Research ArticleEfficient Model Order Reduction of Structural DynamicSystems with Local Nonlinearities under Periodic Motion
M. Mohammadali and H. Ahmadian
Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering,Iran University of Science and Technology, Narmak, Tehran 16844, Iran
Correspondence should be addressed to M. Mohammadali; [email protected]
Received 20 October 2012; Accepted 19 November 2012; Published 9 June 2014
Academic Editor: Hamid Mehdigholi
Copyright Β© 2014 M. Mohammadali and H. Ahmadian. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
In many nonlinear structural systems, compared with the local regions with induced nonlinear effects, the main portions of thestructures are linear. An exact condensation technique based on the harmonic balance method (HBM) in conjunction with themodal expansion technique is employed to convert the motion equations of such a system to a set of nonlinear algebraic equationsthat are considerably small and adequately accurate to determine periodic responses. To demonstrate the capability of the suggestedmethod, few case studies consisting of discrete systems with weak and essential nonlinearities are studied, and the results arecompared to other methodologies results.
1. Introduction
Compared with the local regions with induced nonlineareffects, the main portions of the structural systems are linear.Typical engineering examples of these localized nonlinearsources are friction and vibroimpact in joints, local buckling,crack, nonlinear isolator, dead zone (gap), and squeeze filmdampers in mechanical systems. Consequently, the dynamicresponse of such a system is nonlinear. A dynamic model forthis structural system is useful for detailed study, predictionof response, identification of unknown parameters, or healthmonitoring of the overall system. Therefore, much time andeffort are consumed in the development of this dynamicmodel that includes desired effects. Usually, the dynamicproperties of the structure components are known (analysedor tested) as separate components, and they are used fordevelopment of the dynamic model for the total assembledsystem.
Anynonlinear response analysis involves significant com-putational effort, especially if the full dynamic model isused and a set of performance characteristics related to tem-perature, preload, deflection, and so forth characterize thenonlinear elements. Although there is significant motivationto develop a set of model order reduction (MOR) techniques
to project the system model to a condensed space andconsiderably reduce time of nonlinear response computation,accuracy of these MOR techniques and solution algorithmsare still issues. For reduction of nonlinear discrete modelsone can apply linear MOR techniques [1] such as Guyanreduction [2, 3], improved reduced system (IRS) [4], iter-ated improved reduced system (IIRS) [5], system equivalentreduction expansion process (SEREP), [6] or componentmode synthesis (CMS) [4, 7].
Other reduction methods that can be applied to reducecontinuous and discrete models size are commonly usedlinear normal modes (LNMs) of linearized system andnonlinear normal modes (NNMs). Generation of NNMs fora small system requires considerable effort [8, 9], especiallyin the presence of internal resonance [10] and/or externalexcitation [11]. In the case of a large system, it is possible touse other reduction techniques and then generate NNMs ofthe reduced order model [12].
Reduction techniques based on linear MOR techniques,in best, will lead to an acceptable answer for weakly nonlinearsystem, andby growing the nonlinearity their accuracywill belost due to their linear nature. Local equivalent linear stiffnessmethod (LELMS) was proposed [13] to reduce the nonlineardiscrete models size. In this method, an equivalent linear
Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 152145, 5 pageshttp://dx.doi.org/10.1155/2014/152145
2 Shock and Vibration
system of the nonlinear system is iteratively calculated whosemode shapes are used to reduce the system size.
There aremany different algorithms for solving nonlinearsystems. A semianalytical method that is generally valid overa much larger domain compared to the other methodologiesis the harmonic balance method (HBM) [14]. The HBMcan solve many types of nonlinear problems in the form ofperiodic response. The HBM is used for handling nonlinearproblems consisting of forced excited, strongly nonlinearresponse, chaotic behaviour, and internal resonance and hasa great potential for identification and health monitoring.Furthermore, this technique is applied to estimate NNMs ofnonlinear systems [8].HBMexpands the periodic response ofthe nonlinear system in the form of truncated Fourier series,whose coefficients are estimated by solving a set of nonlinearalgebraic equations. In the absence of internal resonancein many cases, the solution of one term harmonic balancehas good correlation with the recorded experimental results[15]. Typically, more accurate approximation to the solutionobtained by using a Fourier series with a higher number ofterms and, in general, the amplitudes of the higher termsgradually decrease. Consequently, in experiment if the highertermβs amplitudes are small, then inclusion of measurementnoise will result in the difference between these terms in theanalysis and experiment.
