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Keywords
(11)
Analytic Solution
Analytical Model
Constitutive Relation
Energy Dissipation
Equation of Motion
Harmonic Balance
Material Properties
Quality Factor
Resonant Frequency
Spatial Dependence
Spatial Variation
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Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution
Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution,10.1016/j.wavemoti.2010.12.002,Wave Motion,Claudio Pecorari,Daniel
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Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution
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Claudio Pecorari
,
Daniel A. Mendelsohn
This work investigates the nonlinear longitudinal forced vibrations of a bar with hysteretic dynamics typical of rocks and manmade geomaterials. The
material properties
are described by a
constitutive relation
that includes
energy dissipation
effects consistently with hysteretic dynamics. The
harmonic balance
method and a perturbation technique that uses the material's modulus defect as perturbation parameter are employed to solve the equation of motion. The
spatial dependence
of the modulus defect, which is disregarded in earlier solutions of this problem, is duly accounted for. According to this model, the resonance frequency shift is proportional to the amplitude of the excitation, while the inverse of the
quality factor
increases as the square root of the latter. Further, the
spatial variation
of the modulus defect is shown to affect the modulation of the fundamental component along the bar. The nonlinear spectral components are of odd order, with amplitude proportional to the maximum value of the modulus defect and to the square of the excitation's amplitude, and decreases nonmonotonically with increasing harmonic order. The motivation for this work is twofold. Analytical models may improve our understanding of the dynamics of hysteretic materials in general, and of the mutual interaction of material defects in particular. Secondly, this work establishes a benchmark result on a mathematically simple hysteretic system against which alternative mathematical approaches, possibly concerning material with complex constitutive relations, could be tested.
Journal:
Wave Motion
, vol. 48, no. 4, pp. 345357, 2011
DOI:
10.1016/j.wavemoti.2010.12.002
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