- Email: [email protected]
On free vibration analysis of circular annular plates with non-uniform thickness by the differential quadrature method
Journal of Sound and Vibration (1995) 184(3), 547–551
LETTER TO THE EDITOR ON FREE VIBRATION ANALYSIS OF CIRCULAR ANNULAR PLATES WITH NON-UNIFORM THICKNESS BY THE DIFFERENTIAL QUADRATURE METHOD X. W, J. Y J. X Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China (Received 20 July 1994, and in final form 30 September 1994)
1. Recently, differential quadrature (DQ) has been used to obtain the fundamental frequency of annular plates with uniform thickness and shown to be computationally efficient [1]. As is pointed out by Larrondo et al. [2], the methodology seems ideal for dealing with circular annular plates of non-uniform thickness as well as elastic stability problems. In view of the fact that in reference [1] no results were given for the fundamental frequency of circular plates with non-uniform thickness or with elastic restraint against rotation along the edges, the writers have computed these fundamental frequencies using both DQ and HDQ (harmonic differential quadrature) methods [3], and equally good results have been obtained by both methods. The shortcomings in reference [1] (i.e. that the results do not agree well with the existing data for cases with a free inner boundary (F–C and F–SS) and small b/a (the ratio of the inner radius to the outer radius) of 0·1 or 0·2) have been removed by simply using a different grid spacing. The problem considered in this note involves plates of isotropic material only. It should be pointed out that the DQ results for elastic stability problems also agree well with data in the open literature [4]. For completeness, the formula for directly computing the weighting coefficients of the first derivative is also given here; details may be found in reference [5]. 2. It is found that the Vandermonde matrix is ill-conditioned when the dimension is greater than 15. Thus, the number of grid points cannot be greater than 15 in the applications of the DQ method formulated in the standard DQ way. To overcome this shortcoming, and in the intensity of further reduction in computational efforts, the weighting coefficients of the derivative (Aij ) can be directly computed by the formula [5] N
t
Aij =
(xi − xk )
k = 1 k $ i,j
>
N
t
(xj − xk ),
i $ j,
k=1 k$j
N
s
Aii =
1/(xi − xk ),
i, j = 1, 2, . . . , N,
(1)
k=1 k$i
where N is the total number of grid points xj and xj = {(1 + b/a) + (1 − b/a)yj }/2,
yj = [−1, −cos(2j − 3)(p/(2N − 4)), 1],
(2)
in which b and a are the inner and outer radii of the circular annular plate, respectively. Let h0 , h1 , E and n be the thickness at the inner and outer edges, Young’s modulus of elasticity and the Poisson ratio, respectively. Assume that 547 0022–460X/95/280547 + 05 $08.00/0
7 1995 Academic Press Limited
548
h(r) = h0 (r/a)b,
D1 = Eh13 /12(1 − n 2).
(3)
It is obvious that b = 0 corresponds to the case of plates with uniform thickness. Applying differential quadrature, or HDQ, to the governing differential equation for free vibration of an annular plate with non-uniform thickness as given by equations (3) yields M2
M2
j = M1
j = M1
M2
xi2b s Dij Wj + (6b + 2)xi(2b − 1) s Cij Wj + (9b 2 + 3b − 1 + 3nb)xi(2b − 2) s j = M1 M2
× Bij Wj + 9b 2n − 2bn + 1 − 3b)xi2b − 3 s Aij Wj = v¯ 2Wi , j = M1
i = M1 , M2 + 1, . . . , M2 ,
(4)
where Dij , Cij , Bij and Wj are, respectively, the weighting coefficients of the fourth, third and second derivatives, and the deflection at grid j; v¯ = a 2vzrh1 /D1 is the non-dimensionalized frequency. If the displacement is zero at both inner and outer boundaries, M1 = 2 and M2 = N − 2. For plates elastically restrained against rotation along two edges, the boundary conditions in terms of the differential quadrature are N−1
N−1
j=2
j=2
fb s A1j Wj = s [B1j + n(a/b)A1j ]Wj ,
W1 = WN = 0, N−1
N−1
j=2
j=2
fa s ANj Wj = − s [BNj + nANj ]Wj ,
