Approximation of a closely spaced mode subspace by a repeated mode subspace in vibration control of flexible structures

The necessary conditions under which a closely spaced mode subspace might be substituted by a repeated mode subspace from the point of view of the control and dynamics are presented in this paper. The definitions of the controllability degree and observability degree are first suggested. Based on these definitions and dynamical analysis, the necessary conditions are obtained for the approximation of a closely spaced mode subspace by a repeated mode subspace, namely not only the deviation of the natural frequencies should be small, but also should be small enough, where ξ0 is the relative coefficient of the damping.