Torque modeling and characteristic analysis of electromagnetic piezoelectric hybrid-driven 3-degree-of-freedom motor

Abstract

The electromagnetic piezoelectric hybrid-driven 3-degree-of-freedom motor is a new multi-degree-of-freedom motor. To further analyze the torque characteristics of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor. First, the principle and basic structure of the hybrid-drive motor are introduced, and the displacement and pressure distribution of the stator–rotor contact surface are obtained by analytical method. Based on this, the torque model of the piezoelectric stator-drive motor is obtained. Then, the air-gap magnetic field model of the permanent magnet rotor is obtained by analytical method, and the electromagnetic stator-torque model is obtained. Finally, the torque model of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor is established by vector synthesis. The effects of piezoelectric stator mounting position angle, stator–rotor contact materials, and preload on motor torque are analyzed by simulation. The advantages of electromagnetic piezoelectric hybrid drive are analyzed, and the rationality of the model is preliminarily verified. It lays the foundation for further optimization design and performance improvement of electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor.

Keywords

Torque model piezoelectric motor hybrid drive 3 -degree-of-freedom electromagnetic characteristics

Introduction

With the development of modern intelligent industry, the requirements for precision of multi-degree-of-freedom drive devices are constantly increasing. The multi-degree-of-freedom motion actuator composed of the traditional single-degree-of-freedom motor has complicated mechanical structure,¹ large volume, and low positioning control precision, which cannot meet the requirements of intelligent precision driving. The multi-degree-of-freedom motor has a simple structure and high integration, which can be applied in a precise drive system.² There are two main driving methods for multiple free motors: piezoelectric and electromagnetic.

Piezoelectric-driven multi-degree-of-freedom ultrasonic motors were first proposed by the Tokyo University of Agriculture and Technology.³ In recent years, international scholars have conducted extensive research on ultrasonic multi-degree-of-freedom motors.⁴ Yamagata University proposed a small cylindrical 3-degree-of-freedom motor. The hollow cylinder of the stator of the motor acts to conduct amplifying vibration, and the vibration is provided by a piezoelectric layer below the cylinder. Muroran Institute of Technology proposed a disk-type multi-degree-of-freedom ultrasonic motor and conducted in-depth comparative studies on different bearing lubrication methods.^4,5 Zhejiang University conducts in-depth research on mathematical models and motion control of traveling wave multi-degree-of-freedom motors.^6,7 Shanghai University proposed a sandwich motor of more than 10 degrees of freedom⁸ and established a fixed-rotor contact model. Nanjing University of Aeronautics and Astronautics proposed a circular standing wave multi-degree-of-freedom motor⁹ and reviewed the current problems in the design of this type of motor. Harbin Institute of Technology proposed a sandwich ultrasonic multi-degree-of-freedom hybrid-drive motor,¹⁰ and they produced a prototype and analyzed the characteristics of the variable excitation mode of multi-degree-of-freedom ultrasonic motor. Southeast University has designed a multi-degree-of-freedom ultrasonic motor with a double stator clamping structure,¹¹ which has improved drive torque and improved output performance and stability. Piezoelectric-driven motors have received the widest attention because of their high power density, stable efficiency at very small dimensions, high positioning accuracy, fast response, and power-off interlocking.¹² However, since the piezoelectric-driven motor is based on friction transmission, its performance and life depend on the performance of the friction interface.¹³

Electromagnetic-drive multi-degree-of-freedom motors were developed first by the Soviets. In recent years, international scholars have done a lot of research on electromagnetically driven multi-degree-of-freedom motors. For example, Tianjin University applied Halnach array structure to spherical multi-degree-of-freedom motors to improve the complexity of motor magnetic field distribution and motor performance.¹⁴ Hefei University of Technology proposed a new spherical multi-degree-of-freedom stepper motor¹⁵ and analyzed the magnetic field, torque, and mechanical model of the motor in detail. On the basis of summarizing the advantages and disadvantages of the predecessors, Hebei University of Science and Technology proposed a new 3-degree-of-freedom motor with double-layer coil rotor single-layer permanent magnet structure and optimized the motor structure by particle swarm optimization.¹⁶ Hanyang University proposed a new 3-degree-of-freedom motor with a double-excitation source of sandwich structure.¹⁷ This motor greatly reduces the cost of the motor because the rotor no longer uses permanent magnets. Nanyang Technological University proposed a spherical motor for conformal printing,¹⁸ which greatly improves the reliability of the printing system. Although the multi-degree-of-freedom motor has high mechanical integration and simple transmission, the structure of electromagnetic-drive multi-degree-of-freedom motor is relatively complicated, and the magnetic field is difficult to control. The improvement of control and positioning accuracy is always a difficult point. The torque output produced by the rotor is very difficult and the loss is large during torque output. At present, the structure of electromagnetic-drive multi-degree-of-freedom motor is more complicated, and the reliability still falls short of the required state.¹⁹

The development of electromagnetic piezoelectric hybrid-drive motors has gradually become a development trend.^20–23 Piezoelectric-drive multi-degree-of-freedom motor is immune to external electromagnetic interference, which provides the possibility of electromagnetic piezoelectric hybrid-drive multi-degree-of-freedom motor development. The electromagnetic-drive multi-degree-of-freedom motor^24,25 has relatively complicated structure, the magnetic field is difficult to control, the control and positioning accuracy is low, the torque output by the rotor is difficult, and the torque loss is large. Piezoelectric-drive multi-degree-of-freedom motors have problems such as low lifetime and performance affected by fixed-rotor friction materials. The multi-degree-of-freedom motor with electromagnetic drive is suitable for working at high speed and low torque, but its structure is complicated and it is difficult to achieve miniaturization, and the control-positioning accuracy is low, which is not suitable for precise work. Multi-degree-of-freedom motor with piezoelectric drive is compact in structure, flexible in design, easy to realize miniaturization, self-locking, fast response, and free from external electromagnetic interference. It is especially suitable for low-speed and high-torque applications; but the life of piezoelectric-driven motor is short, and it is not suitable for continuous operation. Therefore, combining the electromagnetic drive and the piezoelectric drive can not only avoid the problems of the two drive modes but also improve the speed, accuracy, and torque of the motor.

