Vibration suppression of a rotating cantilever beam under magnetic excitations by applying the magnetostrictive material

Abstract

The dynamic model and vibration suppression of a rotating cantilever beam under magnetic excitations are investigated in this article. The nonlinear constitutive relation of magnetostrictive materials is presented. The layout of the control system is demonstrated and explained. The kinetic energy, potential energy of the system, and work done by the electromagnetic force are obtained. The dynamic equations of the system are obtained and discretized by the Hamilton principle and Galerkin approach, respectively. Based on the negative feedback control method, the control scheme is implemented by the magnetostrictive layer. The dynamic model and control method are validated by the references. Various parameter values of the magnetic excitations and rotating beam systems are investigated to reveal their effects on the control behaviors of the bending vibration. Results illustrate that the magnetic excitations bring negative stiffness in the system and increase the responses of beam greatly. The magnetostrictive suppression is effective and can be regarded as the damping effect in the dynamic equations. Increasing the control gain, bias magnetic field and width ratio of the magnetostrictive layer to the controlled layer are beneficial to the vibration control. However, enlarging the angular velocity and pre-stress is harmful to the vibration suppression.

Keywords

Vibration control magnetostrictive material rotating cantilever beam magnetic field

Introduction

Rotating beams are of great importance in engineering applications such as aircraft rotary wings, helicopter rotor blades, spacecrafts with flexible appendages (Malgaca et al., 2015). Although investigations about the rotating beams are extensive, the studied working conditions rarely involved the magnetic field. With the development of modern aerospace technology, traditional metallic material cannot meet the needs of practical applications. Key progress may depend on the adoption of smart materials among which the attention of soft ferromagnetic material or magnetoelastic material is growing rapidly from the research and industrial communities. The magnetic field is almost everywhere with the popularity of numerous electromagnetic devices. When the ferromagnetic structures are placed in a magnetic field, they are correspondingly subjected to magnetic forces resulting from the interaction between the applied magnetic field and the magnetization in the structures. This magnetic force leads to deformation or vibration of structures and in return changes itself.

More and more highly flexible beams are applied due to their excellent performances in both mass and properties. Therefore, the undesirable vibration caused by their flexible inherency is unavoidable during the rotating process (Kim et al., 2013). The strong vibration will affect the operating accuracy and stability of the structures. The noise which is accompanying with the vibration can be harmful to the environment. Moreover, long-term vibration may cause fatigue damage to the structures and reduce their service life (Wu et al., 2014). The reduction of such vibration may improve the dynamic performances of the beam. Therefore, approaches that suppress or control the vibration of the rotating beam are helpful for the engineering applications. Investigations of appropriate control scheme are very important and necessary.

Modeling of rotating beams are interesting research topics and have seen investigated extensively. In 1987, some simulation results of the linear model for structures with large overall motions were revealed (Kane et al., 1987). Since then, much work has been dedicated to modeling of the geometric stiffening effect (Yang et al., 2004; Yoo et al., 1995). Among them, an improvement was made by taking the motion-induced stiffness into account (Yoo et al., 1995). In the last decade, many dynamic models which include the stiffening and Coriolis effects due to beam rotation have been presented. The modeling methods may be classified into three categories. The elastic deformation described in the first category (Cai et al., 2004; Chung and Yoo, 2002; Hamza-Cherif, 2005) is based on the hybrid set of deformation variables (stretch, chordwise and flapwise deformations). In the modeling methods of the second category (Al-Qaisia and Al-Bedoor, 2005; Banerjee and Kennedy, 2014; Kim et al., 2013), the authors took the stiffening effect of a beam into account by adding the elastic potential energy resulting from centrifugal force to the total potential energy. In order to consider the stiffening effect of rotating beams, the authors in the third category (Arvin et al., 2011; Huang et al., 2010; Li et al., 2014) adopted the nonlinear strain or stress.

In general, vibration control systems are classified according to their dynamics and energy requirements into three categories: (1) the passive control system, (2) the semi-active control system, and (3) the active control system (Abdeljaber et al., 2016). The control theory has been investigated extensively and many approaches such as the classical control (Khot et al., 2012), the adaptive control (Mahmoodi et al., 2010), the intelligent control (Qiu et al., 2009), the robust control (Hu, 2012) have been developed accordingly. Although the passive damping reduces vibration, it is well recognized that active vibration control should be explored to achieve significant improvements in the overall performance of flexible rotating beams. Nowadays, the active vibration control technique has become a research hotspot, due to its advantages of high adaptability and sound control performance.

In the past decades, there has been an increasing interest in the application of smart material and structure technologies in the active vibration control of flexible structures. Magnetostrictive materials are appearing as a highly attractive material for smart structure applications because they show certain unique advantages over other materials, such as the ability to produce large forces at a voltage lower than that of piezoelectric transducer (PZT) ceramic material, to react more rapidly than shape memory alloy (SMA), to keep their properties even after ground into particles (Moon et al., 2007). Magnetostriction is a phenomenon of strong coupling between magnetic properties and mechanical properties of some ferromagnetic materials: strains are generated in response to an applied magnetic field, while, conversely, mechanical stresses in the materials produce measurable changes in magnetization. Although many ferromagnetic materials exhibit this property, the strain that can be obtained from them is very low and not usable for practical applications (Yan et al., 2004). On the other hand, giant magnetostrictive material (GMM) is known to exhibit considerable deformations or changes in magnetization when subjected to magnetic fields or strains, which makes it a viable option to be used in smart structures actuation and sensing applications.

The design and theoretical research in active control of flexible beams with GMM have received considerable attention in recent years (Kumar et al., 2003; Subramanian, 2002). But the linear constitute relations for GMM are usually assumed in their studies and the control system could be modeled by the linear theory. However, it has been shown by the experiment that the magnetomechanical responses are inherently nonlinear and coupled with the applied magnetic field, pre-stress, and induced strain (Dapino et al., 2006). Young’s modulus changes with the stress and the magnetic field nonlinearly. This means the control system based on the linear constitutive relations is only tenable and effective in a narrow approximately linear region of the deformation versus applied magnetic field curves of GMM. Later, an analytically theoretical model to fully describe the experimental results was developed (Zheng and Liu, 2005). To utilize the full potential of GMM, based on this nonlinear and coupled constitutive model, some extended investigations have been carried out and the coupling behavior of Terfenol-D rods was studied (Sun and Zheng, 2006), In addition, the active vibration control of Terfenol-D rods and laminated composite beams was investigated (Zhou et al., 2006; Zhou and Zhou, 2007). The constitutive model for multiferroic composites based on the nonlinear reversible and irreversible ferromagnetic behaviors was established (Avakian and Ricoeur, 2016).

The vibration control of a rotating cantilever beam is hard when compared with the stationary structures. The arrangements of sensor and control force devices are relatively complicated. In fact, many engineering applications for power system are in the rotational state. The vibration characteristics of the rotating beams are considerably different from those of beams without rotational motion due to the coupling of flexible deformations and rigid motions. On the other hand, as investigated in our previous publication (Xu et al., 2017), the surrounding magnetic field will enlarge the vibration of the rotating cantilever beam. Therefore, the vibration suppression for vibration control of rotating beams under magnetic excitation is rather important and necessary. For accurate operation and precise control of such rotating beams, an accurate and reliable dynamic control scheme needs to be investigated.

