Dry friction between two components in rubbing contact may often be the cause of vibration. This type of behaviour was investigated by means of a model apparatus comprising a disc and a cantilever. A mathematical theory for the oscillation of the system was developed: Lagrange's equations were used and the deflections were expressed in normal co-ordinates. Numerical analysis showed that variation of the coefficient of friction with relative velocity was insufficient to cause the vibration. The instability was due to the manner in which the motions of the components were coupled. The amplitude would grow exponentially until limited by a breakdown of the continuity of contact between the components. A refinement of the theory took account of the flexibility and dissipation of the friction material with which the cantilever was tipped. The experimental work verified the existence of stable and unstable regions as predicted by the theory. The measured amplitudes were so great that continuity of contact could not be maintained, as was to be expected if the suggested mechanism of amplitude limitation was actual. A subsidiary investigation was made of the effect of tightness of the cantilever clamp. The investigation has shown that it should be possible to avoid unwanted vibration in any system merely by careful choice of dimensions in the design.

Part 1: Simple struts have been impacted between travelling and stationary masses and the collapse found to consist of two distinct phases. The pre-failure phase, being primarily an elastic region, has been analysed by stability criteria and the post-failure phase by consideration of the plastic collapse of the strut. Associated with the pre-failure phase is a large transient deceleration which is found, among other parameters, to be a function of the strut shape. Part 2: A series of sheet-metal structures have been impacted and, like struts, the collapse found to consist of pre- and post-failure phases of separate identities. The pre-failure phase is again associated with the shape of the structure and may give rise to a large transient deceleration. This analysis has led to the consideration of more complex elements, such as might be found in an automobile frontal structure, and finally to a complete automobile analogue. Conclusions are drawn about the structural parameters required to bring about specific deceleration characteristics in a vehicle crash.

This paper, which examines the effect of gravity on the performance of an impact damper, extends a previous paper (I)$ concerned with the horizontal system. T h e analysis is in two parts: Part 1 which concerns itself with the calculation of all the theoretically possible steady-state motions with two impacts of the free mass, one at each end of its container, per cycle of the main mass; and Part 2 s which examines the dynamic stability and kinematic viability of the various steady-state solutions obtained in Part 1.The impact damper consists of a small mass free to move within a rigid container attached to the main vibrating system. Vibration suppression is achieved through energy dissipation during impacts and through cancellation of the effect of the external exciting force by the cyclical reaction of the free mass on the main system.T h e analysis shows that, in general, several different steady-state motions are theoretically possible for a given exciting frequency and that the effect of gravity on the motion of the system is to reduce the frequency range of two-impact-per-cycle motion. It is also shown that under the practically important condition of resonance the motion consists of two impacts of the free mass per cycle of the main mass. Outside the resonant condition, where the motion of the main mass is weak, the free mass may impact more or less than twice per cycle. This condition is not studied. I N T R O D U C T I O NTHE impact damper consists of a small loose mass which rattles inside a container attached to the vibrating system. Repetitive impacts of the free mass against the ends of the container have the effect of attenuating the motion of the main system. This is achieved partly through the loss of kinetic energy on impact and partly through the phasing and magnitude of the fundamental Fourier component of the cyclical impulsive forces with respect to the external exciting force. The device is, in effect, a vibration absorber in the sense that an auxiliary system is fixed to the main system in order to suppress a resonance but differs from the conventional tuned vibration absorber because the free mass has an indeterminate natural frequency of oscillation within the container. The device is most effective when the dominant Fourier component of the impulsive forces has the same frequency as the external exciting force, that is, when there is one impact at each end of the container during each cycle of the main system. It is this condition which is treated in the present analysis.The M S . of this paper was received at the Institution on 13th November 1969 and accepted for publication on 5th M a y 1970. 44