A feedback control system for vibration of magnetostrictive plate subjected to follower force using sinusoidal shear deformation theory

Ain Shams Engineering Journal (2016) 7, 361–369

Ain Shams University

Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com

MECHANICAL ENGINEERING

A feedback control system for vibration of magnetostrictive plate subjected to follower force using sinusoidal shear deformation theory A. Ghorbanpour Arani *, Z. Khoddami Maraghi Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran Received 8 November 2014; revised 22 March 2015; accepted 29 April 2015 Available online 9 June 2015

KEYWORDS Free vibration; Magnetostrictive plate; Sinusoidal shear deformation theory; Follower force; Velocity feedback gain

Abstract In this research, the vibrational behavior of magnetostrictive plate (MsP) as a smart component is studied. The plate is subjected to an external follower force and a magnetic field in which the vibration response of MsP has been investigated for both loading combinations. The velocity feedback gain parameter is evaluated to study the effect of magnetic field which is generated by the coil. Sinusoidal shear deformation theory is utilized due to its accuracy of polynomial function with respect to other plate theories. Equations of motion are derived using Hamilton’s principle and solved by differential quadrature method (DQM) considering general boundary conditions. The effects of aspect ratio, thickness ratio, follower force and velocity feedback gain are investigated on the frequency response of MsP. Results indicate that magneto-mechanical coupling in MsM helps to control vibrational behaviors of systems such as electro-hydraulic actuator, wireless linear Motors and sensors.  2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Joule [1] is the first person who discovered the magnetostrictive effect or Joule’s effect in 1842. He studied length change in * Corresponding author. Tel.: +98 31 55912450; fax: +98 31 55912424. E-mail addresses: [email protected], a_ghorbanpour@yahoo. com (A. Ghorbanpour Arani), [email protected] (Z. Khoddami Maraghi). Peer review under responsibility of Ain Shams University.

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iron and measured it as a result of magnetization. Villari in 1864 observed that the permeability of ferromagnetic materials depends on the stress state [2]. This phenomenon is the inverse of Joule effect and now called Villari’s effect [3]. After the discovery of Joule and Villari effects, investigation about magnetostriction began. All of ferromagnetic materials have inherent properties that generate magnetostrictive effects due to motion of electrons. In the structure of an atom, orbital magnetic moment is due to electron revolution about the nucleus and spin magnetic moment is generated by electron spinning about its own axis. The superposition of orbital and spin magnetic moment is called atomic magnetic moment. Magnetic domain is a small region containing 109–1015 atoms, in which the orientations of all atomic magnetic moments are the same because of the spontaneous magnetization.

http://dx.doi.org/10.1016/j.asej.2015.04.010 2090-4479  2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

362 Ferromagnetic materials exhibit magnetism on macroscopic scale when they are subjected to external magnetic field [4]. Scientists use this property in design of new structures or to control the disruptive behaviors of the systems and many other applications (see Fig. 1). The brief reports cited above, show the complexity and interesting aspects of this topic. In this regard, some published papers have been collected to review before introducing present work. Free vibration problem of two-dimensional magnetoelectro-elastic laminate plate was investigated by Ramirez et al. [6]. In this work, the composite plate was made of linear homogeneous elastic, piezoelectric, or magnetostrictive layers considering perfect bonding between two layers. They used Ritz method to obtain displacements, electric potential, and magnetic potential with combining discrete layers. Liu [7] investigated an exact deformation analysis for the magnetoelectro-elastic (MEE) fiber-reinforced thin plate. In this regard some characteristics such as elastic displacements, electric potential and magnetic induction for MEE rectangular plate were studied using Kirchhoff’s thin-plate theory. Research about nonlinear principal resonance frequency of an orthotropic and magneto-elastic rectangular plate was done by Xue et al. [8]. Applying a transverse magnetic field and a transverse harmonic mechanical load, the nonlinear vibrational equation for an orthotropic thin plate was derived based on the von Karman plate theory. In this research, the effect of magnetic field, orthotropic material property, plate thickness and mechanical load on the principal resonance behavior was investigated. Based on the nonlinear constitutive equation for giant MsM and linear constitutive relationships for piezoelectric material, a theoretical model was proposed for nonlinear magneto-electric (ME) response in trilayer laminated composites by Yu et al. [9]. The equivalent circuit method was utilized to characterize the ME response considering mechanical losses. In continuation of related papers, Hong [10,11] presented the transient response of MsPs with and without shear effects in two separate papers. In these works, the variation of thermal stresses and center displacement was studied on transient response of thin and thick plates. His results indicated that some parameters could be controlled to desirable values with suitable velocity feedback control gain value. From now on, the referred papers focus on high-order shear deformation plate theory.

