Vibration analysis of a magneto-elastic beam with general boundary conditions subjected to axial load and external force
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.
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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.
34 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.
Abstract: An H∞ method for the vibration control of an iron cantilever beam with axial velocity using the noncontact force by permanent magnets is proposed in the paper. The transverse vibration equation of the axially moving cantilever beam with a tip mass is derived by D'Alembert's principle and then updated by experiments. An experimental platform and a magnetic control system are introduced. The properties of the force between the magnet and the beam have been determined by theoretic analysis and tests. The H∞ control strategy for the suppression of the beam transverse vibration by initial deformation excitations is put forward. The control method can be used for the beam with constant length or varying length. Numerical simulation and actual experiments are implemented. The results show that the control method is effective and the simulations fit well with the experiments.
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 methods from "Vibration analysis of a magneto-ela..."
...(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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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.
20 citations
TL;DR: In this paper, a nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented, and the Galerkin method is adopted to discretize the dynamic equations.
Abstract: The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.
20 citations
References
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Book•
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01 Jan 1978
TL;DR: This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis, and describes the accuracy, efficiency and robustness of these algorithms.
Abstract: Introduction. Mathematical approximations have been used since ancient times to estimate solutions, but with the rise of digital computing the field of numerical analysis has become a discipline in its own right. Numerical analysts develop and study algorithms that provide approximate solutions to various types of numerical problems, and they analyze the accuracy, efficiency and robustness of these algorithms. As technology becomes ever more essential for the study of mathematics, learning algorithms that provide approximate solutions to mathematical problems and understanding the accuracy of such approximations becomes increasingly important. This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis.
7,315 citations
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...Generation of characteristic orthogonal polynomials Given a polynomial f1ðxÞ, an orthogonal set of polynomials in the interval apxpb can be generated by using a Gram–Schmidt process stated below [10,11]: f2ðxÞ 1⁄4 ðx B2Þf1ðxÞ, (3) fkðxÞ 1⁄4 ðx BkÞfk 1ðxÞ Ckfk 2ðxÞ, (4) Bk 1⁄4 Z L...
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3,151 citations
Additional excerpts
...Generation of characteristic orthogonal polynomials Given a polynomial f1ðxÞ, an orthogonal set of polynomials in the interval apxpb can be generated by using a Gram–Schmidt process stated below [10,11]: f2ðxÞ 1⁄4 ðx B2Þf1ðxÞ, (3) fkðxÞ 1⁄4 ðx BkÞfk 1ðxÞ Ckfk 2ðxÞ, (4) Bk 1⁄4 Z L...
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Book•
01 Dec 1967
TL;DR: In this article, the authors present a vector analysis of the Dirac Delta Function, which is a function of a complex variable and can be expressed as follows: 1. Vector Analysis. 2. Vector Differential Operators.
Abstract: 1. Vector Analysis. 2. Electrostatistics. 3. Solution of Electrostatic Problems. 4. The Electrostatic Field in Dielectric Media. 5. Microscopic Theory of Dielectrics. 6. Electrostatic Energy. 7. Electric Current. 8. The Magnetic Field of Steady Currents. 9. Magnetic Properties of Matter. 10. Microscopic Theory of Magnetism. 11. Electromagnetic Induction. 12. Magnetic Energy. 13. Slowly Varying Currents. 14. Physics of Plasmas. 15. Electromagnetic Properties of Superconductors. 16. Maxwell's Equations. 17. Propagation of Monochromatic. 18. Monochromatic Waves in Bounded Regions. 19. Dispersion and Oscillating Fields in Dispersive Media. 20. The Emission of Radiation. 21. Electrodynamics. 22. The Special Theory of Relativity. Appendix I: Computers and Electromagnetic Theory. Appendix II: Coordinate Transformations, Vectors, and Tensors. Appendix III: Systems of Units. Appendix IV: Vector Differential Operators. Appendix V: The Dirac Delta Function. Appendix VI: Fourier Integrals and Fourier Transforms. Appendix VII: Functions of a Complex Variable. Answers to Odd-Numbered Problems. Index.
805 citations
Book•
01 Jan 1962
432 citations
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...Mathematical modeling Hamilton’s principle [9] is adopted to derive the equation of motion [8] of the beam as follows: m qy qt2 þ Cd qy qt þ EI q y qx4 þ ky þ P q y qx2 1⁄4 f ðx; tÞ þ qc qx þ q qx Z x...
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315 citations