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Direct methods are presented, in state space, for design of control matrices for structures with and without initial viscous damping. given the desired changes in the eigenvalues and eigenvectors. The equations with full-state variable feedback are   M   z  ¨   +  C   z  ˙   +  K  z  =  −  G   z  ˙   −  H  z   . Advantage is taken of the special state-space form of the structural plant and eigenvector matrices to develop efficient numerical procedures. Essential relationships are derived between the upper and lower portions of the left and right eigenvector matrices for systems with simple eigenvalues. We distinguish between displacement mode shapes and eigenvectors in state space. A unique method is presented by which the    [   C  +  G   ]    matrices are designed to achieve specified real parts of the eigenvalues, with no change in mode shapes or eigenvectors. In addition, two general methods are discussed. suitable for systems with nonproportional damping, where the eigenvalues are specified with or without change in mode shapes.

Design of Damping and Control Matrices by Modification of Eigenvalues and Eigenvectors