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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 77

Eigenmotions of a One Degree of Freedom Viscoelastically Damped System

P. Muller

Laboratory of Modelling, Materials and Structures (LM2S), University Pierre and Marie Curie, Paris, France

Full Bibliographic Reference for this paper
P. Muller, "Eigenmotions of a One Degree of Freedom Viscoelastically Damped System", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 77, 2006. doi:10.4203/ccp.83.77
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:

d (8)

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:

d (9)

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 .
Figure 1: One DOF oscillator with Biot's viscoelastic damping.

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.

S. Adhikari, Eigenrelations for Nonviscously Damped Systems, A.I.A.A. Journal, 39, 8, 2001, 1624-1630. doi:10.2514/2.1490
S. Adhikari, Dynamics of Nonviscously Damped Linear Systems, J. Engng Mech., 2002, 328-339. doi:10.1061/(ASCE)0733-9399(2002)128:3(328)
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
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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