HBM is improved in different ways because of its capa-bility. Incremental harmonic balance (IHB) [16] suggests anew formulation based on HBM for achieving frequencyresponses of nonlinear systems. With the aim of reducingthe computational cost and preserving the accuracy, adaptiveHBM (AHBM) [17] has proposed selection algorithms for thenumber of Fourier series terms. Recently, a new exact reduc-tion technique [18, 19] was proposed for discrete systems thatcondense the problem into the nonlinear DOFs only.
The aim of this work is to study the technique suggestedby KIM [18, 19] and extend this by expanding the nonlinearDOFsmotions by natural modes of the condensed system. Todemonstrate the capability of the suggested method, few casestudies consisting of a discrete systemwithweak and essentialnonlinearity are studied and the results are compared.
2. Formulation
The general form of a nonlinear dynamic system with π
degrees of freedom (DOFs) is shown below:
Mq + Cq + Kq + fnl = fπΈ, (1)
where q is the field deformation vector,M,C, andK aremass,damping, and stiffness matrices, fnl is nonlinear force, andfπΈis periodic excitation force. q, fnl, and f
πΈare function of
time which time argument, π‘, is dropped for convenience. Ifthe steady-state solution of above equation with the period ofπ = 2π/π is requested, the deformation field can be expandedwith the aid of HBM in the form of
q =π
β
π=0
(ππcos (πππ‘) + π
πsin (πππ‘)) , (2)
where ππand π
πare vectors of vibration amplitude at the
frequency of ππ. By substituting (2) into (1) and usingGalerkin method, the equation of motion is decoupled in theform below:
βπ2π2Mππ+ππCπ
π+Kππ+ fnl,ππ=fπΈ,ππ ,
β π2π2MππβππCπ
π+Kππ+fnl,π π=fπΈ,π π ,
π = 0, 1, 2, . . . , π,
{fπ,ππ,fπ,π π
} = (1 + sgn (π))
Γπ
2πβ«
2π/π
0
{fπcos (πππ‘) , f
πsin (πππ‘)} ππ‘,
(3)
where π is replaced by arbitrary subscripts of force vec-tors. Equation (3) expresses one static equilibrium and 2πdynamic equations. By using imaginary unit π = ββ1, (3) canbe put together in the form
(βπ2π2MβπππC+K)π
π+ fnlπ=fπΈπ
π = 0, 1, 2, . . . , π,
(4)
where
ππ= ππ+ πππ,
fππ
= fπ,ππ+ πfπ,π π
.
(5)
In the case of localized nonlinearity, it is possible to writenonlinear force vector in the form
fπnl = {πnl1, . . . , πnlπ, 0, 0, . . . , 0} ,
fnl = {fnlπΌ0 } .
(6)
Furthermore, the deformation field can be divided intotwo vectors with subscripts πΌ and π½, which stand for activeor master coordinates and deleted or slave coordinates,respectively. Active coordinates are experiencing nonlinearforces and deleted coordinates are only subjected to linearforces. Also, f
πΈvector and M, C, and K matrices can be
divided appropriately with subscripts πΌ and π½. Therefore, themotion equation in (4) can be written in the form
(βπ2π2[MπΌπΌ
MπΌπ½
Mπ½πΌ
Mπ½π½
]βπππ [CπΌπΌ
CπΌπ½
Cπ½πΌ
Cπ½π½
]
+ [KπΌπΌ
KπΌπ½
Kπ½πΌ
Kπ½π½
]){
ππΌπ
ππ½π
} + {fnlπΌ0 }
={fπΈπΌ
fπΈπ½
} , π = 0, 1, 2, . . . , π.
(7)
Shock and Vibration 3
k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12
c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11
knl8 knl9knl10 knl11
f8
knl12
Figure 1: The discrete system model.
