This paper studies the transient dynamics of a linear dynamical system with elastic barriers excited by a deterministic transient force whose Fourier Transform has a bounded frequency narrow band. The system is then non- linear. In order to measure the degree of non-linearity of the system, one looks for the mechanical energy transferred outside the frequency band of excitation. In addition, it is well known that this type of non-linear dynamical system is very sensitive to uncertainties. For this reason one studies the system as being deterministic, and also stochastic in order to take into account random uncertainties. The proposed analysis is then applied to a Timoshenko beam having its motion constrained by a symmetric elastic barrier at its free end. In particular, one shows the confidence region of the random mechanical energy transferred outside the excitation band.