Influence of Chebyshev Collocation Points on the Convergence of Results in the Analysis of Annular Plate Vibrations

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 8, No 1, pp (2005) 57 Influence of Chebyshev Collocation Points on the Convergence of Results in the Analysis of Annular Plate Vibrations R. P. Singh 1 * and S. K. Jain 2 1 Department of Applied Mechanics, Birla Institute of Technology Mesra, Ranchi , India 2 Department of Applied Mathematics, Birla Institute of Technology Mesra, Ranchi , India Abstract The Chebyshev collocation points have been used as the interpolating nodes in the analysis of vibrations of an annular plate by an interpolation approximation and their influence on the convergence of the results has been investigated. For the purpose, two different interpolating approximate methods the spline and the collocation have been considered and, for a typical value of the plate parameters and boundary conditions in each one, the frequency parameters of the plate have been computed by taking both the equi-spaced nodes as well as the Chebyshev points as the interpolating nodes. A faster rate of convergence and appreciably improved results have been obtained at Chebyshev collocation points as compared to the results obtained at equi-spaced nodes (Fourier points) in each of the methods for the lower values of radii ratio. However, no appreciable influence of the Chebyshev collocation points on the results has been observed for the higher values of the radii ratio in any of the methods. Key Words: Chebyshev Points, Annular Points, Interpolation Approximation, Spline and Collocation Methods, Radii Ratio 1. Introduction From the literature, it has been noted that in the analysis of plate vibration by an interpolation method, the original deflection function is approximated by an interpolation polynomial, taking appropriate number of equidistant nodes for a given interval, for example, Kim and Dickinson [2], Liew et al [3], Soni and Amba-Rao [4], Toshihiro and Yamada [5], Gupta and Ansari [7], Wong et al [8]. While computing the eigenvalues of a nonuniform annular plate using the spline interpolation approximate method, Toshihiro and Yamada [5] have observed that when the radii ratio is larger than or equal to 0.5, sufficiently convergent values (almost equal to the exact values) are obtained by taking thirty or forty equidistant nodes at the most along the radius of the plate. But *Corresponding author. saralkjain@yahoo.com by taking the same number of nodes, the convergence becomes quite slower for small radii ratios. It has also been observed from the reference [5] that sufficiently large number of nodes (more than 60) have been considered for the eigenvalues obtained for small radii ratios and even than the results are not sufficiently close to the exact values. As per reference [1], if a given function is interpolated with the equi-spaced nodes, the interpolation error is much larger near the end points than in the middle of the interval and this larger error at the end points can be reduced by taking Chebyshev collocation points as an interpolating nodes. In this note, the Chebyshev collocation points have been used as the interpolating nodes in the analysis of vibration of an annular plate by an interpolation approximate method and their influence on the convergence of the results has been investigated. For the purpose, two different

