΣ(s) DL(s)...

.... . .

......

DΣ(s) DΣ(s) · · · Dλ(s) +DΣ(s) DL(s)

∈ Rn×(n+1) (107)



large gear’s steady-state amplitude of vibration speed (SAVS)are about 0.002 mm

s and 0.3×10−3 mms respectively. Thus, the

steady-state vibration speed of pinions and the large gear issmall, and it could be inferred that the working states of theshield TBM cutterhead driving system can be reflected by pin-ions and large gear’s vibration.

The Fourier Transform (FT) of vibration speeds for pinionsand the large gear is presented in Figs. 8–9 The frequencyspectrum of vibration speeds of pinions and the large gear inFigs. 8–9 reveals that vibration frequencies of the pinions andthe large gear do not locate at the high frequency range anddo locate at the low frequency range. The vibration speed’smaximum frequency range (MFR) is less than 400 Hz. Thefrequency spectrum in Figs. 8–9 shows that the noise signalis smaller and mainly focuses on the high frequency range.In addition, the frequency spectrum in Figs. 8–9 shows thatthe vibration speed of pinions and the large gear contains con-stant components and harmonic components. Pinions 1, 3, 5,6, 8, and 10 have the same frequency spectrum and their cor-responding frequency of maximum signal intensity (MSI) isabout the 0.2883 Hz. However, pinions 2, 4, 7, and 9 have thesame frequency spectrum and their corresponding frequency ofMSI is about 0.2883 Hz. Furthermore, the frequency of MSIfor the large gear’s vibration speed is about 0.2883 Hz and itis equal to the pinions’ frequency of MSI. In the frequencyrange of 0–10 Hz, it concentrates about 90 % of the signalenergy in this frequency range, but it only concentrates about10 % of the signal energy in the frequency range 10–400 Hz.Therefore, the pinions and the large gear’s vibration frequencyspectrums are located at low frequency ranges, and the shieldTBM cutterhead driving system’s vibration is also located ata low frequency range. In addition, the vibration speed andits frequency spectrum of pinions and the large gear impliesthat the shield TBM’s health status and work conditions can bereflected by detecting the vibration of the cutterhead drivingsystem.

4.1. Physical parameters’ effects ondynamic vibration response

The proposed LTVMA dynamic vibration model of shieldTBM cutterhead driving system is simulated and studied underdifferent physical parameters, and the physical parameters’ ef-fects on dynamic vibration response are investigated under var-ious conditions. When discussing a specific physical parame-ter’s effects on the dynamic vibration response, other physicalparameters are fixed. Physical parameters such as motor ro-tor inertia and motor rotor viscous damped and their effectson the vibration response of dynamic vibration model are in-vestigated. The simulation results of the dynamic vibrationmodel under different parameters are shown in Figs. 10–21.The motor rotor inertia’s effects on the shield TBM cutterheaddriving system’s vibration response are investigated. Vibrationresponse results of the dynamic vibration model under vari-ous different motor rotor inertia (in unload case) are shownin Figs. 10–15. The dynamic vibration response results inFigs. 10–15 obviously show that the vibration speed of pinionsand the large gear contains harmonic components and oscil-lation behaviours in the initial stage. Figs. 10–11 show thatthe MVSA of pinions and the large gear is reduced when mo-tor rotor inertia is increased from 2.1 kgm2 to 3.1 kgm2. The

MVSA of pinions and the large gear is about 0.8911 mms and

0.0744 mms , respectively, when the motor rotor inertia is in-

creased to 3.1 kgm2.

The MVSA of pinions and the large gear appears in the ini-tial stage. The pinions and large gear’s ATMVS and VST areincreased when motor rotor inertia is increased to 3.1 kgm2.The SAVS of pinions and the large gear is also reduced, andthe SAVS of pinions and the large gear is about 1.5×10−3 mm

sand 0.18×10−3 mm

s , respectively, when the motor rotor iner-tia is increased to 3.1 kgm2. The vibration response resultsin Figs. 12–13 show that pinions and large gear’s MVSA andSAVS are reduced when the motor rotor inertia is further in-creased from 3.1 kgm2 to 4.1 kgm2. The MVSA of pinions andthe large gear is about 0.6937 mm

s and 0.0577 mms , respectively,

when the motor rotor inertia is increased to 4.1 kgm2. TheMVSA of pinions and the large gear appears as oscillation be-haviours in the initial stage. The vibration results in Figs. 12–13 also obviously show that the pinions and large gear’s VSTand ATMVS are increased considerably when the motor rotorinertia is increased from 3.1 kgm2 to 4.1 kgm2.

When the motor rotor inertia is further increased from4.1 kgm2 to 5.1 kgm2, the corresponding vibration responseresults in Figs. 14–15 obviously show that the vibration speedof the pinions and large gear appears as continuously oscillat-ing behaviours. The vibration response results in Figs. 14–15show that the pinions and large gear’s MVSA and SAVS are re-duced when the motor rotor inertia is increased from 4.1 kgm2

to 5.1 kgm2. The MVSA of pinions and large gear is about0.5679 mm

s and 0.0475 mms , respectively, when the motor rotor

inertia is increased to 5.1 kgm2. The MVSA of the pinionsand large gear appears as oscillation behaviours in the initialstage. The vibration results in Figs. 14–15 obviously show thatthe pinions and large gear’s VST and ATMVS are increasedconsiderable when the motor rotor inertia is increased from4.1 kgm2 to 5.1 kgm2. Therefore, motor rotor inertia has im-portant effects on the vibration response of the dynamic vibra-tion model. The pinions and large gear’s MVSA and SAVSwill be reduced, and pinions and the large gear’s VST and AT-MVS will be increased when motor rotor inertia is increased.On the contrary, the MVSA and SAVS will be increased, andVST and ATMVS will be reduced when the motor rotor inertiais reduced.

The motor rotor viscous damp’s effects on the vibration re-sponse of the shield TBM cutterhead driving system are in-vestigated, and the vibration response results of the dynamicvibration model under various motor rotor viscous damp (inthe unload case) are shown in Figs. 16–21. The dynamic vi-bration response results in Figs. 16–21 obviously show thatthe vibration speeds of the pinions and large gear contain har-monic components and oscillation behaviours in the initialstage. Figs. 16–17 show that the MVSA of the pinions andlarge gear is reduced when the motor rotor viscous damp isincreased from 0.25 kgm2

rad s−1 to 0.35 kgm2

rad s−1 . The MVSA of thepinions and large gear is about 0.222 mm

s and 0.0277 mms re-

spectively when the motor rotor viscous damp is increased to0.35 kgm2

rad s−1 . The MVSA of the pinions and large gear appearin the initial stage. The SAVS of the pinions and large gearis reduced, and the SAVS of pinions and large gear is about0.46×10−3 mm

s and 0.09×10−3 mms respectively when the mo-



tor rotor viscous damp is increased to 0.35 kgm2

rad s−1 .The pinions and large gear’s ATMVS and VST are also

reduced when the motor rotor viscous damp is increased to0.35 kgm2

rad s−1 . The vibration response results in Figs. 18–19show that the pinion and large gear’s MVSA and SAVS are re-duced when the motor rotor viscous damp is further increasedfrom 0.35 kgm2


rad s−1 . The MVSA of the pinions andlarge gear is about 0.2059 mm

s and 0.0254 mms , respectively,

when the motor rotor viscous damp is increased to 0.45 kgm2

rad s−1 .The MVSA of the pinions and large gear appears as oscillationbehaviours in the initial stage. The vibration response resultsin Figs. 18–19 obviously show that pinions and large gear’sVST and ATMVS are reduced when the motor rotor viscousdamp is increased from 0.35 kgm2


rad s−1 .When motor rotor viscous damp is further increased from

0.45 kgm2


rad s−1 , the vibration response results inFigs. 20–21 obviously show that the vibration speed of thepinions and large gear appears as continuously oscillatory be-haviours. The vibration response results in Figs. 20–21 obvi-ously show that the pinions and large gear’s MVSA and SAVSare reduced when the motor rotor viscous damp is increased to0.55 kgm2

rad s−1 . The MVSA of the pinions and large gear is about0.192 mm

s and 0.0231 mms , respectively, when the motor rotor

viscous damp is increased to 0.55 kgm2

rad s−1 . The MVSA of thepinions and large gear appears as oscillation behaviours in theinitial stage.

The vibration response results in Figs. 20–21 obviouslyshow that the pinions and large gear’s VST and ATMVS arereduced when the motor rotor viscous damp is increased from0.45 kgm2


rad s−1 .Therefore, the motor rotor viscousdamp has important effects on the vibration response of thedynamic vibration model. The pinion and large gear’s MVSAand SAVS will be reduced, and the pinions and large gear’sVST and ATMVS will be reduced too when the motor rotorviscous damp is increased. On the contrary, the MVSA andSAVS will be increased, and the VST and ATMVS will be in-creased as well when the motor rotor viscous damp is reduced.Large MVSA and SAVS will cause damage to the cutterheaddriving system in the long run. The vibration behaviours of thecutterhead driving system will reflect the actual working statesand environment of the shield TBM to some degree.

5. CONCLUSIONS

A general LTVMA dynamic vibration model is establishedfor the shield TBM cutterhead driving system, and the corre-sponding MIMO state-space model is also presented. The lin-ear vibration model is analysed to obtain the vibration-torquetransfer function matrix and vibration-torque static gain ma-trix. The LTVMA vibration model is simulated under vari-ous physical parameters conditions. Physical parameters suchas motor rotor inertia and motor rotor viscous damp and theireffects on the dynamic vibration response of linear vibrationmodel are investigated and analysed. A preliminary approachis proposed to reduce the vibration amplitude and vibration in-tensity of the cutterhead driving system by increasing motorrotor inertia and motor rotor viscous damp. Through numer-ically studying and analysing the LTVMA vibration model,the simulation results reveal the following: (1) The vibration

speeds of pinions and the large gear appear as complex har-monic components. The frequency spectrum of the vibrationspeed reveals that vibration frequencies of pinions and largegears are located at a low frequency range, and the shield TBMcutterhead driving system’s vibration is located at a low fre-quency range. Vibration noises concentrate on relatively highfrequency ranges. For the vibration frequency spectrum ofthe pinions and large gear, the vibration signal concentratesabout 90 % of signal energy in a low frequency range. Thevibration speed reveals that health status and working statesof the shield TBM can be reflected by detecting the cutter-head driving system’s vibrations. (2) The motor rotor inertiahas important effects on the dynamic vibration response of theLTVMA vibration model. The pinions and large gear’s MVSAand SAVS will be reduced when the motor rotor inertia is in-creased; however pinions and large gear’s VST and ATMVSwill be increased when the motor rotor inertia is increased. Onthe contrary, the MVSA and the SAVS will be increased whenthe motor rotor inertia is reduced; however, VST and ATMVSwill be reduced when the motor rotor inertia is reduced. (3)The motor rotor viscous damp has important effects on thedynamic vibration responses of the LTVMA vibration model.The pinions and large gear’s MVSA and SAVS will be reducedwhen motor rotor viscous damp is increased, and the pinionsand large gear’s VST and ATMVS will be reduced too whenthe motor rotor viscous damp is increased. On the contrary,the MVSA and SAVS will be increased when the motor rotorviscous damp is reduced, and the VST and ATMVS will be in-creased when the motor rotor viscous damp is reduced. In all,the vibration response of the cutterhead driving system will re-veal actual working states and environment of the shield TBMto some degree. The LTVMA vibration model provides a basisfor FDD and healthy monitoring of the shield TBM cutterheaddriving system.

ACKNOWLEDGMENT

This work is supported by research projectsNos.2007BAF09B01, 2009AA04Z155, and KGCX2-EW-104.

APPENDIX 1: MATRICES Aρ, Aδ, Bρ, Bδ, Cρ,AND Cδ FOR LTVMA DYNAMIC VIBRATIONMODEL

Aρ11 = 0(n+1)×(n+1), Aρ12 = I(n+1)×(n+1),

Aρ21 =

(A11

21 A1221

A2121 A22

21

)∈ R(n+1)×(n+1),

Aρ22 =

(A11

22 A1222

A2122 A22

22

)∈ R(n+1)×(n+1),

A1121 = diag

(−kt,1J1− kt,2

J2· · · − kt,n

Jn

)T∈ Rn×n,

A2221 = −

∑nk=1 i

2m,kkt,k

Jm∈ R

A1221 =

(im,1kt,1J1

im,2kt,2J2

· · · im,nkt,nJn

)T∈ Rn,



Figure 6. Pinion vibration speed of cutterhead driving system.

Figure 7. Large gear vibration speed of cutterhead driving system.

A2121 =

(im,1kt,1Jm

im,2kt,2Jm

· · · im,nkt,nJm

)T∈ Rn,

A1122 =

diag(−b1 + ct,1

J1− b2 + ct,2

J2· · · − bn + ct,n

Jn

)T∈ Rn×n,

A2222 = −

bm +n∑k=1

i2m,kct,k

Jm∈ R

A1222 =

(im,1ct,1J1

im,2ct,2J2

· · · im,nct,nJn

)T∈ Rn,

A2122 =

(im,1ct,1Jm

im,2ct,2Jm

· · · im,nct,nJm

)T∈ R

B11ρ =

(B11

11

B2111

)∈ R(2n+2)×(n+1),

B12ρ =

(B11

12 B1212 B13

12 B1412

B2112 B22

12 B2312 B24

12

)∈ R(2n+2)×4n

B1111 = 0(n+1)×(n+1),

B2111 = diag

(q

J1

q

J2· · · q

Jn− q

Jm

)∈ R(n+1)×(n+1)

B1112 = B12

12 = B1312 = B14

12 = 0(n+1)×n,

B2112 =

(Ω11

12

Ω2112

)∈ R(n+1)×n,

B2212 =

(Λ11

12

Λ2112

)∈ R(n+1)×n,

B2312 =

(Z11

12

Z2112

)∈ R(n+1)×n,

B2412 =

(S11

12

S2112

)∈ R(n+1)×n,

Ω1112 = diag

(− kt,1J1r1

− kt,2J2r2

· · · − kt,nJnrn

)∈ Rn×n,

Λ1112 = −Ω11

12 ∈ Rn×n

Ω2112 =

(im,1kt,1Jmr1

im,2kt,2Jmr2

· · · im,nkt,nJmrn

)∈ Rn,

Λ2112 = −Ω21

12 ∈ Rn, S1112 = −Z11

12 ∈ Rn×n,

S2112 = −Z21

12 ∈ Rn

Z1112 = diag

(− ct,1J1r1

− ct,2J2r2

· · · − ct,nJnrn

)∈ Rn×n,

Z2112 =

(im,1ct,1Jmr1

im,2ct,2Jmr2

· · · im,nct,nJmrn

)∈ Rn

Cρ11 = 0(n+1)×(n+1),

Cρ12 = I(n+1)×(n+1),

Cρ =(Cρ11 Cρ12

)∈ R(n+1)×(2n+2)

Aδ11 = 02n×2n,

Aδ12 = I2n×2n,

Aδ21 =

(V 11

21 V 1221

V 2121 V 22

21

)∈ R2n×2n

Aδ22 =

(V 11

22 V 1222

V 2122 V 22

22

)∈ R2n×2n



Figure 8. Frequency spectrum of each pinion’s vibration speed for the cutterhead driving system.

Figure 9. Frequency spectrum of large gear’s vibration speed for the cutterhead driving system.

Figure 10. Pinion vibration speed with motor rotor inertia 3.1 kgm2.

Figure 11. Large gear vibration speed with motor rotor inertia 3.1 kgm2.



Figure 16. Pinion vibration speed with motor rotor viscous damped 0.35 kgm2

rad s−1 .

Figure 17. Large gear vibration speed with motor rotor viscous damped 0.35 kgm2

rad s−1 .


rad s−1 .


rad s−1 .




rad s−1 .


rad s−1 .

V 1121 = diag

(−ky,1r1 + kf,1

mp,1r1− ky,2r2 + kf,2

mp,2r2

· · · − ky,nrn + kf,nmp,nrn

)∈ Rn×n,

V 1221 = diag

(kf,1mp,1r1

kf,2mp,2r2

· · · kf,nmp,nrn

)∈ Rn×n,

V 2121 = diag

(kf,1mgr1

kf,2mgr2

· · · kf,nmgrn

)∈ Rn×n,

V 2221 = diag

(−ky,mr1 + kf,1

mgr1− ky,mr2 + kf,2

mgr2

· · · − ky,mrn + kf,nmgrn

)∈ Rn×n,

V 1122 = diag

(−cy,1r1 + cf,1

mp,1r1− cy,2r2 + cf,2

mp,2r2

· · · − cy,nrn + cf,nmp,nrn

)∈ Rn×n,

V 1222 = diag

(cf,1

mp,1r1

cf,2mp,2r2

· · · cf,nmp,nrn

)∈ Rn×n,

V 2122 = diag

(cf,1mgr1

cf,2mgr2

· · · cf,nmgrn

)∈ Rn×n,

V 2222 = diag

(−cy,mr1 + cf,1

mgr1− cy,mr2 + cf,2

mgr2

· · · − cy,mrn + cf,nmgrn

)∈ Rn×n

B11δ =

Ψ11

11

Ψ2111

Ψ3111

Ψ4111

∈ R4n×(n+1), B12δ =

Ψ11

12

Ψ2112

Ψ3112

Ψ4112

∈ R4n×(n+1),

Ψ1111 = Ψ21

11 = Ψ1112 = Ψ21

12 = 0n×(n+1)

Ψ3111 =

(Φ11

11Φ1112

)∈ Rn×(n+1),

Φ1111 = diag

(− kf,1mp,1

− kf,2mp,2

· · · − kf,nmp,n

)∈ Rn×n,

Φ1211 =

(im,1kf,1mp,1

im,2kf,2mp,2

· · · im,nkf,nmp,n

)T∈ Rn

Ψ4111 =

(Π11

11Π1112

)∈ Rn×(n+1),

Π1111 = diag

(kf,1mg

kf,2mg· · · kf,n

mg

)∈ Rn×n,

Π1211 =

(− im,1kf,1

mg− im,2kf,2

mg· · · − im,nkf,n

mg

)T∈ Rn

Ψ3112 =

(Φ11

12Φ1212

)∈ Rn×(n+1),

Φ1112 = diag

(− cf,1mp,1

− cf,2mp,2

· · · − cf,nmp,n

)∈ Rn×n,

Φ1212 =

(im,1cf,1mp,1

im,2cf,2mp,2

· · · im,ncf,nmp,n

)T∈ Rn

Ψ4112 =

(Π11

12Π1212

)∈ Rn×(n+1),

Π1112 = diag

(cf,1mg

cf,2mg· · · cf,n

mg

)∈ Rn×n,



Π1212 =

(− im,1cf,1

mg− im,2cf,2

mg· · · − im,ncf,n

mg

)T∈ Rn

Cδ11 = I2n×2n, Cδ12 = 02n×2n, Cδ =

(Cδ11C

δ12

)∈ R2n×4n,

A0 = I4n×4n, A1 = I(2n+2)×(2n+2),

A2 =(0(n+1)×(n+1)I(n+1)×(n+1)

)∈ R(n+1)×(2n+2),

Ji =(q2Jd,i + Jc,i

), bi =

(q2bd,i + bc,i

), (i = 1, 2, ..., n)

APPENDIX 2: SIMULATION PARAMETERS

n=10, q=165.30, Jc,i=1.000 kgm2 (i=1,2,...,10)Jd,i=2.100 kgm2 (i=1,3,5,6,8,10), Jd,k=2.110 kgm2

(k=2,4,7,9) bd,i=0.225 kgm2

rad s−1 (i=1,3,5,6,8,10),

bd,k=0.250 kgm2

rad s−1 (k=2,4,7,9) bc,i=0.115 kgm2

rad s−1

(i=1,3,5,6,8,10), bc,k=0.125 kgm2

rad s−1 (k=2,4,7,9) kt,i = 105 Nmrad

(i=1,2,...,10), ct,i=50 Nmrad s−1 (i=1,2,...,10) Jm=56.93 kgm2,

bm=0.921 kgm2

rad s−1 , im=5.0, ri=r=0.20 m (i=1,2,...,10),rm=1.0 m TL=0 KNm, kf,i = 5 ∗ 105 N

rad , cf,i = 250 Nrad s−1

(i=1,2,....,10) mp,i=10.75 kg, ky,i = 2.5 ∗ 106 Nm ,

cy,i = 2.0 ∗ 103 Nm s−1 (i=1,2,....,10) mg=65.78 kg,

ky,m = 2.0 ∗ 107 Nm , cy,m = 1.125 ∗ 104 N

m s−1

REFERENCES1 Chen, Y.P, and Lim, T.C, Dynamics of hypoid gear trans-

mission with nonlinear time-varying mesh characteristics,ASME Transactions on the Journal of Mechanical Design,125(2), 373–382, (2003).

2 Alshyyab, A., and Kahraman, A. Nonlinear dynamic anal-ysis of a multi-mesh gear train using multi-term harmonicbalance method: period-one motions, Journal of Sound andVibration, 284, 151–172, (2005).

