This chapter analyses the dynamic properties of a system with a damped elastic strip. The strip can vibrate freely or it may be subjected to external harmonic or stochastic excitation. The dynamic systems under consideration can be generally described by a common formula in the form of equation. Dynamic models reflecting permanent joints, such as a strip damped at one side or a system of two beams pressed against each other reveal good damping properties within a certain range of the structural and operating parameters. Investigations of these properties and the obtained results can be used in a process of design of machines in which such systems can be purposely used for limiting harmful vibrations. Mechanical systems with permanent joints exhibit good damping properties if the structural and operational parameters are held within certain limits. Nonlinear eliminators with permanent joints can be successively used in machines where limitation of the vibration amplitude is of great importance.