Some interesting results on vibration and stability of distributed structural systems, vibration-based damage detection, and infinitely variable transmission will be reviewed. Vibration and stability of translating media with time-varying lengths and/or velocities will be addressed. Two types of dynamic stability problems are considered: dynamic stability of translating media during extension and retraction, and parametric instabilities in distributed structural systems with periodically varying velocities. The incremental harmonic balance method is used for high-dimensional models of nonlinear distributed systems with general nonlinearities. A new spatial discretization and substructure method, which ensures that all the matching conditions of distributed components are satisfied, and hence uniform convergence of the solutions, will be discussed. The method overcomes drawbacks of the classical assumed modes and component mode synthesis methods. A new nonlinear model of a slack cable with bending stiffness and arbitrarily moving ends is developed. The new methodologies are applied to elevator and other systems. The model-based damage detection will address two major challenges in model-based damage detection: accurate modeling of structures and development of a robustness algorithm for identifying locations and extent of damage. Non-model-based methods using scanning laser vibrometry and digital image correlation will be presented. Finally, design, analysis, and control of a novel infinitely variable transmission will be discussed. Experimental results are presented to validate theoretical predictions.