Figure 1 (a) Spontaneous magnetism and random orientation of magnetic moments without external field and (b) the alignment of moments under an external magnetic field [4,5].

A. Ghorbanpour Arani, Z. Khoddami Maraghi Vibration and buckling equations of symmetric laminated plates with various boundary conditions were developed by Nosier and Reddy [12]. In the first part, they derived Levinson’s third-order shear deformation theory from Reddy’s third-order theory to study a laminated plate composed of transversely isotropic layers. In the second part, they derived the first-order shear deformation theory and the thirdorder theory of Reddy to study the vibration and buckling of plates. Free vibration of a functionally graded (FG) piezoelectric rectangular plate was investigated by Chen and Ding [13]. Two independent state equations with variable coefficients were derived based on three-dimensional elasticity theory. Considering different boundary conditions at four edges, the exact closed-form solutions were obtained. Free vibration analysis of composite plate assemblies was studied using symbolic computation by Fazzolari et al. [14]. For the first time they developed an exact dynamic stiffness method based on higher-order shear deformation theory. Mantari and Guedes Soares [15] developed a new trigonometric higher-order theory by considering stretching effect. They investigated a FG plate subjected to transverse bi-sinusoidal load and used Naviertype solution for simply supported boundary conditions. To evaluate the accuracy of results, they compared their conclusion with 3D exact solution and other higher-order shear deformation theories. Thai and Kim [16] developed a simple quasi-3D sinusoidal shear deformation theory for bending of FG plates with considering both shear deformation and thickness stretching effects. They used analytical solutions for simplified equations with simply supported edges. Also they compared the accuracy of results with 3D and quasi-3D solutions and those predicted by higher-order shear deformation theories. They concluded that the obtained results are more accurate than those obtained by higher-order shear deformation theories. Hamidi et al. [17] presented a new four variable refined plate theory for thermo-mechanical bending analysis of FG sandwich plates. In their work, the number of unknown functions was only four, unlike any other shear deformation theory. This theory did not require shear correction factor and satisfied shear stress free surface conditions. They validated the result by the classical, the first-order and the other higher-order theories. They concluded that the proposed theory is accurate and simple in solving such a problem. Despite the above mentioned researches, free vibration analysis of MsP using sinusoidal shear deformation theory is a novel topic that cannot be found in the literature. Moreover, MsP is subjected to a tangential surface force which follows the geometry of the system that is called follower force. This is used to control the vibrational behavior of system. Another control parameter is velocity feedback gain which significantly reduces the frequency to desired value. It is assumed that magnetic field is generated by the electric coil and its magnitude varies with coil constant and coil current. In this regard, the velocity feedback gain is defined as a control parameter. Considering above conditions, the effect of various parameters such as the thickness ratio and aspect ratio is studied and results are presented in the form of tables and figures. The results of this work introduce important control factors in vibrational behaviors of MsPs that can help the engineers to design and control new structures.

A feedback control system for vibration of magnetostrictive plate 2. Magnetostrictive materials (MsMs) MsMs are a class of compounds which deform when exposed to magnetic field [4]. There is a magneto-mechanical coupling in this materials due to its reciprocal nature where magnetostrictive effect or Joule effect is deformation response to the applied magnetic field. Fig. 2 shows the MsP subjected to uniform magnetic field that was generated by an electric coil in z direction (Fig. 2a) while an external sub-follower force applied on it, in the opposite direction to x-axis (Fig. 2b). The stress–strain relation considering the magnetic field effect is written for isotropic MsM [10,11] as follows: 2

3 2 32 3 rxx exx Q11 Q12 0 0 0 32 3 6r 7 6 76 e 7 2 0 0 76 yy 7 0 0 e31 0 6 yy 7 6 Q21 Q22 0 6 7 6 76 7 6 76 7 6 rxy 7 ¼ 6 0 6 7 7 e 0 Q 0 5; 0 0  0 0 e 4 5 4 xy 32 44 6 7 6 76 7 6 7 6 76 7 Hz 0 0 Q55 0 54 exz 5 0 0 e34 4 rxz 5 4 0 ryz eyz 0 0 0 0 Q66

