An insight into the behaviour of oscillators with a periodically piecewise-defined time-varying mass

Free vibrations of a linear, single-degree-of-freedom oscillator with a periodic piecewise-defined time-varying mass are studied. Two different cases of this variation are investigated: first, the mass increases and then decreases linearly in time, i.e. it changes triangularly, and the second, when the mass changes trapezoidically, which, unlike the previous case, includes the period when it remains constant. Exact solutions for motion are obtained analytically directly from the equations of motion, and the criteria of stability are derived and used to plot stability charts for different system parameters. In addition, the transformed equation of motion is utilized in conjunction with harmonic balancing to derive approximate analytical expressions for the boundaries of the first and second instability region.