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Nonlinear Vibrations and Frequency Response Analysisof a Cantilever Beam Under Periodically VaryingMagnetic FieldBarun Pratiher a & Santosha K. Dwivedy ba Department of Mechanical Engineering and MME , Indian School of Mines Dhanbad , Indiab Department of Mechanical Engineering , Indian Institute of Technology Guwahati , IndiaPublished online: 19 May 2011.

To cite this article: Barun Pratiher & Santosha K. Dwivedy (2011) Nonlinear Vibrations and Frequency Response Analysisof a Cantilever Beam Under Periodically Varying Magnetic Field, Mechanics Based Design of Structures and Machines: AnInternational Journal, 39:3, 378-391, DOI: 10.1080/15397734.2011.557972

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Mechanics Based Design of Structures and Machines, 39: 378–391, 2011Copyright © Taylor & Francis Group, LLCISSN: 1539-7734 print/1539-7742 onlineDOI: 10.1080/15397734.2011.557972

NONLINEAR VIBRATIONS AND FREQUENCYRESPONSE ANALYSIS OF A CANTILEVER BEAMUNDER PERIODICALLY VARYING MAGNETIC FIELD#

Barun Pratiher1 and Santosha K. Dwivedy21Department of Mechanical Engineering and MME,Indian School of Mines Dhanbad, India2Department of Mechanical Engineering, Indian Institute of TechnologyGuwahati, India

In this paper, nonlinear vibration of a cantilever beam with tip mass subjected toperiodically varying axial load and magnetic field has been studied. The temporalequation of motion of the system containing linear and nonlinear parametric excitationterms along with nonlinear damping, geometric and inertial types of nonlinear terms hasbeen derived and solved using method of multiple scales. The stability and bifurcationanalysis for three different resonance conditions were investigated. The numericalresults demonstrate that while in simple resonance case with increase in magneticfield strength, the system becomes unstable, in principal parametric or simultaneousresonance cases, the vibration can be reduced significantly by increasing the magneticfield strength. The present work will be very useful for feed forward vibration controlof magnetoelastic beams which are used nowadays in many industrial applications.

Keywords: Bifurcation; Cantilever beam; Magnetic field; Method of multiple scales; Stability.

INTRODUCTION

Flexible cantilever beam with end mass can find applications in the fieldof micro-robotic manipulators, elements of machine parts used in nuclear powerplants, aerospace industries, micro-surgery instrumentations, space exploration,micro-switches, and in many other precision industrial applications for their lightweight, high speed, and low cost. In most of these applications, the mass at thefree end of the cantilever beam may be subjected to a harmonic force excitation.Due to low stiffness of the beam, for some amplitude and frequency of the axiallyapplied periodic load, the system undergoes a large transverse vibration even if theexcitation force is well below the Euler buckling load, and the excitation frequencyis away from the natural frequencies. This type of excitation is commonly knownas parametric excitation. To control the vibration of such a system one may applymagnetic field. As the application of magnetic field changes the stiffness of thesystem, hence, one can actively control the vibration of the system by applying

Received May 11, 2010; Accepted October 19, 2010#Communicated by S. Sinha.Correspondence: Barun Pratiher, Department of Mechanical Engineering and MME, Indian

School of Mines Dhanbad, 826004, India; E-mail: pratiher.b.memme@ismdhanbad.ac.in

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 379

magnetic field. A brief review of literature related to the stability analysis of suchsystems is carried out later.

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 transversemagnetic field. In all these cases authors have studied only the trivial state responsesof the system and parametric instability regions were determined. However,practically, most of the engineering structures exhibit nonlinear behavior whichcannot 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 andDwivedy (2009) have studied the nonlinear response of the elastic beams subjectedto alternating electromagnetic field. For a more detailed review on parametricallyexcited beam subjected to magnetic field one may refer authors earlier works ofPratiher and Dwivedy (2007, 2009).

