We propose a novel mathematical framework to examine the free damped
 transverse vibration of a nanobeam by using the nonlocal theory of Eringen
 and fractional derivative viscoelasticity. The motion equation of a nanobeam
 with arbitrary attached nanoparticle is derived by considering the nonlocal
 viscoelastic constitutive equation involving fractional order derivatives and
 using the Euler-Bernoulli beam theory. The solution is proposed by using the
 method of separation of variables. Eigenvalues and mode shapes are determined
 for three typical boundary conditions. The fractional order differential
 equation in terms of a time function is solved by using the Laplace transform
 method. Time dependent behavior is examined by observing the time function
 for different values of fractional order parameter and different ratios of
 other parameters in the model. Validation study is performed by comparing the
 obtained results for a special case of our model with corresponding molecular
 dynamics simulation results found in the literature. [Projekat Ministarstva
 nauke Republike Srbije, br. OI 174001, br. OI 174011 i br. TR 35006]

/pdf/nonlocal-vibration-of-a-fractional-order-viscoelastic-4o5gw2ifpz.pdf

Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle

Vibration analysis of pipes conveying fluid resting on a fractional Kelvin-Voigt viscoelastic foundation with general boundary conditions

Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation

Stability approach to the fractional variational iteration method used for the dynamic analysis of viscoelastic beams

This paper presents a new numerical method to solve the constitutive equations of fractional-order viscoelastic Euler–Bernoulli beams. Firstly, the constitutive equation of Euler–Bernoulli beams is established by analyzing the constitutive relation between the fractional viscoelastic materials. Secondly, the constitutive equation of the beams is transformed into a matrix equation by skillfully using a Quasi-Legendre polynomial in the time domain, which can greatly simplify the solution process. Then the matrix equation is discretized and solved, and the numerical solutions are obtained. Finally, dynamic analysis of two different fractional viscoelastic materials is carried out by numerical experiments, and the influences of time on displacements are considered for the first time. With the change of time and position, displacements under different external loads are obtained for the polybutadiene beams and butyl B252 beams, and the change law of displacements is found. In addition, the performances of the two materials are compared and analyzed.

/pdf/a-numerical-method-for-solving-fractional-order-viscoelastic-1wqqm6vfc0.pdf

A numerical method for solving fractional-order viscoelastic Euler–Bernoulli beams

The paper presents vibration analysis of a simply supported beam with a fractional order viscoelastic material model The Bernoulli-Euler beam model is considered The beam is excited by the supports movement The Riemann - Liouville frac- tional derivative of order 0 <α 1 is applied In the first stage, the steady-state vibrations of the beam are analyzed and therefore the Riemann - Liouville fractional derivative with lower terminal at −∞ is assumed This assumption simplifies solution of the fractional differential equations and enables us to directly obtain amplitude-frequency characteristics of the examined system The characteristics are obtained for various values of fractional derivative of order α and values of the Voigt material model parameters The studies show that the selection of appropriate damping coefficients and fractional derivative order of damping model enables us to fit more accurately dynamic characteristic of the beam in comparison with using integer order derivative damping model

https://downloads.hindawi.com/journals/sv/2013/126735.pdf

Vibrations of a simply supported beam with a fractional viscoelastic material model - supports movement excitation

In the paper, the dynamic response of a simply supported viscoelastic beam of the fractional derivative type to a moving force load is studied. The Bernoulli-Euler beam with the fractional derivative viscoelastic Kelvin-Voigt material model is considered. The Riemann-Liouville fractional derivative of the order 0 < α ˂ 1 is used. The forced-vibration solution of the beam is determined using the mode superposition method. A convolution integral of fractional Green’s function and forcing function is used to achieve the beam response. Green’s function is formulated by two terms. The first term describes damped vibrations around the drifting equilibrium position, while the second term describes the drift of the equilibrium position. The solution is obtained analytically whereas dynamic responses are calculated numerically. A comparison between the results obtained using the fractional and integer viscoelastic material models is performed. Next, the effects of the order of the fractional derivative and velocity of the moving force on the dynamic response of the beam are studied. In the analysed system, the effect of the term describing the drift of the equilibrium position on the beam deflection is negligible in comparison with the first term and therefore can be omitted. The calculated responses of the beam with the fractional material model are similar to those presented in works of other authors.

http://jtam.pl/pdf-101937-33498?filename=Dynamic response of a.pdf

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

The presented work deals with nonlinear dynamics of a three degree of freedom system with a spherical pendulum and a damper of the fractional type. Vibrations in the vicinity of the internal and external resonance are considered. The system consists of a block suspended from a linear spring and a fractional damper, and a spherical pendulum suspended from the block. The viscoelastic properties of the damper are described using the Caputo fractional derivative. The fractional derivative of an order of $$0 < \alpha \le 1$$
 is assumed. The impact of a fractional order derivative on the system with a spherical pendulum is studied. Time histories, the internal and external resonance, bifurcation diagrams, Poincare maps and the Lyapunov exponents have been calculated for various orders of a fractional derivative. Chaotic motion has been found for some system parameters.

/pdf/dynamics-of-a-coupled-mechanical-system-containing-a-3b4a9diudt.pdf

Dynamics of a coupled mechanical system containing a spherical pendulum and a fractional damper

The nonlinear response of a three degree of freedom vibratory system with spherical pendulum in the neighbourhood internal and external resonance is investigated. It was assumed that spherical pendulum is suspended to the main body which is suspended by the element characterized by elasticity and damping and is excited harmonically in the vertical direction. The equation of motion have bean solved numerically. In this type system one mode of vibration may excite or damp another one, and for except different kinds of periodic vibrations there may also appear chaotic vibration.

/pdf/the-dynamics-of-a-coupled-mechanical-system-with-spherical-ttzoauq227.pdf

Jan Freundlich

Papers

Vibrations of a simply supported beam with a fractional viscoelastic material model - supports movement excitation

Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation

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

Dynamics of a coupled mechanical system containing a spherical pendulum and a fractional damper

The Dynamics of a Coupled Mechanical System with Spherical Pendulum