Quasi-Static Analysis of Beam Described by Fractional Derivative Kelvin Viscoelastic Model under Lateral Load

2011 ◽  
Vol 189-193 ◽  
pp. 3391-3394 ◽  
Author(s):  
Qing Zhao Yao ◽  
Lin Chao Liu ◽  
Qi Fang Yan
Keyword(s):  
Fractional Derivative ◽  
Mechanical Behavior ◽  
Constitutive Relations ◽  
Dynamical Behavior ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Bernoulli Beam ◽  
The Laplace Transform ◽  
Static Mechanical ◽  
Euler Bernoulli Beam

The beam is assumed to obey a three-dimensional viscoelastic fractional derivative constitutive relations, the mathematical model and governing equations of the quasi-static and dynamical behavior of a viscoelastic Euler-Bernoulli beam are established, the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative model is investigated, and the analytical solution is obtained by considering the properties of the Laplace transform of Mittag-Leffler function and the properties of fractional derivative. The result indicate that the quasi-static mechanical behavior of Euler-Bernoulli beam described by fractional derivative viscoelastic model can reduced to the cases of classic viscoelastic and elastic, the order of fractional derivative has great effect on the quasi-static mechanical behavior of Euler-Bernoulli beam.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

Active vibration control of a nonlinear three-dimensional Euler–Bernoulli beam

2016 ◽  
Vol 23 (19) ◽  
pp. 3196-3215 ◽  
Author(s):  
Wei He ◽  
Chuan Yang ◽  
Juxing Zhu ◽  
Jin-Kun Liu ◽  
Xiuyu He
Keyword(s):  
Differential Equations ◽  
Boundary Control ◽  
Coupling Effect ◽  
Three Dimensional ◽  
Uniform Boundedness ◽  
Transverse Displacement ◽  
Axial Deformation ◽  
Bernoulli Beam ◽  
Parameter Uncertainties ◽  
Euler Bernoulli Beam

In this paper, boundary control is designed to suppress the vibration of a nonlinear three-dimensional Euler–Bernoulli beam. Considering the coupling effect between the axial deformation and the transverse displacement, the dynamics of the beam are modeled as a distributed parameter system described by three partial differential equations (PDEs) and 12 ordinary differential equations (ODEs). Firstly, model-based boundary control is designed based on a mathematical model of the system. Subsequently, adaptive control is proposed when there are parameter uncertainties in the model. The uniform boundedness and uniform ultimate boundedness are proved under the proposed control laws. Finally, numerical simulations illustrate the effectiveness of the results.

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

2020 ◽  
Vol 234 (24) ◽  
pp. 4801-4812 ◽  
Author(s):  
Rajendra K Praharaj ◽  
Nabanita Datta
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Load ◽  
Order Derivative ◽  
Viscoelastic Foundation ◽  
Bernoulli Beam ◽  
Integer Order ◽  
Foundation Model ◽  
Euler Bernoulli Beam

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.

scholarly journals An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework

2017 ◽  
Vol 24 (2) ◽  
pp. 361-376 ◽  
Author(s):  
Cagan Diyaroglu ◽  
Erkan Oterkus ◽  
Selda Oterkus
Keyword(s):  
Three Dimensional ◽  
Equation Of Motion ◽  
Computational Time ◽  
Benchmark Problems ◽  
Geometrical Shape ◽  
Bernoulli Beam ◽  
Lagrange Equations ◽  
The World ◽  
Euler Bernoulli Beam ◽  
Model Structures

Every object in the world has a three-dimensional geometrical shape and it is usually possible to model structures in a three-dimensional fashion, although this approach can be computationally expensive. In order to reduce computational time, the three-dimensional geometry can be simplified as a beam, plate or shell type of structure depending on the geometry and loading. This simplification should also be accurately reflected in the formulation that is used for the analysis. In this study, such an approach is presented by developing an Euler–Bernoulli beam formulation within ordinary state-based peridynamic framework. The equation of motion is obtained by utilizing Euler–Lagrange equations. The accuracy of the formulation is validated by considering various benchmark problems subjected to different loading and displacement/rotation boundary conditions.

Three-Dimensional Ballistic Equations of Rocket Considering the Elastic Effects

2014 ◽  
Vol 628 ◽  
pp. 157-160
Author(s):  
Fu Liu
Keyword(s):  
Body Length ◽  
Aerodynamic Force ◽  
Three Dimensional ◽  
Diameter Ratio ◽  
The Body ◽  
Dynamic Equations ◽  
Bernoulli Beam ◽  
Ratio Increase ◽  
Transversal Vibration ◽  
Euler Bernoulli Beam

Because the shell of modern rocket is more and more thin and the body length to diameter ratio increase constantly, influence to trajectory caused by elasticity effect cannot be ignored any more. Based on the model of Euler-Bernoulli beam, the dynamic equations of flexible rocket are derived by using of Hamilton principle. The influence of distributing mass and aerodynamic force and the coupling between longitudinal and transversal vibration are considered. Example shows ballistic range and altitude of rocket are influenced by elastic effects observably.

Vibration control for the payload at the end of a nonlinear three-dimensional Euler–Bernoulli beam with input constraints

2017 ◽  
Vol 40 (10) ◽  
pp. 3088-3094 ◽  
Author(s):  
Ning Ji ◽  
Jinkun Liu
Keyword(s):  
Control Problem ◽  
Vibration Control ◽  
Three Dimensional ◽  
Elastic Vibration ◽  
Hyperbolic Function ◽  
Input Constraints ◽  
Bernoulli Beam ◽  
Physical Conditions ◽  
Control Scheme ◽  
Euler Bernoulli Beam

In this paper, the vibration control problem for the payload at the end of a three-dimensional Euler–Bernoulli beam in the presence of input constraints and input disturbances is addressed. Disturbance observers are designed to estimate the disturbances on the tip payload. Based on the disturbance observers, a boundary control scheme is designed to suppress elastic vibration for the payload at the end of the beam. The smooth hyperbolic function is applied for the proposed control scheme, which can satisfy physical conditions and input constraints. It is proved that the proposed control scheme can be guaranteed in handling input constraints and disturbances. Finally, numerical simulations illustrate the effectiveness of the results.

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