Com~urers & S/rucrures Vol. 54. No. 3, pp. 531-540. 1995 Copyright @ 1995 El&ier Science Ltd Printed in Great Bntain. All rights reserved ) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Pergamon 00457949(94Mo347-5 \ , 0045.7949/95-$9.50 + 0.00 zyxwvut VIBRATION CONFINEMENT IN FLEXIBLE STRUCTURES BY DISTRIBUTED FEEDBA CK S. A. Chourat and A. S. YipitS tDepartment of Mechanical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia $Department of Mechanical Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait (Received 7 August 1993) zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Abstract-A method of confining the vibrations in parts of a flexible structure is proposed. The confinement is achieved by applying appropriate distributed feedback forces. The method of vibration confinement consists of converting the original mode shapes of continuous systems to exponentially decaying functions. It has been shown that the strategy of selecting the feedback forces, which lead to the vibration confinement, is dual to that of pole placement in the time domain control design. The method is applied to various one- and two-dimensional structures such as a string, a simply supported beam and a membrane. Simulations show that vibrations due to an impulse can be confined in a small region close to the location of the impulse. Finite element solution for the beam example is also obtained and compared to the analytical solution. It is anticipated that the proposed strategy may be a feasible method for eliminating unwanted vibrations from certain parts of a flexible structure more than others. 1. INTRODUCTION Recently, a wide spectrum of literature has studied the phenomenon of mode localization in structural dynamics [l-8]. In the structural dynamics area, the term ‘mode localization’ refers to the confinement of energy in certain modes of the structure. Hodges [l] has shown theoretically the phenomenon in a chain of coupled pendula with randomly varying natural frequencies. He has concluded that strong localiz- ation occurs when coupling is weak. Later, Ander- son [2, 31 studied the mode localization phenomenon in structures of periodicity-breaking disorder or im- perfections. A mathematical tool for examining the behavior of mode localization is devised by Pierre and Dowel1 [4]. They have developed a perturbation method for studying the mode localization and ap- plied it in a chain of single-degree-of-freedom coupled oscillators. Wei and Pierre [5,6], employing the per- turbation technique developed in Pierre and Dow- ell[4], have investigated localized free and forced vibrations in a mistuned cyclic assembly of coupled structures. The relative effects of dry friction damping and viscous damping on the occurrence of strong localization in a mistuned cyclic assembly of coupled structures have also been studied in Wei and Pierre [7]. Bendiksen [8] has studied the mode localization phenomenon in large space structures. He found that the confinement of modal amplitudes may lead to serious implications for the control problem. He stated that if the mode localization exists in a struc- ture due to manufacturing imperfections or disorders, the regular features of the mode can be destroyed such that the nodal points or lines are not regularly spaced, and their amplitudes are not sinusoidally modulated. Consequently, any controller designed for the nominal system may lead to unsatisfactory performance. He concluded that it is important to consider mode localization at the stage of control design. Instead of examining the mode localization due to irregularities, this study concentrates on the vibration confinement produced by applying appropriate feed- back forces that lead to exponentially decaying mode shapes. The vibration confinement proposed in this study is different from the classical mode localization phenomenon studied in [l-8] in the sense that the alteration of the modes is achieved through feedback forces as opposed to varying the parameters of the structure. Despite this difference, the behavior of the modified modes using feedback resembles those of the disordered periodic structures. The localized modes in disordered periodic structures are exponentially decaying functions in an average sense, while those resulting from feedback are pure exponentials. In Section 2, the proposed strategy for obtaining vibration confinement in flexible structures is devel- oped. In Sections 3 and 4, studies of the vibration confinement with examples are given for one- and two-dimensional structures. Conclusions are given in Section 5. 2. VIBRATION CONFINEMENT IN FLEXIBLE STRUCTURES In this section a novel strategy for confining the vibrations in parts of a flexible structure is developed. 531