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 non-linear behavior of ice-induced vibration of an offshore platform with four legs is investigated in this paper. The equations of motion of the system are derived by using the Hamiltonian Principle The force of moving ice based on the self-excitation and locking is used to model the phenomenon of contact between the ice and the platform. By using the approach of multiple scales, the primary resonance of the ice-induced vibration of the platform is analyzed. The numerical results show that there exist several kinds of combination resonances, including self-excited vibration and locking vibration. These results coincide with those observed from an offshore platform in the North China Sea, and hence enable one to gain insight into the ice-induced vibration of offshore platforms.

The paper presents a method of assessing the stability of high dimensional vibrating systems under state feedback control with a short time delay. It is first proved that if the time delay is sufficiently short, an n-degree-of-freedom system with feedback delay maintains 2n eigenvalues near those of the corresponding system without feedback delay. A perturbation approach is then proposed to determine the first order variation of an arbitrary eigenvalue and corresponding eigenvector of the system with feedback delay by solving a set of linear algebraic equations only. The computation in this approach can be simplified to a matrix multiplication provided that the product of the time delay and the modulus of the eigenvalue is much less than 1. Furthermore, the approach is directly related to the Newton-Raphson iteration in the continuation of eigenvalues for long time delay. The efficacy of the approach is demonstrated via a number of case studies on two feedback delay systems of two degrees of freedom and ten degrees of freedom respectively.

On the basis of generalized Sturm theory, this paper presents a simple, but systematic approach to the delay-independent stability analysis of the linear dynamic systems involving multiple degrees of freedom and possibly two time delays. The approach enables one to complete the stability analysis in a much simpler way than before through the use of a MAPLE routine given in the paper. To demonstrate the approach, the paper gives a detailed analysis, the corresponding sufficient and necessary conditions as well as stable regions in parameter space of concern for the delay-independent stability of a vibrating system with time delays in state feedback, an active-tendon for a tall structure and an active suspension of a quarter car model.

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.