Vibration studies on skew plates: Treatment of internal line supports

Perganion

Computers & Srructwes Vol. 49, No. 6, pp. 941-951. 1993 0 1994 Eisevier Science Ltd

Primed in Gnat Britain. co45.7949/93 s6.00 + 0.00

VIBRATION STUDIES ON SKEW PLATES: TREATMENT OF INTERNAL LINE SUPPORTS

K. M. LIEW~ and C. M. WANC$ tDynatnics and Vibration Centre, School of Mechanical and production Engineering,

Nanyang T~hnolo~~l Unive~ity, Nanyang Avenue, Singapore 2263 @epartrnent of Civil Engineering, National University of Singapore, Kent Ridge, Singapore 0511

(Received 28 Augusf 1992)

Abstract-A comprehensive literature survey on the vibration of thin skew plates is presented and a few virgin areas on this subject are identified. As an initiat part of a research plan to fill these gaps, the paper focuses on vibrating skew plates with internal line supports. For arudysis, the pb-2 Rayleigh-Ritz method is used. The Ritz function is defined by the product of (I) a two-dimensional polynomial function, (2) the equations of the boundaries with each equation raised to the power of 0, 1, or 2 corresponding to a free, simply supported or clamped edge and (3) the equations of the internal line supports. Since the p&-2 Ritz function satisfies the kinematic boundary conditions at the outset, the analyst need not be inconvenient by having to search for the appropriate function; especially when dealing with any arbitrary shaped plate of various combinations of supporting edge conditions. Based on this simple and accurate pb-2 Rayleigh- Ritz method, tabulated vibration solutions are presented for skew plates with different edge conditions, skew angles, aspect ratios and internal line support positions.

,

1. INTRODUCTION

Free vibration of thin skew (or paralleio~am/ oblique) plates have been studied since the late 1940s. The salient features (such as the kind of boundary conditions, aspect ratios, a/b, skew angles, 8, and methods for analysis) of these studies found in the open literature have been summarized in Table I [ 1-3 11. There are, however, some papers cited in Leissa [32-341, excellent reviews and other cross references which have been left out in this literature survey because they are unavailable to us. In the table, the boundary conditions are abbreviated by four letters where the first letter denotes the support condition at edge AB (see Fig. 1) and the subsequent letters for the other sides in a counterclockwise direction. The letters F, S and C denote free, simply supported or clamped edge, respectively. Note that the solutions for simply supported skew plates are also solutions for the corresponding skew membranes when the square root of the fr~uencies are taken [35].

Based on the literature survey, we can identify a few skew plate vibration problems, which hitherto have not been treated; namely (1) skew plates with internal curved or line supports, (2) skew plates with a patch or concentrated mass, (3) skew plates with a hole or holes; (4) optimal thickness variation for maximum fundamental frequency.

As the first part of an intended series of papers to fill the purported gaps, this study considers the vibration of skew plates with internal line supports, The pb-2 Rayleigh-Ritz method [31,36J will be

employed for the analysis. In the method, the Ritz function is taken as the product of a two~~mensional polynomial function (p-2) and a basic function (b). The basic function is formed from the product of the equations of the boundaries (with each equation raised to the power of 0, 1, or 2 corresponding to a free, simply supported or clamped edge) and the equations of the internal line/curved supports. In this way,’ the geometric boundary conditions are satisfied Q priori and thus eliminate the need for the analyst to search for the appropriate Ritz function for any arbitrarily shaped plate with various combinations of supporting edge conditions. Using this efficient and accurate ~5-2 Rayleigh-Ritz method, vibration solutions are obtained and tabulated for skew plates with different edge conditions, skew angles, aspect ratios and internal line support configurations.

2. PROBLEM DEFINITION

Consider a flat, thin, isotropic and elastic skew plate of constant thickness, h. The dimensions of the plate are described by a and b and the skewness of the plate indicated by the angle 0 with respect to the vertical axis. The plate can have any combination of prescribed supporting edges and some internal line supports that span between any two edges of the plate (Fig. 1). The problem is to determine the natural frequencies of vibration.

Note that special cases of skew plates include square plates (a = b, e = O'), rectangular plates (a # b, f? = 0’) and rhombic plates (a = b, f? # 0’).

941

942 K. M. Lmw and C. M. WANG

Table 1. Literature survey of vibration of thin skew plates

Presented results

Researchers Boundary cond. /aspect ratio, a : b

skew angle @“/number of modes/remarks Method of solution

Seth [I]

Barton [2]

Kaul and Cadambe [3]

Hasegawa [4]

Hamada [5]

Conway and Farnham [6]

Argyris and Buck [7]

Monforton f8]

Laura and Grosson [9]

Durvasula [IO]

Durvasula [ 111

Roberts [ 121

Chopra and Durvasula f 19

Thangam Babu and Reddy [14]

Fried and Schmitt [ 151

Nair and Durvasula [ 161

Nair and Durvasula [ 171

Licari and Warner [I 81

Durvasuia and Nair [ f 9]