1st harmonic3rd harmonic
5th harmonic7th harmonic
0
2
4
6
8
10
12
Peak
ampl
itude
1 1.1 1.2 1.3
πβ
πβ = 1.21
(a)
1st harmonic3rd harmonic
5th harmonic7th harmonic
0
1
2
3
4
5
6
Peak
ampl
itude
1 1.1 1.2 1.3 1.4 1.5 1.6
πβ
πβ = 1.36
(b)
Figure 2: The response peak amplitudes of each harmonic at 8th DOF: (a) weakly nonlinear system; (b) essentially nonlinear system.
In above equation, one can eliminate ππ½π
from the firstrow with the aid of the second row; this will result in exactreduced form of motion equation
(βπ2π2 MπΌπΌβπππC
πΌπΌ+KπΌπΌβKππ)ππΌπ+ fnlπΌ π
= fπΈπΌπβfπΈππ½π
,
Aπ= (βπ
2π2MπΌπ½βπππC
πΌπ½+KπΌπ½)
Γ (βπ2π2Mπ½π½βπππC
π½π½+Kπ½π½)β1
Kππ
= Aπ(βπ2π2Mπ½πΌβπππC
π½πΌ+Kπ½πΌ) ,
fπΈππ½π
=AπfπΈπ½π.
(8)
The deformation field vectors, ππΌπ, can be expanded by
one of its basis. Orthogonal bases which consist of arrangedvectors with minimum energy required for excitation will besuitable bases to approximate the deformation field vectorsand reduce the size of condense system. To find such bases,
Kππ
can be considered as a constant stiffness matrix, whosedependence on frequency, ππ, is neglected; then
(βπ2
ππMπΌπΌβππππCπΌπΌ+KπΌπΌβKππ)πππ= 0 (9)
is the eigenvalue problem of the linearized condensed system,where π
ππis the πth mass normalized, maximum real right
eigenvector and πππis its companion eigenvalue. The π
ππare
orthogonal basis of ππΌπ
and by increasing the π index therequired energy for excitation π
ππin linearized condensed
system will grow. Considering that the motion of the non-linear condensed system is close to linearized system, thedeformation field in (8) can be approximated by few vectors ofthese orthogonal bases. Therefore, the deformation field canbe written in the form
ππΌπβ Ξ¦π(πππ+ πππ π) , (10)
where πππand π
π πare unknown real vectors and Ξ¦
πis the
modal matrix in which collaborated mode shapes, πππ, are
arranged in column fashion. After substituting (10) into (8)
4 Shock and Vibration
10th DOF HBM11th DOF HBM7th DOF proposed method8th DOF proposed method9th DOF proposed method10th DOF proposed method11th DOF proposed method
0
5
10
15
0
7th DOF HBM8th DOF HBM9th DOF HBM
7th DOF numerical method8th DOF numerical method9th DOF numerical method10th DOF numerical method11th DOF numerical method
Disp
lace
men
t
Time
β5
β10
β15T/4 T/2 3T/4 T
(a)
0
2
4
6
8
Disp
lace
men
t
β2
β4
β6
β80
TimeT/4 T/2 3T/4 T
10th DOF HBM11th DOF HBM7th DOF proposed method8th DOF proposed method9th DOF proposed method10th DOF proposed method11th DOF proposed method
7th DOF HBM8th DOF HBM9th DOF HBM
7th DOF numerical method8th DOF numerical method9th DOF numerical method10th DOF numerical method11th DOF numerical method
(b)
Figure 3: Time response of nonlinear systems at their resonance: (a) weakly nonlinear system πβ= 1.21; (b) essentially nonlinear system
πβ= 1.36.
and using Galerkinmethod themotion equation will becomein the form
Ξπ(πππ+ πππ π) + β¨fnlπΌ π,Ξ¦πβ©=β¨fπΈπΌ πβfπΈππ½π ,Ξ¦πβ© ,
Ξπ= β¨(βπ
2π2MπΌπΌβπππC
πΌπΌ+KπΌπΌβKππ)Ξ¦π,Ξ¦πβ© ,
β¨a, bβ© = ba,
(11)
where β is transpose operator. The real and imaginary partof above equation will result in two different nonlinearequations. Therefore,
[Re (Ξ
π) β Im (Ξ
π)
Im (Ξπ) Re (Ξ
π)]{πππ
ππ π
} +{
{
{
Re (β¨fnlπΌπ ,Ξ¦πβ©)
Im (β¨fnlπΌπ ,Ξ¦πβ©)}
}
}
={
{
{
Re (β¨fπΈπΌπβfπΈππ½π
,Ξ¦πβ©)
Im (β¨fπΈπΌπβfπΈππ½π
,Ξ¦πβ©)
}
}
}
.
