Polynomial Shape Functions and Numerical Methods for Flexible Multibody Dynamics*

The use of Taylor polynomials as shape functions in a Rayleigh–Ritz discretization of flexible beams in dynamic multibody systems has been previously investigated [[1] Valembois, R. E., Fisette, P. and Samin, J. C. 1997. Comparison of Various Techniques for Modelling Flexible Beams in Multibody Dynamics. Nonlinear Dynam., 12: 367–397. [Crossref], [Web of Science ®] , [Google Scholar] [2] Saad, M., Piedboeuf, J.-C. and Akhrif, O. in press. Comparison of Different Shape Functions in Assumed-Mode Models of a Flexible Slewing Beam. Mechanism Mach. Theory, [Google Scholar]]. This approach is relatively simple, but it was found to cause some ill-conditioning problems in the numerical solution of the system equations. In this paper, two solutions to these problems are presented. First, better-behaved system equations can be obtained by using orthogonal Chebyshev or Legendre polynomials in place of Taylor polynomials. Secondly, a judicious choice of numerical solver can reduce or eliminate the ill-conditioning problems. Two examples are used to demonstrate these findings.