From these literatures it has been observed that no research has been carriedout to find the frequency response for magnetoelastic cantilever beam with tipmass subjected to periodic axial load. Hence, in the present work an attempt hasbeen made to obtain the frequency response curves for such systems. Here, thegoverning temporal equation of motion of the system has been obtained whichcontains nonlinear damping, linear and nonlinear parametric excitation terms, inaddition to the geometric and inertial types of nonlinear terms. By neglecting theeffect of periodically varying axial load, the present system is similar to that ofPratiher and Dwivedy (2009) and by neglecting the geometric and inertial nonlinearterms, 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 fieldstrength, attached tip mass, and static and dynamic amplitude of axial load on thefrequency response curves have been investigated.

MATHEMATICAL MODELING

Figure 1 shows a flexible cantilever beam of length L, cross sectional area A,moment of inertia I , density �, modulus of elasticity E, and with a tip mass M .It is subjected to an alternating magnetic field B0 = Bm cos�1t and an axial forceof P = P0 + P1 cos�2t. The cantilever beam with tip mass has been modeled as an

Figure 1 Schematic diagram of magneto-elastic cantilever beam with tip mass subjected harmonicaxial load.

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380 PRATIHER AND DWIVEDY

Euler–Bernoulli beam with a point mass at the tip. Similar to Pratiher and Dwivedy(2009), using D’Alembert’s principle the following governing differential equationof motion of the system has been obtained in terms of transverse displacement v.

EI

(vssss +

12v2s vssss + 3vsvssvsss + v3ss

)+ �Avs

∫ s

0

(v2� + v�v�

)d�

+ vsvss

∫ L

s

(�Av+ cdv

)d� + vsvssv

∫ L

0M��s − L�d�

− vss

∫ L

s�A

∫ �

0

(v2� + v�v�

)d�d�−

∫ L

0M��s − L�

∫ s

0�v2� + v�v��d� d�

+ P�t�vss −(vss

∫ L

s�pmd��− pmvs

)

−(dcmds

(1− 1

2v2s

)− vsvss

(1+ 1

2v2s

)cm

)+

(1− 1

2v2s

)��Av+ cdv� = 0� (1)

Here, cd is the coefficient of viscous damping. The mathematical expressionsfor the electromagnetic body force �pm� and electromagnetic couple �cm� due to themagnetic field can be given as

pm = −hdB20

(1− 1

2v2s

) ∫ �

0

(vsvs −

12vsv

2s vs

)d� and cm = �mhdB

20

�0�r

vs� (2)

Following Pratiher and Dwivedy (2009), and using single modeapproximation, i.e.,

v�s t� = r �s�q�t� (3)

the temporal equation of motion is obtained by using generalized Galerkin’smethod. Here, r is the scaling factor, q�t� is the time modulation, and �s� is anadmissible function which is the eigenfunction of a cantilever beam with tip mass(Pratiher and Dwivedy, 2009). Following nondimensional parameters are used forfurther analysis.

x = s

L � = �et �1 =

�1

�e

�2 =�2

�e

r = r

L m = M

�AL

�P0 =P0

Pc

�P1 =P1

Pc

� = EI

�AL4� (4)

Here, Pc is the critical Euler buckling load for a cantilever beam, which isequal to ��2EI�/�4L2� and �e is the fundamental frequency of the system whoseexpression is given in the Appendix. Substituting Eqs. (2)–(4) into Eq. (1) and usingthe generalized Galerkin’s method, one may obtain the following nondimensionaltemporal equation of motion:

q + 2��q + q + ���1q3 + �2q

2q + �3q2q − k1qq

2�

− �f1 cos�2�1��q − �k1 cos�2�1��qq2 + �f2 cos��2��q = 0� (5)

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 381

The expressions for the coefficients (i.e., � �1 �2 �3 f1 f2 k1� in this equationare given in the appendix. It has been observed from Eq. (5) that the nondimensionaltemporal equation of motion has parametric terms f1 cos�2�1��q and f2 cos��2��qand nonlinear damping term k1�1+ cos�2�1���qq

2 along with cubic geometric��1q

3� and inertial ��2q2q + �3q

2q� nonlinear terms. It may also be noted that byneglecting parametric term f2 cos��2��q due to the presence of force excitation, thepresent system can be reduced to that of Pratiher and Dwivedy (2009). It has alsobeen observed that the equation of motion contains additional nonlinear terms ofgeometric and inertial type than those obtained in Shih et al. (1998) and Liu andChang (2005). Here the approximate solution of this equation is obtained using thefirst-order method of multiple scales as given later.

