A time‐integration algorithm for non‐linear dynamic systems of first and second order

Helmut J Holl

doi:10.1002/pamm.201900397

Abstract

AbstractNon‐linear dynamic systems are defined by mechanical models involving differential equations of second order where also first order differential equations are present frequently. If some degrees of freedom have no corresponding mass, if the stiffness parameters vanish in a part of the equations or if a controller is implemented in the system, then differential equations of first order result. For linear systems of first and second order a variety of solutions is presented in [1] and various numerical procedures for solving the differential equations in [2]. A semi‐analytical method is presented which is exact for the linear dynamic and decoupled systems of first and second order. A modal transformation of the equations is necessary after a suitable partitioning of the system equations. After a discretisation in the time‐domain the relevant equations for a suitable and effective time‐integration algorithm are defined taking the non‐linearity into account. The resulting procedure is derived and the formulation is analogous to the BEM‐formulation in time, described in [3] for a system of second order. The method is extended to the coupled non‐linear differential equations of first order and is applied to a system with two degrees of freedom.

Highlights

Non-linear dynamic systems are defined by mechanical models involving differential equations of second order where first order differential equations are present frequently
C 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co
Some examples are the temperature in a body with heat exchange, the balance of mass with inflow and outflow and control of dynamic systems

Summary

Introduction - Dynamic system of first and second order

Equations for dynamic mechatronic systems frequently include first and second order differential equations. Some examples are the temperature in a body with heat exchange, the balance of mass with inflow and outflow and control of dynamic systems. The solution is computed after a transformation into a system of differential equations of first order. The non-linear equations of motion M X +DX +KX+FN = F(t) with the mass matrix M, the damping matrix D, the stiffness matrix K, the vector of non-linear reaction force FN , the vector of excitation force F(t) and the vector of degrees of freedom X(t) with n components. The equations of motion can be partitioned and with the assumptions that M12 = 012, M21 = and M22 = after additional manipulations it follows

Semi-analytic time-integration algorithm - Computational procedure

Section 5: Nonlinear oscillations

Mechanical model and computational results