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

01 Jan 1998-Mechanics of Structures and Machines (Taylor & Francis Group)-Vol. 26, Iss: 2, pp 115-130

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.

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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

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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

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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.

29 citations

### Cites background from "Transient vibrations of a simply-su..."

...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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...[8] have studied the transient vibrations of a simply supported beam with axial loads and transverse magnetic fields....

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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

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TL;DR: In this paper, the dynamic instability of a pinned beam subjected to an alternating magnetic field and thermal load with the nonlinear strain, and made of physically nonlinear thermoelastic material has been studied.

Abstract: The dynamic instability of a pinned beam subjected to an alternating magnetic field and thermal load with the nonlinear strain, and made of physically nonlinear thermoelastic material has been studied. Applying the Hamilton's principle, the equation of motion with damping factor, induced current and thermal load is derived. Using the Galerkin's method, the governing equation is reduced to a time-dependent Mathieu equation. The incremental harmonic balance (IHB) method is applied to analyse the dynamic instability. The effects of non-dimensional parameters frequency ratio (Ω), load factor (ϕ), amplitude ( ‖ a ‖ ), damping factor (k1) and temperature increment (ΔT) on the dynamic instability are obtained and discussed. Results show that the characteristics of magnetoelastic instability for the beam having the large deformation are distinctive from those of the beam having small deformation.

23 citations

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166 citations

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TL;DR: In this paper, it is shown that most of the necessary quantities for this subsidiary computation are available as computed by-products in the preceding finite element solution procedure, which is shown to be a particular form of a procedure for which superconvergent theoretical error estimates have been proven elsewhere.

Abstract: A technique is considered whereby very accurate derivatives of a finite element solution can be calculated efficiently. It is demonstrated here that most of the necessary quantities for this subsidiary computation are available as computed by-products in the preceding finite element solution procedure. The calculation is shown in this note to be a particular form of a procedure for which superconvergent theoretical error estimates have been proven elsewhere. Numerical experiments confirm the superior accuracy in the computed derivative (stress or flux).

80 citations

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TL;DR: In this article, the authors derived basic equations and boundary conditions for thin elastic plates (carrying static bias E-M fields) and disturbed by dynamical perturbations, both flexural and in-plane motions are considered.

Abstract: Field equations are obtained for elastic solids (carrying static bias fields) subject to small dynamical loads and electromagnetic (E-M) fields. The general theory is then used to derive basic equations and boundary conditions for thin elastic plates (carrying static bias E-M fields) and disturbed by dynamical perturbations. Both flexural and in-plane motions are considered. Field equations are solved to determine critical transverse electric field and surface charge that cause buckling of a plate.

54 citations