Parametric instability of a cantilever beam with magnetic field and periodic axial load
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.
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TL;DR: In this article, the dynamic and energy properties of a multi-stable bimorph cantilever energy harvester with magnetic attraction effect have been investigated and the mechanism that governs the formation of this multi-stability is thoroughly identified and examined thorough a bifurcation analysis performed on the system's equilibrium solutions.
Abstract: A theoretical investigation is conducted on the dynamic and energetic characteristics of a multi-stable bimorph cantilever energy harvester that uses magnetic attraction effect. The multi-stable energy harvester under study is composed of a bimorph cantilever beam with soft magnetic tip and two externally fixed permanent magnets that are arranged in series. With this configuration, the magnetic force and the moment that are exerted on the cantilever tip tend to be highly dependent on the magnetic field induced by the external magnets. Such an energy harvester can possess multi-stable potential functions, ranging from mono-stable to penta-stable. The mechanism that governs the formation of this multi-stability is thoroughly identified and examined thorough a bifurcation analysis performed on the system׳s equilibrium solutions. From this analysis, it is found that the transitions between these multi-stable states occur through very complicated bifurcation scenarios that include degenerate pitchfork bifurcations and mergers of pitchfork bifurcations or saddle-node bifurcations. Bifurcation set diagram is obtained, which is composed of five separate parametric regions, from mono- to penta-stability. The resulting stability map satisfactorily describes the multi-stable characteristics of the present energy harvester. In addition, the dynamic and energetic characteristics of the present multi-stable energy harvester are more thoroughly examined using its potential energy diagrams and a series of numerical simulations, and the obtained results are compared with those for the equivalent bi-stable cases.
159 citations
TL;DR: In this paper, the primary, sub- and superharmonic resonant behaviors of a cantilever beam-type micro-scale device are analytically solved and examined, and the effects of parameters/operating conditions on the resonant characteristics of the device are thoroughly investigated.
Abstract: In this study, the primary, sub- and super-harmonic resonant behaviors of a cantilever beam-type micro-scale device are analytically solved and examined The device under study includes a tip mass and is subjected to an axial force and electrostatic excitement An appropriate derivation of orthogonality conditions and their application enable us to properly discretize the governing nonlinear field equation along with its boundary conditions to an equation form suitable for ‘single mode approximation’ This procedure results in a Mathieu–Hill type differential equation and causes associated parametric instability problems Using a Taylor series expansion with an electrostatic forcing term, a quadratic nonlinear term naturally appears in the resulting differential equation This term often requires more rigorous mathematical treatment than other conventional approaches To resolve this problem, the concept of nonlinear normal mode is introduced in this study A perturbation technique and asymptotic expansions of modal displacement are employed to accurately solve the resulting nonlinear differential equation by applying an appropriate ordering scheme Finally, the effects of parameters/operating conditions on the resonant characteristics of the device are thoroughly investigated, and the associated parametric instability issue is also discussed
50 citations
TL;DR: The results suggest the existence of two critical damping values, giving rise to the onset and the disappearance of SR, respectively, and the appropriate choice of triple-well potential function and damping coefficient can improve the response of the system to an external periodic excitation according to the damping-induced resonance effect.
Abstract: In this paper, stochastic resonance (SR) in an underdamped triple-well potential system driven by Gaussian white noise and a parametric harmonic excitation is investigated. The analytical expressions of the output signal-to-noise ratio (SNR), together with the mean first-passage times (MFPTs), are derived for the triple-well potential system involving damping in adiabatic limit. The effects of noise intensity, damping coefficient and triple-well potential on MFPTs and SNR are analyzed. The results suggest the existence of two critical damping values, giving rise to the onset and the disappearance of SR, respectively. Since the system is unstable in weak damping regime, emergence of SR is prohibited. Under the weak noise level, SNR exhibits a prominent resonance-like behavior at the optimal value of damping coefficient. Moreover, the two nonlinear stiffness coefficients of restoring force play an opposite role in the enhancement of SR. Thus, the SR effect significantly depends on the change of the two-side potential wells. Particularly, the appropriate choice of triple-well potential function and damping coefficient can improve the response of the system to an external periodic excitation according to the damping-induced resonance effect. Finally, the numerical results confirm the effectiveness of the theoretical analyses.
26 citations
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.
22 citations
Cites background or methods from "Parametric instability of a cantile..."
...The expressions for h1 h2 · · ·h14 are same as those given in Pratiher and Dwivedy (2009)....
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...Following Pratiher and Dwivedy (2009), and using single mode approximation, i....
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...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....
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...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....
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References
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Book•
01 Jan 1995
TL;DR: Perturbation Methods Dynamical Systems and Equilibrium Solutions Dynamic Solutions Tools to Characterize Different Motions Two-to-One Internal Resonance Combination Internal Resonances Three-toone Internal ResonANCE Combination internal Resonances Systems with Quadratic and Cubic Nonlinearities Gyroscopic Systems Systems with More than One internal Resonance Random Excitations
Abstract: Perturbation Methods Dynamical Systems and Equilibrium Solutions Dynamic Solutions Tools to Characterize Different Motions Two-to-One Internal Resonances Combination Internal Resonances Three-to-One Internal Resonances Combination Internal Resonances Systems with Quadratic and Cubic Nonlinearities Gyroscopic Systems Systems with More than One Internal Resonance Random Excitations
1,030 citations
91 citations
TL;DR: In this article, the stability of an axially oscillating cantilever beam is investigated and the stability diagrams of the first and second order approximate solutions are obtained by using the multiple scale perturbation method.
Abstract: Dynamic stability of an axially oscillating cantilever beam is investigated in this paper. Equations of motion for the axially oscillating beam are derived and transformed into dimensionless forms. The equations include harmonically oscillating parameters which are related to the motion-induced stiffness variation. Stability diagrams of the first and the second order approximate solutions are obtained by using the multiple scale perturbation method. The stability diagrams show that there exist significant difference between the first and the second order approximate solutions. It is also found that relatively large unstable regions exist around the first bending natural frequency, twice the first bending natural frequency, and twice the second being natural frequency. The validity of the stability diagram is verified by direct numerical integrations of the equations of motion of the system.
45 citations
TL;DR: In this paper, the dynamic instability and transient vibrations of a pinned beam with transverse magnetic fields and thermal loads are studied. And the authors show that the instability and the transient vibratory behaviors of the beam are influenced by the magnetic fields, thermal loads, and the frequencies of oscillation of the transverse magnetoelastic field.
Abstract: Dynamic instability and transient vibrations of a pinned beam with transverse magnetic fields and thermal loads are studied. The magnetoelastic model, whose beam thickness and the deflection are very small compared with the length, is taken for analysis. Applying the Hamilton's principle, the equation of motion with damping factor is derived. The governing equation is reduced to the Mathieu equation by Galerkin's method with the assumed mode shape. The incremental harmonic balance (IHB) method is applied to analyze the dynamic instability. The amplitude versus time behavior of the system is investigated by using the Runge–Kutta method. The study shows that the instability and transient vibratory behaviors of the beam are influenced by the magnetic fields, thermal loads, and the frequencies of oscillation of the transverse magnetic field. The beat phenomenon and primary resonance are presented and discussed when the frequencies of the oscillating transverse magnetic field are close to the fundamental natural frequency of the system.
35 citations
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.
32 citations