Interaction of internal and external resonances during force driven vibrations of a nonlinear thin plate embedded into a fractional derivative medium

Shitikova, Marina V.; Rossikhin, Yury A.; Kandu, V. V.

doi:10.1016/j.proeng.2017.09.008

Cited by 6 publications

(3 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…q + E r c l RL 0 D α t q + k l q = −m b vb , (C. 22) with the coefficients c l = C l M , k l = K l M and m n = M b M . We note that although we previously assumed M = J = 0 for the boundary conditions, we push the effects of the lumped mass with M = J = 1 through the choice of our spatial eigenfunctions φ(s) (see Appendix D).…”

Section: Appendix C1 Nondimensionalization Of Linearized Equation Of ...mentioning

confidence: 99%

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Structures

Suzuki¹,

Pegah²,

Kharazmi³

et al. 2020

Preprint

View full text Add to dashboard Cite

Fractional models and their parameters are sensitive to changes in the intrinsic micro-structures of anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we analyze the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelastic response is general through a distributed-order fractional model. We employ Hamilton's principle to obtain the corresponding equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, in which the linear counterpart is numerically integrated by employing a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, which yields a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law amplitude decay rates, super-sensitivity of amplitude response at free vibration, and bifurcation in steady-state amplitude at primary resonance.

show abstract

Section: Appendix C1 Nondimensionalization Of Linearized Equation Of ...mentioning

confidence: 99%

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Structures

Suzuki¹,

Pegah²,

Kharazmi³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The fractional derivative standard linear solid model has been utilized in [44] for a viscoelastic layer for active damping of geometrically nonlinear vibrations of smart composite plates using the higher order plate theory and finite element method with discretizing the plate by eight-node isoparametric quadrilateral elements. Recently the approaches suggested in [19,20] for solving the problem on free nonlinear vibrations of elastic plates in a viscoelastic medium, damping features of which are governed by the Riemann-Liouville derivatives of the fractional order, and in [45] for studying the dynamic response of the fractional Duffing oscillator subjected to harmonic loading have been generalized for the case of forced vibrations of a simply-supported nonlinear thin elastic plate under the conditions of different internal resonances, when two or three natural modes corresponding to mutually orthogonal displacements are coupled [46][47][48][49]. In the present paper, the procedure proposed in [20] for solving the problem of free nonlinear vibrations of elastic plates in a fractional derivative viscoelastic medium, when the damped motion is described by a set of three nonlinear equations, has been extended for the case of free vibrations of a simply-supported fractionally damped nonlinear thin elastic plate, the motion of which is described by five equations involving shear deformations and rotary inertia.…”

Section: Introductionmentioning

confidence: 99%

Classification of Internal Resonances in Nonlinear Fractionaly Damped Uflyand-Mindlin Plates

Shitikova¹,

Osipova²

2020

IJCCSE

View full text Add to dashboard Cite

In the present paper, the nonlinear free vibrations of fractionally damped plates are studied, equations of motion of which take the rotary inertia and shear deformations into account and involve fivecoupled nonlinear differentialequations in terms of three mutually orthogonal displacements and two angles of rotation. The procedure resulting in decoupling linear parts of equations has been proposed with further utilization of the generalized method of multiple time scales for solving nonlinear governing equations of motion, in so doing the amplitude functions have been expanded into power series in terms of the small parameter and depend on differenttime scales. The occurrence of the internal or combinational resonances in Uflyand-Mindlinplates has been revealed and classified

show abstract

“…Considering the power law, we can use Riemann Liouville fractional integral for constitutive law relating deformation-stress, and Caputo fractional derivative for stress-strain constitutive law [14,57]. Several works investigated such modeling for bio engineering [36,37,40], visco-elasto-plastic modeling for power law dependent stress-strain [55] and more [45,52]. In the present paper, we consider a general case of fractional constitutive laws through distributed order differential equations (DODEs), where employing specific material distributions functions, we recover a fractional Kelvin-Voigt viscoelastic constitutive model to study time dependent frequency response of the system.…”

Section: Introductionmentioning

confidence: 99%

Vibration Analysis of Geometrically Nonlinear and Fractional Viscoelastic Cantilever Beams

Pegah¹,

Kharazmi²,

Suzuki³

et al. 2019

Preprint

View full text Add to dashboard Cite

We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize the extended Hamilton's principle to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model. By spectral decomposition in space, the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs. We use direct numerical integration in the decoupled system, in which we observe the anomalous power-law decay rate of amplitude in the linearized model. We further develop a semi-analytical scheme to solve the nonlinear equations, using method of multiple scales as a perturbation technique. We replace the expensive numerical time integration with a cubic algebraic equation to solve for frequency response of the system. We observe the super sensitivity of response amplitude to the fractional model parameters at free vibration, and bifurcation in steady-state amplitude at primary resonance.

show abstract

Interaction of internal and external resonances during force driven vibrations of a nonlinear thin plate embedded into a fractional derivative medium

Cited by 6 publications

References 5 publications

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Structures

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Structures

Classification of Internal Resonances in Nonlinear Fractionaly Damped Uflyand-Mindlin Plates

Vibration Analysis of Geometrically Nonlinear and Fractional Viscoelastic Cantilever Beams

Contact Info

Product

Resources

About