This article aims to investigate the possibility and effectiveness of the magnetostrictive control method for a rotating cantilever beam under magnetic excitations. The studied object and control system are illustrated and explained in section “System descriptions.” Then the nonlinear constitutive relation of the GMM is discussed and the dynamic equations of the system are obtained in section “Dynamic model.” Next the control approach is developed and validations are presented in sections “Control method” and “Validations,” respectively. Moreover, the scheme for numerical solutions is demonstrated in section “Numerical simulations.” Section “Results and discussion” displays the simulation results and control effects for different surrounding magnetic fields and system parameters. Finally, some main conclusions are unfolded.

System descriptions

A flexible laminated cantilever beam made of ferromagnetic material with length L, width b, and thickness h is shown in Figure 1, which is fixed to a rotating rigid hub with radius R. A static coordinate system OXYZ and a dynamic coordinate system oxyz which is stationary with respect to the rotating beam are established. The axial and chordwise deformations of the rotating cantilever beam are u and v, respectively. The deformation in the z-direction is neglected here. The surrounding magnetic field is B which depicts its strength and direction. The whole beam can be decomposed into two parts: the magnetostrictive layer and the controlled layer. A magnetostrictive layer with width b_m adheres to the controlled layer closely. The magnetostrictive layer which is often made of Terfenol-D material is applied as the giant magnetostrictive actuator here. The induced magnetic field of magnetostrictive layer is produced and adjusted by solenoid coils. The main goal of this layer is to suppress or control the chordwise deformation.

The layout of the main control system around the rotating cantilever beam is illustrated in Figure 2. The magnetostrictive layer covers the whole side surface (L × h). The magnetostrictive material behaves excellent properties especially the scalability. Consequently, the magnetostrictive layer can work well when it is imposed in the rotating situation and undergoes the centrifugal load. The solenoid coils are supported by four rods which are fixed to the hub and rotate with the cantilever beam. The radius of the solenoid coil is r. To improve the control performances of the system, the multi-input and multi-output approaches are adopted. As demonstrated in Figure 2, a number of solenoid coils are evenly distributed along the x-direction. The midpoint coordinates of the solenoid coils are x₁, x₂, x₃,…, x_k, respectively, and the length of each solenoid coil is l₀. Each solenoid coil is equipped with a velocity sensor which can provide the instructions for the control current. Therefore, the currents of various solenoid coils are independent, which makes it possible to track the vibration information of any position along the beam meticulously and achieve the continuous control.

Next, the control current in the solenoid coil needs to be looked further to figure out how it implements the control process. The short solenoid coil can be regarded as a simple coil with certain current. The center of coil is defined x = 0. According to Biot–Savart’s law, the magnetic induction intensity along the axis (the straight line passing through the center of the circle and perpendicular to the plane of the coil) is

B = \frac{μ_{0} nI r^{2}}{2 {(r^{2} + x^{2})}^{3 / 2}}

(1)

where μ₀ = 4π × 10⁻⁷ H/m is the vacuum permeability and n is the turns of the coil. I is the current in the coil and x is the investigation point in the axis.

Equation (1) can be as well rewritten as follows

B = \frac{μ_{0} nI / r}{2 {[1 + {(x / r)}^{2}]}^{3 / 2}}

(2)

Without losing generality, the parameters in equation (2) can be simplified as μ₀nI/r = 10⁻³. As displayed in Figure 3, the length of each solenoid coil is set as l₀ = 0.1r. Then midpoints of the coils are x_k = (0.1k – 0.05)r (k = 0, 1, 2,…), respectively, and the magnetic field distributions inside the solenoid coils are plotted. From the figure, the magnetic field is distributed approximately uniform and shows wonderful consistency. The blue lines in the figure demonstrate that the currents in solenoid coils are constant, while the red lines show that the current increases successively by 1% from the constant current.

It is well known that the linear velocity increases when the observation point moves to the free end of the rotating beam even if the angular velocity is the same. The zone near the free end of the rotating cantilever beam is the key positions of energy conversion. In addition, the control equipment is hard to work efficiently near the free end due to the high linear velocity and large centrifugal force. As a result, the solenoid coils are usually arranged around the fixed end in engineering applications. The vibration amplitude of the cantilever beam increases from the fixed end to the free end. When other conditions are the same, the control current should be enlarged gradually to adapt to the dynamic responses. Therefore, this control layout is very applicable for the flexible and continuous rotating beam.

Before the dynamic model is established, some assumptions are put forward first: (1) the magnetostrictive layer and the controlled layer adheres to each other perfectly, and no slip occurs during the rotating process of the beam, (2) the induced magnetic field and the surrounding magnetic field are independent, (3) the shear deformation of the beam is negligible, (4) the magnetostrictive effect is performed by the currents in the solenoid coils which remain good contact during the motion, (5) the masses of the solenoid coils and related supporting devices are ignored, and (6) the surrounding solenoid coils do not limit the deformation of the beam.

Dynamic model

The velocity vector of the rotating cantilever beam in the dynamic coordinate system can be expressed as follows

v^{Ω} = \overset{\cdot}{u} i^{Ω} + \overset{\cdot}{v} j^{Ω}

(3)

where the superposed dot represents differentiation with respect to time and i^Ω, j^Ω are the unit vectors of the oxyz coordinate system.

The velocity in the static coordinate system can be obtained through the principle of velocity synthesis

v = v^{Ω} + \bar{Ω} \times e

(4)

where $\bar{Ω} = {\begin{matrix} 0 & 0 & Ω \end{matrix}}^{T}$ and Ω is the rotational angular velocity around the z-axis. e is the position vector of the object in the static coordinate system.

Point C on the beam is selected as the investigated point and it moves to point $C'$ due to motion in Figure 1. The position vector e can be divided into two parts

e = e_{0} + d

(5)

where e₀ ={R + x, 0, 0}^T is the initial position vector along the x-direction and d = {u, v, 0}^T is the displacement vector of the cantilever beam at the selected point.

Therefore, the velocity vector in the static coordinate can be obtained by substituting equations (3) and (5) into equation (4)

v = (\overset{\cdot}{u} - Ω v) i^{Ω} + [\overset{\cdot}{v} + Ω (R + x + u)] j^{Ω}

(6)

The kinetic energy of the whole beam can be written as the form of integration along the x-direction

K = \frac{1}{2} ρ_{e} A \int_{0}^{L} v \cdot v dx

(7)

where ρ_e is the equivalent mass density of the beam and ρ_e = (ρ_mb_m + ρ_b(b – b_m))/b in which ρ_m and ρ_b are the mass densities of the magnetostrictive layer and the controlled layer, respectively. A is cross-sectional area of the beam and A = bh.

