JE Journal of Engineering 2314-4912 2314-4904 Hindawi Publishing Corporation 937596 10.1155/2013/937596 937596 Research Article Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading Bambill D. V. 1, 2 Rossit C. A. 1, 2 Milani Gabriele 1 Department of Engineering Institute of Applied Mechanics (IMA) Universidad Nacional del Sur (UNS) Alem 1253, B8000CPB Bahía Blanca Argentina uns.edu.ar 2 Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Bahía Blanca Argentina conicet.gov.ar 2013 19 2 2013 2013 27 11 2012 31 01 2013 2013 Copyright © 2013 D. V. Bambill and C. A. Rossit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

1. Introduction

The transverse-free vibrations and buckling of plates which are subjected to edge loads acting in their middle planes are areas of research which have received a great deal of attention in the past century.

As it was stated experimentally by Hearmon  for the case of a beam, bifurcation buckling may be regarded as a special case of the vibration problem, that is, determining the in-plane stresses which cause vibration frequencies to reduce to zero.

Most of the work has dealt with rectangular plates having uniformly distributed in-plane edge loads. In that case, the governing differential equations of motion and equilibrium have constant coefficients, yielding exact solutions for frequencies and buckling loads straightforwardly when two opposite edges of the plates are simply supported.

Many researchers have analyzed both the buckling and vibration of rectangular plates subjected to in-plane stress field. Among them, one can mention Kang and Leissa ; Leissa and Kang ; Bassily and Dickinson ; Dickinson ; Kielb and Han ; Kaldas and Dickinson .

For the linearly varying loading, the governing differential equations have variable coefficients.

Leissa and Kang  found exact solutions for the free vibration and buckling problems of the SS-C-SS-C isotropic plate loaded at its simply supported edges by linearly varying in-plane stresses.

They also found the exact solution  for the buckling of rectangular plates having linearly varying in-plane loading on two opposite simply supported edges, with different boundary conditions at the other opposite edges.

Within the realm of the classical theory of plates, the case of buckling and vibrations problems for all the possibilities of boundary conditions and linearly varying in-plane forces offers considerable difficulty. This is the reason why it is quite common to make use of the Ritz variational method.

2. Approximate Analytical Solution

In the case of a transversely vibrating, thin, isotropic plate subjected to in-plane forces Nx, Ny, and Nxy, (Figure 1 and (5)), the maximum value of the potential energy due to bending deformation is (1)Umax=12DA[(2Wx-2+2Wy-2)2+2(1-ν)×(2Wx-22Wy-2-(2Wx-y-)2)]dx-dy-, where W=W(x,y) is the deflection amplitude of the middle plane of the plate, D is the well known flexural rigidity D=Eh3/12(1-ν2), E is the Young modulus, and ν is the Poisson coefficient.

Rectangular vibrating plate subjected to in-plane loads: N-x, N-y, and N-xy.

While the maximum of the kinetic energy is (2)Tmax=12ρω2AhW2dx-dy-, where ρ is the density of the plate material, ω is the circular frequency, and h is the thickness of the plate.

And the maximum potential energy of the internal stresses caused by the in-plane loading is (3)τN=12AP  (N-x(Wx-)2+N-y(Wy-)2+2N-xyWx-Wy-)dx-dy-.

The lengths of the sides of the rectangular plate are a in the x direction and b in the y direction. The coordinates are written in the dimensionless form as follows: (4)x=x-a,y=y-b.

And the in-plane forces are expressed as (Bambill et al. ): (5)Nx=N-xb2D,Ny=N-yb2D,Nxy=N-xyb2D.

Then, the governing functional of the system is (6)J[W]=12D[ba3ANx(Wx)2dxdy+1abANy(Wy)2dxdy+2a2ANxyWxWydxdy(Wx)]+12D[ba3A(2Wx2)2dxdy+2νabA2Wx22Wy2dxdy+ab3A(2Wy2)2dxdy+2(1-ν)abA(2Wxy)2dxdy]-12ρabω2AhW2dxdy.

Equation (6) satisfies, if W is the exact solution, the condition: (7)δJ[W]=0.

Following the Ritz method, the expression of the deflection of the plate is approximated in the form of a truncated series: (8)WWa(x,y)=m=1Mn=1NAmnXm(x)Yn(y), where Xm(x) and Yn(y) are the characteristic functions for the normal modes of vibration of beams with end conditions nominally similar to those of the opposite edges of the plate in each coordinate direction . Consequently, they satisfy the essential boundary conditions, as the method requires.

