Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field

doi:10.1080/15397734.2011.557972

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Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field

Journal Article•DOI•

Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field

Barun Pratiher¹, Santosha K. Dwivedy²•Institutions (2)

Indian Institute of Technology Dhanbad¹, Indian Institute of Technology Guwahati²

19 May 2011-Mechanics Based Design of Structures and Machines (Taylor & Francis)-Vol. 39, Iss: 3, pp 378-391

TL;DR: In this paper, the nonlinear vibration of a cantilever beam with tip mass subjected to periodically varying axial load and magnetic field has been studied and the temporal equation of motion of the system containing linear and nonlinear parametric excitation terms along with nonlinear damping, geometric and inertial types of nonlinear terms has been derived and solved using method of multiple scales.

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Abstract: In this paper, nonlinear vibration of a cantilever beam with tip mass subjected to periodically varying axial load and magnetic field has been studied. The temporal equation of motion of the system containing linear and nonlinear parametric excitation terms along with nonlinear damping, geometric and inertial types of nonlinear terms has been derived and solved using method of multiple scales. The stability and bifurcation analysis for three different resonance conditions were investigated. The numerical results demonstrate that while in simple resonance case with increase in magnetic field strength, the system becomes unstable, in principal parametric or simultaneous resonance cases, the vibration can be reduced significantly by increasing the magnetic field strength. The present work will be very useful for feed forward vibration control of magnetoelastic beams which are used nowadays in many industrial applications.

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Journal Article•DOI•

Prediction of resonantly forced vibrations for suspended footbridges

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Mariella Diaferio¹•Institutions (1)

Instituto Politécnico Nacional¹

02 Sep 2020-Mechanics Based Design of Structures and Machines

TL;DR: In this article, the performance of suspended footbridges under pedestrian loads was investigated and the authors pointed out the possible activation of large amplitude oscillations in the pedestrian load on the bridges.

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Abstract: The aim of the present paper is to investigate the performance of suspended footbridges under pedestrian loads. Indeed, several Authors have underlined the possible activation of large amplitude os...

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3 citations

Journal Article•DOI•

The magneto-elastic internal resonances of rectangular conductive thin plate with different size ratios

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J. Li¹, Y. D. Hu¹, Yanan Wang²•Institutions (2)

Yanshan University¹, Deakin University²

01 Oct 2018-Journal of Mechanics

3 citations

Journal Article•DOI•

Study on Cantilever Beam Tip Response with Various Harmonic Frequencies by Using EDISON Co-rotational Plane Beam-Dynamic Tip Load

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Chul-Woo Park, HyunShig Joo, Hanyeol Ryu¹, SangJoon Shin¹•Institutions (1)

Seoul National University¹

01 Oct 2015

TL;DR: In this paper, Euler-Bernoulli beam theories (EB-beam) are used, and Fast Fourier Transformation (FFT) analysis is then employed to extract their natural frequencies using both analytical approach and Co-rotational plane beam(CR-beam), EDISON program.

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Abstract: In this paper, Euler-Bernoulli beam theories(EB-beam) are used, and Fast Fourier Transformation(FFT) analysis is then employed to extract their natural frequencies using both analytical approach and Co-rotational plane beam(CR-beam) EDISON program. EB-beam is used to analyze a spring-mass system with a single degree of freedom. Sinusoidal force with various frequencies and constant magnitude are applied to tip of each beam. After the oscillatory tip response is observed in EB-beam, it decreases and finally converges to the so-called ‘steady-state.’ The decreasing rate of the tip deflection with respect to time is reduced when the forcing frequency is increased. Although the tip deflection is found to be independent of the excitation frequency, it turns out that time to reach the steady state response is dependent on the forcing frequency.

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1 citations

Journal Article•DOI•

Study on Vibration Characteristics in Terms of Airfoil Cross-Sectional Shape by using Co-Rotational Plane Beam Transient Analysis

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Se-Ill Kim, Yong-Se Kim, Chul-Woo Park, SangJoon Shin

01 Oct 2016

TL;DR: In this paper, the EDISON co-rotational plane beam-transient analysis is used for large rotation and small strain analysis of an aircraft wing as a cantilevered beam, and the natural frequencies of each airfoil cross-sectional shape were estimated using VABS program and fast Fourier transformation (FFT).

