| Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. Parametric instability regions of a second-order non-dispersive distributed structural system in this work are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. The parametric instability regions are classified as period-1 and period- i (i > 1) instability regions, where the former is analytically obtained, and the latter can be numerically calculated using bifurcation diagrams. The parametric instability phenomenon is characterized by a bounded displacement and an unbounded vibratory energy, due to formation of infinitely compressed shock-like waves. Parametric instability in a taut string with a periodically moving boundary is then investigated. The free linear vibration of the taut string is studied first, and three corresponding nonlinear models are introduced next. It is shown that the responses and vibratory energies of the nonlinear models are close to those of the linear model, which indicates that the parametric instability in the linear model can also exist in the nonlinear models.;A new global spatial discretization method for one- and two-dimensional continuous systems is investigated. General formulations for one- and two-dimensional systems that can achieve uniform convergence are established, whose displacements are divided into internal terms and boundary-induced terms. For one-dimensional systems, natural frequencies, mode shapes, harmonic steady-state responses, and transient responses of a rod and a tensioned Euler-Bernoulli beam are calculated using the new method and the assumed modes method, and results are compared with those from exact analyses. The new method gives better results than the assumed modes method in calculating eigensolutions and responses of a system, and it can use sinusoidal functions as trial functions for the internal term rather than possibly complicated eigenfunctions in exact analyses. For two-dimensional systems, natural frequencies and dynamic responses of a rectangular Kirchhoff plate that has three simply-supported boundaries and one free boundary with an attached Euler-Bernoulli beam are calculated using both the new method and the assumed modes method, and compared with results from the finite element method and the finite difference method, respectively. Advantages of the new method over local spatial discretization methods are fewer degrees of freedom and less computational effort, and those over the assumed modes method are better numerical property, a faster calculation speed, and much higher accuracy in calculation of high-order spatial derivatives of the displacement. |
