| Three types of impact systems are modeled and analyzed (a) a sinusoidally driven damped simple harmonic oscillator (SHO) constrained between two stationary walls placed symmetrically about its equilibrium position, (b) a nonlinear pendulum impacting a sinusoidally driven wall and (c) 1-D chain of globally driven impact coupled SHOs constrained by a wall at each end. Dynamical behavior of all model systems are investigated for the underdamped case.;Based on qualitative analysis, we understand the physical origin of hysteresis reported for the case of low amplitude values in terms of the similar hysteretic behavior of a driven nonlinear oscillator with a 'stiff spring'. Using analytical results, we determine the dependence of hysteretic region in parameter space on other system parameters. In particular, we find, for fixed value of coefficient of restitution (K) and viscous damping (beta), the hysteretic region, in the amplitude-frequency ( A -- o) space, increases with increasing A and with increasing o above resonant frequency. The effect of increasing damping (increasing beta or decreasing K) decreases the area of hysteretic region in A -- o space.;We rigorously find that the grazing collision at low drive amplitude is accompanied by the mutual annihilation of a stable and an unstable orbit of the system. From the examples studied at higher drive amplitude, we conjecture that a grazing impact of a stable orbit is always accompanied by a simultaneous grazing of an unstable. Therefore, the saddle-node bifurcation and the grazing bifurcation are similar in the sense that a stable-unstable pair of periodic orbits collide to annihilate each other.;We use the method first used by Nordmark (J. Sound Vib., 145, 279 (1991)) to derive an analytical expression for the Jacobian of the constant phase Poincare map for all system studied. The results were used to calculate the Lyapunov spectra of the systems as a function of the system parameters.;We successfully controlled and tracked low dimensional impact model systems. The system was automatically tracked while maintaining control into regions of parameter values where the eigenvalue in the unstable direction was as high as 45. This is important from the engineering application point of view. For instance, the control and tracking method may be used to stabilize the system on a selected trajectory with a desirable sequence of impacts and avoid the irregular nature of the chaotic motion.;Following the coupled map lattice (CML) approach, in the study of spatially extended systems, we analyze the behavior of 1-D chain of globally driven impact coupled SHO systems constrained by a stationary wall at the two ends. Using different ways of data representation (space-time-amplitude plot, shock wave plot), we study the phenomenon of multistability, periodic behavior, intermittent bursts of periodic and aperiodic response, chaotic motion and hyperchaotic response. Extending the local stability analysis performed for single impact system to the case of many such impact coupled systems, we numerically calculate the Lyapunov spectra of N coupled impact systems. Based on above calculations we find that the Lyapunov dimension of the impact coupled SHO system increases as dissipation was decreased. |