Q11 ¼

E ; ð1  t2 Þ

Q12 ¼ Q21 ¼

tE ; ð1  t2 Þ

E ¼ Q66 ¼ ; 2ð1 þ tÞ

Q44 ¼ Q55 ð1Þ

where rij, eij, E, t and eij are stress, strain, Young modulus, Poisson’s ratio and magnetostrictive coupling module that is determined as follow: e31 ¼ e~31 cos2 h þ e~32 sin2 h; e32 ¼ e~31 sin2 h þ e~32 cos2 h; e34 ¼ ð~ e31  e32 Þ sin h sin h;

ð2Þ

in which h represents the direction along which a given magnetic anisotropy may have been induced. Hz is the magnetic field intensity and can be expressed as follows [10]: Hz ¼ Kc Iðx; y; tÞ ¼ Kc CðtÞ

@Wðx; y; tÞ ; @t

ð3Þ

where Kc is the coil constant, I(t) is the coil current, C(t) is the control gain and W(x, y, t) is the transverse deflection of MsP.

363 e y; z; tÞ ¼ Uðx; y; tÞ  z @Wðx; y; tÞ þ fðzÞ/x ; Uðx; @x @Wðx; y; tÞ ð4Þ e þ fðzÞ/y ; Vðx; y; z; tÞ ¼ Vðx; y; tÞ  z @y f y; z; tÞ ¼ Wðx; y; tÞ; Wðx;   when fðzÞ ¼ ph sin pzh , and (U, V, W) are displacement along (x, y, z) direction and (ux, uy) are the rotation of middle surface in the (x, y) direction. Also h is the thickness of MsP. The linear strains using sinusoidal theory can be expressed using strain–displacement relations as follows: e @U ; @x e @V eyy ¼ ; @x ! e @U e @V cxy ¼ ; þ @x @y ! f @U e @W þ ; cxz ¼ @x @z ! e @W f @V : þ cyz ¼ @z @y exx ¼

ð5Þ

Strains are obtained by substituting Eq. (4) into Eq. (5). 4. Motion equation using energy method The strain energy of an elastic body is expressed as [23]: Z h Z Z 1 2 b a U¼ ðrxx exx þ ryy eyy þ sxy cxy þ sxz cxz 2 h2 0 0  þ syz cyz dx dy dz;

ð6Þ

in which r, e are normal stress and strain and s, c are shear stress and shear strain respectively. To obtain the strain energy of MsP, it is enough to substitute Eq. (5) into Eq. (6). The kinetic energy of the MsP is [23]: 2 !2 !2 !2 3 Z Z e e f qm h b a 4 @ U @V @W 5dx dy; ð7Þ K¼ þ þ @t @t @t 2 0 0 where qm is the mass density of MsP.

3. Sinusoidal shear deformation theory 4.1. Follower force For the first time, Levy used sinusoidal functions to introduce a plate theory [18]. After him, researchers like Stein [19], Touratier [20] and Zenkour [21] extended this theory. Simplicity and accuracy of sinusoidal functions are the superiority over the other polynomial functions. Touratier [22] proposed the sinusoidal theory with following assumption:

In one of the classifications, external forces are divided into two groups [24]:

– Transverse shear stress vanishes on the top and bottom surfaces of the plate and is nonzero elsewhere [16].

– Conservative forces such as in-plane forces which maintain their direction when deformation occurs. – Non-conservative forces such as follower force which changes their direction according to the deformation such as hydraulic pressure, gas flow or magnetic field interactive forces acting on micro-structures.

Thus shear correction factors are not introduced in spite of first order shear deformation theory (FSDT). The displacement field for Touratier’s theory is given as [21]:

Flutter and divergence instability are discussed when the follower force acts on the structure. A follower force follows the geometry as it deflects. It is applicable to problems where

364

A. Ghorbanpour Arani, Z. Khoddami Maraghi

Figure 2

Schematic figure of MsP: (a) subjected to transverse magnetic field generated by the electric coil, (b) subjected to follower force.

geometrical deflection is relatively large. In the other hand, the load direction is continuously updated based on the deflected geometry [25]. Variation of the work done by the follower force is written as follows: dXFF ¼ qða  xÞ

