The GHM Method provides viscoelastic finite elements derived from the commonly used elastic finite elements. Moreover, these GHM elements are used directly and conveniently in second-order structural models jut like their elastic counterparts. The forms of the GHM element matrices preserve the definiteness properties usually associated with finite element matrices—namely, the mass matrix is positive definite, the stiffness matrix is nonnegative definite, and the damping matrix is positive semi-definite. In the Laplace domain, material properties are modeled phenomenologically as a sum of second-order rational functions dubbed mini-oscillator terms. Developed originally as a tool for the analysis of damping in large flexible space structures, the GHM method is applicable to any structure which incorporates viscoelastic materials.

Mathematical models of elastic structures have become very sophisticated: given the crucial material properties (mass density and the several elastic moduli), computer-based techniques can be used to construct exotic finite element models. By contrast, the modeling of damping is usually very primitive, often consisting of no more than mere guesses at “modal damping factors.” The aim of this paper is to raise the modeling of viscoelastic structures to a level consistent with the modeling of elastic structures. Appropriate material properties are identified which permit the standard finite element formulations used for undamped structures to be extended to viscoelastic structures. Through the use of “dissipation” coordinates, the canonical “M, K” form of the undamped motion equations is expanded to encompass viscoelastic damping. With this formulation finite element analysis can be used to model viscoelastic damping accurately.

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This paper introduces the idea of distributed gyricity, in which each volume element of a continuum possesses an infinitesimal quantity of stored angular momentum. The continuum is also assumed to be linear-elastic. Using operator notation, a partial differential equation is derived that governs the small displacements of this gyroelastic continuum. Gyroelastic vibration modes are derived and used as basis functions in terms of which the general motion can be expressed. A discretized approximation is also developed using the method of Rayleigh-Ritz. The paper concludes with a numerical example of gyroelastic modes.

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