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Transverse vibrations of annular plates of variable thickness with rigid mass on inside
JournalofSound and Vibration (1981) 79(2), 311-315
TRANSVERSE VIBRATIONS OF ANNULAR PLATES OF VARIABLE THICKNESS WITH RIGID MASS ON INSIDE 1.
INTRODUCTION
In an interesting study [l] Handelman and Cohen have determined the fundamental frequency of vibration of a circular annular plate clamped on the outside edge and with a rigid mass attached to the inside edge. In view of the complexity of the resulting frequency equation numerical evaluation was limited to a few cases and a minimal principle was used in order to calculate approximate eigenvalues, which are upper bounds [I, 21. The work reported in what follows here constitutes an extension of that of Handelman and Cohen [l] in the sense that a plate of variable thickness is considered, and it is assumed that the outer edge is elastically restrained against rotation. The dynamic structural system under study is shown in Figure 1. When the flexibility parameter 4 is equal to zero, one has the case of an annular plate rigidly clamped on the outside and when 4 approaches infinity the simply supported boundary condition is attained there.
Figure 1. Vibrating mechanical system under study.
The numerical values obtained when 4 is zero and the plate is of uniform thickness are in excellent agreement with the exact eigenvalues reported in reference [l] and in reasonable agreement with the approximate eigenvalues presented there [l] for b/a < O-40, for a wide range of values of the parameter y (= mass per unit area of the rigid insert/plate mass per unit area). 311 0022--460X/81/220311
+05$02.00/O
@ 1981 Academic Press Inc. (London) Limited
312
LETTERS
TO
THE
EDITOR
In this study the displacement amplitude function was approximated in terms of polynomial co-ordinate functions which satisfy the essential boundary conditions and one of the natural boundary requirements. The Ritz method was employed to generate the fundamental equation. 2.
STATEMENT
OF THE
PROBLEM
AND
ITS APPROXIMATE
SOLUTION
To solve the title problem by means of the Ritz method one must minimize the functional J( W) given by the well known expression J[ W] = 27
1a D(P)[(d2 W/dF2)’ + (2k,f)(d
W,di)(d2 W,di2) + (l/r’)
(d W;di)2]i dP -2rraD(a)[(d’W/dP’)(a) + (&a)(d W/dT)(a)](d W/df)(u) n - 21rpw2 h(i)W2fdP-Mo2W2(b), Ib
(1)
where, for axisymmetric modes, w(Y,.t) = W(i) e’“‘. The governing boundary tions are W(u) = 0, (d W/df)(u) = -rjD(u)[(d2 W/dr2)(u) + (k/u)(d W/dP)(u)], (d W/dF)(b) = 0, 2&D(b)(d/df)[(d*
W/dr2) + (l/T)(dW/di)](i=b
= MU* W(b),
condi(2a, b) (2~ d)
where (see Figure 1) h(?)=[(h,-hb)/(a-b)](f-b)+hb, D(u) =
Eh: 12(1-p*)’
e=-.
hb ha
Following the same approach as in previous studies by the senior author and coworkers [3,4], one approximates the displacement function by means of a linear combination of polynomial co-ordinate functions: N
W, = C A,(r4+(y3nr3+(y2nr2+~ln~+l)r2”
n=O
N
= C A&(r), n=O
r=‘. U
(3)
The coefficients ain (i = 3,2, 1) are evaluated by requiring that each functional relation R,(r) satisfies identically conditions (2a), (2b) and (2~). It was decided not to comply with condition (2d) since this condition considerably complicates the generation of the algorithm in view of the fact that the desired eigenvalue is contained in one of the boundary conditions. On the other hand, condition (2d) constitutes a natural boundary condition and when using the Ritz method it is not necessary to satisfy it. As a first approximation one takes iV = 0 in equation (3) and substitution in equation (1) yields, after application of the minimization requirement U[ Wa]/aAo = 0, the desired frequency equation. Figure 2 shows a comparison of results obtained (a) by Handelman and Cohen [l] (both their exact and ap roximate values) and (b) those obtained in this investigation. It was decided to plot ? 0 (instead of L?) as a function of A and for several values of y in order to present the same graphical functional relation as that available in reference [l] and then to include there the eigenvalues calculated in the present study. (y = (M/A1)/(Mp/A,) = mass per unit area of the rigid insert/plate mass per unit area.)