According to the von Karman strain theory, the normal strain along the x-direction at point (x, y) in the oxyz system is given by

ε_{x} = \frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial v}{\partial x})}^{2} - y \frac{\partial^{2} v}{\partial x^{2}}

(8)

Due to the inherent nonlinearities of the magnetostrictive characteristics, the relationship between the output displacement of magnetostrictive materials and input magnetic field is nonlinear and influenced by the pre-stress and bias magnetic field. The nonlinear magnetostrictive constitutive relationship is described as follows (Zheng and Liu, 2005)

\begin{matrix} ε = \frac{σ}{E_{s}} \\ + {\begin{matrix} λ_{s} \tanh (\frac{σ}{σ_{s}}) + [1 - \tanh (\frac{σ}{σ_{s}})] \frac{λ_{s}}{M_{s}^{2}} M^{2} & \frac{σ}{σ_{s}} ⩾ 0 \\ \frac{λ_{s}}{2} \tanh (\frac{2 σ}{σ_{s}}) + [1 - \frac{1}{2} \tanh (\frac{2 σ}{σ_{s}})] \frac{λ_{s}}{M_{s}^{2}} M^{2} & \frac{σ}{σ_{s}} < 0 \end{matrix} \end{matrix}

(9)

\begin{matrix} H = \frac{1}{k} f^{- 1} (\frac{M}{M_{s}}) \\ - {\begin{matrix} 2 {σ - σ_{s} \ln [\cosh (\frac{σ}{σ_{s}})]} \frac{λ_{s}}{μ_{0} M_{s}^{2}} M & \frac{σ}{σ_{s}} ⩾ 0 \\ 2 {σ - \frac{σ_{s}}{4} \ln [\cosh (\frac{2 σ}{σ_{s}})]} \frac{λ_{s}}{μ_{0} M_{s}^{2}} M & \frac{σ}{σ_{s}} < 0 \end{matrix} \end{matrix}

(10)

where k = 3χ_m/M_s is the relaxation factor, χ_m represents the linear magnetic susceptibility, M is the magnetization, M_s is the saturation magnetization, f(x) =coth(x) – 1/x, λ_s is the saturation magnetostrictive coefficient, E₀ stands for the initial Young’s modulus when σ = 0, H = 0, E_s is the saturation Young’s modulus with the relation σ_s = λ_sE_sE₀/(E_s – E₀).

It can be observed from equation (9) that the first two terms are independent of magnetic field and can be combined as one expression. The last term containing λ_s is associated with the magnetostrictive strain λ (σ, H) induced by the magnetic field. Therefore, equation (9) can be rewritten as

ε = \frac{σ}{E (σ)} + λ (σ, H)

(11)

and

\frac{σ}{E (σ)} = \frac{σ}{E_{s}} + {\begin{matrix} λ_{s} \tanh (\frac{σ}{σ_{s}}) & \frac{σ}{σ_{s}} ⩾ 0 \\ \frac{λ_{s}}{2} \tanh (\frac{2 σ}{σ_{s}}) & \frac{σ}{σ_{s}} < 0 \end{matrix}

(12)

λ (σ, H) = {\begin{matrix} [1 - \tanh (\frac{σ}{σ_{s}})] \frac{λ_{s}}{M_{s}^{2}} M^{2} & \frac{σ}{σ_{s}} ⩾ 0 \\ [1 - \frac{1}{2} \tanh (\frac{2 σ}{σ_{s}})] \frac{λ_{s}}{M_{s}^{2}} M^{2} & \frac{σ}{σ_{s}} < 0 \end{matrix}

(13)

A unified expression of the whole cantilever beam can be obtained by taking the properties of the controlled layer and the magnetostrictive layer into account

σ = E ε - E λ (σ, H)

(14)

where E = E(σ) is suitable for the magnetostrictive layer and E = E_b is suitable for the controlled layer. Eλ (σ, H) is the equivalent stress caused by the magnetostrictive layer and this term is only suitable for the magnetostrictive layer.

The potential energy for the whole system can be obtained

\begin{matrix} P = \frac{1}{2} \int_{0}^{L} EA {(\frac{\partial u}{\partial x})}^{2} dx + \frac{1}{2} \int_{0}^{L} E I_{z} {(\frac{\partial^{2} v}{\partial x^{2}})}^{2} dx \\ + \frac{1}{2} \int_{0}^{L} EA \frac{\partial u}{\partial x} {(\frac{\partial v}{\partial x})}^{2} dx + \frac{1}{8} \int_{0}^{L} EA {(\frac{\partial v}{\partial x})}^{4} dx \\ - \int_{0}^{L / 2} E A_{m} λ (σ, H) \frac{\partial u}{\partial x} dx \\ + \int_{0}^{L / 2} E (\int_{- b / 2}^{- b / 2 + b_{m}} hydy) λ (σ, H) \frac{\partial^{2} v}{\partial x^{2}} dx \end{matrix}

(15)

where A = A_m is applicable to the magnetostrictive layer and A_m = b_mh. A = A_b is applicable to the controlled layer and A_b = (b – b_m)h. It is defined that $Λ = \int_{- b / 2}^{- b / 2 + b_{m}} h ydy$ and $Λ = (h / 2) b_{m} (b_{m} - b)$ .

The electromagnetic force acting on a volume (here refers to the rotating cantilever beam) in the surrounding magnetic field is (Wu, 2015)

F = \int [μ (E_{e} + \overset{\cdot}{r} \times B) \times B + M \cdot (\nabla B)] dV

(16)

where μ is the conductivity and $\overset{\cdot}{r}$ is the velocity of the volume element. E_e is the induced electric field and M is the volume density of magnetization in the body.

In this specific study, the magnetic field is considered to be uniform and the induced electric field is equal to zero due to the structural symmetry of the beam. Therefore, the electromagnetic force on a volume element described in equation (16) can be simplified as

f = μ (\overset{\cdot}{r} \times B) \times B

(17)

The magnetic field is uniform within the motion area of the rotating cantilever beam and the surrounding magnetic field can be generally described as follows

B = B_{X} i + B_{Y} j + B_{Z} k

(18)

where B_X, B_Y, and B_Z are the components of B in the static coordinate axes, respectively; i, j, k are the unit vectors of the OXYZ coordinate system.

The magnetic field is stationary when determined, while the cantilever beam will rotate with time. The relative motion and coupling effects will produce the magnetic excitations which enlarge the vibration of the beam. A further investigation about the relationship between the static and dynamic parameters is needed consequently. The magnetic field in the static coordinate system can be transformed to the dynamic coordinate system by a transformation matrix

B^{Ω} = [\begin{matrix} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}] B

(19)

where θ is the angle between the static and dynamic coordinate systems. θ can also be regarded as the angular displacement of the rotating cantilever beam when the two coordinate systems coincide initially.

Substituting equation (18) into equation (19), the expression of B^Ω can be derived

\begin{matrix} B_{x}^{Ω} = B_{X} \cos θ - B_{Y} \sin θ \\ B_{y}^{Ω} = B_{X} \sin θ + B_{Y} \cos θ \\ B_{z}^{Ω} = B_{Z} \end{matrix}

(20)

Substituting equations (6) and (20) into equation (17), the electromagnetic forces in the x-direction and y-direction can be obtained, respectively (Xu et al., 2017)

\begin{matrix} f_{x} = μ {[\overset{\cdot}{v} + Ω (R + x + u)] B_{x}^{Ω} B_{y}^{Ω} - (\overset{\cdot}{u} - Ω v) (B_{y}^{Ω^{2}} + B_{z}^{Ω^{2}})} \end{matrix}

(21)

\begin{matrix} f_{y} = μ {(\overset{\cdot}{u} - Ω v) B_{x}^{Ω} B_{y}^{Ω} - [\overset{\cdot}{v} + Ω (R + x + u)] (B_{x}^{Ω^{2}} + B_{z}^{Ω^{2}})} \end{matrix}

(22)

where $B_{x}^{Ω}$ , $B_{y}^{Ω}$ , and $B_{z}^{Ω}$ are the components of B^Ω in the dynamic coordinate axes, respectively.