Nair and Durvasula 1201

Srinivasan and Munaswamy [211

Srinivasan and Ramachandran [22]

Sakata [23]

Mizusawa et ai. [24]

Kuttler and Sigillito [ZS]

SSSSll~3O/any num~r/isotropic

FFFC/l/O, 1 S, 30,45/2 (with nodal patterns)] isotropic plates

CCCC, CCCS, CCSS/l/O, 15,30,45/8/upper and lower bound solutions CCCS and CCSS solutions are erroneous, isotropic plates

CCCC/ljlS, 20,30, 35/i/isotropic plates

CCCCj l/ l5,3O,45,6O/l/isotropic plates plates, lower bound solutions

SSSSjl, 1.5,2/o, 15,30,45,60/1/cccc/1/0, 10, l&20,25, 30,35,40,45,50,60/1/ isotropic plates

CCCC/ I /isotropic pfates

CCCCjl 10, 15,45/i/isotropic plates

SSSS/l /O-+60/ I /isotropic plates

CCCCjl, 0.667,0.5,0.333~0~~/8~isotropic plates

SSSS/l, 0.667,0.5,0.333/O, 15,20, 30, 35,45/ B/isotropic plates

SSSS/l/O, 9, 18, 27, 36,45/i/isotropic plates

SSSSjllO, 10, 15, 30,45/2/isotropic plates

SFSF/l, 0.667,0.5]0, 15, 30,45/6 (with some mode shapes given)/orthotropic plates

FFFF/l/O, 15, 30,45/8; CCCC/l/O, 15, 30,45/l isotropic plates

ccsc, cscs, cssc, ccss, csss, SSSS~l , 0.667,0.5/o, 5, 10, IS, 20,2s, 30,35,40,45, 50/‘8/isotropic plates

ssss, cccc, cscs, cscc/o. 1-+1.0/o-+50/4 ( with nodal patterns)/isotropic plates

SSSSjl, 1.5/0+6O/l/isotropic plates

CCCC/l, 0.667,0.X 0.333/O, 15, 30,3X 45161 isotropic plates

SSSS, CFCF/I/O, 30,45/8 (with nodal patterns)/orthotropic (boron-epoxy) plates

FFFF with symmetrically located four point suppo~s/l~O, 15,30,45/nodal patterns/ orthotropic plates

SSSS, CCCC/l /O, 30,45/4 (with no&l patterns) /orthotropic (boron-epoxy) plates under uniaxial in-plane forces

SSSS/ a range of a : b/O+36.6/l/orthotropic plates

SSSS, SFSF/l/O, 15, 30,45, ~/2/isotropic plates

CCCCjijO, 15,20,25,30,35,40,45/4 (with nodal patterns for 0 = 1 S)/isotropic plates

Exact solution

Rayleigh-Ritz method/ hyperbolic-trigonometric series

Rayleigh-Ritz method/single product approx.

Rayleigh-Ritz method/ polynomials

TretTtz’s energy method/ trigonometric series

Specially adapted point-matching method (or boundary collocation method)

Finite element method

Finite element method/net degrees method/ N 196

Conformal mapping transformation and Galerkin’s method

Galerkin’s method/ double series of beam characteristic functions in terms of skew coordinates

Rayleigh-Ritz method/double Fourier sine series

Conformal mapping method

Rayleigi-Ritz method/double Fourier series

Finite strip method

Finite element method (total number of degree of freedom equals 784)

Rayleigh-Ritz method/double series of beam characteristic functions in skew coordinates

Perturbation method study focusses on frequency crossings

Parameter differentiation method of Joseph (1967)

Partition (or su~omain~ method with matrix order 32 x 32

Rayleigh-Ritz method with products of beam characteristic functions

Finite strip method

Integral method

Reduction method

RayIeigh-Rib method with Lagrangian multipliers

Rayleigh quotient method with . potynomtals

(continued

Vibrational studies on skew plates 943

Table l-coatinued

Presented results

Researchers

Nagaya ]261

Mizusawa and Kajita 1271

Mizusawa and Kajita [28]

Cheung et al. [29]

Mizusawa and Leonard I301

Liew and Lam I3 l]

Boundary Cond./Aspect ratio, a : b skew angle B”/number of modes/remarks

SSSS, CCCC/ l/O, 10,30,50/4/isotropic plates

SSSS and two other sides elastically restrained against rotation/O.5,0.75, I, 1.25, 1.5, 1.75, 2/0,30,60/3/isotropic

CFFC, SFFC, SFFS, FFFC, FFFS with a comer point support at acute intersection of free edges; FFFF with (a) four comer point supports (b) point supports at mid-points of its sides and (c) eight point supports along edges/ l/O, 15,30,45, (iO/S/isotropic plates

SSSS, CCCC/l/lS, 30,45/6 (with nodal pattems)/isotropic plates

Mixed boundary conditions, special cases SSSS, CCCC/ l/O, l&30,45, ~/5~~otropic plates

