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Abstract

This article presents the dynamic model of three eccentric rotors driven by induction motors and respectively derives the response equations of the three directions with different frequency in a vibrating system. On account of the difficulty to realize multifrequency self-synchronization, the controlling method is used to realize the synchronization of the speed and phase. In this article, the fuzzy PID method is introduced to control the vibrating system which is based on a master–slave controlling strategy. And a tracking method based on the phase ratio is proposed. The effectiveness and stability are shown in the numerical simulation and the experiment validation is given to certify the accuracy of the theory. In the meanwhile, the arbitrary of the multifrequency is also demonstrated. Finally, some conclusions are summarized to present the significant in the engineering.

Keywords

Controlled synchronization multifrequency fuzzy PID phase ratio eccentric rotors

Introduction

Recently, with the development of the industrial technology, people are not satisfied with the traditional industrial equipment and the manufacture requirements. So, some more cheaper and intelligent industrial equipment are bought in. For example, the traditional structure of two shafts needs gears to connect them together to reach the same speed or special speed ratio wanted. This structure increases the cost and weight of the system.^1–3 To improve this structure, the concept of synchronization is proposed by Blekhman who illustrates the self-synchronization theory of two eccentric rotors (ERs).⁴ Moreover, he also shows the generalized stability theory of self-synchronization and controlled synchronization.^5,6 Wen et al.⁷ study the theory of self-synchronization in detail. They establish the dynamical model of two motors in a vibrating system and finally show the stability criteria and synchronous conditions. Meanwhile, they present the concept of multifrequency synchronization and realize 2 and 3 times frequency self-synchronization in a vibrating system which exists non-linear springs. Junkichi⁸ studies the multifrequency self-synchronization and also acquires the results. In their work, four motors are installed on a rigid body with four springs. The motors are divided into two groups. One group is on the top and the other is on the bottom. Each group is symmetric along the y axis. However, the two groups are not symmetric along the x axis. Due to this unsymmetrical installation, the vibrating system realizes multifrequency self-synchronization. Zhao et al.^9–11 establish the electromechanical coupling dynamics model of the induction motors. They use the small parameter method to obtain the eigenvalues of the dynamic equation. And the stability criterion is replaced by the eigenvalues when the vibrating system is at a stable state. Hou and Fang¹² finish the research about the synchronization and stability of two unbalanced rotors. Not only the theory of the self-synchronization get a development but also the experiment. Zhang et al.^13–15 do a deep research. Their work shows three kinds of situations which are respectively near resonance, over resonance and far resonance. Most of all, the experiment of self-synchronization is studied, which can effectively certify the accuracy of the theory. Moreover, the self-synchronization can also be realized with a disturbance which is named synchronous transmission. In this work, when the power of one motor is cut off, the two motors can still reach the synchronization state.

Although the self-synchronization can be realized, it always needs some strict conditions. With the development of the intelligent control theory, many kinds of control method are bought in the motor control.^16–18 Zhang et al. and Zhao et al.^19,20 finish the speed synchronization with the controlling method. Kong et al.^21–24 study the base frequency-controlled synchronization and finally realize the speed and phase synchronization. In their work, the adaptive sliding mode control (ASMC) algorithm is applied via a modified master–slave structure to realize the synchronization with a zero phase difference.

The base frequency synchronization can only realize the motion by straight line (the motors rotate with the opposite direction) or cycle (the motors rotate with the same direction). This phenomenon restricts the motion mode of the vibrating screen and has an influence of the production efficiency. However, the multifrequency synchronization has many different kinds of motion modes and can enlarge the amplitude which can improve the effectiveness of the vibrating screen in the engineering. Although the multifrequency self-synchronization can be realized, it can only realize one multifrequency motion in one working condition. When the speed ratio is changed, the vibrating system cannot reach the synchronization state. This kind of model is lack of arbitrariness with the speed ratio and phase ratio and is very inconvenient. So, in this article, the multifrequency-controlled synchronization is studied to solve the problem above. In section ‘Dynamic model of the vibration system’, the mechanical–electromagnetic coupling model of the vibration system is given and the responses of the three directions are shown. In section ‘Design of the controlling system’, a modified master–slave strategy and a method of phase ratio are used into the vibration system and the fuzzy PID method is bought in to control the speed and phase of the motors. In section ‘Results and discussion’, the numerical simulation and experimental verification are given to certify the effectiveness and stability. Finally, in section ‘Conclusions’, some conclusions are listed.

Dynamic model of the vibration system

In this section, the model of the vibrating system is given firstly. And then, the electromechanical coupling dynamics model of the induction motor is shown. Finally, the amplitude responses of three directions are derived.