SOLUTION OF TEMPORAL EQUATION

In the method of multiple scales the displacement q can be represented interms of different time scales �T0 T1� and a book keeping parameter � as follows:

q��� �� = q0�T0 T1�+ �q1�T0 T1�+ O��2�� (6)

Here, T0 = �, and T1 = ��.Substituting Eqs. (6) into Eq. (5) and equating the coefficient of like powers

of �, yields the following equations:

Order �0 � D20q0 + q0 = 0 (7)

Order �1 � D20q1 + q1 = −2D0D1q0 − 2�D0q0 − �1q

30 − �2�D

20q0�q0 − �3�D0q0�

2q20

+ k1�D0q0�q20 + f1 cos�2�1T0�q0 + k1 cos�2�1T0��D0q0�q

20 − f2 cos��2T0�q0� (8)

General solutions of Eq. (7) can be written as

q0 = A�T1 T2� exp�iT0�+�A�T1 T2� exp�−iT0�� (9)

Here, �A�T1 T2� is the complex conjugate of A�T1 T2�.Substituting Eq. (9) into Eq. (8) leads to the following equation:

D20q1 + q1 = −2iA′ exp�iT0�− 2i�A exp�iT0�− �3�1 − 3�2 + �3 − ik1�A

2�A exp�iT0�

+ �A3 exp�3iT0�+ ik1A3 exp�3iT0�

+ f12�A exp i�2�1 − 1�T0 +�A exp i�2�1 − 1�T0�

− f22�A exp i��2 − 1�T0 +�A exp i��2 − 1�T0�+

ik12A3 exp i�2�1 + 3�T0

+ ik12A2�A exp i�2�1 + 1�T0 −

ik12�A2A exp i�2�1 − 1�T0

+ ik12A3 exp i�3− 2�1�T0 + cc� (10)

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382 PRATIHER AND DWIVEDY

Here � = −�1 + �2 + �3 and cc stands for the complex conjugate of thepreceding terms. The secular or small divisor terms in Eq. (10) should be removedto have a bounded solution. It may be noted from Eq. (10) that the system possessthree different resonance conditions viz. (i) �1 ≈ 1 and �2 is away from 2, (ii) �2 ≈ 2and �1 is away from 1, and (iii) �1 ≈ 1 and �2 ≈ 2. In the following subsections,these resonance cases are discussed.

Simple Resonance Case Due to the Magnetic Field(�1 ≈ 1 and �2 Away from 2)

In this case, one may use a small positive quantity to express the nearnessof � to 1, as

� = 1+ � and = O�1�� (11)

Substituting Eq. (11) into Eq. (10), one may obtain the following secular orsmall divisor terms which has been equated to zero to eliminate these terms:

−2iA′ exp�iT0�− 2i�A exp�iT0�− �3�1 − 3�2 + �3 − ik1�A2�A exp�iT0�

+ f12�A exp�2T1�+ i

k12A3 exp�−2T1�− i

k12�A2A exp�2T1� = 0� (12)

Putting A equal to 12a�T1�e

�i��T1�� in Eq. (12) and separating the real and imaginaryterms, one may find the reduced equations as given below.

a = −�a+ k18a3 + f1

4a sin � (13)

a� = 2a(�− 1�

)− 3

4

(�1 − �2 +

�33

)a3 + 1

4a3k1 sin �+

f12a cos �� (14)

One may observe from Eqs. (13) and (14) that the system possesses both trivialand nontrivial responses. Hence one may determine the nontrivial responses bysolving Eqs. (13) and (14) simultaneously. To find the stability of the steady stateresponses, one may perturb the above Eqs. (13) and (14), by substituting a = ao +a1 and � = �0 + �1, where a0 �0 are the singular points, and then investigate theeigenvalues of the Jacobian matrix (J ), which is given below.