The work done by the electromagnetic force can also be obtained

W = \int \int \int (u f_{x} + v f_{y}) dxdydz

(23)

In order to obtain the dynamic equations of the system, the Hamilton principle is adopted here

\int_{t_{1}}^{t_{2}} (δ K + δ W - δ P) dt = 0

(24)

Therefore, the dynamic equations for stretching motion and bending motion in the axial direction and chordwise direction are finally presented as follows

\begin{matrix} ρ_{e} A (\overset{\cdot\cdot}{u} - 2 Ω \overset{\cdot}{v} - \overset{\cdot}{Ω} v - Ω^{2} u) - EA (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial x^{2}}) \\ - 2 A μ B_{x}^{Ω} B_{y}^{Ω} Ω u - A σ (B_{y}^{Ω^{2}} - B_{x}^{Ω^{2}}) Ω v \\ + A σ ({\overset{\cdot}{B}}_{x}^{Ω} B_{y}^{Ω} + B_{x}^{Ω} {\overset{\cdot}{B}}_{y}^{Ω}) v - 2 A σ B_{y}^{Ω} {\overset{\cdot}{B}}_{y}^{Ω} u + E A_{m} \frac{\partial λ (σ, H)}{\partial x} \\ = ρ_{e} A Ω^{2} (R + x) + A σ B_{x}^{Ω} B_{y}^{Ω} Ω (R + x) \end{matrix}

(25)

\begin{matrix} ρ_{e} A (\overset{\cdot\cdot}{v} + 2 Ω \overset{\cdot}{u} - Ω^{2} v + \overset{\cdot}{Ω} u) + E I_{z} \frac{\partial^{4} v}{\partial x^{4}} \\ - EA [\frac{3}{2} {(\frac{\partial v}{\partial x})}^{2} \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} u}{\partial x^{2}} \frac{\partial v}{\partial x} + \frac{\partial u}{\partial x} \frac{\partial^{2} v}{\partial x^{2}}] \\ + 2 A μ B_{x}^{Ω} B_{y}^{Ω} Ω v - A σ (B_{y}^{Ω^{2}} - B_{x}^{Ω^{2}}) Ω u \\ + A σ ({\overset{\cdot}{B}}_{x}^{Ω} B_{y}^{Ω} + B_{x}^{Ω} {\overset{\cdot}{B}}_{y}^{Ω}) u \\ - 2 A σ B_{x}^{Ω} {\overset{\cdot}{B}}_{x}^{Ω} v + E Λ \frac{\partial^{2} λ (σ, H)}{\partial x^{2}} \\ = - ρ_{e} A \overset{\cdot}{Ω} (R + x) - A σ (B_{x}^{Ω^{2}} + B_{z}^{Ω^{2}}) Ω (R + x) \end{matrix}

(26)

In equations (25) and (26), the differentials of the magnetic field intensity in the dynamic coordinate system can be written as follows

{\overset{\cdot}{B}}_{x}^{Ω} = - Ω B_{X} \sin θ - Ω B_{Y} \cos θ + {\overset{\cdot}{B}}_{X} \cos θ - {\overset{\cdot}{B}}_{Y} \sin θ

(27)

{\overset{\cdot}{B}}_{y}^{Ω} = Ω B_{X} \cos θ - Ω B_{Y} \sin θ + {\overset{\cdot}{B}}_{X} \sin θ + {\overset{\cdot}{B}}_{Y} \cos θ

(28)

If the magnetic field is constant with respect to time, equations (27) and (28) can be further degenerated into simpler expressions

{\overset{\cdot}{B}}_{x}^{Ω} = - Ω B_{X} \sin θ - Ω B_{Y} \cos θ

(29)

{\overset{\cdot}{B}}_{y}^{Ω} = Ω B_{X} \cos θ - Ω B_{Y} \sin θ

(30)

Finally, the corresponding boundary conditions are given by

u = v = \frac{\partial v}{\partial x} = 0 (x = 0)

(31)

\frac{\partial u}{\partial x} = \frac{\partial^{2} v}{\partial x^{2}} = \frac{\partial^{3} v}{\partial x^{3}} = 0 (x = L)

(32)

Control method

The induced magnetic field intensity in equation (10) is composed of the bias magnetic field and the control magnetic field produced by the solenoids. Moreover, the axial component of the surrounding magnetic field should be taken into consideration. Therefore, the total magnetic field intensity for the constitutive relationship of GMM can be expressed as follows

H (x, t) = H_{c} (x, t) + H_{b} + H_{x}

(33)

where H_b, H_c(x, t) are the intensities of the bias magnetic field and control magnetic field, respectively. H_x is the intensity components of the surrounding magnetic field in the x-direction. The total magnetic field in the x-direction is applied to the magnetostrictive constitutive equation and the magnetostrictive strain in other direction is ignored.

The controlled magnetic field is determined by the solenoid and current parameters, and the differential of the magnetic induction intensity can be obtained based on equation (1)

dB = \frac{μ_{0}}{2} \frac{n r^{2}}{{(r^{2} + x^{2})}^{3 / 2}} I (x, t) dx

(34)

where I(x, t) stands for the input current of solenoid.

Then the equations $B = \int dB$ and $H = B / μ_{0}$ are adopted to express the generated magnetic field

\begin{matrix} H_{c} (x, t) = \\ \frac{1}{2} n (\frac{l_{0} / 2 + x_{k}}{\sqrt{r^{2} + {(l_{0} / 2 + x_{k})}^{2}}} + \frac{l_{0} / 2 - x_{k}}{\sqrt{r^{2} + {(l_{0} / 2 - x_{k})}^{2}}}) I (x, t) \end{matrix}

(35)

The magnetic field in the solenoid can be assumed to be evenly distributed. As a result, equation (35) can be simplified as

H_{c} (x, t) = cI (x, t)

(36)

where $c = (1 / 2) n (l_{0} / \sqrt{r^{2} + {(l_{0} / 2)}^{2}})$ is the coil factor of the solenoid.

The current in the coil is proportional to the velocity of the bending vibration based on the fact that the closed-loop negative velocity feedback control is adopted in this article. The feedback current of the solenoid is presented as

I (x, t) = - G \overset{\cdot}{v} (x, t)

(37)

where G is the control gain.

The control magnetic field intensity can be obtained

H_{c} (x, t) = - cG \overset{\cdot}{v} (x, t)

(38)

By adopting the Galerkin method, the stretching vibration and bending vibration are approximated by linear combinations of the basis functions

u (x, t) = \sum_{i = 1}^{N} φ_{i} (x) U_{i} (t), v (x, t) = \sum_{i = 1}^{N} ϕ_{i} (x) V_{i} (t)

(39)

where U_i(t) and V_i(t) are the stretching response and bending response with respect to time, respectively.

The mode functions for the stretching vibration and transverse vibration of a cantilever beam are respectively given by

φ_{i} (x) = \sin (\frac{2 i - 1}{2} \frac{π x}{L})

(40)

\begin{matrix} ϕ_{i} (x) = \cosh λ_{i} x - \cos λ_{i} x \\ - \frac{\sinh λ_{i} L - \sin λ_{i} L}{\cosh λ_{i} L + \cos λ_{i} L} (\sinh λ_{i} x - \sin λ_{i} x) \end{matrix}

(41)

where λ_i should meet the equation cos λ_i L cosh λ_iL + 1 = 0.

It can be observed from equations (9), (10), and (38) that the strain, induced magnetic field, and vibration displacement are related. In addition, two steps are needed to obtain the partial differential of strain with respect to the displacement

\frac{\partial λ (σ, H)}{\partial x} = \frac{\partial λ (σ, H)}{\partial H (x, t)} \frac{\partial H (x, t)}{\partial x}

(42)

The first term in the right side of equation (42) is determined by the nonlinear constitutive relation of the magnetostrictive layer. And the second term is related to the displacement and solenoid coil current.