The variational equation (7) is replaced by the homogeneous linear system of equations: (9)J[Wa]Aql=0,q={1,2,M};l={1,2,N}, and using nondimensional variables, becomes (10)abDJ[Wa]Aql=0.

Finally, one obtains a homogeneous linear system of equations in terms of the Aql’s parameters.

The nontriviality condition of the system (10) requires the determinant to be zero: (11)|S·[τqlmn]+[uqlmn]-Ω2·[tqlmn]|=0, where Ωmn=ωmna2ρh/D are the frequency coefficients.

The elements of the matrices involved in (11) are given by (12)τqlmn=1λ2AnNx*(dXqdxdXmdx)(YlYn)dxdy+AnNy*(XqXm)(dYldydYndy)dxdy×1λAnNxy*(dXqdxXqdXmdxdYldyYn+dXqdxXmYldYndy)dxdy, where λ=a/b is the aspect ratio (13)Nx=S·Nx*,Ny=S·Ny*,Nxy=S·Nxy*, where S is a factor that indicates the magnitude of the in-plane loading system, regarding the relative value of the forces. Consider (14)uqlmn=λ-2A(d2Xqdx2d2Xmdx2)(YlYn)dxdy+νA  [(Xqd2Xmdx2)(d2Yldy2Yn)+(d2Xqdx2Xm)(Yld2Yndy2)]dxdy+λ2A(XqXm)(d2Yldy2d2Yndy2)dxdy+2(1-ν)A(dXqdxdXmdx)(dYldydYndy)dxdy,tqlmn=λ-2A(XqXm)(YlYn)dxdy.

As it is known, the condition Ω=0 in (11) yields the critical value of the in-plane loading.

3. Numerical Evaluations

Hearmon  has experimented on a fixed-free strip. Admittedly, the problem is analytically simpler in the case of one-dimensional domains. As an example, let us try with a pinned-pinned transversely vibrating beam, subjected to an axial compressive force P. The expression of the frequency coefficient is (15)Ωn=ρA0EIL2ωn=(nπ)2[1-PL2EI(nπ)2];withn=1,2,3,4, where ρ is the density of the material, A0 is the cross-section, L is the length, and EI the flexural rigidity of the beam.

All the Euler buckling loads are determined making zero expression (15). For n=1, the critical buckling load of the beam, Pcrit, is obtained.

Plotting the values Ωn=(nπ)2(1-(P/n2Pcrit)) of the first three frequency coefficients depending upon the ratio P/Pcrit yield regular curves as is shown in Figure 2. The presence of the compressive axial load P does not alter the order of the modal shapes of the beam.

Curves of the frequency coefficients of transverse vibration for a pinned-pinned beam under axial compressive load P.

In the case of a plate, in general, and due to the bidimensional behavior induced by the torsional rigidity, the compressive in-plane load may alter both the order and shape of the modal shapes associated to each natural frequency.

This situation has an important technological signification from the point of view of vibration control.

Certainly, the modal shape of a natural resonant frequency must be known in order to suppress it. In the case of in-plane loading, this shape can be different from the expected one.

Due to the quantity and variability of the parameters involved in the description of the behaviour of these kinds of structures, just a few representative cases will be considered to demonstrate the convenience of the procedure and the importance of the situation.

All the values are determined taking M=N=15 in (8).

Table 1 shows the values of the first natural frequency coefficients for a CCFF plate subjected to a general in-plane loading: linear load in x direction (α=2)—bending moment, constant load in y direction (β=0), and constant shear force Nxy=N1=N2=N).

The first six natural frequency coefficients Ωi for a C-C-F-F plate under general in-plane loading.

N / N crit
0 0.25 0.5 0.75 1
λ = a / b = 0.75 ; Ncrit=5.95168

Ω 1 5.39789 4.75442 3.95041 2.84503 0
Ω 2 15.7547 15.2037 14.5835 13.8967 13.1443
Ω 3 23.8207 23.9071 24.0071 24.1145 24.2241
Ω 4 36.3999 36.2826 36.1113 35.7654 35.1888
Ω 5 38.1446 37.5193 36.9334 36.5072 36.2961
Ω 6 59.7605 59.5098 59.2367 58.9439 58.6327

