Dynamics of unsymmetric piecewise-linear/non-linear systems using finite elements in time

Abstract

The dynamic response and stability of a single-degree-of-freedom system with unsymmetric piecewise-linear/non-linear stiffness are analyzed using the finite element method in the time domain. Based on a Hamilton's weak principle, this method provides a simple and efficient approach for predicting all possible fundamental and sub-periodic responses. The stability of the steady state response is determined by using Floquet's theory without any special effort for calculating transition matrices. This method is applied to a number of examples, demonstrating its effectiveness even for a strongly non-linear problem involving both clearance and continuous stiffness non-linearities. Close agreement is found between available published findings and the predictions of the finite element in time approach, which appears to be an efficient and reliable alternative technique for non-linear dynamic response and stability analysis of periodic systems.

Unbalance Responses of Rotor/Stator Systems with Nonlinear Bearings by the Time Finite Element Method

“…The principle of this method is to interpolate the displacement of all spatial degrees of freedom between given instants t i and t i+1 by polynomials (Wang, 1995(Wang, , 1997. In this article, the Lagrange polynomials of order k are used, however all kinds of polynomials may be used (Park, 1996) …”

Vibrations of an Axial Bar Experiencing Periodic Unilateral Contact Using the Wavelet Balance Method

“…Taking the inner product of Eq. 7with a weighting function stored column-wise in v results in the strong integral form of the equation: find u 2 (8) and the superscript T denotes a transpose. This strong form of the equation is not necessarily the best framework for obtaining a solution [14]; for this example involving a vector ordinary differential equation of order 2, the solution must be at least H 2 , limiting the permissible basis of trial functions.…”

“…This method involves approximating the solution using a set of time-dependent basis functions, called trial functions, and enforcing the respective residual error to be orthogonal to an independent set of weighting functions [6,7]. Unlike the shooting method which can become numerically sensitive to possible jumps in the velocity field, weighted residual techniques directly enforce the periodicity conditions while the remaining unilateral contact constraints and governing local equations of motion are satisfied in a weak integral sense [8]. It is worth noting that the well-known Harmonic Balance Method (HBM) is a weighted residual approach where both the trial and weighting functions are identical Fourier series.…”

Forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method

“…The orthogonality is enforced using an inner product in the sense of a scalar product adequately defined on a functional Hilbert space. Unlike the shooting method, which can become numerically sensitive to possible jumps in the velocity field, weighted residual techniques directly enforce the periodicity conditions, while the remaining possibly regularized unilateral contact constraints and governing local equations of motion are satisfied in a weak integral sense . It is worth noting that the well‐known harmonic balance method (HBM) is a weighted residual approach where both the trial and weighting functions are Fourier series.…”