Free vibration of SDOF systems with arbitrary time-varying coefficients

A new analytical approach for determining the exact solutions for free vibration of single-degree-of-freedom (SDOF) systems with non-periodically time-varying coefficients (mass and stiffness) is presented herein. In this paper, the function for describing the variation of mass of a SDOF system with time is an arbitrary one, and the variation of the stiffness is expressed as a functional relation with the mass function and vice versa. Using appropriate functional transform, the governing differential equation for the title problem is reduced to a Bessel’s equation or other analytically solvable equations. Exact solutions for free vibration of SDOF systems with non-periodically varying coefficients are obtained for six important cases. In order to simplify the free vibration analysis of a SDOF system with multi-step time-varying coefficients, the fundamental solutions that satisfy the normalization conditions are constructed based on the exact solutions derived. It is more convenient to determine the displacement response of the SDOF system by using the fundamental solutions and a recurrence formula developed in this paper. Numerical example shows that the proposed procedure is a simple, efficient and exact method.