In earlier investigations, the author used extensions of two theorems of G. Strang to derive bounds on the displacements of a symmetric damped linear mechanical system subject to prescribed periodic forces. This work is extended in the current investigation to obtain bounds under prescribed periodic motions. For prescribed periodic forces, the bounds were expressed in terms of the extreme eigenvalues of several symmetric, positive definite matrices. In contrast, in the current case the bounds also depend on several nonsymmetric matrices. The bounds under prescribed motion are evaluated in an example and comparison is made with an exact result. The results reported here are new.

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