Inverse structural modification using constraints

In a structural modification problem the mass and stiffness matrices are modified to obtain a desired spectrum. In this paper, this is done by imposing constraints on the structure. The undamped natural vibrations of a constrained linear structure are calculated by solving a generalized eigenvalue problem derived from the equations of motion for the constrained system involving Lagrangian multipliers. The coefficients of the constraint matrix are taken as design variables and a set of equations defining the inverse structural modification problem is formulated. This modification problem requires an iterative method for its solution. An algorithm based on Newton's method is employed. Each iteration step involves the calculation of a rectangular Jacobian and the solving of an associated underdetermined system of linear equations. The system can be solved by using the Moore–Penrose inverse. The method is demonstrated in some numerical examples.