SSSS, SSSC, CSSC, ~FFF/l/O~~l4 (with nodal patterns)

Method of solution

Using exact solution of equation of motion and boundary conditions satisfied by Fourier expansion method

Spline strip method

Rayleigh-Ritz discretization procedure with B-spline function

Spline finite strip method

Spline element method

pb -2 ~ylei~-~~ method

3. pb-2 RAYLEIGBRITZ METHOD

In view of the skew geometry of the plate, it is expedient to use the oblique coordinates (&jr) instead of the usual orthogonal coordinates (x, y) (see Fig. 1). The coordinate transformation relations are

2=x-ytan@ (la)

y=y sece (lb)

and therefore

am W) -=- ax aa (2a)

. CW

For convenience and generality, the following nondimensional oblique coordinates are adopted

Fig. 1. Vibration of skew plate with internal line supports.

f Y

/

wl i_!h internal line support

The strain energy, U of the considered plate, in nondimensional oblique coordinates is thus given by

where D = LW/[12(1 - v*)] = flexural rigidity of the plate; v = Poisson’s ratio; ~(5, q) = the deflection of the middle plane of the plate perpendicular to the xy plane.

The kinetic energy of the plate, T, after eliminating the time component is given by

I 1 T = {phabw ’ cos fl

IS w2 dt drt, (5)

0 0

where p is the mass density per unit area of plate, and w the angular frequency of vibration.

The energy functional, F, is given by

F=U-T. (6)

The transverse displacement surface may be parameterized by

944 K. M. LIEW and C. M. WANG

/!D/!cQ

+--a@--+-~-+ ii ~al2--+-ai2-_i g )+-----a-_ci ’

(a) (b) (a)

Fig. 2. Various orientations of internal line supports.

where p is the degree of the polynomial space, c, are the unknown coefficients to be varied with the subscript r given by

r;=(4+w7+2)_i ? i.

and

4, = (TV 356, (9)

in which the basic function, Cp,(& n), is taken as the product of the boundary integral line support equations, i.e.

where n is the number of internal line supports, Aj is the equation of the jth internal line support and Sz,, depending on the support edge condition, takes on

ti, = 0 if the kth edge is free

Q, = 1 if the kth edge is simply supported

0, = 2 if the k th edge is clamped (11)

and the subscripts of Q k = 1,2,3,4 refer to the supporting edges AB, BC, CD, DA, respectively (see Fig. 1).

Note that Cp, ensures that the Ritz functions satisfy the kinematic boundary conditions and that the m number of potynomial terms in a pth degree set is equal to @ + l)@ + 2)/2.

Applying the Rayleigh-Ritz method

8F %,=O; i=1,2 ,..., m. (12)

I

Substituting eqns (4)~(11) into eqn (12) yields

(W-l - Wf)(c) = fO1, (13)

where R2 = phw2a4/D and the elements of the matrices are given by

x (M;?20 + M;!“2) +

x (4 sin2 # + 2(1 - v)cos* e)iw$”

Table 2. Frequency parameters, Jphw2a4/(lr4D), for FSFS skew plates with an oblique central transverse internal line support

Modes

aI6 6 1 2 3 4 5 6

0” 3.946 4.736 6.220 6.847 7.168 8.933

;;: 4.204 4.942 6.640 6.654 8.086 8.890

1 5.123 5.708 7.064 8.040 9.450 10.17 45” 7.332 7.605 8.782 10.23 12.76 14.19 60” 12.27 12.53 14.70 15.94 20.98 22.41

0 3.962 4.325 5.496 6.234 6.520 7.417

::r 4.229 4.570 5.663 6.558 6.992 7.310

213 5.185 5.443 6.414 7.359 8.384 8.621 45” 7.494 7.546 8.521 9.234 10.77 12.65 60” 12.63 12.95 15.18 15.32 17.89 20.68

0” 3.971 4.174 4.860 5.985 6.241 6.404 15” 4.242 4.431 5.104 6.092 6.722 6.741

l/2 30 5.216 5.347 6.008 6.804 7.610 8.330 45” 7.561 7.569 8.358 8.870 9.703 11.05 60 12.95 13.25 15.35 15.41 16.87 18.47

0” 3.979 4.066 4.384 4.905 5.630 6.248 15” 4.255 4.333 4.653 5.161 5.843 6.511

l/3 30 5.246 5.289 5.634 6.104 6.699 7.298 45” 7.610 7.640 8.178 8.518 8.994 9.511 60” 13.42 13.78 15.56 15.69 16.31 16.94


Table 3. Frequency parameters, dm, for SSSS skew plates with an oblique central transverse internal line support

Modes

0 0 I 2 3 4 5 6

0” 5.000 7.045 8.000 9.636 13.00 14.29 15” 5.366 7.360 8.674 9.910 13.89 14.40

1 30 6.695 8.684 10.91 11.16 15.44 16.60 45” 10.07 12.27 14.80 16.10 19.36 21.67 60 20.17 22.98 26.22 28.99 34.44 34.57