λ = a / b = 1 ; Ncrit=3.69137

Ω 1 6.93254 6.06065 4.99441 3.56413 0
Ω 2 23.9780 23.9778 23.6766 23.0314 22.1751
Ω 3 26.6265 26.0033 25.6318 25.5511 25.6238
Ω 4 47.7179 47.4465 47.1607 46.8601 46.5440
Ω 5 62.8830 62.6774 62.3443 61.9069 61.3970
Ω 6 65.6818 65.4396 65.3185 65.2954 65.3384

λ = a / b = 2 ; Ncrit=1.85222

Ω 1 17.1515 14.9296 12.2458 8.69504 0
Ω 2 36.4410 36.0647 35.6415 35.1668 34.6357
Ω 3 73.5943 73.8555 74.0906 74.2962 74.4680
Ω 4 91.1002 89.5702 88.0081 86.4144 84.7901
Ω 5 115.363 114.262 113.147 112.021 110.886
Ω 6 131.962 132.267 132.566 132.855 133.130

In order to show the influence of the in-plane loading, the next two examples are presented.

Table 2 shows the natural frequency coefficients for a C-C-SS-SS square plate under uniform compression in the x direction (Nx=N, α=0, and Ny=Nxy=0).

The first six natural frequency coefficients Ωi for a C-C-SS-SS plate under uniform Nx loading (α=0).

λ = a / b = 1 Ncrit=61.4151 N / N crit
0 0.1 0.25 0.5 0.75 1
Ω 1 27.0542 25.7489 23.6368 19.5326 14.0587 0
Ω 2 60.5387 58.4465 54.9733 48.6153 41.2865 32.4139
Ω 3 60.7863 60.1238 59.2827 57.8385 56.3769 54.8441
Ω 4 92.8371 91.4269 89.2672 85.5386 81.6286 77.5081
Ω 5 114.557 112.082 108.152 101.257 93.8564 85.8259
Ω 6 114.704 114.350 113.923 113.207 112.485 111.755

Figure 3 shows that a minimal presence of in-plane loading (10% of the critical value) dramatically modifies the mode shapes, while changes in the values of frequencies may not be noticed (0.3% in the sixth frequency). It is important to point out that the small load can be originated by thermal variations and restrictions on plane displacements imposed by the external supports.

Normal modes of vibration of a square C-C-SS-SS plate under uniform Nx loading (λ=a/b=1, Ncrit=61.4151).

Finally, Table 3 shows the results for a rectangular C-C-SS-SS plate subjected to shear in-plane forces.

The first six natural frequency coefficients Ωi for a C-C-SS-SS plate under uniform shear Nxy loading.

λ = a / b = 1.5 Ncrit=89.8137 N / N crit
0 0.1 0.25 0.4 0.5 0.75 0.9 1
Ω 1 44.8904 44.7005 43.7991 42.4070 40.2822 32.4323 22.3996 0
Ω 2 76.5451 76.1864 74.3165 70.7817 67.4142 54.4376 41.3590 27.7956
Ω 3 122.319 122.349 122.548 121.465 117.487 104.882 95.5807 88.7020
Ω 4 129.393 128.871 126.129 122.871 123.101 123.415 123.108 122.473
Ω 5 152.529 152.707 153.680 155.033 155.912 157.429 157.633 157.273
Ω 6 202.615 200.101 194.748 188.611 184.103 171.463 163.073 214.018

Figure 4 shows that the third and fourth natural frequencies interchange their normal modes as Nxy increases. This situation is noticeable from Figure 5.

Normal modes of vibration of a rectangular C-C-SS-SS plate under uniform Nxy loading (λ=a/b=1.5, Ncrit=89.8137).

The first six natural frequency coefficients Ωi for a C-C-SS-SS plate under uniform shear Nxy loading (λ=a/b=1.5, Ncrit=89.8137).

This means that for a given value of Nxy, between 0.25 and 0.4 of the critical value, there are two normal modes for the same natural frequency (repeated frequency). This is an important point in vibration control, since when repeated frequencies arise in a system, the related vibration mode shape cannot be uniquely determined. Any linear combination of the modes is still valid for the repeated frequency.

In order to evaluate the accuracy of the expounded procedure, comparison is made with the results obtained in  for a SS-C-SS-C plate loaded at its simply supported edges by linearly varying in-plane stresses (Tables 4 and 5).

Comparison of nondimensional critical buckling loads Ncrit=N-critb2/D for a SS-C-SS-C plate.

