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Natural frequencies of spinning annular plates
Journal
of Soundand Vibration
(1981) 74(Z), 303-310
LETTER NATURAL
FREQUENCIES
TO THE EDITOR OF SPINNING
ANNULAR
PLATES
The dynamics of spinning discs has been studied by many investigators because of their commercial use as saw blades, turbine wheels and modern high speed computer storage discs. Lamb and Southwell [l] examined the problem of the completely free circular plate spinning about its cylindrical axis with a constant angular velocity. As a sequel to this work, Southwell [2] analyzed the problem of a circular disc which is clamped at its centre, is free at its outer edge and is rotating with a constant angular velocity. These two investigations carried out about six decades ago and a corroboration of these works by Prescott [3] are probably the earliest and the most frequently cited work on vibrations of spinning discs. Since then, there have been several analytical and experimental studies on isotropic, polar orthotropic and variable thickness spinning discs. On the other hand, there are only a few investigations reported in the literature on vibrations of spinning annular plates [4-lo] and in fact all of these have been carried out only during the last few years. Loh and Carney [4,5] and Dyka and Carney [6] have presented a general solution by adopting the Frobenius method for determining the natural frequencies of spinning annular plates reinforced with surrounding edge beams at the inner and outer edges. In the remaining four investigations [7-lo], some form of finite element method has been employed for plates with free outer and clamped inner edges. Numerical results in all this work are confined to those for some typical hole ratios and either a particular angular speed or some specific values of spinning parameter (R”yhb”/D). In the work reported here, the problem of vibrations of spinning annular plates has been analyzed in detail by the Rayleigh-Ritz method for eight different combination of boundary conditions. Accurate estimates of eigenfrequencies corresponding to the axisymmetric mode as well as the modes with one and two nodal diameters have been obtained for various values of hole sizes and for various values of the non-dimensional spinning parameter (R2rh64/D). The numerical results are presented in the form of tables and graphs for direct use by design engineers. A thin annular plate of constant thickness h with a and b as the radii of the inner and outer edges, respectively, is considered. It is assumed that the material of the plate is homogeneous and isotropic and the plate is rotating about its cylindrical axis with a constant angular velocity R (radians per second). The influence of rotation is to induce stresses which are assumed to be distributed symmetrically about the axis of rotation. These stresses in the absence of any in-plane pressure at the inner and outer edges are given by b2 - r7 - a2b’/r2], gr = [(3 + ~)/8]yf?*[a’+ il! u,j =[(3+v)/8]~~2[a’+bZ-;1+3~)/(3+v)r’+a’b2/r2], in which material.
12)
v is the Poisson’s ratio and y is the mass density per unit volume of the plate The flexural mode W is assumed to be expressible in the usual form w=
W,,(~)COS(ne+~)cos(ot+~),
r3i
in which n is the number of nodal diameters and w is the frequency in radians per second. The potential energy due to flexure is composed of two parts: namely, (1) that due to the strain energy of bending, V, and (2) that due to the centrifugal forces, U. The kinetic 303 r~072-~~0.~,/81/020.703
+0x
$02.00/0
0
19X I Academic Press Inc. / London I I .Imited
304
LETTER
TO THE EDITOR
energy, T, remains the same as for a stationary plate. The expressions for these quantities are as follows: b d*W, 1 dW n2 ’ v=;(l+&“)DJ w r dr r* n1 n
((-g+--l--
d2W
-2(1-v)$
1dW
-2
( r dr
-7
n*
W” +2(1-v)-i_ F* (F-T)*]r
~=5~l+~~~)~[~~b{~,(~)2+~~(~
[I(
I(
a2+b2-r2-T
b
+
a
W”)‘]rdr],
b
3+v U=;(1+SO”)eyn2h
a2b2
T=t(
I+s,.)
o*yh
dW, >(
1+3v” 2 * 3+v r2+7-arb > (s W,,)r
a2+b2--
(4)
dr,
>
* >
r dr
dr], dr
(5)
Jab W”,r dr,