The model of the mechanical system

In this chapter, the structure of the vibration system is shown in Figure 1. Usually, the theory of elastic deformation should be considered in the engineering. But in this article, the rigid body is connected with four steel plates and the thickness of the plates is about 1 cm. When the vibrating system is in a stable state, the working frequencies are much higher than the natural frequency of the rigid motions of structures and lower than those of elastic vibration of steel plates. Therefore, the elastic deformation and elastic vibration can be neglected. The structure can be simplified as a rigid body.

Figure 1.

Dynamic model of the vibratory system.

As demonstrated, the model of the mechanical system consists of one rigid body, four springs, a base support and three ERs driven by inductor motors. Three inductor motors which rotate in the counter-clockwise direction are installed on the rigid body. Motor 1 is the master motor and motors 2 and 3 are slave motors. Motor 2 is installed on the center line. Motors 1 and 3 are symmetrically installed based on the center line. The left is motor 3 and the right is motor 1. $o_{1}$ , $o_{2}$ and $o_{3}$ are respectively the rotating center of the three motors. r is the eccentric radius of the three motors. o is the center of the rigid body. l₁ represents the distance between the center of the body and the rotating center of motor 1. It also can be expressed as $o o_{1} = l_{1}$ . Simply, $o o_{2} = l_{2}$ and $o o_{3} = l_{3}$ , where $l_{1} = l_{3} > l_{2} ≫ r$ . $θ$ is the installation angle. Four springs provide the damping and stiffness of the vibrating system in the x, y and $ψ$ directions. Finally, the rigid body is installed on the base support. According to the document,²² the dynamic model of three motors rotating in the same direction can be expressed as

\begin{array}{l} M \ddot{x} + f_{x} \dot{x} + k_{x} x = \sum_{i =1}^{3} m_{i} r ({\dot{φ}}_{i}^{2} \cos φ_{i} + {\ddot{φ}}_{i} \sin φ_{i}) \\ M \ddot{y} + f_{y} \dot{y} + k_{y} y = \sum_{i =1}^{3} m_{i} r ({\dot{φ}}_{i}^{2} \cos φ_{i} - {\ddot{φ}}_{i} \sin φ_{i}) \\ J \ddot{ψ} + f_{ψ} \dot{ψ} + k_{ψ} ψ = m_{1} r l_{1} [{\dot{φ}}_{1}^{2} \sin (φ_{1} - θ) - {\ddot{φ}}_{1} \cos (φ_{1} - θ)] \\ - m_{2} r l_{2} [{\dot{φ}}_{2}^{2} \cos φ_{2} + {\ddot{φ}}_{2} \sin φ_{2}] \\ - m_{3} r l_{3} [{\dot{φ}}_{3}^{2} \sin (φ_{3} + θ) + {\ddot{φ}}_{3} \cos (φ_{3} + θ)] \\ J_{1} {\ddot{φ}}_{1} + f_{1} {\dot{φ}}_{1} = T_{e 1} - T_{L 1} \\ J_{2} {\ddot{φ}}_{2} + f_{2} {\dot{φ}}_{2} = T_{e 2} - T_{L 2} \\ J_{3} {\ddot{φ}}_{3} + f_{3} {\dot{φ}}_{3} = T_{e 3} - T_{L 3} \end{array}

(1)

where

\dot{}

and

\ddot{}

respectively represent

d () / d t

and

d^{2} () / d t^{2}

. M is the mass of the whole vibrating system. M = m+m₁+m₂+m₃. m is the mass of the rigid body. m_i=η_im₀ (i = 1,2,3) are the mass of the ERs. r is the eccentric radius. J is the moment of inertia of the vibration system,

J \approx J_{p} + \sum_{i = 1}^{3} m_{i} (l_{i}^{2} + r^{2})

. J_p is the moment of inertia of the rigid body. J_i is the moment of inertia of the inductor motor i concluding the eccentric mass, J_i=m_ir² (i = 1,2,3). f_x, f_x, f_ψ and k_x, k_y, k_ψ are respectively the damping coefficients and stiffnesses of the vibrating system in the x, y and ψ directions,

f_{ψ} = l_{0}^{2} (f_{x} \sin^{2} β + f_{y} \cos^{2} β)

k_{ψ} = l_{0}^{2} (k_{x} \sin^{2} β + k_{y} \cos^{2} β)