LETTERS
313
TO THE EDITOR
5,
21 0
I 0.1
1 0.2
I 0.3
I 0.4
t 0.5
I
Figure 2. Values of 6 for a clamped annular plate of uniform thickness: comparison of results. q5= 0, e=l;-, reference [l]; 0, reference [l]; - --, present study.
For A Q O-40 the present approach yields reasonable agreement with the values of & presented in reference [l]. Actually the two exact values [l] fall right on the approximate curve for a determined in the present study. The relative accuracy of the fundamental frequency coefficients as determined by the present approach deteriorates as A increases. It was therefore decided to obtain further frequency coefficients only for A C 0.40. This range is still very useful for many engineering applications.
Figure 3. Values of L?, for a simply supported annular plate of uniform thickness; I$ = CO,e = 1.
LETTERS TO THE EDITOR
314
TABLE 1
Values of ROO e =O-9 o-2
0.4
0.01 0.1 1 10 100 co
12.03 11.78 10.20 6.97 5.83 5.68 5.66
(y=l)
0 0.01 0.1 1 10 100 I co
e=l
I
18.21 17.70 14.78 10.08 8.70 8.53 8.51
12.42 12.13 10.42 7.29 6.30 6-17 6.16
18.69 18.15 15.12 10.49 9.20 9.05 9.03
11.29 11.06 9.61 6.73 5.75 5.62 5.61
13.98 13.65 11.67 8.21 7.14 7.01 7.00
11.52 11.27 9.75 6.91 5.98 5.87 5.85
14944 14.08 12.00 8.59 7.59 7.47 7.45
11.40 11.17 9.69 6.80 5.82 5.70 5.68
10.56 10.35 9.06 6.41 5.48 5.36 5.35
11.77 11.51 9.95 7.11 6.20 6.09 6.08
10.793 10.572 9.204 6.581 5.714 5.605 5.592
12.19 11.90 10.25 7.45 6.60 6.50 6.49
6.22 6-11 5.40 3.90 3.36 3.30 3.29
7.46 7.34 6.57 4.82 4.17 4.08 4.07
6.45 6.33 5.57 4.10 3.60 3.54 3.53
7.67 7.54 6.71 4.97 4.36 4.28 4.27
6.71 6-57 5.76 4.31 3.85 3.79 3.78
17.81 17.33 14.52 9.70 8.21 8.02 8.00
12.20 11.93 10.29 7.12 6.06 5.92 5.91
11.10 10.89 9.52 6.58 5.52 5-39 5.37
13.58 13.28 11.40 7.87 6.71 6.57 6.55
(y=2)
0 0.01 0.1 1 10 100 I co
10.37 10.18 8.95 6.26 5.26 5.13 5.12
(y=lO)
0 0.01 0.1 1 10 100 1 0;)
7.28 7.18 6.46 4.69 3.99 3.89 3.88
1
e = 1.1 o-4
0.4
(y=O)
I
o-2
0.2
0
\
Figure 3 shows results for a simply supported plate of constant thickness, a case not previously treated by other authors. wOOa for y = 0, Table 1 contains values of the fundamental coefficient Jph,/D(a) 1, 2 and 10 respectively. The eigenvalues are tabulated, in each case, for e = 0.9, 1.0 and 1.1, for 4 = qSD(a)/a varying from zero to infinity, and for two particular values of A, namely A = 0.20 and 0.40, where A = b/a. An advantage of the methodology used to obtain these results is the fact that it can be easily implemented on a microcomputer. Institute of Applied Mechanics, Puerto Belgrano Naval Base, 8111 -Argentina
P. A. A. LAURA R. H. GUTIBRREZ
(Received 23 March 1981, and in revised form 14 July 1981) REFERENCES 1. G. HANDELMAN and H. COHEN 1957 Proceedings 7th International Congress of Applied Mechanics VII, 509-518. On the effects of the addition of mass to vibrating systems.
LEI-IERS
TO THE EDITOR
315
2. A. W. LEISSA 1969 NASA SP-160. Vibration of plates. 3. P.A. A. LAURA, C.FILIPICH and R. D. SANTOS 1977 Journal of Sound and Vibration 52, Static and dynamic behavior of circular plates of variable thickness elastically 243-251. restrained along the edges. 4. R. H. GLJTIBRREZ, P.A. A. LAURA and R. 0. GROSSI 1980 Journal of Sound and Vibration 69, 285-295. Transverse vibrations of plates with stepped thickness over a concentric circular region.
