On measures of nonlinearity effects for uncertain dynamical systems—Application to a vibro-impact system

The transient dynamics of a linear dynamical system with elastic barriers is studied. The system is excited by a deterministic transient force whose Fourier transform is limited to a narrow frequency band. As the system responds it may impact the elastic barrier, therefore, the system behavior is nonlinear. In order to measure the degree of nonlinearity of the system, one looks for the mechanical energy transferred outside the frequency band of excitation as a function of the parameter η defined by ε / a , in which ε is the size of the barrier gap and a is the amplitude of the excitation force. The mechanical energy transferred outside the frequency band of excitation can potentially be a source of excitation for other subsystems. Consequently, quantification of this energy transfer is an important step in developing an understanding of the nonlinear dynamical system behavior. In addition, it is well known that this type of nonlinear dynamical system is very sensitive to uncertainties. For this reason the system is considered to be deterministic and also stochastic in order to take into account random uncertainties. The proposed analysis is applied to a Timoshenko beam having its motion constrained by a symmetric elastic barrier at its free end. The confidence region of the random mechanical energy transferred outside the excitation band is shown as a function of η for several levels of model and data uncertainties. From this, the robustness of the predictions can be analyzed with respect to model and data uncertainties.