(12)
By solving (12) with an arbitrarymethod the final solutionof motion will be achieved.
3. Numerical Results
In what follows, to prove the power of the proposed reductiontechnique discrete vibrating systems are studied with weakand essential nonlinearities of the cubic form. These systems
have 11 DOFs in which 5 of its DOFs are connected togetherwith nonlinear stiffness (Figure 1).
The steady-state responses of the system under periodicexcitation are determined by the numerical method, HBM,and proposed method. The resultant nonlinear algebraicequations of HBM and proposed method are solved byNewton-Raphson algorithm. The computational complexityof thismethod at each step can be approximated by the squareof the number of equations.
The motion of these systems is governed by
ποΏ½οΏ½π+ π (2οΏ½οΏ½
πβ οΏ½οΏ½π+1
β οΏ½οΏ½πβ1
) + π (2π₯πβ π₯π+1
β π₯πβ1)
+ πnlπ(π₯π β π₯πβ1)3+ πnlπ+1(π₯π β π₯π+1)
3= ππ,
(13)
where π = 1, 2, . . . , 11,π = 1, π = 1/10, π = 1, and
ππ= {
0 π = 8
cos (ππ‘) π = 8.(14)
For the case of weak and essential nonlinearities πnlπ is takenas
{
{
{
0 π < 7
1
5π β₯ 7,
{0 π < 7
2 π β₯ 7,(15)
respectively. π1= 0.26Hz, real part of the first resonance
frequency of linearized system, is used to normalize theexcitation frequency πβ = π/π
1. Peak amplitudes of each
Shock and Vibration 5
harmonic at 8th DOF are shown in Figure 2 for differentfrequencies.
The steady-state responses of the systems at their reso-nances are approximated by first 7 harmonies.
Thenonlinear systemswith 11DOFs are reduced to 5mas-ter DOFs and solved using HBM, which causes around 80%decrease in computational cost. By applying the proposedmethod to the condensed system and expanding the responseusing its first three mode shapes, another 64% computationalcost reduction is achieved due to reduction of systems sizefrom 5 DOFs to 3 DOFs. The results are shown in Figure 3.
In both cases, there are negligible differences betweennumerical method and the HBMdue to harmonic truncationorder; therefore, this error is present in the proposed methodresults.
Compared to the results of the HBM and numericalmethod, phase of the DOFs vibration computed by theproposed method is accurate in both nonlinear systems.However, the difference between the calculated amplitudeswith the proposed and other methods is increased withgrowing the nonlinearity, but the captured response is stillacceptable. These differences are due to reduction of exactcondensed system size, from 5 DOFs to 3 DOFs, which leadsto a stiffer system.
4. Conclusion
In this paper, a new condensation technique is suggested fordiscrete systems by the combination of exact condensationtechniques based on HBM and modal expansion techniqueto reduce the computational cost and preserve the accuracy.The proposed method is used to reduce the size of two non-linear systems with weak and essential nonlinearity. At last,the predicted periodic responses are compared with othermethodologies results. The correlation between the results ofthe proposed method and other techniques is appropriate.Using the suggested technique, the reduced form of thesystem is independent of system nonlinearities; therefore,this technique has a great potential for the identification ofsystem nonlinearities. The main drawback of the suggestedtechnique is its frequency dependency.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
[1] Z. Qu,Model Order Reduction Techniques, 2004.[2] R. J. Guyan, βReduction of stiffness and mass matrices,β AIAA
Journal, vol. 3, no. 2, pp. 380β380, 1965.[3] R. W. Stephenson and K. W. Rouch, βModeling rotating shafts
using axisymmetric solid finite elements with matrix reduc-tion,β Journal of Vibration and Acoustics, Transactions of theASME, vol. 115, no. 4, pp. 484β489, 1993.
