Optimum vibration absorbers for continuous and discrete mechanical systems

Abstract

In this thesis, the optimum parameters of passive vibration absorbers for continuous and discrete mechanical systems are investigated. Two approaches are undertaken to obtain the optimum absorber parameters. The first approach is built on the continuous system analysis for minimization of vibration amplitudes and the related methodology, motivated by Lagrange’s equation and the Lyapunov equation, is derived. The second approach, based on the receptance concept, uses a new methodology which employs the Sherman- Morrison formula to minimize the vibration amplitudes in discrete systems. A method is developed for calculating the optimum parameters of n absorbers attached to a uniform beam or rectangular plate, the vibrations of which are characterized by the first M modes where n and M can be any positive integers. For the most general case, dissipation due to damping, kinetic and potential energies, and the effects of external forces are all taken into account. Lagrange’s equation is used to generate a state space representation of the system, wherein the Lyapunov equation is used to obtain the H2 norm of the transfer function. This norm is used to construct the objective function for optimization. The optimal response is compared to cases without an absorber and with randomly selected absorber parameters. Focusing on the minimization of the vibration amplitudes using the concept of receptance, a method for calculating the receptance of a generic translational mass-springdamper system with m masses and n absorbers is developed. The dynamic stiffness of the entire system is derived both directly from the equations of motion and through a linear graph representation of the system. The receptance of the combined system, in terms of the parameters of the main and absorber systems is obtained by applying Sherman-Morrison formula sequentially. The optimal absorber parameters of the system are easily obtained once the receptance is known. The complexity of the formulation is simplified with the proposed approach. Two basic models, both the continuous and discrete system modeling for suppressing vibration, are built in this thesis. It is shown that continuous and discrete systems are interrelated and the corresponding developed methods can be applied to each type of system mutually.