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In earlier studies of multiple-degree-of-freedom (MDOF) systems, an impact had always been assumed to affect the velocity of the lumped mass at the point of impact. Experiments show that the velocities of all points in a continuous system are influenced by the impact of the damper. In this study, the stead7-state behavior of an impact-damped beam is modeled by applying conservation of momentum between .the tmpact damper and the modal masses of the beam. Friction and gravity are considered for the case of honzontal and vertical motions, respectively. The impact damper exhibits a wide range of impact patterns and its optimum behavior occurs at two impacts per cycle. Three types of impact are identified for the case of two impacts per cycle, one of which expands the limit cycle of the system and therefore is detrimental.. Within limi~s, the effectiveness of the impact damper improves with an increase in its mass, with a decrease In the coeffictent of restitution, and when it is at a point of large amplitude. An optimum gap exists for each of the first three modes of vibrations. The damper expands the limit cycle of the system at higher excitation amplitudes. The phenomena of bouncedown and stuck impacts are shown to exist in continuous systems. as previously suggested (Masri 1973b). Therefore, there is a need to understand the particle's motion, at varying parame­ ters, as it travels between the container's boundaries to explore the potential of the impact damper fully. Studies mentioned above were very crucial in the develop­ ment of the theory presented in this paper, though limited in their scope. This study employs the consistent mass matrix to develop a finite-element model of a continuous beam, thus considering rotational and translational inertia at the nodes. The response of the system between impacts is obtained through the mode-superposition method. It is assumed, as had been suggested by Masri (1970), that the impact occurs during an infinitesimal period and that the displacement of the system does not change immediately before and after the impact. Pre­ vious experimental work shows that the duration of impact is indeed infinitesimal (Akl and Butt 1994). Momentum and res­ titution equations in terms of the principal coordinates are used to model the behavior of the impacting mass and the oscillat­ ing system. By using principal coordinates, we can determine the velocity of the system at all the nodes simultaneously after each impact without oversimplifying the model. The exact time of impact is determined by employing an optimization algorithm. Eventually, frequency response functions are ob­ tained by transforming the time domain data to the frequency domain by performing fast Fourier transforms (Weaver 1989).

Statistical analysis of the phenomenon of impact damping is performed on 60 steady-state vibration tests of a flexible beam to relate the system's modal damping to system parameters. Compared with other measures of effectiveness of the impact damper, modal damping ratios are useful in predicting the response of continuous systems under arbitrary forcing functions. Contour plots show that the modal damping ratio is nonlinearly dependent on the damper's mass and gap. However, these are not the only parameters that control the effectiveness of the impact damper. Multiple nonlinear regression analysis of the data is used to test the dependence of the system's modal damping on a number of parameters. It is hypothesized that the system's parameters are the mass and gap of the impact damper, the excitation frequency, the peak of the frequency response function, and the modal amplitude at the location of the impact damper. Multiple nonlinear regression analysis shows the validity of the hypothesis and the ability of the derived formula to predict the modal damping ratio of the beam.