Dynamics of a two-degree-of-freedom cantilever beam with impacts

Impacts in mechanical systems are an object of interest for many scientists in the world. In this paper, we present detailed investigations of the dynamical behavior of the system consisting of a massless cantilever beam with two concentrated masses. The maximum displacement of one of the masses is limited to the threshold value by a rigid stop, which gives rise to non-linearity in the system. Impacts between the mass and the basis are described by a coefficient of restitution. The conducted calculations show a good agreement of the results obtained with two qualitatively different methods of behavior analysis of the system under consideration, namely: the Peterka’s method and the method of numerical integration of motion equations. It has been observed that stable solutions describing the motion with impacts of a two-degree-of freedom mechanical system exist in significantly large regions of the parameters that describe this system. The location and size of periodic motion regions depend strongly on mutual relations between the excitation force frequency and the system eigenvalues. In order to obtain stable and periodic motion with impacts, the system parameters should be selected in such a way as to make the excitation force frequency an even multiple of the fundamental eigenvalue and to make the higher eigenvalue an even multiple of the excitation force frequency. These two conditions can be applied in designing mechanical systems with impacts. This information is even of more significance since it has turned out that the system exhibits some adaptability, owing to which stable solutions exist even if the above-mentioned conditions are satisfied only approximately.