Engineering Transactions, 31, 2, pp. 213-240, 1983

Comparison of Some Numerical Integration Methods for The Equations of Motion of Systems with a Finite Number of Degrees of Freedom

Z. Kacprzyk
Technical University of Warsaw, Warszawa

T. Lewiński
Technical University of Warsaw, Warszawa

The subject of this paper are the numerical integration methods of the differential equations of vibration (in matrix form) of a physical system with a finite number of degrees of freedom. Comparison was made between finite difference method. Newmark's method and Space-Time Finite Element method. The compared methods have been brought to a unified form, disclosing far-reaching similarities concerning the calculation methods of displacement and velocity. Analogies have been found between the recurrence formulae relating the displacement and velocity vectors in different methods. It was proved that these analogies allow for a comparative analysis of convergence of the three methods and confine it to the analysis of approximation of initial conditions and of external loads. Thirteen methods of approximation of initial conditions are discussed. New formulation is given of the Space: Time Finite Elements. The comparative analysis of convergence is limited to free oscillation of a system with a single degree of freedom.

Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).


A. A. SAMARSKIJ, Introduction to the theory of difference schemes [in Russian], Nauka, Moskwa 1971.

N. M. NEWMARK, A method of computation for structural dynamics, J. Eng. Mech. Div., ASCE, 85, EM 3, pp. 67-94, 1959.

Z. KĄCZKOWSKI, GeneraJ formulation of stiffness matrix for space-time finite elements, Arch. Inż. Ląd., 25, 3, 351-357, 1979.

R. D. RICHTMYER, K. W. MORTON, Difference methods for initial value problems, New York, London, Sydney 1967.

K. J. BATHE, E. WILSON, Numerical methods in finite element analysis, Prentice-Hall, New Jersey 1976.

O. C. ZIENKIEWICZ, Finite element method, Mc Graw Hill, 1977.

O. C. ZIENKIEWICZ, A new look at the Newmark, Haubolt and other time stepping formulas. A weighted residual approach, Earth. Eng. and Struct. Dyn., 5, 413-418, 1977.

J. LANGER, Parasitic damping in computer-aided solutions of equations of motion [in Polish], Arch. Inż. Ląd., 25, 3, 351-369, 1979.

I. KOŹNIEWSKA, Recurrence equations [in Polish], PWN, Warszawa 1972.

T. LEWIŃSKI, Stability analysis of a difference scheme for the vibration equation with a finite number of degrees of freedom, Applicationes Mathematicae, 18, 3, 1982.

Z. KĄCZKOWSKI, J. LANGER, Synthesis of the space-time finite element method, Arch. Inż. Ląd., 26, 1, 11-17, 1980.

Z. KACPRZYK, On certain modifications of the time-space FEM, Doctoral thesis [in Polish], Warszawa 1982.