Affiliations: Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802
Abstract: Dissipation of energy in vibrating structures can be accomplished with a combination of passive damping and active, constant gain, closed loop control forces. The matrix equations are $M\ddot{z}+C\dot{z}+Kz=-G\dot{z}$. With conventional viscous damping, the damping force is proportional to relative velocity, with $F_{i}=C_{ii}\dot{z}_{i}-C_{ij}\dot{z}_{j}$, where Cii=Cij but the subscripts show the position of the number in the C matrix. For a dashpot connected directly to ground, $F_{i}=C_{ii}\dot{z}_{i}$. Thus there is a definite pattern to the positions of numbers in the ith and jth rows of the C matrix, that is, a positive number on the diagonal is paired with an equal negative number, or zero, off the diagonal. With the control matrix G, it is here assumed that the positioning of individual controllers and sensors is flexible, with $F_{i}=G_{ii}\dot{z}_{i}$ or $F_{i}=G_{ij}\dot{z}_{j}$, the latter meaning that the control force at i is proportional to the velocity sensed at j. Thus the problem addressed herein is how the individual elements in the G matrix affect the modal eigenvalues. Two methods are discussed for finding the sensitivities, the classical method based on the products of eigenvectors and a new method, derived during the present study, involving the derivatives of the invariants in the similarity transformation. Examples are presented for the sensitivities of the complex eigenvalues of the form λr=−ζrωr+iωDr to individual elements in the G matrix, to combinations of elements, and to a combination of passive damping and active control. Systems with two, three, and eight degrees of freedom are investigated.