In order to further study the electromagnetic piezoelectric hybrid-drive multi-degree-of-freedom motor, a 3-degree-of-freedom electromagnetic piezoelectric hybrid-drive motor was modeled and analyzed. First, the piezoelectric-driven stator–rotor contact model and the electromagnetically driven air-gap magnetic field model are established by combining the piezoelectric drive and electromagnetic-drive principle. Then the torque model of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor is established by vector synthesis method. The simulation software analyzes the influence of piezoelectric-drive stator position, stator–rotor contact material, and preload on motor torque and verifies the rationality of the analysis results by finite element analysis.

Electromagnetic piezoelectric hybrid-driven 3-degree-of-freedom motor structure and operation mechanism

The electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor is mainly composed of four stators and one rotor. The structure diagram is shown in Figure 1. As shown in Figure 2, the stator is composed of three piezoelectric stators (No. 1 stator, No. 2 stator, and No. 3 stator) and one electromagnetically driven stator (No. 4 stator) in Figure 3. The piezoelectric ceramic piece on the piezoelectric stator is applied with an alternating voltage. This voltage phase difference is 90°, the frequency is the same, and the amplitude is the same. Due to the inverse effect principle of piezoelectric ceramics, two-phase standing waves are reflected. The two-phase standing wave amplitudes are equal and the phase difference in time and space is 90°. The two-phase standing waves are superimposed to form a circular traveling wave. At this time, the rotor of the motor rotates in the tangential direction of the particle surface of the stator surface, and the direction of rotation of the rotor is opposite to the direction of the traveling wave.

Figure 1.

Electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor overall structure diagram.

Figure 2.

Electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor rotor and piezoelectric stator.

Figure 3.

Electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor rotor and electromagnetic stator.

As shown in Figure 3, the spherical rotor surface is mounted with four permanent magnets. The electromagnetic-drive torque is generated by the interaction of the surface-mounted permanent magnets and the fourth stator. Through the coordinated control of the four stators and the two drive modes being sequential in this process, the motor can finally achieve 3-degree-of-freedom motion.

Piezoelectric stator-drive model

Stator traveling wave generation

The piezoelectric ceramic piece on the piezoelectric stator is applied with an alternating voltage. This voltage phase difference is 90°, the frequency is the same, and the amplitude is the same. Due to the inverse effect principle of piezoelectric ceramics, two-phase standing waves are reflected. The two-phase standing wave amplitudes are equal and the phase difference in time and space is 90°. The two-phase standing waves are superimposed to form a circular traveling wave. The inverse piezoelectric effect can be expressed by the piezoelectric equation²⁶

X = c^{E} x - e \cdot E

(1)

D = e^{T} . x + ε^{S} E

(2)

where e is the piezoelectric constant matrix (e^T is the transposition of the piezoelectric constant matrix), X is the stress vector, x is the strain vector, c^E is the piezoelectric ceramic short-circuit stiffness matrix, ε^s is the clamping dielectric constant, D is the electricity displacement vector

x = {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} \end{matrix}]}^{T}

(3)

X = {[\begin{matrix} X_{1} & X_{2} & X_{3} & X_{4} & X_{5} & X_{6} \end{matrix}]}^{T}

(4)

D = {[\begin{matrix} D_{1} & D_{2} & D_{3} \end{matrix}]}^{T}

(5)

E = {[\begin{matrix} E_{1} & E_{2} & E_{3} \end{matrix}]}^{T}

(6)

u is a position function of the point on the ceramic piece, and (x, y, z) is the position vector. There is no displacement of the ceramic piece in the Y-axis direction, only the force in the X₁ direction is considered, and the electric field obtained by the ceramic piece is a uniform electric field. The ceramic sheets are polarized in the thickness direction, and the length of the ceramic sheets is greater than the radial width.

From the above assumptions

\begin{array}{l} D_{1} = D_{2} = 0, E_{1} = E_{2} = 0, x_{4} = x_{5} = x_{6} = 0, \\ E_{3} = E_{0} s i n ω t \end{array}

The above assumptions are brought into the piezoelectric equation to obtain

X_{1} = c_{11} x_{1} - e_{13} E_{3}

(7)

According to Newton’s second law, the equation of motion of ceramic sheets can be known

ρ \frac{\partial^{2} u}{\partial t^{2}} = c_{11} \frac{\partial X_{1}}{\partial x^{2}}

(8)

Equation (7) is brought into equation (8) to obtain

ρ \frac{\partial^{2} u}{\partial t^{2}} = c_{11} \frac{\partial u}{\partial x^{2}} - e_{13} \frac{\partial E_{3}}{\partial x}

(9)

The electric field component E₃ is a uniform electric field

\frac{\partial E_{3}}{\partial x_{3}} = 0

(10)

Therefore, the piezoelectric wave equation is

\frac{\partial^{2} u}{\partial t^{2}} = \frac{c_{11}}{ρ} \cdot \frac{\partial u}{\partial x^{2}}

(11)

where ρ is the density of the piezoelectric ceramic.

The special solution for obtaining the traveling wave equation is

u_{a} = Asin (kx) \sin (ω t)

(12)

u_{b} = Acos (kx) \cos (ω t)

(13)

where, $k = (2 π / λ)$ , $λ$ is the wavelength, and the displacement at the 0 point of the ceramic piece is 0.

Equations (12) and (13) are brought into equation (7): $A = | e_{13} E_{0} / c_{11} k |$

The traveling wave is obtained by superposition of the two-phase standing waves of a and b

u = Acos (kx - ω t)

(14)

Contact model of stator and rotor

It is assumed that the contact layer of the stator is an elastomer. The stator is displaced to the maximum displacement in the Z-axis direction when the stator is in contact. The stator is a rigid body and does not displace, and the friction between the stator and rotor is ignored. The stator–rotor contact is equivalent to the contact of the parabolic indenter with the contact layer (elastomer) as shown in Figure 4.

Figure 4.

Stator and rotor contact diagram.

The preload applied by each stator is F, then the one-dimensional pre-stress experienced by each peak is $P = (F / nb)$ , the number of waves is n, and b is the equivalent width of the fixed-rotor contact.

The waveform of the stator waveform at a certain time is $u_{1} = Acos (kx - wt)$ , and the radius of curvature of the waveform

R = | \frac{{[1 + {(\frac{d u_{1}}{dx})}^{2}]}^{\frac{3}{2}}}{\frac{d^{2} u_{1}}{d x^{2}}} |

(15)

The stator–rotor contact zone displacement function can be set to

w = - \frac{x^{2}}{2 R} + d_{0}, z = 0, | x | \leq a

(16)

where d₀ is an unknown constant.