[4] M. I. Friswell, J. E. T. Penny, and S. D. Garvey, βUsing linearmodel reduction to investigate the dynamics of structures with
local non-linearities,βMechanical Systems and Signal Processing,vol. 9, no. 3, pp. 317β328, 1995.
[5] M. I. Friswell, S. D. Garvey, and J. E. T. Penny, βModel reductionusing dynamic and iterated IRS techniques,β Journal of Soundand Vibration, vol. 186, no. 2, pp. 311β323, 1995.
[6] J. OβCallahan, P. Avitabile, and R. Riemer, βSystem equivalentreduction expansion process (SEREP),β Tech. Rep., IMAC, 1989.
[7] M. C. C. Bampton and R. R. Craig, βCoupling of substructuresfor dynamic analyses,β AIAA Journal, vol. 6, no. 7, pp. 1313β1319,1968.
[8] G. Kerschen, M. Peeters, J. C. Golinval, and A. F. Vakakis,βNonlinear normal modes, Part I: a useful framework for thestructural dynamicist,β Mechanical Systems and Signal Process-ing, vol. 23, no. 1, pp. 170β194, 2009.
[9] M. Peeters, R. Viguie, G. Serandour, G. Kerschen, and J.-C.Golinval, βNonlinear normal modes, Part II: toward a prac-tical computation using numerical continuation techniques,βMechanical Systems and Signal Processing, vol. 23, no. 1, pp. 195β216, 2009.
[10] D. Jiang, C. Pierre, and S. W. Shaw, βThe construction of non-linear normal modes for systems with internal resonance,βInternational Journal of Non-LinearMechanics, vol. 40, no. 5, pp.729β746, 2005.
[11] D. Jiang, C. Pierre, and S. W. Shaw, βNonlinear normal modesfor vibratory systems under harmonic excitation,β Journal ofSound and Vibration, vol. 288, no. 4-5, pp. 791β812, 2005.
[12] M. Legrand, D. Jiang, C. Pierre, and S. Shaw, βNonlinear normalmodes of a rotating shaft based on the invariant manifoldmethod,β The International Journal of Rotating Machinery, vol.10, no. 4, pp. 319β335, 2004.
[13] E. A. Butcher, βClearance effects on bilinear normal modefrequencies,β Journal of Sound and Vibration, vol. 224, no. 2, pp.305β328, 1999.
[14] M. Peeters, G. Kerschen, and J. C. Golinval, βDynamic testing ofnonlinear vibrating structures using nonlinear normal modes,βJournal of Sound andVibration, vol. 330, no. 3, pp. 486β509, 2011.
[15] T. A. Doughty, P. Davies, andA. K. Bajaj, βA comparison of threetechniques using steady state data to identify non-linear modalbehavior of an externally excited cantilever beam,β Journal ofSound and Vibration, vol. 249, no. 4, pp. 785β813, 2002.
[16] A. Grolet and F. Thouverez, βOn a new harmonic selectiontechnique for harmonic balance method,β Mechanical Systemsand Signal Processing, vol. 30, pp. 43β60, 2012.
[17] Y. B. Kim and S. T. Noah, βResponse and bifurcation analysisof a MDOF rotor system with a strong nonlinearity,β NonlinearDynamics, vol. 2, no. 3, pp. 215β234, 1991.
[18] Y. B. Kim and S. T. Noah, βStability and bifurcation analysisof oscillators with piecewise-linear characteristics. A generalapproach,β Journal of Applied Mechanics, Transactions ASME,vol. 58, no. 2, pp. 545β553, 1991.
[19] S. L. Lau and Y. K. Cheung, βAmplitude incremental variationalprinciple for nonlinear vibration of elastic systems,β Journal ofApplied Mechanics, vol. 48, no. 4, pp. 959β964, 1981.
Submit your manuscripts athttp://www.hindawi.com
VLSI Design
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mechanical Engineering
Advances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Electrical and Computer Engineering
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Distributed Sensor Networks
International Journal of
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Active and Passive Electronic Components
Advances inOptoElectronics
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Chemical EngineeringInternational Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Antennas andPropagation
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Navigation and Observation
International Journal of