Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Hozhabrossadati, Seyed Mojtaba

doi:10.1080/19373260.2015.1054957

Cited by 9 publications

(5 citation statements)

References 23 publications

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“…Firstly, the Euler-Bernoulli beam theory and the extended Hamilton's principle are adopted to obtain the equation of motion of the cantilever [2]. Then, the model reduction is conducted by the Galerkin's method along with the mode function given in [34]. After that, the equation similar to Eq.…”

Section: 4mentioning

confidence: 99%

Constrained parameter-splitting perturbation method for the improved solutions of nonlinear vibrations of Euler-Bernoulli cantilever

et al. 2022

Preprint

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In this paper, a new method named constrained parameter-splitting perturbation method for improving the solutions obtained from the parameter-splitting perturbation method is proposed for solving the problems in some extremal cases, such as the strongly nonlinear vibration of Euler-Bernoulli cantilever. The proposed method takes the advantages of both the perturbation method and the harmonic balance method. The idea is that the solution obtained by the parameter-splitting perturbation method is substituted into the equation of motion and then the accumulative error of the equation is minimized for determining the unknown splitting parameters under the constrains constructed under the frame of harmonic balance method. The forced vibration of an oscillator with cubic geometric nonlinearity and inertia nonlinearity and the forced vibration of a planar micro cantilever beam with a lumped tip mass are studied as examples to reveal the efficacy of the proposed method. The inspection of the steady-state response is conducted by means of comparing the frequency-response curves obtained by the proposed method with those obtained by the numerical continuation method and harmonic balance method, respectively, to show the efficacy and the advantages of the proposed method. Meanwhile, the nonlinear ordering effect on the solutions of the proposed method is also studied by comparing the results obtained by using different nonlinear orderings in the systems.

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Section: 4mentioning

confidence: 99%

Constrained parameter-splitting perturbation method for the improved solutions of nonlinear vibrations of Euler-Bernoulli cantilever

et al. 2022

Preprint

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“…Chebyshev polynomials are used in [14] for the solution to a beam of exponentially varying width clamped at one end with the other end attached to a translational or torsional spring. Unfortunately, it is not always possible to determine an exact analytical solution or express the solution in terms of these special functions with exception of very specific types of nonuniformities some of which are discussed in [7][8][9][10][11][12][13][14]. This is a major difficulty with obtaining a dynamic solution for spatially varying beams.…”

Section: Introductionmentioning

confidence: 99%

“…For example, [7,8] considered a beam with an exponentially varying width. It is shown that for this type of geometry the spatial ordinary differential equation (ODE), which is used to determine the natural frequencies and mode shapes, reduces to one with constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

Martin

Salehian

2018

Shock and Vibration

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The Lindstedt-Poincaré method is applied to a nonuniform Euler-Bernoulli beam model for the free transverse vibrations of the system. The nonuniformities in the system include spatially varying and piecewise continuous bending stiffness and mass per unit length. The expression for the natural frequencies is obtained up to second-order and the expression for the mode shapes is obtained up to first-order. The explicit dependence of the natural frequencies and mode shapes on reference values for the bending stiffness and the mass per unit length of the system is determined. Multiple methods for choosing these reference values are presented and are compared using numerical examples.

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“…In this study, the TB under axial load rests on the arbitrary variable elastic foundation. On the other hand, Hozhabrossadati [22] presented the exact solution for free vibration of the cantilever EBB with exponentially varying width. The mass-spring system is joined to EBB at the free end.…”

Section: Introductionmentioning

confidence: 99%

Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Ghannadiasl

Zamiri

Borhanifar

2020

J Braz. Soc. Mech. Sci. Eng.

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Natural frequencies and free vibration are important characteristics of beams with non-uniform cross section. Hence, the solution for free vibrations of non-uniform beams is presented using a Laguerre collocation method. The elastically restrained beam model is based on the Euler-Bernoulli theory. Also, the non-uniform beam is rested on a non-uniform foundation (Winkler type). The Laguerre collocation method is introduced for solving the differential equation. This approach reduces the governing differential equation to a system of algebraic equations, and finally, the problem is greatly simplified. Properties of Laguerre polynomials and the operational matrix of derivation are first presented. Eventually, the proposed method is applied for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.

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Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

Cited by 9 publications

References 23 publications

Constrained parameter-splitting perturbation method for the improved solutions of nonlinear vibrations of Euler-Bernoulli cantilever

Constrained parameter-splitting perturbation method for the improved solutions of nonlinear vibrations of Euler-Bernoulli cantilever

Reference Value Selection in a Perturbation Theory Applied to Nonuniform Beams

Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

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