A Thermodynamically Consistent Entropic Variational Formulation for Nonlinear Damped Vibrations

Abstract

The proposed framework is based on a path-dependent entropic variational construction for nonlinear damped vibration systems. Accumulated mechanical spending and heat-transfer power are represented as history contributions to the action. Each irreversible mechanism is supplied as a force–power pair: the force acts on arbitrary mechanical test directions, while its realized non-negative power is assigned through a separate receiver-channel variation to the entropy/heat receiver. The damping law and the admissibility condition on its realized power are constitutive inputs; the proposed formulation supplies the common directional stationarity structure that links those inputs to the mechanical and thermodynamic balances. The continuum form is reviewed for thermoelastic vibration, and a finite-dimensional vibration formulation is presented separately. A frequency-resolved version based on response PSDs is derived for asymptotically stable linear systems under stationary random excitation, allowing response spectra to be converted into frequency-resolved contributions to mean dissipated power and entropy production. Numerical examples cover a linear oscillator, a Duffing oscillator with nonlinear damping, an oscillator with regularized Coulomb friction, a tuned mass damper with deterministic and spectral excitation, and a piezoelectric RC oscillator with resistive loading. The responses are also compared with analytical solutions or independently integrated classical equations. The computations verify energy closure and dissipative power accounting for the selected admissible models.