The approximate solution of second order differential equations using interpolation integration
by
Bruce W Golley
School of Civil Engineering, Australian Defence Force Academy
Coauthors: Xiaomin Chen (Department of Mechanical Engineering, University of Wollongong), Michael P West (Department of Mechanical Engineering, University of Wollongong)

A new method for solving the equations arising in structural dynamics is discussed. For the purpose of this abstract, a SDOF linear system is considered, but extension to MDOF linear systems is straightforward, and the method has potential for the solution of non-linear problems.
We address the solution of

m  ×× x   (t)+c  × x   (t)+k x(t)=p(t)

(1)

where m, c and k are constants, and p(t) is a prescribed force. The displacement x(t) is to be determined subject to the initial conditions that when t=0, x(0) and [x\dot](0) are prescibed.
The MDOF form of equation (1) is normally solved using time stepping, in which an approximate solution during a time step \Deltat is determined. The displacement and velocity at t=\Deltat then become initial values for the next time step.
Introducing a new variable [x\dot](t)=v(t), we can then solve the two first order equations

× x   (t)=v(t)

(2)

×× x   (t)= × v   (t)=-  c m v(t)-  k m  x(t)+  p(t) m

(3)

In the paper, x(t) and v(t) are approximated by polynomials of various orders, and two methods for determining the coefficients are discussed. Using linear polynomial interpolation, the approximate forms of equations (2) and (3) are integrated directly, a procedure described as interpolation integration. In this lowest order case, it is shown that this is identical to collocation of the equations at the midpoint of the time step, or at the Gauss point for one point Gauss quadrature. With higher order polynomial interpolation, collocation at Gauss points is considered directly, so that two Gauss points are considered with quadratic polynomials, three Gauss points for cubic polynomials etc.
The algorithms so developed are unconditionally stable, and truncation errors are O(\Deltat³), O(\Deltat⁵) and O(\Deltat⁷) for linear, quadratic and cubic interpolation respectively. In the paper, the development of the algorithm is described, and stability and accuracy are discussed in detail. Computational aspects are also considered.

Date received: July 23, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-40.