On the use of a coordinate transformation for analysis of axisymmetric vibration of polar orthotropic annular plates

Journal of Sound and Vibration (1972) 24 (2), 165-175

ON THE USE OF A COORDINATE ANALYSIS OF AXISYMMETRIC ORTHOTROPIC

TRANSFORMATION VIBRATION

FOR

OF POLAR

ANNULAR PLATES

K. VIJAYAKUMAR ANDG. K. RAMAIAH Department of Aeronautical Engineering, Indian Institute of Science, Bangalore 12, India (Received 29 February 1972, and in revised form 22 June 1972)

Estimates of natural frequencies corresponding to axisymmetric modes of flexural vibration of polar orthotropic annular plates have been obtained for various combinations of clamped, simply supported and free edge conditions. A coordinate transformation in the radial direction has been used to obtain effective solutions by the classical RayleighRitz method. The analysis with this transformation has heen found to be advantageous in computations, particularly for large hole sizes, over direct analysis. Numerical results have been obtained for various values of hole sizes and rigidity ratio. The eigenvalue parameter has been found to vary more or less linearly with the rigidity ratio. A comparison with the results for isotropic plates has brought out some interesting features. 1. INTRODUCTION Thin circular plates are widely used in many important engineering structures and machines. Elastic behaviour of these plates as derived from polar orthotropic theory is of great utility in analysing structural components of practical importance like plates stiffened in radial and circumferential directions and plates fabricated out of modern laminated composites. Flexural vibration of polar orthotropic circular plates (without holes) has been treated earlier in the literature [l, 2, 31. A detailed review of these investigations together with results and numerical data may be found in reference [4]. In the case of annular plates, Pyesyennikova and Sakharov [S] obtained one term approximate solutions for axisymmetric modes by the Boobnov-Galerkin method. They considered the two cases of a free inner edge and a clamped or simply supported outer edge and obtained results for various hole sizes and rigidity ratios. However, the procedure adopted involves the use of Bessel functions and solution of a transcendental equation, which is rather tedious and cumbersome. Recently, Vijayakumar and Joga Rao [6] used the classical Rayleigh-Ritz method with simple polynomials as admissible functions. For a hole ratio of l/2, estimates of the least eigenvalues in the axisymmetric case were obtained for various combinations of clamped, simply supported and free edge conditions. During the computations, it was observed that the simple polynomials used resulted in a set of ill-conditioned equations. Consequently, double precision arithmetic was required even for a three term approximation. On a m-examination of the above mentioned feature of the polynomials as a part of the present investigations, it was found that single precision arithmetic would be quite adequate for small hole sizes. To achieve corresponding accuracies for large hole sizes, a modification in the form of a coordinate transformation was introduced in evaluating the energy integrals. The details of this modification and the results obtained thereby are presented in this paper. Estimates to the least eigenvalues are obtained for various values of hole sizes and rigidity ratio. 165

166

K. VIJAYAKUMAR

AND G. K. RAMAIAH

2. MATHEMATICAL

ANALYSIS

Consider a thin annular plate of uniform thickness h. Let a and b denote radii of inner and outer edges, respectively (see Figure 1). It is assumed that the material of the plate is homogeneous and cylindrically (polar) orthotropic. In the axisymmetric modes of flexural vibration, the expressions for the maximum strain energy and maximum kinetic energy within the classical thin plate theory when the effects of rotary inertia and transverse shear deformations are neglected [6] are b

vmax

=

7c

S(

D,W,$+2~H$.,FV,+~W’f r

(1)

(I

and

b

Tmax= no2yh

W’rdr. s s

The suffix after a comma denotes differentiation Appendix.

(2)

and the symbols are as defined in the

Figure 1. Annular plate of constant thickness.

The problem is to find solutions such that WInI,, - T,,,) = 0

(3)

for arbitrary variations of W satisfying relevant geometric boundary conditions. In the Rayleigh-Ritz method, the axisymmetric mode W is expressed as W = A,u, + A,u, f A,u,

+ ...

