On the natural vibrations of linear structures with constraints

The undamped natural vibrations of a constrained linear structure are given by the solutions to a generalized eigenvalue problem derived from the equations of motion for the constrained system involving Lagrangian multipliers. The eigenvalue problem derived is defined by the mass matrix of the unconstrained structure and a non-symmetric and singular stiffness matrix for the constrained system. The character of the solution of the eigenvalue problem of the constrained system is stated and proved in a Theorem. Applications of the constrained eigenvalue problem to some simple structures are demonstrated. Finally a condition for the calculation of the damped natural vibrations for the constrained structure in terms of the undamped mode shapes is formulated.