Modal modification of structural damping applied to increase the stability and convergence of numerical integration

Krzysztof Lipiński

Modal modification of structural damping applied to increase the stability and convergence of numerical integration

Abstract

The presented paper refers to numerical tests done on systems fused of multibody and finite-element parts. The appearance of its multibody part gives rise to significant nonlinear components, i.e., second-order nonlinear differential equations express the dynamics. We usually solve these equations by “step-by-step” integration methods. When using the currently available integration algorithms, we approximate these initial systems of equations by their associated sets of difference equations. The resulting approximated answers enclose superfluous solutions not related to the true one. Calculation that does not magnify these approximation errors is called numerically stable. The initial task is to find robust algorithms, i.e., such that do not produce extraordinarily different results for tiny changes in the initial data. We should underline that numerical instability is a phenomenon due to the employed numerical method but not due to the problem itself. We apply a modal modification of the damping matrix to eliminate these instabilities. The idea of modal modification is not a novelty. We should recall the modal reduction of system degrees of freedom as the example (suppression of its highest damped modes). With the Author's proposition, we do not eliminate these modes but reduce their damping to correlate them with the user-optimal time step. The highest damped modes of the resulting model diminish slower but still under the supposition of vanishing in a period lower than the assumed interval of the observations.