A New Fourier series solution for free vibration of non-uniform beams, resting on variable elastic foundation

Motaghian, Seyedemad; Mofid, Massood; Akin, J.E.

doi:10.24200/sci.2017.4239

Cited by 3 publications

(4 citation statements)

References 30 publications

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“…This is the single most important feature which yields the fast convergence to mode shape determinations. As is well known, it usually requires an order of 10,000 grid points (at least 2D model needed to be drawn) in a finite element type simulation (Ghannadiasl and Golmogany, 2017; Kuo et al, 2006) and about 50 basis functions to converge the Fourier expansion to the current calculation precision (Jovanovic, 2011; Motaghian et al, 2018). As both methods have nonlinear scaling in the computational costs, the crucial factor is trying to reduce the number of grid points or the basis function representations.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…Although it is relatively easy to obtain accurate resonance frequencies, determining the corresponding mode shapes is more difficult. Most of the solution methods such as Fourier basis-set expansion method (Jovanovic, 2011; Motaghian et al, 2018), perturbation methods using Green’s functions (Zhao et al, 2017), and numerical finite element methods (Ghannadiasl and Golmogany, 2017; Kuo et al, 2006) involve very tedious formulations and complicated calculations. For example, it requires a very large number of sinusoidal (sine and cosine) functions in the Fourier analysis to achieve a prescribed accuracy.…”

Section: Introductionmentioning

confidence: 99%

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Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

Chiu

Chao

2021

Journal of Vibration and Control

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We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.

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Section: Numerical Results and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

Chiu

Chao

2021

Journal of Vibration and Control

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“…Ö Z et al [2] take the accelerating moving beam as the research object, and use the perturbation method to solve the problem, analyze the vibration of the system under different moving speed frequencies, and demonstrate the stability of the steady-state solution and the time-bounded amplitude. MOTAGHIAN et al [3] developed an analysis method based on superposition of sine and cosine Fourier series to analyze the free vibration of Euler-Bernoulli beams with variable cross-section, and verified that this method can be applied to different boundary conditions. YAYLI [4] Based on the elastic theory in differential form, the free transverse vibration behavior of an elastic beam with rotational restraint boundary condition is studied and the influence of rotational restraint on the natural frequency is obtained.…”

Section: Introductionmentioning

confidence: 99%

Research on transverse vibration characteristics of beam-strut composite system based on analytical method

Liang

2023

Third International Conference on Mechanical Design and Simulation (MDS 2023)

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In this paper, the beam-strut composite system is taken as a simplified theoretical model, and the transverse vibration of the model is modeled by analytical method, and the reliability of the model is verified by numerical substitution. Based on this model, the influence of each mode of the system on the final displacement, the lateral vibration response of the system at different excitation frequencies and the influence of excitation displacement velocity on the lateral vibration of the system are analyzed. The results show that the higher the mode order, the smaller the influence of the mode on the final displacement of the system; the system will not resonate at the actual working frequency; the lateral vibration displacement of the system will increase with the increase of excitation displacement speed.

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“…In practical projects, when using beams with variable thickness, the system not only is lighter, but also the aesthetics of the structures can be enhanced. Motaghian et al [22] developed a new Fourier series solution for free vibration analysis of non-uniform beams resting on elastic foundation using Euler-Bernoulli beam theory. Shahba et al [23] applied the differential transform element method and differential quadrature element method of lowest order to simulate the free vibration analysis of tapered Euler-Bernoulli beams made of axially FGM.…”

Section: Introductionmentioning

confidence: 99%

A New Beam Model for Simulation of the Mechanical Behaviour of Variable Thickness Functionally Graded Material Beams Based on Modified First Order Shear Deformation Theory

et al. 2019

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There are many beam models to simulate the variable thickness functionally graded material (FGM) beam, each model has advantages and disadvantages in computer aided engineering of the mechanical behavior of this beam. In this work, a new model of beam is presented to study the mechanical static bending, free vibration, and buckling behavior of the variable thickness functionally graded material beams. The formulations are based on modified first order shear deformation theory and interpolating polynomials. This new beam model is free of shear-locking for both thick and thin beams, is easy to apply in computation, and has efficiency in simulating the variable thickness beams. The effects of some parameters, such as the power-law material index, degree of non-uniformity index, and the length-to-height ratio, on the mechanical behavior of the variable thickness FGM beam are considered.

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A New Fourier series solution for free vibration of non-uniform beams, resting on variable elastic foundation

Cited by 3 publications

References 30 publications

Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

Research on transverse vibration characteristics of beam-strut composite system based on analytical method

A New Beam Model for Simulation of the Mechanical Behaviour of Variable Thickness Functionally Graded Material Beams Based on Modified First Order Shear Deformation Theory

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