(5)
where fb and fa are the non-dimensionalized rotational stiffnesses at the inner and outer boundary, respectively. 3. The free vibration of uniform annular plates with one or two free edges has been re-analyzed by the DQ method for small inner to outer radius ratio. The results are listed in Table 1 and compared with data in the open literature [1, 2, 6]. As can be seen, the DQ T 1 Non-dimensional fundamental frequency of flexural vibration of annular plates with various boundary conditions (v¯ = va 2zrh/D; n = 1/3; N = 9) Boundary conditions ZXXXXXXXXXXXCXXXXXXXXXXXV F–C F–SS F–F SS–F C–F
b/a
Methods
0·1
Leissa [6] DQM [1] DQM Larrondo et al. [2]
10·18 13·41 10·13 10·13
4·933 7·138 5·198 4·890
5·203 5·077 5·191 NA
3·516 3·447 3·436 3·440
4·235 4·230 4·254 4·263
0·2
Leissa [6] DQM [1] DQM Larrondo et al. [2]
10·34 10·04 10·32 10·35
4·726 4·481 4·757 4·733
5·053 5·011 5·051 NA
3·312 3·313 3·313 3·313
5·244 5·212 5·212 5·214
0·3
Leissa [6] DQM [1] DQM Larrondo et al. [2]
11·37 11·31 11·34 11·34
4·654 4·633 4·661 4·659
4·822 4·818 4·821 NA
3·378 3·386 3·387 3·387
6·739 6·702 6·701 6·701
549
Figure 1. Variation of frequency with rotational stiffness; E–SS, N = 11.
results are much improved for the two worst cases (F–C and F–SS), over those in reference [1] by simply using a different grid spacing: i.e., equations (2). Note that both DQ and HDQ methods give equally good results for all problems considered in this note and an increase of the number of grid points improves the results. No difficulty will be encountered for a large number of grid points if one computes the weighting coefficients by using equations (1). It is found, however, that little improvement is achieved by using a large number of grid points (say, N q 21) because of the influence of the round-off errors. The DQ (or HDQ) method has also been used to obtain the fundamental frequency of uniform annular plates with one edge elastically restrained against rotation (denoted by the symbol E) and the other edge clamped, simply supported or elastically restrained against rotation (denoted by symbols C, SS or E). There are five combinations of boundary constraints; namely, E–C, E–SS, E–E, C–E and SS–E. For brevity, results for only two of them are reported here. The variations of the non-dimensionalized frequency with rotational stiffness are shown in Figures 1 and 2 for E–SS and E–E uniform annular plates. Various ratios of inner radius to outer radius are considered and the Poisson ratio is 1/3. As can be seen from Figure 1, the results change greatly when the non-dimensionalized
Figure 2. As Figure 1, but for E(10)–E(uniform thickness) boundary conditions.
550
Figure 3. Variation of frequency for annular plates of linearly varying thickness with the ratio of inner to outer radii.
rotational stiffness is in the range of 0·1–1000. Therefore, in the case of E–E (see Figure 2), the rotational stiffness at innner edge is chosen as 10. Similar variations as in the case of E–SS (see Figure 1) can be observed. It is seen that f E 0·1 is equivalent to a simply supported boundary and f e 1000 is equivalent to a clamped edge. Similar results have been obtained for all other combinations of boundary conditions. The results compare well with the data in reference [7]. The free vibration of annular plates of varying thickness (b = 1, b = −1) with various boundary conditions has been analyzed by the DQ (or HDQ) method. The results are shown in Figure 3 for the variations of frequency with b/a. In Figure 3, the values 1 or −1 in the parentheses are the values of b in equations (3). It should be mentioned that only a small number of grid points is necessary to obtain accurate solutions for large values of b/a (say, b/a q 0.3), in contrast to the situation for the Rayleigh–Ritz method. When b/a Q 0·3, however, a larger number of grid points is required to obtain very accurate
Figure 4. As Figure 1, but for E(20)–E boundary conditions.