J =[ −�+ 3k1

8 a20 + f1

4 sin �0f14 a0 cos �0

− 32

(�1 − �2 + �3

3

)a0 + 1

2a0k1 sin �014a

20k1 cos �0 − f1

2 sin �0

]� (15)

It may be noted that the system will be stable if and only if all the real parts of theeigenvalues are negative. Now the first-order nontrivial steady-state response of thesystem can be given by

q = a cos��1�− ��� (16)

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 383

Principle Parametric Resonance Due to Axial Load(�2 ≈ 2 and �1 Away from 1)

In this resonance, one may introduce the detuning parameter to express thenearness of �2 to 2, as

� = 2+ 2� and = O�1�� (17)

Substituting Eq. (17) into Eq. (10), one may obtain the following secular or smalldivisor terms, which should be eliminated:

−�2iA′ + 2i�A� exp�iT0�− �3�1 − 3�2 + �3 − ik1�A2�A exp�iT0�−

f22�A exp�2T1� = 0�

(18)

Substituting A equal to 12a�T1�e

�i��T1�� into Eq. (18) and separating the real andimaginary terms, one may obtain the following expressions.

a = −�a+ k18a3 − f2

4a sin � (19)

a� = a

(�− 2�

)− 3

4

(�1 − �2 +

�33

)a3 − f2

2a cos �� (20)

Similar to the previous case, here it may be noted from Eqs. (19) and (20) thatthe system has both trivial and nontrivial responses. The response of the system canbe determined by numerically solving the Eqs. (19) and (20), simultaneously. Similarto the simple resonance case, here the stability of the system can be determined byinvestigating the eigenvalues of the Jacobian matrix (J ) which is given by

J =[−�+ 3k1

8 a20 − f2

4 sin �0 − f24 a0 cos �0

− 32 ��1 − �2 + �3

3 �a0f22 sin �0

]� (21)

Now the first-order nontrivial steady-state response of the system in thisresonance condition can be given by

q = a cos(12��2�− ��

)� (22)

Simultaneous Resonance Due to Magnetic Field and Axial Load(�1 ≈ 1 and �2 ≈ 2)

For this resonance condition, one may use the detuning parameters 1 and 2

to express the nearness of �1 to 1 and �2 to 2 as given below.

�1 = 1+ �1 �2 = 2+ 2�2 and 1 2 = O�1�� (23)

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384 PRATIHER AND DWIVEDY

Substituting Eq. (23) into Eq. (10), one may obtain the following secular orsmall divisor terms which has been eliminated by equating these terms to zero:

−2iA′ exp�iT0�− 2i�A exp�iT0�− �3�1 − 3�2 + �3 − ik1�A2�A exp�iT0�

+ f12�A exp�21T1�+ i

k12A3 exp�−21T1�− i

k12�A2A exp�21T1�

− f22�A exp�22T1� = 0� (24)

Similar to previous cases, substituting Aequal to 12a�T1�e

�i��T1�� into Eq. (24)and separating the real and imaginary terms, one may obtain the following reducedequations.

a = −�a+ k18a3 + f1

4a sin �− f2

4a sin�2�+ �� (25)

a� = 2a(�− 1�

)− 3

4

(�1 − �2 +

�33

− 14k1 sin �

)a3 + f1

2a cos �

− f22a cos�2�+ ��� (26)

Here, � = 1�− �, and � = �2 − 1�T1. It may be noted that, � is a measureof the phase difference between the application of the axial load and the magneticfield. In this case also, one may observe that the system possesses both trivial andnontrivial type of responses. Using similar procedure as in previous sections, thestability can be studied using the following Jacobian matrix (J ):

J =

−�+ 3k18 a2

0 + f14 sin �0

f14 a0 cos �0

f24 sin�2�0 + �0� − f2

4 a0 cos�2�0 + �0�

− 32

(�1 − �2 + �3

3

)a0 + 1

2a0k1 sin �0 − f12 sin �0 + f2

2 sin�2�0 + �0�

� (27)