When the stator friction is neglected, the stator and rotor boundary shear force is 0: $x = 0$ , $τ_{xy} = 0$ .

The normal stress distribution of the fixed rotor is: $p (x)$ , $z = 0,$ $| x | \leq a$ . Where E is the elastic modulus of the contact layer.

According to the balance of the stator and rotor

\int_{- a}^{a} p (x) dx = P

(17)

a is half of the fixed-rotor contact distance for one wavelength of the stator.

When the stator’s normal contact force $p (ξ)$ acts, the vertical displacement expression²⁷

w = \frac{- 1}{π E} \int_{- a}^{a} \ln {(x - ξ)}^{2} p (ξ) d ξ + B ″

(18)

Suppose the rigid body does not shift, so B" = 0.

So we get

\frac{- 1}{π E} \int_{- a}^{a} \ln {(x - ξ)}^{2} p (ξ) d ξ = - \frac{x^{2}}{2 R} + d_{0}

(19)

The method of solving the first type of Freehold equation by the Chebyshev polynomial of Popov^28,29

- \frac{1}{π} \int_{- 1}^{1} {(1 - τ^{2})}^{- \frac{1}{2}} \ln | τ - x | T_{n} (τ) d τ = \int_{- 1}^{1} U_{n - 1} (x) dx

(20)

{\begin{matrix} T_{0} (x) = 1 \\ T_{1} (x) = x \\ T_{2} (x) = 2 x^{2} - 1 \\ \begin{matrix} ⋮ \\ T_{n + 1} (x) = 2 T_{n} (x) - T_{n - 1} (x) \end{matrix} \end{matrix}

(21)

{\begin{matrix} T_{n} (x) - {T_{n}}_{+ 2} (x) = 2 (1 - x^{2}) U_{n} (x) \\ {T_{n}}^{'} (x) = n U_{n - 1} (x) \end{matrix}

(22)

- \frac{1}{π} \int_{- 1}^{1} {(1 - τ^{2})}^{- \frac{1}{2}} \ln | τ - x | T_{n} (τ) d τ = n^{- 1} T_{n} (x)

(23)

Form obtained $p (ξ)$ according to equation (23)

\begin{array}{l} p (ξ) = \frac{E}{2} a_{0} a {(a^{2} - ξ^{2})}^{- \frac{1}{2}} T_{0} (\frac{ξ}{a}) + \frac{E}{2} a_{1} a {(a^{2} - ξ^{2})}^{- \frac{1}{2}} \\ T_{1} (\frac{ξ}{a}) + \frac{E}{2} a_{2} a {(a^{2} - ξ^{2})}^{- \frac{1}{2}} T_{2} (\frac{ξ}{a}) \end{array}

(24)

The contact pressure of the contact point of the contact zone is 0, so, $p (\pm a) = 0$ , $a_{0} = - a_{2}$ , $a_{1} = 0$

Equation (24) is taken into equation (17) to

a_{0} = \frac{2 P}{π a E}

(25)

Equation (24) is taken into equation (19) to

[a_{0} a \ln 2 + a_{2} a \frac{1}{2} (\frac{2 x^{2}}{a^{2}} - 1)] = - \frac{x^{2}}{2 R} + d_{0}

(26)

Therefore, $a_{2} = - (1 / 2 R)$ , $d_{0} = (1 / 2) a (2 a_{0} \ln 2 - a_{2})$ , it is further known that $P = (π E a^{2}) / 4 R$ ,

The stator-torque contact surface force distribution equation

p (x) = {(\frac{P E}{π R a^{2}})}^{\frac{1}{2}} {(a^{2} - x^{2})}^{\frac{1}{2}}, z = 0, | x | \leq a

(27)

The displacement equation of the stator–rotor contact surface

u = - \frac{x^{2}}{2 R} + \frac{P}{π E} (2 \ln 2 + 1), z = 0, | x | \leq a

(28)

Friction model of stator and rotor

According to the Coulomb friction model^30,31

F < μ_{s} p (x), v_{rel} = 0

(29)

F = μ_{d} p (x) sign (v_{rel}), v_{rel} \neq 0

(30)

where μ_d is the dynamic friction factor; μ_s is the static friction factor; v_rel is the relative sliding speed of the stator; p(x) is the normal pressure, and F is the tangential thrust $sign (v_{rel}) = {\begin{matrix} 1 & v_{rel} > 0 \\ - 1 & v_{rel} < 0 \end{matrix}$ .

In the contact region within the normal contact pressure, on the cut synthetic rotor thrust F

F = \int_{- a}^{a} sign (v_{rel}) μ_{d} p (x) dx = 2 μ_{d} [\int_{0}^{c} p (x) dx - \int_{c}^{a} p (x) d x]

(31)

where c is the point at which the relative sliding speed of the rotor is zero. When c = a, the maximum tangential thrust is obtained, and the single stator obtains the maximum output torque

T_{j} = n R_{1} F

(32)

where n is the number of traveling wave peaks; R₁ is the radius of the stator rotor contact surface; j = 1, 2, 3.

Electromagnetic stator-drive model

Electromagnetic-drive stator–rotor air-gap magnetic field

According to the boundary condition of the magnetic medium, the magnetic field generated by the rotor permanent magnet can be divided into three parts

B = {\begin{matrix} B_{1} = μ_{0} H_{1} & 0 \leq r \leq R_{i} \\ B_{2} = μ_{0} μ_{m} H_{2} + μ_{0} M_{r} & R_{i} \leq r < R_{o} \\ B_{3} = μ_{0} μ_{r} H_{3} & r \geq R_{o} \end{matrix}

(33)

where $μ_{0}$ is the magnetic permeability value in the air $(μ_{0} = 1)$ , $μ_{m}$ is the relative magnetic permeability $(μ_{m} = 1.02)$ , the permanent magnet of the motor is NdFeB, and M_r, $(M_{r} = B_{r} / μ_{0})$ ,is the residual magnetization, and $B_{r}$ is the residual magnetic induction (B_r = 1 T), which is the relative magnetic permeability of the rotor material. $μ_{r}$ is the relative magnetic permeability $(μ_{r} = 1)$ .