. f_i (i = 1,2,3) respectively represents damping coefficients of the inductor motor. T_ei (i = 1,2,3) is the electromagnetic torques of the inductor motor. T_Li (i = 1,2,3) is the time varying load torque which can be expressed as

\begin{array}{l} T_{L 1} = m_{1} r [\ddot{y} \cos φ_{1} - \ddot{x} \sin φ_{1} + l_{1} {\dot{ψ}}^{2} \sin (φ_{1} - θ) + l_{1} \ddot{ψ} \cos (φ_{1} - θ)] \\ T_{L 2} = m_{2} r [\ddot{y} \cos φ_{2} - \ddot{x} \sin φ_{2} - l_{2} {\dot{ψ}}^{2} \cos φ_{2} + l_{2} \ddot{ψ} \sin φ_{2}] \\ T_{L 3} = m_{3} r [\ddot{y} \cos φ_{3} - \ddot{x} \sin φ_{3} - l_{3} {\dot{ψ}}^{2} \sin (φ_{3} + θ) - l_{3} \ddot{ψ} \cos (φ_{3} + θ)] \end{array}

(2)

The model of the electromagnetic system

The electromechanical coupling model of the inductor motor is divided into two parts. One is the mechanical system which is shown in section ‘The model of the mechanical system’. The other is the electromagnetic system. According to the electrical machinery theory, the model of the induction motor is given as below.

\begin{array}{l} ϕ_{s d} = L_{s} i_{s d} + L_{m} i_{r d} \\ ϕ_{s q} = L_{s} i_{s q} + L_{m} i_{r q} \\ ϕ_{r d} = L_{m} i_{s d} + L_{r} i_{r d} \\ ϕ_{r q} = L_{m} i_{s q} + L_{r} i_{r q} \end{array}

(3)

\begin{array}{l} {\dot{ϕ}}_{s d} = - R_{s} i_{s d} + ω_{s} ϕ_{s q} - u_{s d} \\ {\dot{ϕ}}_{s q} = - R_{s} i_{s q} + ω_{s} ϕ_{s d} - u_{s q} \\ {\dot{ϕ}}_{r d} = - R_{s} i_{s q} + (ω_{s} - ω) ϕ_{r q} \\ {\dot{ϕ}}_{r q} = - R_{r} i_{r q} - (ω_{s} - ω) ϕ_{r d} \end{array}

(4)

Equations (3) and (4) respectively represent the flux-linkage equations and voltage equations of the induction motor in the synchronous reference frame of the d and q axis. Due to the feature of the squirrel cage motor, its inner structure forms a phenomenon of short out. Hence, $u_{r d} = u_{r q}$ . When the system is at a steady state, $ϕ_{r d}$ = constant and $ϕ_{r q} = 0$ . There are nine variables in the synchronous reference frame of the d and q axis totally. Five of them are chosen to establish the state equation of an induction motor in the rotor field-oriented coordinate. They are respectively $ω$ , $i_{s d}$ , $i_{s q}$ , $ϕ_{r d}$ and $ϕ_{r q}$ . Then the equation can be expressed as

\begin{array}{l} {\dot{ϕ}}_{r d} = - ϕ_{r d} / T_{r} + (ω - s n_{p} ω) / ϕ_{r q} + L_{m} i_{s d} / T_{r} \\ {\dot{ϕ}}_{r q} = - (ω - s n_{p} ω) / ϕ_{r d} - ϕ_{r q} / T_{r} + L_{m} i_{s q} / T_{r} \\ {\dot{i}}_{s d} = L ϕ m r d / (σ L_{s} L_{r} T_{r}) + L n m p ω ϕ_{r q} / (σ L_{s} L_{r}) - (L_{m}^{2} + R_{s} L_{r} T_{r}) i_{s d} / (σ L_{s} L_{r} T_{r}) + ω i s s q + u_{s d} / (σ L_{s}) \\ {\dot{i}}_{s q} = - L n m p ω ϕ_{r d} / (σ L_{s} L_{r}) + L ϕ m r q / (σ L_{s} L_{r} T_{r}) - ω i s s d - (L_{m}^{2} + R_{s} L_{r} T_{r}) i_{s q} / (σ L_{s} L_{r} T_{r}) + u_{s q} / (σ L_{s}) \end{array}

(5)

where the subscript s and r respectively represent stator and rotor. The subscripts d and q respectively represent the d axis and q axis in the rotor field-oriented coordinate; i represents current. u represents voltage. R represents resistance. L_s and L_r are respectively self-inductance of the stator and rotor. R_s and R_r are respectively the stator and rotor resistance. L_m is the mutual inductance of the stator and rotor. Tr is a rotor time constant, Tr= L_r/Rr. In the similar way, T_s=L_s/R_s. σ is leakage factor, σ = 1−L_m/(L_sL_r). n_p is the number of pole-pairs of the induction motor. ω is the mechanical speed. ω_s is the synchronous electric angular speed. According to the model of the induction motor, the following equation can be obtained.