551
results, especially when the outer edge is simply supported. In other words, the convergence of the DQ (or HDQ) method is slower for a circular plate with a very small central hole. In Figure 4 are shown the variations of frequency with the rotational stiffness at the outer edge when the inner edge is elastically constrained against rotation. The non-dimensionalized rotational stiffness is taken as 20. From Figure 4 one can see that the variation is similar to that in the cases of annular plates with uniform thickness (see Figures 1 and 2). 4. Based on the results reported in this note, several conclusions may be drawn. The DQ method and HDQ method are equally good for free vibration analysis of annular plates and yield convergent solutions for the fundamental frequency. Using equations (1) to compute directly the weighting coefficients improves the accuracy and efficiency of the method. A non-dimensionalized rotational stiffness f E 0·1 is equivalent to a simply supported boundary, and f e 1000 is equivalent to a clamped edge. A This work has been supported by the Aeronautical Science Foundation and Jiangsu Natural Science Foundation of China. 1. X. W, A. G. S and C. W. B 1993 Journal of Sound and Vibration 164, 173–175. Free vibration analysis of annular plates by the DQ method. 2. H. L, V. T, D. R. A and P. A. A. L 1994 Journal of Sound and Vibration 177, 137–139. Comments on ‘‘Free vibration analysis of annular plates by the DQ method’’. 3. A. G. S, X. W and C. W. B 1994 Acta Mechanica (in press). Harmonic differential quadrature method and applications to structural components. 4. R. Y, Y. W and X. W 1994 Journal of Nanjing University of Aeronautics & Astronautics (English edition) 94(10), 74–81. Buckling analysis of polar orthotropic annular plates under uniform pressures. 5. B. H and X. W 1994 Journal of Nanjing University of Aeronautics & Astronautics (English edition) (in press). Error analysis in differential quadrature method. 6. A. W. L 1969 Vibration of Plates (NASA SP-160). Washington, D.C.: U.S. Government Printing Office. 7. D. R. A and P. A. A. L 1979 Journal of Sound and Vibration 66, 63–67. A note on transverse vibrations of annular plates elastically restrained against rotation along the edges.
LETTER TO THE EDITOR ON FREE VIBRATION ANALYSIS OF CIRCULAR ANNULAR PLATES WITH NON-UNIFORM THICKNESS BY THE DIFFERENTIAL QUADRATURE METHOD X. W, J. Y J. X Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, People’s Republic of China (Received 20 July 1994, and in final form 30 September 1994)
1. Recently, differential quadrature (DQ) has been used to obtain the fundamental frequency of annular plates with uniform thickness and shown to be computationally efficient [1]. As is pointed out by Larrondo et al. [2], the methodology seems ideal for dealing with circular annular plates of non-uniform thickness as well as elastic stability problems. In view of the fact that in reference [1] no results were given for the fundamental frequency of circular plates with non-uniform thickness or with elastic restraint against rotation along the edges, the writers have computed these fundamental frequencies using both DQ and HDQ (harmonic differential quadrature) methods [3], and equally good results have been obtained by both methods. The shortcomings in reference [1] (i.e. that the results do not agree well with the existing data for cases with a free inner boundary (F–C and F–SS) and small b/a (the ratio of the inner radius to the outer radius) of 0·1 or 0·2) have been removed by simply using a different grid spacing. The problem considered in this note involves plates of isotropic material only. It should be pointed out that the DQ results for elastic stability problems also agree well with data in the open literature [4]. For completeness, the formula for directly computing the weighting coefficients of the first derivative is also given here; details may be found in reference [5]. 2. It is found that the Vandermonde matrix is ill-conditioned when the dimension is greater than 15. Thus, the number of grid points cannot be greater than 15 in the applications of the DQ method formulated in the standard DQ way. To overcome this shortcoming, and in the intensity of further reduction in computational efforts, the weighting coefficients of the derivative (Aij ) can be directly computed by the formula [5] N
t
Aij =
(xi − xk )
k = 1 k $ i,j
>
N
t
(xj − xk ),
i $ j,
k=1 k$j
N
s
Aii =
1/(xi − xk ),
i, j = 1, 2, . . . , N,
(1)
k=1 k$i
where N is the total number of grid points xj and xj = {(1 + b/a) + (1 − b/a)yj }/2,
yj = [−1, −cos(2j − 3)(p/(2N − 4)), 1],
(2)
in which b and a are the inner and outer radii of the circular annular plate, respectively. Let h0 , h1 , E and n be the thickness at the inner and outer edges, Young’s modulus of elasticity and the Poisson ratio, respectively. Assume that 547 0022–460X/95/280547 + 05 $08.00/0
7 1995 Academic Press Limited
548
h(r) = h0 (r/a)b,
D1 = Eh13 /12(1 − n 2).