According to the principle of the motor, there is no current excitation in the rotor area, which can be obtained

\nabla \times H_{k} = 0, k = 1, 2, 3

(34)

The negative value of the gradient of the scalar magnetic position can represent the magnetic field strength

H_{k} = - \nabla V_{k}, k = 1, 2, 3

(35)

According to the structure of the motor, the permanent magnets are symmetrically distributed, so

\nabla \cdot M_{r} = 0

(36)

Then the scalar magnetic positions of the three regions satisfy the Laplace equation

{\begin{matrix} \nabla^{2} V_{1} = 0 \\ \nabla^{2} V_{2} = \frac{1}{μ_{0}} \nabla \cdot M_{r} = 0 \\ \nabla^{2} V_{3} = 0 \end{matrix}

(37)

According to the structure of the permanent magnet, the residual magnetization of the motor in the spherical coordinate system can be expressed as

M_{r} = {\begin{matrix} M_{rr} \\ M_{r θ} \\ M_{r φ} \end{matrix}} = (- 1)^{p - 1} | M_{r} | {\begin{matrix} \cos (ϕ - α_{p}) \sin θ \\ \cos (ϕ - α_{p}) \cos θ \\ - \sin (ϕ - α_{p}) \end{matrix}}

(38)

where p (p = 1, 2, 3, 4) is the permanent magnet number, L is the total number of permanent magnets. $α_{p} = α / 2 + 2 π (p - 1) / L$ , $π / 2 - β / 2 \leq θ \leq π / 2 + β / 2$ , $π (p - 1) / L \leq ϕ \leq α + π (p - 1) / L$ , α is a permanent magnet with a longitude angle of 90°, and β is a permanent magnet with a latitude angle of 50°.

In the spherical coordinate system, the magnetic position V can be expressed as a function of the spatial point (r, θ, φ), that is, V(r, θ, φ), where r is the coordinate radial distance of the point, θ is the latitude, and φ is longitude.

The Laplace equation of the magnetic position V can be expressed as the Laplace equation of the scalar magnetic position

\frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial V}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial V}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2} V}{\partial ϕ^{2}} = 0

(39)

Assume $V (r, θ, ϕ) = R (r) Y (θ, ϕ)$ , brought it into equation (39)

\begin{array}{l} \frac{Y (θ, ϕ)}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial R (r)}{\partial r}) + \frac{R (r)}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial Y (θ, ϕ)}{\partial θ}) \\ + \frac{R (r)}{r^{2} \sin^{2} θ} \frac{\partial^{2} Y (θ, ϕ)}{\partial ϕ^{2}} = 0 \end{array}

(40)

Equation (40) is divided by R(r) Y(θ, φ)

\begin{array}{l} \frac{1}{r^{2} R (r)} \frac{\partial}{\partial r} (r^{2} \frac{\partial R (r)}{\partial r}) + \frac{1}{Y (θ, ϕ) r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial Y (θ, ϕ)}{\partial θ}) \\ + \frac{1}{Y (θ, ϕ) r^{2} \sin^{2} θ} \frac{\partial^{2} Y (θ, ϕ)}{\partial ϕ^{2}} = 0 \end{array}

(41)

Equation (41) is multiplied by r² on both sides, and the variables are adjusted

\begin{array}{l} \frac{1}{R (r)} \frac{\partial}{\partial r} (r^{2} \frac{\partial R (r)}{\partial r}) = - \frac{1}{Y (θ, ϕ) \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial Y (θ, ϕ)}{\partial θ}) \\ - \frac{1}{Y (θ, ϕ) \sin^{2} θ} \frac{\partial^{2} Y (θ, ϕ)}{\partial ϕ^{2}} = μ \end{array}

(42)

r^{2} \frac{d R^{2}}{d r^{2}} + 2 r \frac{dR}{dr} - μ R = 0

(43)

\frac{1}{\sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial Y}{\partial θ}) + \frac{1}{\sin^{2} θ} \frac{\partial^{2} Y}{\partial ϕ^{2}} + μ Y = 0

(44)

where μ is the parameter introduced by the separation variable method.

The spherical harmonic equation further separates the variables, assume $Y (θ, ϕ) = Θ (θ) Φ (ϕ)$ , substituting equation (44)

\frac{1}{\sin θ} \frac{d}{d θ} (\sin θ \frac{d Θ}{d θ}) + (u - \frac{λ}{\sin^{2} θ}) Θ = 0

(45)

\frac{d^{2} Φ}{d ϕ^{2}} + λ Φ = 0

(46)

where λ is the parameter introduced by the separation variable method.

Obtained by simultaneous equations (43), (45), and (46)

\begin{matrix} V (r, θ, ϕ) = R (r) Y (θ, ϕ) \\ = \sum_{n = 0}^{+ \infty} \sum_{m = 0}^{n} (C_{n} r^{n} + \frac{D_{n}}{r^{n + 1}}) P_{n}^{m} (\cos θ) \\ (A_{m} \cos m ϕ + B_{m} \sin m ϕ) \\ = \sum_{n = 0}^{+ \infty} \sum_{m = - n}^{n} (C_{n} r^{n} + \frac{D_{n}}{r^{n + 1}}) Y_{n}^{m} (θ, ϕ) \end{matrix}

(47)

where $Y_{n}^{m} (θ, ϕ) = S_{n}^{m} P_{n}^{m} (\cos θ) e^{im ϕ}$ $m = 0, 1, \dots, n$ ; $S_{n}^{m} = \sqrt{\frac{2 n + 1}{4 π} \frac{(n - m)!}{(n - m)!}}$ ; $(x = \cos θ)$ , $Y_{n}^{m} (θ, ϕ)$ is spherical harmonic function. $P_{n}^{m} (\cos θ)$ is a Legendre function.

The permanent magnet rotor boundary conditions are:

The magnetic mark is 0 at infinity

V_{1} |_{r \to \infty} = 0

(48)

so, $C_{n, 1} = 0$ .

The magnetic mark is finite at the origin of the magnetic field

V_{3} |_{r \to 0} \neq \infty

(49)

so, $D_{n, 3} = 0$ .

