Asymptotic method and wave theory of motion in studying the effect of periodic impulse forces on systems characterized by longitudinal motion velocity

A methodology for researching dynamic processes of one-dimensional systems with distributed parameters that are characterized by longitudinal component of motion velocity and are under the effect of periodic impulse forces has been developed.   The boundary problem for the generalized non-linear differential Klein–Gordon equation is the mathematical model of dynamics of the systems under study in Euler variables.  Its specific feature is that the unexcited analogue does not allow applying the known classical Fourier and D'Alembert methods for building a solution.  Non-regularity of the right part for the excited non-linear analogue is another problem.

This study shows that the dynamic process of the respective unexcited motion can be treated as overlapping of the direct and reflected waves of different lengths but equal frequencies.  Analytical dependencies have been obtained for describing the aforesaid parameters of the waves.  They show that the dynamic process in such mechanical systems depends not only on their main physical and mechanical parameters and boundary conditions, but also on the longitudinal motion velocity (relative momentum).  As relative momentum increases, the frequency of the process decreases.

To describe excited motion, we use the principle of single frequency of oscillations in non-linear systems with concentrated masses and distributed parameters as well as regularization of periodic impulse excitations.  The main idea of asymptotic integration of systems with small non-linearity into the class of dynamic systems under study has been generalized.  A standard equation for the resonance and non-resonance cases has been obtained.  It has been established that for the first approximation, in the non-resonance case, impulse excitation affects only the partial change of the form of oscillations.  Resonance processes are possible at a specific relation between the impulse excitation period, the motion velocity of the medium, and physical-mechanical features of the body.  The amplitude of transition through resonance becomes higher when impulse actions are applied closer to the middle of the body.  As the longitudinal motion velocity increases, it initially increases and then decreases.