(4)

in which ul, v2, v3, . . . are chosen admissible functions such that W satisfies the relevant geometric boundary conditions for arbitrary values of the linear parameters A, (m = 1, 2,3, * . .). These parameters are to be determined from the stationary condition (3) for arbitrary variations of these parameters. This process leads to a set of linear, homogeneous, simultaneous equations in the A,‘s. For non-trivial solutions, one obtains the characteristic equation for the determination of eigenvalues as det (V,, - w2yhTm,) = 0 in which

(5)

b vtn,

D,.vit.$’ -t Dl(u$A

= !I

0

+ vbvy)/r

(6)

VJBRATION OF ANNULAR PLATES

167

b

and T,, =

s

v,v,,r dr,

0)

where ’ denotes differentiation with respect to r. It is to be noted that the elements V, and T,, are symmetric in m and IZ. The solution of equation (5) is, in practice, obtained approximately by retaining only a finite number of terms in equation (4). TABLE 1

Admissible functions W = q,(r) [A, + A2r + A,r2 -t- . . .]

(4)

w = u,(y) [B, + B,y + B,y2 + . . .]

(11)

Boundary condition /

.

Outer

Inner

Free Simply supported FIX%? Fixed Simply supported Simply supported Fixed Fixed

Simply supported Free Fixed FlW Simply supported Fixed Simply supported Fixed

Designation

00(r)

F-S S-F F-C C-F S-S S-C

JksY)

(r - 4 (b - r) (r - a)2

Y (1 -v) Y2

(b - r)’ r) (r {i 1 r) (r - r)2 (r - r)’ (r -

;I;

(1 - Y)2 4

~7)~ a) a)’

Yi’

- Y)

;(1’”- -$ Y2(I - Y)l

In reference [6] the admissible functions v,,, were constructed from ordinary power series and numerical results were obtained for a hole ratio of 3. In the present investigation, the same set of admissible functions are used for small hole sizes (see Table 1). 2.1.

COORDINATE TRANSFORMATION

As mentioned earlier, the direct analysis described above is not convenient for large hole sizes due to rounding errors in numerical work. It is felt that one of the reasons for these rounding errors is due to the difficulty in accurate evaluation of the energy integrals, more so in the case of both edges clamped. To overcome this difficulty, a coordinate transformation r2 - a2 y = b2 _ =2 is proposed. When this transformation form

is used, the expressions for V,, and T,,

take the

(9) b2 - a2 T,,, = ?I2 in which c = a2/(b2 - a’).

7

1

K. VIJAYAKUMAR AND G. K. RAMAIAH

168

Expressing the mode W as W = B,u, + B,u, + B,u, + . . .

(11)

and proceeding as before, one obtains the characteristic equation as det (J,,,,, - 02yhl,,)

= 0,

(12)

in which J,,,, = J,,, I,,,,, = I,,,,,,

J~.=4~(~+c)iu.u:-dy+(~-l)j

u:u; dv + 2(1 + ue) [(y + c) u’ u’lyI’0 mny

0

(13)

0

and I,,

=

@24;a2’2\umumdy, ,

(14)

0

where.’ denotes differentiation with respect to y. The admissible functions u,,,are constructed from ordinary power series in y (see Table 1) and in computations four to six terms are used depending on the combination of edge conditions. The above mentioned set of admissible functions is found to be quite adequate to obtain accurate estimates to the least eigenvalues for large hole sizes and for all combinations of edge conditions under investigation. However in the case of a clamped or simply supported inner edge, use of these functions for small hole sizes results in slow convergence of the solutions. This is to be expected from the fact that the assumed mode tends to satisfy higher order constraint conditions as the inner radius u tends to be zero. To clarify this point consider, for example, the case of a free outer edge and simply supported inner edge. The mode is represented by W = B,y + B2y2 + . . . ,

(15)

satisfying only the relevant geometric boundary condition, namely, W = 0 at y = 0, i.e. W = 0 at r = a. From equation (8) it can be seen that y + (r/b)2 for a + 0. Hence for .a approaching zero, the mode represented by equation (15) tends to become W = B, (r/b)’ + B, (r/b)4 + . . .

(16)

which satisfies, besides W = 0, an additional constraint condition, namely dW/dr = 0, at r = 0. Thus the solution represented by equation (15) for a simply supported inner edge tends to represent the solution for a clamped inner edge as the hole size tends to zero. In view of the above, it is to be expected that a large number of terms in equation (15) would be required to achieve results of reasonable accuracy for small hole sizes. However, one may overcome this unfavourable requirement by a suitable choice of admissible functions. 2.2.