TRANSVERSE VIBRATIONS OF ANNULAR PLATES OF VARIABLE THICKNESS WITH RIGID MASS ON INSIDE 1.
INTRODUCTION
In an interesting study [l] Handelman and Cohen have determined the fundamental frequency of vibration of a circular annular plate clamped on the outside edge and with a rigid mass attached to the inside edge. In view of the complexity of the resulting frequency equation numerical evaluation was limited to a few cases and a minimal principle was used in order to calculate approximate eigenvalues, which are upper bounds [I, 21. The work reported in what follows here constitutes an extension of that of Handelman and Cohen [l] in the sense that a plate of variable thickness is considered, and it is assumed that the outer edge is elastically restrained against rotation. The dynamic structural system under study is shown in Figure 1. When the flexibility parameter 4 is equal to zero, one has the case of an annular plate rigidly clamped on the outside and when 4 approaches infinity the simply supported boundary condition is attained there.
Figure 1. Vibrating mechanical system under study.
The numerical values obtained when 4 is zero and the plate is of uniform thickness are in excellent agreement with the exact eigenvalues reported in reference [l] and in reasonable agreement with the approximate eigenvalues presented there [l] for b/a < O-40, for a wide range of values of the parameter y (= mass per unit area of the rigid insert/plate mass per unit area). 311 0022--460X/81/220311
+05$02.00/O
@ 1981 Academic Press Inc. (London) Limited
312
LETTERS
TO
THE
EDITOR
In this study the displacement amplitude function was approximated in terms of polynomial co-ordinate functions which satisfy the essential boundary conditions and one of the natural boundary requirements. The Ritz method was employed to generate the fundamental equation. 2.
STATEMENT
OF THE
PROBLEM
AND
ITS APPROXIMATE
SOLUTION
To solve the title problem by means of the Ritz method one must minimize the functional J( W) given by the well known expression J[ W] = 27
1a D(P)[(d2 W/dF2)’ + (2k,f)(d
W,di)(d2 W,di2) + (l/r’)
(d W;di)2]i dP -2rraD(a)[(d’W/dP’)(a) + (&a)(d W/dT)(a)](d W/df)(u) n - 21rpw2 h(i)W2fdP-Mo2W2(b), Ib
(1)
where, for axisymmetric modes, w(Y,.t) = W(i) e’“‘. The governing boundary tions are W(u) = 0, (d W/df)(u) = -rjD(u)[(d2 W/dr2)(u) + (k/u)(d W/dP)(u)], (d W/dF)(b) = 0, 2&D(b)(d/df)[(d*
W/dr2) + (l/T)(dW/di)](i=b
= MU* W(b),
condi(2a, b) (2~ d)
where (see Figure 1) h(?)=[(h,-hb)/(a-b)](f-b)+hb, D(u) =
Eh: 12(1-p*)’
e=-.
hb ha
Following the same approach as in previous studies by the senior author and coworkers [3,4], one approximates the displacement function by means of a linear combination of polynomial co-ordinate functions: N
W, = C A,(r4+(y3nr3+(y2nr2+~ln~+l)r2”
n=O
N
= C A&(r), n=O
r=‘. U
(3)
The coefficients ain (i = 3,2, 1) are evaluated by requiring that each functional relation R,(r) satisfies identically conditions (2a), (2b) and (2~). It was decided not to comply with condition (2d) since this condition considerably complicates the generation of the algorithm in view of the fact that the desired eigenvalue is contained in one of the boundary conditions. On the other hand, condition (2d) constitutes a natural boundary condition and when using the Ritz method it is not necessary to satisfy it. As a first approximation one takes iV = 0 in equation (3) and substitution in equation (1) yields, after application of the minimization requirement U[ Wa]/aAo = 0, the desired frequency equation. Figure 2 shows a comparison of results obtained (a) by Handelman and Cohen [l] (both their exact and ap roximate values) and (b) those obtained in this investigation. It was decided to plot ? 0 (instead of L?) as a function of A and for several values of y in order to present the same graphical functional relation as that available in reference [l] and then to include there the eigenvalues calculated in the present study. (y = (M/A1)/(Mp/A,) = mass per unit area of the rigid insert/plate mass per unit area.)
