Free vibration stability of SDoF systems undergoing variable mass-induced damping

Abstract

Due to its application in various dynamic systems, the free vibration case of an SDoF oscillator system with non-periodic time varying parameters has been the subject of recent interest. When dealing with this subject, past investigators have often isolated two special cases of free vibration from the governing equation of motion. Naturally, this leads to the examination of solutions to two separate equations, making it a tedious and tiresome process given that each equation is different and equally complicated to solve. In this paper, a single equation that does not discriminate between the two special free vibration cases, during the determination of the solution, is established from the existing equation of motion. Most importantly, the equation allows the description of all possible free vibration cases, and the two commonly investigated cases become special cases of this general equation. To illustrate the procedure, a general solution for the free vibration case of an SDoF oscillator system subjected to a linear time varying mass, constant stiffness and damping is presented in terms of Bessel functions. The general solution enables one to investigate a wide range of free vibration scenarios and draw some conclusions on the stability of motion. Two unstable states are identified and based on these states, a minimum damping criteria that ensures stability, is defined in terms of the natural frequency and the rate of change of mass of the system. Finally, the solution is used further to develop a numerical procedure for analyzing systems with arbitrary time varying mass. A numerical example demonstrates that the proposed method converges to the exact solution more rapidly compared to other generic time stepping techniques. © 2009 Society for Experimental Mechanics Inc.