3 Kahraman, A., and Singh, R. Nonlinear dynamics of a spurgear pair, Journal of Sound and Vibration, 142, 49–75,(1990).

4 Kahraman, A., Lim, J., and Ding, H. A dynamic model ofa spur gear pair with friction, Proceedings of 12th IFToMMWorld Congress, Besancon (France), June 18–21, (2007).

5 Li, R.F. and Wang, J.J. Gear system dynamic [M], ChinaScience Press, 105–110 (in Chinese), (1995).

6 Chen, Q.W., Guo, Y., Hu, W.L., and Wang, J.X., Dynamicsanalysis and modeling of multi-motor synchronized drivingsystem, Journal of Southeast University (Natural ScienceEdition), 34, 135–140 (in Chinese), (2004).

7 Jodl, H.G, and Stempkowski, R. Operation research aspectof TBM drive-case study of the Wienerwald tunnel, Pro-ceedings of International Lecture Series TBM TunnelingTrends, Hagenberg, Austria, 14–15, 69–79, (1995).

8 Zhu, W. and Chen, R.J. Shield tunnel construction tech-nology situation and prospect, Geotechnical EngineeringWorld, 4(11) (in Chinese), (2001).

9 Wang, S.Y. Cutterhead driving study of shield machine,Urban Roads and Flood Control, 12, 95–96 (in Chinese),(2007).

10 Kovri, K. and Ramoni, M. Urban tunneling in soft groundusing TBM, Proceedings of the International Congress onMechanized Tunneling: Challenging Case Histories, Po-litecnico Ditorino, Italy, 16–19, (2004).

11 Yukinori, K. Present status and technology of shield tun-neling method in Japan, Tunneling and Underground SpaceTechnology, 18, 145–159, (2003).

12 Acaroglu, O., Ozdemir, L., and Asbury, B. A fuzzy logicmodel to predict specific energy requirement for TBM per-formance prediction, Tunneling and Underground SpaceTechnology, 23, 600–608, (2008).

13 Bernhard, M., Martin, H., and Lothar, A., Mechanisedshield tunnelling, Ernst Sohn, Berlin, (1996).

14 Guglielmentti, V., Grasso, P., Mahtab, A., and Shulin, X.Mechanized tunneling in urban areas, Taylor Francis, Lon-don, UK, (2008).

15 Mashimo, H. and Ishimura, T. Evaluation of the load onshield tunneling in gravel, Tunneling and UndergroundSpace Technology, 18, 233–241, (2003).

16 Zhang, Z.X. Estimate of loading rate for a TBM machinebased on measured cutter forces, Rock Mechanics and RockEngineering, 37(3), 239–248, (2004).

17 Yu, Y., Xu, B.F., Xing, Y., and Song, M.S. Torque momentanalysis of earth pressure balanced shield cutter head undersoft foundation, Chinese Journal of Construction Machin-ery, 3(2), 314–318 (in Chinese), 2004.

18 Guan, H.S. and Gao, B. Theoretical model for estimation ofcutter head torque in shield tunneling, Journal of SouthwestJiao Tong University, 2(43), 213–217 (in Chinese), (2008).

19 Zhang, K., Xia, Y.M., Tan, Q., Wang, K., and Yi, N.E. Es-tablishment of TBM disc cutter dynamic model for verti-cal vibration, Proceedings of the 2nd International Con-ference on Intelligent Robotics and Applications (ICIRA-2009), 374–382, (2009).

20 Li, X.H., Yu, H.B., Yuan, M.Z., Wang, J., and Yin, Y.Dynamic modeling and analysis of shield TBM cutter-head driving system, ASME Transactions on the Journal ofDynamic Systems, Measurement, and Control, 132, 1–14,(2010).

21 Li, X.H., Yu, H.B.,Yuan, M.Z., Yin, Y., and Pan, H. Studyon the linear dynamic model of shield TBM cutterhead driv-ing system, Proceedings of the 37th Annual Conference ofthe IEEE Industrial Electronics Society, Melbourne, VIC,Australia, 7–10, 3733–3740, (2011).

22 Li, X.H., Yu, H.B., Yuan, M.Z., and Zhao, Y. Research ondynamic models and performances of shield tunnel boringmachine cutterhead driving system, Advances in Mechani-cal Engineering, 1–29, (2013).

23 Li, X.H., Yu, H.B., Zeng, P., Yuan, M.Z., Sun, L.X.,and Zhao, Y. Dynamic two-dimensional nonlinear vibrationmodeling and analysis for shield TBM cutterhead drivingsystem, Transactions of the Canadian Society for Mechan-ical Engineering (Accepted), (2013).

24 Hu, G.L., Gong, G.F., and Yang, H.Y. Electro-hydrauliccontrol system of shied tunnel boring machine for simula-tor stand, The 6th International Conference on Fluid PowerTransmission and Control, Hangzhou, 94–99, (2005).


An Algorithm for Solving Torsional VibrationProblems Based on the InvariantImbedding MethodAntonio Lopes GamaDepartment of Mechanical Engineering, Universidade Federal Fluminense-UFF, 24210-000, Niteroi, RJ, Brazil.

Rafael Soares de OliveiraNational Institute of Metrology-INMETRO, 25250-020, Duque de Caxias, RJ, Brazil.

(Received 25 September 2012; provisionally accepted 20 August 2013; accepted 18 September 2013)

In this work the invariant imbedding method has been used to develop an algorithm to study the torsional vibrationof non-uniform systems. The algorithm is based on the propagation, reflection, and transmission of waves in astepped waveguide and is part of a procedure to transform two-point boundary value problems in initial valueproblems. Based on this approach, a continuous model has been developed and a simple, versatile, and robustalgorithm has been constructed to solve torsional vibration problems of non-uniform shafts with circular cross-sections. The proposed solution algorithm was extensively evaluated through comparisons with analytical solutionsand the finite element method. The results show that the proposed method can provide the exact solution foruniform shafts with concentrated elements and accurate results for a wide variety of torsional vibration problems.Systems with continuously varying geometry may be approximated by stepped shafts. The proposed method canalso be used to study the dynamic behaviour of others stepped systems.

1. INTRODUCTION

In the present paper, an approach usually applied to investi-gate the wave propagation in layered media is used to developa continuous model to study the torsional vibration of non-uniform systems.1 The formulation of the proposed methodconsiders that partial torsional waves propagate in opposite di-rections in a system with stepped changes in its properties asshown in Fig. 1. In the frequency domain, the governing equa-tions are written in a state space form where the state variablesare the angular displacement and twisting moment. The statematrix varies in a piece-wise constant fashion according to theproperties of each segment of the rod. The part of the rod withcontinuously varying geometry (e.g., a conical part) is approx-imated by thin uniform segments. The solution of the statespace equation is obtained by employing a discrete version ofthe Riccati transformation, which is a key ingredient in the in-variant imbedding approach.2 This technique is used to trans-form two-point boundary value problems in initial value prob-lems, and is also known as the method of sweeps, the Riccatimethod, or the factorization method.3, 4 Based on this transfor-mation, a recursive algorithm has been constructed for the so-lution, providing a simple and powerful computational methodcapable of solving problems of torsional vibration in circularnon-uniform rods.5, 6 Comparisons to analytical solutions andfinite element results show that the proposed method can pro-vide the exact solution for the torsional vibration of uniformshafts with concentrated elements and an approximated solu-tion for shafts with a continuously varying geometry.

It must be pointed out that previous works have consid-ered the propagation, reflection, and transmission of waves tosolve vibration problems in finite inhomogeneous systems.7, 8

It must also be mentioned that other methods to obtain the solu-

Figure 1. Stepped shaft with a conical part approximated by cylindricalsegments.

tion for the torsional vibration of non-uniform rods have beenpresented in the literature. An analytical solution was pro-vided by Pouyet and Lataillade9 for specific profiles of non-uniform rods, and the exact solution for more general caseshas been obtained by Qiao et al.10 and Li.11 A continuousmodel for stepped shafts was proposed by Mioduchowski,12

and a general approach for stepped systems governed by theone-dimensional wave equation was presented by Bapat andBhutani.13 A new exact approach for the analysis of torsionalvibration of a non-uniform shaft carrying an arbitrary numberof rigid disks has been proposed by Chen.14 Xiang et al.15

used the modified Riccati torsional transfer matrix method tocalculate the torsional natural frequencies of a shaft systemmodelled as a chain consisting of an elastic spring with con-centrated mass points.

As described in the foregoing sections, the main contribu-tion of the present work is to provide a simple and concisealgorithm that is able to solve a great variety of vibration prob-lems. The algorithm can be easily implemented and used tosolve the forced torsional vibration of non-uniform systemswith classical or non-classical boundary conditions. Althoughthe proposed method has been developed to study the vibra-


A. L. Gama, et al.: AN ALGORITHM FOR SOLVING TORSIONAL VIBRATION PROBLEMS BASED ON THE INVARIANT...

tion of systems governed by the one-dimensional wave equa-tion, the algorithm presented here can also be applied to studythe dynamic behaviour of other stepped systems. In fact, thismethod has been used by the author to investigate the high fre-quency response of stepped layered composite beams, wherethe discrete form of the Riccati transformation has been usedto solve a stiff system of ordinary differential equations.6

2. THE STATE SPACE EQUATION

The algorithm presented in this work to study the vibrationof stepped systems will be used for solving the torsional vibra-tion of non-uniform rods. In the proposed method, the rod istreated as one formed by a series of cylindrical segments. Thestate space equation is formulated assuming that each cylin-drical part is elastic, homogeneous, and isotropic. It is alsoconsidered that the angular motion occurs as a rotation of thecross-sectional area as a whole; i.e., all the points of a cross-section present the same angular displacement.

From mechanics of solids, the relationship between the an-gular displacement θ(x, t) and the twisting moment M(x, t) isgiven by:

∂θ(x, t)

∂x=M(x, t)

GJ(x); (1)

where G is the shear modulus of elasticity and J is the polararea moment of inertia of the shaft cross-section. The equationof motion for the free torsional vibration is written as

∂M(x, t)

∂x= I(x)

∂2θ(x, t)

∂t2; (2)

where I(x) is the polar mass moment of inertia per unit length.If the time dependence of θ(x, t) and M(x, t) is harmonic andrepresented by functions of the form θ(x, t) = θ(x)e−iωt andM(x, t) = M(x)e−iωt, Eqs. (1) and (2) reduce to:

dθ(x)

dx=

M(x)

GJ(x); (3a)

dM(x)

dx= −ω2I(x)θ(x). (3b)

Using Eqs. (3a) and (3b), a state space equation is written in amatrix form as

dζ

dx= Tζ; (4)

where ζ is the state vector and T is the state matrix given by:

ζ =

θ(x)M(x)

; (5a)

T =

[0 1

GJ−ω2I 0

]. (5b)

3. SOLUTION OF THE STATE SPACEEQUATION

The solution of the state space equation is based on the in-variant imbedding method where a two-point boundary valueproblem is transformed in an initial value problem.1–4 The pur-pose of the following procedure is to find a solution for Eq. (4)in the form

M(x) = K(x)θ(x); (6)

where K(x) is called here the global torsional stiffness. Ofcourse, in general K(x) depends on the material and geometryof the rod, as well as frequency.

The solution of Eq. (4) may be written as

ζ(x) = N(x)ζ(0); (7)

where N is the transfer matrix that relates the state vector in aposition x to its value in the initial position x = 0, and has theform

N(x) = eTx. (8)

Writing the matrix T in function of their eigenvalues andeigenvectors, one can rewrite Eq. (8) as

N(x) = V

diag[ek1x, ek2x]V−1; (9)

where V is the matrix whose columns are the eigenvectors, andkα, α = 1, 2, are the eigenvalues of the state matrix T. Theeigenvalues of T are the wave numbers of partial waves thatpropagate in the positive and negative direction of the x-axisand are related as k2 = −k1. Separating the eigenvalues andeigenvectors according to waves that propagate in the positiveand negative direction of the x-axis, the matrix v is decom-posed as

V =

[A1 A2

L1 L2

]; (10)

where Aα and Lα, α = 1, 2, are components of the eigenvec-tors of T. The subscripts 1 and 2 are associated to the wavesthat propagate in the positive and negative direction of the x-axis, respectively.

The state vector in the initial position x = 0 may be ex-pressed as a linear combination of the eigenvectors of matrixT as

ζ(0) = Vc =

[A1 A2

L1 L2

]c1c2

; (11)

where c is a constant vector. Substituting Eqs. (9) and (11) intoEq. (7) yields

ζ(x) =

[A1 A2

L1 L2

] [W1(x) 0

0 W2(x)

]c1c2

; (12)

where W1(x) = W−12 (x) = ek1x are called here the propaga-

tor functions. At this point, the state variables θ(x) and M(x)will be separated in two parts, corresponding to the contribu-tions to the total fields of the partial waves that propagate in op-posite directions. This is a common practice in the descriptionof wave motion in layered media and is an important featureof the method of solution presented in this work.5 As men-tioned before, the subscripts 1 and 2 are related respectively tothe waves that propagate in the positive and negative directionof the x-axis. The angular displacement θ(x) and the twistingmoment M(x) are now represented as a result of waves thatpropagate in the positive and negative direction of the x-axisas

θ(x) = θ1(x) + θ2(x) (13a)

andM(x) = M1(x) +M2(x). (13b)

According to Eq. (12), each component of the variables θ(x)and M(x) in Eqs. (13a) and (13b) are given by

θα(x) = AαWα(x)cα (14a)



Figure 2. Partial waves propagating in a rod.

andMα(x) = LαWα(x)cα; (14b)

where α = 1, 2. The constants cα = θα(0)/Aα are determinedfrom Eq. (14a) at x = 0; therefore, Eqs. (14a) and (14b) arerewritten as

θα(x) = Wα(x)θα(0) (15a)

and

Mα(x) = LαWα(x)θα(0)

Aα. (15b)

The propagator functions Wα(x) relate the positive (α = 1)and negative (α = 2) partial waves with its value at x = 0.From Eq. (15a), θα(0) is obtained and substituted in Eq. (15b)as

Mα(x) = Sαθα(x); (16)

where Sα = LαAα

is called here the local torsional stiffness ofthe rod. It must be pointed out that both Wα(x) and Sα arefunctions of the geometry and material properties of each uni-form part of the rod and of the pair (k, ω). Substitution ofEq. (16) in Eq. (13b) results in

M(x) = S1θ1(x) + S2θ2(x). (17)

Consider now a torsional wave θ2(x) propagating in the neg-ative direction of the x-axis, next impinging on the left end ofthe rod (x = 0), and being reflected as a wave θ1(x) propagat-ing in the positive direction of the x-axis, as shown in Fig. 2.At x = 0, these waves are related by the reflection coefficientR1 (Fig. 2):

θ1(0) = R1θ2(0). (18)

From the relationship between the partial waves given byEq. (15a) and Eq. (18), Eq. (13a) may be written as

θ(x) = [H(x) + 1]θ2(x); (19)

where

H(x) =W1(x)R1

W2(x). (20)

Using Eqs. (19) and (20), Eq. (17) becomes

M(x) = S1H(x)θ2(x) + S2θ2(x). (21)

Finally, obtaining θ2(x) from Eq. (19) and substituting inEq. (21) one has the expression relating the twisting momentand the angular displacement in the desired form of Eq. (6) as

M(x) =S1H(x) + S2

H(x) + 1θ(x) = K(x)θ(x). (22)

Figure 3. Rod composed of two uniform segments.

Therefore, the global torsional stiffness K(x) of Eq. (6) is

K(x) =S1H(x) + S2

H(x) + 1. (23)

Observe that Eq. (22) provides a relationship between the to-tal twisting moment M(x) and the total angular displacementθ(x), whereas the local torsional stiffness Sα in Eq. (16) re-lates the partial twisting moment Mα(x) with the partial tor-sional waves θα(x). The local torsional stiffness Sα is a func-tion of the material property only, while the global torsionalstiffness K(x) depends on the reflection coefficient R1, andconsequently on the boundary conditions.

4. NON-UNIFORM ROD SUBJECTED TOEXCITATION TORQUE

The torsional waves propagating in a non-uniform rodformed by uniform segments will be reflected and transmittedat the interfaces between the segments. Therefore, it is neces-sary to determine the expression for the reflection coefficientR of the interfaces.

Consider first the case of a rod composed of two parts, andsubjected to a torque m at the interface (x = L1) betweenthe two segments, as shown in Fig. 3. It should be emphasizedthat if the torque is applied within a uniform segment, one mustdivide this segment to create an interface at the section wherethe torque is applied. From Eqs. (13a) and (22), one has forx = L−

1

θ(L−1 ) = θ1(L−

1 ) + θ2(L−1 ) (24a)

andM(L−

1 ) = K(L−1 )θ(L−

1 ) −m. (24b)

Now, writing Eqs. (17) and (18) for x = L+1 yields

M(L+1 ) = S1θ1(L+

1 ) + S2θ2(L+1 ) (25a)

andθ1(L+

1 ) = R2θ2(L+1 ) +Qm. (25b)

In Eqs. (25a) and (25b), R2 is the coefficient of reflection atx = L1 and Q is the term that transmits the effect of the exter-nal torque applied at x = L1 to the other sections of the rod.Note that Sα, α = 1, 2, in Eq. (25a), are the local torsionalstiffness of segment 2. Substituting Eq. (25b) into Eqs. (13a)and (25a) yields:

θ(L+1 ) = (1 +R2)θ2(L+

1 ) +Qm. (26a)

and

M(L+1 ) = (S1R2 + S2)θ2(L+

1 ) + S1Qm. (26b)



The condition of continuity of angular displacement and twist-ing moment at x = L1, θ(L−

1 ) = θ(L+1 ), and M(L−

1 ) =M(L+

1 ), and substitution of Eq. (26a) into Eq. (24b) leads to

K(L−1 )(I +R2)θ2(L+

1 ) +K(L−1 )Qm−m =

(S1R2 + S2)θ2(L+1 ) + S1Qm. (27)

Therefore, from Eq. (27), one can conclude that the expres-sions for R2 and Q are

R2 =K(L−

1 ) − S2

S1 −K(L−1 )

(28a)

andQ =

1

K(L−1 ) − S1

. (28b)

Repeating the same procedure presented in the preceding sec-tion to obtain Eq. (22), but now also employing Eqs. (25b) to(26b), that take into account the external torque applied to therod, one can finally find

M(x) = K(x)θ(x) + h(x)m; (29)

whereh(x) = [S1 −K(x)]W1(x)Q (30)

is the function that transfers to the other sections of the rod theeffect of the moment m applied at x = L1. The proceduredescribed in this section will be used in the next section in asolution algorithm to obtain the torsional natural frequenciesand distribution of angular displacement of stepped rods.

5. SOLUTION ALGORITHM FOR THETORSIONAL VIBRATION OF STEPPEDSHAFTS

The concepts presented in the preceding sections are nowgeneralized to the case of non-uniform rods with arbitrary ge-ometry, as illustrated in Fig. 4. Each uniform segment of therod is labelled by an index j = 1 . . . N . The interfaces betweensegments are also labelled by j, running from 1 to N + 1. It isassumed that the rod is connected at its left end to an elementof known stiffness K1. The stiffness of this element representsthe boundary condition of the rod at x = 0. For example, if therod is free one takes K1 = 0; whereas, if its end is clampedone lets 1/K1 = 0. Non-classical boundary conditions canbe implemented by using an appropriated value of K1. Con-centrated moments and concentrated elements such as rotaryinertias and torsional springs applied along the rod can also betaken into account. For instance, consider a twisting momentm1 at x = 0. The algorithm is composed of two parts, the firstto determine the frequency response function and the secondto obtain the distribution of angular displacement.

5.1. Frequency Response Function of theRod

In the first part of the algorithm, the frequency responsefunction of the rod is obtained as detailed below. Start fromthe left end of the rod, where the stiffness K1 is known, thenevaluate the stiffness K2 at the second interface between seg-ments 1 and 2 and so on until finally evaluating the stiffness

Figure 4. Shaft divided in N uniform segments.

KN+1 at the right end of the rod. The procedure is summa-rized in the following algorithm:

GIVEN ω, K1 AND m1 REPEAT FROM j = 1 TO N

Rj =(Kj − S

(j)2

)/(S(j)−Kj1

);

Qj = 1/(Kj − S

(j)1

);

Hj = W(j)1 (Lj)Rj/

(W

(j)2 (Lj)

);

hj =(S(j)1 −Kj

)W

(j)1 (Lj)Qj ;

Kj+1 =(S(j)1 Hj + S

(j)2

)/ (Hj + 1) ;

mj+1 = hjmj ;

END.

Note that the superscripts (j) of Sα andWα, α = 1, 2, corre-spond to the segments of the rod, while the subscript j is usedto designate the interfaces between the segments. To obtain,for example, the frequency response function of the angulardisplacement of the rod at x = L, one evaluates θN+1 for eachfrequency ω using the algorithm and the following expression:

MN+1 = KN+1θN+1 +mN+1; (31)

where MN+1 is the twisting moment at x = L. Also observethat mN+1 can also represent the contribution of the concen-trated twisting moments applied along the rod to the angulardisplacement of the rod at x = L. The natural frequencies ofthe rod can be obtained from its frequency response or fromthe variation of the global torsional stiffness KN+1 in functionof frequency.

