A cantilever beam chattering against a stop

Contact chatter or bouncing of a cantilever beam with the free end pressed against a stop was studied. The problem was treated according to the Bernoulli-Euler beam theory and the resulting integral equation was solved by the small time increment technique. Deflections, contact force and chatter were calculated. The number of modes included in the solution, which depends on the fixed contact stiffness, was found to have a great effect on the fine details of chatter, but to have little effect on the overall pattern of chatter or the deflection. Deflections were measured photographically by using a multiflash strobe light and the chatter was measured with an oscilloscope. The overall chatter pattern and the deflection agreed with the calculated values remarkably well. Finally, an approximate method was developed by using kinetic energies based on the beam theory and contact open or closure times based on an equivalent mass-spring system. Kinetic energies for vibrational modes other than the fundamental mode for a propped cantilever beam (clamped-supported) were assumed to be damped out rapidly. Deflections, and the chatter pattern, based on this approximate method agreed well with the observed values. The time for the chatter to stop was also calculated, and it agreed reasonably well with the measured time.