@2W dW: @x2

ð8Þ

In this work, follower force is applied on MsP as an external tangential surface force and it is not affected by magnetic field. Utilizing Hamilton’s principle the variational form of motion equations can be written as [23]: Z d

t2

½K  ðU  XFF Þdt ¼ 0:

ð9Þ

t1

Putting the coefficients of dU, dV, dW, dux, duy equal to zero, the dimensionless motion equations are obtained as follows, respectively:

 2   2   2  1 @ m 1 @ m 1 @ m  Q21 a  Q12 a  Q44 a 4 @g@f 2 @g@f 2 @g@f  2   2 2 1 @ w 1 @ w þ S31 a þ S34 ab 4 @s@g 2 @s@f  2   2   2  1 @ u @ u @ u  Q11 a þa  Q44 a 4 @s2 @f2 @f2 ¼ 0;

ð10aÞ

 2   2   2  1 @ u 1 @ u 1 @ u  Q21 b  Q12 b  Q44 b 4 @g@f 2 @g@f 2 @g@f  2   2 2  2  1 @ w 1 @ w @ m þ S34 a  Q22 b þ S32 ab 2 @s@g 4 @f@s @g2  2    1 @ m a @2m  Q44 b þ 4 @g2 c @s2 ¼ 0;

ð10bÞ

A feedback control system for vibration of magnetostrictive plate 







@4w @4w 2 2 þ 1=12a b Q 12 2 @g2 @f @g2 @f2  2   2  @ u @ w þ 1=2S31 a2 þ 1=4S34 b @s@g @s@f  2   4  @ v @ w þ 1=12a4 Q11 þ 1=2S32 a @s@g @f4  4   4  @ w @ w 2 2 þ 1=12b4 Q22 þ 1=12a b Q 44 @g4 @g2 @f2 ! ! @ 3 ug @ 3 uf 1 1 2 2  3 Q44 a b  3 Q44 ab p p @g2 @f @g@f2 ! ! @ 3 uf Q11 3 @ 3 uf 1 2  3 Q12 ab 2 3 a p p @g2 @f @f3 ! ! @ 3 ug @ 3 ug 1 1 2 Q a b   3 Q12 a2 b p p3 21 @g@f2 @g@f2 ! ! 3 @ 3 uf 1 1 2 3 @ ug  2 3 Q22 b  3 Q21 ab @g2 @f @g3 p p  4   4   2  @ w @ w 2 2 2 @ w b  1=12a þ a  1=12a4 @s2 @g2 @s2 @s2 @f2 ! ! @ 3 ug @ 3 uf 1 1 þ 2 3 a3 þ 2 3 a2 b 2 p p @s @g @s2 @f

1=12a2 b2 Q21

¼ 0; 

in which, the dimensionless parameters are introduced as follows:   x y U V W ; ðf; gÞ ¼ ; ; ðu; v; wÞ ¼ ; ; a b a b h sffiffiffiffiffiffi   h h a eij CðtÞKc t E ; ; ; ; Sij ¼ pffiffiffiffiffiffiffiffiffi ; s ¼ ða; b; cÞ ¼ a b b a qm Eqm q qb ¼ : ð11Þ E 5. Solving procedure In this study, DQ method is utilized to solve motion equations to obtain the frequency of MsP. The displacement field is the function of x, y, t, by considering the following mode shapes: uðx; y; tÞ ¼ UðxÞ cosðmpyÞext ; vðx; y; tÞ ¼ VðxÞ sinðmpyÞext ; wðx; y; tÞ ¼ WðxÞ cosðmpyÞext ; /x ðx; y; tÞ ¼ ux ðxÞ cosðmpyÞext ; /y ðx; y; tÞ ¼ uy ðxÞ sinðmpyÞext ;

ð10cÞ

! ! 1 Q44 ab @ 2 uf Q12 ab @ 2 uf  1=4 8 p2 p2 @g@f @g@f ! 2 Q ab @ uf 1 þ Q66 ug  1=4 212 p 8 @g@f ! ! 2 2 2 1 Q22 b @ ug 1 Q44 2 @ ug  a  2 p2 8 p2 @g2 @f2     Q a2 b @ 3 w Q12 a2 b @ 3 w þ þ 44 3 p p3 @g@f2 @g@f2  3  3 3  2 Q ab @ w Q b @ w þ 2 223 þ 21 3 p @g3 p @g@f2 !   2 a2 b @ 3 w 1 a2 @ ug þ 2 3 2 2 @s2 p @s @g 2p ¼ 0;