5.2. Distribution of Angular DisplacementTo evaluate the angular displacement along the rod, one has

to march backwards using the following recursive algorithm:

REPEAT FROM j = N TO 1

θj2 = (1 +Hj)−1θj+1 − (1 +Hj)

−1W1jQjmj ;

θj = (I +Rj)W−12j θ

j2 +Qjmj ;

END.

Note that the determination of the angular displacement dis-tribution begins at x = L, where the angular displacementθN+1 (j = N ), determined using the first part of the al-gorithm and Eq. (31), is used to calculate the angular dis-placement θN at x = L − LN . Then, proceed to θN−1 at



x = L− (LN−1 +LN ), and so on, until determining the angu-lar displacement θ1 at x = 0. It should be pointed out that eachuniform segment of the rod can be arbitrarily subdivided intosmaller segments to obtain a detailed description of a vibrationmode. Note that one should record the values of Hj , Qj , Wαj ,andmj when performing the first part of the algorithm in orderto use the recursive algorithm to determine the distribution ofangular displacement.

6. EXAMPLE RESULTS

In this section the proposed method is demonstrated and theresults are compared with the results obtained with other meth-ods in the existing literature. Simulations using the commer-cial finite element (FEM) software COSMOSTM/SolidWorkswere also employed to verify the proposed method.

6.1. Uniform Rod with ConcentratedElements

The case of a uniform rod with concentrated elements waschosen to perform the first demonstration and verification ofthe proposed method. Natural frequencies obtained by Chenfor a clamped free shaft with five rotary inertia using a nu-merical method that provides the exact solution for uniformcircular shafts carrying multiple concentrated elements wasused for comparison.14 The problem is described as follows,using the nomenclature and units used by Chen14 shown be-tween parentheses: a circular shaft with length L = 1016 mm(40 in), diameter d = 25.4 mm (1.0 in), shear modulus ofshaft material G = 82.74 × 109 Pa (1.2 × 107 psi), massdensity of shaft material ρ = 7, 839 kg/m3 (0.283 lbm/in3),with five non-dimensional concentrated rotary inertias givenby I∗0v = JpL/I0v , where Jp is the mass moment of inertiaabout the rotational axis (x) per unit length and I0v representsthe vth attached rotary inertia. The locations of the rotary iner-tias are: ξ1 = 0.1, ξ2 = 0.3, ξ3 = 0.5, ξ4 = 0.7, and ξ5 = 0.9,where ξj = xj/L, j = 1 to 5. To determine the natural fre-quencies using the method proposed here, one should use thealgorithm presented in subsection 5.1.

Starting from the clamped end, a large value to the stiff-ness K1 should be taken to represent this boundary condition.If one is interested only in the natural frequencies, a uniquesegment is necessary for each uniform part of the shaft; oth-erwise, if the distribution of angular displacement must alsobe known, the uniform parts of the shaft should be subdividedarbitrarily in smaller segments. In this problem the concen-trated rotary inertia was represented by a thin large diameterdisc. The diameters are determined according to the values ofthe rotary inertias. A comparison between the results obtainedby Chen14 and the corresponding ones obtained using the pro-posed algorithm is presented in Table 1 for the lowest five nat-ural torsional frequencies. As the length of the rotary inertia isreduced, the values of the natural frequencies converge to theresults obtained by Chen.14 An almost perfect agreement ofresults is observed in Table 1 using disks with 0.005 in lengthwith large diameters as concentrated rotary inertias. Neverthe-less, it should be emphasized that the method does not accountfor the Poisson’s effect or for any cross-section deformation,and therefore cannot model accurately the high frequency vi-bration.

Table 1. Comparison with the results obtained by Chen14 for the lowest fivenatural torsional frequencies of the clamped-free shaft carrying five rotaryinertia.

Method Torsional Natural Frequencies (rad/s)ω1 ω2 ω3 ω4 ω5

Chen14 104.09671 304.98929 482.88004 619.40419 694.81096Proposed 104.10 305.09 483.06 619.65 695.16Method

Table 2. Comparison between the lowest five torsional natural frequenciesobtained with the proposed method and the FEM for a stepped shaft with fivesegments.

B.C. Method Torsional Natural Frequencies (rad/s)ω1 ω2 ω3 ω4 ω5

Proposed 9432 54823 74995 111400 140890MethodC-F FEM 9356 53956 73739 110070 139630

Difference 0.81 1.61 1.70 1.21 0.90(%)Proposed 41023 65849 90354 123760 152650Method

F-F FEM 40188 64610 88530 122500 150320Difference 2.08 1.92 2.06 1.03 1.55(%)

6.2. Stepped ShaftThe proposed method will now be evaluated through com-

parisons with the FEM results. First, to check the finite ele-ment model and analysis, the exact values of natural frequen-cies of a uniform shaft were calculated and compared to FEMresults. In the following FEM analysis, we have adopted tetra-hedral mesh model. The mesh was refined around geometricaldetails. Convergence tests have been performed to ensure thecalculated natural frequencies.

FEM results obtained for the torsional vibration of an ar-bitrary stepped shaft composed of five segments with lengthsequal to L1 = 60, L2 = L3 = 50, L4 = 80, and L5 = 70 mmand diameters D1 = 30, D2 = 35, D3 = 40, D4 = 50, andD5 = 40 mm, shear modulus G = 77 GPa, and mass den-sity ρ = 7900 kg/m3, are compared to the results obtainedwith the proposed method. The lowest five natural frequenciesfor the free-free (F-F) and clamped-free (C-F) boundary condi-tions (B.C.) are presented in Table 2. The natural frequencieswere obtained using the algorithm presented in subsection 5.1.The FEM analysis was implemented using solid elements anda modal shape analysis.

Table 2 also shows the difference between the values of thetorsional natural frequencies obtained by the proposed methodand the FEM. One can note that the results are in good agree-ment, with a maximum difference of approximately 2%.

6.3. Rod with a Continuously VaryingGeometry

In this section, the proposed method is evaluated, as well asan approximated solution for the torsional vibration of rodswith continuously varying geometry. Consider a rod withcontinuously-varying diameter, as for example a conical rod.To address this problem, a stepped cone, i.e., a cone composedof uniform segments with different diameters, is used as an ap-proximation to the conical rod. In order to verify the accuracyof the method for determining the torsional natural frequen-cies of continuously varying systems, the natural frequenciesof truncated cones (Fig. 5) with different angles, shear modulus



Figure 5. Truncated cone.

G = 77 GPa, and mass density ρ = 7900 kg/m3 were deter-mined and compared to the corresponding ones obtained usingthe FEM. The diameter D1 = 60 mm and the cone lengthL = 30 mm were kept constant while the angle α was in-creased from α = 5 to α = 30 in order to obtain differentvalues of ∆ = 1 −D2/D1.

The lowest two torsional natural frequencies for a clamped-free boundary condition were determined for each cone usingan increasing number of segments until the value of each natu-ral frequency converged. The segments of the stepped rod hadthe same length (L/N ) and the diameter of each segment wasequal to the diameter of the mid-segment. Table 3 presents acomparison between the results obtained using commercial fi-nite element software and the method proposed in this paper.Solid elements were employed to model the cone. A modalshape analysis was performed and the values of torsional nat-ural frequencies were identified among the results provided bythe FEM. As expected, a good agreement between the resultswas observed for small values of ∆; i.e., the accuracy of theproposed method decreases if the rod geometry becomes verydifferent from a cylindrical rod. Even if the number of seg-ments are increased, the proposed method cannot provide agood accuracy for large values of ∆.

6.4. Shaft with Uniform Parts and a ConicalSegment

A shaft with a geometry composed of two uniform parts anda conical section, as depicted in Fig. 6, with shear modulusG = 77 GPa and mass density ρ = 7900 kg/m3, was alsoanalysed using the proposed method. A comparison betweenthe lowest two torsional natural frequencies for a free-free andclamped-free boundary condition obtained with the FEM andthe proposed method are shown in Table 4. Figure 7 presents acomparison between the angular displacements along the shaftdetermined using the method presented in this work and theFEM. According to Table 4 and Figs. 7(a) to 7(d), the resultsobtained by the proposed method and the FEM are in goodagreement. To determine the distribution of angular displace-ment, the uniform parts of the rod can be subdivided into anynumber of smaller segments. The distribution of angular dis-placement is obtained using the second part of the solution al-gorithm described in section 5. The rod should be subdividedinto smaller segments, as short as necessary, to obtain the com-plete description of a vibration mode. A stepped cone was usedas an approximation to the conical part of the shaft. In this ex-ample, the uniform parts and the conical part were subdividedinto segments of 1.0 mm length (N = 500).

Table 4. Lowest two torsional natural frequencies (rad/s) for the shaft shownin Fig. 6.

Boundary Condition Method ω1 ω2

FEM 25445 38184Free-free Proposed Method 25950 37910

Difference (%) 1.9 0.7FEM 15093 26178

Clamped-free Proposed Method 14990 27010Difference (%) 0.7 3.1

Figure 6. Shaft with a conical part and uniform segments (dimensions inmillimetres).

7. CONCLUSIONS

A new method to study the torsional vibration of non-uniform rods has been developed using a theory usually ap-plied to investigate the wave propagation in layered media. Asimple and efficient algorithm based on the discrete form ofthe Riccati transformation has been proposed to determine thetorsional natural frequencies and angular displacement distri-bution of stepped shafts. The algorithm is computationally ef-ficient and very easy to implement. The proposed method wasevaluated through comparisons with analytical and finite el-ement method. The results show that the proposed solutionalgorithm may provide exact results for uniform shafts withconcentrated elements and accurate results for stepped shafts.Systems with continuously varying geometry can be properlyrepresented by stepped geometries; however, the accuracy ofthe method decreases if the continuous system becomes verydifferent from a uniform rod.

REFERENCES1 Chin, R. C. Y., Hedstorm, G. W., and Thigpen, L. Matrix

methods in synthetic seismograms, Geophysical Journal ofthe Royal Astronomical Society, 77 (2), 483–502, (1984).

2 Keller, H. B. and Lentine, M. Invariant imbedding, the boxscheme and an equivalence between them, SIAM Journalon Numerical Analysis, 19 (5), 942–962, (1982).

3 Dieci, L., Osborne, M. R., and Russel, R. D. A Riccatitransformation method for solving linear BVPs. I: theoret-ical aspects, SIAM Journal on Numerical Analysis, 25 (5),1055–1073, (1988).

4 Dieci, L., Osborne, M. R., and Russel, R. D. A Riccatitransformation method for solving linear BVPs. II: com-putational aspects, SIAM Journal on Numerical Analysis,25 (5), 1074–1092, (1988).

5 Braga, A. M. B., Honein, B., Barbone, P. E., and Her-rmann, G. Suppression of Sound Reflected from a Piezo-electric Plate, Journal of Intelligent Material Systems andStructures, 3 (2), 209–223, (1992).



Table 3. Lowest two torsional natural frequencies (rad/s) for truncated cones with different ∆ = 1 −D2/D1 (α = 5, 10, 20 and 30).

Method ∆ = 0.087 (α = 5) ∆ = 0.19 (α = 10) ∆ = 0.364 (α = 20) ∆ = 0.577 (α = 30)ω1 ω2 ω1 ω2 ω1 ω2 ω1 ω2

Proposed Method 175800 494780 190038 500484 287115 558220 287115 558220FEM 175740 492480 188800 494500 220260 499870 265800 514540

Difference (%) 0.03 0.5 0.8 1.2 3.3 3.8 8.0 8.5

Figure 7. Comparison between the distribution of angular displacement obtained by the proposed method (PM) and the FEM of the shaft described in Fig. 6 fordifferent boundary conditions: (a) and (b) first and second vibration modes free-free, (c) and (d) first and second vibration modes clamped-free, respectively.

6 Braga, A. M. B., de Barros, L. P. F., and Gama, A. L.Models for the high frequency response of active piezoelec-tric composite beams, Smart Materials and Structures, Pro-ceedings of the 4th European and 2nd MIMR Conference,115–122, (1998).

7 Langley, R. S. Wave evolution, reflection, and transmissionalong inhomogeneous waveguides, Journal of Sound andVibration, 227 (1), 131–158, (1999).

8 Lee, S. K., Mace, B. R., and Brennan, M. J. Wave prop-agation, reflection and transmission in non-uniform one-dimensional waveguides, Journal of Sound and Vibration,304 (1–2), 31–49, (2007).

9 Pouyet, J. M. and Lataillade, J. L. Torsional vibrations of ashaft with non-uniform cross section, Journal of Sound andVibration, 76 (1), 13–22, (1981).

10 Qiao, H., Li, Q. S., and Li, G. Q. Torsional vibration ofnon-uniform shafts carrying an arbitrary number of rigiddisks, Journal of Vibration and Acoustics, 124 (4), 656–659, (2002).

11 Li, Q. S. Torsional vibration of multi-step non-uniform rodswith various concentrated elements, Journal of Sound andVibration, 260 (4), 637–651, (2003).

12 Mioduchowski, A. Torsional waves and free vibrations ofdrive systems with stepped shafts, Archive of Applied Me-chanics (Ingenieur Archiv), 56 (4), 314–320, (1986).

13 Bapat, C. N., and Bhutani, N. General approach for freeand forced vibrations of stepped systems governed by theone-dimensional wave equation with non-classical bound-ary conditions, Journal of Sound and Vibration, 172 (1),1–22, (1994).

14 Chen, D.-W. An exact solution for free torsional vibrationof a uniform circular shaft carrying multiple concentratedelements, Journal of Sound and Vibration, 291 (3–5), 627–643, (2006).

15 Xiang, L., Yang, S., and Gan, C. Torsional vibration mea-surements on rotating shaft system using laser Doppler vi-brometer, Optics and Lasers in Engineering, 50, 1596–1601, (2012).


Free Flexural Vibration Response of Integrally-Stiffened and/or Stepped-Thickness CompositePlates or PanelsJaber Javanshir, Touraj Farsadi and Umur YuceogluDepartment of Aerospace Engineering, Middle East Technical University, Ankara, 06531, Turkey

(Received 29 September 2012; accepted 21 May 2013)

This study is mainly concerned with a general approach to the theoretical analysis and the solution of the freevibration response of integrally-stiffened and/or stepped-thickness plates or panels with one or more integral platestiffeners. In general, the Stiffened System is considered to be composed of dissimilar Orthotropic Mindlin Plateswith unequal thicknesses. The dynamic governing equations of the individual plate elements of the system andthe stress resultant-displacement expressions are combined and algebraically manipulated. These operations leadto the new Governing System of the First Order Ordinary Differential Equations in state vector forms. The newgoverning system of equations facilitates the direct application of the present method of solution, namely, theModified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). As shown in the present study, theMTMM is sufficiently general to handle the free vibration response of the stiffened system (with, at least, one or upto three or four Integral Plate Stiffeners). The present analysis and the method of solution are applied to the typicalstiffened plate or panel system with two integral plate stiffeners. The mode shapes with their natural frequenciesare presented for orthotropic composite cases and for several sets of support conditions. As an additional example,the case of the stiffened plate or panel system with three integral plate stiffeners is also considered and is shownin terms of the mode shapes and their natural frequencies for several sets of the boundary conditions. Also, someparametric studies of the natural frequencies versus the aspect ratio, stiffener thickness ratio, stiffener length (orwidth) ratio and the bending stiffness ratio are investigated and are graphically presented.

1. INTRODUCTORY REMARKS AND BRIEFREVIEW

The so-called integrally-stiffened and/or stepped-thicknessplate or panel systems of various configurations are primarilyused in air and space flight vehicle structures and substructures.They may also be utilized in high speed hydrodynamic vehi-cle structural systems.1–3 Their applications in engineering aredue to their advantageous properties of light-dead-weight, free-dom from mechanical (riveted, bolted, or welded) connections,and of their favourable stiffness characteristics in appropriateplaces. A typical and well-known example is their utilizationas aircraft wing cover panels with multi-stiffeners.3

These integrally-stiffened plate or panel systems are man-ufactured by means of the CAD-CAM process as one-pieceplate systems out of a solid (or raw stock) of advanced Metal-Alloy plates. In some applications, they may also be manu-factured as one-piece advanced composite plate systems withsome stepped thicknesses.2, 3

The integrally-stiffened plate or panel systems may gener-ally be categorized or grouped in terms of four main groupsas shown in Fig. 1. For the purposes of the present study, theGroup I systems may further be organized in terms of the num-ber of steps in their configurations, as shown in Fig. 2. In thepresent study, the integrally-stiffened plate or panel systems(one-step, two-step, three-step, four-step, etc., and all in onedirection) are to be analysed by means of a general approachto their free dynamic response.

Some significant research studies on the aforementionedintegrally-stiffened and/or stepped-thickness plates are avail-able in world-wide scientific and engineering literature.4–19 In

this brief literature survey, one may include the analytic so-lutions.4, 5, 10, 17, 19 The Raleigh-Ritz Method,7, 9, 11, 12 the FiniteElement Method (FEM),18 the Finite Strip Method (FSM),13, 15

the Kantrovich Method,8 and the Superposition Method,16

have all been studied. More recently, there appeared somestudies by Yuceoglu et al.,20–24 which employed the ModifiedTransfer Matrix Method (MTMM).20–33

The main concern of this study is to present a general ap-proach to the free vibration response of integrally-stiffenedand/or stepped-thickness, rectangular Mindlin plates or panels.Thus, the present work aims to achieve two objectives: (1) Ageneral theoretical analysis, and (2) A general method of solu-tion consistent with (1) for the various types of the integrally-stiffened plate or panel systems as defined and shown in Fig. 2.

In the present theoretical analysis, as a general approach,the integrally-stiffened and/or stepped-thickness plate or panelsystem of Group I, Type 4 (or Four-Step case) in Fig. 3 will beconsidered. Later on, it will be shown that the Lower-Step orHigher-Step cases may easily be obtained from the aforemen-tioned case. Thus, the Group I, Type 4 (or Four-Step case) ispresented in terms of its general configuration, material direc-tions, coordinate systems, and the longitudinal cross-section inFig. 3. Each stiffened plate system is assumed to be a com-bination of Mindlin plates34 with dissimilar orthotropic mate-rial properties and unequal thicknesses. In both figures, eachplate or panel system is considered to be simply supported atx = 0, a, while the boundary conditions in the y-directioncan be arbitrarily specified within the Mindlin plate theory,34

which is one of the First Order Shear Deformation Plate The-ories (FSDPTs).34, 35

114 (pp. 114–126) International Journal of Acoustics and Vibration, Vol. 19, No. 2, 2014

J. Javanshir, et al.: FREE FLEXURAL VIBRATION RESPONSE OF INTEGRALLY-STIFFENED AND/OR STEPPED-THICKNESS COMPOSITE. . .

Figure 1. Groups (or classes) of stiffened plate or panel systems.22

Figure 2. Various types or classes of integrally-stiffened and/or stepped-thickness plates or panels.23

The importance of the transverse shear deformations on thedynamic response of plates are pointed out by Whitney andPagano,36 and also by Librescu et al.37 A brief review of someof the various Higher Order Shear Deformation Plate Theories(HSDPTs) can be found in Subramanian.38

2. GENERAL ANALYSIS AND DERIVATIONOF GOVERNING EQUATIONS

The first step in the present theoretical analysis is to applythe Domain Decomposition technique for the entire plate orpanel system under consideration. Thus, depending on a par-ticular type, the plate elements of the system are consideredto form Part I, Part II, Part III, etc. regions (or domains) asshown in the longitudinal cross-section of the stiffened systemof Fig. 3.

In the second step, the sets of the dynamic equations ofthe orthotropic Mindlin plates and also the stress resultants-displacement expressions are taken into account as given in

Appendix A. The entire system of the governing partial differ-ential equations (PDEs), in a special form (which is very suit-able for the present method of solution—i.e., MTMM), is pre-sented in Appendix B. The aforementioned governing PDEsmay be written in a compact matrix form, in terms of the statevectors of each part in a type or problem under considerationhere.

Hence, in this study, the Group I and Type 4 (or Four-StepCase) is to be considered (see also Fig. 3). Then, referring toAppendix B:

• In Part I region (or Left Plate Stiffener)

1

lI

∂

∂ξI

Y(1)

=

[C′(∂

∂η,∂2

∂t2, . . .

)]Y(1)(ξI, η)

; (1)

• In Part II region (or Right Plate Stiffener)

1

lII

∂

∂ξII

Y(2)

=

[D′(∂

∂η,∂2

∂t2, . . .

)]Y(2)(ξII, η)

; (2)



Figure 3. Integrally-stiffened and/or stepped-thickness plate or panel systemwith two plate stiffeners: (a) general configuration, geometry, coordinate, andmaterial directions; (b) longitudinal cross-section with Parts I, II, III, IV, V andcoordinate systems.

• In Part III region (or Far Left Plate Element)

1

lIII

∂

∂ξIII

Y(3)

=

[E′(∂

∂η,∂2

∂t2, . . .

)]Y(3)(ξIII, η)

; (3)

• In Part IV region (or Far Right Plate Element)

1

lIV

∂

∂ξIV

Y(4)

=

[F′(∂

∂η,∂2

∂t2, . . .