The relative magnetization factor is defined as ζ = M/M_s and $ζ \in [- 1, 1]$ . Then the differentials of equations (10) and (13) are conducted respectively

\frac{\partial λ}{\partial H} = \frac{\partial λ}{\partial ζ} / \frac{\partial H}{\partial ζ}

(43)

Substituting equation (39) into equation (38), the induced magnetic field can be obtained by making derivations. The differential of magnetic field with respect to location is written as follows

\frac{\partial H (x, t)}{\partial x} = \frac{\partial H_{c} (x, t)}{\partial x} = - cG \sum_{i = 1}^{N} ϕ_{i}' (x) {\overset{\cdot}{V}}_{i} (t)

(44)

Substituting equations (39) to (44) into equations (25) and (26), the discrete dynamic equations of the system can be obtained

\begin{matrix} \sum_{i = 1}^{N_{u}} (m_{ij}^{u} {\overset{\cdot\cdot}{U}}_{i} + g_{ij}^{uv} {\overset{\cdot}{V}}_{i} + k_{ij}^{u} U_{i} + k_{ij}^{uv} V_{i} + κ_{ikj} V_{i} V_{k}) = F_{j}^{u} \\ j = 1, 2, \dots, N_{u}, k = 1, 2, \dots, N_{v} \end{matrix}

(45)

\begin{matrix} \sum_{i = 1}^{N_{v}} (m_{in}^{v} {\overset{\cdot\cdot}{V}}_{i} + g_{in}^{vu} {\overset{\cdot}{U}}_{i} + g_{in}^{v} {\overset{\cdot}{V}}_{i} + k_{in}^{v} V_{i} + k_{in}^{vu} U_{i} \\ + κ_{ijkn} V_{i} V_{j} V_{k} + κ_{ikn} U_{i} V_{k}) = F_{n}^{v} \\ n = 1, 2, \dots, N_{v}, j = 1, 2, \dots, N_{v} \end{matrix}

(46)

where $m_{ij}^{u} = ρ_{e} A \int_{0}^{L} φ_{i} (x) φ_{j} (x) dx$ , $g_{ij}^{uv} = - 2 ρ_{e} A Ω \int_{0}^{L} ϕ_{i} (x) φ_{j} (x) dx + cGE A_{m} \frac{\partial λ}{\partial H} \int_{0}^{L} {ϕ'}_{i} (x) φ_{j} (x) dx$ , $k_{ij}^{uv} = - ρ_{e} A \overset{\cdot}{Ω} \int_{0}^{L} ϕ_{i} (x) φ_{j} (x) dx - A μ (B_{y}^{Ω^{2}} - B_{x}^{Ω^{2}}) Ω \int_{0}^{L} ϕ_{i} (x) φ_{j} (x) dx + A μ ({\overset{\cdot}{B}}_{x}^{Ω} B_{y}^{Ω} + B_{x}^{Ω} {\overset{\cdot}{B}}_{y}^{Ω}) \int_{0}^{L} ϕ_{i} (x) φ_{j} (x) dx$ , $k_{ij}^{u} = - ρ_{e} A Ω^{2} \int_{0}^{L} φ_{i} (x) φ_{j} (x) dx - EA \int_{0}^{L} φ_{i}^{″} (x) φ_{j} (x) dx - 2 A μ B_{x}^{Ω} B_{y}^{Ω} \int_{0}^{L} φ_{i} (x) φ_{j} (x) dx - 2 A μ B_{y}^{Ω} {\overset{\cdot}{B}}_{y}^{Ω} \int_{0}^{L} φ_{i} (x) φ_{j} (x) dx$ , $κ_{ikj} = - EA \int_{0}^{L} {ϕ'}_{i} (x) ϕ_{k}^{″} (x) φ_{j} (x) dx$ , $F_{j}^{u} = ρ_{e} A Ω^{2} \int_{0}^{L} (R + x) φ_{j} (x) dx + A μ B_{x}^{Ω} B_{y}^{Ω} Ω \int_{0}^{L} (R + x) φ_{j} (x) dx$ , $m_{in}^{v} = ρ_{e} A \int_{0}^{L} ϕ_{i} (x) ϕ_{j} (x) dx$ , $g_{in}^{vu} = 2 ρ_{e} A Ω \int_{0}^{L} φ_{i} (x) ϕ_{j} (x) dx$ , $g_{in}^{v} = cGE Λ \frac{\partial λ}{\partial H} \int_{0}^{L} ϕ_{i}^{″} (x) ϕ_{j} (x) dx$ , $k_{in}^{v} = - ρ_{e} A Ω^{2} \int_{0}^{L} ϕ_{i} (x) ϕ_{n} (x) dx + E I_{z} \int_{0}^{L} {ϕ_{i}}^{(4)} (x) ϕ_{n} (x) dx + 2 A μ {\hat{B}}_{x} {\hat{B}}_{y} Ω \int_{0}^{L} ϕ_{i} (x) ϕ_{n} (x) dx - 2 A μ {\hat{B}}_{x} \overset{\cdot}{\hat{}} B_{x} \int_{0}^{L} ϕ_{i} (x) ϕ_{n} (x) dx$ , $k_{in}^{vu} = ρ_{e} A \overset{\cdot}{Ω} \int_{0}^{L} φ_{i} (x) ϕ_{n} (x) dx - A μ ({\hat{B}}_{y}^{2} - {\hat{B}}_{x}^{2}) Ω \int_{0}^{L} φ_{i} (x) ϕ_{n} (x) dx + A μ (\overset{\cdot}{\hat{}} B_{x} {\hat{B}}_{y} + {\hat{B}}_{x} \overset{\cdot}{\hat{}} B_{y}) \int_{0}^{L} φ_{i} (x) ϕ_{n} (x) dx$ , $κ_{ijkn} = - \frac{3}{2} EA \int_{0}^{L} {ϕ'}_{i} (x) {ϕ'}_{j} (x) ϕ_{k}^{″} (x) ϕ_{n} (x) dx$ , $κ_{ikn} = - EA \int_{0}^{L} φ_{i}^{″} (x) {ϕ'}_{k} (x) ϕ_{n} (x) dx - EA \int_{0}^{L} {φ'}_{i} (x) ϕ_{k}^{″} (x) ϕ_{n} (x) dx$ , $F_{n}^{v} = - ρ_{e} A \overset{\cdot}{Ω} \int_{0}^{L} (R + x) ϕ_{n} (x) dx - A μ ({\hat{B}}_{x}^{2} + {\hat{B}}_{z}^{2}) Ω \int_{0}^{L} (R + x) ϕ_{n} (x) dx$ and $I_{z} = A b^{2} / 12$ .

