This paper presents the stability analysis of linear, damped SDOF vibration control systems with two time delays, one in the displacement feedback and the other in the velocity feedback. First, a sufficient and necessary algebraic criterion is proved to check the system stability independent of time delays. According to this criterion, all possible combinations of the feedback gains that guarantee the delay-independent stability are given. Then, the effect of the feedback gains on the system stability is discussed when the time delays are finite. The most dangerous case is found when the time delay in the displacement feedback is much longer than that in the velocity feedback.

The primary resonance and the stability of a harmonically forced oscillator with a pair of symmetric set-up elastic stops are studied by means of the average approach. It is found that the set-up elastic stops greatly increase the complexities of the primary resonance as follows. Four kinds of persistent primary resonance and three critical cases exist. The motion of many of the persistent resonances becomes unstable when it begins to touch the set-up elastic stops with decrease of the excitation frequency. Moreover, there coexist three stable periodic motions and two unstable periodic motions at certain combinations of system parameters.

The paper presents an adaptive control scheme for recovering the original dynamics of a nonlinear system after a sudden disturbance in a system parameter by controlling the system parameter through linear feedback. The key problem is to choose the control stiffness in the feedback by assigning the poles of the linearized controlled system in an extended state space, which consists of the system state and the system parameter. The paper gives the simulations of recovering the fixed point of the logistic map and the periodic orbit of a harmonically forced Duffing oscillator from the chaos due to a large disturbance in the parameter. The simulations demonstrate well the efficacy and the advantages of the proposed adaptive control scheme.

A critical phenomenon in a continuous, piecewise-linear oscillator subjected to periodic excitation is considered, in which a periodic orbit happens to graze a switching plane between two linear regions of the oscillator at zero velocity. The analysis shows that if the piecewise linearity of the oscillator changes dramatically on the switching plane, the phenomenon turns the stability trend of the periodic orbit so abruptly that it may be hard to predict an incident bifurcation, although it is of a local type, according to the concepts of smooth dynamical systems. To detect the critical phenomenon induced by the non-smooth nature of the oscillator, a numerical scheme of searching for all periodic grazing orbits is presented, as well as a test function for grazing phenomena in tracing a branch of a periodic orbit. There follows a numerical simulation based on the scheme and the test function, which reveals an abundance of grazing phenomena and incident bifurcations of the oscillator.

The vector field of a continuously piecewise-linear oscillator under periodic excitation is by nature nonsmooth. However, the nonsmoothness in studying the oscillator dynamics has drawn little attention. The paper presents the nonsmoothness effect on the differentiability of the Poincar