Eigenvector Scaling for Mode Localization in Vibrating Systems

A new closed-loop mode localization technique known as eigenvector scaling is presented. Eigenvector scaling is a form of eigenstructure placement that uses feedback to alter portions of the original system eigenvectors. When properly applied, this technique restricts energy propagation in a vibrating system. Active eigenvector scaling also permits the development of an analytic solution for the closed-loop mode-localized discrete system. The ability to produce analytic solutions for both the controlled and the uncontrolled systems permits a direct comparison of the absolute displacements between the two cases. It is demonstrated that closed-loop mode localization can be used to restrict vibrations in certain regions of a discrete structure. Three examples are provided that show how this feedback control technique may be applied to multi-degree-of-freedom simple spring-mass and spring-mass- damper systems, as well as to lumped models of more complicated flexible structures. IBRATION of flexible structures is a common problem in dy- namics and controls. The relative ease with which vibrational energy may propagate through complex systems can lead to exces- sive displacements at critical locations, which in turn can lead to reduced performance or failure. It is not always necessary to control vibrations throughout an entire structure, but rather controlling cer- tain areas within the structure that contain sensitive instruments or machines may be most critical. The motivation for this work, then, is to seek vibration control methods that produce quiet areas within a vibrating system subjected to arbitrary disturbances. One way of controlling flexible structure vibration is through mode localization. Mode localization restricts motion by contain- ing vibrational energy within a small area of the total system. This phenomenon was originally described in solid state physics1 but also exists in flexible structures.2'3 Mode-localized structures exhibit ar- eas of comparatively high displacements, with amplitudes in these parts potentially several magnitudes greater than in the rest of the system. In a large number of studies, mode localization has been considered to be unfavorable, as in mistuned rotor blades.4 How- ever, some studies maintain that mode localization may be useful, since it confines system motion within a specific area, preventing the transmission of potentially harmful vibrational energy.5 Addition- ally, if vibrational energy is contained, then it may be more easily controlled with damping or cancellation. Loosely coupled, slightly mistuned flexible structures are espe- cially susceptible to mode localization.2'6'7 As a result, studies have concentrated on the passive mistuning of this type of structure. It is, however, also possible to create mode localization with active, closed-loop feedback control using a technique known as eigen- structure placement, which is the approach used in this paper.