0 4.444 5.778 6.605 7.700 8.000 9.638

:;I 4.759 6.119 7.099 7.850 8.780 9.820

213 5.897 7.385 8.859 9.015 10.91 10.99 45 8.765 10.60 12.38 13.24 14.28 16.08 60 17.23 19.98 22.50 24.67 26.39 27.87

0 4.250 5.000 6.250 6.455 7.059 8.000 15” 4.549 5.325 6.566 6.909 7.573 8.082

l/2 30” 5.635 6.500 7.781 8.541 9.227 9.357 45” 8.379 9.436 10.92 12.26 12.98 13.53 60 16.53 17.99 20.09 22.05 24.32 25.57

0” 4.111 4.444 5.000 5.778 6.349 6.617 15” 4.402 4.745 5.314 6.095 6.803 6.990

l/3 30 5.460 5.834 6.447 7.259 8.199 8.447 45” 8.147 8.580 9.284 10.20 11.30 12.29 60” 16.18 16.71 17.60 18.76 20.62 22.19

4. NUMERICAL RESULTS

Numerical calculations have been made for some (14) combinations of side ratios, a/b = 1,2/3, l/2, l/3 and skew angles 6 = o”, 15”, 30”, 45”, 60”. The single

G, = (cos 6)My

whereda=drdn,i,j=l,2 ,..., m.

(1% internal line support considered is placed in an oblique central transverse direction (Fig. 2a) while the two line supports are placed either on the oblique central transverse and longitudinal directions (Fig. 2b) or on the plate diagonals (Fig. 2~). The four combinations of supporting edge conditions considered are

Table 4. Frequency parameters ,/m, for FCFC skew plates with an oblique central transverse internal line SUDDOrt

Modes

alb 9 1 2 3 4 5 6

0" 6.200 6.809 8.879 9.052 9.527 11.21 15” 6.570 7.153 9.169 9.560 10.16 11.40

1 30” 7.860 8.378 10.47 11.09 12.14 12.91 45” 10.86 11.28 14.01 14.35 16.79 17.89 60” 18.85 18.90 24.58 24.74 28.65 30.82

0”

213 ::I 45” 60”

6.215 6.488 7.433 9.064 9.090 9.281 6.606 6.846 7.832 9.344 9.740 9.751 7.967 8.110 9.350 10.66 11.20 11.80

11.11 11.14 13.39 14.14 14.58 16.35 19.25 19.66 24.82 25.03 26.02 28.44

0” 15”

112 30” 45” 60”

0” 15”

l/3 30” 45” 60

6.223 6.374 6.913 6.624 6.746 7.315 8.014 8.055 8.840

11.18 11.26 12.92 19.74 20.16 25.03

7.842 8.253 9.802

13.81 25.23

9.070 9.195 9.458 9.629

10.78 11.21 14.30 14.71 25.47 26.60

6.231 6.643 8.052

11.34 20.57

6.294 6.684 8.061

11.44 21.13

6.538 6.949 7.543 8.331 6.951 7.368 7.969 8.745 8.514 8.937 9.556 10.29

12.65 13.05 13.66 14.17 25.13 25.35 25.61 25.64

946 K. M. LIEW and C. M. WANG

Table 5. Frequency parameters ,,/wj , for CCCC skew plates with an oblique central transverse internal line support

Modes

sib e I 2 3 4 5 6

0 7.436 9.988 10.96 12.95 16.72 18.22 15” 7.972 10.45 11.89 13.38 17.86 18.49

I 30 9.911 12.36 14.93 15.22 20.18 21.39 45” 14.80 17.54 20.39 21.95 25.75 28.27 60” 29.25 33.12 36.37 39.85 45.84 45.99

0

213 :;I 45” 60”

6.700 8.086 7.173 8.594 8.884 10.44

13.20 15.11 25.99 28.69

9.426 10.11 12.38 17.25 31.89

10.45 10.52 12.51 10.65 11.60 12.81 12.56 14.46 14.50 18.60 19.44 21.23 34.16 37.17 37.33

0” 6.483 7.202 8.437 9.264 9.825 10.21 15 6.941 1.682 8.926 9.911 10.51 10.60

112 30” 8.604 9.418 10.72 12.12 12.50 12.92 45” 12.82 13.78 15.28 16.93 18.36 18.48 60 25.40 26.61 28.57 30.90 33.19 35.16

0” 6.345 6.638 7.137 7.856 8.805 9.161 15” 6.796 7.096 7.605 8.330 9.270 9.814

113 30 8.438 8.758 9.299 10.05 11.01 12.02 45” 12.62 12.97 13.57 14.40 15.44 16.58 60” 25.13 25.54 26.24 27.19 28.60 30.21

FSFS, SSSS, FCFC and CCCC as shown in Fig. 2. For accurate solutions, the degree set of polynomial of p = 11 is taken.