ð12Þ

where x (rad/s) is the frequency of MsP and m is an integer number known as wave number. After this simplification, one-dimensional DQM is used to approximate the partial derivative of function. In this case, two edges of plate are simply supported. Let F be a function representing u w, v, ux and uy with respect to variables n in the domain of (0 < n < L) [26]: N @kF X ¼ AðKÞ pq Fðfi Þ; k @f k¼1

ð13Þ

where AðkÞ pq is the weighting coefficients associated with kthorder partial derivative of F, and N is the number of grid points in longitudinal direction. Chebyshev polynomials [26] are selected for positions of the grid points. Substituting Eq. (12) into Eq. (10) and using Eq. (13) a system of algebraic equations are achieved as follows: MY€ þ CY_ þ KY ¼ 0;

ð10dÞ

! ! 2 2 1 Q44 ab @ ug 1 Q12 ab @ ug   @g@f @g@f 8 p2 4 p2 ! ! 2 1 Q21 ab @ ug 1 Q44 b2 @ 2 uf   4 p2 8 p2 @g@f @g2 !   1 Q11 a2 @ 2 uf 1 Q44 b2 a @ 3 w þ Q u þ  2 p2 8 55 f p3 @g2 @f @f2     Q b2 a @ 3 w Q b2 a @ 3 w þ 21 3 þ 12 3 2 p @g @f p @g2 @f !     Q11 a3 @ 3 w 1 a2 @ 2 uf a3 @ 3 w þ  2 þ2 3 2 p2 @s2 p p3 @s2 @f @f3 ¼ 0;

365

ð14Þ

where Y is the displacement vector, M is the mass matrix, C is the damping matrix and K is the stiffness matrix. Applying boundary conditions into the final equations, the standard form of motion equations is obtained. Dimensionless frequencies are the eigenvalue of following matrix ([A]) that is named state-space matrix:

½0 ½I ; ð15Þ ½A ¼ ½M1 K ½M1 C where [I] and [0] are the unitary and zero matrixes. Results of Eq. (14) are complex values that contain real and imaginary parts. 6. Numerical results and discussion

ð10eÞ

In this research, vibrational behavior of isotropic MsP is investigated using sinusoidal shear deformation theory. Follower force and velocity feedback gain are two parameters studied in order to control the system frequency.

366

A. Ghorbanpour Arani, Z. Khoddami Maraghi -3

8

x 10

Dimensionless frequency ( )

7

Third mode Second mode First mode

6 5 4 3 2 1 0 0

0.05

0.1

0.15

0.2

0.25

0.3

qb

Dimensionless frequency versus follower force for three

Figure 3 modes.

-3

2.5

x 10

KcC(t)=1e8

2

Dimensionless frequency ( )

In this regard, the effect of various parameters such as thickness ratio and aspect ratio for MsP with mechanical and magnetic properties of E = 30e9 Pa, t = 0.25, qm = 9.25 · 103 kg/m3, e31 = e32 = (26.5 · 109) · (1.67 · 108)N/(mA) is . investigated. The unit of KcC(t) is As m2 The first three frequencies for different values of a = h/a have been reported in Table 1. The standard dimension for thick plate is defined as a P 0:1. The sinusoidal shear deformation theory is valid for both thick and thin plates. Sinusoidal shear deformation theory is an appropriate theory   to analyze thick plates, because the term fðzÞ ¼ ph sin pzh satisfies zero shear stress at top and bottom surfaces of the plate while it is nonzero elsewhere. It can be concluded from the reported results that increasing a will increase the frequency because it is related to the mass and inertia term. Similarly, second and third frequencies also grow with increasing a. Table 2 reports the first three frequencies for different aspect ratios (c = a/b). MsP subjected to uniform magnetic field that generated by electric current passing through the coil. In this case, a constant value for velocity feedback gain is assumed which is equal to KcC(t) = 108. Result indicates that increasing the length to width ratios leads to increase first, second and third frequencies. Results of Tables 1 and 2 have been obtained without considering the follower force. Fig. 3 shows the effect of follower force on dimensionless frequency for three different modes. Obtained frequencies from motion equations are complex values the imaginary part of which is indicated in Figs. 3–6. The flutter instability for a non-conservative force (follower force) occurs when two of the frequencies from two different modes coincide and become complex conjugate as it is shown in Fig. 2. As can be seen, when the system is stable, Im(x) decreases with increasing qb, while the Re(x) is zero. The critical force is the point in which the frequency becomes zero, after it, the stability of system loses. In fact, the system at critical point becomes susceptible to buckling. In Fig. 3 the critical value for follower force is qb  0.11 in the first mode and this value increases in higher modes. In the special interval of follower