LETTERS
313
TO THE EDITOR
5,
21 0
I 0.1
1 0.2
I 0.3
I 0.4
t 0.5
I
Figure 2. Values of 6 for a clamped annular plate of uniform thickness: comparison of results. q5= 0, e=l;-, reference [l]; 0, reference [l]; - --, present study.
For A Q O-40 the present approach yields reasonable agreement with the values of & presented in reference [l]. Actually the two exact values [l] fall right on the approximate curve for a determined in the present study. The relative accuracy of the fundamental frequency coefficients as determined by the present approach deteriorates as A increases. It was therefore decided to obtain further frequency coefficients only for A C 0.40. This range is still very useful for many engineering applications.
Figure 3. Values of L?, for a simply supported annular plate of uniform thickness; I$ = CO,e = 1.
LETTERS TO THE EDITOR
314
TABLE 1
Values of ROO e =O-9 o-2
0.4
0.01 0.1 1 10 100 co
12.03 11.78 10.20 6.97 5.83 5.68 5.66
(y=l)
0 0.01 0.1 1 10 100 I co
e=l
I
18.21 17.70 14.78 10.08 8.70 8.53 8.51
12.42 12.13 10.42 7.29 6.30 6-17 6.16
18.69 18.15 15.12 10.49 9.20 9.05 9.03
11.29 11.06 9.61 6.73 5.75 5.62 5.61
13.98 13.65 11.67 8.21 7.14 7.01 7.00
11.52 11.27 9.75 6.91 5.98 5.87 5.85
14944 14.08 12.00 8.59 7.59 7.47 7.45
11.40 11.17 9.69 6.80 5.82 5.70 5.68
10.56 10.35 9.06 6.41 5.48 5.36 5.35
11.77 11.51 9.95 7.11 6.20 6.09 6.08
10.793 10.572 9.204 6.581 5.714 5.605 5.592
12.19 11.90 10.25 7.45 6.60 6.50 6.49
6.22 6-11 5.40 3.90 3.36 3.30 3.29
7.46 7.34 6.57 4.82 4.17 4.08 4.07
6.45 6.33 5.57 4.10 3.60 3.54 3.53
7.67 7.54 6.71 4.97 4.36 4.28 4.27
6.71 6-57 5.76 4.31 3.85 3.79 3.78
17.81 17.33 14.52 9.70 8.21 8.02 8.00
12.20 11.93 10.29 7.12 6.06 5.92 5.91
11.10 10.89 9.52 6.58 5.52 5-39 5.37
13.58 13.28 11.40 7.87 6.71 6.57 6.55
(y=2)
0 0.01 0.1 1 10 100 I co
10.37 10.18 8.95 6.26 5.26 5.13 5.12
(y=lO)
0 0.01 0.1 1 10 100 1 0;)
7.28 7.18 6.46 4.69 3.99 3.89 3.88
1
e = 1.1 o-4
0.4
(y=O)
I
o-2
0.2
0
\
Figure 3 shows results for a simply supported plate of constant thickness, a case not previously treated by other authors. wOOa for y = 0, Table 1 contains values of the fundamental coefficient Jph,/D(a) 1, 2 and 10 respectively. The eigenvalues are tabulated, in each case, for e = 0.9, 1.0 and 1.1, for 4 = qSD(a)/a varying from zero to infinity, and for two particular values of A, namely A = 0.20 and 0.40, where A = b/a. An advantage of the methodology used to obtain these results is the fact that it can be easily implemented on a microcomputer. Institute of Applied Mechanics, Puerto Belgrano Naval Base, 8111 -Argentina
P. A. A. LAURA R. H. GUTIBRREZ
(Received 23 March 1981, and in revised form 14 July 1981) REFERENCES 1. G. HANDELMAN and H. COHEN 1957 Proceedings 7th International Congress of Applied Mechanics VII, 509-518. On the effects of the addition of mass to vibrating systems.
LEI-IERS
TO THE EDITOR
315
2. A. W. LEISSA 1969 NASA SP-160. Vibration of plates. 3. P.A. A. LAURA, C.FILIPICH and R. D. SANTOS 1977 Journal of Sound and Vibration 52, Static and dynamic behavior of circular plates of variable thickness elastically 243-251. restrained along the edges. 4. R. H. GLJTIBRREZ, P.A. A. LAURA and R. 0. GROSSI 1980 Journal of Sound and Vibration 69, 285-295. Transverse vibrations of plates with stepped thickness over a concentric circular region.