Nonlinear Vibration of a Magneto-Elastic Cantilever Beam With Tip Mass
01 Apr 2009-Journal of Vibration and Acoustics (American Society of Mechanical Engineers)-Vol. 131, Iss: 2, pp 021011
TL;DR: In this article, the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated using D'Alembert's principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method.
Abstract: In this work the effect of the application of an alternating magnetic field on the large transverse vibration of a cantilever beam with tip mass is investigated. The governing equation of motion is derived using D'Alembert's principle, which is reduced to its nondimensional temporal form by using the generalized Galerkin method. The temporal equation of motion of the system contains nonlinearities of geometric and inertial types along with parametric excitation and nonlinear damping terms. Method of multiple scales is used to determine the instability region and frequency response curves of the system. The influences of the damping, tip mass, amplitude of magnetic field strength, permeability, and conductivity of the beam material on the frequency response curves are investigated. These perturbation results are found to be in good agreement with those obtained by numerically solving the temporal equation of motion and experimental results. This work will find extensive applications for controlling vibration inflexible structures using a magnetic field.
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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.
50 citations
TL;DR: In this article, the dynamics of a single-link flexible joint (SLFJ) robot manipulator were analyzed using phase portrait, Lyapunov spectrum, instantaneous phase plot, Poincare map, parameter space, bifurcation diagram, 0-1 test and frequency spectrum plot.
Abstract: This paper reports various chaotic phenomena that occur in a single-link flexible joint (SLFJ) robot manipulator. Four different cases along with subcases are considered here to show different types of chaotic behaviour in a flexible manipulator dynamics. In the first three cases, a partial state feedback as joint velocity and motor rotor velocity feedback is considered, and the resultant autonomous dynamics is considered for analyses. In the fourth case, the manipulator dynamics is considered as a non-autonomous system. The system has (1) one stable spiral and one saddle-node foci, (2) two saddle-node foci and (3) one marginally stable nature of equilibrium points. We found single- and multi-scroll chaotic orbits in these cases. However, with the motor rotor velocity feedback, the system has two unstable equilibria. One of them has an index-4 spiral repellor. In the non-autonomous case, the SLFJ robot manipulator system has an inverse crisis route to chaos and exhibits (1) transient chaos with a stable limit cycle and (2) chaotic behaviour. In all the four cases, the SLFJ manipulator dynamics exhibits coexistence of chaotic orbits, i.e. multi-stability. The various dynamical behaviours of the system are analysed using available methods like phase portrait, Lyapunov spectrum, instantaneous phase plot, Poincare map, parameter space, bifurcation diagram, 0–1 test and frequency spectrum plot. The MATLAB simulation results support various claims made about the system. These claims are further confirmed and validated by circuit implementation using NI Multisim.
47 citations
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.
27 citations
TL;DR: In this article, an H∞ method for the vibration control of an iron cantilever beam with axial velocity using the noncontact force by permanent magnets is proposed, which can be used for the beam with constant length or varying length.
25 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 from "Nonlinear Vibration of a Magneto-El..."
...Here, r is the scaling factor, q t is the time modulation, and s is an admissible function which is the eigenfunction of a cantilever beam with tip mass (Pratiher and Dwivedy, 2009)....
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References
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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
TL;DR: A survey of the literature related to dynamic analyses of flexible robotic manipulators has been carried out in this article, where both link and joint flexibility are considered in this work and an effort has been made to critically examine the methods used in these analyses, their advantages and shortcomings and possible extension of these methods to be applied to a general class of problems.
791 citations
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30 Jun 1990
TL;DR: In this paper, the conservative single degree of freedom model and the non-conservative single degree-of-freedom model are used for single-degree of freedom (SDF) systems.
Abstract: Part 1 Linear vibrations in mechanical engineering: classification of vibration problems fundamentals of vibrating mechanical systems the conservative single degree of freedom model the non-conservative single degree of freedom model practical uses for single degree of freedom models transients and impulses in single degree of freedom systems systems with more than one degree of freedom continuous systems. Part 2 Parametric vibrations in linear vibrating systems: stability considerations some simple physical systems analysis of governing equations further investigations into parametric phenomena. Part 3 Non-linear vibrations in forced and parametrically excited systems large deflection non-linearities material and structural configuration non-linearities. Part 4 Non-linear vibrations in autoparametric systems: two mode interaction in a coupled beam system application to vibration absorption multi-modal autoparametric interactions. Part 5 Phase plane concepts and chaotic vibrations: an introduction to the phase plane a summary of fundamental phase plane concepts chaos in vibrating systems.
246 citations
TL;DR: In this article, the Euler-Bernoulli theory for a slender beam is used to derive the governing non-linear partial differential equation for an arbitrary position of the lumped mass.
Abstract: The non-linear response of a slender cantilever beam carrying a lumped mass to a principal parametric base excitation is investigated theoretically and experimentally. The Euler-Bernoulli theory for a slender beam is used to derive the governing non-linear partial differential equation for an arbitrary position of the lumped mass. The non-linear terms arising from inertia, curvature and axial displacement caused by large transverse deflections are retained up to third order. The linear eigenvalues and eigenfunctions are determined. The governing equation is discretized by Galerkin's method, and the coefficients of the temporal equation—comprised of integral representations of the eigenfunctions and their derivatives—are computed using the linear eigenfunctions. The method of multiple scales is used to determine an approximate solution of the temporal equation for the case of a single mode. Experiments were performed on metallic beams and later on composite beams because all of the metallic beams failed prematurely due to the very large response amplitudes. The results of the experiment show very good qualitative agreement with the theory.
174 citations
91 citations