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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY 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 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.Summary
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.
References
- 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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