The tangential and radial components of the magnetic induction at the interface of the medium are continuous

B_{1 r} = B_{2 r}, B_{2 r} = B_{3 r}

(50)

H_{1 θ} = H_{2 θ}, H_{2 θ} = H_{3 θ}

(51)

H_{1 ϕ} = H_{2 ϕ}, H_{2 ϕ} = H_{3 ϕ}

(52)

The expression of magnetic induction in spherical coordinates is

H = [H_{r} e_{r}, H_{θ} e_{θ}, H_{ϕ} e_{ϕ}]^{T} = {[- \frac{\partial V}{\partial r}, - \frac{1}{r} \frac{\partial V}{\partial θ}, - \frac{1}{r \sin θ} \frac{\partial V}{\partial ϕ}]}^{T}

(53)

The radial component of the residual magnetization is expressed in the form of a spherical harmonic function. Any single-valued and finite function $f (θ, φ)$ in the ( $0 \leq θ \leq π$ , $0 \leq φ \leq 2 π$ ,) interval can be expanded into the generalized Fourier series form of $Y_{n}^{m} (θ, ϕ)$ form

f (θ, ϕ) = \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} C_{n}^{m} Y_{n}^{m} (θ, ϕ)

(54)

where, $C_{n}^{m} = \int_{0}^{π} \int_{0}^{2 π} f (θ, ϕ) Y_{n}^{m *} (θ, ϕ) \sin θ d ϕ d θ$ , $Y_{n}^{m *} (θ, ϕ)$ is the conjugate complex number of $Y_{n}^{m} (θ, ϕ)$

C_{n}^{m} = | M_{r} | c_{mn} (a_{m} \pm b_{m} i)

(55)

a_{m} \pm b_{m} i = \sum_{j = 1}^{2 p} \int_{2 π (p - 1) / L}^{α + 2 π (p - 1) / L} \cos (ϕ - α_{p}) e^{- im ϕ} d ϕ

(56)

c_{mn} = \int_{\frac{π}{2} - \frac{β}{2}}^{\frac{π}{2} + \frac{β}{2}} S_{n}^{m} P_{n}^{m} \cos θ \sin^{2} θ d θ

(57)

B_{1} = [\begin{matrix} B_{1 r} \\ B_{1 θ} \\ B_{1 ϕ} \end{matrix}] = [\begin{matrix} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} ((n + 1) μ_{r} D_{n 1}^{m} \frac{1}{r^{n + 2}}) Y_{n}^{m} (θ, ϕ) \\ \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (- μ_{r} D_{n 1}^{m} \frac{1}{r^{n + 2}}) \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial θ} \\ \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (- μ_{r} D_{n 1}^{m} \frac{1}{r^{n + 2}}) \frac{1}{\sin θ} \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial ϕ} \end{matrix}]

(58)

\begin{array}{l} B_{2} = [\begin{matrix} B_{2 r} \\ B_{2 θ} \\ B_{r ϕ} \end{matrix}] = [\begin{array}{l} - μ_{0} μ_{m} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (n C_{n, 2}^{m} r^{n - 1} - (n + 1) D_{n, 2}^{m} \frac{1}{r^{n + 2}}) Y_{n}^{m} (θ, ϕ) \\ - μ_{0} μ_{m} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (C_{n, 2}^{m} r^{n - 1} + D_{n, 2}^{m} \frac{1}{r^{n + 2}}) \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial θ} \\ - μ_{0} μ_{m} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (C_{n, 2}^{m} r^{n - 1} + D_{n, 2}^{m} \frac{1}{r^{n + 2}}) \frac{1}{\sin θ} \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial ϕ} \end{array}] \\ + [\begin{array}{l} μ_{0} | M_{r} | {(- 1)}^{p - 1} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} C_{n}^{m} Y_{n}^{m} (θ, ϕ) \\ μ_{0} | M_{r} | {(- 1)}^{p - 1} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} C_{n}^{m} \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial θ} \\ μ_{0} | M_{r} | {(- 1)}^{p - 1} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} C_{n}^{m} \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial ϕ} \end{array}] \end{array}

(59)

\begin{array}{l} B_{3} = [\begin{matrix} B_{3 r} \\ B_{3 θ} \\ B_{3 ϕ} \end{matrix}] \\ = [\begin{array}{l} \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (- n μ_{0} μ_{m} C_{n, 3}^{m} r^{n - 1}) Y_{n}^{m} (θ, ϕ) \\ \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (- μ_{0} μ_{m} C_{n, 3}^{m} r^{n - 1}) \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial θ} \\ \sum_{n = 0}^{\infty} \sum_{m = - n}^{n} (- μ_{0} μ_{m} C_{n, 3}^{m} r^{n - 1}) \frac{1}{\sin θ} \frac{\partial Y_{n}^{m} (θ, ϕ)}{\partial ϕ} \end{array}] \end{array}

(60)

The special solution of the specific magnetic mark can be obtained by the above boundary conditions. The air-gap magnetic-tight fundamental wave of the motor can be obtained by calculation

B_{11} = [\begin{matrix} B_{11 r} \\ B_{11 θ} \\ B_{11 ϕ} \end{matrix}] = μ_{r} r^{- 4} D_{n 1}^{m} S_{n} [\begin{matrix} 3 {(\sin θ)}^{2} \sin 2 ϕ \\ - 2 \sin θ \cos θ \sin 2 ϕ \\ - 2 \sin θ \cos 2 ϕ \end{matrix}]

(61)

Electromagnetic coil–drive torque model

The magnetic field generated by the energized coil (radial length q (mm)) interacts with the magnetic field generated by the permanent magnet, which produces the Lorentz force of the interaction (the force is macroscopically expressed as Ampere); the electromagnetic force generated by a certain coil dl micro-element that can be obtained by the law^32,33

d F_{i} = - J_{i} rdrd δ dl \times B_{11 r} (r, θ, ϕ)

(62)

The electromagnetic torque generated by the dl micro-element

d T_{i} = [- J_{i} rdrd δ dl \times B_{11 r} (r, θ, ϕ)] \times r e_{r}

(63)

The torque generated by the single coil is obtained by integrating the torque micro-element on a single coil

T_{i} = - J_{i} \int_{r_{o}}^{r_{1}} \int_{δ_{0}}^{δ_{1}} {\int_{C} r e_{r} \times [B_{11 r} (r, θ, ϕ) \times dl]} rdrd δ

(64)

T_{i} = J_{i} \int_{r_{o}}^{r_{1}} \int_{δ_{0}}^{δ_{1}} \int_{C} r B_{11 r} (r, θ, ϕ) dl rdrd δ

(65)

where J_i is the surface current density of the coil, and r_o r₁ is the coil radial length, $δ_{0}$ $δ 1$ is the coil thickness. To simplify the calculation, the current surface is equivalent to that shown in Figure 5.

