# Nonlinear vibrations and frequency response analysis of a cantilever beam under periodically varying magnetic field

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### Cites background from "Nonlinear vibrations and frequency ..."

...Communicated by Chandrika Prakash Vyasarayani 2020 Taylor & Francis Group, LLC Kumar and Pratiher 2020; Mehrjooee, Dehkordi, and Korayem 2020; Piedboeuf and Moore 2002; Pratiher and Dwivedy 2011) and hence significant literature is available....

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4 citations

### Cites background from "Nonlinear vibrations and frequency ..."

...Twin-roll strip casting-rolling process. electric field intensity E, and electric displacement vector D (Pratiher and Dwivedy 2011)....

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##### References

91 citations

### "Nonlinear vibrations and frequency ..." refers background in this paper

...It may also be noted that by neglecting parametric term f2 cos ̄2 q due to the presence of force excitation, the present system can be reduced to that of Pratiher and Dwivedy (2009). It has also been observed that the equation of motion contains additional nonlinear terms of geometric and inertial type than those obtained in Shih et al....

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...It may also be noted that by neglecting parametric term f2 cos ̄2 q due to the presence of force excitation, the present system can be reduced to that of Pratiher and Dwivedy (2009). It has also been observed that the equation of motion contains additional nonlinear terms of geometric and inertial type than those obtained in Shih et al. (1998) and Liu and Chang (2005)....

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34 citations

### "Nonlinear vibrations and frequency ..." refers background or methods in this paper

...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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32 citations

### "Nonlinear vibrations and frequency ..." refers background in this paper

...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field. In all these cases authors have studied only the trivial state responses of the system and parametric instability regions were determined. However, practically, most of the engineering structures exhibit nonlinear behavior which cannot be predicted from these analyses. Few authors’ viz., Kojima and Nagaya (1985), Lu et al. (1995), Shih et al. (1998), Liu and Chang (2005), and Pratiher and Dwivedy (2009) have studied the nonlinear response of the elastic beams subjected to alternating electromagnetic field. For a more detailed review on parametrically excited beam subjected to magnetic field one may refer authors earlier works of Pratiher and Dwivedy (2007, 2009). From these literatures it has been observed that no research has been carried out to find the frequency response for magnetoelastic cantilever beam with tip mass subjected to periodic axial load. Hence, in the present work an attempt has been made to obtain the frequency response curves for such systems. Here, the governing temporal equation of motion of the system has been obtained which contains nonlinear damping, linear and nonlinear parametric excitation terms, in addition to the geometric and inertial types of nonlinear terms. By neglecting the effect of periodically varying axial load, the present system is similar to that of Pratiher and Dwivedy (2009) and by neglecting the geometric and inertial nonlinear terms, the present equation of motion is similar to that of Lu et al. (1995), Shih et al. (1998), and Liu and Chang (2005). The influences of the amplitude of magnetic field strength, attached tip mass, and static and dynamic amplitude of axial load on the frequency response curves have been investigated....

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...Moon and Pao (1969), Wu et al. (2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field....

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...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field. In all these cases authors have studied only the trivial state responses of the system and parametric instability regions were determined. However, practically, most of the engineering structures exhibit nonlinear behavior which cannot be predicted from these analyses. Few authors’ viz., Kojima and Nagaya (1985), Lu et al. (1995), Shih et al. (1998), Liu and Chang (2005), and Pratiher and Dwivedy (2009) have studied the nonlinear response of the elastic beams subjected to alternating electromagnetic field. For a more detailed review on parametrically excited beam subjected to magnetic field one may refer authors earlier works of Pratiher and Dwivedy (2007, 2009). From these literatures it has been observed that no research has been carried out to find the frequency response for magnetoelastic cantilever beam with tip mass subjected to periodic axial load. Hence, in the present work an attempt has been made to obtain the frequency response curves for such systems. Here, the governing temporal equation of motion of the system has been obtained which contains nonlinear damping, linear and nonlinear parametric excitation terms, in addition to the geometric and inertial types of nonlinear terms. By neglecting the effect of periodically varying axial load, the present system is similar to that of Pratiher and Dwivedy (2009) and by neglecting the geometric and inertial nonlinear terms, the present equation of motion is similar to that of Lu et al....

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...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field....

[...]

...(2000), Chen and Yah (2001), Wu (2005), and Pratiher and Dwivedy (2007) studied beam-plate systems subjected to transverse magnetic field. In all these cases authors have studied only the trivial state responses of the system and parametric instability regions were determined. However, practically, most of the engineering structures exhibit nonlinear behavior which cannot be predicted from these analyses. Few authors’ viz., Kojima and Nagaya (1985), Lu et al. (1995), Shih et al. (1998), Liu and Chang (2005), and Pratiher and Dwivedy (2009) have studied the nonlinear response of the elastic beams subjected to alternating electromagnetic field....

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29 citations

### "Nonlinear vibrations and frequency ..." refers methods in this paper

...(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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28 citations