Romanian Reports in Physics, Vol. 57, No. 2, P. 239–248, 2005

THEORETICAL AND EXPERIMENTAL STUDIESON THE VIBRATIONS OCCURRING THROUGH

COMPOSITE CROSSBARS

D. BOLCU, M. URSACHE, G. STANESCU

University of Craiova, Str. A.I. Cuza, nr. 13E-mail:stanescu_gelica@yahoo.com;

Received November 11, 2003

Abstract. This work presents the mathematical model of the vibrations within a compositecrossbar, with a constant and symmetrical section, as is deduced from the hypotheses of thefirst-order deformation theory. The mathematical model is solved for the case of a bar performing amovement of plane translation, by the use of the Laplace transformation in respect to time and theFourier sine finite transformation in respect to the space variable. In the end of the paper, theoreticalresults are compared to the experimental ones that have been measured through the help of thevibrations analyzer B&K 2515.

Key words: vibration, composite, mathematical model.

1. INTRODUCTION

As composite materials have been widely introduced in the technical practiceand due to the fact that these materials are relatively new, the researches into thisfield have known a particular development. Many theories were elaborated, whichtried to take into account the highest possible number of parameters, from thethickness of the structure to the type and distribution of the reinforcer.

Laminated composite bars could be analyzed through many theories whichdiffer one from another by inclusion or neglect of the effects of angulardeformation, respectively of the rotation inertia. Among them is the classical one,symbolized as CPT, successfully employed by Hashin [1], also by Whitney andPagano [2], which relies on the Bernoulli hypothesis according to which a planesection, normal to the average fibre before deformation, should remain plane andnormal to the average fibre during the deformation. So it has been noticed that, forthe laminated pieces at which the ratio between the elasticity modulus E and thesliding modulus G reaches values from 25 to 40, the CPT theory overestimates thenatural frequencies of the structure.

Another theory, the “First-order Shear Deformation Theory”, symbolized asFSDT, developed by Yan and Dowell [3], relies on a linear distribution of

240 D. Bolcu, M. Ursache, G. Stanescu 2

tangential tensions and states that a plane and normal section to the average fibrebefore deformation should remain plane, but would no more keep, during thedeformation, the previously mentioned perpendicularity.

Exact theories rely on a non-linear distribution of tangential tensionsthroughout the bar’s thickness. The use of superior order terms involves the inclu-sions of supplementary unknown factors, bringing about mathematical complications.The third-order theory is the most employed, in the specialized literature.

Using the principle of virtual mechanical work, Reddy [4, 5] constructed atheory, symbolized as HSDT (High-Order Shear Deformation Theory) in which itis accepted that tensions and deformations, on the direction perpendicular to themedian plane, are nil. Another HSDT, for boards that are anisotropic, but whichalso take into account normal tensions to the median plane, was developed byLibrescu [6, 7]. This theory eliminates some of the contradictions which occur inthe previous theories, by the fact that it accepts the non-linearity of the tangentialtension throughout the thickness.

Studies on the vibrations of sandwich-type composite bars were performedby Douglas and Young [8], also by Miles and Reinhall [9], who admitted avariation of the transversal deflection into this section, variation which producesthe appearance of compression efforts on thickness, reaching the conclusion thatsuch a sollicitating mechanism has its influence only at high frequencies. A similarmodel was proposed by Rao [10], in order to investigate the answer ofunsymmetrical sandwich bars under a stationary regime. He stated that externalstrata should be elastic bars of the Euler-Bernoulli type, which endure only normalefforts while the viscid-elastic rein forcer also acts like Timoshenkv bars, able toendure apart from tangential efforts, normal tensions simultaneously. One yearlater, Rao implemented a simplified model of short bars, in which all strata aremeant to endure tangential as well as normal tensions, his analysis also includingthe longitudinal, transversal and rotational inertia effects.

A more complex model was elaborated by Di Sciuva [11], as he stated thatthe constant angular deformations dot appear in every stratum, but they differ fromone stratum to another, a model that he named a linear zig-zag.