in which So0 = 1, Son = 0 for n # 0 and D =Eh3/12(1 -v*) is the flexural rigidity of the plate. The total potential (V + U - T) is minimized by the Rayleigh-Ritz method to obtain relevant characteristic equations for the determination of eigenfrequencies (w2rhb4/D) for specified values of spinning parameter (0*rhb4/D). In the method, the mode shape W was expressed as a linear combination of simple least order polynomials in r [ 1 l] satisfying relevant geometric boundary conditions. For large values of the hole size, the quantities V, U and T were modified prior to the application of the Rayleigh-Ritz method by the transformations given earlier in reference [l l] for the axisymmetric mode and in reference [12] for the asymmetric modes. The number of terms used in the Rayleigh-Ritz method was up to five. In the computations, the parameter a/b was varied from O-1 to O-9 in steps of 0.2. The Poisson’s ratio v was fixed at 0.3. All computations were carried out with double precision arithmetic (about 16 significant digits) on an IBM 370/168 digital computer at the Technische Hochschule Darmstadt. Estimates of the frequency parameter ((2wb2/h) a) arepresented in Table 1 for several different values of the non-dimensional spinning parameter (LJ2rhb4/D). The TABLE
&&!$I f or rotating annular plates; v = O-3
Estimates of frequency parameter (= 2(wb*/h)
Values of (RZyhb4/D)
Values of (R2yhb4/D) h
I a/b
I*
0
0.1
0 1 2
0.3
0 1 2
2.823 7.758 14.60
0.5
0 1 2
3.073 7.025 13.53
25
1
10000
r
100
250
6.399 13.52 21.49
9.106 18.40 27.86
16.34 32.38 47.10
45.29 91.69 130.9
4.149 9.408 16.81
6.504 12.99 21.85
9.305 17.80 28.86
16.63 31.43 49.04
46.06 89.58 136.8
2.071 2.042 3.681
4.293 8.626 15.91
6.663 12.15 21.38
9.638 16.96 29.06
17.40 47.86 30.41 86.92 SO.75 142.6
2.494 2.942 4.833
(1) SF Case: 2.938 4.173 8.396 9.983 15.39 17.17
1000
,
0
25
(2) FS Case: 2.089 2.945 1.483 3.420 3.288 5.736
100
250
1000
10000
4.466 6.323 9.953
6.314 9.809 15.20
11.33 19.39 29.84
33.16 61.02 93.78
3.064 3.832 6.221
4.871 6.779 10.67
7.086 10.38 16.25
13.02 20.32 31.82
37.42 63.33 99.91
3.491 4.545 7.283
5.445 7.514 11.92
7.940 11.29 17.88
14.65 21.79 34.76
41.65 67.04 108.8
\
LETTER TABLE
TO THE EDITOR
1 (continued) Values of (R2yhb4/D)
Values of (f12ykb4/D) A
r
250
(1) SF Case: 4,195 5.099 1 8,056 9,261 14.72 16.58 2
7.146 12.17 21.17
10.03 16.50 28.17
18.40 29.62 49.86
51.82 85.93 144.8
0.9
0 1 2
12.09 19.90 33.92
13.89 22.42 37.92
20.67 32.21 53.60
56.76 86.21 141,7
0.1
0
9.895 18.14 27.33
13.15 23.52 34.28
21.85 39.13 55.37
58.16 107.3 150.6
(I.3
0
n
0.7
0
0
10.72 18.02 30.97
25
\
100
a/b
Il.08 18.51 31.73
(3) CF Case: 6.152 7,368 1 12.83 14.40 2 20.90 22.74
1000
10000
2 0.5
0 1 2
10.72 13.32 19.44
11.42 14.54 21.54
13.26 17.64 26.78
16.18 22.43 34.73
25.17 37.05 58.45
61.61 98.87 157.8
0.7
0
26.11 27.43 31.22
26.38 27.93 32.27
27.17 29.36 35.25
28.66 32.01 40.52
35.03 42.71 60.13
72.45 101.7 159.2
1 2 0.9
0
218.1
1 218.8
2
220.9
0 1 2
0.3
0 1 2
18.14 19.00 21.94
0.5
0
36.20 36.91 39.12
1 2
3.744 5.054 8.126 10.36 15.67 25.79
250
1000
10000
4.552 6,270 10.05
6,383 8,978 14.35
8.968 12.76 20.39
16.51 23.89 38.26
47.41 72.41 118.4
10.70 16.16 26.58
Il.68 17.56 28.80
13.42 20.07 32.79
19.97 29.56 48.00
54.86 80.73 130.5
5,263 6.651 9,991
7.443 10.14 15.23
13.45 19.80 29.8.5
40.11 61.97 03.80
13.84 22.56 35.15
22.73 37.79 57.50
58.39 102.4 154.0
4,031 3,966 4,817
4.892 5.275 6.953
6.761 7,967 Il.12
9.260 11.53 16.56
16.14 21.62 32.03
45.61 65.86 100.1
7.882 8,043 8.899
8,453 8.903 10.47
9.945 11.07 14.16
12.32 14.42 19.55
lY.76 24.86 35.89
51.35 71.58 109.8
22.60 23.07 24.54
23.30 24.17 26.71
24.62 26.22 30.59
30.27 34.60 45.21
64.04 82.98 123.1
218.1 218.8 221.0
218.2 219.0 221.4
218.4 219.3 221.1
219.3 220.9 225.7
229.5 238.9 265.0
22.36 22.69 23.77 208.4 208.9 210.3
16.81 20.19 28.09
26.84 33.77 46.98
71.91 94.15 130.8
18.70 19.76 23.16
20.26 21.86 26.41
22.99 25.46 31.98
32.76 38.14 50.52
80.05 99.38 137.4
20.44 21.73 25.86
36.51 37.32 39.82
37.40 38.53 41.87
39.11 40.81 45.68
46.51 92.61 SO.54 109.7 61.07 148 :
38.72 39.63 42.45
102.1 102.7 104.6
102.4 103.2 105.4
103.1 104.1 107.1
106.4 108.5 114.8
138.5 150.8 182.9
208.5 208.9 210.4
(6) CS Case: 13.74 14.57 15.30 16.58 21.43 23.17
13.61 15.78 21.91
105.5 106.2 108.2
0.1
0
(7)SS Case: 8.767 9.589 1 10.16 11.46 2 15.70 17.42
11.64 14.59 21.64
14.75 19.19 27.93
24.13 32.91 46.95
64.61 92.66 130.9
0.3
0 1 2
12.76 14.11 18.32
13.39 15.05 19.84
15.10 17.54 23.75
17.97 21.60 29.92
27.55 34.83 40.38
70.76 95.15 136.9
27.44 28.23 30.95
0.5
0 1 2
24.23 25.30 28.50
24.58 25.82 29.47
25.60 27.33 32.19
27.50 30.11 37.00
35.28 41.08 54.77
78.98 100.6 144.7
54.02 54.61 56.48
0.7
0
66.61 67.45 69.95
66.74 67.64 70.34
67.13 68.23 71.50
67.89 69.39 73.75
71.57 74.89 84.10
105.0 121.9 162.2
Notes:
100
10.53 17.23 27.40
0 1 2
2
25
(4) FC Case: 2.568 3.506 2.117 3.813 3,413 5.806
0.7
1
102.0 102.6 104.3
\ 0
8.041 13.45 22.00
(5) SC Case: lg.77 11.57 11.75 12.90 16.19 17.83
0.1
, (2) FS Case:
6,914 11.83 19.73
1
305
208.7 209.4 211.4
209.6 210.8 214.5
210.4 227.3 249.3
16.74 19.82 27.60
