Free Vibration States of an Oscillator With a Linear Time-Varying Mass

Due to its application in various dynamic systems, the free vibration behavior of a single-degree-of-freedom (SDoF) oscillator system with nonperiodic time-varying parameters has been the subject of recent interest. When dealing with this subject, past investigators have often isolated two special states 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 process given that each equation is different and equally complicated to solve. This paper presents a single equation that does not discriminate between the two special free vibration cases. The equation allows the description of all possible free vibration states and the two commonly investigated cases become its special cases. The procedure is illustrated by analyzing the free vibration behavior of a SDoF oscillator system subjected to a linear time-varying mass, constant stiffness, and damping. The solution established here highlights the critical role played by negative damping on the instability of the system. Based on this role, a minimum damping criterion that ensures stability is defined in terms of the natural frequency and the rate of change of mass of the principal 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 with other generic time stepping techniques.