(3)
It is obvious that b = 0 corresponds to the case of plates with uniform thickness. Applying differential quadrature, or HDQ, to the governing differential equation for free vibration of an annular plate with non-uniform thickness as given by equations (3) yields M2
M2
j = M1
j = M1
M2
xi2b s Dij Wj + (6b + 2)xi(2b − 1) s Cij Wj + (9b 2 + 3b − 1 + 3nb)xi(2b − 2) s j = M1 M2
× Bij Wj + 9b 2n − 2bn + 1 − 3b)xi2b − 3 s Aij Wj = v¯ 2Wi , j = M1
i = M1 , M2 + 1, . . . , M2 ,
(4)
where Dij , Cij , Bij and Wj are, respectively, the weighting coefficients of the fourth, third and second derivatives, and the deflection at grid j; v¯ = a 2vzrh1 /D1 is the non-dimensionalized frequency. If the displacement is zero at both inner and outer boundaries, M1 = 2 and M2 = N − 2. For plates elastically restrained against rotation along two edges, the boundary conditions in terms of the differential quadrature are N−1
N−1
j=2
j=2
fb s A1j Wj = s [B1j + n(a/b)A1j ]Wj ,
W1 = WN = 0, N−1
N−1
j=2
j=2
fa s ANj Wj = − s [BNj + nANj ]Wj ,
(5)
where fb and fa are the non-dimensionalized rotational stiffnesses at the inner and outer boundary, respectively. 3. The free vibration of uniform annular plates with one or two free edges has been re-analyzed by the DQ method for small inner to outer radius ratio. The results are listed in Table 1 and compared with data in the open literature [1, 2, 6]. As can be seen, the DQ T 1 Non-dimensional fundamental frequency of flexural vibration of annular plates with various boundary conditions (v¯ = va 2zrh/D; n = 1/3; N = 9) Boundary conditions ZXXXXXXXXXXXCXXXXXXXXXXXV F–C F–SS F–F SS–F C–F
b/a
Methods
0·1
Leissa [6] DQM [1] DQM Larrondo et al. [2]
10·18 13·41 10·13 10·13
4·933 7·138 5·198 4·890
5·203 5·077 5·191 NA
3·516 3·447 3·436 3·440
4·235 4·230 4·254 4·263
0·2
Leissa [6] DQM [1] DQM Larrondo et al. [2]
10·34 10·04 10·32 10·35
4·726 4·481 4·757 4·733
5·053 5·011 5·051 NA
3·312 3·313 3·313 3·313
5·244 5·212 5·212 5·214
0·3
Leissa [6] DQM [1] DQM Larrondo et al. [2]
11·37 11·31 11·34 11·34
4·654 4·633 4·661 4·659
4·822 4·818 4·821 NA
3·378 3·386 3·387 3·387
6·739 6·702 6·701 6·701
549
Figure 1. Variation of frequency with rotational stiffness; E–SS, N = 11.
