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Stability of one-dimensional nonlinear hereditary system

Vibration reduction is an important issue in design of modern machines. One of the present trends to reduce vibration is toward the use of viscoelastic materials. To present a good design, an accurate as well as efficient mathematical model of viscoelasticity is required. In this paper, we proposed a weak singular integro-differential equation to analyze the vibratory behavior of a viscoelastic body. This equation includes nonlinear and hereditary terms to model realistic vibration. Then, we solve the mathematical model by the numerical method which employs an integration method with the elimination of weak singularity. As demonstrated in the numerical experiments of one-dimensional rotor blade, the vibration along a creep curve under constant force and resonant vibration under harmonic force are well presented when the viscoelastic material is considered. These vibrations are properly reduced by using the viscosity parameter of the hereditary term in nonlinear and linear cases.
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