)]Y(4)(ξIV, η)

; (4)

• In Part V region (or Middle Plate Element)

1

lV

∂

∂ξV

Y(5)

=

[G′(∂

∂η,∂2

∂t2, . . .

)]Y(5)(ξV, η)

. (5)

In the above, ξk = yklk

(k = I, II, III, IV,V) and η = xa , and the

fundamental dependent variables (or the state vectors) of thestiffened system are given as

Y(j)(ξk, η)

=ψ(j)x , ψ(j)

y ,W (j);M(j)yk ,M

(j)y , Q(j)

y

T;

(j = 1, 2, 3, 4, 5, k = I, II, III, IV,V) . (6)

It should be pointed out that the above Eqs. (1)–(5) are cou-pled through the continuity conditions at each intersection be-tween the plate elements of the entire stiffened plate system.It is also important to note here that, although the above equa-tions are written for Type 4 (or the Four-Step case), they caneasily be reduced to Type 2 (or the Two-Step case). They canalso easily be extended to Type 6 (or the Six-Step case) or withthe appropriate continuity conditions at the interfaces of theindividual plate elements. Therefore, the analysis as presentedhere is quite general and it can be further extended, if desired,to other higher order-step cases in a similar manner.

The next step in the analysis is the Non-Dimensionalizationprocedure of the governing PDEs of Eqs. (1)–(5) in each oftheir respective regions. For this purpose the plate elementof Part I is chosen as the reference plate. Therefore, B11(1),h1, ρ1, and a are chosen as the main (or reference) quanti-ties (or parameters). Additionally, l′I, l

′II, l′III, l

′IV, l′V are se-

lected as the length reference parameters in the y-direction ineach part (or region), respectively. All other quantities are non-dimensionalized with respect to these parameters.

The dimensionless coordinates in Part I, Part II, . . . , Part Vregions, respectively, are

η = x/a; ξI = yI/l′I; (in Part I)

η = x/a; ξII = yII/l′II; (in Part II)

η = x/a; ξIII = yIII/l′III; (in Part III)

η = x/a; ξIV = yIV/l′IV; (in Part IV)

η = x/a; ξV = yV/l′V. (in Part V) (7)

The dimensionless parameters related to the orthotropic elasticconstant of plate elements are

B(j)

ik = B(j)ik /B

(1)11 ; (j = 1, 2, 3, . . . and i, k = 1, 2, . . .);

B(j)

ll = B(j)ll /B

(1)11 ; (j = 1, 2, 3, . . . and l = 1, 2, . . .); (8)

and the dimensionless parameters related to the densities andthe geometry of the plates are

ρ2 =ρ2ρ1

; ρ3 =ρ3ρ1

; ρ4 =ρ4ρ1

; ρ5 =ρ5ρ1

; ρ1 =ρ1ρ1

= 1;

LII =l′IIa

; LIII =l′IIIa

; LIV =l′IVa

; LV =l′Va

; LI =l′Ia

;

h2 =h2h1

; h3 =h3h1

; h4 =h4h1

; h5 =h5h1

; h1 =h1h1

= 1;

(9)

where a is the width of the entire plate system.The dimensionless natural frequency parameter ωmn of the

entire stiffened plate or panel system, (i.e., Type 4 (or Four-Step case)) is

ωmn =ρ1a

4ω2mn

h21B(1)11

; Ω = ωmn; (m,n = 1, 2, 3, . . .) .

(10)Here, the non-dimensional natural frequencies ωmn are ar-ranged in the ascending order of Ω1 < Ω2 < Ω3 < . . . Theymay be obtained once m is assigned and n is calculated ac-cordingly.

Now, considering the simple support conditions at x = 0, a,in the x-direction, then the generalized displacements andthe stress resultants can be expressed in the Classical Levy’sMethod in Fourier series in each part (or region for each plateelement). These series are given in Appendix C. Thus, by sim-ply substituting Classical Levy’s Solutions into the system ofEqs. (1)–(5), and making use of some algebraic manipulationsand combinations, the governing system of the first order or-dinary differential equations in the state-vector forms are ob-tained. Thus, for the Type 4 (or Four-Step case):

• In Part I region (or Left Plate Stiffener)

d

dξI

Y

(1)

mn

=[C′]

Y(1)

mn

; (0 ≤ ξI ≤ 1) ; (11)

with the continuity conditions at ξI = 0, 1.



• In Part II region (or Right Plate Stiffener)

d

dξII

Y

(2)

mn

=[D′]

Y(2)

mn

; (0 ≤ ξII ≤ 1) ; (12)

with the continuity conditions at ξII = 0, 1.

• In Part III region (or Far Left Plate Element)

d

dξIII

Y

(3)

mn

=[E′]

Y(3)

mn

; (0 ≤ ξIII ≤ 1) ; (13)

with the arbitrary boundary conditions at ξIII = 0 and thecontinuity conditions at ξIII = 1.

• In Part IV region (or Far Right Plate Element)

d

dξIV

Y

(4)

mn

=[F′]

Y(4)

mn

; (0 ≤ ξIV ≤ 1) ; (14)

with the arbitrary boundary conditions at ξIV = 1 and thecontinuity conditions at ξIV = 0.

• In Part V region (or Middle Plate Element)

d

dξV

Y

(5)

mn

=[G′]

Y(5)

mn

; (0 ≤ ξV ≤ 1) ; (15)

with the continuity conditions at ξV = 0, 1.

In the above, the dash (—) over the matrices indicates thatthese matrices are non-dimensionalized. The dimensionlessfundamental dependent variables or the state vectors of theproblem are now given as

Y(j)

mn (ξk)

=ψ(j)

mnx, ψ(j)

mny,W(j)

mn,M(j)

mnyx,M(j)

mny, Q(j)

mny

T;

(j = 1, 2, 3, 4, 5; k = I, II, III, IV,V) . (16)

The above dimensionless coefficient matrices[C′],[D′], . . . ,[

G′]

are of dimensions (6 × 6). They include the dimension-less geometric and material characteristics of the individualplate elements of the entire stiffened system. They also in-clude the unknown dimensionless frequency parameter ωmnof the system.

It is important to recall that the reduced system of Eqs. (11)–(15) is coupled by means of the continuity conditions at the in-terfaces of the plate elements of the system. At this point, theInitial Value and Boundary Value Problem of the free dynamicresponse of the stiffened plate or panel system under study arenow reduced or converted to the so-called Two-Point BoundaryValue Problem of Mechanics and Physics in terms of the gov-erning system of Eqs. (11)–(15). This is a very important stepin the theoretical analysis of the problems considered here.

In order to show that the present analytical formulation isquite general, the Group I and Type 2 (or Two-Step case), asshown in Fig. 4, may be considered. For this purpose, one mayreplace the following quantities as

l′I → l′I and[C′]→ [

C′]

;

l′III → l′II and[E′]→ [

D′]

;

l′IV → l′III and[F′]→ [

E′]

;

l′V → 0 and[G′]→ 0; (17)

Figure 4. Integrally-stiffened and/or stepped-thickness plate or panel systemwith a non-central plate stiffener: (a) general configuration, geometry, coordi-nate, and material directions; (b) longitudinal cross-section with Parts I, II, IIIand coordinate systems.

and the dimensionless coordinates as

ξI → ξI; η = x/a;

ξIII → ξII; η = x/a;

ξIV → ξIII; η = x/a;

ξV → 0; η = x/a. (18)

Making use of Eqs. (17) and (18) and inserting these intothe governing system of the First Order Ordinary DifferentialEquations (ODEs) in Eqs. (11)–(15), they can be reduced tothe Group I and Type 2 (or Two-Step case) of Fig. 4. Hence:

• In Part I region (or Non-Central Plate Stiffener)

d

dξI

Y

(1)

mn

=[C′]

Y(1)

mn

; (0 ≤ ξI ≤ 1) ; (19)

with the continuity conditions at ξI = 0, 1.

• In Part II region (or Left Plate Element)

d

dξII

Y

(2)

mn

=[D′]

Y(2)

mn

; (0 ≤ ξII ≤ 1) ; (20)

with the arbitrary boundary conditions at ξII = 0 and thecontinuity conditions at ξII = 1.

• In Part III region (or Right Plate Element)

d

dξIII

Y

(3)

mn

=[E′]

Y(3)

mn

; (0 ≤ ξIII ≤ 1) ; (21)

with the arbitrary boundary conditions at ξIII = 1 and thecontinuity conditions at ξIII = 0.



In the above, the dimensionless fundamental dependent vari-ables or the state vectors are now expressed as

Y(j)

mn (ξk)

=ψ(j)

mnx, ψ(j)

mny,W(j)

mn,M(j)

mnyx,M(j)

mny, Q(j)

mny

T;

(j = 1, 2, 3; k = I, II, III) . (22)

Again, it should be noted that the Initial Value and Bound-ary Value Problem corresponding to the configuration in Fig. 4is finally reduced to a Two-Point Boundary Value Problem ofMechanics and Physics in terms of the new set of the govern-ing systems of the First Order ODEs in Eqs. (19)–(21) for theGroup I and Type 2 (or Two-Step case); see also Yuceoglu etal.25–33

In a similar manner, one can write the governing system ofthe First Order ODEs for the Group I Type 6 (Six-Step case)(or higher order cases).

3. GENERAL METHOD OF SOLUTION ANDNUMERICAL PROCEDURE

The solution technique to be employed is the ModifiedTransfer Matrix Method (MTMM) (with Interpolation Polyno-mials). The state vector forms of the governing system of equa-tions as given in Eqs. (11)–(15), facilitates the present methodof solution. This semi-analytical and numerical solution pro-cedure is a combination of the Classical Levy’s Method, theTransfer Matrix Method, and the Integrating Matrix Method(with Interpolation Polynomials).

The aforementioned method has been developed and beenefficiently utilized for some classes of plate and shallow shellvibration problems by Yuceoglu et al.25–33, 39, 40 It can be foundin more detail in Yuceoglu and Ozerciyes,25–33 Yuceoglu etal.,39 and in earlier versions in Yuceoglu et al.40

The two other more recently developed versions of thismethod are the MTMM (with Chebyshev Polynomials) andthe MTMM (with Eigenvalue Approach). These recent ver-sions by Yuceoglu and Ozerciyes39 are also highly accurateand more suitable for computing the higher natural frequen-cies and modes (higher than the sixth mode and up to fifteen orhigher modes).

In the present study, in connection with the stiffened platesor panels, the MTMM (with Interpolation Polynomials) is tobe used. Referring to the previous studies on the integrally-stiffened and/or stepped-thickness plates by Yuceoglu etal.,20–24 the main steps of the method are briefly explainednext. The very first step is to discretise the state vectors andthe Coefficient Matrices as defined in Eqs. (11)–(15). Thediscretization procedure is simply to divide Part I, Part II, . . . ,Part V regions, into sufficient number of points (or stations)along ξI, ξII, . . . , ξV directions, respectively. Following the dis-cretization, in the second step, the discretized versions of thegoverning system of Eqs. (11)–(15), are pre-multiplied by theappropriate Global Integrating Matrices [LI] , [LII] , . . . , [LV]in their respective parts (or regions). The global integratingmatrices include integrating sub-matrices [l] for each part, re-spectively. Then, in the Part I region (Left Plate Stiffener), thefollowing equation is obtained

Y(1)

mn

−Y

(1)

ξI=0

= [LI]

[C′]

Y(1)

mn

. (23)

In Eq. (23), the discretizations are shown by the · sign on ma-trices along the ξ-directions; the subscript ξI = 0 means thatthe matrix is evaluated at the initial end point at ξI = 0. Theabove expression can be further rearranged between the initialend point ξI = 0 and a general station ξI > 0 in the Part I re-gion. Then, dropping the mn subscripts for convenience, oneobtains:

• In Part I region (Left Plate Stiffener)Y

(1)

ξI

=[U]

Y(1)

ξI=0

;

[U]

=

([I]− [LI]

[C′])−1

; (24)

where[U]

is the discretized Global Modified Transfer Ma-trix for Part I region and [I] is the unit matrix. Similarly, onecan write the following expressions for the Part II, Part III, . . . ,Part V regions:

• In Part II region (Right Plate Stiffener)Y

(2)

ξII

=[V]

Y(2)

ξII=0

;

[V]

=

([I]− [LII]

[D′])−1

; (25)

• In Part III region (Far Left Plate)Y

(3)

ξIII

=[W]

Y(3)

ξIII=0

;

[W]

=

([I]− [LIII]

[E′])−1

; (26)

• In Part IV region (Right Plate)Y

(4)

ξIV

=[S]

Y(4)

ξIV=0

;

[S]

=

([I]− [LIV]

[F′])−1

; (27)

• In Part V region (Middle Plate)Y

(5)

ξV

=[T]

Y(5)

ξV=0

;

[T]

=

([I]− [LV]

[G′])−1

; (28)

where[U],[V],[W],[S], and

[T]

are the discretizedforms of the Global Modified Transfer Matrices for their re-spective parts (or regions), the matrices [LI], [LII], [LIII], [LIV],and [LV] are the Integrating Matrices with roman subscriptsindicating the particular part (or region) in which they oper-ate. As mentioned before, they include the Integrating Sub-Matrices [l].

Here, one can express the relations between the state vectorsat the initial end points ξI, ξII, ξIII, ξIV, ξV = 0, and the statevectors at the final end points ξI, ξII, ξIII, ξIV, ξV = 1 in eachPart I, Part II, . . . , Part V regions, respectively. Thus, droppingthe · sign for convenience, the following equation systems areobtained:



Table 1. Material constants and dimensions of the stiffened plate or panel system.

Orthotropic Composite System with Two Plate Stiffeners Orthotropic Composite System with Three Plate StiffenersGraphite-Epoxy Kevlar-Epoxy Kevlar-Epoxy Graphite-Epoxy Kevlar-Epoxy Kevlar-EpoxyPlate 1 = Plate 2 Plate 3 = Plate 4 Plate 5 Plate 1 = Plate 2 = Plate 3 Plate 4 = Plate 5 Plate 6 = Plate 7

(j = 1, 2) (j = 3, 4) (j = 5) (j = 1, 2, 3) (j = 4, 5) (j = 6, 7)E

(j)x = 11.71 GPa E

(j)x = 5.5 GPa E

(j)x = 5.5 GPa E

(j)x = 11.71 GPa E

(j)x = 5.5 GPa E

(j)x = 5.5 GPa

E(j)y = 137.8 GPa E

(j)y = 76.0 GPa E

(j)y = 76.0 GPa E

(j)y = 137.8 GPa E

(j)y = 76.0 GPa E

(j)y = 76.0 GPa

G(j)xy = 5.51 GPa G

(j)xy = 2.10 GPa G

(j)xy = 2.10 GPa G

(j)xy = 5.51 GPa G

(j)xy = 2.10 GPa G

(j)xy = 2.10 GPa

G(j)xz = 2.5 GPa G

(j)xz = 1.5 GPa G

(j)xz = 1.5 GPa G

(j)xz = 2.5 GPa G

(j)xz = 1.5 GPa G

(j)xz = 1.5 GPa

G(j)yz = 3.0 GPa G

(j)yz = 2.0 GPa G

(j)yz = 2.0 GPa G

(j)yz = 3.0 GPa G

(j)yz = 2.0 GPa G

(j)yz = 2.0 GPa

ν(j)xy = 0.0213 ν

(j)xy = 0.024 ν

(j)xy = 0.024 ν

(j)xy = 0.0213 ν

(j)xy = 0.024 ν

(j)xy = 0.024

ν(j)yx = 0.25 ν

(j)yx = 0.34 ν

(j)yx = 0.34 ν

(j)yx = 0.25 ν

(j)yx = 0.34 ν

(j)yx = 0.34

ρ(j) = 1.6 g/cm2 ρ(j) = 1.3 g/cm2 ρ(j) = 1.3 g/cm2 ρ(j) = 1.6 g/cm2 ρ(j) = 1.3 g/cm2 ρ(j) = 1.3 g/cm2

h1 = 0.04 m h2 = h3 = 0.02 m h2 = h3 = 0.02 m h1 = 0.04 m h2 = h3 = 0.02 m h2 = h3 = 0.02 ma = 0.50 m a = 0.50 m a = 0.50 m a = 0.50 m a = 0.50 m a = 0.50 m

• In Part I region (Left Plate Stiffener)Y

(1)

ξI=1

=[U1,1

]01

Y

(1)

ξI=0

; (0 ≤ ξI ≤ 1) ; (29)

• In Part II region (Right Plate Stiffener)Y

(2)

ξII=1

=[V1,1

]01

Y

(2)

ξII=0

; (0 ≤ ξII ≤ 1) ; (30)

• In Part III region (Far Left Plate Element)Y

(3)

ξIII=1

=[W1,1

]01

Y

(3)

ξIII=0

; (0 ≤ ξIII ≤ 1) ; (31)

• In Part IV region (Far Right Plate Element)Y

(4)

ξIV=1

=[S1,1

]01

Y

(4)

ξIV=0

; (0 ≤ ξIV ≤ 1) ; (32)

• In Part V region (Middle Plate Element)Y

(5)

ξV=1

=[T1,1

]01

Y

(5)

ξV=0

; (0 ≤ ξV ≤ 1) ; (33)

where the subscripts 01 mean that the final forms of the dis-cretized Global Modified Transfer Matrices

[U],[V],[W],[

S], and

[T]

are transferring the state vectors from the initialend point 0 to the final end point 1 in each part (or region),respectively.

At this point, the continuity conditions in terms of the statevectors between (Part III, Part I), (Part I, Part V), (Part I,Part II), etc. are inserted into the above Eqs. (29) through (33).Some further combinations and manipulations yield the finalequation for the entire panel system as

Y(4)

ξIV=1

=[

S] [

V] [

T] [

U] [

W]

01

Y

(3)

ξIII=0

;

Y(4)

ξIV=1

=[

Q]

01

Y

(3)

ξIII=0

. (34)

It is important to observe here that[Q]01

is the final form ofthe discretized Overall Global Modified Transfer Matrix. Thismatrix transfers the above state variables from the initial endpoint (far left end support) ξIII = 0 to the final end point (farright end support) ξIV = 1 of the entire plate or panel systemof Group I and Type 4 (or Four-Step case). The above OverallGlobal Modified Transfer Matrix

[Q]01

can further be reduced

to a (3×3) matrix by making use of or inserting of the Bound-ary Conditions at the initial end point (or far left end support)ξII = 0 and the final right end point (far right end support)ξIII = 1. This operation yields the following:

[C0 (ωmn)]Y0

= 0 ;

Determinant of Coeff. Matrix∣∣ [C0 (ωmn)]

∣∣ = 0; (35)

where the above Coefficient Matrix [C0] implicitly includesthe unknown natural frequency parameter (ωmn). Thus, Y0is not necessarily zero, then the determinant of the CoefficientMatrix must be equal to zero, yielding the polynomial whoseroots are the natural frequencies of the entire plate or panelsystem:

Ω = ωmn; (m,n = 1, 2, 3, . . .) ;

Ω1 < Ω2 < Ω3 < . . . ; (36)

where the dimensionless natural frequencies ωmn are com-puted by searching the roots numerically on the basis of thegiven m and the assigned n values. After then, they are se-quenced or reorganized (or renamed) according to their mag-nitudes as shown above with a single subscript in Eq. (36).

4. SOME NUMERICAL RESULTSAND CONCLUSIONS

The present general approach to the analytical formulationand the method of solution are applied to the problem of thefree dynamic response of integrally-stiffened and/or stepped-thickness rectangular plate or panel system with two plate stiff-eners given in Fig. 3; i.e., Group I and Type 4 (or Four-Stepcase). The present analytical formulation can easily be ex-tended to the Type 6 (or the Six-Step case).

In the set of the present numerical examples, the entire plateor panel system is assumed to be made of orthotropic compos-ite system (Orthotropic case). The two and three plate stiff-eners are considered as made of Graphite-Epoxy and the plateelements of the system are chosen as Kevlar-Epoxy. Thus, it ispossible to compare, in terms of the mode shapes and the cor-responding natural frequencies, and consequently reach someimportant conclusions.

The material and the dimensional characteristics of or-thotropic composite cases are given in Table 1. The bound-ary conditions on all figures are shown only as the far leftand the far right support conditions of the whole stiffened sys-tem. They read from left to right along the y-direction as C =Clamped, S = Simple, and F = Free support conditions.



Figure 5. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness rectangular plate or panel system with two platestiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.15 m, lII = 0.15 m, lIII = 0.233 m,lIV = 0.233 m, lV = 0.234 m; a = 0.50 m, h1 = h2 = 0.04 m, h3 = h4 = h5 = 0.02 m, L = 1.00 m; boundary conditions in y-direction CC.

In Figs. 5–8 for the Orthotropic Composite Plate or PanelSystem, the mode shapes and their natural frequencies areshown for the (CC) and (CS) boundary conditions, respec-tively. The symmetric and/or skew-symmetric mode shapesare observed when the Complete Symmetry conditions hold,as in Fig. 5. Otherwise, in Fig. 6, this is not the case.

The significant effects of several important parameters onthe dimensionless natural frequencies of the present stiffenedsystem are also investigated. These parameters are the aspectratio a/L, the stiffeners length (or width) ratio lI(= lII)/L, thestiffener thickness ratio h3(= h4 = h5)/h1(= h2) and alsothe bending stiffness ratio D(3)

22 /D(1)22 (= D

(2)22 ). Each individ-

ual parameter versus the dimensionless natural frequencies arecomputed and presented in the next set of Figs. 9–12.

