Analytical solution of the longitudinal wave propagation due to the single impact

In the present work, the rod response due to the impact of a ball is studied in order to analyze the longitudinal wave propagation during the impact and evaluate the velocity of points over the rod length at the end of impact. An analytical solution is given for the governing equation of rod vibration by assuming the impact force as an equivalent sinusoidal function of time. The impact force is defined by a Fourier series to be zero at the rod length except at the impacted end. An analysis of wave propagation will give information about the rod-ball interaction. The evaluated velocity profile at the end of impact can be used as the initial condition for after-impact solution of the rod vibration. It will modify the common simple assumption of constant initial velocity for impacted end.