Analytical Solution for an Infinite Euler-Bernoulli Beam on a Viscoelastic Foundation Subjected to Arbitrary Dynamic Loads

SUMMARYIn this research, using an approximate analytical method, vibration analysis of a 3-PRP (active prismatic—P, passive revolute—R, passive prismatic—P) planar parallel robot having a flexible moving platform is presented. A specific architecture of the 3-PRP parallel robot, also known as the ST (Star-Triangle) parallel robot, is considered. The moving platform of the robot, called the star, is assumed to be made of three flexible beams shaped like a star. For analytical modeling, each of the three beams of the star is assumed to be a discrete Euler–Bernoulli beam with a passive prismatic joint. Continuity equations at the center of the star are used to relate vibrations of the three beams. The vibration behavior of each beam is modeled using previously developed constrained motion equations for a planar Euler–Bernoulli beam having a prismatic joint. In this paper, previously presented “constrained assumed modes method” is further developed to solve the constrained motion equation for the ST parallel robot. The solution method is used to obtain the vibration of the robot for the inverse dynamics problem and simultaneously provides generalized constraint forces. Furthermore, the solution method can be used for the direct dynamics problem of the ST robot. Several input trajectories are considered to investigate the different behavior for the center of the star. For each of the trajectories, three different groups of mode shapes are considered and their vibrational responses are compared. In this research, for the first time, effects of the passive prismatic joint parameters such as mass, rotational moment of inertia, and its actual length are considered in an analytical model. Finally, the analytical solution and a FEM (Finite Element Method) software solution are compared.

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Analytical solutions for steady state responses of an infinite Euler-Bernoulli beam on a nonlinear viscoelastic foundation subjected to a harmonic moving load

Journal of Sound and Vibration ◽

10.1016/j.jsv.2020.115271 ◽

2020 ◽

Vol 476 ◽

pp. 115271 ◽

Cited By ~ 2

Author(s):

Bin Zhen ◽

Jian Xu ◽

Jianqiao Sun

Keyword(s):

Steady State ◽

Analytical Solutions ◽

Moving Load ◽

Viscoelastic Foundation ◽

Bernoulli Beam ◽

Nonlinear Viscoelastic ◽

Steady State Responses ◽

Euler Bernoulli Beam ◽

State Responses

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

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406220932597 ◽

2020 ◽

Vol 234 (24) ◽

pp. 4801-4812 ◽

Cited By ~ 1

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.

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On the Dynamics of Fractional Visco-Elastic Beams

Volume 4: Dynamics, Control and Uncertainty, Parts A and B ◽

10.1115/imece2012-86566 ◽

2012 ◽

Cited By ~ 7

Author(s):

Salvatore Di Lorenzo ◽

Francesco P. Pinnola ◽

Antonina Pirrotta

Keyword(s):

Elastic Beam ◽

Constitutive Law ◽

Dynamic Loads ◽

Elastic Behavior ◽

Advanced Manufacturing ◽

Suitable Model ◽

Bernoulli Beam ◽

Static Behavior ◽

Creep Tests ◽

Euler Bernoulli Beam

With increasing advanced manufacturing process, visco-elastic materials are very attractive for mitigation of vibrations, provided that you may have advanced studies for capturing the realistic behavior of such materials. Experimental verification of the visco-elastic behavior is limited to some well-known low order models as the Maxwell or Kelvin models. However, both models are not sufficient to model the visco-elastic behavior of real materials, since only the Maxwell type can capture the relaxation tests and the Kelvin the creep tests, respectively. Very recently, it has been stressed that the most suitable model for capturing the visco-elastic behavior is the spring-pot, characterized by a fractional constitutive law. Based on this assumption, the quasi-static behavior has been investigated very recently, however for noise control there is a need of exploiting the dynamic behavior of such a fractional visco-elastic beam. The present paper introduces the dynamic response of fractional visco-elastic Euler-Bernoulli beam under dynamic loads.

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A Wave-Based Analytical Solution to Free Vibrations in a Combined Euler–Bernoulli Beam/Frame and a Two-Degree-of-Freedom Spring–Mass System

Journal of Vibration and Acoustics ◽

10.1115/1.4039961 ◽

2018 ◽

Vol 140 (6) ◽

Cited By ~ 1

Author(s):

C. Mei

Keyword(s):

Analytical Solution ◽

Coupling Effect ◽

Free Vibrations ◽

Degree Of Freedom ◽

Transverse Motion ◽

Bernoulli Beam ◽

Combined System ◽

Mass System ◽

Euler Bernoulli Beam ◽

Two Degree Of Freedom

In this paper, natural frequencies and modeshapes of a transversely vibrating Euler–Bernoulli beam carrying a discrete two-degree-of-freedom (2DOF) spring–mass system are obtained from a wave vibration point of view in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external forces applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam and 2DOF spring–mass system, the vibrating discrete spring–mass system injects waves into the distributed beam through the spring forces at the two spring attached points. Assembling these wave relations in the beam provides an analytical solution to vibrations of the combined system. Accuracy of the proposed wave analysis approach is validated through comparisons to available results. This wave-based approach is further extended to analyze vibrations in a planar portal frame that carries a discrete 2DOF spring–mass system, where in addition to the transverse motion, the axial motion must be included due to the coupling effect at the angled joint of the frame. The wave vibration approach is seen to provide a systematic and concise technique for solving vibration problems in combined distributed and discrete systems.

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