The dynamics of a multidegree impact vibrator subject to harmonic loading is investigated. The system is represented by a lumped mass model which hits and rebounds from a rigid wall during vibration. The periodic solution to the equations of motion with N forcing cycles and P impacts is formulated. The variational equations and the resulting transition matrix for investigating local stability of the periodic solutions are derived. A two-degree-of-freedom example is analysed, and a variety of motion types are found. Chaotic windows are present between regions of periodic response, and at these boundaries N-P motions are prevalent. Low velocity impacts are evident at exciting frequencies away from the natural frequencies. Two basins of attraction are computed, and the sensitivity to initial conditions is noted. The quality of the N-P motion is discussed from an engineering application perspective.

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Real-life structures often possess piecewise stiffness because of clearances or interference between subassemblies. Such an aspect can alter a system's fundamental free vibration response and leads to complex mode interaction. The free vibration behaviour of an L-shaped beam with a limit stop is analyzed by using the frequency response function and the incremental harmonic balance method. The presence of multiple internal resonances, which involve interactions among the first five modes and are extremely complex, have been discovered by including higher harmonics in the analysis. The results show that mode interaction may occur if the higher harmonics of a vibration mode are close to the natural frequency of a higher mode. The conditions for the existence of internal resonance are explored, and it is shown that a prerequisite is the presence of bifurcation points in the form of intersecting backbone curves. A method to compute such intersections by using only one harmonic in the free vibration solution is proposed.

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The problem of robust stability in the state space model of linear time-varying systems with time- varying parameter uncertainties is considered. The Lyapunov approach is employed to obtain bounds on the uncertain time-varying parameters to guarantee the stability of the system. The robust stability bounds obtained are not necessarily symmetric with respect to the origin in the parameter space and can significantly reduce the conservatism in the previous results. Sufficient conditions for the stability of the steady-state responses of nonlinear systems (especially of piecewise linear systems) subjected to periodic excitations are proposed by using the results derived. Two numerical examples are used to demonstrate the accuracy of the formulation.