# ACTIVE CONTROL OF STRUCTURAL VIBRATION AND ACOUSTIC RADIATION VIA LEFT AND RIGHT EIGENVECTOR ASSIGNMENT

Open Access

- Graduate Program:
- Mechanical Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- December 05, 2006
- Committee Members:
- Kon Well Wang, Committee Chair
- Christopher Rahn, Committee Member
- Mary I Frecker, Committee Member
- Heath Hofmann, Committee Member

- Keywords:
- Active Control
- Structural Vibration
- Acoustic Radiation
- Piezoelectric Material
- Modal Confinement
- Eigenvector Assignment
- Disturbance Rejection
- Modal Radiation Reduction.

- Abstract:
- The goal of this thesis research is to investigate the feasibility of utilizing the left and right eigenvector assignment concept for active control of structural vibration and acoustic radiation. The right eigenvector assignment approach is directly related to mode shape tailoring. It therefore can be utilized to achieve structural vibration confinement and acoustic radiation reduction. The control strategy of vibration confinement is to alter the right eigenvectors through active action, such that the modal components corresponding to the concerned region have relatively small amplitude. Similarly, the control strategy for reducing acoustic radiation is to alter the right eigenvectors through active action so that the modal velocity distributions cause as small radiation as possible. Reciprocally, the concept of left eigenvector assignment is to alter the left eigenvectors so that the effects of the exogenous disturbances on the system responses can be modified. Therefore the left eigenvector assignment can be conceptually used to reduce the effects of external excitations, and thus achieve disturbance rejection. The design goal is to alter the left eigenvectors through active action such that the forcing vectors are as closely orthogonal to the left eigenvectors as possible. Because of these clear physical meanings, the proposed left-right eigenvector assignment concept can target the nature of the structural vibration-acoustics problem with more physical insight as compared to many more classical control schemes. With such an approach, one can achieve both disturbance rejection and modal confinement (vibration control purpose) or modal radiation reduction (noise reduction purpose) concurrently for forced vibration-acoustics problems. In this research, simultaneous left-right eigenvector assignment and partial left-right eigenvector assignment approaches are synthesized for structural vibration control (discussed in Chapters 2 and 3 of this thesis) and acoustic radiation reduction (Chapter 5), respectively. With the simultaneous left-right eigenvector assignment approach, the feedback gain matrix is derived based on the generalized inverse procedure. In such a method, all the left and right eigenvectors of the closed-loop system are determined to best-match the desired eigenvectors through a least square approximation. On the other hand, the partial left-right eigenvector assignment method can exactly assign the selected left and right eigenvectors of the closed-loop system as the desired optimal ones. With this algorithm, both the left and right eigenvectors can be determined accurately from the achievable subspaces through solving generalized eigenvalue problems. Numerical simulations are performed to evaluate the effectiveness of the proposed methods on a clamped-clamped beam structure example for the vibration and noise control problem. Frequency responses of different case studies in the selected frequency range are illustrated. It is shown that with the simultaneous left-right eigenvector assignment or the partial left-right eigenvector assignment techniques, both disturbance rejection and modal confinement or modal radiation reduction can be achieved, and thus the vibration amplitude in the concerned region or the sound pressure radiation at the receiver can be reduced significantly. Experimental efforts are performed to implement the new active control concepts for structural vibration control (Chapter 4), where the test results demonstrate the effectiveness of the proposed approaches. Finally, in the last chapter of this thesis, the research efforts and achievements are summarized, and recommendations for possible future investigations are discussed.