This model states, in fact, that all strata act like Timoshenkv bars, thevariables of the problem being reduced by considering tangential tensions ascontinuous, and adding them to the usual continuity conditions for deplacementswithin the bar’s section. Di Sciuva’s model was taken over and developed by Lee[12] who took into account a cubic variation for longitudinal deplacements, eachstratum disposing of its own deplacements field and a parabolic distribution for thetangential tension along the transversal section of the bar, including the continuityof these tensions at the separation surfaces between its constructive parts. Thismodel was used for bars which owned a middle stratum of a considerablethickness, for the study of the occurring deplacements and tensions, and it reachedthe conclusion that the mentioned theory was overrating them. This model was also

3 Vibrations occurring through composite crossbars 241

used for the study of the dynamical response of the sandwich structures, thefrequencies being evaluated as they suited the category of the component materialsand the geometric parameters of the bar’s section.

The numerical results that were obtained and presented stood for thestatement that, in most of the presented cases, natural frequencies and the dampingout factor predicted by the zig-zag theory have, generally, smaller values than theones obtained through the other theories.

2. THEORETICAL CONSIDERATIONS –THE MATHEMATICAL MODEL

Let us consider a composite bar with a constant section and a uniformdistribution on its length of the constructive parts, which, through a kinematicchain, performs a movement of spatial roto-translation. Due to this movement, andunder the action of the linking forces within kinematic couples, the bar will deformitself and will vibrate. The deduction of the mathematical model for the compositebar’s vibrations is made in [13], under the following hypotheses:

– on the external surface of the bar distributed forces and couples do notappear;

– during the movement supplementary supports or any other types of linkswhich might suppose occurring shocks do not appear;

– the initial state of the bar is considered to be untensed, so the tension-freebar is straight;

– a section that is plane and normal to the bar’s axis before the deformationshould remain plane but without preserving the previously mentioned perpendicularity.

Though the considered section might deplane itself during deformation, itwill however be approximated as a plane, according to the preceding hypothesis.The approximation modality is presented in [14], so chosen that deformations andtensions should be the closest reachable to the real ones. This way, the deplacementof a bar’s section might be regarded as a solid and rigid type of deplacement.

If its own reference system should be chosen as to have the origin in one ofthe bar’s ends and the x1 axis to be precisely the one of the bar, the mathematicalmodel of the bar’s vibrations, through a symmetrical section, would acquire theform that has been given in [13]:

2

21

[ ] [ ] [ ] [ ] M q C q K q V q Fx

∂+ + + =∂

(1)

where:

31 21 2 3

1 1 1 ; ; ; ; ;

t

q u u ux x x

∂θ∂θ ∂θ⎧ ⎫= ⎨ ⎬⎩ ⎭

(2)

242 D. Bolcu, M. Ursache, G. Stanescu 4

1

2

3

0 0 0 0 00 0 0 0 00 0 0 0 0

[ ]0 0 0 0 00 0 0 00 0 0 0

AA

AM

IA I

A I

⟨ρ ⟩⎡ ⎤⎢ ⎥⟨ρ ⟩⎢ ⎥

⟨ρ ⟩⎢ ⎥= ⎢ ⎥⟨ρ ⟩⎢ ⎥

⟨ρ ⟩ ⟨ρ ⟩⎢ ⎥⎢ ⎥−⟨ρ ⟩ ⟨ρ ⟩⎣ ⎦

(3)

3 2

3 1

2 1

2 3 3 2

2 2 2 3

3 1 3 2

0 2 2 0 0 0

2 0 2 0 0 0

2 2 0 0 0 0[ ]