NUMERICAL RESULTS AND DISCUSSIONS

For numerical simulation, a steel beam has been taken similar to thatconsidered 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 beamper unit length m = 0�03965 kg, damping constant cd = 0�01N-s/m, relativepermeability �r = 3000, material conductivity = 107 Vm−1, and the permeabilityof the vacuum, �0 = 1�26× 10−6 Hm−1. Numerically solving the reduced equationsfor different resonance conditions the frequency response curves have been plottedfor different system parameters such as amplitude of magnetic field strength (Bm),tip mass (M), static (�P0), and dynamic (�P1) amplitude of axial load. In the frequencyresponse curves dotted and solid lines represent, respectively, the unstable andstable responses of the system. In the following subsections the results obtained fordifferent resonance conditions have been critically analyzed.

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 385

Simple Resonance Due to Magnetic Field(�1 ≈ 1 and �2 Away from 2)

In this case, the cantilever beam is subjected to the transverse magnetic fieldwith a frequency closely equal to the first natural frequency of the system and thefrequency of the applied axial load is away from twice the natural frequency of thesystem. In Fig. 2(a), the frequency response curve of the system has been plottedwith amplitude of magnetic field strength equal to 0.30T for M = 0�02 kg�P0 =0�2507N. It has been observed that the system has both trivial response andnontrivial responses and hence the system will not vibrate if the frequency ofthe magnetic field is below a frequency marked by point R1 (subcritical pitchforkbifurcation point) or if it operates at a frequency which is above the point markedby R2 (supercritical pitchfork bifurcation point). With increase in the frequencyof the magnetic field, at R1 the cantilever beam suddenly experiences a jump upphenomenon which may lead to the catastrophic failure of the system, or thecantilever beam may start vibrating with amplitude equal to that of the nontrivialstate marked by point R′

1. With further increase in the frequency of the magneticfield, the beam will vibrate with amplitude equal to that of the nontrivial stablestate corresponding to �1. This vibration will continue till it reaches R2. Again,at this stage, if one decreases the frequency of the magnetic field, the systemresponse will go on increasing and follow the path R2R

′1 BD. At D, which is a

saddle node (S–N) bifurcation point, with further decrease in the frequency �1, thesystem will experience a jump down phenomenon. Similar to the previous jump upphenomenon, here also, either the system will fail or the vibration of the systemwill reduce to zero. Also, it should be noted that the system has a bistable regionbetween frequency range D′R1. Hence to completely suppress the vibration in thisresonance condition, it is recommended that one should apply a magnetic field witha frequency which is well below D′ or above R2. It may be recalled from the workof Pratiher and Dwivedy (2009) where only the trivial state instability regions wereplotted, that the system is prone to vibration only in the region R1R2.

To check the accuracy of the perturbation results, one may compare theseresults by numerically solving the temporal equation of motion (Eq. (5)). Here,Fig. 2(b) shows the time responses and Fig. 2(c) shows the phase portraits obtainednumerically by solving Eq. (5) corresponding to three distinct points viz., A, B,and C marked in Fig. 2(a). While the dotted line represents the steady-state timeresponse and phase portrait for point B, solid and solid-dash lines, respectively,represent the steady-state responses for point C and A. Clearly, the response curvecorresponding to point A shows to have zero amplitude of oscillation and points Band C have amplitude of 1.75 and 0.79, respectively. Hence, the results obtained bythe perturbation method are in good agreement with those obtained by solving thetemporal equation of motion.

In Figs. 3–6, only the nontrivial states have been plotted and one shouldnote that though the trivial state has not been plotted it exists and is unstableonly between the sub- and supercritical pitchfork bifurcation points as discussed inFig. 2(a).