The values of magnetic field are usually around 10⁵ in the unit of A/m according to the references (Abdeljaber et al., 2016; Arvin et al., 2011; Khot et al., 2012; Li et al., 2014), and the magnitude of pre-stress is 10 MPa in the investigations. Therefore, reference values H₀ = 10⁵A/m and σ₀ = 10 MPa may be introduced here for the dimensionless process. The following dimensionless parameters are applied for numerical computations

ξ = \frac{x}{L}, τ = \frac{t}{T}, δ = \frac{R}{L}, χ = \frac{b_{m}}{b}, κ = \frac{b}{L}, α = \frac{cGL}{T H_{0}}

(47)

η = T Ω, γ = T^{2} \overset{\cdot}{Ω}

(48)

ε_{b} = \frac{H_{b}}{H_{0}}, ε_{c} = \frac{H_{c}}{H_{0}}, ε = \frac{H}{H_{0}}, ζ = \frac{σ}{σ_{0}}

(49)

β_{X} = B_{X} \sqrt{\frac{μ T}{ρ_{e}}}, β_{Y} = B_{Y} \sqrt{\frac{μ T}{ρ_{e}}}, β_{Z} = B_{Z} \sqrt{\frac{μ T}{ρ_{e}}}

(50)

β = \sqrt{β_{X}^{2} + β_{Y}^{2} + β_{Z}^{2}}

(51)

where $T = L^{2} \sqrt{ρ A / E I_{z}}$ , ξ is the dimensionless position, τ is the dimensionless time, δ is the dimensionless hub radius, χ is the width ratio of the magnetostrictive layer to the controlled layer, $κ$ is the slenderness of the beam, α is the dimensionless control gain, η is the dimensionless rotating velocity, γ is the dimensionless rotating acceleration, ε_b is the dimensionless bias magnetic field intensity, ε_c is the dimensionless control magnetic field intensity, ε is the dimensionless total magnetic field intensity, ζ is the dimensionless pre-stress, and β_X, β_Y, and β_Z are the dimensionless ambient magnetic field components in the X, Y, Z directions, respectively. β is the dimensionless ambient magnetic field intensity.

Validations

The dynamic model of a rotating cantilever beam in the surrounding magnetic field can be verified from two aspects. The rotating cantilever beam without the surrounding of magnetic field has been studied (Kim et al., 2013). On the other hand, there exist some investigations about the structural dynamics of a pinned beam in the transverse magnetic field (Qiu et al., 2009). But the study about clamped-free (C-F) boundary condition of a beam in the magnetic field is rare. Here the ANSYS software is adopted to compute the dynamic responses of the C-F case. The proposed model is validated through comparisons of the dynamic responses with reference models.

The simulation parameters in Kim et al. (2013) are adopted and the parameters related to the magnetic field are defined as zero for the purpose of results comparison. Figure 4(a) and (b) illustrates the dynamic responses in the axial and chordwise directions. The results of Kim et al. (2013) are also given in the figure for comparisons. It is observed that the non-dimensional responses are consistent with that of Kim et al. (2013). Therefore, the dynamic model is verified well for these cases.

It is clear that the bending vibration is much larger than the stretching vibration and the bending vibration is evaluated in Figure 4(c). The magnetic field intensity is 0.5 T in the y-direction. The step is 2.0 × 10⁻⁵ and the initial condition is set to be v/h = 1.0 for τ = 0. Figure 4(c) displays the dynamic responses of a C-F beam from τ = 0.1 s so as to remove the effects of initial conditions. The results obtained by the proposed model agree well with the FEM results. The slight difference on the curves may be due to the different calculation methods.

Moreover, the control method also needs to be validated to evaluate its effectiveness. A laminate composite beam with a magnetostrictive layer in Khot et al. (2012) is taken as the reference. The bias magnetic field H_b = 0.1912kOe and pre-stress σ = –5 MPa are applied in the computation process. The surrounding magnetic field and rotating angular velocity are defined to be zero for the same condition with the reference case. The present and reference control results of the dimensionless bending deformations are illustrated in Figure 5.

Figure 5 shows that the control responses of the bending vibration attenuate fast when compared with the uncontrolled case. When the control scheme is applied, the damped natural frequency should be considered. This explains the phenomenon that the controlled response curves shift to right when compared with the uncontrolled cases. Furthermore, the present results are in accordance with the references well. The analytical approach is adopted here while the finite element method is applied in the reference, which may explain for the slight difference in the initial attenuation stage. Consequently, the control method is verified to be effective.

Numerical simulations

The dynamic responses of the cantilever beam are investigated in the following. First, the material characteristics of the magnetostrictive layer and the controlled layer are provided: χ_m = 80, E_s = 110 GPa, σ_s = 200 MPa, λ_s = 1.3 × 10⁻³, μ₀M_s = 0.8 T, E_b = 148.6 GPa, ρ_m = 9250 kg/m³, ρ_b = 7860 kg/m, E_b = 148.6 GPa. Second, the constant dimensionless parameters are given: δ = 0.01 and κ = 0.05. Then, the rotational angular velocity (η) and angular acceleration (γ) need to be determined. The common motion process of the cantilever beam which includes the startup and operation is investigated here. The expressions of the rotational angular velocity and angular acceleration are depicted as follows

η = {\begin{matrix} τ - \frac{5}{π} \sin (\frac{π τ}{5}) & 0 ⩽ τ ⩽ 10 \\ 10 & 10 < τ ⩽ 20 \end{matrix}

(52)

γ = {\begin{matrix} 1 - \cos (\frac{π τ}{5}) & 0 ⩽ τ ⩽ 10 \\ 0 & 10 < τ ⩽ 20 \end{matrix}

(53)

Furthermore, if only the uniform motion is studied, the velocity and acceleration are defined as η = 10 and γ = 0, respectively. Next the adjustable parameters are presented: α = 6, χ = 0.1, ζ = –40.4, and ε_b = 1.5. If not specially mentioned, the initial parameters above are adopted in the numerical simulations.

Finally, the dimensionless magnetic field components (β_X, β_Y, and β_Z) in the three coordinate axes should be provided. The kinds of surrounding magnetic field may be different and should be investigated separately. The surrounding magnetic field can be generally studied from simple to complex: the in-plane magnetic field, the spatial magnetic field, and the time-varying magnetic field. The parameters that describe the magnetic field are displayed in Figure 6.

As displayed in Figure 6(a), the in-plane magnetic field can be characterized by two parameters: the amplitude β and the plane angle Ф. And the components in the X-axis and Y-axis are β_X = β cos Ф and β_Y = β sin Ф, respectively. Three parameters are needed to describe a spatial magnetic field in Figure 6(b): the amplitude β, the spatial angle ψ, and the plane angle Ф. When these parameters are determined, the magnetic field components in the X-direction, Y-direction, and Z-direction can be obtained, respectively, and β_X = β sin ψ cos Ф, β_Y = β sin ψ sin Ф, and β_Z =β cos ψ.

As displayed in Figure 7, the magnetostrictive strain increases with the magnetic field intensity and this relationship is nonlinear. The results are in accordance with the widely accepted results in the reference (Zheng and Liu, 2005). When the magnetic field intensity increases continually and reaches a certain value, all of the magnetic poles have been arranged orderly and the strain tends to be saturated at these cases. It can be observed from Figure 7 that the strain is a function of both the pre-stress and the magnetic field intensity. If the pre-stress is determined in the beginning, the accurate value of strain can be obtained through the magnetic field intensity which is related to the dynamic responses according to the control method. Therefore, this relationship lays a reliable theoretical foundation for the implementation of the vibration control.

Before solving the dynamic equations, the interactive way between the magnetostrictive parameters and the dynamic responses should be revealed. The procedures that handle the calculation of the magnetostrictive layer can be divided into five steps:

Provide the pre-stress σ and bias magnetic field intensity H_b first; in addition, the control magnetic field in the beginning is zero, that is, H_c = 0.

Calculate the relative magnetization factor ζ = M/M_s.

Compute the nonlinear relationship $\partial λ / \partial H$ and substitute it into the dynamic model.

Obtain the dynamic responses and the control magnetic field intensity H_c.

Acquire the total magnetic field H = H_b +H_c + H_x and go back to step 2.