KcC(t)=0.9e8 KcC(t)=0.8e8 KcC(t)=0.7e8

1.5

1

0.5

0 0

0.05

0.1

qb

0.15

0.2

0.25

Figure 4 Dimensionless frequency versus follower force for different velocity feedback gains.

Table 1 The first three frequencies of MsP for different thickness ratios. -3

First frequency

2d frequency

3th frequency

6

a = 0.05 a = 0.1 a = 0.2 a = 0.3

0.0004 0.0008 0.0015 0.0020

0.0011 0.0020 0.0035 0.0050

0.0022 0.0040 0.0071 0.0102

5

Table 2 ratios.

The first three frequencies of MsP for different aspect

x 10

=2 =2.1 =2.2 =2.3

4

3

2

1 8

Kc CðtÞ ¼ 10 ; a ¼ 0:2 c = 0.5 c=1 c = 1.5 c=2

Dimensionless frequency ( )

Kc CðtÞ ¼ 108 ; c ¼ 1

First frequency

2d frequency

3th frequency

0.0010 0.0015 0.0027 0.0045

0.0019 0.0035 0.0070 0.0121

0.0036 0.0071 0.0143 0.0246

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

qb

Figure 5 Dimensionless frequency versus follower force for different aspect ratios.

A feedback control system for vibration of magnetostrictive plate

367

-3

0.05

x 10

2

=0.2 =0.3

0.04

1 0.5

0x 10-4

0.05

0.1

0.15

0.2

0.25

8

=0.05 =0.1

6

Dimensionless frequency ( )

Dimensionless frequency ( )

1.5

0

0.045

4

0.035 0.03

=0.5 =1 =1.5 =2

0.025 0.02 0.015 0.01 0.005

2

0 0

0.5

1

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

1.5

2

2.5

3 8

KcC(t)

0.016

x 10

qb

Figure 6 Dimensionless frequency versus follower force for different thickness ratios.

Figure 8 Dimensionless frequency versus velocity feedback gain for different aspect ratios.

-3

9

0.025

x 10

0.015

Dimensionless frequency ( )

Dimensionless frequency ( )

8 0.02

=0.05 =0.1 =0.2 =0.3

0.01

0.005

0

7 6

CSCS CSSS SSSS FSFS SSFS

5 4 3 2 1

0

0.5

1

1.5

KcC(t)

2

2.5

3 8

x 10

0

0

0.5

1

1.5

KcC(t)

Figure 7 Dimensionless frequency versus velocity feedback gain for different thickness ratios.

force, first and second modes combine together and flutter occurs. The effect of velocity feedback gain on dimensionless frequency is illustrated in Fig. 4. Velocity feedback gain as a control parameter can be used to reduce the frequency of MsP. On the other hand, magnetic field significantly influences vibrational behavior of system where in the presence of follower force the minimal changes in velocity feedback gain cause a considerable change in frequency. Increasing the velocity feedback gain leads to decrease the dimensionless frequency significantly. Fig. 4 has been plotted for four values of velocity feedback gain which are close together. It worth to mention that KcC(t) = 0 means uncontrolled system. Fig. 5 indicates the effect of aspect ratio on dimensionless frequency in presence of the follower force. The curves show that vibration response of system is dependent on dimension of MsP so that the change in aspect ratio (c = a/b) from 2 to 2.3 significantly increases the dimensionless frequency. For this reason the small range of aspect ratio has been selected. So it is attempted to report the dimensionless frequency for

2

2.5

3 8

x 10

Figure 9 Dimensionless frequency versus velocity feedback gain for different boundary conditions.