Figure 5.

Stator and rotor section equivalent diagram.

Since the integral operation of the equation (65) cannot be directly performed in the spherical coordinate system, it is necessary to convert the components in the spherical coordinate system to the Cartesian coordinate system

\cos ϕ = \frac{x}{r \sin θ}, \sin ϕ = \frac{y}{r \sin θ}

(66)

After converting each component of the spherical coordinate system into a Cartesian coordinate system, the expression of each unit of the coil

\begin{array}{l} x = r \cos δ \sin θ_{i} \cos ϕ_{i} - r \sin δ \cos φ \cos θ_{i} \cos ϕ_{i} \\ + r \sin δ \sin φ \sin ϕ_{i} \end{array}

(67)

\begin{array}{l} y = r \cos δ \sin θ_{i} \cos ϕ_{i} - r \sin δ \cos φ \cos θ_{i} \cos ϕ_{i} \\ - r \sin δ \sin φ \sin ϕ_{i} \end{array}

(68)

z = r \cos δ \cos θ_{i} + r \sin δ \cos φ \sin θ_{i}

(69)

where r is the distance from the point on the coil boundary to the center of the sphere. θ_i is the projection of the coil boundary point below the spherical coordinate θ axis. φ_i is the projection of the coil boundary point on the spherical coordinate φ axis. δ is the angle between the line from the point on the coil to the center of the ball and the line from the center of the coil to the center of the ball. $φ$ is the angle between the radius of the coil boundary point in the local coordinate system and the negative half-axis of e _θi. Electromagnetic stator position and projection on the rotor are as shown in Figure 6.

Figure 6.

Electromagnetic stator projected on the rotor.

In the local rectangular coordinate system, the resulting derivative is converted to dl element: $d l = r \sin δ d ϕ (\cos φ e_{ϕ_{i}} - \sin φ e_{θ_{i}})$

The electromagnetic torque can be obtained by combining the above formulas

\begin{matrix} T_{i} = J_{i} \int_{r_{0}}^{r_{1}} \int_{δ_{0}}^{δ_{1}} \int_{0}^{2 π} B_{11 r} (r, θ_{i}, ϕ_{i}, φ, δ) \sin δ \\ (\cos φ e_{φ_{i}} - \sin φ e_{θ_{i}}) d φ drd δ \end{matrix}

(70)

The torque decomposition can be used to decompose the electromagnetic torque to obtain the single-coil rotation torque.

Rotational torque

T_{i} = J_{i} \int_{r_{1}}^{r_{2}} \int_{δ_{0}}^{δ_{1}} \int_{0}^{2 π} B_{11 r} (r, θ_{i}, ϕ_{i}, φ, δ) \sin δ \sin φ d φ drd δ

(71)

where i is the coil number (i = 1, 2, 3,…,18)

As shown in Figure 7, the total torque produced by all coils is calculated from the distribution of the coils

T_{4} = \sum_{i = 1}^{18} T_{i}

(72)

Figure 7.

Electromagnetic stator equatorial plane distribution map.

Electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor torque synthesis

The line passing through the No. 2 stator is defined as the Y-axis, the axis perpendicular to the Y-axis is the X-axis, and the line perpendicular to the X-Y plane and passing through the origin is defined as the Z axis. The three traveling wave stators are mounted at 120° and are α1° to the X-Y plane as shown in Figures 8 –10. The electromagnetically driven stator is mounted outside the spherical rotor to provide torque to the rotor about the Z-axis. Therefore, the hybrid-drive motor realizes 3-degree-of-freedom motion by multi-degree-of-freedom motion through four stator synthesis. Tx, Ty, and Tz are vectors of the rotor rotating around the X, Y, and Z axes, respectively. The torque of the 3-degree-of-freedom motor is T, and the torque vectors T₁, T₂, T₃, and T₄ of the four stators are combined to obtain^34,35

T = [\begin{matrix} T_{x} \\ T_{y} \\ T_{z} \end{matrix}] = [\begin{matrix} \frac{\sqrt{3}}{2} T_{1} \cos α_{1} - \frac{\sqrt{3}}{2} T_{2} \cos α_{1} \\ - \frac{1}{2} T_{1} \cos α_{1} + T_{2} \cos α_{1} - \frac{1}{2} T_{3} \cos α_{1} \\ T_{1} \sin α_{1} + T_{2} \sin α_{1} + T_{3} \sin α_{1} + T_{4} \end{matrix}]

(73)

Figure 8.

Piezoelectric stator–rotor structure top view.

Figure 9.

Front view of the piezoelectric stator–rotor structure.

Figure 10.

Top view of the electromagnetic stator–rotor structure.

Numerical simulation and analysis

It can be seen from the formula (71) that the yaw torque T_X is obtained by the torque synthesis generated by the No.1 and No.2 stators. The deflection torque Ty is obtained by the torque synthesis generated by the No.1, No.2, and No.3 stators. The rotation torque Tz is obtained by torque synthesis of the No.1, No.2, No.3, and No.4 stators. The magnitude of the torque of the 3-degree-of-freedom motor driven by the electromagnetic piezoelectric hybrid is affected by the mounting position angle a1 of the No.2 stator of the No.1 stator of the No.1 stator. According to the stator material parameters of Tables 1 and 2, the torque characteristics were analyzed by MATLAB.

Table 1.

Piezoelectric stator structure parameters.

Parameter	value (unit)
R ₁	30 mm
b	2 mm
h	0.5 mm
λ	12π
ω	37286 Hz
V	110 V
J_i	0.5mA/mm²
$δ_{0}$	2°
$δ_{1}$	7°
q	30 mm

Table 2.

Friction material parameters of piezoelectric stator and rotor contact layer.

Contact layermaterial	Dynamic frictionfactor (μd)	Elasticmodulus
Steel	0.3	210 GPa
Nylon	0.25	2.6 GPa
PTFE	0.04	0.4 GPa

PTFE: polytetrafluoroethylene.

Piezoelectric ceramic piezoelectric constant matrix: $[e] = [\begin{matrix} 0 & 0 & - 4.3 \\ 0 & 0 & - 4.3 \\ 0 & 0 & 14.1 \\ 0 & 0 & 0 \\ 0 & 10.5 & 0 \\ 10.5 & 0 & 0 \end{matrix}] C / m^{2}$ .

