SYMMETRIZATION OF SOME LINEAR CONSERVATIVE. NONSELF-ADJOINT SYSTEMS

B. L. LY

AECL, Mississauga, Ontario, Canada

Abstract

We derive here equivalent self-adjoint systems for conservative systems of the second kind. Ex­istence of the symmetrized systems confirms that certain conservative systems of the second kind behave as a true conservative system. In this way, study of stability can be carried out on the symmet­rized system. In general, it is easier to study a self-adjoint system than a nonself-adjoint system. For the conservative system of the second kind, including the Pfluger column, we also presented a lower bound self-adjoint system. For a linear conservative gyroscopic system, we gave a zero parameter sufficient condition for instability and one for stability. The criteria depend only on the characteristics of the system. For a simple 2-DOF system, the present criteria yield the exact solutions.

1. Introduction

Two types of conservative nonself-adjoint systems are studied. One is the so-called conservative system of the second kind. The other one is a gyroscopic system.

Certain nonself-adjoint systems have only divergent type of instability, despite the presence of a polygenic force. Pfluger’s column and Greenhill’s shaft are two such systems. Leipholz (1974a, 1974b) called a true divergent nonself-adjoint system which has dynamic properties very similar to those of a self-adjoint system a conservative system of the second kind. He showed that such a system is self-adjoint with respect to an assigned self-adjoint operator, hence it is self-adjoint in a generalized sense. For such a system there exists a Lyapunov for predicting stability (Walker, 1972; Leipholz, 1974a) and a generalized Rayleigh quotient for determining the buckling load (Leipholz, 1974a). Inman and Olsen (1988) included velocity dependent forces in conservative systems of the second kind and proved the generalized self-adjointness and the existence of the eigenfunctions. In this way, the solution can be obtained by a modal analysis.

For certain asymmetric discrete systems, Inman (1983) demonstrated that there exists a similarity transformation that transforms the asymmetric system into an equivalent symmetric one, one that has the same eigenvalues.

Here we show for certain conservative systems of the second kind, an equivalent self-adjoint system can be derived. In this way, a conservative system of the second kind is symmetrized, sim­ilar to the symmetrization of an asymmetric discrete system. Existence of the symmetrized system confirms a conservative system of the second kind behaves like a self-adjoint system.

A gyroscopic system is nonself-adjoint because of the presence of a skew-symmetric operator. Walker (1991) presented a Lyapunov containing undetermined parameters for the study of the sta­bility. By determining the parameters by trial and error, he was able to optimize the stability region. Because a zero-parameter stability criterion is easier to use, we derive a zero-parameter sufficient condition for instability and one for stability, all via symmetrization. For the example considered by Walker, the present method also yields the exact solutions.

563

M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 563-570.

© 2006 Springer. Printed in the Netherlands.