In this paper, properties of operators of the kind A + Λ are investigated, where A is a selfadjoint, positive definite differential operator with compact inverse on a Hilbert space, and Λ is assumed to be A-bounded, A-symmetric and A-positive semidefinite. A general theory on the existence and uniqueness of solutions of second order evolution equations involving the operator A + Λ is developed. The new findings obtained in this paper have potential applications in modeling and control of flexible structures.