The aspect ratio a/L is the parameter which significantlyaffects the natural frequencies of the entire system. Thus, inFig. 9, the natural frequency curves are highly non-linear andthey increase sharply as the aspect ratio a/L increases, regard-less of the support conditions of the system.

The stiffeners length (or width) ratio lI(= lII)/L versus thedimensionless natural frequencies for each set of the support

conditions are shown in Fig. 10. Regardless of the supportconditions, the main characteristic observed is that the dimen-sionless natural frequencies exhibit linear behavior and theyincrease gradually with the increasing parameter under study.The bending stiffness ratio D(3)

22 /D(1)22 (= D

(2)22 ) versus the di-

mensionless natural frequencies are shown in Fig. 12. Re-gardless of the boundary conditions the natural frequencies in-crease gradually and almost linearly.

On the basis of the present Governing Equations and thepresent Solution Method, it can further be concluded that theseare fairly general for the Integrally-Stiffened and/or Stepped-Thickness Plates Systems considered in this study.

REFERENCES1 Hoskins, B. C. and Baker, A. A. Composite Materials for

Aircraft Structures, AIAA Educational Series, New York,(1986).

2 Marshall, I. H. and Demuts, E. (Editors) Supportabilityof Composite Airframes and Aero-Structures, Elsevier Ap-plied Science Publihers, New York, (1988).



Figure 6. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness rectangular plate or panel system with two platestiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.15 m, lII = 0.15 m, lIII = 0.233 m,lIV = 0.233 m, lV = 0.234 m; a = 0.50 m, h1 = h2 = 0.04 m, h3 = h4 = h5 = 0.02 m, L = 1.00 m; boundary conditions in y-direction CS.

3 Niu, M. C.-Y. Airframe Structural Design, Commilit PresLtd., Hong Kong, (1988).

4 Chopra, I. Vibration of Stepped-Thickness Plates, Interna-tional Journal of Mechanical Science, 16, 337–344, (1974).

5 Warburton, G. B. Comment on Vibration of Stepped-Thickness Plates by I. Chopra, International Journal of Me-chanical Sciences, 16, 239–239, (1975).

6 Laura, O. A. A. and Filipich, A. Fundamental Frequencyof Vibration of Stepped-Thickness Plates, Journal of Soundand Vibration, 50 (1), 157–158, (1977).

7 Bambil, D. V., et al. Fundamental Frequency of Trans-verse Vibration of Symmetrically Stepped Simply Sup-ported Rectangular Plates, Journal of Sound and Vibration,150, 167–169, (1991).

8 Cortinez, V. H. and Laura, P. A. A. Analysis of VibratingRectangular Plates of Discontinuously varying thickness bythe Cantorovich Method, Journal of Sound and Vibrations,137 (3), 457–461, (1990).

9 Yuan, J. and Dickson, S. M. The Flexural Vibration of Rect-angular Plate Systems Approached by Artificial Springs inthe Rayleigh-Ritz Method, Journal of Sound and Vibration,159 (1), 39–55, (1992).

10 Harik, I. E., Liu, X., and Balakrishnan, N. Analytic Solutionfor Free Vibration of Rectangular Plates, Journal of Soundand Vibration, 153 (1), 51–62, (1992).

11 Takabatake, H., Imaizumi, T., and Okatomi, K. SimplifiedAnalysis of Rectangular Plates with Stepped Thickness,ASCE Journal of Structural Engineering, 117, 1759–1779,(1995).

12 Bambill, D. V., Rossi, R. E., et al. Vibrations of a Rect-angular Plate with Free Edge in the Case of Discontinu-ously Varying Thickness, Ocean Engineering, 29 (1), 45–49, (1996).

13 Guo, S. J., Khane, A. I., and Moshreff-Torbati, M. VibrationAnalysis of Stepped-Thickness Plates, Journal of Soundand Vibration, 204 (4), 645–657, (1997).



Figure 7. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness plate or panel system with three plate stiffeners(orthotropic case). Plate 1 = Plate 2 = Plate 3 = Graphite-Epoxy, Plate 4 = Plate 5 = Plate 6 = Plate 7 = Kevlar-Epoxy; lI = lII = lIII = 0.140 m,lIV = lV = lVI = lVII = 0.145 m; a = 0.50 m, h1 = h2 = h3 = 0.04 m, h4 = h5 = h6 = h7 = 0.02 m, L = 1.00 m; boundary conditions in y-directionCC.

14 Li, Q. S. Exact Solutions for Free Vibration of Multi-StepOrthotropic Shear Plates, Journal of Structural Engineeringand Mechanics, 9, 269–288, (2000).

15 Cheung, Y. K., Au, F. T. K., and Zheng, D. V. FiniteStrip Method for the Free Vibrations and Buckling Analysisof Plates with Abrupt Changes in Thickness, Thin WalledStructures, 36, 89–110, (2000).

16 Gorman, D. J. and Singhal, R. Free Vibration Analysisof Catilever Plates with Step Discontinuities in Propertiesby Superposition Method, Journal of Sound and Vibration,253 (3), 631–652, (2002).

17 Xiang, Y. and Wang, C. M. Exact Buckling and VibrationAnalysis for Stepped Rectangular Plates, Journal of Soundand Vibration, 250 (3), 503–517, (2002).

18 Hull, P. V. and Buchanan, G. R. Vibration of ModeratelyThick Square Orthotropic Stepped-Thickness Plates, Jour-nal of Applied Acoustics, 64, 753–763, (2003).

19 Xiang, Y. and Wei, G. W. Exact Solutions for Buckling andVibration of Stepped-Rectangular Mindlin Plates, Interna-tional Journal of Solids and Structures, 41 (3), 279–294,(2004).

20 Yuceoglu, U., et al. Further Generalization of Analysisand Solution for Free Flexural Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels, VIKayseri Havacilik Sempozyumu (VIth Kayseri Symposiumon Aeronautics and Astronautics), Erciyes University, Kay-seri, Turkey, (2006).

21 Yuceoglu, U., Gemalmayan, N., and Sunar, O. FreeFlexural Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with a Central Plate Stiffener,48th AIAA/ASME/ASCE/AHS/ASC/ Structures, Struct. Dy-namics, Materials (SDM) Conference and Exhibit, Waikiki,Hawaii, (2007).

22 Yuceoglu, U., Gemalmayan, N., and Sunar, O. FreeFlexural Vibrations of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with a Non-Central Plate Stiff-



Figure 8. Mode shapes and dimensionless natural frequencies of integrally-stiffened and/or stepped-thickness plate or panel system with three plate stiffeners(orthotropic case). Plate 1 = Plate 2 = Plate 3 = Graphite-Epoxy, Plate 4 = Plate 5 = Plate 6 = Plate 7 = Kevlar-Epoxy; lI = lII = lIII = 0.140 m,lIV = lV = lVI = lVII = 0.145 m; a = 0.50 m, h1 = h2 = h3 = 0.04 m, h4 = h5 = h6 = h7 = 0.02 m, L = 1.00 m; boundary conditions in y-directionCS.

ener, ASME international Mech. Engineering Congress andExposition on Advanced Structures and Materials for Light-Weight Design, Seattle, Washington, (2007).

23 Yuceoglu, U., Javanshir J., and Eyi S. Free Vi-bration analysis of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with Two Side Stiffeners, 49th

AIAA/ASME/ASCE/AHS/ASC Structures, Struc. Dynamicsand Materials (SDM) Conference and Exhibit, Schaum-burg, Illinois, (2008).

24 Yuceoglu, U., Guvendiko O, Gemalmayan, N., andSunar, O. Effects of Central and Non-Central PlateStiffeners on Free Vibrations Response of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels, 49th

AIAA/ASME/ASCE/ASC/AHS Structural, Struct. Dynamicsand Materials (SDM) Conference and Exibit, Schaumburg,Illinois, (2008).

25 Yuceoglu, U. and Ozerciyes, V. Free Bending Vibrations ofPartially-Stiffened, Stepped-Thickness Composite Plates,

Advanced Materials for Vibro-Acoustic Applications, 23,191–202, (1996).

26 Yuceoglu, U. and Ozerciyes, V. Natural Frequencies andMode Shapes of Composite Plates or Panels with a Cen-tral Stiffening Plate Strip, Vibroacoustic Methods in Pro-cessing and Characterization of Advanced Materials andStructures, 24, 185–196, (1997).

27 Yuceoglu, U. and Ozerciyes, V. Free Bending Vibrations ofComposite Base Plates or Panels Reinforced with a Non-Central Stiffening Plate Strip, Vibroacoustic Characteriza-tion of Advanced Materials and Structures, 25, 233–243,(1998).

28 Yuceoglu, U. and Ozerciyes, V. Sudden Drop Phe-nomena in Natural Frequencies of Partially Stiffened,Stepped-Thickness, Composite Plates or Panels, 40th

AIAA/ASME/ASCE/AHS/ASC Structures, Struct. Dynamics,and Materials (SDM) Conference and Exhibit, 2336–2347,(1999).



Figure 9. Dimensionless natural frequencies Ω versus aspect ratio a/L inintegrally-stiffened and/or stepped thickness plate or panel system with twostiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 =Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.2 m, lII = 0.2 m, lIII = 0.2 m,lIV = 0.2 m, lV = 0.2 m; a varies, h1 = h2 = 0.04 m, h3 = h4 =h5 = 0.02 m, L = 1.00 m; (a) boundary conditions in y-directions CC, (b)boundary conditions in y-directions CS.

29 Yuceoglu, U. and Ozerciyes, V. Natural Frequenciesand Modes in Free Transverse Vibrations of Stepped-Thickness and/or Stiffened Plates and Panels, 41st

AIAA/ASME/ASCE/AHS/ASC Structures, Struct. Dynamics,and Materials (SDM) Conference and Exhibit, ???, (2000).

30 Yuceoglu, U. and Ozerciyes, V. Sudden Drop Phenomenain Natural Frequencies of Composite Plates or Panels witha Central Stiffening Plate Strip, International Journal ofComputers and Structures, 76 (1–3), 247–262, (2000).

31 Yuceoglu, U. and Ozerciyes, V. Orthotropic CompositeBase Plates or Panels with a Bonded Non-Central (or Ec-centric) Stiffening Plate Strip, ASME Journal of Vibrationand Acoustics, 125, 228–243, (2003).

32 Yuceoglu, U., Ozerciyes, V., and Cil, K. Free Flexural Vi-brations of Bonded Centrally Doubly Stiffened CompositeBase Plates or Panels, ASME International Mechanical En-gineering Congress and Exposition, Anaheim, California,(2004).

33 Yuceoglu, U., Guvendik, O., and Ozerciyes, V. Free Flex-ural Vibrations Response of Composite Mindlin Plates orPanels with a Centrally Bonded Symmetric Double LapJoint (or Symmetric Double Doubler Joint), ASME Interna-tional Mechanical Engineering Congress and Exposition,Orlando, Florida, doi: 10.2514/6.2000-1348, (2005).

Figure 10. Dimensionless natural frequencies Ω versus stiffeners length ra-tio lI(= lII)/L in integrally-stiffened and/or stepped thickness plate or panelsystem with two stiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = lII varies ⇒ lIII =lIV = lV varies; a = 0.5 m, h1 = h2 = 0.04 m, h3 = h4 = h5 = 0.02 m,L = 1.00 m; (a) boundary conditions in y-directions CC, (b) boundary con-ditions in y-directions CS.

34 Mindlin, R. D. Infuence of Rotatory Inertia and Shear onFlexural Motions of Isotropic, Elastic Plates, ASME Journalof Applied Mechanics, 18, 31–38, (1951).

35 Reissner, E. The Effect of Transverse Shear Deformationson the Bending of Elastic Plates, ASME Journal of AppliedMechanics, 12 (2), 69–77, (1945).

36 Whitney, J. M. and Pagano, N. J. Shear Deformation in Het-erogeneous Anisotropic Plates, ASME Journal of AppliedMechanics, 37, 1031–1036, (1970).

37 Khdeir, A. A., Librescu, L., and Reddy, J. N. Analysisof Solutions of a Refined Shear Deformation Plate Theoryfor Rectangular Composite Plates, International Journal ofSolids and Structures, 23 (10), 1447–1463, (1987).

38 Subramanian, P. Flexural Analysis of Laminated CompositePlates, Composite Structures, 45, 51–69, (1999).

39 Yuceoglu, U. and Ozerciyes, V. Free Vibrations of BoundedSingle Lap Joints in Composite Shallow Cylindrical ShellPanels, AIAA Journal, 43 (12), 2537–2548, (2005).

40 Yuceoglu, U., Toghi, F., and Tekinalp, O. Free Bending Vi-brations of Adhesively-Bonded, Orthotropic Plates with aSingle Lap Joint, ASME Journal of Vibration and Acous-tics, 118, 122–134, (1996).



Figure 11. Dimensionless natural frequencies Ω versus thickness ratio h3(=h4 = h5)/h1(= h2 = 0.04) in integrally-stiffened and/or stepped thicknessplate or panel system with two stiffeners (orthotropic case). Plate 1 = Plate 2= Graphite-Epoxy, Plate 3 = Plate 4 = Plate 5 = Kevlar-Epoxy; lI = 0.2 m,lII = 0.2 m, lIII = 0.2 m, lIV = 0.2 m, lV = 0.2 m; a = 0.50 m, h1 =h2 = 0.04 m, h3 = h4 = h5 varies, L = 1.00 m; (a) boundary conditionsin y-directions CC, (b) boundary conditions in y-directions CS.

APPENDIX A: MINDLIN PLATE THEORY ASAPPLIED TO ORTHOTROPIC PLATES

Equations of Motion of Mindlin Plates

∂Mx

∂x+∂Myx

∂y−Qx +

h

2

(q+zx + q−zx

)=ρh3

12

∂2ψx∂t2

;

∂Myx

∂x+∂My

∂y−Qy +

h

2

(q+zy + q−zy

)=ρh3

12

∂2ψy∂t2

;

∂Qx∂x

+∂Qy∂y

+(q+z + q−z

)= ρh

∂2w

∂t2; (A.1)

where q-s are upper and lower loads or surface stresses, respec-tively.

Stress Resultant-Displacement Relations(in terms of Elastic Constants)

Mx =h3

12

(B11

∂ψx∂x

+B12∂ψy∂y

);

Qx = κ2xhB55

(ψx +

∂w

∂x

);

My =h3

12

(B21

∂ψx∂x

+B22∂ψy∂y

);

Figure 12. Dimensionless natural frequencies Ω versus bending stiffness ra-tio in integrally-stiffened and/or stepped thickness plate or panel system withtwo stiffeners (orthotropic case). Plate 1 = Plate 2 = Graphite-Epoxy, Plate3 = Plate 4 = Plate 5 varies; lI = 0.2 m, lII = 0.2 m, lIII = 0.2 m,lIV = 0.2 m, lV = 0.2 m; E(1)

y = E(2)y constant, E(3)

y = E(4)y = E

(5)y

varies; a = 0.50 m, h1 = h2 = 0.04 m, h3 = h4 = h5 varies,L = 1.00 m;(a) boundary conditions in y-directions CC, (b) boundary conditions in y-directions CS.

Qy = κ2yhB44

(ψy +

∂w

∂y

);

Myx =h3

12B66

(∂ψx∂y

+∂ψy∂x

); (A.2)

where κ2 terms are the Shear Correction Factors of the MindlinPlate Theory, and B-s are material coefficients in the or-thotropic stress-strain relations (Hooke’s Law) such that

B11 =Ex

1− νxyνyx; B44 = Gyz;

B22 =Ey

1− νyxνxy; B55 = Gxz;

B12 = B21 = νyxB11 = νxyB22; B66 = Gxy. (A.3)

Stress Resultant-Displacement Relations(in terms of Stiffnesses)

Mx = D11∂ψx∂x

+D12∂ψy∂y

;

Qx = A55

(ψx +

∂w

∂x

);

My = D21∂ψx∂x

+D22∂ψy∂y

;



Qy = A44

(ψy +

∂w

∂y

);

Myx = D66

(∂ψx∂y

+∂ψy∂x

);

D21 = D12; (A.4)

where the bending stiffness D-s and the shear stiffness A-s are

Dik =h3Bik

12; (i, k = 1, 2);

D66 =h3B66

12;

A44 = κ2yhB44; A55 = κ2xhB55. (A.5)

Mindlin Boundary Conditions

F (free) : Mnt = Mn = Qn = 0;

S (simply supported) : w = ψt = Mn = 0;

F (clamped) : w = ψn = ψt = 0; (A.6)

where n and t are normal and tangential directions.

APPENDIX B: SPECIAL FORM OF GOVERN-ING PDES FOR PARTS I, II, III, IV, V (OR-THOTROPIC MINDLIN PLATE THEORY)

1

l′k

∂ψ(j)y

∂ξk=

1

B(j)22

(12

h3jM (j)y −B

(j)12

1

a

∂ψ(j)x

∂η

);

1

l′k

∂ψ(j)x

∂ξk=

12

h3jB(j)66

M (j)yx −

1

a

∂ψ(j)y

∂η;

1

l′k

∂w(j)

∂ξk=

1

κ2yhjB(j)44

Q(j)y − ∂ψ(j)

y ;

1

l′k

∂M(j)yx

∂ξk=ρjh

3j

12

∂ψ(j)x

∂t2− 1

a

∂M(j)x

∂η+

Q(j)x −

hj2

(q(+)zx + q(−)zx

);

1

l′k

∂M(j)y

∂ξk=ρjh

3j

12

∂ψ(j)y

∂t2− 1

a

∂M(j)yx

∂η+

Q(j)y −

hj2

(q(+)zy + q(−)zy

);

1

l′k

∂Q(j)y

∂ξk= ρjhj

∂2w(j)

∂t2− 1

a

∂Q(j)x

∂η−(q(+)zy − q(−)zy

);

(B.1)

where

j = 1; k = I; (in Part I)j = 2; k = II; (in Part II)j = 3; k = III; (in Part III)j = 4; k = IV; (in Part IV)j = 5; k = V; (in Part V) (B.2)

and where q-s are the surface loads (or stresses) which are iden-tically zero in this case. In the above system of equations, the

state vectors of the problem are given as column vectors forParts I, II, III, . . . ,V, respectively

Y(j) (ξk, η)

=ψ(j)x , ψ(j)

y ,W (j),M(j)yk ,M

(j)y , Q(j)

y

T;

(j = 1, 2, 3, 4, 5; k = I, II, III, IV,V) . (B.3)

APPENDIX C: CLASSICAL LEVY’SSOLUTIONS

Following the Domain Decompositions Technique, theClassical Levy’s Solutions in the x-direction, corresponding,to each part or region are given below.

Displacement and Angles of Rotation

w(j) (η, ξk, t) = h1

∞∑m=1

∞∑n=1

W(j)

mn (ξk) sin(mπη)eiωmnt;

ψ(j)x (η, ξk, t) =

∞∑m=1

∞∑n=1

ψ(j)

mnx (ξk) cos(mπη)eiωmnt;

ψ(j)y (η, ξk, t) =

∞∑m=1

∞∑n=1

ψ(j)

mny (ξk) sin(mπη)eiωmnt; (C.1)

(0 < ξI < 1); (0 < ξII < 1);

(0 < ξIII < 1); (0 < ξIV < 1);

(0 < ξV < 1);

where j = 1 for Part I, j = 2 for Part II, j = 3 for Part III,j = 4 for Part IV, j = 5 for Part V, k = I for Part I, k = IIfor Part II, k = III for Part III, k = IV for Part IV, k = V forPart V.

Stress Resultants

M (j)x (η, ξk, t) =

h51B(1)11

a3

∞∑m=1

∞∑n=1

M(j)

mnx (ξk) sin(mπη)eiωmnt;

M (j)yx (η, ξk, t) =

h51B(1)11

a3

∞∑m=1

∞∑n=1

M(j)

mnyx (ξk) cos(mπη)eiωmnt;

M (j)y (η, ξk, t) =

h51B(1)11

a3

∞∑m=1

∞∑n=1

M(j)

mny (ξk) sin(mπη)eiωmnt;

Q(j)y (η, ξk, t) =

h41B(1)11

a3

∞∑m=1

∞∑n=1

Q(j)

mny (ξk) sin(mπη)eiωmnt;

Q(j)x (η, ξk, t) =

h41B(1)11

a3

∞∑m=1

∞∑n=1

Q(j)

mnx (ξk) cos(mπη)eiωmnt;

(C.2)

(0 < ξI < 1); (0 < ξII < 1);

(0 < ξIII < 1); (0 < ξIV < 1);

(0 < ξV < 1);

where j = 1 for Part I, j = 2 for Part II, j = 3 for Part III,j = 4 for Part IV, j = 5 for Part V, k = I for Part I, k = IIfor Part II, k = III for Part III, k = IV for Part IV, k = V forPart V.


Vibrational Power Flow Analysis of a CylindricalShell Using a Four-Point TechniqueH. Salimi-MofradDepartment of Mechanical Engineering, IOOC, Tehran, Iran.

