Note on vibrations of point supported rectangular plates

Journal of Sound and Vibration (1984) 93(4) 593-597

NOTE

ON VIBRATIONS OF POINT SUPPORTED RECTANGULAR PLATES 1.

INTRODUCTION

A considerable number of investigations of the free vibrations of point-supported rectangular plates have been reported [l-15], but some discrepancies exist among the numerical results, significant enough to merit a thorough discussion. In this note a detailed comparison of natural frequencies obtained in the foregoing references is presented, and the accuracy of the frequency values is discussed. Moreover, a method is introduced for analyzing vibration of orthotropic rectangular plates having point supports of arbitrary location. The problem is solved by a classical Ritz method, with a trial function expressed in terms of double power series, wherein the constraint conditions of the supports are taken into account by Lagrange multipliers.

2. The maximum strain and kinetic Figure 1, are expressed by

ANALYSIS

energies

for a thin orthotropic

plate,

as shown

in

respectively, where w(x, y) is the deflection of the plate, the D’s are the flexural rigidities definedas DX=EXh3/12(1-v,u,), D,=E,h3/12(1v,v,) and Dk = Gh”/12 with E,, E, and G being Young’s moduli and shear modulus, respectively, h the uniform plate thickness and v,, vy Poisson’s ratios; p is the mass density per unit volume, and w is the radian frequency of the plate.

Figure

1. Orthotropic

rectangular

plate with symmetric

point supports.

593 0022-460X/84/080593+05

$03.00/O

@ 1984 Academic

Press Inc. (London)

L.imited

594

LETTERS

TO THE EDITOR

Equations (1) and (2) are written in non-dimensional form by using 5 = x/(a/2), n = y/( b/2), and for applying the Ritz method the trial function denoting the deflection of plate is expressed in the polynomial form: (3)

w,,(& 7) = : : &X”r/“. rn=”n=O

When the deflection is constrained rigidly at the points pi (tip n,), equation (3) must satisfy the constraint conditions: w,fl(&, Vi) = 0.

(4)

After introducing the Lagrange undetermined multipliers into the constraint equations (4) and then adding them to the total potential energy, the resulting functional is then minimized with respect to the coefficients A,,,,,. This yields an eigenvalue determinant of order (M+ 1) X (N+ 1) + (number of constraint equations), the zeros of which are the non-dimensional frequency parameters 0 = wa’Jph/ H, with H = Dxvy + 2Dk. The orthotropy is described by the three parameters 0,/H, D,/H and V, (or yY), which reduce to DJH = D,/H = 1 for an isotropic plate. 3.

NUMERICAL

RESULTS

AND

DISCUSSIONS

3.1. Comparison of the results For a square plate symmetrically point-supported, four types of vibration mode exist: i.e., SS, SA, AS and AA modes (S: symmetric, A: antisymmetric). In the frequency equation, m and n are odd or even integers depending upon the mode. For example, the SA mode is symmetric about the y axis and antisymmetric about the x axis, so one hasm=O,2 ,... andn=1,3 ,.... TABLE

Comparison and convergence

Determinant (a)

size

study of frequency parameters 0 for corner point supported square plates ss-1

SA (AS)-1

ss-2

AA-l

ss-3

7.11180 7.11094 7.11089 7.11089 7.117 7.12 7.11 7.14 7.11088 7.15

15.7716 15.7703 15.7703 15.7702 15.73 15.77 15.43 15.79 15.7702 15.64

19.7257 19.5963 19.5961 19.5961 19.13 19.60 -

38.4586 38.4323 38.4316 38.4316 38.42 38.44 -

19.69 19.5961 19.49

38.73

45.5599 44.3762 44.3697 44.3696 43.55 44.4 43.91 44.44 -

38.62

43.89

7.10274 7.10196 7.10192 7.10192 7.180 7.10178 7.1018

15.5438 15.5426 15.5426 15.5426 15.56 15.5402 15.540

19.3473 19.2240 19.2239 19.2239 19.22 19.2200 19.220

38.2055 38.1796 38.1791 38.1790 38.36 38.1762 38.176

45.1979 44.0490 44.0429 44.0428 44.28 44.0395 44.040

v=O.3

Present

17x17 26x26 (37x37

work

Cox and Boxer [l] Reed [2] Johns and Nataraja [4] Srinivasan and Munaswamy [6] Venkateswara Rao et al. [8] Kerstens [lo] (b)