The effect of the amplitude of magnetic field strength Bm on the frequencyresponse has been shown in Fig. 3. It has been observed that with increase in Bm,the trivial unstable region R1R2 increases and in this region, the nontrivial response

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386 PRATIHER AND DWIVEDY

Figure 2 (a) Frequency response curves; (b) Time responses; (c) Phase portraits corresponding to thepoints A, B, and C, for M = 0�02 kg, �P0 = 0�2507, Bm = 0�30T .

amplitude marginally increases. For example, from Fig. 3, it may be observed thata point P ′ corresponding to �1 equal to 0.95, which is in a stable trivial region forthe value of Bm equal to 0.20T, will be in the unstable region for Bm equal to 0.35T.Also, it is observed that the response amplitude which corresponds to the S–N

Figure 3 Effect of the amplitude of magnetic field strength (Bm) on the frequency response curves forM = 0�02 kg, and �P0 = 0�2507.

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 387

bifurcation point decreases with increase in Bm and for higher values of Bm (e.g.,Bm = 0�3, 0.35), this amplitude remains almost constant. Similarly, the frequency�1 corresponding to this S–N bifurcation point which gives the lower limit of thefrequency of the magnetic field above which the system may vibrate, increases withincrease in Bm and almost remains unchanged for higher values Bm. Hence, tocontrol the vibration of the cantilever beam it is advisable to apply lower amplitudeof magnetic field strength with a frequency higher than the supercritical pitchforkbifurcation point. Also, one may control the vibration by applying a higher valueof amplitude of magnetic field strength with a frequency lower than the frequencycorresponding to the S–N bifurcation point. Further, it may be noted from theexpression of the natural frequency �e that there exists a critical value of Bm (B′

m =1+ P ′

0) beyond which the system is unstable as �2e will be negative. Hence, while

operating at a lower frequency, one should not increase the Bm value arbitrarily andshould operate within this critical limit. In the present work, all Bm values have beentaken below the critical limit.

It may be noted that the figures for the effects of amplitude of static force (�P0)and tip mass (M) on the frequency response curves are not shown here. However,it has been studied that with increase in �P0, the response amplitude increases fora frequency �1 less than 1 and decreases for �1 greater than 1. Hence, �P0 has astabilizing effect for frequency �1 greater than 1 and has a destabilizing effect for�1 less than 1. Also, with increase in tip mass (M), both the response amplitudeand unstable trivial region increases. Therefore, one may control the vibration ofthe system by decreasing the tip mass.

Principle Parametric Resonance Case Due to the Axial Loading(�2 ≈ 2 and �1 Away from 1)

Figure 4 shows the effect of magnetic field on the frequency response curvesin this resonance condition for tip mass M = 0�02 kg, �P0 = 0�2507N , and �P1 =0�2507N . Similar to the previous case, here also the system has both trivial and

Figure 4 Effect of the magnetic field strength (Bm) on the frequency response curves for M = 0�02 kg,�P0 = 0�2507, �P1 = 0�2507.

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388 PRATIHER AND DWIVEDY

nontrivial responses and may observe jump up (R1�, jump down (R2�, and S–Nbifurcation points. It may be observed that with an increase in Bm, the responseamplitude decreases significantly for a frequency �2 less than 2 and increasesmarginally for �2 greater than 2. For example, with �2 equal to 1.6, when themagnetic field is equal to zero, the response amplitude is equal to 2.3 and for Bm

equal to 0.40 T the response amplitude decreases significantly to 1.8. Here, thedecrease in the response amplitude is about 21.7%. Hence, unlike in the simpleresonance case, here the system can be stabilized with increase in Bm for �2 lessthan 2. However, for �2 greater than 2, the magnetic field has a destabilizing effectas the response amplitude increases with increase in Bm. Further, it may be notedthat the trivial state unstable range increases with increase in Bm. Hence, it may beobserved that if the system is vibrating with a frequency �2 in between R1 and R2,it is difficult to bring the response to its trivial state only by changing the amplitudeof the magnetic field.

It has been observed though the figure not shown here that with increase in �P1,both the response amplitude and unstable trivial region gets increased. Also it hasbeen found that with increase in�P0, the response amplitude increases for a frequency�2 less than 2 and decreases for �2 greater than 2. Hence, �P0 has a stabilizing effectfor �2 greater than 2 and has a destabilizing effect for �2 less than 2. Unlike inthe case of Bm, with increase in �P0 as the unstable trivial range decreases, there isa possibility that the system response can be brought to the stable trivial state byincreasing �P0. The tip mass has similar effect as in the previous resonance condition.

