Hamilton’s law of varying action: Part I: Assumed-time-modes method

Abstract

Without consideration of force equilibrium and differential equations of motion, a general derivation of algebraic equations for dynamic systems is presented. This is achieved by an assumed-time-modes approach in conjunction with a direct application of Hamilton’s law of varying action. By assumed-time-modes, it is implied that the dependent variables of the general dynamics problem can be expanded in terms of admissible basis functions in time. This approach allows explicit a prioriintegration in time of the energy related integrals, leading to the general algebraic equations of motion (AEM) in which the constant expansion coefficients constitute the generalized states of motion. Essential features of the AEM are noted and a unified end result is presented. Identification of system matrices is made in the algebraic form. The general motion is considered to be non-linear, but linear contributions and algebraic reductions to associated linear systems are also shown. A simple demonstration is included to show the basic steps to obtain the AEM.