S. Ziaei-Rad and M. MoradiDepartment of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran.

(Received 20 October 2012; provisionally accepted 31 March 2013; accepted 25 April 2013)

The aim of studying and analysing vibrational characteristics of structures using power flow is to control the noiseand vibration within the structure and prevent it from being transmitted into the environment. The power flowwithin a cylindrical shell is investigated because of its importance in designing spacecraft and marine structures.In this paper, a four-point power flow technique was used to examine the effects of flexural and shear forces on thetotal power flow in a cylindrical shell in free-free conditions. To obtain better results, the exponential window wasused considering the use of a hammer for the excitation of the structure. The examining of the results and obtaineddiagrams determined that the effect of shear forces on the total power flow was more than the effect of flexuralforces. Moreover, the precision of two-point and four-point techniques was compared.

1. INTRODUCTION

Nowadays, the attention of many researchers and scholarsin the field of vibration science is focused on energy reductionentered from a source, which generates noise to a structure andimpedes its circulation into the environment. The Power Flowmethod is deemed proper for the measurement of vibrationalenergy in the structure. This method is applied as well in deter-mining noise production source position within the structure.Vibration intensity is defined in rigid bodies and its measure-ment is performed at the level of structure. The reason for thislogic comes from the fact that in thin structures, wave propa-gation within the structure is normally estimated in a definiteway by its propagation in the structure.1

Several studies have been conducted on the cylindrical shell.One was made by Noisex. He discovered that the peak powerflow of a structure happens in its vibrational mode.2 Fullerand Fahy inspected the free wave propagation in cylindricalshells. They studied the physical interpretation of wave prop-agation equations in a coupled system.3 Langley tested thefrequency modification effect and changes of material specifi-cations on the power flow.4 Zhang investigated the input powerflow effect on the shells. He used different external forces andfrequency during his study.5 In his investigation, a cylindri-cal shell filled infinitely with the fluid was considered. Fangproved that the effect of coupling of the fluid and shell on theresponse is important.6

Another problem to be solved was evaluating energy prop-agation in an arbitrary thin-walled structure yet unsolved.7For the purpose of studying the vibrational behaviour of thinshells, many techniques were developed and applied. Arnoldand Warburton applied an energy method and used Lagrange’sequation as well as Love’s first approximation theory. Theystudied the free vibration of a thin cylindrical shell with freelysupported ends.8 Lam and Loy applied beam functions as theaxial modal functions in the Ritz process for examining bound-ary condition effects on the free vibration characteristics for acylindrical shell which was layered with nine different bound-ary conditions.9

Zhang and White inspected the power input of a shell due

to point force excitation. A good degree of correlation wasobserved by comparing the driving and transfer points’ test ac-celeration in the frequency range of study with the theoreticalresults.10 Merkulov et al. merely tested the point force ex-citation of an infinite thin-walled cylindrical shell filled withfluid.11 Energy flow through an arbitrary cross-section of aninfinitely long shell was formulated by Sorokin at various cir-cumferential mode numbers. During other studies performed,the inspected issue was “the energy redistribution among sev-eral transmission routes in a near-field and the influence of ex-citation conditions on steady fluctuations of the overall energyflow in a far field.”12

In a study examining the dynamic characteristics of cylin-drical shells, Heckl obtained the input impedance of a simply-supported cylindrical shell; however, the analysis was not cov-ering the influence of the bending stiffness of the shell. Hisstudy had restrictions in estimating input power in practical ap-plications.13 Harari offered a general formula, which assessedthe loss of transmitted energy or power based on the structuralimpedance of finite and semi-infinite cylindrical shells.14 Mis-saoui et al. performed another study on the free and forced vi-brations of a cylindrical shell with a floor partition, accordingto variational formulation. In this study, the structural cou-pling was simulated using artificial spring systems.15 As fordescribing structural coupling between the shell and plate, Liet al. applied artificial stiffness, presenting a methodology onthe structural acoustic coupling characteristics of a cylindricalshell with an internal floor partition.16

Wang and Xing investigated the dynamic characteristics ofpower flow in a coupled plate-cylindrical shell system.17 Man-dal and his colleagues applied the two-transducer method fornaturally-orthotropic plates, in order to estimate vibration en-ergy transfer in technically-orthotropic plates. They concludedthat the higher the rigidity of the plate, the lower the vibrationenergy transmission. As the rigidity of the plates increases, thevibration amplitudes of it decrease, thereby reducing acous-tic radiation.18 Zhu et al. have also studied the vibrationalpower flow of a thin cylindrical shell with a circumferentialsurface crack. They showed that changes obviously depend on


H. Salimi-Mofrad, et al.: VIBRATIONAL POWER FLOW ANALYSIS OF A CYLINDRICAL SHELL USING A FOUR-POINT TECHNIQUE

the depth and the location of the crack.19

This paper’s objective was to inspect the flexural and shearpower flow effect on the total power flow using the four-pointtechnique. Moreover, this was then compared with the two-point technique for a cylindrical shell.

2. GOVERNING EQUATIONS

The assumption is that the dimension, d, in the direction ofdisplacement, is noticeably smaller than the wavelength (i.e.,λ < 6d), which allows the thin theory to be applied while be-ing written in terms of the product of forces (Q) and moments(M ) with their corresponding normal velocity (ξ) and angularvelocity (θ) as2

Ix =⟨Qxξ

⟩t

+⟨Mxθx

⟩t

+⟨Mxy θy

⟩t; (1)

where the subscripts x and y denote direction and t shows time.The first term of Eq. (1) is the intensity due to shear forces,which can be written by means of the bending stiffness B as2

Ix,sf = −B⟨∂

∂x

(∂2ξ

∂x2+∂2ξ

∂y2

)ξ

⟩t

. (2)

The second term in Eq. (1) is the component due to the bendingmoment, which can be written as2

Ix,bm = B

⟨(∂2ξ

∂x2+ µ

∂2ξ

∂y2

)(∂ξ

∂x

)⟩t

. (3)

The third term in Eq. (1) is the term for intensity due to thetwisting moment, which can be written as2

Ix,tm = B (1− µ)

⟨∂2ξ

∂x∂y

∂ξ

∂y

⟩t

. (4)

A finite difference approximation was applied by Linjama toprovide a simplified description only by means of the velocitysignals in a lateral direction at two points as20

Ix = +2√Bm′

∆ImG23. (5)

In Eq. (5), above, G23 is the cross spectrum between the ve-locity signals measured in a lateral direction at the two pointsdemonstrated by subscripts, ∆ is their space, and m′ is thematerial surface density.

In a similar vein, the third order spatial derivatives ofEqs. (2) and (3) can be estimated by applying finite differ-ence approximations and the measured velocity at four equallyspaced co-linear measurement points. The resulting equationscan be shown as20

Ix,sf =B

2ω∆3(Im6G23 −G13 −G24 +G21 −G34)

(6)and

Ix,bm =B

2ω∆3(Im2G23 −G13 −G24 −G21 +G34) ;

(7)where ω is the angular frequency. The imaginary part of thecross-spectrum in Eqs. (6) and (7) can be obtained from theFFT analyser, which is then post-processed using Matlab orExcel. Furthermore, in Eqs. (6) and (7) the direction of the

Figure 1. The direction of the flow power in the cylinder.

power flow is from point 1 toward point 4, as shown in Fig. 1.The total intensity is the sum of the two components is20

Ix = Ix,sf + Ix,bm (8)

andIx =

B

ω∆3(Im4G23 −G13 −G24) . (9)

For the two-point technique, Ix was calculated from Eq. (5).20

3. TECHNICAL CHARACTERISTICS OF THEMEASURED SYSTEM

To measure power flow, a portable vibration analysis device(PVAT) was used, which was produced by Pooya Pajoh Pishro.Also, an impulse force hammer manufactured by Kistler (se-rial number: 2044659 and model: 724A5000) and an AC 102-A piezoelectric accelerometer sensor manufactured by CTCwith the calibration form code of A1-A4 (serial number 81678-81681) were used for measurement.

4. GEOMETRICAL CHARACTERISTICS OFTHE MODEL

The selected model was a thin cylindrical shell which wasset to be in free-free conditions, as seen in Fig. 2. The lengthof cylinder was 90 cm with an internal radius and thicknessof 16.988 mm and 2 mm, respectively. The cylinder was heldusing four internal rings with thickness and width of 5 mm andtwo rings at the two ends with thickness and width of 5 mmand 15 mm, respectively.

5. TEST CONDITIONS AND STIMULATIONMETHOD

To calculate power flow in the cylindrical shell, the sensorswere installed at four points, 20 cm far from each other andin line with the cylinder shaft. After providing the appropriatelaboratory conditions, the shell was hung from a firm anchorusing an elastic string, in order to provide the free anchor con-ditions required for conducting the test. Then, the force wasapplied on the position of Sensor 1 using the hammer and, afterdoing the necessary adjustments (calculating the spectral mul-tiplication of the domain and phase of the mentioned points),the data were collected in the FFT analyser system in order



Figure 2. The test rig.

to calculate the power flow. After that, using algebraic calcu-lations, the amount of power flow in the four-point techniquewas compared in shear, flexural, and total conditions. More-over, the information obtained from the four-point techniqueand the data collected using the two-point technique were anal-ysed.

6. EXPERIMENTAL RESULTS

Figure 3 depicts the power flow of the cylinder versus fre-quency in the two-point technique. The power flow for thefour-point technique is displayed in Fig. 4. In these figures,the power flow is in mW. A comparison between the two fig-ures revealed that the effect of noise on the two-point tech-nique is greater than on the four-point technique. In otherwords, one can say that the two-point technique is more proneto noise than the four-point technique. Since the two-pointtechnique rank is lower than the four-point technique rank, ithas less precision for determining the total amount of powerflow. Therefore, as can be observed in Fig. 5, the total amountof power flow is noticeably different in the two-point and four-point techniques.

Figure 6 shows both the flexural and shear power flow ofthe cylinder. The graph indicates that both power flows are in-creasing drastically in the vicinity of the natural frequencies. Itmeans that both power flows are increased near the natural fre-quency and by distancing from it, they both decrease rapidly.As can be seen in Fig. 6, the total amount of power flow prop-agated in the shell was equal to the shear power flow, and theflexural power flow did not have a large influence on the totalamount of power flow. Therefore, it can be concluded that flex-ural (bending) power flow can be disregarded when calculatingpower flow. The phase mismatch is an important source of er-

Figure 3. Power flow in the cylinder versus the frequency calculated from thetwo-point technique.

Figure 4. Power flow in the cylinder versus the frequency calculated from thefour-point technique.

ror in both the two-point and four-point approximations. It iswell known that there is an optimal spacing between measure-ment points that makes this error minimum. Also, it is recog-nized that the optimal spacing is a function of the wavelength.For two accelerometers and a dual channel FFT analyser, ear-lier studies by Redmon-White21 have proposed an operatingrange of 0.15λ < ∆ < 0.2λ.

However, a systematic study of point spacing for the two-point and four-point methods has not been made for generalcases. For both methods there is a bias and the choice of spac-ing between measurement points is critical in order to obtainaccurate intensity estimations. A very small spacing causesan overestimation, while underestimation will occur if a largespacing between accelerometers is used. Incorrect estimates ofthe intensity direction are another difficulty for large spacingbetween measurement points. It should be noted that the four-point method is more sensitive to the spacing between pointsand can produce large errors when a value outside the optimalrange is used. In this study, a trial-error method is used to findout the optimal value for spacing between the four accelerom-eters.

7. CONCLUSION

The power flow within a cylindrical shell was investigatedbecause of its importance in designing spacecraft and marinestructures. A four-point power flow technique was used to ex-amine the effects of flexural and shear forces on the total powerflow of the cylindrical shell. To obtain better results, the expo-nential window was used, considering the use of a hammer forthe excitation of the structure. The experimental results indi-



Figure 5. A comparison between power flows calculated from the two-pointand four-point techniques.

Figure 6. Flexural and shear power flow in the cylinder.

cate that the effect of shear forces on the total power flow wasmore than the effect of flexural forces. Moreover, the preci-sion of the two-point and four-point techniques was comparedwith each other. Also, noise affects the results of the two-pointtechnique more than it does the four-point technique.

REFERENCES1 Pavic, G. Determination of sound power-flow in structures:

principles and problems of realization, Proceeding of In-ternational Congress on Recent Developments in Acous-tic Intensity Measuring, CETIM, Senlis, France, 209–215,(1981).

2 Noiseux, D. U., Measurement of power flow in uniformbeam and plates, Journal of the Acoustical Society of Amer-ica, 47 (1), 238–247, (1970).

3 Fuller, C. R. and Fahy, F. J. Characteristics of wave prop-agation and energy distribution in cylindrical elastic shellsfilled with fluid, Journal of Sound and Vibration, 81 (4),501–518, (1982).

4 Langley, R. S. Wave motion and energy flow in cylindri-cal shells, Journal of Sound and Vibration, 169 (1), 29–42,(1994).

5 Zhang, W. H. and Zhang, X. M. Vibrational power flow ina cylindrical shell with periodic stiffeners, Proceeding ofASME PVP, Atlanta, U.S.A, (1991).

6 Feng, L. Acoustic propagates of fluid-filled elastic pipes,Journal of Sound and Vibration, 176 (3), 399–413, (1994).

7 Garvic, L. and Pavic, G. A finite element method for com-putational of structural intensity by the normal mode ap-proach, Journal of Sound and Vibration, 164 (1), 29–43,(1993).

8 Arnold, R. N. and Warburton, G. B. Flexural vibrations ofthe walls of thin cylindrical shells having freely supportedends, Proceedings of the Royal Society London A086, 127–145, (1949).

9 Lam, K. Y. and Loy, C. T. Effects of boundary conditionson frequencies of a multi-layered cylindrical shell, Journalof Sound and Vibration, 188 (3), 363–384, (1995).

10 Zhang, X. M. and White, R. G. Vibrational power input to acylindrical shell due to point force excitation, Proceedingsof the 4th International Congress on Intensity Techniques,France, (1993).

11 Merkulov, V. N., Prikhodko, V. Y., and Tyutekin, V. V. Nor-mal modes in a thin cylindrical elastic shell filled with fluidand driven by forces specified on its surface, Soviet PhysicsAcoustics, 25 (1), 51–54, (1979).

12 Sorokin, S. V., Nielsen, J. B., and Olhoff, N. Green’s matrixand the boundary integral equation method for the analysisof vibration and energy flow in cylindrical shells with andwithout internal fluid loading, Journal of Sound and Vibra-tion, 273 (3–5), 815–847, (2004).

13 Heckl, M. Vibrations of point-driven cylindrical shell, Jour-nal of the Acoustical Society of America, 34 (10), 1553–1557, (1962).

14 Harari, A. Wave propagation in cylindrical shells with finiteregions of structural discontinuity, Journal of the AcousticalSociety of America, 62 (5), 1196–1205, (1977).

15 Missaoui, J., Cheng, L., and Richard, M. J. Free and forcedvibration of a cylindrical shell with a floor partition, Journalof Sound and Vibration, 190 (1), 21–40, (1996).

16 Li, D. S., Cheng, L., and Gosselin, C. M. Analysis of struc-tural acoustic coupling of a cylindrical shell with an inter-nal floor partition, Journal of Sound and Vibration, 250 (5),903–921, (2002).

17 Wang, Z. H., Xing, J. T., and Price, W. G. A study of powerflow in a coupled plate-cylindrical shell system, Journal ofSound and Vibration, 271 (3–5), 863–882, (2004).

18 Mandal, N. K., Rahman, R. A., and Leong, M. S. Experi-mental investigation of vibration power flow in thin techni-cal orthotropic plates by the method of vibration intensity,Journal of Sound and Vibration, 285 (1), 669–695, (2005).

19 Zhu, X., Li, T. Y., Zhao, Y., and Yan, J. Vibrational powerflow analysis of thin cylindrical shell with a circumferentialsurface crack, Journal of Sound and Vibration, 302 (1–2),332–349, (2007).

20 Linjama, J. and Lahti, T. Estimation of bending wave in-tensity in beams using the frequency response technique,Journal of Sound and Vibration, 153 (1), 21–36, (1992).

21 Redmon-White, W. The experimental measurement of flex-ural wave power flow in structures, Institute of Sound andVibration Research, University of Southampton, 467–474,(1983).


Non-Linear Thickness Variation on the Thermally-Induced Vibration of a Rectangular Plate:A Spline TechniqueArun Kumar Gupta and MamtaDepartment of Mathematics, M.S. College, Saharanpur 247001, UP, India

The fourth order differential equation, governing the transverse motion of an elastic rectangular plate of a non-linear thickness variation with a thermal gradient, has been analysed on the basis of classical plate theory employingthe Quintic spline interpolation technique. An algorithm for computing the solution of this differential equationis presented for the case of equal intervals. The effect of the thermal gradient, together with taper constants onthe natural frequencies of vibration, is illustrated for the first three modes of vibration for C-S-C-S and S-S-S-Srectangular plates.

1. INTRODUCTION

Plates of variable thickness having non-linear nature are ofimmense importance to engineers. For the design of aircraftwing sections, earthquake resistant structures, ship control de-vices, telephone receivers, radio telescopes, and other struc-tures prone to vibration, a construction engineer needs the firstfew modes of vibration before finalizing a design.

With the advancement of technology, plates of varyingthickness are being extensively used in civil, electronic, me-chanical, aerospace, and marine engineering applications. Itis necessary, therefore, to study the vibration behaviour ofplates in order to avoid resonance excited by internal or ex-ternal forces. Modern engineering structures are based on dif-ferent types of design that involve various anisotropic and non-homogeneous materials in the form of their structure compo-nents. Depending upon the requirement, durability, and re-liability, materials are being developed so that they can pro-vide better strength and efficiency. A number of researchershave worked on the free vibration analyses of plates of dif-ferent shapes and varying thicknesses. Rectangular plates ofnon-linear varying thickness are widely used in different struc-tures; however, they have been poorly studied, unlike linearly-varying thickness. Rectangular plates of non-linear varyingthickness with thermal gradients find various applications inthe construction of modern high speed air craft. The vibrationcharacteristics of such plates are of interest to the designers.

Various numerical techniques such as the Frobeniusmethod,1 the finite-difference method,2 the simple polyno-mial approximation,3 the Galerkin method,4, 5 the Rayleigh-Ritz method,6–8 characteristic orthogonal polynomials,9 thefinite element method,10, 11 and the Chebyshev collocationmethod,12, 13 etc., have been employed to study the vibrationcharacteristics of plates of various geometries. The above nu-merical methods, such as finite difference and finite element,require fine mesh size to obtain accurate results and are compu-tationally expensive. The method of characteristic orthogonalpolynomials and the Frobenius method require an appreciablenumber of terms for plates of varying thickness.

Here, a Quintic splines procedure is developed for obtain-ing the natural frequencies of rectangular plates of nonlinear

varying thickness with the thermal gradient effect. The con-sideration of the present type of thickness variation was takenearlier by Gupta et al. for circular plates.14 The plate typestructural components in aircraft and rockets have to operateunder elevated temperatures, which cause non-homogeneity inthe plate material; i.e., elastic constants of the material becomefunctions of the space variables. In an up-to-date survey of theliterature, these authors have come across various models toaccount for non-homogeneity in plate materials, as proposedby researchers dealing with vibration.

Akiyama and Kuroda discussed the fundamental frequen-cies of rectangular plates with linearly-varying thickness.15

Civalek discussed the fundamental frequency of isotropic andorthotropic rectangular plates with linearly-varying thicknessby the discrete singular convolution method.16 Gupta et al.studied the thermal gradient effect on vibration of a non-homogeneous orthotropic rectangular plate having bi-directionlinear thickness variation.17 Gupta et al. did the vibration anal-ysis of visco-elastic orthotropic parallelogram plate with linearthickness variation in both directions.18 Lal et al. studied thetransverse vibrations of non-uniform orthotropic rectangularplates by the Quintic splines method.19 Gupta and Kaur stud-ied the effect of the thermal gradient on the free vibration ofclamped visco-elastic rectangular plates with linear thicknessvariation in both directions.20 Gupta and Khanna studied thevibration of visco-elastic rectangular plates with linear thick-ness variations in both directions.21

Gupta et al. observed the thermal effect on the vibrationof a non-homogeneous orthotropic rectangular plate havingbi-directional, parabolically-varying thickness.22 Tomar andGupta studied the effects of thermal gradient on the frequen-cies of orthotropic rectangular plates of varying thickness inone and two directions.23, 24 Gupta et al. studied the ther-mal effect on vibration of a parallelogram plate of linearly-varying thickness and bi-directional linearly-varying thick-ness.25, 26 Gupta et al. did the vibration study of a visco-elasticparallelogram plate of linearly-varying thickness.27 Raju stud-ied the vibration of thin elastic plates of linearly-varying thick-ness.28 Grossi and Laura discussed the transverse vibrationsof circular plates of linearly-varying thicknesses.29 Gupta andAnsari studied the effect of elastic foundation on the asym-


A. K. Gupta, et al.: NON-LINEAR THICKNESS VARIATION ON THE THERMALLY-INDUCED VIBRATION OF A RECTANGULAR PLATE. . .

metric vibration of polar, orthotropic, linearly-tapered circu-lar plates.30 Laura et al. worked on the vibrations and elasticstability of polar orthotropic circular plates of linearly-varyingthickness.31

As linear thickness variation is not perfect linear thought andneither is the quadratic, the non-linear variation in thicknessis therefore very useful for scientists and engineers in study-ing the vibration of plates to finalize the modes of vibrations.Since no work is available on the non-linear thickness variationof the thermally-induced vibration of rectangular plates, thispaper examines the thermal effect on the vibration of a rectan-gular plate of non-linear varying thickness. The vibration of arectangular plate of non-linear varying thickness under steadylinear temperature distribution has been studied. The effect oftemperature on the modulus of elasticity is assumed to varylinearly along the x-axis. The non-linear thickness variation istaken as a combination of linear and parabolic variation fac-tors. The differential equation of motion has been solved bythe Quintic spline interpolation technique. The two edges par-allel to the x-axis (y = 0 and y = b) are assumed to be simplysupported. A different set of boundary conditions has been im-posed at the other two edges. The frequency parameters forthe first three modes of vibrations for C-S-C-S and S-S-S-Sboundary conditions and for various values of taper constants,the thermal constant, and the fixed value of length to breadthratio are obtained. The results are presented in tabular form.

