Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load

Freundlich, Jan

doi:10.15632/jtam-pl.54.4.1433

Cited by 22 publications

(15 citation statements)

References 19 publications

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“…The identity above simplifies the solution of the system of differential equations Eq. (35) Due to the presence of the fractional derivative in the equations of motion, the use of the method presented in the author's previous paper [32] was practically impossible, due to the considerable difficulties in computing the convolution of the fractional Green function and the forcing function [31,32].…”

Section: Problem Formulationmentioning

confidence: 99%

“…In their paper, the transient response of the beam to step loading using a continuum formulation and the finite element method was presented. Later, the dynamics of beams with viscoelastic properties described by the fractional Kelvin-Voigt model were studied in various works [28,[30][31][32][33]. It is the most commonly used fractional model of a viscoelastic material in modeling the dynamics of beams.…”

Section: Introductionmentioning

confidence: 99%

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Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Freundlich

2021

Meccanica

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The presented work concerns the kinematically excited transient vibrations of a cantilever beam with a mass element fixed to its free end. The Euler–Bernoulli beam theory and the fractional Zener model of the beam material are assumed. A fractional Caputo derivative is used to formulate a viscoelastic material law. A characteristic equation, modal frequencies, eigenfunction and orthogonality conditions are achieved for the beam considered. The equations of motion of the system are solved numerically. A numerical solution of a multi-term fractional differential equation is obtained by means of a conversion to a mixed system of ordinary and fractional differential equations, each of the order of $$0 < \gamma \le 1$$ 0 < γ ≤ 1 . The transient time histories of the beam vibrations during the passage through resonance are calculated. A comparison between the beam responses obtained with a fractional and an integer viscoelastic material model is presented. The calculations performed reveal that use of the fractional damping affects on the time histories of the system. The calculated beam responses show that for some values of the order of the fractional derivative $$\gamma$$ γ , the amplitudes occurring in the area of the second resonance are greater than those obtained in the area of the first resonance, which does not occur in the case of the integer order of the fractional derivative. Moreover, an evaluation is made of the difference between the results obtained for the calculations using the fractional Zener model and the fractional Kelvin model. It is shown that for some physical beam parameters, the calculation results obtained using both models are virtually the same for both models, which means that the the simpler, fractional Kelvin–Voigt material can be used instead of the fractional Zener material model. This simplifies the solution and decreases the time needed to make the numerical calculations.

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Section: Problem Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Freundlich

2021

Meccanica

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“…They have used both the mode superposition method and Fourier transform method for calculating the dynamic response of the system. Freundlich [14] has obtained the dynamic response of an Euler-Bernoulli simply supported beam subjected to a moving force load using Green's function. The author has included viscoelastic damping using Kelvin-Voigt model.…”

Section: Review On Viscoelastic Dampingmentioning

confidence: 99%

Hybrid vibration control of a Two-Link Flexible manipulator

Mishra

Singh

2019

SN Appl. Sci.

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This research work deals with tip vibration control of a Two-Link Flexible manipulator using hybrid control technique. This technique involves the implementation of unconstrained viscoelastic damping layer on the links in conjunction with active damping using piezoelectric sensors and actuators. Mathematical modelling of the complete system is done using the finite element approach in the inertial frame. Viscoelastic damping is modelled using Kelvin-Voigt elements for which a damping matrix is derived. Active damping is modelled as time-dependent uniformly distributed load applied by the piezoelectric actuator on the flexible link working under feedback control. The angular and linear velocities of the tips of flexible links are used for direct feedback. The unconstrained viscoelastic damping layer effectively reduces the vibrations of the system. The effectiveness of the active control depends upon the relative position of sensors and actuators on the links. The novelty of the work lies in control of torsional and flexural vibrations through the application of passive and active damping methods to the non-inertial frames represented by the manipulator links.

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“…33 studied the forced vibration of an elastic rod loaded by an axial compressive force, resting on a fractionally-damped Kelvin–Voigt foundation. Freundlich 34 studied the dynamic behaviour of a fractionally-damped beam, subject to a moving concentrated force. The effects of the order of the fractional derivative on natural frequency, damping coefficient, loss factor and dynamic displacement are discussed.…”

Section: Introductionmentioning

confidence: 99%

Dynamic response of Euler–Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load

Praharaj

Datta

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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The dynamic behaviour of an Euler–Bernoulli beam resting on the fractionally damped viscoelastic foundation subjected to a moving point load is investigated. The fractional-order derivative-based Kelvin–Voigt model describes the rheological properties of the viscoelastic foundation. The Riemann–Liouville fractional derivative model is applied for a fractional derivative order. The modal superposition method and Triangular strip matrix approach are applied to solve the fractional differential equation of motion. The dependence of the modal convergence on the system parameters is studied. The influences of (a) the fractional order of derivative, (b) the speed of the moving point load and (c) the foundation parameters on the dynamic response of the system are studied and conclusions are drawn. The damping of the beam-foundation system increases with increasing the order of derivative, leading to a decrease in the dynamic amplification factor. The results are compared with those using the classical integer-order derivative-based foundation model. The classical foundation model over-predicts the damping and under-predicts the dynamic deflections and stresses. The results of the classical (integer-order) foundation model are verified with literature.

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Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load

Cited by 22 publications

References 19 publications

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Hybrid vibration control of a Two-Link Flexible manipulator

Dynamic response of Euler–Bernoulli beam resting on fractionally damped viscoelastic foundation subjected to a moving point load

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