Clamping dielectric constant matrix: $[ε^{s}] = [\begin{matrix} 7.124 & 0 & 0 \\ 0 & 7.124 & 0 \\ 0 & 0 & 5.841 \end{matrix}] \times 10^{- 9} F / m$ .

Piezoelectric ceramic short-circuit stiffness matrix: $[c^{E}] = [\begin{matrix} 13.2 & 7.1 & 7.3 & 0 & 0 & 0 \\ 7.1 & 13.2 & 7.3 & 0 & 0 & 0 \\ 7.3 & 7.3 & 11.5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2.6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2.6 \end{matrix}] \times 10^{10} Pa$ .

Influence of materials on torque characteristics of 3-degree-of-freedom motor driven by electromagnetic piezoelectric hybrid drive

Influence of material on deflection torque characteristics

In order to further analyze the torque of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor, the maximum output deflection torque characteristics of different preload, different contact materials and piezoelectric stator mounting position angle rotors were calculated using MATLAB, as shown in Figures 11 –16. When the piezoelectric stator mounting position angle is constant, the deflection torque T_X increases as the preload F increases. At this time, the maximum deflection torque can reach 4.871 Nm; when the preload is constant, the deflection torque Tx of the motor decreases as the installation angle of the piezoelectric stator increases; when the installation angle of the rotor is constant, the deflection torque Ty is pre-adjusted. The pressure F increases as the pressure increases. At this time, the maximum yaw torque can reach 5.625 Nm; when the preload is constant, the yaw torque Ty decreases with the increase of the piezoelectric stator mounting position angle; when the material is the same, the yaw torque Tx is greater than the yaw torque Ty.

Figure 11.

Maximum output deflection torque Tx when the rotor contact is nylon.

Figure 12.

Maximum output deflection torque Tx when the rotor contact is PTFE.

Figure 13.

Maximum output deflection torque Tx when the rotor contact is steel.

Figure 14.

Maximum output deflection torque Ty when the rotor contact is nylon.

Figure 15.

Maximum output deflection torque Ty when the rotor contact is PTFE.

Figure 16.

Maximum output deflection torque Ty when the rotor contact is steel.

Influence of material on rotation torque

In order to further analyze the torque of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor, the maximum output rotation torque characteristics of the rotor with different preload, different contact materials and stator installation position angle are calculated using MATLAB, as shown in Figures 17 –19. When the rotor-mounting position angle is constant, the rotation torque Tz increases as the preload F increases. When the preload is constant, the rotation torque Tz increases as the angular position of the piezoelectric stator is increased. The maximum rotation torque reaches 15.1 Nm.

Figure 17.

Maximum output rotation torque Tz when the rotor contact is PTFE.

Figure 18.

Maximum output rotation torque Tz when the rotor contact is nylon.

Figure 19.

Maximum output rotation torque Tz when the rotor contact is steel.

Influence of driving mode on rotation torque

When the stator contact material is nylon, the rotation torque of the motor when piezoelectric drive is obtained (as shown in Figure 20) and the rotation torque of the motor when electromagnetic piezoelectric hybrid drive are performed (Figure 21). By comparison, it can be seen that the maximum rotational torque of the piezoelectric single drive can reach 7.03 Nm, and the maximum rotational torque of the electromagnetic piezoelectric hybrid-drive motor can reach 13.6 Nm. In addition, the piezoelectric-drive motor has a compact structure, flexible design, easy miniaturization, self-locking, and fast response. The motor with electromagnetic drive has a complicated structure, and it is difficult to achieve miniaturization, and it has low control-positioning accuracy, and it is difficult to be applied in a precision working place. It is found by calculation that the combination of electromagnetic-drive and piezoelectric drive increases the torque of the motor.

Figure 20.

Output rotation torque Tz when the piezoelectric stator is separately driven to contact nylon.

Figure 21.

Electromagnetic piezoelectric hybrid-drive contact is nylon when output rotation torque Tz.

The reliability of the motor is improved by a combination of two drive methods. In order to further verify the rationality of the analytical calculation, the influence of the piezoelectric stator position angle on the rotation torque was verified by the finite element software for the nylon material at the preload of 25 N, as shown in Figure 22.

Figure 22.

Comparison of analytic method and finite element method when the rotor contact is nylon.

Conclusion

First, the principle and basic structure of the hybrid-drive motor are introduced and the displacement and pressure distribution of the stator–rotor contact surface are obtained by analytical method. Based on this, the torque model of the piezoelectric stator-drive motor is obtained. Then, the air-gap magnetic field model of the permanent magnet rotor is obtained by analytical method, and the electromagnetic stator-torque model is obtained. Finally, the torque model of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor is established by vector synthesis.

The influence of different contact layer materials on the contact pressure distribution of the stator and rotor and the contact distance on the torque characteristics were analyzed by MATLAB. It can be seen from the analysis that the larger the position angle of the No.1, No.2, and No.3 stators, the larger the deflection torque. The larger the position angle of the No.1 and No.2 piezoelectric stators and No.3 stator, the smaller the self-rotation torque. In order to achieve a higher precision in the output reliability of the motor, the positional angle of the No.1, No.2, and No.3 stators can not be 0°. When the position angle of the No.1, No.2, and No.3 stators is 0, the motor rotates only for the electromagnetically driven rotor, which greatly reduces the precision of the rotor drive. By comparing and analyzing the piezoelectric driving motor and the electromagnetic piezoelectric hybrid driving motor, it is known that the combination of piezoelectric and electromagnetic driving not only improves the torque of the motor but also improves the reliability of the motor. It provides a theoretical basis for the design and torque optimization and improvement of the electromagnetic piezoelectric hybrid-drive 3-degree-of-freedom motor.

Footnotes

Handling Editor: Xiaodong Sun

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: This work was supported by the National Natural Science Foundation of China (grant no.: 51577048, 51877070, and 51637001), the Natural Science Foundation of Hebei Province of China (grant no.: E2018208155), the Overseas Students Science and Technology Activities Funding Project of Hebei Province (grant no.: C2015003044), the Hebei Industrial Technology Research Institute of Additive Manufacturing (Hebei University of Science and Technology) open projects funding, the National Engineering Laboratory of Energy-saving Motor & Control Technique, Anhui University (grant no.: KFKT201804), and Key project of science and technology research in Hebei provincial colleges and universities (grant no.: ZD2018228).

