Quadratic representation of a nonlinear dynamic stiffness matrix and related eigenvalue problems Original Research Article

The approximate representation of an exact dynamic stiffness matrix K(ρ) by a quadratic matrix formulation, A-ρB-ρ2C, is studied theoretically in this paper. The matrix formulation is formed by expressing the elements of K(ρ) as parabolic functions based on choosing three fixed values of the eigenparameter p. The general bounding properties of the approximate eigenvalues provided by the quadratic matrix formulation are shown to exist, provided that the three fixed values are below the lowest pole of the nonlinear stiffness matrix and that the three coefficient matrices, A, B and C, are positive definite. It is shown theoretically in this paper that the approximate eigenvalues are either upper or lower bounds of the corresponding exact ones of the exact dynamic stiffness matrix.