Keywords: Biot, damping, eigenmotions, rheology, vibrations, viscoelasticity.

In a recent series of papers, Adhikari et al. [1,2], have shown that, for an -degree of freedom viscoelastically damped system governed by:

the search of eigensolutions in the form leads to values for . These values give rise to independent solutions which are damped with or without oscillations. But the distribution between these two types of solutions has not actually been made explicit.

For a system with DOFs with classical viscous damping, Bulatovic [3] has given conditions for the eigenmotions to be all damped with oscillations (so-called heavily damped systems).

Systems with DOF with a viscoelastic spring, the rheology of which is represented by a so-called three parameters model, have been studied by Muller [4].

For a system with DOF consisting of a mass associated with a viscoelastic spring with relaxation modulus , the governing equation is:

and the search of eigensolutions in the form leads to the following equation for :

where denotes the Laplace-Carson transform of the relaxation modulus .

When this viscoelastic spring is represented by a Biot's model with parameters the relaxation modulus takes the form of a so-called "Prony's series":

where the relaxation times and the physical constants and are .

The Laplace-Carson transform of may then be written in the form:

where and where are the retardation times, and it may then be proven by a simple graphical discussion of the roots of equation (10) that there are independent eigenmotions of the following nature: two eigenmotions damped with or without oscillations (as in the case of classical viscous damping) and eigenmotions damped without oscillations which are induced by the viscoelastic rheology.

1

S. Adhikari, Eigenrelations for Nonviscously Damped Systems, A.I.A.A. Journal, 39, 8, 2001, 1624-1630. doi:10.2514/2.1490

2

S. Adhikari, Dynamics of Nonviscously Damped Linear Systems, J. Engng Mech., 2002, 328-339. doi:10.1061/(ASCE)0733-9399(2002)128:3(328)

3

R.M. Bulatovic, On the Heavily Damped Response in Viscously Damped Dynamic Systems, J. Appl. Mech., 71, 2004, 131-134.doi:10.1115/1.1629108

4

P. Muller, Are the Eigensolutions of a 1-d.o.f. system with Viscoelastic Damping Oscillatory or not?, J. of Sound and Vibration, 285, 2005, 501-509. doi:10.1016/j.jsv.2004.09.007

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