Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms

Abstract

This paper deals with the free nonlinear dynamical system [xdot](t)=A x(t)+h(t, x(t)), t≥t ₀, x(t ₀)=x ₀, where A is an n×n-matrix and h a nonlinear vector function with h(t, u)=o(∥u∥). As the first novel point, a lower bound for the asymptotic behaviour on the solution x(t) is derived. Two methods are applied to determine the optimal two-sided bounds, where one of the methods is the differential calculus of norms. In this context, the second novel point enters; it consists of a new strategy to significantly reduce the computation time for the determination of the optimal constants in the two-sided bounds. The obtained results are especially of interest in engineering and cannot be obtained by the methods used so far.

Keywords:

• nonlinear dynamical system with vibration behaviour
• multi-mass vibration model
• differential calculus of norms
• two-sided bounds
• reduction of computation time

2000 AMS Subject Classification :

• 65L05
• 65L20
• 65L70