0 0 0 0 2 2

2 2 0 2 0 0

2 0 2 2 0 0

A A

A A

A AC

I I

A A I

A A I

− ⟨ρ ⟩ω ⟨ρ ⟩ω⎡ ⎤⎢ ⎥⟨ρ ⟩ω − ⟨ρ ⟩ω⎢ ⎥⎢ ⎥− ⟨ρ ⟩ω ⟨ρ ⟩ω

= ⎢ ⎥− ⟨ρ ⟩ω − ⟨ρ ⟩ω⎢ ⎥⎢ ⎥− ⟨ρ ⟩ω − ⟨ρ ⟩ω ⟨ρ ⟩ω⎢ ⎥

− ⟨ρ ⟩ω ⟨ρ ⟩ω − ⟨ρ ⟩ω⎢ ⎥⎣ ⎦

(4)

( ) ( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )

2 22 3 1 2 3 1 3 3

21 2 3 3 2 3 1

21 3 2 2 3 1 2

21 3 2 2 3 1 2

2 2 21 2 3 3 2 3 1

23 2

[ ]0 0 0

0 0 00 00 0

A A A

A A A

A A AK

A A A

A A A

AGAG

I

⎡ −⟨ρ ⟩ ω + ω ⟨ρ ⟩ ω ω − ε ⟨ρ ⟩ ω ω + ε⎢

⟨ρ ⟩ ω ω + ε −⟨ρ ⟩ω ⟨ρ ⟩ ω ω − ε⎢⎢ ⟨ρ ⟩ ω ω − ε ⟨ρ ⟩ ω ω + ε −⟨ρ ⟩ω⎢=⎢⎢

⟨ρ ⟩ ω ω − ε ⟨ρ ⟩ ω ω + ε −⟨ρ ⟩ω⎢⎢−⟨ρ ⟩ ω ω + ε ⟨ρ ⟩ω −⟨ρ ⟩ ω ω − ε⎢⎣

⟨ ⟩−⟨ ⟩

−ω ⟨ρ ⟩ ( ) ( )( ) ( )( ) ( )

22 3 1 2 3 2 1 3 2 3

2 21 2 3 2 2 3 2

2 21 3 2 3 2 3 3

0

0

I I I

I I

I I

⎤⎥⎥⎥⎥− ω ⟨ρ ⟩ − ω ω + ε ⟨ρ ⟩ − ω ω − ε ⟨ρ ⟩⎥

− ω ω − ε ⟨ρ ⟩ − ω + ω ⟨ρ ⟩ ⎥⎥

− ω ω + ε ⟨ρ ⟩ − ω + ω ⟨ρ ⟩ ⎥⎦

(5)

2

3

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0[ ]

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

EA

AG

AGV

GI

EI

EI

−⟨ ⟩⎡ ⎤⎢ ⎥−⟨ ⟩⎢ ⎥⎢ ⎥−⟨ ⟩

= ⎢ ⎥−⟨ ⟩⎢ ⎥⎢ ⎥−⟨ ⟩⎢ ⎥

−⟨ ⟩⎢ ⎥⎣ ⎦

(6)

5 Vibrations occurring through composite crossbars 243

( )( )( )

( )

( )

2 21 01 2 3 1

2 02 1 2 3 1

3 03 1 3 2 1

1

1

13 03 1 3 2 1

1

12 02 1 2 3 1

1

p A a x

p A a x

p A a x

mFx

mp A a x

x

mp A a x

x

⎧ ⎫⎡ ⎤− ρ − ω + ω⎣ ⎦⎪ ⎪⎪ ⎪− ρ + ω ω + ε⎡ ⎤⎣ ⎦⎪ ⎪⎪ ⎪− ρ + ω ω − ε⎡ ⎤⎣ ⎦⎪ ⎪⎪ ⎪∂= ⎨ ⎬∂⎪ ⎪⎪ ⎪∂

+ − ρ + ω ω − ε⎡ ⎤⎪ ⎪⎣ ⎦∂⎪ ⎪⎪ ⎪∂

− + ρ + ω ω + ε⎡ ⎤⎪ ⎪⎣ ⎦∂⎩ ⎭

(7)

where:u1, u2, u3 – the deplacements of the elastic centre of the current section of the

bar, in regard to its own reference point;θ1, θ2, θ3, – the rotations of the current section of the bar, in regard to its

initial status;

1,ϖ 2 ,ϖ 3ϖ – the projections of the angular speed of the bar on the axes ofits reference system;

ε1, ε2, ε3 – the projections of the angular acceleration of the bar on the axes ofits reference system;

a01, a02, a03 – the projections of the acceleration of the origin of thisreference system on the axes of this system;

p1, p2, p3 – the components of the external charge, as it is distributed on thebar’s axis;

m1, m2, m3 – the components of the external couple, as it is distributed on thebar’s axis.