COMPUTATION

Numerical work is carried out for hole sizes O-1 to 0.9 in steps of 0.1. The Poisson’s ratio v0 is fixed at O-3. The rigidity ratio parameters Do/D, and D,/D* are varied from 0.3 to 1-Oin steps of 0.1. Estimates to .eigenvalues are obtained for all possible combinations of clamped, simply supported and free edge conditions except the case of both edges free. The numbers of terms used in equations (5) and (12) are indicated in Table 2.

169

VIBRATIONOF ANNULAR PLATES TABLE 2

Comparison of estimates of frequency parameter [2(ob’/h)J(y/E)] exact values. Isotropic annularplate: v0 = v, = v = 0.3

with

Hole size a/b N

/ 0.1

0.3

0.5

0.7

Y 0.9

(16.5) 17.0 16.5

28.0 27.4 27.4

54.2 (10.8) 54.0

150.5 150.0

1356.0 1354.0

9.5 8.8 8.8

12.8 12.8 12.8

24.3 24.2 24.2

equation (12) equation (5) reference [7]

4

C-C

S-S

equation (12) equation (5) reference [7]

5 3

F-S

equation (12) equation (5) reference [7]

6 3

2.25 2.10 2.09

2.08 2.07 2.07

2.5 2.49 2*49

3.75 3.74 3.74

10*4 10.4 10.4

F-C

equation (12) equation (5) reference [7]

5 3

3.06 2.60 2.56

4.13 4.04 4.03

7.91 7.89 7.88

22.33 (17.9) 22.4

208.73 91.77

S-F

equation (12) equation (5) reference [7]

6 3

2.99 2.97 2.94

2.86 2.85 2.82

3.07 3.08 3.07

4.20 4.20 4.20

S-C

equation (12) equation (5) reference [7J

(10.1) 10.8 10.8

equation (12) equation (5) reference [7]

6.26 6.19 6.15

C-F

C-S

2

14.7 13.8 13.7

equation (12) equation (5) reference [7]

(18.05) 36.3 18.1 (17.2) 18.1 36.2

(zz) 66.6

-598.3 598.0

10.7 10.7 10.7

102.1 102

929 218.4 218 939.3 938

6.97 6.93 6.91

10.73 10.74 10.7

26.15 (16.7) 26.1

20.46 (20.37) 20.4

38.8 (7.94) 38.7

106 106

929.8

N = number of terms used. t Value is incorrect. Note: values in parentheses indicate the presence of rounding errors in computations.

All the computations are carried out with single precision arithmetic (6 significant digits) on an IBM 360/44 Computer at the Indian Institute of Science, Bangalore. 3. PRESENTATION AND DISCUSSION OF RESULTS 3.1.

ACCURACY

OF NUMERICAL

RESULTS

In order to assess the relative merits of solving equations (5) and (12), the estimates of the frequency parameter obtained for isotropic plates are compared with the corresponding exact values [7] for various hole sixes and for different combinations of edge conditions (see Table 2). It may be mentioned here that, from Rayleigh’s principle, these estimates are upper bounds to the exact values. The numbers in parentheses in Table 2 are those which are less than the exact values and are therefore in error due to rounding errors.

170

K. VIJAYAKUMAR AND G. K. RAMAIAH

It is evident, as can be seen from the number of terms used in each of the equations (4) and (1 l), that equation (5) is more ill-conditioned than equation (12). In the former case, the conditioning of the equation deteriorates with increasing hole sizes. The hole sizes at which the rounding errors become predominant are indicated by enclosing the corresponding estimates to the frequencies in parentheses. It is seen that the direct analysis is not convenient for hole sizes beyond O-3 except in the S-F and F-S cases for which the three term solutions yield accurate results for all hole sizes. For small hole sizes, however, the estimates are in close agreement with the exact values in all cases and are far superior to those obtained by the proposed modification in the analysis. The fact that these accurate estimates are obtained by two or three term solutions adds further strength to the suitability of the direct analysis for small hole sizes. For hole sizes greater than 0.3, the frequency values obtained by the procedure with modification coincide very well with exact values. The values differ, depending on the edge

0.6 1

(b)

0.4

0.6

0.8

I.0

0.6

.fZg/G

Figure 2. Variation of eigenvalue parameter Ee < E, :

L = dD/lm; D,.