20.17 24.80 34.42
30.88 40.07 55.40
76.55 106.3 147.7
21.03 22.59 27.28
22.71 24.97 31.10
25.64 29.04 37.43
36.09 43.19 58.38
84.89 109.3 153.3
39.04 40.08 43.26
39.97 41.41 45.60
41.76 43.91 49.92
49.52 54.51 61.18
97.51 117.9 162.2
105.7 106.4 108.5
(8) CC Case: 16.51 17.31 17.51 18.63 22.18 23.82
150.0 150.8 152.2
208.6 209.1 210.7
106.0 106.9 109.4
106.7 107.8 111.2
110~1 112.5 119.6
143.0 156.9 192.4
19.43 21.57 28.06
22.90 26.28 34.71
34.06 41.26 55.52
82.75 107.0 146.8
27.96 28.94 32.09
29.53 30.95 35.27
32.31 34.55 40.78
42.79 47.91 6@35
94.17 113.9 154.0
54.30 54.98 57.09
55.15 56.08 58.90
56.80 58.20 62.33
64.19 67,61 77.01
114.0 130.0 168.0
150.4 150.9 152.4
150.8 151.3 153.1
151.4 152.1 154.4
354.5 156.1 161.0
186.2 196.3 223.8
1. C = clamped, S = simply supported, F = free; the first letter denotes the edge support conditions at r = h. 2. n = number of nodal diameters.
306
LETTER
TO THE
EDITOR
300
I00
200
100
FS
9 0 0 4 x . 600,
“3
,
Y/d ‘““vLIP n=l,a/b=O.l
0,0.5
1
I
0
,/A-
I
10
50
&
1
100
b
1
I50
11
’
200
250
0
50
100
150
200
250
a2 yhb4/D Figure 1. Variation of ~*yhb~/D with R2yhb4/D. n = number of nodal diameters. supported and F = free. The first letter denotes the edge support conditions at r = b.
C= clamped,
S= simply
LETTER
cl
z9 c
TO
THE
EDITOR
307
SC 300
Figure 2. Variation of 02yhb4/D with R2yhb4/D. n = number of nodal diameters. C = clamped, S = simply supported and F = free. The first letter denotes the edge support conditions at r = b.
308
LETTER
TO THE EDITOR
variation of the eigenvalue parameter (w*+~~/D) with the spinning parameter (0’$zb4/D) is shown in Figures 1 and 2 for some typical values of hole sizes and for different modes. The data in these figures suggest that w2 varies more or less linearly with 0’ in the form W2= I& + rnf12.
(7)
w,~in equation (7) corresponds to the frequency of the stationary plate (0 = 0) and m is a constant depending on the flexural mode, the value of a/b and the boundary condition. Approximate values of m can be calculated by using the data available in Table 1. The accuracy of the formula (7) improves with increasing values of the hole size but the formula suffers from inaccuracy when R tends to infinity. However, for finite values of 0 equation (7) yields reasonably accurate frequencies for all hole sizes and for all flexural modes. This is clearly exhibited by the data presented in Table 2 where the frequencies reported by Ginesu et al. [lo] for a steel plate rotating at 0 = 400 radians/s are compared with the corresponding frequencies obtained by using equation (7). Ginesu er al. [lo] employed the finite element program EACO for an annular plate (u/b = 0.5) with free outer and clamped inner edges. The constant m in equation (7) was numerically evaluated at each stage from computed values of 02yhb4/D by varying the parameter R2yhb4/D from 0 to 250 in steps of 25. The average value of m thus obtained was used in equation (7). TABLE
2
Comparison of frequencies obtained by using the formula o2 = wzl + rnf12 (equation (7)) with the corresponding accurate estimates reported by Ginesu et al. [lo] using the finite element program EA CO Plate dimensions: a/b = O-5, a = 101.6 mm, b = 203.2 mm, h = 1.016 mm; rotational
speed: 0 = 400 radians/s; plate material: steel disc, values of E’s and y are not given in reference [lo] and these values are not necessary when using equation (7); boundary condition: free outer and clamped inner edges; wSfis the frequency of the stationary plate and w is the frequency of the rotating plate. Frequency
n
s
0
0
0 1 1 2 2
1 0 1 0 1
Notes:
m 0.998 6,293 1.578 7.231 3.313 10.06
in radians/s
for
Stationary plate as given in reference [lo]
Rotating plate as given in reference [lo]
. Rotating plate as calculated from equation (7)
497 3244 507 3308 561 3500
638 3396 714 3479 919 3723
637.7 3395.6 713.8 3478.5 919.1 3722.8
1. n = number of nodal diameters, s = number of nodal circles. 2. m is a constant and it was evaluated numerically at each stage from computed values of 02yhb4/D by varying R2yhb4/D from 0 to 250 in steps of 25. The average value of m thus obtained was used in equation (7). 3. v=O.3.
The strain energy component V can be neglected if it is assumed that the plate is very thin and the rotation is very rapid. The solution of the resulting variational problem S( U - T) = 0 then yields the ratio cr = u2/R2. The values of this ratio (Yare presented in Table 3. The value of o calculated for an annular plate by using Q (instead of m) in equation (7) for a finite R would be a lower bound to the true eigenfrequency.