T_{e} = n_{p} L_{m} (i_{s q} ϕ_{r d} - i_{s d} ϕ_{r q}) / L_{r}

(6)

Because of $ϕ_{r d}$ =constant and $ϕ_{r q} = 0$ , equation (6) can be expressed as

T_{e} = n_{p} L_{m} i_{s q} ϕ_{r d} / L_{r}

(7)

As shown above, n_p, L_m, $ϕ_{r d}$ and L_r are constants. Hence, equation (7) is an equation with the variable i_sq when the induction motor is at the steady state.

Responses of the vibrating system

With the theory of non-linear dynamics, the phase difference of motors in the multifrequency synchronization can be expressed as $p φ_{1} - q φ_{2}$ =constant. p and q are prime numbers. When the constant equals to zero, $p φ_{1} - q φ_{2}$ =0. The vibrating system can obtain the maximum amplitude which is needed in the engineering. And then, $p φ_{1} - q φ_{2}$ =0 can be expressed as $φ_{2} / φ_{1} = p / q$ . Use n₁ to replace $p_{1} / q_{1} (p_{1} \in p, q_{1} \in q)$ . Similarly, n₂ can be obtained. So, the problem of phase difference is converted to the problem of phase ratio. The phase of three motors can be assumed as¹³

\begin{array}{l} φ_{1} = φ + 4 / 3 α_{1} + 2 / 3 α_{2} \\ φ_{2} = n_{1} (φ - 2 / 3 α_{1} + 2 / 3 α_{2}) \\ φ_{3} = n_{2} (φ - 2 / 3 α_{1} - 4 / 3 α_{2}) \end{array}

(8)

where n₁ and n₂ are the positive real numbers,

α_{1}

and

α_{2}

are the initial angles of the phase difference. Then, take the derivative of ϕ₁, ϕ₂ and ϕ₃, it can be derived as

\begin{array}{l} {\dot{φ}}_{1} = \dot{φ} + 4 / 3 {\dot{α}}_{1} + 2 / 3 {\dot{α}}_{2} \\ {\dot{φ}}_{2} = n_{1} (\dot{φ} - 2 / 3 {\dot{α}}_{1} + 2 / 3 {\dot{α}}_{2}) \\ {\dot{φ}}_{3} = n_{2} (\dot{φ} - 2 / 3 {\dot{α}}_{1} - 4 / 3 {\dot{α}}_{2}) \end{array}

(9)

According to equation (1), a conclusion can be obtained that the motion of the vibrating system is periodic when the system is at the synchronous state. So, the change of the mechanical angular velocity of a motor is also periodic. The rotating periods of the motors are respectively T₁, T₂ and T₃. There must be a positive real number T which is the least common multiple period of the motors to satisfy $T = m T_{1} = n T_{2} = r T_{3}$ . Where the parameters m, n and r are co-prime integrals. Then, the mean value of the angular velocity $ω_{0}$ is a constant over the interval of T. It can be expressed as

ω_{0} = \int_{0}^{T} ({\dot{φ}}_{1} + {\dot{φ}}_{2} / n_{1} + {\dot{φ}}_{3} / n_{2}) d t / 3 T = constant

(10)

When the vibrating system is at the synchronous state, equation (10) can also be derived as $ω_{0} = \int_{0}^{T} {\dot{φ}}_{1} d t / T = \int_{0}^{T} ({\dot{φ}}_{2} / n_{1}) d t / T = \int_{0}^{T} ({\dot{φ}}_{3} / n_{2}) d t / T$ =constant. So, each angular velocity of the motors in the period T equals a constant. With the small parameter averaging method, the higher-order items are neglected. Then, $\dot{φ} = (1 + ε_{0}) ω_{0}$ and ${\dot{α}}_{i} = ε_{i} ω_{0}$ (i = 1,2) can be acquired, where $ε_{1}$ and $ε_{2}$ are small parameters. Take them into equation (9), it can be derived as

\begin{array}{l} {\dot{φ}}_{1} = (1 + ε_{0} + 4 / 3 ε_{1} + 2 / 3 ε_{2}) ω_{0} \\ {\dot{φ}}_{2} = n_{1} (1 + ε_{0} - 2 / 3 ε_{1} + 2 / 3 ε_{2}) ω_{0} \\ {\dot{φ}}_{3} = n_{2} (1 + ε_{0} - 2 / 3 ε_{1} - 4 / 3 ε_{2}) ω_{0} \end{array}

