Approximate frequency domain analysis for linear damped space structures

Abstract

A method is presented for the analysis of damped structures in which the structural components are represented by impedance models and analyzed in the frequency domain. Methods are presented to assemble and condense system impedance matrices and then to identify approximate mass, stiffness, and viscous damping matrices for systems whose impedances are complicated functions of frequency. Formulas are derived for determination of approximate values for the natural frequencies and damping of systems represented by mass, stiffness, and viscous damping matrices. The sensitivities of these approximate values to system parameter changes are analyzed. The implementation of these analysis tools is discussed and applied to a simple mechanical system. Nomenclature C = damping matrix E = Young's modulus F = vector of global forces K — stiffness matrix M = mass matrix P = real power T = transformation matrix for reduction of coordinates v = vector of global velocities dv ?= first variation of the velocity mode shape W(s) — complex weighting function X = reactive power Z = impedance matrix &m,pn = viscoelastic material parameters f = damping ratio rj = loss factor a>n = natural frequency Subscripts. E = eliminated degrees of freedom (DOF) EXT = external / = imaginary L = linear NL — nonlinear R = real or retained DOF V = viscoelastic