( )

dS

A sρ = ρ∫∫ 22 3

( )

dS

I x sρ = ρ∫∫

( )

dS

EA E s= ∫∫ 23 2

( )

dS

I x sρ = ρ∫∫

23 2

( )

dS

EI E x s= ⋅∫∫ ( )2 21 2 3

( )

dS

I x x sρ = + ρ∫∫ (8)

22 3

( )

dS

EI E x s= ⋅∫∫( )

dS

AG G s= ∫∫

( )2 22 3

( )

dS

GI G x x s= +∫∫

244 D. Bolcu, M. Ursache, G. Stanescu 6

E, G, ρ – respectively, the elasticity modulus, the shearing modulus and thespecific mass into the current point of this section.

However, if the bar’s section should not be symmetrical, then thelongitudinal or transversal deplacements into a plane would generate phenomena oftorsion, respectively bending, into a perpendicular plane, the occurring phenomenawould gain in complexity and, by that, the possibilities of solving them would bemuch reduced. A case very often met is the one of the bar in a plane movement.The mathematical model of this bar’s vibrations within the movement planes isobtained from the general form (1), but by keeping into account only the termscorresponding to this plane. The same form of the equation for the movement isobtained, where:

( )( )

31 2

1

3

2

2

2 23 3

21 01 1

2 02 1

3

; ;

0 0 0 2 0

[ ] 0 0 [ ] 2 0 0

0 2 0 0

0 0 0

[ ] [ ] 0 0

0 0

m

t

q u ux

A A

M A C A

A I A

A A EA

K A A AG V AG

A A I EI

p A a x

p A a xF

∂θ⎧ ⎫= ⎨ ⎬∂⎩ ⎭⎡ ρ ⎤ − ρ ω⎡ ⎤⎢ ⎥ ⎢ ⎥= ρ = ρ ω⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− ρ ρ − ρ ω⎣ ⎦⎣ ⎦

⎡ ⎤− ρ ω − ρ ε ⎡− ⎤⎢ ⎥ ⎢ ⎥= ρ ε − ρ ω = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− ρ ε ρ ω − ρ ω −⎣ ⎦⎣ ⎦

− ⟨ρ ⟩ − ω

− ⟨ρ ⟩ + ε=∂∂ ( )2 02 1

1

xp A a x

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪

− + ⟨ρ ⟩ + ε⎪ ⎪⎩ ⎭

(9)

in which ω and ε are the angular speed and the angular acceleration of the bar.The matrices [M] and [V] have constant elements, while the matrices [C] and

[K] depend upon the speed and acceleration, both angular, of the bar, so they haveelements which vary in time. If the bar should be in a translation movement, orshould have a constant angular speed, then all matrices given by (9) would haveconstant elements, the mathematical model becoming, in fact, a system ofequations with partial derivatives, the coefficients of which are constant.

3. EXPERIMENT, RESULTS AND CONCLUSIONS

The numerical computations and the experimental attempts were performedfor the case of a composite bar with a matrix of epoxide resin, while the reinforcer

7 Vibrations occurring through composite crossbars 245

is made of glass fibres stripes, which constitute a tissue disposed in strata, thevolume ratio of the fibres being Vf = 0.5.

The main assets of the matrix (denoted by the index ‘m’) and of the fibres(denoted by the index ‘f’) are:– Young’s modulus: Em = 4 500 MPa; Ef = 73 000 MPa;– the transversal elasticity modulus: Gm = 1 600 MPa; Gf = 30 000 MPa;– the specific mass: ρm = 1 400 kg/m3; ρf =2 500 kg/m3;– the transversal contraction coefficient: υm = 0.35; υf = 0.25;– the compressibility coefficient: Km = 7 500 MPa; Kf = 49 000 MPa.

