Vehicle clutch dampers belong to a family of torsional devices or isolators that contain multi-staged torsional springs, pre-load features, clearances, and multi-staged dry friction elements. Estimation of elastic and dissipative parameters is usually carried out under quasi-static loading and then these static parameters are often assumed when predicting dynamic responses. For the purpose of comparison, this article proposes a new time domain parameter estimation method under dynamic, transient loading conditions. The proposed method assumes a priori knowledge of few nonlinear features based on the design and quasi-static characterization. Angular motion measurements from a component-level laboratory experiment under dynamic loading are utilized. Elastic parameters are first estimated through an instantaneous stochastic linearization technique. A model-based approach and energy balance principle are employed to estimate a combination of viscous and Coulomb damping parameters for seven local (stage-dependent) and global damping formulations for a practical device. The proposed method is validated by comparing time domain predictions from nonlinear models to dynamic measurements. Nonlinear models that utilize the proposed damping formulations are found to be superior to those that solely rely on parameters from a quasi-static experiment.