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Języki publikacji
Abstrakty
The paper deals with the application of the differential quadrature method based on a piecewise polynomial to the nonlinear vibration analysis of beams. The initial–boundary-value problem is solved to study the computational stability of the method. The results are compared with those, obtained by the conventional differential quadrature. The effects of the spline degree, the number of nodes and the distribution of sampling points on the convergence and stability is also presented. The nonlinear free vibration analysis is carried out to verify the accuracy of the method.
Rocznik
Tom
Strony
1--13
Opis fizyczny
Bibliogr. 26 poz., tab., wykr.
Twórcy
autor
- Cracow University of Technology. Institute of Computing Science, Al. Jana Pawła II 37, 31-864 Cracow
Bibliografia
- [1] R. Bellman, J. Casti. Differential quadrature and long term integration. Journal of Mathematical Analysis and Application, 34: 235-238, 1971.
- [2] R. Bellman, B.G Kashef, J. Casti. Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics, 10: 40-52, 1972.
- [3] C.W. Bert, M. Malik. Differential quadrature method in computational mechanics. Applied Mechanics Review, 49: 1-28, 1996.
- [4] C.W. Bert, X. Wang, A.G. Striz. Differential quadrature for static and free vibration analysis of anisotropic plates. International Journal of Solid Structures, 30: 1737-1744, 1993.
- [5] G.R. Bhashyam, G. Prathap. Galerkin finite element method for non-linear beam vibrations. Journal of Sound and Vibration, 72: 191-203, 1980.
- [6] W. Chen, T.X. Zhong. On the DQ analysis of geometrically non-linear vibration of immovably simply-supported beams. Journal of Sound and Vibration, 206: 745-748, 1997.
- [7] E. Fehlberg. Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsproblcme. Computing, 6: 61-71, 1970.
- [8] Y. Feng, C.W. Bert. Application of the Quadrature Method to Flexural Vibration Analysis of a Geometrically Nonlinear Beam. Nonlinear Dynamics, 3: 13-18, 1992.
- [9] T.C. Fung. Solving initial value problems by differential quadrature method — Part 1: first-order equations. International Journal for Numerical Methods in Engineering, 50: 1411-1427, 2001.
- [10] T.C. Fung. Solving initial value problems by differential quadrature method — Part 2: second- and higher-order equations. International Journal for Numerical Methods in Engineering, 50: 1429-1454, 2001.
- [11] Q. Guo, H. Zhong. Non-linear vibration analysis of beams by a spline-based differential quadrature method. Journal of Sound and Vibration, 269: 413-420, 2004.
- [12] A. Krowiak. Symbolic computing in spline-based differential quadrature method. Communications in Numerical Methods in Engineering, 22: 1097-1107, 2006.
- [13] A. Krowiak. The convergence and stability of the spline-based differential quadrature method applied to the vibration analysis of rectangular plates with free corners. Proceedings of the 22nd Symposium: Vibrations in Physical Systems, 197-202. Poznan-Będlewo, 2006.
- [14] R. Lewandowski. Free vibration of structures with cubic non-linearity-remarks on amplitude equation and Rayleigh quotient. Computer Methods in Applied Mechanics and Engineering, 192: 1681-1709, 2003.
- [15] M. Malik, F. Civan. A comparative study of differential quadrature and cubature method vis-a-vis some conventional techniques in context of convection-diffusion-reaction problems. Chemical Engineering Science, 50: 531-547, 1994.
- [16] C. Mei. Finite element displacement method for large amplitude free oscillations of beams and plates. Computers and Structures, 3: 163-174, 1972.
- [17] J.R. Quan, C.T. Chang. New insight in solving distributed system equations by the quadrature methods — I. Computer and Chemical Engineering, 13: 779-788, 1989.
- [18] B.S. Sarma, T.K. Varadan. Lagrange-type formulation for finite element analysis of non-linear beam vibrations. Journal of Sound and Vibration, 86: 61-70, 1983.
- [19] M. Sathyamoorthy. Nonlinear analysis of beams, Part I: A survey of recent advances. Shock and Vibration Digest, 14: 19-35, Aug. 1982.
- [20] M. Sathyamoorthy. Nonlinear analysis of beams, Part II: Finite element methods. Shock and Vibration Digest, 14: 7-18, Sept. 1982.
- [21] C. Shu. Differential quadrature and its application in engineering. Springer-Verlag, London, 2000.
- [22] C. Shu, H. Du. Implementation of clamped and simply supported boundary conditions in GDQ free vibration analysis of beams and plates. International Journal of Solids and Structures, 34: 819-835, 1997.
- [23] C. Shu, B.E. Richards. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 15: 791-798, 1992.
- [24] G. Singh, A.K. Sharma, G.V. Rao. Large-amplitude free vibration of beams — a discussion on various formulations and assumptions. Journal of Sound and Vibration, 142: 77-85, 1990.
- [25] X. Wang, H. Gu. Static analysis of frame structures by the differential quadrature element method. International Journal for Numerical Methods in Engineering, 40: 759-772, 1997.
- [26] H. Zhong. Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates. Applied Mathematical Modelling, 28: 353-366, 2004.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BPB1-0035-0018