Tables 2-5 present the first six natural frequency parameters, 1’ = (pI~~a’/(rr’D))“~, for the four different support conditions with an oblique central transverse internal line support. The equation for this internal line support used in eqn (10) is

A, = 5 - 0.5. (17)

Tables 6-9 present the first six frequency parameters for skew plates with two internal line supports placed in oblique central transverse and longitudinal directions. The equations of these line supports used in eqn (10) are given by

A,=5 -0.5 and A,=q -0.5. (18)

Tables lo-13 furnish the first six frequency parameters for skew plates with two internal line supports placed on their diagonals. The equations for

Table 6. Frequency parameters Jm, for FSFS skew plates with oblique central longitudinal and transverse internal line supports

Modes

a/b e 1 2 3 4 5 6

0 4.736 5.149 6.834 7.163 11.25 12.70 15” 5.066 5.388 7.199 7.296 12.21 12.71

1 30” 6.138 6.251 8.106 8.413 13.77 14.81 45” 8.186 8.260 10.86 11.27 17.35 17.49 60” 12.47 12.80 19.63 19.67 27.24 27.46

0” 4.325 4.454 6.509 6.618 7.417 8.311

:;I 4.612 4.707 6.667 6.740 8.265 8.861

213 5.580 5.602 7.444 7.601 10.52 10.58 45” 7.660 7.728 9.825 9.874 13.28 13.72 60” 12.42 12.70 16.87 17.06 22.21 22.29

0” 4.174 4.230 5.985 6.393 6.438 6.448 15” 4.450 4.489 6.133 6.356 7.090 7.223

f/2 30 5.406 5.406 6.997 7.089 8.812 9.147 45” 7.566 7.626 9.230 9.243 11.83 12.12 60” 12.70 12.98 15.65 15.82 21.09 21.11

0” 4.066 4.084 4.905 5.075 6.312 6.338

;;: 4.340 4.350 5.210 5.334 6.563 6.660

113 5.300 5.306 6.268 6.296 7.489 7.585 45” 7.580 7.623 8.690 8.720 10.03 10.09 60” 13.15 13.44 15.46 15.61 18.09 18.11


Table 7. Frequency parameters ,/‘m, for SSSS skew plates with oblique central longitudinal and transverse internal line supports

Modes

a/b e 1 2 3 4 5 6

0” moo 9.617 9.617 11.07 20.00 20.00 15” 8.709 10.12 10.44 11.49 19.89 21.04

I 30” 11.23 12.44 12.96 13.30 21.33 24.06 45” 17.31 18.46 18.79 19.20 27.79 30.68 60” 34.52 36.10 36.45 37.94 50.44 53.32

0” 5.778 6.336 7.685 8.162 11.11 12.50

:;I 6.270 6.781 8.181 8.480 12.02 12.82

213 8.037 8.450 9.855 9.918 14.53 15.05 45” 12.44 12.81 13.85 14.16 19.51 19.60 60” 25.31 26.02 26.64 26.71 33.81 34.78

0” 5.000 5.253 7.046 7.264 8.008 8.818

::I 5.401 5.621 7.466 7.513 8.787 9.417

l/2 6.842 6.985 8.771 8.942 11.26 11.57 45” 10.46 10.52 12.37 12.61 15.31 15.88 60” 21.24 21.25 23.47 23.52 27.15 28.09

0” 4.444 4.527 3.778 6.081 6.606 6.688

:s 4.776 4.841 6.149 6.403 7.137 7.190

l/3 5.963 5.988 7.492 7.648 8.997 9.010 45” 8.911 8.917 10.89 10.91 12.51 12.86 60° 17.59 17.62 20.70 20.91 23.25 23.40

these line supports to be used in eqn (10) are given

by

A=q--{ and A2=tf+(--1. (19)

Referring to the results in Tables Z-13, it can be seen that the frequency parameters increase with respect to increasing skew angles, 0 for a given aspect ratio, a/b, in all the boundary conditions considered. For a given skew angle, the frequency parameters decrease with decreasing aspect ratios

except for the fundamental frequencies of the FSFS and FCFC cases with an oblique central transverse internal line support which show the opposite trend.

5. CONCLUDING REMARKS

Based on a comprehensive literature survey of vibration of thin skew plates, some virgin areas of research in this subject are identified. This study

Table 8. Frequency parameters ,,/m , for FCFC skew plates with oblique central longitudinal and transverse internal line supports

Modes

c/b e 1 2 3 4 5 6

0” 6.809 7.118 9.511 9.743 12.61 14.51 15” 7.230 7.491 9.910 10.01 13.66 14.91

1 30” 8.691 8.849 11.16 11.23 16.93 17.03 43” 12.09 12.14 14.43 14.44 21.62 21.84 60” 20.78 21.24 24.56 24.91 34.82 34.93

0” 6.488 6.579 9.090 9.267 9.332 9.865

:;I 6.874 6.940 9.371 9.377 10.17 10.63

213 8.203 8.217 10.79 10.88 12.56 13.10 45” 11.22 11.31 14.25 14.28 17.79 18.38 60” 19.28 19.76 24.69 24.87 30.73 31.04