LETTER
TO THE TABLE
Estimates
of a = w2/02 obtained
309
EDITOR 3
by solving the variational problem S(T_J- T) = 0; v = 0.3 Hole size a/b
Case
n
0.1
0.3
0.5
0.7
SF
0
0.475 2.054 4.172 0.290 I.015 2,398 0,842 2.939 5.792 0.426 1.044 2.398 1,041 1.959 3.905 1.310 2.745 5.380 0.877 1.985 4.014 1,471 2,639 5.105
0.457 1.902 4.409 0.347 1.083 2.720 0,789 2.575 5.824 0.530 1.163 2.729 1.173 2.042 4.175 1,351 2.616 5.473 0.938 1.973 4.265 1.600 2,650 5.250
0,441 1.727 4,672 0,376 1.175 3.217 0.748 2,216 5.756 0.590 1,306 3,249 1,237 2,130 4.563 1.347 2.504 5,566 0.958 1.970 4.593 1.579 2.598 5.398
0.428 1.575 4.744 0.394 1.273 3,760 0.714 1.943 5,417 0.660 1.480 3.856 1,295 2.222 4.920 1,320 2.390 5.487 0.964 1.967 4.838 1.669 2.665 5.574
1 FS
CF
FC 2 0 1 2 0 1 2 0 1 2 0 1 2
SC
cs
ss
CC
Notes:
0.9 ~--I_ 0.413 I.454 4557 0.41 1 1,371 4,234 0,678 I.740 4,912 0.669 1,611 4.424 1,305 2,283 5,208 1,312 2,334 5,389 0.970 1.969 4.955 1.672 2.672 5,663
1. C = clamped, S = simply supported, F = free; the first letter denotes the edge support conditions at the outer edge (r = b). 2. n = number of nodal diameters. 3. These solutions are found to be highly sensitive to the number of terms used in the Rayleigh-Ritz method. They have been obtained here by taking five terms in the method.
TABLE
4
Comparison of a = w2/f12 obtained by the present analysis for an annular plate clamped at the inner edge and free at the outer edge and having a very small hole size (a/b = 0.001) with the corresponding estimates reported by Southwell [2] f or a solid circular plate clamped at its center and free at its outer edge; v = O-3
Estimates
from
Southwell [2] Present analysis Note:
n = number
n=O s=o
n=O S=l
n=l s=o
n=l s=l
n=2 .F= 0
n=2 s=l
0 0.252
3.3 4,211
1 1.001
5.95 5.962
2.35 2.350
8.95 8,950
of nodal diameters;
s = number
of nodal circles.
Southwell [2] has reported the values of (Yfor a solid plate clamped at its at its outer edge. In the present analysis values of (Y have been obtained plate with the same boundary conditions but having a very small hole size A comparison of the values of cz obtained by the present analysis with the values reported by Southwell [2] is presented in Table 4.
center and free for an annular (a/b = 0.001). corresponding
310
LETTER TO THE EDITOR ACKNOWLEDGMENTS
The author is grateful to the Alexander von Humboldt Foundation for the award of an AvH fellowship during the course of this work. The help provided by Professor W. Schnell in arranging this fellowship is also gratefully acknowledged. Institut fiir Mechanik, Technische Hochschule Darmstadt, D-6100 Darmstadt, West Germany
G. K. RAMAIAH
(Received 4 September 1980)
REFERENCES 1. H. LAMB and R.V. SOUTHWELL 1921Proceedings of the Royal Society of London, Series A 99, 272-280. The vibrations of a spinning disk. 2. R. V. SOUTHWELL 1922 Proceedings of the Royal Society of London, Series A 101,133-153. On the free vibrations of an uniform circular disk clamped at its centre and on the effect of rotation. 3. J. PRESCOTT 1961 Applied Elasticity. New York: Dover Publications, Inc. 4. H. C. LOH and J.F. CARNEY III1976 Proceedings of the Eighth Southeastern conference on Theoretical and Applied Mechanics 8, 365-375. Vibration and stability of beam reinforced spinning annular plates. 5. H. C. LOH and J.F. CARNEY III1977 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 44, 499-501. Vibration and stability of spinning annular plates reinforced with edge beams. 6. C. T. DYKA andJ.F.CARNEY III1979Journalof Sound and Vibration 64,223-231. Vibration and stability of spinning polar orthotropic annular plates reinforced with edge beams. 7. J. KIRKHOPE and G. J.WILSON 1976Journal of Sound and Vibration 44,461-474. Vibration and stress analysis of thin rotating discs using annular finite elements. 8. W. KENNEDY and D. GORMAN 1977Journal of Sound and Vibration 53, 83-101. Vibration analysis of variable thickness discs subjected to centrifugal and thermal stresses. and W. KENNEDY 1979 Journal of Sound and Vibration 62, 51-64. 9. D. G. GORMAN Membrane effects upon the transverse vibration of linearly varying thickness discs. 10. F. GINESU, B. PICASSO and P. PRIOLO 1979 Journal of Sound and Vibration 65, 97-105. Vibration analysis of polar orthotropic annular discs. 11. K. VIJAYAKUMAR and G. K. RAMAIAH 1972 Journal of Sound and Vibration 24,165-175. On the use of a coordinate transformation for analysis of axisymmetric vibration of polar orthotropic annular plates. 12. G. K. RAMAIAH and K. VIJAYAKUMAR 1973 Journal of Sound and Vibration 26,517-531. Natural frequencies of polar orthotropic annular plates.