The number of basis function is an important value in the Galerkin method, because it may affect the accuracy of responses. It can be assumed that the number of basis function satisfies the equation N = N_u = N_v for convenience. The convergence test for the dynamic responses is performed as the number of basis functions increases. The dimensionless magnetic field intensity is 0.2 as an example and ψ = Ф = 0. Results indicate that the dynamic responses at the free end converge fast as N increases. To find out the differences, the root mean square (RMS) of the dynamic responses is adopted. Results shows that the maximum difference between the RMS values for N = 20 and N = 30 in stretching and bending vibrations is below 1%. Moreover, as the number of the basis functions increases, the increment of RMS becomes less and less. For the perfect results, the basis function number N should be taken as many as possible. However, it may bring about more complexity and higher time cost. Therefore, N = 30 is adopted in this article based on the balance between the accuracy and calculation cost.

The zero initial conditions are imposed on the stretching and bending vibration. The dynamic responses of the system are computed by the Runge–Kutta approach in the software MATLAB at ξ = 1 with the parameters and initial conditions prescribed above. Moreover, considering the fact that the stretching vibration is weaker than the bending vibration for a cantilever beam with high ratio of length to width, the control of bending vibration is focused in this article.

Results and discussion

The in-plane magnetic field is investigated first and the parameters are β = 0.2, Ф = π/4. Figure 8(a) shows the motion process that the rotating cantilever beam starts up from the initial equilibrium position, accelerates gradually, and maintains a constant angular velocity finally. The dynamic responses are the dimensionless displacement of the bending vibration at the free end. It can be observed that the amplitude decreases apparently with control and the deflection returns to the neighbor of equilibrium position quickly. Figure 8(c) illustrates the corresponding dimensionless control magnetic field intensity which is self-adaptive to the vibration displacement. The bending vibration during this process is controlled well. Figure 8(b) displays the dynamic responses of the beam with and without control when the angular velocity of the beam is constant. Figure 8(d) is the control magnetic field intensity. The vibration response is periodic without the control while the response attenuates rapidly when the magnetostrictive control approach is applied. The amplitude of initial periodic response attenuates over 80% within 5 s and the amplitude of controlled responses remains at a small range. It implies that the magnetostrictive control plays an equivalent role of damping and the control method is effective. When comparing the final responses in Figures 5 and 8, it can be noticed that the controlled responses tend to be zero for the cases without surrounding magnetic field and the amplitudes of final responses for the system under magnetic excitations are larger than zero. It implies that the magnetic excitations act as the negative stiffness in the dynamic equations, which can also be found from equation (48). This effect will increase the dynamic responses of the system, which has been investigated in our previous publication (Xu et al., 2017). Therefore, applying the control force of GMM is effective for the vibration attenuation of rotating beams under magnetic excitations. The following content mainly discusses the control performances for different parameters including the magnetic field environment and control conditions in the case of constant angular velocity.

Control effects for different magnetic excitations

The parameters that affect the control results for the case of in-plane magnetic field are the dimensionless magnetic field intensity β and the plane angle Ф. The dynamic responses of the rotating cantilever beam for different magnetic field intensities are illustrated in Figure 9. And in the analysis, the plane angle can be assumed to be π/4.

As shown in Figure 9, the responses without control are periodic and the controlled responses decrease gradually. It can be observed from Figure 9(a) that when the magnetic field intensity increases, the response amplitudes and the average value of the responses increase. When the GMM control force is applied, the bending vibration of different situations attenuates and remains at a small range ultimately. Furthermore, Figure 9(b) displays that the controlled responses are around 0.003 for β = 0.2 while the controlled responses are about 0.02 for β = 0.5. It illustrates that the average of the fluctuation zone gradually deviates the initial equilibrium position as the magnetic field intensity increases. The damping control effect can suppress the fluctuation, but it cannot alter the final responses zone. At the same time, the fluctuation scope is expanded by the increasing magnetic field intensity. The fluctuation attenuation is harder for a large response amplitude; thence, the fluctuation scope increases with the pace of magnetic field intensity. A magnetic field with high intensity may increase the control cost and weaken the preciseness of the control simultaneously.

Then the effects of plane angle are investigated when the magnetic field intensity maintains constant (β = 0.2), and the control responses of bending vibration are displayed in Figure 10.

The plane angles within the range [0, 2π] are studied in detail and the control effects for different plane angles can be obtained from corresponding lines in Figure 10. The responses around τ = 0.1 stand for the initial bending vibrations without control. The vibration amplitudes are the biggest when the plane angle is near π/2 and 3π/2 in the uncontrolled state. On the other hand, the vibration amplitudes are the smallest when the plane angle is near π/4 and 5π/4. This phenomenon reveals the effects of plane angle on the uncontrolled dynamic responses. The responses attenuate fast with time and reach stable responses with small fluctuations within 5 s when the control method is applied. In addition, the final dynamic responses of the controlled beam for different plane angles are almost the same, which demonstrates that the steady-state responses of the controlled beam are not influenced by the initial plane angles and the control method is effective for the whole range of the plane angle.

Next, the cases of spatial magnetic field are investigated and three main parameters namely the magnetic field intensity β, spatial angle Ψ as well as the plane angle Ф are involved. Based on the discussions of the in-plane magnetic field, we need to study the differences between the spatial and plane distributions of the magnetic field with the same intensity. Moreover, if the spatial distribution of the magnetic field is determined, the effects of different intensities on control results also need to be considered; the corresponding results are presented in Figure 11.

As displayed in Figure 11, if the magnetic field intensity keeps unchanged (e.g. β = 0.4), the stable responses of the controlled beam for the plane distribution (Ψ = π/2) are around 0.013, while the response values are close to 0.02 for the spatial distribution (Ψ = π/4). The magnetic field component in the Z-direction (β_Z) is nonzero for the cases of spatial distribution. The bending vibration is directly affected by β_Z in the dynamic equations and nonzero β_Z increases the dynamic responses. Although the spatial distribution results in larger vibration amplitude, the control method can still restrain the bending vibration and make the initial responses attenuate in a narrow range. When the spatial distribution remains constant (Ψ = π/4, Ф = π/4), the control effects of magnetic field intensity are the same as the cases of in-plane magnetic field. The increase of magnetic field intensity will add the difficulties in vibration control.

The spatial angle needs to be investigated alone. The other parameters are set and unchanged (β = 0.2, Ф = π/4), and the whole range of spatial angle [0, 2π] is divided into some equal parts. The dynamic responses of the controlled beam for different spatial angles are displayed in Figure 12.

The responses for different spatial angles around τ = 0.1 show the bending vibration of the beam without control, and the periodic feature is apparent. When the control force is applied on the beam, the vibration becomes weak gradually and is suppressed well after the first 5 s. The initial characteristics of responses for spatial angles and plane angles are similar. However, the control effects of spatial angle and plane angle are rather different. As has been discussed, the final responses for different plane angles are almost the same, while the final results for different spatial angles are still unequal. The controlled responses are the largest around Ψ = π/2 and Ψ = 3π/2. Ψ = 0 and Ψ = π lead to the smallest controlled responses. The controlled bending vibration still remains the relative size with the initial state, even though the vibration amplitude has been attenuated greatly by the control approach. If the surrounding magnetic field is determined and cannot be changed, adjusting the installation location of the controlled cantilever beam system is helpful for the vibration control.

In the above analysis, the surrounding magnetic field is assumed to be time-invariant with various values of intensity and spatial angle. If the amplitude of surrounding magnetic field is varying with time periodically, the control results may be different. The common form of a time-varying parameter in industries is a sine or cosine function. Here other parameters are fixed (Ψ = π/2, Ф = π/4) and the magnetic field intensity is assumed to be β = 0.2 cos (ωτ) in which ω is the angular frequency. Figure 13 displays the control effects of the cantilever beam for different angular frequencies of the magnetic field.