2. ANALYSIS AND EQUATION OF MOTION

The equation of motion of an isotropic rectangular plate ofvariable thickness may be written in the form, as by Leissa1

∂2Mx

∂x2+ 2

∂2Mxy

∂x∂y+∂2My

∂y2= ρh

∂2w

∂t2. (1)

Here, Mx, My , and Mxy are moments per unit length of plate,ρ is mass per unit volume, h is the thickness of the plate, andw is the displacement at time t.

The expression for Mx, My , Mxy is given by

Mx = −D[∂2w

∂x2+ ν

∂2w

∂y2

];

My = −D[∂2w

∂y2+ ν

∂2w

∂x2

];

Mxy = −2D(1− ν)∂2w

∂x∂y; (2)

where D = D(x, y) = E(x,y)h3(x,y)12(1−ν2) is the flexural rigidity

of the plate. Here, E is the module of elasticity and ν is thePoisson ratio.

Using Eq. (2) in Eq. (1) and then simplifying, one gets thedifferential equation as follows:

D∇4w + 2∂D

∂x

∂

∂x∇2w + 2

∂D

∂y

∂

∂y∇2w +∇2D∇2w +

(ν − 1)

∂2D

∂x2∂2w

∂y2− 2

∂2D

∂x∂y

∂2w

∂x∂y+∂2D

∂y2∂2w

∂x2

+

ρh∂2w

∂t2= 0. (3)

Assuming that the plate under consideration is subjected toa steady one-dimensional temperature distribution along thelength, one can take T as32

T = T0

(1− x

a

); (4)

where T denotes the temperature excess above the referencetemperature at any point at distance x

a , and T0 denotes the tem-perature excess above the reference temperature at the end;i.e.,x = a. The temperature dependence of the modulus of elas-ticity for most of engineering materials can be expressed in theform33

E(T ) = E0 (1− γT ) ; (5)

where E0 is the value of the Young’s modulus at the referencetemperature; i.e., T = 0 and γ is the slope of the variation ofE with T . Thus, the modulus variation becomes

E = E0

(1− α

(1− x

a

)); (6)

where α = γT0 (0 ≤ α < 1), a parameter known as thermalgradient.

Assume that the thickness varies in the x-direction only.Consequently, the thickness h and flexural rigidity D of theplate become a function of x only. Furthermore, let the twoopposite edges of the plate, y = 0 and y = b, be simply sup-ported so that plate undergoing free transverse vibrations withcircular frequency p, may have the Levy type solution as

w(x, y, t) = W1(x) sin(mπy

b

)eipt; (7)

where m is a positive integer. Then, substitution of Eq. (7) inEq. (3) yields

DW1,xxxx + 2D,xW1,xxx +

[−2

(m2π2

b2

)D+D,xx

]W1,xx +[

−2

(m2π2

b2

)D,x

]W1,x +

[(m4π4

b4

)D−ν

(m2π2

b2

)D,xx

]W1 =

ρhp2W1. (8)

A comma followed by a suffix denotes partial differentiationwith respect to that variable. Thus, Eq. (8) reduces to a formindependent of y and on introducing the non-dimensional vari-ables

H = h/a; W = W1/a; X = x/a; D1 = D/a3; (9)

the differential Eq. (8) reduces to

D1W,xxxx + 2D1,xW,xxx +[D1,xx−2r2D1

]W,xx −

2r2D1,xW,x + r2[r2D1−νD1,xx

]W = ρHa2p2W ; (10)

where r2 = (mπa/b)2.In view of the previous assumption, the present analysis is

restricted to the modes having waves along the x-direction,only where the standing waves will be independent of the y-coordinate. Further, variation in thickness is quadratic, andtherefore one can assume that

H = H0

(1 + β1X + β2X

2)

; (11)

where β1 and β2 are taper constants such that |β1| ≤ 1, |β2| ≤1, and β1 + β2 > −1, H0 is thickness at X = 0. Considering



Eqs. (6) and (11) with the help of Eq. (9), the expression forrigidities D1 comes out as

D1 = D0 [1− α(1−X)](1 + β1X + β2X

2)3

; (12)

where D0 =E0H

30

12(1−ν2) . The substitution of Eqs. (11) and (12)into Eq. (10), the differential equation, takes the form

[1−α(1−X)](1+β1X+β2X

2)2W,xxxx +

2[α(1+β1X+β2X

2)2

+

3 (1−α(1−X))(1+β1X+β2X

2)

(β1+2β2X)]W,xxx +[(

6α(1+β1X+β2X

2)

(β1+2β2X) +

6 (1−α(1−X)) (β1+2β2X)2

+

6 (1−α(1−X))(1+β1X+β2X

2)β2

)−

2r2 (1−α(1−X))(1+β1X+β2X

2)2 ]

W,xx −

2r2[α(1+β1X+β2X

2)2

+

3 (1−α(1−X))(1+β1X+β2X

2)

(β1+2β2X)]W,x +

r2[r2 (1−α(1−X))

(1+β1X+β2X

2)2 −

ν(

6α(1+β1X+β2X

2)

(β1+2β2X) +

6 (1−α(1−X)) (β1+2β2X)2

+

6 (1−α(1−X))(1+β1X+β2X

2)β2

)]W = λ2W ;

(13)

where

λ2 =(p2a2/(E0/ρ)

) (12(1− ν2)/H2

0

)(14)

is a frequency parameter.

3. METHOD OF SOLUTION

Let f(X) be a function with continuous derivatives in therange (0, 1). Choose (n + 1) points X0, X1, X2, . . . , Xn, inthe range 0 ≤ X ≤ l, such that 0 = X0 < X1 < X2 < X3 <. . . < Xn = 1.

Then, let the approximating function W (X) for f(X) be aQuintic spline with the following properties:

(a) W (X) is a Quintic polynomial in each interval(Xk, Xk+1),

(b) W (Xk) = f(Xk), k = 0, 1, . . . n,

(c) W ′(X), W ′′(X), W ′′′(X), and W iv(X) are continuous.

From the definition, a Quintic spline takes the form

W (X) = a0 +4∑i=1

ai (X−X0)i+n−1∑j=1

bj (X−Xj)5+ ; (15)

where

(X−Xj)+ =

0 if X < Xj

X−Xj if X ≥ Xj; (16)

It is also assumed, for simplicity, that the knots Xi are equallyspaced in (0, l) with the spacing interval ∆X , so that

∆X = l/n; Xi = i∆X (i = 0, 1, 2, . . . , n). (17)

The number of unknown constants in Eq. (15) is (n + 5).Satisfaction of the differential Eq. (13) by collocation at the(n + 1) knots in the interval (0, l) together with the bound-ary conditions (to be explained in the next section) gives pre-cisely the requisite number of equations for the determinationof unknown constants. Substituting W (X) from Eq. (15) intoEq. (13) gives, for satisfaction at the mth knot,

B4a0 +[B4(Xq−X0) +B3

]a1 +

[B4(Xq−X0)2 +

2B3(Xq−X0) + 2B2

]a2 +

[B4(Xq−X0)3 +

3B3(Xq−X0)2 + 6B2(Xq−X0) + 6B1

]a3 +[

B4(Xq−X0)4 + 4B3(Xq−X0)3 + 12B2(Xq−X0)2 +

24B1(Xq−X0) + 24B0

]a4 +

n−1∑i=0

[B4(Xq−X0)5 +

5B3(Xq−X0)4 + 20B2(Xq−X0)3 + 60B1(Xq−X0)2 +

120B0(Xq−X0)]bi = 0; (18)

where

B0 = [1−α(1−Xq)](1+β1Xq+β2X

2q

)2;

B1 = 2[α(1+β1Xq+β2X

2q

)2+

3 (1−α(1−Xq))(1+β1Xq+β2X

2q

)(β1+2β2Xq)

];

B2 =[(

6α(1+β1Xq+β2X

2q

)(β1+2β2Xq) +

6 (1−α(1−Xq)) (β1+2β2Xq)2

+

6 (1−α(1−Xq))(1+β1Xq+β2X

2q

)β2

)−

2r2 (1−α(1−Xq))(1+β1Xq+β2X

2q

)2 ];

B3 = − 2r2[α(1+β1Xq+β2X

2q

)2+

3 (1−α(1−Xq))(1+β1Xq+β2X

2q

)(β1+2β2Xq)

];

B4 =[r2[r2 (1−α(1−Xq))

(1+β1Xq+β2X

2q

)2 −ν(

6α(1+β1Xq+β2X

2q

)(β1+2β2Xq) +

6 (1−α(1−Xq)) (β1+2β2Xq)2

+

6 (1−α(1−Xq))(1+β1Xq+β2X

2q

)β2

)]− λ2

].

Thus, one obtains a homogeneous set of equations in terms ofunknown constants a0, a1, a2, a3, a4, b0, b1, . . . , bn−1, which,when written in matrix notation, takes the form

[B][C] = 0; (19)

where [B] is an (n+1)×(n+5) matrix and [C] is an (n+5)×1matrix.



Table 1. The value of the frequency parameter (λ) for different values of thermal constant (α) with different combinations of taper constant (β1, β2) and fixedaspect ratio (a/b = 1.5) for C-S-C-S and S-S-S-S plates for the first three modes of vibrations; m = 1.

β1, β2 αC-S-C-S PLATE S-S-S-S PLATE

FIRST MODE SECOND MODE THIRD MODE FIRST MODE SECOND MODE THIRD MODE0.0 29.3011 63.0984 111.0257 21.4941 53.0078 91.32430.1 28.4986 61.3754 107.9841 20.5862 50.9898 88.2017

β1 = −0.5, 0.2 27.6011 59.1034 103.2275 19.5190 48.5902 85.1101β2 = −0.5 0.3 26.5327 57.0312 99.7452 18.4208 46.3155 81.9869

0.4 25.4217 54.9031 95.6472 17.2904 44.1103 78.88430.5 24.3572 52.6301 91.4845 16.0326 42.0095 75.51900.0 36.0132 72.5490 127.4781 27.3761 63.7524 108.15620.1 35.2301 70.6011 123.4501 26.5012 61.5320 105.0054

β1 = −0.5, 0.2 34.2002 68.5761 119.3776 25.6987 59.3321 101.7532β2 = 0.5 0.3 33.1987 66.3465 115.2108 24.5481 57.2097 98.5107

0.4 32.0998 64.1063 111.1147 23.4311 55.0863 95.32670.5 30.9938 62.0096 107.0064 22.3009 52.7859 92.02130.0 49.4210 106.8851 191.4330 39.7562 97.2203 171.46120.1 48.6231 104.9987 188.1174 38.7431 95.1983 168.3890

β1 = 0.5, 0.2 47.6109 102.6581 185.0021 37.7304 93.1201 165.2993β2 = 0.5 0.3 46.5211 100.3352 181.8697 36.7012 91.0934 162.1873

0.4 45.3991 98.1035 177.6987 35.6891 88.9982 159.08720.5 44.2267 96.0935 173.4367 34.6327 86.7978 155.9898

4. BOUNDARY CONDITIONS ANDFREQUENCY EQUATIONS

The frequency equations for clamped (C) and simply-supported (S) rectangular plates have been obtained by em-ploying the appropriate boundary conditions.

4.1. C-S-C-S PlatesFor a rectangular plate clamped at both the edgesX = 0 and

X = 1 (and simply supported at the remaining two edges),

W |X=0,1 = (∂W/∂X) |X=0,1 = 0. (20)

Applying the boundary conditions in Eq. (20) to the deflectionfunction in Eq. (15), at the two edges X = 0 and X = 1,one obtains a set of four homogeneous equations in terms ofunknown constants, which can be written as

[A1][C] = 0; (21)

where [A1] is an 4× (n+ 5) matrix and [C] is an (n+ 5)× 1matrix. Equation (21), taken together with the Eq. (19), givesa complete set of (n+ 5) equations for a C-S-C-S plate. Thesecan be written as

[B/A1][C] = 0. (22)

For a non-trivial solution of Eq. (22), the characteristic deter-minant must vanish:

|B/A1| = 0. (23)

This is the frequency equation for a C-S-C-S plate.

4.2. S-S-S-S PlatesFor a rectangular plate simply supported at both the edges

X = 0 and X = 1 (and simply supported at the remaining twoedges),

W |X=0,1 =(∂2W/∂X2

)|X=0,1 = 0. (24)

Employing the boundary conditions in Eq. (24), to the deflec-tion function in Eq. (15), at the two edges X = 0 and X = 1,one gets the boundary equations for S-S-S-S plate as

[A2][C] = 0; (25)

where [A2] is an 4× (n+ 5) matrix and [C] is an (n+ 5)× 1matrix. Hence, the frequency equation comes out for S-S-S-Splate as

|B/A2| = 0. (26)

5. RESULTS AND DISCUSSION

The frequency Eqs. (23) and (26) are transcendental equa-tions in λ2, from which roots can be obtained infinitely. Thefrequency parameter λ, corresponding to first three modes ofvibration of C-S-C-S and S-S-S-S rectangular plates, havebeen computed for m = 1 and various values of the thermalconstant (α) and taper constants (β1, β2) for the fixed aspectratio (a/b = 1.5). The value of the Poisson ratio ν has beentaken as 0.3.

To choose the appropriate interpolation interval ∆X , thecomputer programme has been developed for the evaluationof the frequency parameter λ and run for n = 10 (5) 60. Thenumerical values show a consistent improvement with the in-crease of the number of knots. In the above computation, au-thors have fixed n = 50, since a further increase in n does notimprove the results except in the fifth or sixth decimal places.These results have been tabulated in Tables 1, 2, and 3.

Table 1 shows the variation of frequency parameter (λ) withthermal constant (α) for different combinations of the taperconstant (β1, β2) and fixed aspect ratio (a/b = 1.5), corre-sponding the first three modes of vibration for C-S-C-S and S-S-S-S plates. The value of the frequency parameter decreaseswith the increase of the thermal constant for both the boundaryconditions considered here. Further, it can be seen from thetable that the frequency parameter for both the boundary con-ditions decreases gradually in the third mode of vibrations incomparison to the first two modes of vibration.

The results presented in Table 2 show a marked effect ofvariation of taper constant (β1) on the frequency parameter fortaper constant (β2 = 0.5), two values of the thermal constant(α = 0.0, 0.4), and the fixed aspect ratio (a/b = 1.5) corre-sponding to the first three modes of vibration. It is observedthat the frequency parameter increases with the increase of ta-per constant for both the boundary conditions considered here.

In Table 3, the effect of the taper constant (β2) on the fre-quency parameter for the taper constant (β1 = 0.5), two val-



Table 2. The value of the frequency parameter (λ) for different values of taper constant (β1) with different combinations of thermal constant (α) and fixed aspectratio (a/b = 1.5) for C-S-C-S and S-S-S-S plates for the first three modes of vibrations; m = 1 and β2 = 0.5.

α β1C-S-C-S PLATE S-S-S-S PLATE

FIRST MODE SECOND MODE THIRD MODE FIRST MODE SECOND MODE THIRD MODE-0.5 36.0132 72.5490 127.4781 27.3761 63.7524 108.1562-0.3 38.3782 78.1352 137.9481 29.4861 69.2189 118.7010-0.1 40.5601 83.7103 148.5275 31.5908 74.7902 129.0211

0.0 0.0 42.5324 89.0914 158.7945 33.3920 79.8053 138.83860.1 44.8210 94.5601 169.4642 35.3904 85.2100 149.08420.3 47.0572 100.4805 180.3443 37.3026 91.0409 160.04900.5 49.4210 106.8851 191.4330 39.7562 97.2203 171.4612-0.5 32.0998 64.1063 111.1147 23.4311 55.0863 95.3267-0.3 34.2041 69.7266 122.1776 25.5287 60.6320 105.8731-0.1 36.3908 75.4846 133.2310 27.5981 65.8074 116.3851

0.4 0.0 38.2098 80.3568 143.8710 29.2311 70.8962 126.11260.1 40.5023 86.3097 155.0074 31.2909 76.2821 136.69210.3 42.9210 92.0851 166.0330 33.3562 82.4520 147.86010.5 45.3991 98.1035 177.6987 35.6891 88.9982 159.0872

Table 3. The value of the frequency parameter (λ) for different values of taper constant (β2) with different combinations of thermal constant (α) and fixed aspectratio (a/b = 1.5) for C-S-C-S and S-S-S-S plates for the first three modes of vibrations; m = 1 and β1 = 0.5.

α β2C-S-C-S PLATE S-S-S-S PLATE

FIRST MODE SECOND MODE THIRD MODE FIRST MODE SECOND MODE THIRD MODE-0.5 37.5320 77.7213 142.9941 27.8301 65.7224 116.9956-0.3 39.4712 82.2052 151.0998 29.7861 70.8189 126.0010-0.1 41.4011 86.9910 159.2275 31.7408 75.9702 135.1221

0.0 0.0 43.1534 91.1293 167.0042 33.4221 80.7053 143.61860.1 45.1210 96.0601 175.1864 35.3090 86.0030 152.74840.3 47.1572 101.2805 183.2443 37.3026 91.3409 162.00600.5 49.4210 106.8851 191.4330 39.7562 97.2203 171.4612-0.5 33.1623 75.8019 126.6118 23.5301 59.9982 104.6580-0.3 35.1042 79.5146 135.2776 25.5587 64.7321 113.6371-0.1 37.0908 83.2846 143.7710 27.6181 69.5807 122.5985

0.4 0.0 38.7088 86.6456 151.8271 29.3511 74.0019 130.84260.1 40.7702 90.2997 160.7074 31.3009 78.8021 140.00210.3 43.0001 94.0851 168.8321 33.3262 83.4220 149.06210.5 45.3991 98.1035 177.6987 35.6891 88.9982 159.0872

ues of the thermal constant (α = 0.0, 0.4), and the fixed aspectratio (a/b = 1.5), corresponding to the first three modes ofvibration for C-S-C-S and S-S-S-S plates, have been shown.From this table, one can observe that frequency parameter infirst three modes of vibration increases with the increase of thetaper constant for C-S-C-S and S-S-S-S plates.

Further, it can be seen from Tables 2 and 3 that the frequencyparameter for both the boundary conditions increases gradu-ally in the third mode of vibrations in comparison to first twomodes of vibration. Also, one can observe from Tables 1 to 3that frequency parameter of the C-S-C-S plate is higher thanthat of S-S-S-S plate.

6. CONCLUSIONS

It can be concluded from the results that frequency parame-ter increases with an increase in taper constants and decreaseswith increase in thermal gradient. Also, it is evident from Ta-bles 2 and 3 that when β1 = 0.5, the values of frequencyparameter are larger in comparison to β2 = 0.5 for all threemodes of vibration and both the boundary conditions. It is alsoclear from the tables that third mode of vibration changes moresharply than the second and first.

REFERENCES1 Leissa, A. W. Vibration of Plates, NASA-SP, (1969).

2 Singh, B., Hassan, S. M., and Lal, J. Some numericalexperiments on high accuracy fast direct finite difference

methods for elliptic problem, Communications in Numeri-cal Methods in Engineering, 12, 631–641, (1996).

3 Gupta, U. S., Lal, R., and Jain, S. K. Effect of elastic foun-dation on axisymmetric vibration of polar orthotropic cir-cular plates of variable thickness, Journal of Sound and Vi-bration, 139, 503–513, (1999).

4 Avalos, D. R., Hack, H., and Laura, P. A. A. Galerkinmethod and axisymmetric vibrations of polar orthotropiccircular plates, AIAA Journal, 20, 1626–1628, (1982).

5 Ratko, M. Transverse vibration and instability of an eccen-tric rotating circular plate, Journal of Sound and Vibration,280, 467–478, (2005).

6 Gutierrez, R. H., Romanelli, E., and Laura, P. A. A. Vibra-tion and elastic stability of thin circular plates with variableprofile, Journal of Sound and Vibration, 195 (3), 391–399,(1996).

7 Singh, B. and Saxena, V. Transverse vibration of a circu-lar plate with unidirectional quadratic thickness variations,International Journal of Mechanical Science, 38 (4), 423–430, (1996).

