The Impact of a Mass Striking a Beam

Abstract

Abstract The integral equation for the horizontal impact of a mass on a beam was developed by Timoshenko in 1912. He showed that any such impact problem can be solved provided sufficient patience and tenacity are applied in using the tedious numerical method of solution by dividing the time into small increments during which the contact force can be considered constant. This method is too lengthy to be applied in practice, and moreover, a solution is required giving characteristics of the impact as a whole, rather than slowly progressing through its duration with no indication of the future development of the impact. In the first part of this paper a concise approximate solution is developed, making the assumptions that the duration of contact is small in comparison with the period of the fundamental mode of vibration of the beam, and that only the fundamental mode of oscillation of the beam need be considered. Results obtained by this method are checked against those determined by the lengthier numerical method. Good agreement is reached in some examples, but other results show large errors, both in the contact compression force and in the resulting motion. Light was brought to bear on this difficulty by using a solution based on energy and momentum considerations, giving a curve which supplies a rapid determination of the distribution of energy among the various modes of vibration of the beam. This curve gives directly the condition for the vibrational energy developed in the beam, due to the impact, to be confined mainly to the fundamental mode. This is shown by examples to be the required condition for the validity of the previous approximate method. The solution using energy and momentum considerations is developed to give the motions of the beam and mass resulting from the impact (free vibrations in the beam and uniform velocity of the mass) so that the subsequent development of the impact can be ascertained and repeated impacts dealt with. The investigation also gives the deflection curves of the beam produced by the impact; these show a wide deviation from static-deflection curves.