2 58 R. P. Singh and S. K. Jain approximate methods the spline and the collocation have been considered and the frequency parameters of the plate, with a typical value of the plate parameters and boundary conditions in each one, have been computed and compared to the corresponding frequency parameters obtained by taking equidistant nodes in each one of the methods. 2. Analysis 2.1 Equation of Motion Without taking the rotary inertia and shear deformation into account, the equation of motion given by Lekhnitskii [6, chap. ix] for a harmonically vibrating thin plate of uniform thickness possessing cylindrical anisotropy has been extended for the case of variable thickness along the radial direction and by expressing the transverse deflection w(r, ) = (aw( )) cos n (n=0,1,2, ), the equation of motion has been written in the nondimensional form as: F( )d 4 W( )/d 4 +(F 1 / )d 3 W( )/d 3 +(F 2 / 2 )d 2 W( )/d 2 +(F 3 / 3 ) dw( )/d +(F 4 / 4 )W( ) ( f) W( ) = 0,... (1) W( ) is non-dimensional transverse deflection, = r/a is the radii ratio ( = whenr=b)andf( )=f 3 where f is non-dimensional thickness variation function. Herein, f=1+ has been taken for linearly varying thickness where á is the taper parameter. = 2 a 4 h b /D rb is defined as the non-dimensional frequency parameter where D rb =E r h b 3 /12 (1 v r v ) is the flexural rigidity of plate at inner edge and h b is plate thickness at the inner edge. F 1 =2( df( )/d +F( )),.. (2) F 2 =( 2 )d 2 F( )/d 2 + ((2 + v ) ) df( )/d (p + 2n 2 (v +2D kr ))F( ),. (3) F 3 =( 2 v )d 2 F( )/d 2 ( (p+2n 2 (v +2d kr )) df( )/d +(p+2n 2 (v +2D kr ))F( ),.. (4) F 4 = ( 2 v n 2 )d 2 F( )/d 2 +(n 2 (p + 2(v +2D kr ))) df( )/d +(n 2 (pn 2 2p 2(v +2D kr )))F( ), (5) D kr =G r (1 v r v )/E r where G r is the shear modulus and v r, v are the Poisson s ratios and p is the rigidity ratio (E /E r ). E and E r are the Young s modulus. 3. Methods of Solution 3.1 Spline Approximations For this purpose, the interval (, 1) has been divided into subintervals taking m nodes and the displacement function W( ) is approximated by a quintic spline as: W( )=a 1 +a 2 ( 1 )+a 3 ( 1 ) 2 +a 4 ( 1 ) 3 +a 5 ( 1 ) 4 m + bj ( j ) 5 u( j ), (6) j where u( j )={ 0if( ) j 1if( ) 1 j is the unit step function, a 1,a 2,a 3,a 4,a 5 and b j (j = 1,, m) are the (m+5) unknowns to be determined from boundary conditions. 3.2 Collocation Approximations The interval (, 1) has been divided into subintervals taking m nodes and the displacement function W( ) is approximated as: m W( )= Aj (1+ j + j 2 + j 3 + j 4 ) j 1 ).. (3) j 1 where A j, j, j, j, j are unknown coefficients to be determined from boundary conditions. 4. Chebyshev Collocation Points For displacement function W( ) with an interval (, 1), the Chebyshev collocation points are taken as follows j =(( + 1)/2) + (( 1)/2) cos ((2j 1)/2m) where j =1, 2, 3,, m and < <1 5. Results and Discussions The frequency parameters of an annular plate (considering both isotropic as well as orthotropic cases) have been computed using each approximate method by considering both the types of distribution of nodes in each one of the methods. In Table 1, the frequency parameters of an isotropic annular plate clamped at the outer edge and free at the inner edge have been computed via spline technique considering a typical number of nodes m for the radii ratios

3 Influence of Chebyshev Collocation Points on the Convergence of Results in the Analysis of Annular Plate Vibrations 59 equal to 0.1, 0.3, 0.5 and 0.7 and Poisson s ratio equal to 0.3. The results have been compared to the results obtained by Toshihiro and Yamada [5]. In Table 2, the frequency parameters of an isotropic annular plate with the F-C and the F-S boundary conditions have been computed via collocation technique for the radii ratios equal to 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6 and Poisson s ratio equal to 1/3. Herein, an appropriate number of nodes have been taken both at Chebyshev points and Fourier points for the convergence of results. In Table 3, the frequency parameters of a polar orthotropic annular plate of linearly varying thickness (centrally thinner) have been computed via spline technique with a typical number of nodes m for the radii ratios equal to 0.1, 0.3, 0.5 and 0.7 and Poisson s ratio equal to 0.3 considering the taper parameter equal to 0.5 and the rigidity ratio p equal to 0.5 in fundamental mode when both edges of the plate are clamped. In Table 4, the frequency parameters of a polar orthotropic annular plate of linearly varying thickness Table 1. Comparison of frequency parameters for uniform annular plate clamped at outer edge while free at inner edge; Poisson s ratio = 0.3 via spline method (n,s) m Method (0,0) (1,0) (0,1) Toshihiro[5] Present* Present** Toshihiro[5] Present* Present** Toshihiro[5] Present* Present** Exact Value Toshihiro[5] Present* Present** Toshihiro[5] Present* Present** Toshihiro[5] Present* Present** Exact Value Toshihiro[5] Present* Present** Toshihiro[5] Present* Present** Exact Value Toshihiro[5] Present* Present** Toshihiro[5] Present* Present** Exact Value Present* (using equi-spaced nodes); Present** (using Chebyshev points), (n, s) number of nodal diameters and nodal circles.