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Abstract: In this paper, vibration characteristics in terms of the airfoil cross-sectional shape was examined by using the EDISON co-rotational plane beam-transient analysis. Co-Rotational plane beam analysis is appropriate for large rotation and small strain. Assuming aircraft wing as a cantilevered beam, natural frequencies of each airfoil cross-sectional shape were estimated using VABS program and fast Fourier transformation(FFT). VABS conducts finite element analysis on the cross-section including the detailed geometry and material distribution to estimate the beam sectional properties. Under the same airfoil geometric configuration and material selection, variation of material induced difference in the deflection and natural frequencies. It was observed that variation of the natural frequency was dependent on variation of the airfoil shape and material.

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1 citations

Journal Article•DOI•

Study of transient analysis of a conductive beam carrying an electrical current subjected to magnetic field with elastically restrained ends

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Elham Tahmasebi, Nariman Ashrafi Khorasani

01 Feb 2023-Forces in mechanics

TL;DR: In this paper , the nonlinear vibration behavior of a beam with general boundary conditions that carry an electrical current in the magnetic field is considered. And the effect of the rotational and the translational support flexibilities, magnetic field, and other parameters are evaluated.

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Abstract: The present study considers the nonlinear vibration behavior of a beam with general boundary conditions that carry an electrical current in the magnetic field. This paper discusses the magnetic couple, the transverse magnetic force, the electrical current, and the damper. By contrast, the magnetic field is selected as an arbitrary function of time. Under certain hypotheses, Hamilton's principle is used along with Maxwell's equations to derive the governing equation. An elastically restrained beam carrying an electrical current is also solved using Galerkin's method under a magnetic field. Thus, the effect of the rotational and the translational support flexibilities, the magnetic field, and other parameters are evaluated. For a more detailed investigation, some numerical examples are investigated to present the simplicity and efficiency of this formulation. Based on the numerical results, it is clear that the natural frequency of the ferromagnetic beam is sensitive to the angle and magnetic field. By increasing magnetic field intensity, the magnitude of the natural frequency of the beam increases. But with the increase of the angle, the frequency value decreases. Therefore, at larger angles, the impact of the intensity of the magnetic field will be less. Also, it is determined from the results that the beam deflection in various magnetic fields indicates a significant effect of the boundary conditions, not only on the dynamic response of a damped beam but also on the rate of damping of the response. The dynamic response under the magnetic field is decreased when the beam experiences a stiffer constant in its support. The results are shown that the effect of stiffening for the transitional support is more significant than that of the rotational support. Also, the influence of the boundary constraints becomes smaller when the magnetic field becomes smaller.

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Journal Article•DOI•

Vibration and Dynamic Instability of a Beam-Plate in a Transverse Magnetic Field

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Francis C. Moon¹, Yih-Hsing Pao²•Institutions (2)

Princeton University¹, Cornell University²

01 Mar 1969-Journal of Applied Mechanics

91 citations

"Nonlinear vibrations and frequency ..." refers background in this paper

...It may also be noted that by neglecting parametric term f2 cos ̄2 q due to the presence of force excitation, the present system can be reduced to that of Pratiher and Dwivedy (2009). It has also been observed that the equation of motion contains additional nonlinear terms of geometric and inertial type than those obtained in Shih et al....
[...]
...It may also be noted that by neglecting parametric term f2 cos ̄2 q due to the presence of force excitation, the present system can be reduced to that of Pratiher and Dwivedy (2009). It has also been observed that the equation of motion contains additional nonlinear terms of geometric and inertial type than those obtained in Shih et al. (1998) and Liu and Chang (2005)....
[...]

Journal Article•DOI•

Parametric instability of a cantilever beam with magnetic field and periodic axial load

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Barun Pratiher¹, Santosha K. Dwivedy¹•Institutions (1)

Indian Institute of Technology Guwahati¹

11 Sep 2007-Journal of Sound and Vibration

TL;DR: In this paper, the parametric instability regions of a cantilever beam with tip mass subjected to time-varying magnetic field and axial force were investigated using second-order method of multiple scales.