Simultaneous Resonance Case (�1 ≈ 1 and �2 ≈ 2)

This is the most interesting resonance condition where the system is subjectedto a magnetic field with a frequency equal to the natural frequency of the systemand also to an axial loading with a frequency nearly equal to twice the naturalfrequency of the system. Figure 5 shows frequency response curve correspondingto M = 0�02 kg, �P0 = 0�2507N�P1 = 0�2507, for various values of Bm and � = 0.

Figure 5 Effect of the amplitude of magnetic field strength (Bm) on the frequency response curves forM = 0�02 kg, �P0 = 0�2507, �P0 = 0�2507, � = 0.

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 389

Figure 6 Frequency response curves to show (a) effect of the amplitude of dynamic axial load (�P1�

for M = 0�02 kg, �P0 = 0�2507, Bm = 0�25 T, � = 0; (b� effect of the phase angle (�� on the frequencyresponse curves for M = 0�02 kg, �P0 = 0�2507, �P0 = 0�2507, Bm = 0�25T.

These curves are similar to that discussed in Fig. 2. Unlike the simple and principalparametric resonance cases, here the unstable trivial region and amplitude of thestable nontrivial state response of the system decreases with increase in Bm. Hence,in this case, the vibration of the system can be reduced completely (i.e., the responsecan be brought to the stable trivial state) with increase in Bm. Like the principalparametric resonance, similar effects have been observed in this case (Fig. 6(a)) byincreasing �P1 where both the response amplitude and unstable trivial range increaseswith increase in �P1. In all these above cases, the frequency of the axial load wasconsidered to be exactly twice the frequency of the magnetic field. Consideringa phase change � between these two frequencies, the response curves have beenplotted as shown in Fig. 6(b) which shows similar effects that are described inFig. 6(a). It may be observed from Figs. 6(a) and (b) that with Bm equal to 0.25T,while the response amplitude a equals to 0.6654 for � equal to zero, with increasein � to �/4, the amplitude increases to 1.0. In other words, the phase will havea destabilizing effect and both the excitations should be applied simultaneously tohave better vibration control.

CONCLUSIONS

In this work the nonlinear vibration of an elastic cantilever beam with tipmass subjected to the harmonic axial load and time varying magnetic field has beenstudied for three different resonance conditions. In all the resonance conditions,with increase in tip mass and amplitude of dynamic loading the unstable trivialrange and the amplitude of stable nontrivial response have been found to beincreased.

For simple resonance condition, it has been observed that lower amplitudeof magnetic field strength with a frequency higher than the supercritical pitchforkbifurcation point or a higher value of amplitude of magnetic field strength with afrequency lower than the frequency corresponding to the saddle-node bifurcationpoint will be more useful to control the vibration. The amplitude of static loading

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390 PRATIHER AND DWIVEDY

has a stabilizing effect for frequency of magnetic field greater than 1 and hasa destabilizing effect for frequency of magnetic field less than 1. In the case ofprincipal parametric resonance condition, it may be noted that with increase inamplitude of magnetic field strength, the response amplitude decreases significantlyfor a frequency of axial loading less than 2 and increases marginally for frequency ofaxial loading greater than 2. It may be observed that if the system is vibrating witha frequency of axial loading which is in the unstable trivial state, it is difficult tobring the response to its trivial state only by changing the amplitude of the magneticfield.

Unlike the simple and principal parametric resonance cases, in the case ofsimultaneous resonance, the unstable trivial region and amplitude of the stablenontrivial state response of the system decreases significantly with increase inamplitude of magnetic field strength. Hence, in this case, the vibration of the systemcan be reduced completely with increase in amplitude of magnetic field strength.Also, it has been observed that with increase in the phase angle between the twoexcitations has a destabilizing effect and it is better to apply both the excitationssimultaneously to have better vibration control.