of Soundand Vibration
(1981) 74(Z), 303-310
LETTER NATURAL
FREQUENCIES
TO THE EDITOR OF SPINNING
ANNULAR
PLATES
The dynamics of spinning discs has been studied by many investigators because of their commercial use as saw blades, turbine wheels and modern high speed computer storage discs. Lamb and Southwell [l] examined the problem of the completely free circular plate spinning about its cylindrical axis with a constant angular velocity. As a sequel to this work, Southwell [2] analyzed the problem of a circular disc which is clamped at its centre, is free at its outer edge and is rotating with a constant angular velocity. These two investigations carried out about six decades ago and a corroboration of these works by Prescott [3] are probably the earliest and the most frequently cited work on vibrations of spinning discs. Since then, there have been several analytical and experimental studies on isotropic, polar orthotropic and variable thickness spinning discs. On the other hand, there are only a few investigations reported in the literature on vibrations of spinning annular plates [4-lo] and in fact all of these have been carried out only during the last few years. Loh and Carney [4,5] and Dyka and Carney [6] have presented a general solution by adopting the Frobenius method for determining the natural frequencies of spinning annular plates reinforced with surrounding edge beams at the inner and outer edges. In the remaining four investigations [7-lo], some form of finite element method has been employed for plates with free outer and clamped inner edges. Numerical results in all this work are confined to those for some typical hole ratios and either a particular angular speed or some specific values of spinning parameter (R”yhb”/D). In the work reported here, the problem of vibrations of spinning annular plates has been analyzed in detail by the Rayleigh-Ritz method for eight different combination of boundary conditions. Accurate estimates of eigenfrequencies corresponding to the axisymmetric mode as well as the modes with one and two nodal diameters have been obtained for various values of hole sizes and for various values of the non-dimensional spinning parameter (R2rh64/D). The numerical results are presented in the form of tables and graphs for direct use by design engineers. A thin annular plate of constant thickness h with a and b as the radii of the inner and outer edges, respectively, is considered. It is assumed that the material of the plate is homogeneous and isotropic and the plate is rotating about its cylindrical axis with a constant angular velocity R (radians per second). The influence of rotation is to induce stresses which are assumed to be distributed symmetrically about the axis of rotation. These stresses in the absence of any in-plane pressure at the inner and outer edges are given by b2 - r7 - a2b’/r2], gr = [(3 + ~)/8]yf?*[a’+ il! u,j =[(3+v)/8]~~2[a’+bZ-;1+3~)/(3+v)r’+a’b2/r2], in which material.
12)
v is the Poisson’s ratio and y is the mass density per unit volume of the plate The flexural mode W is assumed to be expressible in the usual form w=
W,,(~)COS(ne+~)cos(ot+~),
r3i
in which n is the number of nodal diameters and w is the frequency in radians per second. The potential energy due to flexure is composed of two parts: namely, (1) that due to the strain energy of bending, V, and (2) that due to the centrifugal forces, U. The kinetic 303 r~072-~~0.~,/81/020.703
+0x
$02.00/0
0
19X I Academic Press Inc. / London I I .Imited
304
LETTER
TO THE EDITOR
energy, T, remains the same as for a stationary plate. The expressions for these quantities are as follows: b d*W, 1 dW n2 ’ v=;(l+&“)DJ w r dr r* n1 n
((-g+--l--
d2W
-2(1-v)$
1dW
-2
( r dr
-7
n*
W” +2(1-v)-i_ F* (F-T)*]r
~=5~l+~~~)~[~~b{~,(~)2+~~(~
[I(
I(
a2+b2-r2-T
b
+
a
W”)‘]rdr],
b
3+v U=;(1+SO”)eyn2h
a2b2
T=t(
I+s,.)
o*yh
dW, >(
1+3v” 2 * 3+v r2+7-arb > (s W,,)r
a2+b2--
(4)
dr,
>
* >
r dr
dr], dr
(5)
Jab W”,r dr,
