The objective of this research is to investigate the feasibility of utilizing left eigenvector assignment for vibration disturbance rejection. In the previous study, it has been shown that through right eigenvector assignment and modal confinement, one can enhance the performance of periodic vibration isolators. However, it was also recognized that since vibration mode confinement is based on the concept of modal response, it does not guarantee that vibration will always be reduced in a forced excitation scenario.
In this research, the left eigenvector assignment technique is utilized to achieve vibration suppression throughout a broad frequency range. The principle is to alter the left eigenvectors of the closed-loop system so that the system's forcing vectors are as closely orthogonal to each left eigenvector as possible. With such an approach, one can directly attack the forced response problem. A new formulation is developed so that the desired left eigenvectors of this integrated system are selected through solving a generalized eigenvalue problem, where the orthogonality indices between the forcing vector and the left eigenvectors are minimized. The integrated system with assigned left eigenvectors achieves to reject external disturbance of the complete electromechanical system.
An integrated closed-loop system with state estimator is also developed so that the algorithm can be implemented realistically. Numerical simulations are performed to evaluate the effectiveness of the proposed method on disturbance rejection for an isolator design example. Frequency responses of the isolator in the selected frequency range are illustrated. It is shown that with the left eigenvector assignment technique, the system’s external disturbances are rejected and vibration amplitude of the isolated regions can be effectively suppressed.
The objective of this research is to investigate the feasibility of utilizing eigenvector assignment and piezoelectric networking for enhancing vibration isolator design through energy confinement. For a classical periodic isolator structure, the material discontinuity creates stop bands that could suppress the wave propagation of external excitation in a particular frequency range. While effective, such method can not always create wide enough stop bands such that all the disturbance frequencies are covered. In this study, the eigenvector assignment technique and piezoelectric networks are utilized to reduce the transmissibility of the isolator modes near the boundary of the stop bands, and therefore widen the effective frequency range and enhance the performance of the isolator.
The eigenvector assignment principle is to alter the mode shapes of the system so that the modal components have smaller amplitude in concerned coordinates than in other parts of the system. By applying the eigenvector assignment method on the spatially tailored periodic isolator structure, the attenuated end (the end of the isolator designed to have small vibration) response amplitude at resonant frequencies near the stop band can be reduced, which enhances the vibration isolation performance in the frequency range of interest. On the other hand, piezoelectric networks connecting to the isolator structure increase the degrees of freedom of the integrated system, and enlarge the design space for achievable eigenvectors. The right eigenvectors of this integrated system are selected such that the modal energy in the concerned area is minimized by using the Rayleigh Principle. The integrated system with assigned eigenvectors will re-distribute vibratory energy of the complete electromechanical system. Small vibration at the attenuated end of the isolator is achieved since the energy is confined in the circuitry and other parts of the isolator.
Numerical simulations are performed to evaluate the effectiveness of the proposed method on vibration confinement for isolator designs. Frequency responses of the different generalized coordinates in the selected frequency range are illustrated. It is shown that with the piezoelectric networking and eigenvector assignment, the system energy is redistributed and confined in the unconcerned areas, which can greatly enhance the performance of the vibration isolation system.