ORCID iD

Zheng Li

References

Sun

Shen

Wang

et al . Core losses analysis of a novel 16/10 segmented rotor switched reluctance BSG motor for HEVs using nonlinear lumped parameter equivalent circuit model. IEEE/ASME T Mech 2018; 23: 747–757.

Wan

Yang

Sun

et al . Decoupling control of a 5-degree-of-freedom bearingless induction motor based on least squares support vector machine inverse. Adv Mech Eng 2016; 8: 672120.

Toyama

Hatae

Multi-degree of freedom spherical ultrasonic motor. Berlin: Springer, 1993, pp.243–252.

Aoyagi

Beeby

White

NM.

A novel multi-degree-of-freedom thick-film ultrasonic motor. IEEE T Ultrason Ferr 2002; 49: 151–158.

Aoyagi

Nakajima

Tomikawa

et al . Examination of disk-type multidegree-of-freedom ultrasonic motor. Jpn J Appl Phys 2004; 43: 2884–2890.

Guo

Shen

The driving mathematic model of spherical traveling-wave type ultrasonic motor. Proc CSEE 2006; 26: 115–119.

Research on the structure, model and control of multi-degree of freedom spherical travel ultrasonic motor. PhD Thesis, Zhejiang University, Hangzhou, China, 2010.

Liu

Zhang

An examination on analytical model for contacts of cross-type multi-DOF ultrasonic motor. Machinery 2015; 53: 6–11.

Huang

Zhao

On design of a cylinder-sphere ultrasonic motor with multi-degree of freedom. Mech Sci Technol Aerospace Eng 2004; 23: 157–160.

10.

Yang

Liu

Chen

et al . Sandwich-type multi-degree-of-freedom ultrasonic motor with hybrid excitation. IEEE Access 2017; 4: 905–913.

11.

Jin

et al . Research on a novel 3-DOF ultrasonic motor with two cylinder stators. Micromotors 2009; 42: 27–29.

12.

Chen

Research on the in-plane bending-mode travelling wave ultrasonic motors. PhD Thesis, Tsinghua University, Beijing, China, 2016.

13.

Zhou

Designing and modeling of the bi-vibrator standing wave linear ultrasonic motor. PhD Thesis, Nanjing University of Aeronautics and Astronautics, Nanjing, China, 2014.

14.

3-D magnetic field analysis of a Halbach array PM spherical motor. PhD Thesis, Tianjin University, Tianjin, China, 2008.

15.

Basic research on permanent magnet spherical stepping motor for humanoid robot joint. PhD Thesis, Hefei University of Technology, Hefei, China, 2007.

16.

Guo

Zhang

Magnetic field analysis of a novel 3-DOF deflection type PM motor. J Hebei Univ Sci Technol 2012; 33: 422–428.

17.

Kwon

Kim

YB.

Design and analysis of double excited 3-degree-of-freedom motor for robots. J Electr Eng Technol 2011; 6: 618–625.

18.

Bai

Lee

et al . Design and development of a spherical motor for conformal printing of curved electronics. IEEE T Ind Electron 2018; 65: 9190–9200.

19.

Wang

et al . The summary and latest research of PM spherical M-DOF motor. Micromotors 2011; 44: 66–70.

20.

Sang

Huang

Liu

et al . A vibration-based hybrid energy harvester for wireless sensor systems. IEEE T Magn 2012; 48: 4495–4498.

21.

Siddique

ARM

Mahmud

Heyst

BV.

A comprehensive review on vibration based micro power generators using electromagnetic and piezoelectric transducer mechanisms. Energ Convers Manag 2015; 106: 728–747.

22.

Zhang

Uchiyama

Ishizuka

High-speed, high-precision magnetic/piezoelectric hybrid drive actuator for micro electrical discharge machining. Adv Mater Res 2014; 941–944: 2112–2115.

23.

Khaligh

Zeng

Zheng

Kinetic energy harvesting using piezoelectric and electromagnetic technologies—state of the art. IEEE T Ind Electron 2010; 57: 850–860.

24.

Sun

Chen

Yang

Overview of bearingless permanent-magnet synchronous motors. IEEE T Ind Electron 2013; 60: 5528–5538.

25.

Sun

Chen

et al . Precise control of a four degree-of-freedom permanent magnet biased active magnetic bearing system in a magnetically suspended direct-driven spindle using neural network inverse scheme. Mech Syst Signal Pr 2017; 88: 36–48.

26.

Wang

Zhao

The piezoelectric ferroelectric physical. Alexandria, NSW, Australia: Science Press, 2009.

27.

Kong

Jiang

Jin

Solid contact mechanics. Uitikon: China Railway Publishing House, 1999.

28.

Popov

GI.

Some properties of classical polynomials and their application to contact problems. J Appl Math Mech 1963; 27: 1255–1271.

29.

Popov

GI.

On the method of orthogonal polynomials in contact problems of the theory of elasticity. J Appl Math Mech 1969; 33: 503–517.

30.

Goryacheva

IG.

Contact mechanics in tribology. Dordrecht: Springer, 1998, pp.135–142.

31.

Zhu

Contact analysis and mathematical modeling of traveling wave ultrasonic motors. IEEE T Ultrason Ferr 2004; 51: 668.

32.

Yan

Chen

Lim

et al . Torque modeling of spherical actuators with double-layer poles. In: Proceedings of the 2006 IEEE/RSJ international conference on intelligent robots and systems, Beijing, China, 9–15 October 2006, pp.5447–5452. New York: IEEE.

33.

Liang

Chen

Chee Kian

et al . Torque modeling of a spherical actuator based on Lorentz force law. In: Proceedings of the 2005 IEEE international conference on robotics and automation, Barcelona, 18–22 April 2005, pp.3646–3651. New York: IEEE.

34.

Takesue

Ohara

Ishibashi

et al . Position control methods of spherical ultrasonic motor. In: Proceedings of the IEEE/RSJ international conference on intelligent robots and systems, Taipei, Taiwan, 18–22 October 2010, pp.3061–3066. New York: IEEE.

35.

Mashimo

Awaga

Toyama

. Development of a spherical ultrasonic motor with an attitude sensing system using optical fibers. In: Proceedings of the IEEE international conference on robotics and automation, Roma, 10–14 April 2007, pp.4466–4471. New York: IEEE.