Integral formulas for non-self-adjoint distributed dynamic systems

Abstract

Integral formulas for analytical prediction of the dynamic response of a class of non-self-adjo int distributed systems are studied. Combined non-self-adjoint effects of damping, gyroscopic, and circulatory forces are examined. The response of the distributed system subject to arbitrary external, boundary, and initial disturbances is obtained in a closed-form Green's function integral. The Green's function is expanded in an eigenfunction series, without assuming completeness of system eigenfunctions. In addition, a generalized reciprocal theorem is derived. ISTRIBUTED dynamic systems whose parameters are dependent on the spatial domain are common in many branches of science and technology. The mathematical description of these systems usually leads to boundary-init ial value problems associated with partial differential equations. The development of modern technologies, including optimal design and active control of flexible structures, requires accurate prediction of the dynamic response of distributed systems. This work investigates integral formulas for analytical prediction of the dynamic response of a class of linear distributed systems that have combined non-self-adjoint effects of damping, gyroscopic, and circulatory forces, and are under arbitrary external, boundary, and initial disturbances. Mainly, three issues are addressed: 1) Green's function formulation for transient response prediction, 2) modal expansion of system Green's function, and 3) a generalized reciprocal principle. These issues are important to structural dynamics and structural control and lay a foundation for the development of analytical and numerical solution methods. The concept of Green's functions is by no means new. It was first introduced by G. Green as early as 1828 and since has been applied to various problems in mathematical physics. Several excellent monographs on the subject have been published.14 Instead of its many advantages, the Green's function method has been for formal and