Exact Sensitivity of Nonlinear Dynamic Response with Modal and Rayleigh Damping Formulated with the Tangent Stiffness

Abstract

Derivatives of nonlinear dynamic response are calculated in an exact and efficient manner with respect to material, geometry, mass, and damping parameters. Developments are presented for the Rayleigh and modal damping options that use the updated tangent stiffness. The calculation of exact response sensitivities for those options requires the calculation of derivatives of eigenvalues and eigenvectors; it is also shown that the third-order tensor formed by the derivative of the stiffness matrix with respect to the displacement vector is needed. That tensor, which amends the coefficient matrix of the system of equations that governs the response sensitivities, is nonzero for materials with continuously varying stiffness. The Bouc–Wen material model exhibits that feature and is selected to demonstrate the developments. Correct differentiation and assembly at the material, section, and element levels are highlighted. The results, verified by finite difference, suggest that the sensitivity of the response is influenced by the choice of damping model, strongly for some parameters, particularly when higher modes contribute to the nonlinear structural behavior.