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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

Nonlinear longitudinal forced vibration of a hysteretic bar: An analytical solution  
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This work investigates the nonlinear longitudinal forced vibrations of a bar with hysteretic dynamics typical of rocks and man-made 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 non-monotonically 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 bench-mark 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. 345-357, 2011
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