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Research Papers

Nonlinear Vibration of a Magneto-Elastic Cantilever Beam With Tip Mass

[+] Author and Article Information
Barun Pratiher

Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, Assam 781 039, India

Santosha K. Dwivedy1

Department of Mechanical Engineering, Indian Institute of Technology, Guwahati, Assam 781 039, Indiadwivedy@iitg.ernet.in

1

Corresponding author.

J. Vib. Acoust 131(2), 021011 (Feb 19, 2009) (9 pages) doi:10.1115/1.3025829 History: Received February 04, 2008; Revised September 15, 2008; Published February 19, 2009

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 in flexible structures using a magnetic field.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of a flexible magneto-elastic cantilever beam with a tip mass

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Figure 2

The region of instability of a cantilever beam with tip mass subjected to a magnetic field, Pao’s theoretical result, Pao’s experimental result, and the present result

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Figure 3

(a) Time response for point A and (b) time response for point B (points A and B are marked in Fig. 2)

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Figure 4

Effect of the magnetic field strength (Bm) on the frequency response curves for M=0.02 kg, Cd=0.01 N s/m, μr=3000, σ=107 V m−1: (a) Bm=0.20 A m−1, (b) Bm=0.25 A m−1, (c) Bm=0.30 A m−1, and (d) Bm=0.35 A m−1

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Figure 5

(a) Time response for point A and (b) time response for point B (A and B points are marked in Fig. 4)

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Figure 6

(a) Transient response and (b) steady state time response for point C with a magnetic field (solid line) and without a magnetic field (dotted line). Key the same as in Fig. 4.

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Figure 7

Influence of damping on frequency response curve for M=0.02 kg, μr=3000, σ=107 V m−1, and Bm=0.30 A m−1: (a)Cd=0.02 N s/m and (b)Cd=0.03 N s/m

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Figure 8

Influence of mass ratio (m¯) on frequency response curve for Cd=0.01 N s/m, μr=3000, σ=107 V m−1, and Bm=0.30 A m−1: (a)M=0.01 kg and (b)M=0.015 kg

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Figure 9

Effect of relative permeability (μr) of the material on the frequency response curve M=0.02 kg, Cd=0.01 N s/m, σ=107 V m−1, and Bm=0.30 A m−1: (a)μr=2.0 and (b)μr=7.0

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Figure 10

Effect of material conductivity (σ) on the frequency response curve for M=0.02 kg, Cd=0.01 N s/m, μr=2500, Bm=0.30 A m−1: (a) σ=102 V m−1, (b) σ=106 V m−1

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