https://www.jstage.jst.go.jp/article/souonseigyo1977/18/3/18_3_117/_pdf

On the Vibration Damping

Research and applications of viscoelastic vibration damping materials: A review

A time domain model of linear viscoelasticity is developed based on a decomposition of the total displacement field into two parts: one elastic, the other anelastic. The anelastic displacement field is used to describe that part of the strain that is not instantaneously proportional to stress. General coupled constitutive equations for (1) the total and (2) the anelastic stresses are developed in terms of the total and anelastic strains, and specialized to the case of isotropic materials. A key feature of the model is the absence of explicit time dependence in the constitutive equations. Apparent time-dependent behavior is described instead by differential equations that govern (1) the motion of mass particles and (2) the relaxation of the anelastic displacement field. These coupled governing equations are developed in a parallel fashion, involving the divergence of appropriate stress tensors. Boundary conditions are also treated: the anelastic displacement field is effectively an internal field, as it is driven exclusively through coupling to the total displacement, and cannot be directly affected by applied loads. In order to illustrate the use of the method, model parameters for a commonly-used high damping polymer are developed from available complex modulus data.

Time Domain Modeling of Linear Viscoelasticity Using Anelastic Displacement Fields

A linear electromechanical model is developed for ionic polymer materials. The model is based on an equivalent circuit representation that is related to the mechanical, electrical, and electromechanical properties of the material. Expressions for the quasi-static and dynamic mechanical impedance are derived from beam theory. The Golla-Hughes-McTavish model of viscoelasticity is incorporated into the model to include effects due to a rate dependent modulus. Similar to previous research, the electrical impedance is modeled as a series combination of resistive and capacitive elements. The major contribution of this work is the derivation of an electromechanical coupling term that is related to an effective bending strain coefficient. This parameter is also frequency dependent to model the low-frequency relaxation that has been measured in certain ionic polymer materials. The resulting linear electromechanical model is based on the measurement of the effective permittivity, elastic modulus, and effective stra...

Linear Electromechanical Model of Ionic Polymer Transducers -Part I: Model Development

Damping models for structural vibration

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.

D. J. McTavish

Papers

Modeling of linear viscoelastic space structures