in which So0 = 1, Son = 0 for n # 0 and D =Eh3/12(1 -v*) is the flexural rigidity of the plate. The total potential (V + U - T) is minimized by the Rayleigh-Ritz method to obtain relevant characteristic equations for the determination of eigenfrequencies (w2rhb4/D) for specified values of spinning parameter (0*rhb4/D). In the method, the mode shape W was expressed as a linear combination of simple least order polynomials in r [ 1 l] satisfying relevant geometric boundary conditions. For large values of the hole size, the quantities V, U and T were modified prior to the application of the Rayleigh-Ritz method by the transformations given earlier in reference [l l] for the axisymmetric mode and in reference [12] for the asymmetric modes. The number of terms used in the Rayleigh-Ritz method was up to five. In the computations, the parameter a/b was varied from O-1 to O-9 in steps of 0.2. The Poisson’s ratio v was fixed at 0.3. All computations were carried out with double precision arithmetic (about 16 significant digits) on an IBM 370/168 digital computer at the Technische Hochschule Darmstadt. Estimates of the frequency parameter ((2wb2/h) a) arepresented in Table 1 for several different values of the non-dimensional spinning parameter (LJ2rhb4/D). The TABLE
&&!$I f or rotating annular plates; v = O-3
Estimates of frequency parameter (= 2(wb*/h)
Values of (RZyhb4/D)
Values of (R2yhb4/D) h
I a/b
I*
0
0.1
0 1 2
0.3
0 1 2
2.823 7.758 14.60
0.5
0 1 2
3.073 7.025 13.53
25
1
10000
r
100
250
6.399 13.52 21.49
9.106 18.40 27.86
16.34 32.38 47.10
45.29 91.69 130.9
4.149 9.408 16.81
6.504 12.99 21.85
9.305 17.80 28.86
16.63 31.43 49.04
46.06 89.58 136.8
2.071 2.042 3.681
4.293 8.626 15.91
6.663 12.15 21.38
9.638 16.96 29.06
17.40 47.86 30.41 86.92 SO.75 142.6
2.494 2.942 4.833
(1) SF Case: 2.938 4.173 8.396 9.983 15.39 17.17
1000
,
0
25
(2) FS Case: 2.089 2.945 1.483 3.420 3.288 5.736
100
250
1000
10000
4.466 6.323 9.953
6.314 9.809 15.20
11.33 19.39 29.84
33.16 61.02 93.78
3.064 3.832 6.221
4.871 6.779 10.67
7.086 10.38 16.25
13.02 20.32 31.82
37.42 63.33 99.91
3.491 4.545 7.283
5.445 7.514 11.92
7.940 11.29 17.88
14.65 21.79 34.76
41.65 67.04 108.8
\
LETTER TABLE
TO THE EDITOR
1 (continued) Values of (R2yhb4/D)
Values of (f12ykb4/D) A
r
250
(1) SF Case: 4,195 5.099 1 8,056 9,261 14.72 16.58 2
7.146 12.17 21.17
10.03 16.50 28.17
18.40 29.62 49.86
51.82 85.93 144.8
0.9
0 1 2
12.09 19.90 33.92
13.89 22.42 37.92
20.67 32.21 53.60
56.76 86.21 141,7
0.1
0
9.895 18.14 27.33
13.15 23.52 34.28
21.85 39.13 55.37
58.16 107.3 150.6
(I.3
0
n
0.7
0
0
10.72 18.02 30.97
25
\
100
a/b
Il.08 18.51 31.73
(3) CF Case: 6.152 7,368 1 12.83 14.40 2 20.90 22.74
1000
10000
2 0.5
0 1 2
10.72 13.32 19.44
11.42 14.54 21.54
13.26 17.64 26.78
16.18 22.43 34.73
25.17 37.05 58.45
61.61 98.87 157.8
0.7
0
26.11 27.43 31.22
26.38 27.93 32.27
27.17 29.36 35.25
28.66 32.01 40.52
35.03 42.71 60.13
72.45 101.7 159.2
1 2 0.9
0
218.1
1 218.8
2
220.9
0 1 2
0.3
0 1 2
18.14 19.00 21.94
0.5
0
36.20 36.91 39.12
1 2
3.744 5.054 8.126 10.36 15.67 25.79
250
1000
10000
4.552 6,270 10.05
6,383 8,978 14.35
8.968 12.76 20.39
16.51 23.89 38.26
47.41 72.41 118.4
10.70 16.16 26.58
Il.68 17.56 28.80
13.42 20.07 32.79
19.97 29.56 48.00
54.86 80.73 130.5
5,263 6.651 9,991
7.443 10.14 15.23
13.45 19.80 29.8.5
40.11 61.97 03.80
13.84 22.56 35.15
22.73 37.79 57.50
58.39 102.4 154.0
4,031 3,966 4,817
4.892 5.275 6.953
6.761 7,967 Il.12
9.260 11.53 16.56
16.14 21.62 32.03
45.61 65.86 100.1
7.882 8,043 8.899
8,453 8.903 10.47
9.945 11.07 14.16
12.32 14.42 19.55
lY.76 24.86 35.89
51.35 71.58 109.8
22.60 23.07 24.54
23.30 24.17 26.71
24.62 26.22 30.59
30.27 34.60 45.21
64.04 82.98 123.1
218.1 218.8 221.0
218.2 219.0 221.4
218.4 219.3 221.1
219.3 220.9 225.7
229.5 238.9 265.0
22.36 22.69 23.77 208.4 208.9 210.3
16.81 20.19 28.09
26.84 33.77 46.98
71.91 94.15 130.8
18.70 19.76 23.16
20.26 21.86 26.41
22.99 25.46 31.98
32.76 38.14 50.52
80.05 99.38 137.4
20.44 21.73 25.86
36.51 37.32 39.82
37.40 38.53 41.87
39.11 40.81 45.68
46.51 92.61 SO.54 109.7 61.07 148 :
38.72 39.63 42.45
102.1 102.7 104.6
102.4 103.2 105.4
103.1 104.1 107.1
106.4 108.5 114.8
138.5 150.8 182.9
208.5 208.9 210.4
(6) CS Case: 13.74 14.57 15.30 16.58 21.43 23.17
13.61 15.78 21.91
105.5 106.2 108.2
0.1
0
(7)SS Case: 8.767 9.589 1 10.16 11.46 2 15.70 17.42
11.64 14.59 21.64
14.75 19.19 27.93
24.13 32.91 46.95
64.61 92.66 130.9
0.3
0 1 2
12.76 14.11 18.32
13.39 15.05 19.84
15.10 17.54 23.75
17.97 21.60 29.92
27.55 34.83 40.38
70.76 95.15 136.9
27.44 28.23 30.95
0.5
0 1 2
24.23 25.30 28.50
24.58 25.82 29.47
25.60 27.33 32.19
27.50 30.11 37.00
35.28 41.08 54.77
78.98 100.6 144.7
54.02 54.61 56.48
0.7
0
66.61 67.45 69.95
66.74 67.64 70.34
67.13 68.23 71.50
67.89 69.39 73.75
71.57 74.89 84.10
105.0 121.9 162.2
Notes:
100
10.53 17.23 27.40
0 1 2
2
25
(4) FC Case: 2.568 3.506 2.117 3.813 3,413 5.806
0.7
1
102.0 102.6 104.3
\ 0
8.041 13.45 22.00
(5) SC Case: lg.77 11.57 11.75 12.90 16.19 17.83
0.1
, (2) FS Case:
6,914 11.83 19.73
1
305
208.7 209.4 211.4
209.6 210.8 214.5
210.4 227.3 249.3
16.74 19.82 27.60
20.17 24.80 34.42
30.88 40.07 55.40