Load (α) Solution λ = a / b
0.4 0.5 0.6 0.7 0.8 0.9 1.0
[A]  M = 5 93.3059 75.9452 69.6553 69.1116 72.0966 77.5543 75.9452
[ A]  M = 10 93.2555 75.9146 69.6351 69.0972 72.0859 77.5460 75.9146
0 [ A]  M = 15 93.2477 75.9105 69.6323 69.0954 72.0846 77.5450 75.9105
[ A]  M = 20 93.2476 75.9101 69.6323 69.09531 72.0844 77.5449 75.9101
[ B] 93.247 75.910 69.632 69.095 72.084 77.545 75.910

[A]  M = 5 174.533 145.286 134.809 134.624 140.981 152.024 145.286
[ A]  M = 10 174.395 145.215 134.765 134.593 140.958 152.007 145.215
1 [ A]  M = 15 174.379 145.207 134.760 134.590 140.956 152.005 145.207
[ A]  M = 20 174.377 145.206 134.760 134.590 140.956 152.005 145.206
[ B] 174.4 145.2 134.8 134.6 141.0 152.0 145.2

[A]  M = 5 401.518 392.147 412.162 424.140 401.518 392.143 392.147
[ A]  M = 10 400.478 391.589 411.812 422.594 400.478 391.398 391.589
2 [ A]  M = 15 400.410 391.548 411.790 422.490 400.410 391.351 391.548
[ A]  M = 20 400.399 391.548 411.787 422.472 400.399 391.343 391.547
[ B] 400.4 391.5 411.8 422.5 400.4 391.5

[ A]: present approach with different M=N and [B]: .

Comparison of the first six natural frequency coefficients for a SS-C-SS-C plate under bending moment in  x-direction (α=2).

λ = a / b N / N crit Solution Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 Ω 6
0 [ A] 13.6858 23.6465 38.6942 42.5871 51.6767 58.647
[ B] 13.69 23.65 42.59 51.67
0.5 0.5 [ A] 11.4936 24.152 37.9343 38.8757 52.632 58.7048
[ B] 11.49 24.15 37.93 52.63
0.95 [ A] 3.92178 24.906 27.6496 39.3337 52.6657 58.8581
[ B] 3.926 24.91 27.65 52.66

0 [ A] 28.9509 54.7433 69.3271 94.5862 102.217 129.096
[ B] 28.95 54.74 69.33 94.59 102.2
1 0.5 [ A] 27.4647 45.9744 69.6407 87.2215 96.6079 129.164
[ B] 27.47 45.97 69.65 87.22 96.61
0.95 [ A] 15.6871 23.3557 46.9793 70.3963 99.624 110.599
[ B] 15.70 23.37 46.96 70.41 99.63 110.6

0 [ A] 95.2625 115.803 156.357 218.973 254.138 277.308
[ B] 95.26 115.8 156.4 254.1 277.3
2 0.5 [ A] 94.7438 109.859 137.29 183.897 254.214 254.254
[ B] 94.76 109.9 137.3 183.9 254.2
0.95 [ A] 62.7485 77.444 93.4166 93.4229 101.133 187.917
[ B] 62.82 77.50 93.46 93.47

[A]: present approach and [B]: .

In Table 4, values of Ncrit are compared for three different cases of the x direction load: constant (α=0), linear with null value at one extreme (α=1), and bending moment (α=2), and different aspect radii of the plate. A convergence study is also made. As it can be seen, taking M=N=15 provides an excellent accuracy from an engineering viewpoint.

4. Conclusions

The classical, variational method of Ritz has been successfully used in the present study to obtain an approximate, yet quite accurate, solution to a difficult elastodynamics problem.

Natural frequencies and mode shapes of transverse vibration are obtained for a meaningful combination of the boundary conditions of a thin plate subjected to general in-plane loads. The critical values of the in-plane forces which cause instability of the plates are also obtained.

The obtained values are the outcome of an algorithm, relatively simple to implement,  which allows studying these with only the assistance of a PC.

Additional complexities like orthotropic material characteristics can be taken into account .

The agreement with results available in the literature is excellent. Nevertheless, it is also possible to increase the number of terms in the summation on (8) to increase the accuracy.

No claim of originality is made, but it is hoped that the present work draws the attention to the effect that the presence of plane stress state may have on the effectiveness of vibration control on plates.

Acknowledgments

The present work has been sponsored by the Secretaría General de Ciencia y Tecnología of Universidad Nacional del Sur, UNS, at the Department of Engineering and by Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET. The authors wish to thank Dr. D. H. Felix from Universidad Nacional del Sur.