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A time‐integration algorithm for non‐linear dynamic systems of first and second order
Author(s) -
Holl Helmut J.
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900397
Subject(s) - mathematics , differential equation , discretization , numerical partial differential equations , linear system , independent equation , degrees of freedom (physics and chemistry) , distributed parameter system , linear differential equation , differential algebraic equation , mathematical analysis , ordinary differential equation , physics , quantum mechanics
Non‐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.