Instead of solving the time consuming governing or temporal equation ofmotion or conducting expensive experiments one may use the developed reducedequations in all these resonance conditions for different system parameters toobtain the response of the system with sufficient accuracy. Also, these observationsregarding the application of magnetic field and axial force will find extensiveindustrial applications to control the vibration of elastic beams.

APPENDIX

The expressions for the coefficients (i.e., 2�� ��1 ��2 ��3 �f1 �f2 �k1) aregiven below.

�2e =

EI

mL4

h1

h14

+ P0

mL2

(h15

h14

)− B2

m

2�mhd

�0�rmL2

(h15

h14

)

= EI

mL4

h1

h14

(1+ P0L

2

mEI

(h15

h1

)− B2

m

2�mhdL

2

�0�rEI

(h15

h1

))= �2

L�1+ P ′0 − B′

m�

Here,

�2L = EI

mL4

h1

h14

P ′0 =

P0L2

mEI

(h15

h1

) and B′

m = B2m�mhdL

2

2�0�rEI

(h15

h1

)�

Damping ratio due to the viscous damping to the system, 2�� = cdm�e

.

Coefficient of the nonlinear geometric term q3 = ��1 = EIr2

mL4�2e

(h2h14

+ h32h14

+3 h4h14

).Coefficient of the nonlinear inertia term

q2q = ��2 = r2(h5

h14

+ h6

h14

+ m

(h7

h14

− h9

h14

)− h10

h14

− h8

h14

)�

Coefficient of the nonlinear inertia term q2q = ��3 = r2(h11h14

− h12h14

− m h13h14

).

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CANTILEVER BEAM UNDER VARYING MAGNETIC FIELD 391

Coefficient of the parametric excitation term cos�2�1��q = �f1 = B2r

�2eB

2c, where

B2r = B2

m

2 and B2c = �0�rE1IL

2

�mhd

(h1h14

).

Coefficient of the nonlinear damping terms �1+ cos�2�1���q2q = �k1 =

−B2mhd

2m�er2�− h16

h14+ h17

h14�.

Coefficient of the nonlinear damping terms cos��2��q = �f2 = P1mL2�2

e� h15h14

�.The expressions for h1 h2 · · ·h14 are same as those given in Pratiher and

Dwivedy (2009).

REFERENCES

Chen, C. C., Yah, M. K. (2001). Parametric instability of a beam underelectromagnetic excitation. Journal of Sound and Vibration 240:747–764.

Kojima, H., Nagaya, K. (1985). Nonlinear forced vibration of a beam with amass subjected to alternating electromagnetic force. Japan Society of MechanicalEngineers 28:468–474.

Liu, M. F., Chang, T. P. (2005). Vibration analysis of a magnetoelastic beam withgeneral boundary conditions subjected to axial load and external force. Journal ofSound and Vibration 288:399–411.

Lu, Q. S., To, C. W. S., Huang, K. L. (1995). Dynamic stability and bifurcationof an alternating load and magnetic field excited magnetoelastic beam. Journal ofSound and Vibration 181:873–891.

Moon, F. C., Pao, Y. H. (1969). Vibration and dynamic instability of a beam-platein a transverse magnetic field. Journal of Applied Mechanics 36:92–100.

Pratiher, B., Dwivedy, S. K. (2007). Parametric instability of a cantilever beam withmagnetic field and periodic axial load. Journal of Sound and Vibration 305:904–917.

Pratiher, B., Dwivedy, S. K. (2009). Nonlinear vibration of a magneto-elasticcantilever beam with tip mass. Journal of Vibration and Acoustics 131:091011, 1–9.

Shih, Y. S., Wu, G. Y., Chen, J. S. (1998). Transient vibrations of a simplysupported beam with axial loads and transverse magnetic fields. Mechanics ofStructures and Machines 26:115–130.

Wu, G. Y. (2005). Transient vibration analysis of a pinned beam with transversemagnetic fields and thermal loads. Journal of Vibration and Acoustics 127:247–253.

Wu, G. Y., Tsai, R., Shih, Y. S. (2000). The analysis of dynamic stability andvibration motions of a cantilever beam with axial loads and transverse magneticfields. Journal of Acoustical Society of ROC 4:40–55.

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