The studied case was the one of a composite bar of length L = 0.45 m and ofsection 0.012 m × 0.012 m performing a movement of plane translation, ensuredthrough a parallelogram mechanism, defined by the following geometrical andkinematic parameters:L = 0.45 m, b = 0.04, p = 2.5π s–1.

The bar is considered as inserted into the mechanism through rotation couples.This is why the solution to the mathematical model could be given only under:

– the limit conditions:

1 10 0x x Lq q= == = (10)

– the initial conditions:

( ) 10tq f x= = (11)

( ) 10tq g x= = (12)

For the translation movement, the angular speed and acceleration are nil. Inthis case, the equations with partial derivatives (1) have constant coefficients, and,to solve them, the Laplace transformation in respect to time and the Fourier sinefinite transformation in respect to the space variable x1 are applied. For the planetranslation it comes to:

( )

( )( ) 1

* 11 2 2 1 * *

1

;

2 [ ] [ ] [ ] [ ] sinnn

q x t

n xs M K V F M s f g

L L

∞− −

=

=

π= + − α ⋅ + +∑L

(13)

with 2n

nπα = (14)

The chosen denotations were:

( ) 1*1 1

0

sin dL

n xf f x x

Lπ

= ∫

246 D. Bolcu, M. Ursache, G. Stanescu 8

( ) 1*1 1

0

sin dL

n xg g x x

Lπ

= ∫ (15)

( ) 1*1 1

0 0

; e sin d dL

st n xf F x t t x

L

∞− π

= ∫ ∫

The transversal vibrations own pulses are given by the line:

22 n nn

EI

A IΩ = α

ρ + α ρ (16)

and the longitudinal vibrations own pulses are given by the line:

n nEA

A⟨ ⟩ω = α⟨ρ ⟩

(17)

In Fig. 1 the transversal deformation at the middle of the bar is presented,while in Fig. 2 the transversal deformation at one third of the bar’s length is shown,as they have resulted from the numerical solving.

Fig. 1 – The transversal deforma-tion at the middle of the bar – theo- retical results.

Fig. 2 – The transversal deforma-tion at one third of the bar – theo- retical results.

9 Vibrations occurring through composite crossbars 247

The experimental measurements were performed with the help of the B&K2515 vibrations analyzer, covering as well the time modulus (0–12 s) and thefrequency modulus (0–1000 Hz). The signal of the mechanical deplacement wasrecorded with the help of a piezoelectric translating deice, with a sensitivity of1 + 0.02 pC/ms and processed through the B&K 7616 software, when theproperties of the IEEE-488 interface were employed. In Figs. 3 and 4 the recordingsare presented, at precisely the same experimental points whereform the theoreticalresults were issued.

Fig. 3 – The transversal deformation atthe middle of the bar – experimental results.

Fig. 4 – The transversal deformationat one third of the bar – experi- mental results.

From the study of the theoretical results and from their comparison with theexperimental ones, we might draw the following conclusions:

– The pulses of longitudinal vibrations depend only upon the rein forcer’sproportion, and not on its distribution within the bar’s section.

– The pulses of transversal vibrations depend as well upon the rein forcer’samount as on its distribution in the bar’s section. This is how, through thecontrol upon the distribution knot of the component parts, the line of thetransversal vibrations pulse could be modified in order to avoid theresonance phenomenon.

248 D. Bolcu, M. Ursache, G. Stanescu 10

– It is the transversal vibrations that are effectively dangerous, as wellregarding their size range as through the fact that their own pulses come tobe placed within the usual working range of the tool machines, while thepulses of the longitudinal vibrations have very high values.

– The dynamical response, measured with the help of the B&K 2515 analyzer,does concord with the response analytically deduced, the effectivelyrecorded errors being situated on a beach from 0 to 10%.

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