Abstract
This paper deals with a viscoelastic beam obeying a fractional differentiation constitutive law. The governing equation is derived from the viscoelastic material model. The equation of motion is solved by using the method of multiple scales. Additionally, principal parametric resonances are investigated in detail. The stability boundaries are also analytically determined from the solvability condition. It is concluded that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations.
Highlights
1 Introduction Many researchers have demonstrated the potential of viscoelastic materials to improve the dynamics of fractionally damped structures
5 Conclusion In this study, the effects of the damping term modeled with a fractional derivative on the dynamic analysis of a beam having viscoelastic properties subject to the harmonic external force are investigated
It is concluded that the value of the natural frequency of the beam modeled with a fractional damper is greater than that of the beam modeled with a classical damper
Summary
Many researchers have demonstrated the potential of viscoelastic materials to improve the dynamics of fractionally damped structures. The partial differential equations of fractional order are increasingly used to model problems in the continuum and other areas of application. The numerical solution for the time fractional partial differential equations subject to the initial-boundary value is introduced by Podlubny [ ]. The approximate solution of the beam modeled by a fractional derivative is obtained and an application of the fractional damped model is given. Substituting Eq ( ) into Eq ( ), we obtain the equation φn + ωn φn = iωnD AnXn – aAnXn cos σnT + η(iωn)αAnXn with the boundary conditions φn( ) = , φn( ) =. The supplementary natural frequency from the fractional derivative is given by a ωna
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