76.55 106.3 147.7
21.03 22.59 27.28
22.71 24.97 31.10
25.64 29.04 37.43
36.09 43.19 58.38
84.89 109.3 153.3
39.04 40.08 43.26
39.97 41.41 45.60
41.76 43.91 49.92
49.52 54.51 61.18
97.51 117.9 162.2
105.7 106.4 108.5
(8) CC Case: 16.51 17.31 17.51 18.63 22.18 23.82
150.0 150.8 152.2
208.6 209.1 210.7
106.0 106.9 109.4
106.7 107.8 111.2
110~1 112.5 119.6
143.0 156.9 192.4
19.43 21.57 28.06
22.90 26.28 34.71
34.06 41.26 55.52
82.75 107.0 146.8
27.96 28.94 32.09
29.53 30.95 35.27
32.31 34.55 40.78
42.79 47.91 6@35
94.17 113.9 154.0
54.30 54.98 57.09
55.15 56.08 58.90
56.80 58.20 62.33
64.19 67,61 77.01
114.0 130.0 168.0
150.4 150.9 152.4
150.8 151.3 153.1
151.4 152.1 154.4
354.5 156.1 161.0
186.2 196.3 223.8
1. C = clamped, S = simply supported, F = free; the first letter denotes the edge support conditions at r = h. 2. n = number of nodal diameters.
306
LETTER
TO THE
EDITOR
300
I00
200
100
FS
9 0 0 4 x . 600,
“3
,
Y/d ‘““vLIP n=l,a/b=O.l
0,0.5
1
I
0
,/A-
I
10
50
&
1
100
b
1
I50
11
’
200
250
0
50
100
150
200
250
a2 yhb4/D Figure 1. Variation of ~*yhb~/D with R2yhb4/D. n = number of nodal diameters. supported and F = free. The first letter denotes the edge support conditions at r = b.
C= clamped,
S= simply
LETTER
cl
z9 c
TO
THE
EDITOR
307
SC 300
Figure 2. Variation of 02yhb4/D with R2yhb4/D. n = number of nodal diameters. C = clamped, S = simply supported and F = free. The first letter denotes the edge support conditions at r = b.
308
LETTER
TO THE EDITOR
variation of the eigenvalue parameter (w*+~~/D) with the spinning parameter (0’$zb4/D) is shown in Figures 1 and 2 for some typical values of hole sizes and for different modes. The data in these figures suggest that w2 varies more or less linearly with 0’ in the form W2= I& + rnf12.
(7)
w,~in equation (7) corresponds to the frequency of the stationary plate (0 = 0) and m is a constant depending on the flexural mode, the value of a/b and the boundary condition. Approximate values of m can be calculated by using the data available in Table 1. The accuracy of the formula (7) improves with increasing values of the hole size but the formula suffers from inaccuracy when R tends to infinity. However, for finite values of 0 equation (7) yields reasonably accurate frequencies for all hole sizes and for all flexural modes. This is clearly exhibited by the data presented in Table 2 where the frequencies reported by Ginesu et al. [lo] for a steel plate rotating at 0 = 400 radians/s are compared with the corresponding frequencies obtained by using equation (7). Ginesu er al. [lo] employed the finite element program EACO for an annular plate (u/b = 0.5) with free outer and clamped inner edges. The constant m in equation (7) was numerically evaluated at each stage from computed values of 02yhb4/D by varying the parameter R2yhb4/D from 0 to 250 in steps of 25. The average value of m thus obtained was used in equation (7). TABLE
2
Comparison of frequencies obtained by using the formula o2 = wzl + rnf12 (equation (7)) with the corresponding accurate estimates reported by Ginesu et al. [lo] using the finite element program EA CO Plate dimensions: a/b = O-5, a = 101.6 mm, b = 203.2 mm, h = 1.016 mm; rotational
speed: 0 = 400 radians/s; plate material: steel disc, values of E’s and y are not given in reference [lo] and these values are not necessary when using equation (7); boundary condition: free outer and clamped inner edges; wSfis the frequency of the stationary plate and w is the frequency of the rotating plate. Frequency
n
s
0
0
0 1 1 2 2
1 0 1 0 1
Notes:
m 0.998 6,293 1.578 7.231 3.313 10.06
in radians/s
for
Stationary plate as given in reference [lo]
Rotating plate as given in reference [lo]
. Rotating plate as calculated from equation (7)
497 3244 507 3308 561 3500
638 3396 714 3479 919 3723
637.7 3395.6 713.8 3478.5 919.1 3722.8
1. n = number of nodal diameters, s = number of nodal circles. 2. m is a constant and it was evaluated numerically at each stage from computed values of 02yhb4/D by varying R2yhb4/D from 0 to 250 in steps of 25. The average value of m thus obtained was used in equation (7). 3. v=O.3.
The strain energy component V can be neglected if it is assumed that the plate is very thin and the rotation is very rapid. The solution of the resulting variational problem S( U - T) = 0 then yields the ratio cr = u2/R2. The values of this ratio (Yare presented in Table 3. The value of o calculated for an annular plate by using Q (instead of m) in equation (7) for a finite R would be a lower bound to the true eigenfrequency.