0” 6.374 6.412 7.842 8.215 9.182 9.227

:;I 6.738 6.782 8.339 8.639 9.747 9.748

l/2 8.079 8.084 10.13 10.25 11.37 11.47 45” 11.13 11.21 14.10 14.13 15.70 15.72 60” 19.42 19.81 24.86 25.05 29.26 29.28

0” 6.294 6.306 6.949 7.080 8.331 8.727 15” 6.688 6.693 7.411 7.506 8.812 9.131

l/3 30” 8.044 8.038 9.097 9.110 10.56 10.64 45” 11.25 11.33 13.33 13.36 14.33 14.41 60” 20.12 20.62 25.19 23.41 26.16 26.21

948 K. M. LEW and C. M. WANG

Table 9. Frequency parameters, ,/ pho2a4/(x’D), for CCCC skew plates with oblique central longitudinal and transverse internal line SUDDO~~S

a/b e 1 2 3

Modes

0 10.96 12.94 12.94 15” 11.90 13.60 14.03

1 30 15.22 16.61 17.38 45” 23.25 24.43 24.85 60 45.71 47.49 47.69

4 5

14.69 24.54 15.29 24.67 17.75 26.86 25.51 35.47 49.50 64.83

24.64 25.84 29.76 38.59 67.19

0”

213 ::I 45” 60”

8.086 8.696 10.51 11.02 13.79 15.51 8.743 9.296 11.18 11.48 14.92 15.96

11.10 11.52 13.43 13.54 18.36 18.67 16.99 17.28 18.89 19.27 24.73 24.92 34.37 34.67 35.97 36.14 43.05 44.33

0 7.202 7.451 9.811 10.02 10.21 11.04 15” 7.754 7.967 10.30 10.38 11.31 11.84

l/2 30” 9.736 9.859 12.15 12.40 14.45 14.74 45” 14.68 14.71 17.33 17.57 20.03 20.63 60” 29.20 29.29 33.08 33.12 36.27 36.78

0

l/3 ::I 45” 60

6.638 6.709 7.856 8.129 9.380 9.450 7.121 7.175 8.394 8.624 10.07 10.12 8.858 8.877 10.33 10.45 12.17 12.48

13.20 13.21 15.12 15.12 17.14 17.32 26.05 26.08 29.06 29.08 32.05 32.65

6

on vibrating skew plates with internal line supports forms the initial part of a series of projects to fill the existing research gaps.

In the current work, the application of the pb-2 Rayleigh-Ritz method for the analysis of internally supported skew plates was shown to be effective and relatively simple for the analyst to handle plates with various combinations of edge conditions and any number of internal curved/line supports. The method does not require any discretization or mesh generation

and eliminates the need for seeking appropriate Ritz functions which must satisfy the geometric boundary conditions. The method advantage of not needing any discretization becomes even more evident when the optimal locations of the internal line supports are to be determined so as to maximize the fundamental frequency of the plate.

The tabulated vibration solutions for skew plates with one or two internal line supports should be useful for designers and researchers.

Table 10. Frequency parameters, J pho2a4/(n4D), for FSFS skew plates with two diagonal internal line supports

Modes

sib e 1 2 3 4 5 6

0 4.492 4.722 10.27 10.76 11.66 13.27 15” 4.686 4.918 10.64 10.94 12.38 13.72

1 30 5.367 5.606 11.84 11.84 14.75 15.63 45” 6.979 7.228 14.21 14.44 20.24 20.73 60” 11.39 11.59 20.91 21.21 32.81 33.13

0” 3.711 3.723 8.411 9.111 9.175 9.818 15” 3.874 3.886 8.686 9.140 9.594 10.45

213 30” 4.447 4.455 9.591 9.769 11.25 12.51 45” 5.808 5.816 11.73 11.74 15.49 17.29 60” 9.547 9.637 17.54 17.84 27.18 27.23

0” 3.342 3.344 7.303 7.556 8.450 8.469 15” 3.492 3.493 7.569 7.697 8.725 9.032

l/2 30” 4.017 4.018 8.427 8.431 10.26 10.82 45” 5.267 5.281 10.39 10.47 14.45 14.72 60 8.755 8.866 16.01 16.46 24.67 25.55

0” 2.998 3.000 5.698 5.734 7.499 7.810 15” 3.136 3.140 5.947 5.990 7.982 8.152

113 30” 3.619 3.631 6.831 6.877 9.337 9.404 45” 4.778 4.825 8.898 8.943 12.25 12.57 60” 8.086 8.314 14.63 14.91 21.09 22.33


Table 11. Frequency parameters, ,/m, for SSSS skew plates with two diagonal internal line supports

Modes

a/b 9 1 2 3 4 5 6

0 10.00 12.34 12.34 15.10 20.00 23.42 15” 10.54 12.69 13.33 15.93 20.65 23.84

1 30” 12.44 14.62 16.16 18.89 23.11 26.41 45” 17.05 19.57 22.79 26.25 29.39 33.57 60 29.85 33.87 41.31 48.16 48.31 56.12

