Vibration analysis of high speed and light weight mechanism systems must consider the mechanisms as elastic bodies in order to accurately predict their performance of specified functions. A general model to describe the elastic motion of a mechanism can be properly established with the use of standard spatial finite element methods, which results in a set of second order differential equations. A common assumption in this procedure is that the total motion is comprised of an elastic motion superposed onto the rigid body motion. As a result, the equations of elastic motion have as an important feature time dependent coefficients. If the effects of non-linear elastic deflections and/or non-linear joint characteristics (e.g., clearances) are considered, the equations of motion will be non-linear [1].
The problem of a numerical solution for the formulated differential equations has been the focus of extensive research in the field of dynamics of mechanisms [2]. The dynamic response of a flexible mechanism can be viewed as a transient response and a steady state response. A class of methods for the dynamic analysis is based on the classical numerical integration techniques such as the fourth order Runge-Kutta method and the Newmark method [1,2]. These methods can be made general and flexible for an initial value problem. They are, however, inefficient for the purpose of steady state response.
Another class of methods were developed for the linear systems. In an algorithm presented in reference [3], the time domain is divided into a series of subintervals, in which the coefficient matrices are approximated to be constant. Within each subinterval, a modal analysis approach can be applied, and the transient response can be found step by step. A closed form solution procedure is also devised for the steady state response by applying the boundary conditions for periodic motions and by solving a set of linear algebraic equations [4]. This algorithm is further modified in reference [5] with improved computational efficiency. A similar algorithm for steady state response is also proposed in reference [1] based on a multi-step integration scheme. This class of methods cannot be directly applied to solve the non-linear problems.
This paper presents a new algorithm for solving the general dynamic response of flexible mechanisms. The algorithm is in the same spirit of time domain discretization as of reference [3], but it uses an entirely different scheme. The proposed method is based on a variational approach and employs temporal finite elements. The governing differential equations of motion with time-varying coefficient matrices are first transformed into a weighted integral form in an application of Hamilton's principle. Then the equations of the weak form are discretized in the time domain by applying the finite element in time method. The final set of algebraic equations are obtained, describing system response in terms of a set of temporal nodes of all spatial degrees of freedom of the system.