other aspect ratios (0.5, 1, 1.5 and 2) in Table 1. Fig. 5 proves that the frequency of MsP increases when the length to width of MsP increases. The variation of thickness ratio for MsP is more effective than aspect ratio as Fig. 6 illustrated. The critical value of follower force (when the dimensionless frequency is equal to zero) considerably changes for a = 0.05, 0.1, 0.2, 0.3. The curves with obvious differences prove that the dimensionless frequency increases where the thickness ratio of MsP increases. Figs. 7 and 8 display the effect of velocity feedback gain on the dimensionless frequency of MsP by considering the dimensions of MsP. As it is clear in Figs. 7 and 8, increasing the velocity feedback gain causes convergence of the curves almost to the same values so that for large values of KcC(t), the influence of plate dimensions is negligible. In fact, decrease in frequency continues until the effect of magnetic field is dominated by the other parameters so that no longer a, c have any practically effect on the frequency in comparison with low velocity feedback gain.

368

A. Ghorbanpour Arani, Z. Khoddami Maraghi Fig. 11 displays the first mode shape for different boundary conditions for a = 0.2, c = 2, KcC(t) = 108, q = 0. These figures approve that the boundary conditions are satisfied in this method.

1.4

Present work (Sinusoidal shear deformation theory)

Dimensionless frequency ( )

1.2

Higher order shear deformation theory[14] First order shear deformation theory[12]

1

7. Conclusion

0.8 0.6 0.4 0.2 0

0

1

2

3

4

5

6

7

8

Figure 10 theories.

9

10 5

KcC(t)

x 10

Comparison of the results using three different plate

(a) CSCS

(b) SSSS

(c) FSCS

(d) FSSS

In this study, forced vibration response of an isotropic MsP subjected to follower force and unidirectional magnetic field has been investigated using an accurate theory known as sinusoidal shear deformation theory. This theory satisfies the stress-free boundary conditions on the top and bottom surfaces of the plate and also there is no need to shear correction factor. Follower force and velocity feedback gain can be used as controlling parameters of vibration behavior in MsP. Results indicate that increasing the follower force leads to decrease frequency until it approaches zero, while an intense magnetic field significantly reduces frequency to a constant value. Increasing intensity of the magnetic field using velocity feedback control can change the vibration behavior of MsP and this is an advantage of MsM compared to other smart materials. Increasing aspect ratio and thickness ratio increases the frequency of MsP while these changes are negligible at high velocity feedback gain. In fact, we can study the vibration response of the system to an active noise and specify input in order to counteract an undesirable noise or vibration. In this regard, MsPs are utilized in many smart structures such as magneto-mechanical sensors, actuators, acoustic/ultrasonic transducers, fuel injectors, active noise and vibration cancellation. Acknowledgments

(e) FSFS

The author would like to thank the reviewers for their comments and suggestions to improve the clarity of this article. This work was supported by University of Kashan [grant number 363443/31]. References

Figure 11 Vibrational mode shapes of MsP for different boundary conditions.

Also Fig. 9 shows the same results for five different boundary conditions. Boundary conditions on each edge of the plate are combination of clamped (C), simply (S) and free (F) cases. The cases CSCS and SSSS have the maximum and minimum frequencies, respectively. To check the accuracy of results, the motion equations are solved by three different plate theories for a = 0.2, c = 1. Higher-order shear deformation theory presented in Ref. [14] and first-order shear deformation theory developed in Ref. [12] were used to compare the results with sinusoidal shear deformation theory. As can be seen from Fig. 10, there is a good agreement between the result of sinusoidal shear deformation theory and two other theories with a little difference due to different displacement fields and assumptions.

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Ali Ghorbanpour Arani received his B.Sc. degree from Sharif University of Technology in Tehran, Iran, in 1988. He then received his M.Sc. degree from Amirkabir University of Technology in Tehran, Iran, in 1991 and his Ph.D degree from the Esfahan University of Technology in Esfahan, Iran, in 2001. He is currently a Professor at the Mechanical Engineering faculty of University of Kashan in Kashan, Iran. He has authored more than 140 refereed journal papers and 10 books. His current research interests are stress analyses, stability and vibration of nano/micro structures, composite and FGMs. Zahra Khoddami Maraghi received her B.Sc. degree from the University of Kashan in kashan, Iran, in 2008. She then received her M.Sc. degree from University of Kashan in Kashan, Iran, in 2011. She is currently a Ph.D student at University of Kashan in Kashan, Iran. She has authored more than 15 refereed journal papers and two books. Her research interests are nano/micro-mechanics, vibration, control and instability of smart materials.