0

219 :;I 45 60”

7.280 7.720 9.967 10.12 14.71 15.07 7.695 8.125 10.49 10.73 15.31 15.52 9.187 9.577 12.36 12.93 17.34 17.52

12.90 13.18 16.92 18.40 22.21 23.23 23.79 23.89 30.03 34.00 37.48 41.04

0

l/2 ::I 45” 60

0 15”

l/3 30 45” 60”

6.043 6.080 8.237 9.237 10.99 12.36 6.393 6.423 8.721 9.804 11.48 12.61 7.653 7.659 10.48 11.82 13.28 14.07

10.77 10.82 14.93 16.52 18.19 18.29 20.07 20.34 28.24 29.39 31.76 34.06

4.713 4.724 7.411 7.955 8.156 8.478 4.995 5.010 7.897 8.340 8.540 9.042 6.018 6.046 9.638 9.737 9.973 11.04 8.619 8.699 13.26 13.49 14.29 15.94

16.50 16.77 23.93 24.74 27.89 30.33

Table 12. Frequency parameters, ,/m, for FCFC skew plates with two diagonal internal line supports

Modes

a/b e 1 2 3 4 5 6

0" 4.969 5.058 11.44 11.51 14.40 16.21 15” 5.065 5.422 11.55 11.95 15.39 17.07

1 * 30 5.666 6.455 12.40 13.24 18.70 19.95 45” 7.295 8.769 14.86 16.35 24.41 25.63 60” 12.07 14.87 21.97 25.02 34.93 37.65

0 4.101 4.111 9.870 9.874 10.15 10.33

213 :;’ 4.130 4.474 9.548 9.967 11.20 11.47 4.595 5.399 10.07 10.88 13.73 14.35

45 5.893 7.454 12.05 13.56 18.92 19.68 60 9.745 12.85 18.05 21.47 28.68 31.51

0 3.735 3.750 8.024 8.028 9.176 9.180

:z 3.760 4.087 8.035 8.362 9.700 9.992

l/2 4.172 4.953 8.717 9.376 11.54 12.27 45” 5.315 6.885 10.72 12.02 15.59 16.68 60” 8.718 12.01 16.51 19.78 25.20 27.78

0 3.429 3.435 6.037 6.080 8.547 8.582 15” 3.450 3.747 6.213 6.445 8.791 9.163

l/3 30 3.806 4.559 7.095 7.531 9.892 10.75 45” 4.779 6.395 9.308 10.11 12.74 14.22 60” 7.712 11.43 15.19 17.67 21.66 24.71

950 K. M. LIEW and CT. M. WANG

Table 13. Frequency parameters, ~p~~~o/(n4D), for CCCC skew plates with two diagonal internal line supports

Modes

sib B --I 2 3 4 5 6

0” 13.33 16.06 16.06 19.22 24.54 28.31 15” 14.08 16.59 17.34 20.30 25.43 28.92

1 30 16.75 19.30 21.09 24.19 28.79 32.32 45” 23.25 26.22 29.91 33.81 31.22 41.50 60” 41.50 46.19 55fol 61.78 62.49 69.93

0” 9.422 9.726 12.96 13.53 17.96 18.31 15” 9.911

213 30” 11.94 45” 16.82 60” 31.01

10.26 12.19

13.68 16.25 22.58 40.83

14.36 17.36

16.96 31.20

24.74 44.12

18.64 18.66 20.86 21.32 27.13 28.15 48.57 49.31

0” 7.433 7.441 15” 7.868 7.872

112 30” 9.424 9.430 45” 13.31 13.35 60” 24.88 25.13

11.14 12.53 11.82 13.28 14.28 20.35 36.09

15.58 20.70 36.68

13.27 14.33 13.85 14.66 16.40 16.48 21.78 23.41 40.17 43.66

0” 5.517 5.523 9.130 9.142 10.50 11.66 15” 5.582 5.860 9.566 9.583 11.19 12.41

113 30” 7.062 7.078 11.14 11.17 13.73 14.76 45” 10.12 10.18 15.11 IS.21 20.13 20.19 60” 19.31 19.56 27.24 27.72 36.51 38.02

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Il.

12.

13.

REFERENCES

B. R. Seth, Transverse vibrations of rectilinear plates. Proc. of Indian Academic Science, Sec. A 25, 25-29 (1947). M. V. Barton, Vibration of rectangular and skew cantilever plates. J. appl. Mech., ASME 18, 129-134 (1951). R. K. Kaul and V. Cadambe, The natural frequencies of thin skew plates. Aeronaut. Quart. 7, 337-352 (1956). M. Hasegawa, Vibration of clamped parallelogrammic isotropic flat plates. J. Aeronaut. Sci. 24, 145-146 (1957). M. Hamada, Compressive or shearing buckling load and fundamental frequency of rhomb&dal plate with all edaes clamned. Bull. JSME 2. 520-526 (1959). H. D-Conway and K. A. Famham, The free flexural vibrations of triangular, rhombic and parallelo~m plates and some analogies. ht. J. Mech. Sci. 7,81 l-816 (1965). J. H. Argyris and K. E. Buck, The oscillations of plates or arbitrary shape. Symposium on Numerical Methodr for Vibrations Problems, July 12-15, Institute of Sound and Vibration Research, University of Southampton, England (1966). G. R. Monforton, Finite element, displacement and vibration analysis of skew plates. Case Western Reserve University, Cleveland, Ohio, Division of Solid Mechanics, Structures and Mechanical, Design Report 18 (1968). P. A. Laura and J. Grosson, Fundamental frequency of vibration of rhombic plates. J. Acourt. Sot. Am. 44, 823-824 (19 S. Durvas a, Natural frequencies and modes of f

