Simulation of damped vibrations based on augmented Hooke’s law and elastic modes of vibration

A linear, unconditionally convergent modal vibration response modelling technique is presented. Material damping is simulated using the augmented Hookeʹs law introduced by Dovstam [Dovstam, K., 1995. Augmented Hookeʹs law in frequency domain. A three-dimensional, material damping formulation. International Journal of Solids and Structures 32, 2835±2852]. The method is based on continuous, elastic, displacement modes and vibrational stress modes, dual to the traditional displacement modes. The stress modes are implicitly used to derive a generally convergent modal response model in generally damped cases with boundary traction excitation. The real eigenvalue problem de®ning the stress modes is formulated, and explicit frequency domain modal system equations of motion, for computation of needed stress and displacement mode coecient spectra, are derived. Introduced parameters, accounting for the interdependence of the di€erent modal contributions (modal coupling) to the response, are computable from known material properties (elastic and damping) and geometry by post processing results from three-dimensional, standard ®nite element (FE), eigenvalue calculations. Practical means for predicting whether modal coupling will occur or not are thus provided, as well as means for predicting damped resonance frequencies. When applied to an isotropic material, the new response model, for small damping, approaches the modal receptance model recently introduced and discussed by Dovstam [Dovstam, K., 1997. Receptance model based on isotropic damping functions and elastic displacement modes. International Journal of Solids and Structures 34, 2733±2754]. A close agreement between direct FE calculations and response simulations using the proposed method is obtained for a highly damped three-dimensional cantilever test plate.