LETTER
TO THE TABLE
Estimates
of a = w2/02 obtained
309
EDITOR 3
by solving the variational problem S(T_J- T) = 0; v = 0.3 Hole size a/b
Case
n
0.1
0.3
0.5
0.7
SF
0
0.475 2.054 4.172 0.290 I.015 2,398 0,842 2.939 5.792 0.426 1.044 2.398 1,041 1.959 3.905 1.310 2.745 5.380 0.877 1.985 4.014 1,471 2,639 5.105
0.457 1.902 4.409 0.347 1.083 2.720 0,789 2.575 5.824 0.530 1.163 2.729 1.173 2.042 4.175 1,351 2.616 5.473 0.938 1.973 4.265 1.600 2,650 5.250
0,441 1.727 4,672 0,376 1.175 3.217 0.748 2,216 5.756 0.590 1,306 3,249 1,237 2,130 4.563 1.347 2.504 5,566 0.958 1.970 4.593 1.579 2.598 5.398
0.428 1.575 4.744 0.394 1.273 3,760 0.714 1.943 5,417 0.660 1.480 3.856 1,295 2.222 4.920 1,320 2.390 5.487 0.964 1.967 4.838 1.669 2.665 5.574
1 FS
CF
FC 2 0 1 2 0 1 2 0 1 2 0 1 2
SC
cs
ss
CC
Notes:
0.9 ~--I_ 0.413 I.454 4557 0.41 1 1,371 4,234 0,678 I.740 4,912 0.669 1,611 4.424 1,305 2,283 5,208 1,312 2,334 5,389 0.970 1.969 4.955 1.672 2.672 5,663
1. C = clamped, S = simply supported, F = free; the first letter denotes the edge support conditions at the outer edge (r = b). 2. n = number of nodal diameters. 3. These solutions are found to be highly sensitive to the number of terms used in the Rayleigh-Ritz method. They have been obtained here by taking five terms in the method.
TABLE
4
Comparison of a = w2/f12 obtained by the present analysis for an annular plate clamped at the inner edge and free at the outer edge and having a very small hole size (a/b = 0.001) with the corresponding estimates reported by Southwell [2] f or a solid circular plate clamped at its center and free at its outer edge; v = O-3
Estimates
from
Southwell [2] Present analysis Note:
n = number
n=O s=o
n=O S=l
n=l s=o
n=l s=l
n=2 .F= 0
n=2 s=l
0 0.252
3.3 4,211
1 1.001
5.95 5.962
2.35 2.350
8.95 8,950
of nodal diameters;
s = number
of nodal circles.
Southwell [2] has reported the values of (Yfor a solid plate clamped at its at its outer edge. In the present analysis values of (Y have been obtained plate with the same boundary conditions but having a very small hole size A comparison of the values of cz obtained by the present analysis with the values reported by Southwell [2] is presented in Table 4.
center and free for an annular (a/b = 0.001). corresponding
310
LETTER TO THE EDITOR ACKNOWLEDGMENTS
The author is grateful to the Alexander von Humboldt Foundation for the award of an AvH fellowship during the course of this work. The help provided by Professor W. Schnell in arranging this fellowship is also gratefully acknowledged. Institut fiir Mechanik, Technische Hochschule Darmstadt, D-6100 Darmstadt, West Germany
G. K. RAMAIAH
(Received 4 September 1980)
REFERENCES 1. H. LAMB and R.V. SOUTHWELL 1921Proceedings of the Royal Society of London, Series A 99, 272-280. The vibrations of a spinning disk. 2. R. V. SOUTHWELL 1922 Proceedings of the Royal Society of London, Series A 101,133-153. On the free vibrations of an uniform circular disk clamped at its centre and on the effect of rotation. 3. J. PRESCOTT 1961 Applied Elasticity. New York: Dover Publications, Inc. 4. H. C. LOH and J.F. CARNEY III1976 Proceedings of the Eighth Southeastern conference on Theoretical and Applied Mechanics 8, 365-375. Vibration and stability of beam reinforced spinning annular plates. 5. H. C. LOH and J.F. CARNEY III1977 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 44, 499-501. Vibration and stability of spinning annular plates reinforced with edge beams. 6. C. T. DYKA andJ.F.CARNEY III1979Journalof Sound and Vibration 64,223-231. Vibration and stability of spinning polar orthotropic annular plates reinforced with edge beams. 7. J. KIRKHOPE and G. J.WILSON 1976Journal of Sound and Vibration 44,461-474. Vibration and stress analysis of thin rotating discs using annular finite elements. 8. W. KENNEDY and D. GORMAN 1977Journal of Sound and Vibration 53, 83-101. Vibration analysis of variable thickness discs subjected to centrifugal and thermal stresses. and W. KENNEDY 1979 Journal of Sound and Vibration 62, 51-64. 9. D. G. GORMAN Membrane effects upon the transverse vibration of linearly varying thickness discs. 10. F. GINESU, B. PICASSO and P. PRIOLO 1979 Journal of Sound and Vibration 65, 97-105. Vibration analysis of polar orthotropic annular discs. 11. K. VIJAYAKUMAR and G. K. RAMAIAH 1972 Journal of Sound and Vibration 24,165-175. On the use of a coordinate transformation for analysis of axisymmetric vibration of polar orthotropic annular plates. 12. G. K. RAMAIAH and K. VIJAYAKUMAR 1973 Journal of Sound and Vibration 26,517-531. Natural frequencies of polar orthotropic annular plates.