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Abstract: The present work deals with the parametric instability regions of a cantilever beam with tip mass subjected to time-varying magnetic field and axial force. The nonlinear temporal differential equation of motion having two frequency parametric excitations is solved using second-order method of multiple scales. The closed-form expressions for the parametric instability regions for three different resonance conditions are determined. The influence of magnetic filed, axial load, damping constant and mass ratio on the parametric instability regions are investigated. These results obtained from perturbation analysis are verified by solving the temporal equation of motion using fourth-order Runge–Kutta method. The instability regions obtained using this method is found to be in good agreement with the experimental result.

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34 citations

"Nonlinear vibrations and frequency ..." refers background or methods in this paper

...The expressions for h1 h2 · · ·h14 are same as those given in Pratiher and Dwivedy (2009)....
[...]
...Following Pratiher and Dwivedy (2009), and using single mode approximation, i....
[...]
...Similar to Pratiher and Dwivedy (2009), using D’Alembert’s principle the following governing differential equation of motion of the system has been obtained in terms of transverse displacement v....
[...]
...It may be recalled from the work of Pratiher and Dwivedy (2009) where only the trivial state instability regions were plotted, that the system is prone to vibration only in the region R1R2....
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...For numerical simulation, a steel beam has been taken similar to that considered in Pratiher and Dwivedy (2007) with length L = 0 5m, width d = 0 005m, depth h = 0 001m, Young’s Modulus E = 194GPa, mass of the beam per unit length m = 0 03965 kg, damping constant cd = 0 01N-s/m, relative permeability r = 3000, material conductivity = 107 Vm−1, and the permeability of the vacuum, 0 = 1 26× 10−6 Hm−1....
[...]

Journal Article•DOI•

Parametric instability of a beam under electromagnetic excitation

[...]

C.-C. Chen¹, M.-K. Yeh²•Institutions (2)

Dahan Institute of Technology¹, National Tsing Hua University²

01 Mar 2001-Journal of Sound and Vibration

TL;DR: In this paper, an electromagnetic device acting like a spring with alternating stiffness was designed to parametrically excite the beam, and the frequency and amplitude of the excitation force were accurately controlled by the AC current flowing through the coil of the electromagnetic device.

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Abstract: The parametric instability of a beam under electromagnetic excitation was investigated experimentally and analytically. In experiment an electromagnetic device, acting like a spring with alternating stiffness, was designed to parametrically excite the beam. The frequency and the amplitude of the excitation force were accurately controlled by the AC current flowing through the coil of the electromagnetic device. Since the excitation force is a non-contact electromagnetic force which acts on the beam in the transverse direction, the disturbances induced by the geometric imperfection of the beam, by the eccentricity of the usual axial excitation force, and the coupling effects between the excitation mechanism and the beam were effectively avoided. The dynamic system was analyzed based on the assumed-modes method. The instability regions of the system were found to be the functions of the modal parameters of the beam and the position, the stiffness of the electromagnetic device for various cantilevered beams. The modal damping ratios of the beam specimens were also identified. The experimental results were found to agree well with the analytical ones.