D6

0.4

wcY

with rigidity ratio: (a) F-S casz, (b) S-F case. E, < Eg :

A = cd D/q2De.

VIBRATIONOF ANNULAR

PLATES

171

conditions and hole sizes, at most in the third or fourth significant digit. It is to be observed that the frequency value 91.7 obtained by Vogel and Skinner [7] for the hole size 0.9 in the case of a free outer edge and clamped inner edge is incorrect. From the earlier discussion with regard to the suitability of the admissible functions chosen in equation (11) for small hole sizes, it is to be expected that the errors increase with decreasing hole size in the six cases corresponding to simply supported or clamped inner edges. This is contirmed by the relative errors in the results for hole sizes O-1and O-3 in the above cases and the same behaviour is also exhibited in the remaining two cases of C-F and S-F. The lower values obtained by the five term solution for hole sizes 0.1 and 0.3 in the S-C case are evidently due to the influence of rounding errors in computation. This implies that equation (12) becomes ill-conditioned for small hole sizes. With the assumption that this is also true in the C-C case, the value 16.5 obtained for hole size O-1is believed to be due to the influence of rounding errors though it coincides with the exact value.

1

I

I

I

I

I

(a) I-0

0.8

=

E Ee

0.6

(b)

0.6 0.4

0.6

Figure 3. Variation of frequemcy with rigidity ratio: (a) F-S case, (b) S-F case.

1’72 3.2.

K. VIJAYAKUMAR INFLUENCE OF .iUGIDITY RATIO

AND G. K. RAMAIAH

: &/-D,( = I??,/&)

The two sets of curves (02yhb4/D,) vs. (I&./E,) and (~*yhb~/D,) vs. (I&/E,) for various hole sizes (not presented here) are found to be straight lines for all the edge condition cases except in the S-F case. Even in this case, the first set of curves are again straight lines and the deviation from linearity in the second set is rather small (see Figure 2). Generally, the eigenvalue parameters decrease with decrease in the corresponding rigidity ratios. However in the S-F and F-S cases (see Figure 2), the eigenvalue parameter increases with decreasing value of the rigidity ratio E,/EB for hole sizes greater than 0.7. 3.3.

COMPARISON WITH ISOTROPIC PLATES

The effects of the rigidity ratios E,/E, and E,/E, on the frequencies of orthotropic plates relative to those of the corresponding isotropic plates (v = 0.3 = ve) are shown in Figures 3-6. The curves in these figures exhibit some interesting features noted later.

s-

8-

I

L-o.5

,7 -

-

6-

E-E,

O-7 0.9

6(b

*I -\

o/b 0.9

\

\

-00:X I-I I r-O.3

O\

s-

a.-

c-r,

7.I 0.4..

0.6

;,

-1

f-lEr

0.8

I.0

0.8

..

Figure 4. Variation of frequency with rigidity ratio: (a) GF

0.6

-’

0.4

case, (b) F-C case

173

VIBRATION OF ANNULAR PLATES

3.3.1. E. < E, = E InthecaseofE r = E and E, < E, (i.e., v, > vO),the frequency of the orthotropic plate is lower than that of the isotropic plates for .a11hole sizes in the S-F and F-S cases and decreases with decreasing E, (see Figure 3). In contrast to this behaviour in the above two cases, the frequency is always higher and increases with decreasing E8 in the S-C and C-C cases (see Figures 5(a) and 6(b)), For a constant EB the rate of decrease in the former two cases and the rate of increase in the latter two cases increase with increasing hole size. The above trends in S-C and C-C cases are also observed in the remaining four cases of edge conditions for hole sizes beyond 0.5 (see Figures 4, 5(b) and 6(a)). However, the frequency in these four cases decreases initially up to a certain minimum value and then increases with decreasing E, for small hole sizes up to about 0.3. The maximum decrease occurs in the C-F case and this decrease is always less than 5 per cent.