).

damped spew plates. AIAA Jnl7, II641166 (1969). S. Durvasula, Free vibration of simply supported parallelogrammic plates. J. Aircraft 6, 66-68 (1969). S. B. Roberts, Buckling and vibrations of polygonal and rhombic mates. J. Enann Mech. Div., Proc. ASCE 97, 305-315 (1971). - .. I. Chopra and S. Durvasula, Natural frequencies and modes of tapered skew plates. ht. J. Mech. Sci. 13, 935-944 (1971).

14. P. V. Thangam Babu and D. V. Reddy, Frequency analysis of skew orthotropic plates by the finite strip method. J. Sound Yibr. 18, 465-474 (1971).

15. f. Fried and K. H. Schmitt, Numerical results from the application of gradient iterative techniques to the finite element vibration and stability analysis of skew plates. Aeronauf. JnZ 76, 166-169 (1972).

16. P. S. Nair and S. Durvasula, Vibration of skew plates. 1. Sound Vibr. 26, l-19 (1973).

17. P. S. Nair and S. Durvasula, On quasi-degeneracies in plate vibration problems. Inr. J. Mech. Sci. H&975-986 (1973).

18. J. P. Licari and W. H. Warner, Domain dependence of eigenvalues of vibrating plates. SIAM, J. appl. Math. 24, 383-395 (1973).

19. S. Durvasula and P. S. Nair, Application of partition method to vibration problems of plates. J. Sound Vibr. 37,429-445 (1974).

20. P. S. Nair and S. Durvasula, Vibration of generally orthotropic skew plates. J. ACOW. Sot. Am. 55, 998- 1002 (1974).

21. R. S. Srinivasan and K. Munaswamy, Frequency analysis of skew orthotropic point supported plates. J. Sound Vibr. 39, 207-216 (1975).

22. R. S. Srinivasan and S. V. Ramachandran, Vibration of generally orthotropic skew plates. J. Acoust. Sot. Am. 57, 1113-1118 (1975).

23. T. Sakata, A reduction method for problems of vibration of orthotropic plates. J. Sound Vibr. 48, 405-412 (1976).

24. T. Mizusawa, T. Kajita and M. Naruoka, Analysis of skew date oroblems with various constraints. J. Sound Vibr. ?3, 57.5-584 (1980).

25. J. R. Kuttbr and V. G. Siaillito. Uooer and lower bounds for frequencies of cgmped rhdmbicai plates. J. Sound Vi&r. 68, 597-607 (1980).

26. K. Nagaya, Vibration of a plate with straight line boundaries. J. Sound Vibr. 68, 35-43 (1980).

27. T. Mizusawa and T. Kajita, Vibration and buckling of skew plates with edges elastically restrained against rotation, Camput. Slruct. 22, 987-994 (1986).

28. T. Mizusawa and T. Kajita, Vibration of skew plates resting on point supports. J. Sound Vibr. 115, 243-251 (1987).


29. Y. K. Cheung, L. G. Tham and W. Y. Li, Free vibration and static analysis of general plate by spline finite strip. Computational Mech. 3, 187-197 (1988).

30. T. Mizusawa and J. W. Leonard, Vibration and buckling of plates with mixed boundary conditions. Engng Struct. 12, 285-290 (1990).

31. K. M. Liew and K. Y. Lam, Application of two- dimensional orthogonal plate function to flexural vibration of skew plates. J. Sound Vibr. 139, 241-252 (1990).

32. A. W. Leissa, Vibration of plates, NASA SP-160, Science and Technical Information Division, Office of

Technical Utilities, National Aeronautical and Space Administration, Washington, DC (1969).

33. A. W. Leissa, Recent research in plate vibrations: classical theory. Shock Vibration Digest 9(10), 13-24 (1977).

34. A. W. Leissa, Plate vibration research, 197661980: classical theory. Shock Vibration Digest 13(9), 11-22 (1981).

35. S. Durvasula, Natural frequencies and modes of skew membranes. J. Acoust. Sot. Am. 44, 1636-1646 (1968).

36. K. M. Liew and C. M. Wang, pb-2 Rayleigh-Ritz method for general plate analysis. Engng Struct. 14 (1992).

Documents

Vibration studies on skew plates: Treatment of internal line supports