In the paper, one class of mechanical systems with nonlinear force fields is considered. It is assumed that the system is under influence of potential, dissipative, and gyroscopic forces. Moreover, we suppose that there is a time-depending coefficient at potential forces. The case where this coefficient is piecewise continuous and piecewise monotonous on any finite-time interval is investigated. Thus, the system can be considered as a non-stationary switched system. We study the situation where this coefficient can be unbounded. Stability conditions for such class of systems are obtained. The Lyapunov direct method is used. Both multiple Lyapunov functions and single Lyapunov functions are constructed. Some examples are presented to illustrate the results.