I

I

I

(a)

(b)

_

u/b

E-E, 0.6

I 0.4

I 0.6

1 0.6

,

09

0.6

o-4

Figure 5. Variation of frequency with rigidity ratio: (a) S-c case, (b) GS case.

174

K. VIJAYAKUMAR

ABIDG. K. RAMAIAH

3.3.2. Ef < E, = E In the case of E, = E and E, < E. (i.e., v, < vO), the frequencies are unaffected by changes in E,for hole sires equal to or greater than 0.5 in the F-S case. For hole sixes less than 0.5, the frequency decreases with decrease in E,.The rate of decrease in the frequency for a constant E, decreases with increasing hole size. A similar trend is also noticed in the S-F case. In contrast to this behaviour in the F-S and S-F cases, the rate of decrease in the frequency increases with increasing hole size in the remaining cases. l

I

I

(a)

(b)

I

0.6

0.4

Figure 6. Variation of frequency with rigidity ratio: (a) S-S case, (b) C-C case. 4. CONCLUSIONS

The classical Rayleigh-Ritz method with simple polynomials as admissible functions has been found to be quite satisfactory to obtain the least eigenvalues. The direct analysis is convenient and useful for obtaining solutions for small hole sixes not greater than 0.3. The analysis with the proposed radial coordinate transformation is superior and extremely effective for obtaining solutions for large hole sixes. By introducing similar transformations, the technique is being extended for modes with nodal diameters.

VIBRATION

OF ANNULAR

PLATES

175

The eigenvalue parameter is found to vary more or less linearly with rigidity ratio. The plate behaviour with changes in rigidity ratio in the two cases of one edge free and the other simply supported is distinctly different from that of the plate in the remaining cases of edge conditions. A comparison with the results of isotropic plates has brought out some interesting features. REFERENCES 1. T. AKASAKAand T. TAKAGI~HI 1958 Bulletin of the Japan Society of Mechanical Engineers 1, 215-221. Vibration of corrugated diaphragm. 2. I. A. M~NKARAHand W. H. HOPPMANN,II 1964 Journal of the Acoustical Society of America 36, 470-475. Flexural vibration of cylindrically aelotropic circular plates. 3. K. A. V. PANDALAIand S. A. PATEL 1965 Journal of the American Znstitute of Aeronautics and Astronautics 3, 780-781. Natural frequencies of orthotropic circular plates. 4. A. W. LEISSA 1969 NASA SP. 160. Vibration of Plates. Washington: OtIices of Technology Utilisation, NASA. 5. N. K. PYBYENN~KOVA and I. E. SAKHAROV1959 Izvestiya Akademii Nauk SSSR-Otdelenie Tekhnicheskikh Nauk. Moscow. Mekhanika i Mashinostroenie. (In Russian), 134136. Natural vibration frequencies of the fundamental of annular plates with a cylindrical anisotropy. 6. K. VIJAYAKUMARand C. V. JOGA RAO 1971 (February) Proceedings of the Symposium on ‘Recent Developments in Analytical, Experimental and Constructional Techniques applied to Engineering Structures’, Regional Engineering College, Warangal. India. Axisymmetric vibration and buckling of polar orthotropic armular plates. 7. S. M. VOGEL and D. W. SKINNER 1965 Transactions of the American Society of Mechanical Engineers (Journal of Apphcal Mechanics) 32E, 926-931. Natural frequencies of transversely vibrating uniform annular plates. APPENDIX LIST OF SYMBOLS

a, b radii of inner and outer edges, respectively

4 43 4 4, Ee E, v h r Tmax ulw v* Vmax Y Y v, v0 0 01

a2/(b2 - a’) E, h3/12(1 - v,v,), flexural rigidity in radial direction Ee h3/12(1 - v,v,), flexural rigidity in tangential direction vPe = v, 0, Young’s moduli in radial and tangential directions, respectively Young’s modulus and Poisson’s ratio of isotropic material thickness of plate radial coordinate of a point in mid-plane maximum kinetic energy admissible functions maximum strain energy (r2 - a2)/(b2 - a2) mass density per unit volume Poisson’s ratio defined as strain in tangential direction due to unit strain in radial direction Poisson’s ratio defined as strain in radial direction due to unit strain in tangential direction circular frequency in radians per second circular frequency in isotropic case