A Bilinear Variational Principle Governing Longitudinal Vibration of Rods With Frequency-Dependent Material Damping

Abstract

The equations governing the longitudinal vibration of rods with frequency-dependent material damping are developed as the Euler equations of a bilinear variational principle. Frequency-dependent material damping and modulus are accommodated through the introduction of an augmenting thermodynamic field that interacts with the mechanical displacement field. These two primary dependent fields are supplemented with two corresponding adjoint fields for the purpose of addressing nonconservative system behavior. The variational function is nearly symmetric in the primary and adjoint variables, a formulation which may be particularly useful in computational simulation of system behavior using finite elements. The augmenting thermodynamic field is found to be effectively internal—no boundary conditions involve it alone.