When the controlled responses of the time-varying magnetic field are compared with that of the constant magnetic field, the responses increase obviously. The vibration of controlled beam still tends to be periodic after attenuation. The amplitude decreases apparently, which implies the control approach is effective for the time-varying cases. However, the control results for the time-varying magnetic fields become worse. The time-varying magnetic field will provide the time-varying stiffness continuously. As a result, the scope of the final responses is periodical. On the other hand, when the angular frequency of the magnetic field increases from 0.79 to 1.57, the amplitude of the controlled vibration becomes large and the fluctuation pace tends to be faster. The differential of the time-varying magnetic field with respect to time will bring the angular frequency into the dynamic equations, which can increase the amplitude of the controlled responses. Therefore, more effective control method should be investigated for a rotating cantilever beam under time-varying magnetic excitations.

Control effects for different system parameters

Without losing generality, the surrounding magnetic field is assumed to be characterized by β = 0.2, Ψ = π/2, Ф = π/4. The effects of control gain, width ratio of the magnetostrictive layer, pre-stress, bias magnetic field, and rotational angular velocity on the control behaviors are investigated in this section.

Figure 14(a) illustrates the effects of different dimensionless control gains on dimensionless controlled responses. With the increase of control gain, the control magnetic field intensity induced by the displacement increases. The control reaction is more sensitive and the attenuation speed is rapider for a greater control gain. Therefore, the attenuation time for α = 6 is less than that for α = 4. As has been revealed, the control approach is reflected in the dynamic equations as damping. Consequently, under the premise of a reasonable control gain, increasing the control gain appropriately will be helpful for speeding up the attenuation process.

Figure 14(b) illustrates the bending vibration of the cantilever beam for different width ratios of the magnetostrictive layer to the controlled layer. It is clear that when the width ratio increases, the attenuation time decreases. If the size of controlled layer is fixed, increasing the width ratio means enlarging the width of the magnetostrictive layer. Thus, larger force and torque brought by the magnetostrictive strain appear. When this effect is applied in the control process, better control outcomes are obtained.

Figure 14(c) displays the control results of bending vibration for different pre-stresses. When the pre-stress changes from −40.4 to −65.4 MPa, the attenuation time for the controlled beam increases, which indicates that the control magnetic field becomes weak. It can be known from equations (9) and (10) that the change of pre-stress will adjust the relationship between the strain of the magnetostrictive layer and the induced magnetic field. The pre-stress is determined when the magnetostrictive layer adheres to the controlled layer. When the pre-stress is negative, it means the magnetostrictive layer undergoes compressive stress. Therefore, the control magnetic field should remove the effect of compressive stress before producing the magnetostrictive pull strain. If all the other conditions are the same, the magnetostrictive strain decreases apparently as the pre-stress alters from −40.4 to −65.4 MPa in Figure 7. This effect is reflected in the damping term of the dynamic equations; therefore, the attenuation process slows down.

Figure 14(d) demonstrates the control effects of the controlled beam for different bias magnetic fields. The existence of the bias magnetic field can make the magnetostrictive layer magnetized to some extend for the application of the control magnetic field. From the figure, the attenuation process of bending vibration accelerates with the increase of the bias magnetic field and the control effect behaves better. The bias magnetic field can influence the initial total magnetic field intensity considering H = H_b + H_c + H_x. In the beginning of the control stage, H_c = 0 and H = H_b + H_x. When the total magnetic field is large, it can be deduced from Figure 7 that the corresponding strain increases. Therefore, the initial control strength is enough and the control process enters the good state smoothly. All these reasons result in less time for the attenuation process.

Figure 15 demonstrates the bending vibrations of the controlled and uncontrolled cantilever beam for different rotational angular velocities (η = 8 and η = 10). From the figure, the responses of the uncontrolled beam increase greatly when the angular velocity increases. When the control is conducted, the responses attenuate quickly and reach to a small fluctuation zone. The fluctuation range is not restrained by the angular velocity when comparing the final controlled vibrations, which reflects the wide application scope of the control method for different velocities. On the other hand, when the constant angular velocity increases, the relative strength of negative feedback becomes weaker if other parameters are the same. The attenuation speed of the controlled beam slows down and the whole attenuation time increases.

Now the qualitative analysis and discussion about the main control parameters are revealed. Increasing the control gain, width ratio and bias magnetic field as well as decreasing the angular velocity and pre-stress are benefit for the control behaviors. However, there still exist some problems in evaluating which parameter is more significant because the investigations above are independent. Therefore, quantitative investigations are needed. An idea of how to distinguish the importance of the various parameters is proposed here. When the amplitude of controlled responses first reaches the 10% of the uncontrolled amplitude, the time during this process is defined as the attenuation time τ_s. Because it is hard to assess different parameters simultaneously, the same change percent of each parameters is conducted. The initial attenuation time is τ_s₁ and the attenuation time after change is τ_s₂. An example of how to compare the relative importance is displayed in Table 1.

Table 1.

The relative importance of the control parameters.

Parameters	Initial value	Changes	τ_s ₁	τ_s ₂	(τ_s₁ – τ_s₂)/τ_s₁ (%)
α	6	+20%	4.7325	4.2304	10.61
η	10	–20%	4.7325	4.2768	9.63
χ	0.1	+20%	4.7325	4.0245	14.96
ξ	–40.4	–20%	4.7325	3.5158	25.71
ε_b	0.2	+20%	4.7325	3.6469	22.94

It can be observed that the pre-stress and bias magnetic field are more sensitive than other parameters. By conducting this quantitative analysis, the relative importance of these five parameters can be sorted. Moreover, it can provide some instructions when we want to adjust the attenuation speed and time. It should be pointed out that we just give an idea to evaluate the relative importance of different parameters. The control system is nonlinear and different initial values as well as changes may result in different outcomes. This idea is meaningful because if enough initial parameters values and changes percent are investigated, it will work as a map for us to find the most appropriate control parameters.

Conclusion

The vibration control of a rotating cantilever beam under magnetic excitations was investigated. The magnetostrictive layer was applied as the actuator and the nonlinear constitutive relation was analyzed. The layout of control system was presented and inspected. The kinetic energy, potential energy, and work done by the electromagnetic force were obtained. The Hamilton principle and Galerkin approach were adopted to obtain and discretize the dynamic equations, respectively. The negative feedback control method was utilized in the control system, which was performed by the solenoid coils. The dynamic model and control method were validated. Different magnetic excitations and control parameters were investigated to discover their effects on the control outcomes. Main conclusions can be drawn from the discussions:

The magnetostrictive control method is effective for the bending vibration of a rotating cantilever beam in surrounding magnetic fields and it plays the role of damping in the dynamic equations.

The magnetic excitations act as the negative stiffness for the system and increase the responses amplitude and fluctuation.

Nonlinear constrictive characteristics of the magnetostrictive layer can affect the control results deeply. It can be determined by the values of pre-stress and bias magnetic field.

The magnetostrictive control works mainly by attenuating the dynamic responses. The attenuation strength and speed are influenced by many parameters. Increasing the control gain, width ratio, and bias magnetic field as well as decreasing the angular velocity and pre-stress can reduce the attenuation time.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author(s) gratefully acknowledge the financial support from Natural Science Foundation of China (grant nos 51335006 and 11472147).

ORCID iD

Fulei Chu

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