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

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.
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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TL;DR: In this article, the effect of initial stress and the magnetic field on thermoelastic interactions in an isotropic, thermally and electrically conducting half-space whose surface is subjected to mechanical and thermal loads is investigated.
Abstract: The present paper is aimed at studying the effect of initial stress and the magnetic field on thermoelastic interactions in an isotropic, thermally and electrically conducting half-space whose surface is subjected to mechanical and thermal loads. The formulation is applied under the thermoelasticity theory with three-phase-lag, proposed by Choudhuri (2007). The normal mode analysis is used to obtain the expressions for the variables considered. Numerical and computations are performed for a specific material and the results obtained are represented graphically. Comparisons are made with the results predicted by different theories Lord–Shulman theory (L–S), the theory of thermoelasticity type III (G-N III) and the three-phase-lag model (3PHL) in the absence and presence of the initial stress and magnetic field.

52 citations


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TL;DR: In this paper, the authors apply the differential transformation method (DTM) to solve linear and nonlinear vibration problems of elastically end-restrained beams, which demonstrates many advantages such as rapid convergence, high accuracy, and computational stability.
Abstract: The objective of this paper is to apply the differential transformation method (DTM) to solve linear and nonlinear vibration problems of elastically end-restrained beams. The method demonstrates many advantages such as rapid convergence, high accuracy, and computational stability to determine linear and nonlinear natural frequencies as well as mode shapes of such beams. The mathematical models provided in this paper can be solved easily using symbolic tools in available software packages such as Maple and Matlab. An accuracy of the present solutions is confirmed by comparing with some published results in the open literature. New numerical results of nonlinear frequency ratio of beams supported by various types of elastic boundary conditions are presented and discussed in detail. The significant effects of translational and rotational springs including vibration amplitudes on linear and nonlinear vibration results are also taken into investigation. Based on the numerical exercises, it is revealed that the...

21 citations

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TL;DR: In this article, the transverse vibration response, stability, and bifurcations of an axially moving viscoelastic beam with time-dependent axial speed are investigated.
Abstract: The subject of this article is the investigation of the transverse vibration response, stability, and bifurcations of an axially moving viscoelastic beam with time-dependent axial speed. The force and moment balances as well as constitutive relations are employed to derive the equation of motion. Due to the presence of the time-dependent axial speed and steady dissipation terms, time-dependent coefficients and nonlinear dissipation terms are generated, respectively. The equation of motion is reduced into a set of coupled nonlinear ordinary differential equations with time-dependent coefficients. The subcritical resonant response of the system is obtained using the pseudo-arclength continuation technique, while the bifurcation diagrams of Poincare maps are obtained via direct time integration of the discretized equations of motion.

19 citations


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TL;DR: In this article, a viscoelastic beam supported by vertical springs is proposed with nonrotatable left boundary and freely rotatable right end, and the steady-state responses of the beam excited by a distributed harmonic force are obtained by an approximate analytical method and a numerical approach.
Abstract: Under the conditions of horizontal placement and only considering geometric nonlinearity, depending on the boundary constraints, primary resonances of an elastic beam exhibit either hardening or softening nonlinear behavior. In this paper, the conversion of softening nonlinear characteristics to hardening characteristics is studied by using the multi-scale perturbation method. Therefore, in a local sense, the condition is established for the resonance of the elastic beam exhibits only linear characteristics by finding the balance between asymmetric elastic support and geometric nonlinearity. A viscoelastic beam supported by vertical springs is proposed with nonrotatable left boundary and freely rotatable right end. In order to truncate the continuous system, natural frequencies and modes of the proposed asymmetric beam are analyzed. The steady-state responses of the beam excited by a distributed harmonic force are, respectively, obtained by an approximate analytical method and a numerical approach. Under the condition that the beam is placed horizontally, the transition from the cantilever state to the clamped–pinned state is demonstrated by constructing different asymmetry support conditions. The resonance peak of the first-order primary resonance is used to demonstrate the transition from softening nonlinear characteristics to the hardening characteristics. This research shows that the transformation from softening characteristics to hardening characteristics caused by asymmetric elastic support and geometric nonlinearity exists only in the first-order mode resonance.

19 citations

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TL;DR: The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates, and as the excitation frequency decreases, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinate is different, while this behavior has more similarity after this point.
Abstract: In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the oc...

17 citations


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References
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85 citations


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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.
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.

32 citations


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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.
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.

29 citations


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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.
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.

28 citations


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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.
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.

26 citations