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32 citations

"Nonlinear vibrations and frequency ..." refers background in this paper

...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field. In all these cases authors have studied only the trivial state responses of the system and parametric instability regions were determined. However, practically, most of the engineering structures exhibit nonlinear behavior which cannot be predicted from these analyses. Few authors’ viz., Kojima and Nagaya (1985), Lu et al. (1995), Shih et al. (1998), Liu and Chang (2005), and Pratiher and Dwivedy (2009) have studied the nonlinear response of the elastic beams subjected to alternating electromagnetic field. For a more detailed review on parametrically excited beam subjected to magnetic field one may refer authors earlier works of Pratiher and Dwivedy (2007, 2009). From these literatures it has been observed that no research has been carried out to find the frequency response for magnetoelastic cantilever beam with tip mass subjected to periodic axial load. Hence, in the present work an attempt has been made to obtain the frequency response curves for such systems. Here, the governing temporal equation of motion of the system has been obtained which contains nonlinear damping, linear and nonlinear parametric excitation terms, in addition to the geometric and inertial types of nonlinear terms. By neglecting the effect of periodically varying axial load, the present system is similar to that of Pratiher and Dwivedy (2009) and by neglecting the geometric and inertial nonlinear terms, the present equation of motion is similar to that of Lu et al. (1995), Shih et al. (1998), and Liu and Chang (2005). The influences of the amplitude of magnetic field strength, attached tip mass, and static and dynamic amplitude of axial load on the frequency response curves have been investigated....
[...]
...Moon and Pao (1969), Wu et al. (2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field....
[...]
...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field. In all these cases authors have studied only the trivial state responses of the system and parametric instability regions were determined. However, practically, most of the engineering structures exhibit nonlinear behavior which cannot be predicted from these analyses. Few authors’ viz., Kojima and Nagaya (1985), Lu et al. (1995), Shih et al. (1998), Liu and Chang (2005), and Pratiher and Dwivedy (2009) have studied the nonlinear response of the elastic beams subjected to alternating electromagnetic field. For a more detailed review on parametrically excited beam subjected to magnetic field one may refer authors earlier works of Pratiher and Dwivedy (2007, 2009). From these literatures it has been observed that no research has been carried out to find the frequency response for magnetoelastic cantilever beam with tip mass subjected to periodic axial load. Hence, in the present work an attempt has been made to obtain the frequency response curves for such systems. Here, the governing temporal equation of motion of the system has been obtained which contains nonlinear damping, linear and nonlinear parametric excitation terms, in addition to the geometric and inertial types of nonlinear terms. By neglecting the effect of periodically varying axial load, the present system is similar to that of Pratiher and Dwivedy (2009) and by neglecting the geometric and inertial nonlinear terms, the present equation of motion is similar to that of Lu et al....
[...]
...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field....
[...]
...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field. In all these cases authors have studied only the trivial state responses of the system and parametric instability regions were determined. However, practically, most of the engineering structures exhibit nonlinear behavior which cannot be predicted from these analyses. Few authors’ viz., Kojima and Nagaya (1985), Lu et al. (1995), Shih et al. (1998), Liu and Chang (2005), and Pratiher and Dwivedy (2009) have studied the nonlinear response of the elastic beams subjected to alternating electromagnetic field....
[...]

Journal Article•DOI•

Vibration analysis of a magneto-elastic beam with general boundary conditions subjected to axial load and external force

[...]

M.-F. Liu¹, T.-P. Chang²•Institutions (2)

I-Shou University¹, National Kaohsiung First University of Science and Technology²

22 Nov 2005-Journal of Sound and Vibration

TL;DR: In this paper, the interactive behaviors among transverse magnetic fields, axial loads and external force of a magneto-elastic beam with general boundary conditions are investigated, where axial forces and transverse forces are assumed to be periodic with respect to time and two specified frequencies are applied to the whole system.

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Abstract: In this study, the interactive behaviors among transverse magnetic fields, axial loads and external force of a magneto-elastic beam with general boundary conditions are investigated. The axial force and transverse magnetic force are assumed to be periodic with respect to time and two specified frequencies, one for axial force and the other for oscillating transverse magnetic field, are applied to the whole system. The equation of motion for the physical model is derived by using the Hamilton's principle and the vibration analysis is performed by employing the characteristic orthogonal polynomials as well as the Galerkin's method. The displacement of the beam with the effect of the magnetic force, axial force and spring force are determined from the modal equations by using the Runge–Kutta method. Based on the present study, we can conclude that the effect of the magnetic field not only reduces the deflection but also decreases the natural frequencies of the system, also it should be noted that the specified beam model can be adopted to simulate several structures in mechanical, civil and electronic engineering.

...read moreread less

29 citations

"Nonlinear vibrations and frequency ..." refers methods in this paper

...(1998) and Liu and Chang (2005). Here the approximate solution of this equation is obtained using the first-order method of multiple scales as given later....
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Journal Article•DOI•

Transient vibrations of a simply-supported beam with axial loads and transverse magnetic fields

[...]

Yan-Shin Shih, Guan-Yuan Wu, Edward J. S. Chen

01 Jan 1998-Mechanics of Structures and Machines

TL;DR: In this paper, two frequencies of pulsating axial force and oscillating transverse magnetic field are applied to the system and the amplitude versus time and velocity versus amplitude diagrams for the first mode and the first two modes are determined.

...read moreread less

Abstract: Transient vibrations of a simply supported beam are considered. Including axial force, magnetic force and magnetic couple, the equation of motion is derived by Hamilton's principle. The damping factor is also considered in this study. Two frequencies of pulsating axial force and oscillating transverse magnetic field are applied to the system. Using the Runge-Kutta method, the amplitude versus time and velocity versus amplitude diagrams for the first mode and the first two modes are determined.

...read moreread less

28 citations