4 60 R. P. Singh and S. K. Jain (centrally thinner) have been computed via collocation technique with a typical number of nodes m for the radii ratios equal to 0.1, 0.3, 0.5 and 0.7 and Poisson s ratio equal to 0.3 by considering the taper parameter equal to 0.3 and the rigidity ratio p equal to 5.0 in fundamental mode when both edges of the plate are clamped. In order to show the effect of nodal distribution on the results, the percentage error curves have been plotted taking some results from the Table 4. In Figure 1, the curves have been plotted for the radii ratio = 0.1 (a Table 2. Comparison of frequency parameters of an isotropic annular plate for the case of clamped and simply supported at the outer edge while the inner edge is free via collocation method taking appropriate number of nodes; Poisson s ratio = 1/3 F C F S m Method (0,0) (1,0) (2,0) (0,0) (1,0) (2,0) Present* Present** Exact Present* Present** Exact Present* Present** Exact Present* Present** Exact Present* Present** Exact Present* Present** Exact %Errorin For Chebyshev points [m = 5 (2) 15] % error = (( m 9 )/ 9 ) 100 For equidistant nodes [m = 5 (2) 25] %error=(( m 21 )/ 21 ) Number of Nodes m %Errorin For Chebyshev points [m = 5 (2) 15] % error = (( m 9)/ 9) 100 For equidistant nodes [m = 5 (2) 15] % error = (( m 9)/ 9) Number of Nodes m Figure 1. Percentage error in frequency parameter vs. number of nodes m for a CC annular plate in fundamental mode; radii ratio = 0.1 taper parameter = 0.3; rigidity ratio p = 5.0; Poisson s ratio v = 0.3. Figure 2. Percentage error in frequency parameter vs. number of nodes m for a CC annular plate in fundamental mode; radii ratio = 0.7 taper parameter = 0.3; rigidity ratio p = 5.0; Poisson s ratio v = 0.3.

5 Influence of Chebyshev Collocation Points on the Convergence of Results in the Analysis of Annular Plate Vibrations 61 Table 3. Comparison of frequency parameters of a polar orthotropic annular plate of linearly varying thickness in fundamental mode when both edges are clamped; taper parameter = 0.5, p = 0.5, Poisson s ratio = 0.3; via spline method (for m = 30 ) (for m = 40) (for m = 60) (for m = 70) Present* Present** Present* Present** Present* Present** Present* Present** Table 4. Comparison of frequency parameters of a polar orthotropic annular plate of linearly varying thickness in fundamental mode when both edges are clamped; taper parameter = 0.3, p = 5.0, Poisson s ratio = 0.3; via collocation method m Present* Present** = = = = lower value) for both types of distribution of nodes. In Figure 2, the percentage error curves have been plotted for the radii ratio = 0.7 (a higher value) for both types of distribution of nodes. 6. Conclusion From the results, it has been observed that a larger number of nodes have been required for convergence of the results for the lower values of radii ratio ( 0.4) than that for the higher values of radii ratio ( 0.4). A faster rate of convergence and appreciably improved results have been obtained in both the approximate methods at Chebyshev points as compared to that obtained at Fourier points (equi-spaced nodes) for the lower values of radii ratio. However, no appreciable influence of the Chebyshev collocation points on the results has been observed for the higher values of the radii ratio in any of the methods. References [1] Conte and Boor, Elementary Numerical Analysis (An Algorithmic Approach), Third Edition, Mc Grow -Hill International, NY, U.S.A. Chap. 6 (1985). [2] Kim, C. S. and Dickinson, S. M., The Flexural Vibration of the Isotropic and Polar Orthotropic Annular and Circular Plates with Elastically Restrained Peripheries, Journal of Sound and Vibration, Vol. 143, pp (1990). [3] Liew, K. M., Lam, K. Y. and Chow, S. T., Free Vibration Analysis of Rectangular Plates Using Orthogonal Plate Function Computers and Structures, Vol. 34, pp (1990). [4] Soni, S. R. and Amba-Rao, C. L., Axisymmetric Vibrations of Annular Plates of Variable Thickness, Journal of Sound and Vibration, Vol. 38, pp (1975).

6 62 R. P. Singh and S. K. Jain [5] Toshihiro, I. and Yamada, G., Analysis of Free Vibration of Annular Plate of Variable Thickness by Use of a Spline Technique Method, Bulletin of JSME, Vol. 23, pp (1980). [6] Lekhnitskii, S. G. Anisotropic Plates, Gordon & Breach Science Publishers, New York, NY, U.S.A. (1968). [7] Gupta, U. S. and Ansari A. H., Asymmetric Vibrations and Elastic Stability of Polar Orthotropic Circular Plates of Linearly Varying Profile, Journal of Sound and Vibration, Vol. 215, pp (1998). [8] Wong, W. O., Yam, L. H., Li, Y. Y., Law, L. Y. and Chan, K. T., Vibration Analysis of Annular Plates Using Mode Subtraction Method, Journal of Sound and Vibration, Vol. 232, pp (2000). Manuscript Received: Feb. 11, 2004 Accepted: Aug. 17, 2004