results are much improved for the two worst cases (F–C and F–SS), over those in reference [1] by simply using a different grid spacing: i.e., equations (2). Note that both DQ and HDQ methods give equally good results for all problems considered in this note and an increase of the number of grid points improves the results. No difficulty will be encountered for a large number of grid points if one computes the weighting coefficients by using equations (1). It is found, however, that little improvement is achieved by using a large number of grid points (say, N q 21) because of the influence of the round-off errors. The DQ (or HDQ) method has also been used to obtain the fundamental frequency of uniform annular plates with one edge elastically restrained against rotation (denoted by the symbol E) and the other edge clamped, simply supported or elastically restrained against rotation (denoted by symbols C, SS or E). There are five combinations of boundary constraints; namely, E–C, E–SS, E–E, C–E and SS–E. For brevity, results for only two of them are reported here. The variations of the non-dimensionalized frequency with rotational stiffness are shown in Figures 1 and 2 for E–SS and E–E uniform annular plates. Various ratios of inner radius to outer radius are considered and the Poisson ratio is 1/3. As can be seen from Figure 1, the results change greatly when the non-dimensionalized
Figure 2. As Figure 1, but for E(10)–E(uniform thickness) boundary conditions.
550
Figure 3. Variation of frequency for annular plates of linearly varying thickness with the ratio of inner to outer radii.
rotational stiffness is in the range of 0·1–1000. Therefore, in the case of E–E (see Figure 2), the rotational stiffness at innner edge is chosen as 10. Similar variations as in the case of E–SS (see Figure 1) can be observed. It is seen that f E 0·1 is equivalent to a simply supported boundary and f e 1000 is equivalent to a clamped edge. Similar results have been obtained for all other combinations of boundary conditions. The results compare well with the data in reference [7]. The free vibration of annular plates of varying thickness (b = 1, b = −1) with various boundary conditions has been analyzed by the DQ (or HDQ) method. The results are shown in Figure 3 for the variations of frequency with b/a. In Figure 3, the values 1 or −1 in the parentheses are the values of b in equations (3). It should be mentioned that only a small number of grid points is necessary to obtain accurate solutions for large values of b/a (say, b/a q 0.3), in contrast to the situation for the Rayleigh–Ritz method. When b/a Q 0·3, however, a larger number of grid points is required to obtain very accurate
Figure 4. As Figure 1, but for E(20)–E boundary conditions.
551
results, especially when the outer edge is simply supported. In other words, the convergence of the DQ (or HDQ) method is slower for a circular plate with a very small central hole. In Figure 4 are shown the variations of frequency with the rotational stiffness at the outer edge when the inner edge is elastically constrained against rotation. The non-dimensionalized rotational stiffness is taken as 20. From Figure 4 one can see that the variation is similar to that in the cases of annular plates with uniform thickness (see Figures 1 and 2). 4. Based on the results reported in this note, several conclusions may be drawn. The DQ method and HDQ method are equally good for free vibration analysis of annular plates and yield convergent solutions for the fundamental frequency. Using equations (1) to compute directly the weighting coefficients improves the accuracy and efficiency of the method. A non-dimensionalized rotational stiffness f E 0·1 is equivalent to a simply supported boundary, and f e 1000 is equivalent to a clamped edge. A This work has been supported by the Aeronautical Science Foundation and Jiangsu Natural Science Foundation of China. 1. X. W, A. G. S and C. W. B 1993 Journal of Sound and Vibration 164, 173–175. Free vibration analysis of annular plates by the DQ method. 2. H. L, V. T, D. R. A and P. A. A. L 1994 Journal of Sound and Vibration 177, 137–139. Comments on ‘‘Free vibration analysis of annular plates by the DQ method’’. 3. A. G. S, X. W and C. W. B 1994 Acta Mechanica (in press). Harmonic differential quadrature method and applications to structural components. 4. R. Y, Y. W and X. W 1994 Journal of Nanjing University of Aeronautics & Astronautics (English edition) 94(10), 74–81. Buckling analysis of polar orthotropic annular plates under uniform pressures. 5. B. H and X. W 1994 Journal of Nanjing University of Aeronautics & Astronautics (English edition) (in press). Error analysis in differential quadrature method. 6. A. W. L 1969 Vibration of Plates (NASA SP-160). Washington, D.C.: U.S. Government Printing Office. 7. D. R. A and P. A. A. L 1979 Journal of Sound and Vibration 66, 63–67. A note on transverse vibrations of annular plates elastically restrained against rotation along the edges.