1

v=O.333

(*v=1/3)

Present work Gorman [ 141 Present work* Raju and Amba-Rao*

(10X10 17x17 26x26 (37x37 37x37 [9]

LETTERS

TO THE

EDITOR

595

In Table 1 the calculated frequency parameters L! = wa*Jph/H are compared with those of other authors for the lowest five modes of an isotropic square plate supported on the four corners. This support condition has often been used in numerical examples in the literature. The result of a convergence test is also given in the table, with the respective numbers of terms being m X n = 3 X 3,4 X 4, 5 X 5 and 6 X 6 (the corresponding determinant sizes being 10 X 10, 17X 17, 26X 26, and 37 X 37). It can be seen that the present converged values show excellent agreement with those of references [S, 91: obtained by finite element analysis with a high precision triangular element. In contrast, no common values are found among the other results. Gorman [13,14] proposed a series-type solution by a method of superposition, and obtained natural frequencies for v = 0.333. These values are, however, somewhat higher than the present ones and those of Raju and Amba-Rao [9] for v = l/3. Since the Ritz method always yields upper bounds and the convergence study indicates that the calculated values are converged to within five significant figures, it seems that the present results, as well as those in references [7-Y]. are closer to the exact ones.

Figure 2. Variations of frequency parameters R of an isotropic square plate with position symmetric point supports along the diagonals. u = 0.3; 0, German [ 141 for v = 0,333.

of the four

Figure 2 shows the frequency parameters for an isotropic square plate supported on four symmetric points on the diagonals, as functions of the positions of these points along the diagonals. The point locations are defined by 5, = T,, denoting the position of a point in the first quarter plane ([> 0, 77> 0). For [I = 7, = 1, the square plate is supported at the four corners. As t1 = 71~approaches zero, the four point supports come closer to each other and coalesce at the center in the limit. Although the lowest SA (AS) mode should exhibit a rigid body motion of rotation in the limit, a frequency of R = 10.6 is obtained in the vicinity of 5, = 771=0 because the four point supports, even if they are very close to the center, have a clamping effect by stiffening rotation of the plate. The values obtained by Gorman [14] are indicated by circles in Figure 2. In respect to Gorman’s results, Raju and Amba-Rao [9] suggested that (for the AA mode) “the first frequency may have been

LETTERS

596

TO THE EDITOR

missed for a/a = 0.2,0*3,0*4” (which correspond to & = nl = 0*6,0.4,0*2 in the present definition), and this suggestion is corroborated by the behavior of the AA mode curve shown in the figure (the broken curve). 3.2. Frequencies

of a square plate supported at arbitrary point location

It is evident from Figure 2 that the point support locations have significant effects on the natural frequencies, in particular when the support locations are not at the plate edges. However, the numerical results in the literature have been limited to cases of plates symmetrically supported at points on the diagonals. In Table 2, three sets of the lowest natural frequencies for the SS, SA (AS) and AA modes are presented to illustrate in detail how the point location affects the frequencies. The results are for a square isotropic plate with four point supports symmetrically located with respect to the x and y axes but not necessarily on the diagonals (see Figure 1). In the special cases t1 = 0 or n1 = 0 the plate is supported at two points, and in the case (& , vr) = (0,O) it is supported only at the center. The results given in Table 2 show that the maximum frequency points (i.e., the support points which cause the maximum natural frequency in that mode) for the SS and AA modes are on the diagonals (these points are indicated by **), and the maximum frequency point for the SA (AS) mode is at an off-diagonal position. For the minimum points (marked with *), yielding lowest frequency values, that for the SS mode is at the corner. TABLE

2

Frequency parameters 0 of isotropic square plates supported symmetrically at four poirits (v=o.3;**: maximum point, *: minimum point) 5, 0 (a)

SS

rl’ (b)

(c)

