Receptance model based on isotropic damping functions and elastic displacement modes

Abstract

Starting from a fundamental, continuum mechanical, constitutive material damping description, the augmented Hooke's law (AHL) introduced by Dovstam [Dovstam, K. (1995). Augmented Hooke's law in frequency domain. A three-dimensional, material damping formulation. International Journal of Solids and Structures32, 2835–2852], a linear three-dimensional damped vibration response model for isotropic material is proposed. The derivations, valid for a restricted but important class of cases, are based on general, continuous, elastic vibration modes, Gurtin [Gurtin, M. E. (1972). The linear theory of elasticity. In Encyclopedia of Physics, vol. VIa/2, Mechanics of Solids II (eds Flügge, S. and Truesdell, C.). Springer, Berlin]. Material damping is represented by two complex, frequency dependent, damping functions defined directly by constitutive parameters in the isotropic AHL. Alternatively, for isotropic, viscoelastic solids, the damping functions are derived from two stress relaxation functions. Modal damping functions, which define overall “structural” damping in terms of elastic, modal parameters and the two material damping functions, are introduced in the response model. It is shown that the modal damping functions, indirectly, depend on the geometry and boundary conditions of the piece of material under investigation. Introduced are also new, elastic modal parameters which determine the quantitative contribution to the modal damping functions from the two isotropic, material damping functions. The new modal parameters are easily computed using, slightly modified, standard finite element code, and elastic modes resulting from standard finite element eigenvalue analyses. A close agreement between direct finite element calculations and responses predicted using the proposed modal model is obtained for a studied three-dimensional cantilever test plate.