8 Gupta, U. S. and Ansari, A. H. Transverse vibration of po-lar orthotropic parabolically tapered circular plates, IndianJournal of Pure and Applied Mathematics, 34 (6), 819–830,(2003).



9 Singh, B. and Chakarverty, S. Use of characteristic orthog-onal polynomials in two dimensions for transverse vibra-tion of elliptical and circular plates with variable thickness,Journal of Sound and Vibration, 173 (3), 289–299, (1994).

10 Chen, D. Y. and Ren, B. S. Finite element analysis of the lat-eral vibration of thin annular and circular plates with vari-able thickness, ASME Journal of Vibration and Acoustics,120 (3), 747–752, (1998).

11 Liu, C. F. and Lee, Y. T. Finite element analysis of threedimensional vibrations of thick circular and annular plates,Journal of Sound and Vibration, 233 (1), 63–80, (2000).

12 Gupta, U. S., Lal, R., and Sagar, R. Effect of elastic founda-tion on axisymmetric vibration of polar orthotropic Midlinecircular plate, Indian Journal of Pure and Applied Mathe-matics, 25, 1317–1326, (1994).

13 Lal, R., Gupta, U. S., and Goel, C. Chebyshev polynomialsin the study of transverse vibrations of non-uniform rect-angular orthotropic plates, The Shock and Vibration Digest,33 (2), 103–112, (2001).

14 Gupta, U. S., Lal, R., and Sharma, S. Vibration analy-sis of non-homogeneous circular plates of nonlinear thick-ness variation by differential quadrature method, Journal ofSound and Vibration, 298, 892–906, (2006).

15 Akiyama, K. and Kuroda, M. Fundamental frequencies ofrectangular plates with linearly varying thickness, Journalof Sound and Vibration,205 (3), 380–384, (1997).

16 Civalek, O. Fundamental frequency of isotropic and or-thotropic rectangular plates with linearly varying thicknessby discrete singular convolution method, Applied Mathe-matical Modelling, 33, 3825–3835, (2009).

17 Gupta, A. K., Johri, T., and Vats, R. P. Study of ther-mal gradient effect on vibrations of a non-homogeneousorthotropic rectangular plate having bi-direction linearlythickness variations, Meccanica, 45 (3), 393–400, (2010).

18 Gupta, A. K., Kumar, A., and Kaur, H. Vibration of visco-elastic orthotropic parallelogram plate with linearly thick-ness variation in both directions, International Journal ofAcoustics and Vibration, 16 (2), 72–80, (2011).

19 Lal, R., Gupta, U. S., and Reena, Quintic splines in thestudy of transverse vibrations of non-uniform orthotropicrectangular plates, Journal of Sound and Vibration, 207 (1),1–13, (1997).

20 Gupta, A. K. and Kaur, H. Study of the effect of thermalgradient on free vibration of clamped visco-elastic rectan-gular plates with linearly thickness variation in both direc-tions, Meccanica, 43 (4), 449–458, (2008).

21 Gupta, A. K. and Khanna, A. Vibration of visco-elastic rect-angular plate with linearly thickness variations in both di-rections, Journal of Sound and Vibration, 301 (3–5), 450–457, (2007).

22 Gupta, A. K., Johri, T., and Vats, R. P. Thermal effect on vi-bration of non-homogeneous orthotropic rectangular platehaving bi-directional parabolically varying thickness, Pro-ceeding of International Conference in World Congress onEngineering and Computer Science, San-Francisco, USA,784–787, (2007).

23 Tomar, J .S. and Gupta, A. K. Thermal effect of frequen-cies of an orthotropic rectangular plate of linearly varyingthickness, Journal of Sound and Vibration, 90 (3), 325–331,(1993).

24 Tomar, J. S. and Gupta, A. K. Effect of thermal gradienton frequencies of orthotropic rectangular plate whose thick-ness varies in two directions, Journal of Sound and Vibra-tion, 98 (2), 257–262, (1985).

25 Gupta, A. K., Kumar, M., Kumar, S., and Khanna, A.Thermal effect on vibration of parallelogram plate of bi-direction linearly varying thickness, Applied Mathematics,2 (1), 33–38, (2011).

26 Gupta, A. K., Kumar, M., Khanna, A., and Kumar, S. Ther-mal effect on vibrations of parallelogram plate of linearlyvarying thickness, Advanced Studies of Theoretical Physics,4 (17), 817–826, (2010).

27 Gupta, A. K., Kumar, A., and Gupta, Y. K. Vibrationstudy of visco-elastic parallelogram plate of linearly vary-ing thickness, International Journal of Engineering and In-terdisciplinary Mathematics, 2 (1),21–29, (2010).

28 Raju, B. B. Vibration of thin elastic plates of linearly vari-able thickness, International Journal of Mechanical Sci-ences, 8, 89–100, (1966).

29 Grossi, R. O. and Laura, P. A. A. Transverse vibrationsof circular plates of linearly varying thicknesses, AppliedAcoustics, 13, 7–18, (1980).

30 Gupta, U. S. and Ansari, A. H. Effect of elastic founda-tion on asymmetric vibration of polar orthotropic linearlytapered circular plates, Journal of Sound and Vibration,254 (3), 411–426, (2002).

31 Laura, P. A. A., Avalos, R., and Galles, C. D. Vibrationsand elastic stability of polar orthotropic circular plates oflinearly varying thickness, Journal of Sound and Vibration,82, 151–156, (1982).

32 Olsson, U. On free vibration at temperature dependent ma-terial properties and transient temperature fields, Journal ofApplied Mechanics, 39 (3),723–726, (1972).

33 Nowacki, W. Thermo elasticity, Pergamon Press, NewYork, (1962).


About the Authors

Evangelos G. Ladopoulos is a professor who holds a PhD in engineering and mechanicalengineering. He was included on Cambridge’s list of 2000 Outstanding Scientists of the 20thCentury, as well as Cambridge’s list of 2000 Outstanding Scientists of 21th Century. Hewas also included on the list of the 100 Top Scientists of 2007, by Cambridge. He has over200 publications in high quality scientific journals, and has been a project manager of morethan 500 projects in civil engineering, mechanical engineering and petroleum engineering.He is Chairman and Professor of the Interpaper Research Organization, as well as a visitingprofessor in universities around Europe and the USA.

Luıs Godinho received his MS and PhD from the University of Coimbra, in both cases study-ing the application of boundary elements to acoustic and wave propagation problems. Since2004, he has been an assistant professor in the Department of Civil Engineering at the Univer-sity of Coimbra. He is also a research member of the Construction Sciences Research Unit-CICC there, and within the scope of this research unit he has been developing experimentaland numerical research in the field of acoustics, wave propagation, and building physics. Hehas published around 60 papers in international peer-reviewed scientific journals, three bookchapters, and more than 100 scientific papers in the proceedings of national and internationalcongresses and conferences. He supervised more than 25 MS theses, as well as one PhDthesis; currently, he supervises two PhD students and one MS student. He participated andcoordinated several research projects, which received funding from official institutions andalso from the industry.

Edmundo Guimaraes de Araujo Costa received a degree in Civil Engineering from GamaFilho University in 2003, a Master of Science in Civil Engineering from the Federal Uni-versity of Rio de Janeiro in 2008, and a Doctor of Science from the Federal University ofRio de Janeiro in 2013. He currently holds a Postdoctoral Grant from the Research SupportFoundation of the State of Rio de Janeiro (Fundacao de Amparo a Pesquisa do Estado doRio de Janeiro-FAPERJ), Brazil, and will from 2013 to 2016. He has authored more than tentechnical publications including five international journal papers. He has experience in thearea of Civil Engineering. His research is focused mainly o n the boundary element method,the method of fundamental solution, and Green’s functions.

Jose Antonio Fontes Santiago is an Associate Professor at the COPPE/Federal Universityof Rio de Janeiro, Civil Engineering Programme. He has been a full-time lecturer there formore than fifteen years, specializing in computational mechanics with applications in Numer-ical Methods Formulations and Engineering Problems Modelling. He has a Doctor of Sci-ence degree in Civil Engineering from COPPE/UFRJ and has co-authored more than twentyinternational journal articles and more than seventy technical publications in internationalconferences. He has also supervised several doctoral and master theses in numerical meth-ods applied in engineering. He is currently researching non-linear elasticity, viscoelasticity,elastic and acoustic wave propagation, and cathodic protection.


About the Authors

Andreia Pereira has been an assistant professor in the Department of Civil Engineering atthe University of Coimbra since 2007, during which time she has been developing researchin experimental and numerical acoustics. She has developed her Master and PhD theses onthe use of the Boundary Element Method in the prediction of wave propagation in elastic andfluid media, with applications in building acoustic problems, oceanography, or seismology. Inrecent years she has published, as a co-author, several papers in international peer reviewedjournals with impact factors on the use of numerical methods, such as the Boundary Ele-ment Method and the Method of Fundamental Solution in the prediction of wave propagationproblems. She has been supervising master students in experimental acoustics in the fieldof building acoustics, focusing on room acoustics and on the development of the acousticperformance of materials to provide sound insulation using small sized models. She also hasa number of recent publications in peer reviewed journals.

Paulo Amado Mendes received a Degree in Civil Engineering from the University of Coim-bra, Portugal (UC); an MS in Civil Engineering, specializing in hydraulics and water re-sources, from the University of Coimbra; and a PhD in Civil Engineering, specializing inconstruction sciences, from the University of Coimbra. Currently, he is an Assistant Profes-sor in the Department of Civil Engineering at the University of Coimbra (FCTUC), where hehas been lecturing on construction sciences, acoustics, numerical methods, and hydraulics.He has supervised 14 concluded MS theses and is a research member of the ConstructionSciences Research Unit-CICC at UC. As a research team member, has participated in severalsupported research and applied research projects in the area of Construction Sciences. He isalso co-author to 16 papers published in scientific journals, two book chapters, and 48 paperspublished in conference proceedings.

Li Xianhong obtained his PhD degree in mechanical electronic engineering from the Chi-nese Academy of Sciences. Now, he is an assistant professor at the Shenyang Institute ofAutomation (SIA). His research interests include mechanical systems and their control sys-tems, vibration measurement and the control of huge mechanical rotating equipment, and themodelling, analysis, and control of complex systems and processes.

Yu Haibin obtained his PhD degree in control theory and control engineering from NortheastUniversity. He is currently the Director of Shenyang Institute of Automation (SIA), ChineseAcademy of Sciences. He is an International Society of Automation (ISA) Fellow, and hisresearch interests include industrial control and automation, robust control, process control,and smart grid.


About the Authors

Zeng Peng obtained his PhD degree in mechanical electronic engineering from the ChineseAcademy of Sciences. His research interests include the wireless sensor network, wirelesscommunication, industrial automation, process control, robust control, and energy saving.

Yuan Mingzhe obtained his PhD degree in mechanical electronic engineering from the Chi-nese Academy of Sciences. His research interests include process control, robust control,industrial automation, and energy saving.

Han Jianda obtained his PhD degree in control theory and control engineering from theHarbin Institute of Technology. His research interests include robust control methods formechanical and electrical systems, self-control methods, self-planning methods, intelligent-control methods, and microdrive sensor integration.

Sun Lanxiang obtained her PhD degree in mechanical electronic engineering from the Chi-nese Academy of Sciences. Now, she is an associate professor at the Shenyang Institute ofAutomation (SIA). Her research interests include mechanical systems and their control sys-tems, laser measurement, component detection, and the modelling, analysis, and control ofcomplex systems.

Antonio Lopes Gama received a Mechanical Engineering degree from the Fluminense Fed-eral University in 1987, a MS degree in Mechanical Engineering from the Pontifical CatholicUniversity of Rio de Janeiro in 1991, and a PhD in Mechanical Engineering from the Pontif-ical Catholic University of Rio de Janeiro in 1998. He is currently an associate professor inthe Department of Mechanical Engineering at the Fluminense Federal University, Niteroi, inRio de Janeiro, Brazil. He has experience in mechanical engineering with emphasis on struc-tural dynamics, acting on the following topics: vibration, two-phase flow induced vibration,piezoelectric strain sensors, and structural integrity.


About the Authors

Rafael Soares de Oliveira received a Mechanical Engineering degree from the Universi-dade Federal Fluminense in 2000, and an MS degree in Mechanical Engineering from theUniversidade Federal Fluminense in 2007. He is currently pursuing a PhD in industrialengineering at the Universidade Federal da Bahia. He is a technologist at the NationalMetrology Institute, Xerem, in Rio de Janeiro, Brazil, and has experience in mechanicalengineering with an emphasis in industrial metrology and quality, acting on the followingtopics: torque metrology, force metrology, traceability, and metrology standardization.

Jaber Javanshir Hasbestan has been working on a PhD from the University of Tennessee,Chattanooga, and as a research assistant in the national Sim Center since the fall of 2011.He holds an MS in Aerospace Engineering from the Department of Aerospace Engineering,Middle East Technical University, Ankara, Turkey, and he served as the research assistant inthe department. He obtained his BS in Aerospace Engineering from Azad University, Tehran,Iran. He finished both graduate and undergraduate degrees with high honor and as the topstudent of the department. He has several publications in AIAA and ASME in the field ofstructural dynamics and vibrations.

Touraj Farsadi holds an MS in Aerospace Engineering from the Department of AerospaceEngineering, Sharif University of Technology, in Tehran, Iran. He obtained his BS inAerospace Engineering from Azad University in Tehran, Iran. His areas of interest and ex-pertise include aeroelasticity, structural dynamics, and wind turbines.

Umur Yuceoglu is a retired professor and faculty member of the Department of AerospaceEngineering at the Middle East Technical University, in Ankara, Turkey. He holds a PhDin Mechanics from Lehigh University, in Bethlehem, PA. He also holds an MS Degree inAerospace Engineering and Applied Mechanics from the Polytechnic University in Brook-lyn, NY. He has “ASME, AIAA-(SDM) Conferences”. participated in conferences and hasbeen an active and contributing member of the Structures and Materials Committee of theASME Aerospace Engineering Division. He has also published many papers on the theoreti-cal fracture mechanics of shells and has authored research papers on the bonded (or adhesive)joints of orthotropic plates, shallow shells, and full cylindrical shells.His areas of interest in-clude aircraft structures, stress analysis and the dynamics of bonded joints, vibration andflutter, and the mechanics of composite multilayer plates and shells.

Hadi Salimi was born in Bushehr, Iran. He received his BS from the Persian Gulf Univer-sity in 2007 and his MS degree from Maleke-Ashtar University’s Mechanical EngineeringDepartment in 2009. His research areas of interest include noise, vibration, and acoustics.


About the Authors

Saeed Ziaei-Rad received both his BS and MS degrees in Mechanical Engineering fromIsfahan University of Technology, Isfahan, Iran, in 1988 and 1990, respectively. He alsoearned his PhD from the Department of Mechanical Engineering, Imperial College of Scienceand Technology, in 1997. He is an academic member of Mechanical Engineering Departmentof Isfahan University of Technology and has many publications in international journals andconferences.

Mehran Moradi was born in south of Iran, in 1962. He graduated with BS and MS in Me-chanical Engineering in 1987 and 1989, respectively, from Isfahan University of Technology.He earned his PhD in Computational Mechanics in 1996 from Shinshu University, in Japan.He is Assistant Professor in the Faculty of Mechanical Engineering at the Isfahan Universityof Technology. His research areas are thermoelasticity, composite materials, vibration, andmetal forming.

Arun Kumar Gupta is an associate professor in the Department of Mathematics, M.S. Col-lege, Saharanpur, U.P., India. He has been teaching undergraduate and graduate courses inmathematics since 1983. Other than teaching, his research interests are in the vibration ofplates. He has participated in many national and international seminars and conferences.He has more than 65 research papers published in international journals of repute. He hasreviewed a few international research papers, as well. He is also on the editorial board ofseveral international journals, and has guided more than 17 PhD students. He also worked asa professor in mathematics at the University of Tikreet, in Tikreet, Iraq.

Mamta works in the Department of Mathematics, M.S. College, Saharanpur, U.P., India.Mamta obtained the a Master of Science in Mathematics (First Division) from M.S. Col-lege, Saharanpur, India during the year 1995. Further, she obtained a Master of Philosophy(Computer Applications) from UOR, Roorkee (now IIT, Roorkee), India. At present, she isworking on a PhD in Mathematics under the supervision of Dr. Arun Kumar Gupta, Profes-sor, Department of Mathematics, Maharaj Singh Post Graduate College, Saharanpur, UttarPradesh, India, and under Choudhary Charan Singh University, Meerut, India. Since 2009,she has been giving private coaching to +1, +2 , B. Tech, BCA, BBA, M.Tech, MCA, andMBA students at Ludhiana, India. Also, she is working as freelance ISO auditor.


Book Reviews

Ground Vehicle Dynamics

By: Karl Popp and Werner SchiehlenSpringer Berlin Heidelberg2010, XIV, 352 p. 109 illus.ISBN 978-3-540-24038-9Price: US $169

I became quite interested in re-viewing this book due to my par-ticipation in a new master’s de-gree in automotive engineering,currently offered at my univer-sity. One of the main courses isin vehicle dynamics, where thestudents learn the mathematicalprinciples and modelling behindthe dynamics of passenger vehi-cles regarding suspension design,road handling and comfort, ma-noeuvring, etc. In this context, Irealized this book could be useful for such courses.

The book deals with the dynamic behaviour of ground ve-hicles, such as automotive passenger cars, but also addressesrailway vehicles and magnetically levitated trains. The aim ofthe book is to set the theoretical background for developingmathematical models in a system- oriented approach, in orderto simulate and predict the motion response and the effect ofdifferent parameters prior to the design of a prototype. Thebook is divided into ten chapters with plenty of mathematicalfoundations used to derive the models. The first part of thebook covers the modelling of vehicle subsystems such as kine-matic and dynamic modelling, support and guidance systems,and guideways; then concepts are united into a full vehicle-guideway system. The second part of the book covers topicssuch as ride comfort and safety, longitudinal, lateral and verti-cal motions, as well as computational models. Plenty of exam-ples are supplied in each chapter, which are very useful for thereader as the concepts are applied in typical situations.

The fundamentals of system definitions and modelling arepresented in chapter one, introducing concepts such as multi-body (or lumped parameter) systems, discrete (FEM) models,and continuous models, as well as different forces that can actupon a particular system. Then, several multi-body (or rigidbody) models are introduced in chapter two. This chapter isdivided into the geometry of motion; i.e., the kinematics ofsingle body and multi-body models, as well as the dynamicbehaviour where the inertia of systems is considered. TheNewton-Euler method, D’alembert’s principle, and Jourdain’sprinciple are used to obtain the mathematical models; then theenergy considerations are introduced and the Lagrange equa-tions applied. These concepts are later used in multi-body sys-tems. The chapter ends with a discussion of non-recursive and

recursive formalisms used for large, multi-body systems.Chapter three explores support and guidance systems, both

passive and active. The chapter begins with the models forlinear and non-linear stiffness and damping elements. Typicalconfigurations are studied, such as linear springs and dampersin different arrangements. Non-linear elements, i.e., cubicstiffness, quadratic dampers, dry friction damping, and others,are also studied. Force actuators are then analysed, focusingon magnetic actuators. The properties of passive and activesystems are also discussed. However, most of the chapter isdevoted to the study of contact forces between the wheel andthe guideway, providing many examples regarding the rollingof rigid and deformable wheels, contact forces in elastic rails,and contact forces in rigid roads. Chapter four follows withdescriptions of guideway models. Elastic guideways are anal-ysed, beginning with periodically pillared guideways and con-tinuously bedded beams. A review of bending vibration inbeams is performed, since most pillared guideways can bebased on such models. The modelling of perturbations androad unevenness is also studied. Chapter five assembles theinformation of the previous chapters into one global system.

The assessment of vehicle dynamics is the main topic ofchapter six, exploring longitudinal motion (driving and brak-ing), lateral motion (guidance and steering), and vertical mo-tion (suspension systems). These motions are assessed throughcriteria such as vehicle performance, driving stability, ridecomfort, and safety. Chapter seven presents computationalmethods used for the assessment criteria introduced in the pre-vious chapter; for instance, numerical simulation through dif-ferent solvers such as Runge Kutta. Stability, frequency re-sponse analysis, random vibration, and non-linear systems arealso considered. Chapters eight, nine, and ten explore indi-vidually the assessment of longitudinal, lateral, and verticalmotions. Dynamics of elastic wheels and complete vehiclesare studied in chapter eight. The forces studied include aero-dynamic effects, driving, and braking torques. Road handlingand driving stability are covered in chapter nine, and the prin-ciples of vehicle suspensions are the main topic for chapterten. The book concludes with an appendix regarding optimalcontrol of multivariable systems.

In conclusion, this book is a very useful companion for anadvanced course on automotive vehicle dynamics, althoughmay not function well as a main course book since the topicscover railway vehicles. However, the mathematical depth andprovided examples are adequate for postgraduate courses as anadditional reference material, or for providing the foundationsnecessary to find practical solutions and design new vehicles.

Diego Ledezma-RamirezUniversidad Autónoma de Nuevo LeónFacultad de Ingeniería Mecánica y EléctricaSan Nicolás de los Garza, Nuevo León, México


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