AA

0.4

0.8

1

7.621 10.76 14.10 15.13 13.31

8.529 12.58 17.20 19.15

8.841 12.78

7.111*

18.88 21.17 20.15 16.24 11.71 0

21.30 23.68 22.86 18.68 13.79 0

27.01 30.27 30.01 23.80 16.29 0

33.16 33.92** 32.33 24.34 16.31 0

22.87 23.75 21.30 16.97 12.76 0

15.77 15.54 13.60 10.90 8.137 0

13.47* 13.47* 13.47* 13.47* 13.47* 13.47*

21.76 25.87 24.16 19.85 16.23

28.59 35.67 33.75 25.86

38.16 53.84 51.61

44.73 66.82**

38.43

mode 1 0.8 0.6 0.4 0.2

7.152 10.03 12.90 13.69 12.25

0

11.29

I

SA(AS)

7)l

0.6

0.2

I

9.286 14.49 19.60** (symmetric)

mode

1 0.8 0.6 0.4 0.2 0 mode

(symmetric)

597

LETTERS TO THE EDITOR

In the SA (AS) mode, since the plate shows a rigid body motion of rotation for ql(.$,) = 0. the frequency becomes smaller as vI(&) approaches zero. The minimum point for the AA mode is along the x or y axis, because the supporting points have no effects on the transverse motion of plate due to the nodal lines on the two axes. One can find from the table an optimal point support location to make the natural frequency of the four-point-supported square plate maximum or minimum. It should be mentioned that the frequency values shown in Table 2 may not be completely converged, particularly for the internally constrained cases. For obtaining natural frequencies of orthotropic plates, the orthotropic parameters of the plate material are simply inserted into the frequency equation. Y.

Computer Center, Hokkaido Institute of Technology, Sapporo 061-24, Japan

NARITA

I Received I December 1983) REFERENCES 1960 Aeronautical Quarterly 11, 41-50. Vibration of rectangular plates point-supported at the corners. R. E. REED, JR 1965 NASA TN D-3030. Comparison of methods in calculating frequencies of corner-supported rectangular plates. E. H. DOWELL 1971 Journal of Applied Mechanics 38, 595-600. Free vibrations of a linear structure with arbitrary support conditions. D. J. JOHNS and R. NATARAJA 1972 Journal of Sound and Vibration 25, 75-82. Vibration of a square plate symmetrically supported at four points. SADASIVARAO,G.VENKATESWARA Roland CL. AMBA-RAO 1974 Journal Y.V.K. ofSound and Vibration 32,286-288. Experimental study of vibrations of a four-point supported square plate. 1975 Journal of Sound and Vibration 39,207-216. R. S. SRINIVASAN and K. MUNASWAMY Frequency analysis of skew orthotropic point supported plates. G. VENKATESWARE RAO,I. S.RAJU and C. L. AMBA-RAO 1973 Journal ofSound and Vibration 29, 387-391. Vibrations of point supported plates. G. VENKATESWARARAO,C.L.AMBA-RAO~~~T.V.K.MURTHY 1975JournalofSound and Vibration 40, 561-562. On the fundamental frequency of point supported plates. I. S. RAJU and C. L. AMBA-RAO 1983 Journal of Sound and Vibration 90, 291-297. Free vibrations of a square plate symmetrically supported at four points on the diagonals. J. G. M. KERSTENS 1979 Journal of Sound and Vibration 65, 493-504. Vibration of a rectangular plate supported at an arbitrary number of points. D. J. GORMAN 1978 Journal of the Acoustical Society of America 64, 823-826. Free vibration analysis of rectangular plates with inelastic lateral support on the diagonals. D. J. GORMAN 1979 Journal of Sound and Vibration 66, 239-246. Solutions of the Lev! type for the free vibration analysis of diagonally supported rectangular plates. D. J. GORMAN 1980 Journal of Sound and Vibration 73, 563-574. Free vibration analysis of rectangular plates with symmetrically distributed point supports along the edges. D. J. GORMAN 1981 Journal of Sound and Vibration 79, 561-574. An analytical solution for the free vibration analysis of rectangular plates resting on symmetrically distributed point supports. J.G.M. KERSTENS,~.A.A.LAuRA,R.O.GROSSI and L.ERCOLI 1983JournalofSound and Vibration 89, 291-293. Vibrations of rectangular plates with point supports: comparison of results.

1.H. L. COX and J.BOXER 2. 3. 4. 5.

6. 7. 8. 9. IO.

I 1. 12. 13. 14.

15.