(11)

Take equation (11) into equation (1) and make ${\dot{ε}}_{i} = 0$ (i = 0,1,2). The responses of the x, y and ψ directions can be obtained.

\begin{array}{l} x = - (r_{m} r / μ_{x}) [η_{1} \cos (φ_{1} + γ_{x}) + η_{2} n_{1}^{2} \cos (φ_{2} + γ_{x}) + η_{3} n_{2}^{2} \cos (φ_{3} + γ_{x})] \\ y = - (r_{m} r / μ_{y}) [η_{1} \sin (φ_{1} + γ_{y}) + η_{2} n_{1}^{2} \sin (φ_{2} + γ_{y}) + η_{3} n_{2}^{2} \sin (φ_{3} + γ_{y})] \\ ψ = - (r_{m} r_{l} r / μ_{ψ} l_{e}) [η_{1} ι_{1} \sin (φ_{1} - θ + γ_{ψ}) - ι_{2} η_{2} n_{1}^{2} \cos (φ_{2} + γ_{ψ}) \\ - ι_{3} η_{3} n_{2}^{2} \sin (φ_{3} + θ + γ_{ψ})] \end{array}

(12)

where

ω_{x}^{2} = k_{x} / M

ω_{y}^{2} = k_{y} / M

ω_{ψ}^{2} = k_{ψ} / J

ζ_{x} = f_{x} / (2 \sqrt{k_{x} M})

ζ_{y} = f_{y} / (2 \sqrt{k_{y} M})

ζ_{ψ} = f_{ψ} / (2 \sqrt{k_{ψ} J})

μ_{x} = 1 - ω_{x}^{2} / ω_{0}^{2}

μ_{y} = 1 - ω_{y}^{2} / ω_{0}^{2}

μ_{ψ} = 1 - ω_{ψ}^{2} / ω_{0}^{2}

η_{i} = m_{i} / m_{0}

\tan γ_{x} = 2 ζ_{x} ω_{x} / (μ_{x} ω_{0})

\tan γ_{y} = 2 ζ_{y} ω_{y} / (μ_{y} ω_{0})

\tan γ_{ψ} = 2 ζ_{ψ} ω_{ψ} / (μ_{ψ} ω_{0})

r_{m} = m_{0} / M

r_{l} = l_{0} / l_{e}

ι_{i} = l_{i} / l_{0} (i = 1, 2, 3)

According to the synchronization theory, the trajectory of the vibrating system is decided by responses in the three directions. From equation (12), it can be concluded that the responses change with the parameters n₁ and n₂. With different n₁ and n₂, the trajectory presents various kinds of forms. In the multifrequency synchronization, the parameters n₁ and n₂ can be suitably adjusted to increase the amplitude which is needed to improve the efficiency in the engineering.

Design of the controlling system

Although the average angular velocity $ω_{0}$ is a constant with the small parameter averaging method, this method cannot be used in the load torque. This is the difference between the base frequency synchronization and multifrequency synchronization. As shown in equation 2, when the load torque of the master motor is averaged with a period of $2 π$ , the periods of the slave motors’ are respectively $2 n_{1} π$ and $2 n_{2} π$ . There is not a uniform period to average the load torques. This is the reason why the multifrequency self-synchronization cannot be realized with the dynamical model in Figure 1. So, in this chapter, the ratios n₁ and n₂ are used as target variables to realize the multifrequency-controlled synchronization with Fuzzy PID method.

Design of the electromechanical coupling system

As shown in Figure 2, the controlling law of the master–slave form is adopted. The induction motor 1 is the master motor and the induction motors 2 and 3 are slave motors. The induction motors 2 and 3 trace the induction motor 1 with the phase ratio. The rotor flux oriented control (RFOC) and fuzzy PID method are used on the electromagnetic system of the induction motors. $ω_{t}$ as a given speed is transferred to the master induction motor. With the fuzzy PID controlling method, the speed of motor 1 $ω_{1}$ is obtained. Obviously, $ω_{1}$ is used for two parts. One part is as a feedback value to control the master motor speed with the given speed $ω_{t}$ . The other part is to obtain $φ_{1}$ which is traced by the slave motors. Finally, the speed of the induction motors $ω_{2}$ , $ω_{3}$ and phases $φ_{2}$ , $φ_{3}$ which are respectively fed back to the slave induction motors are obtained.

Figure 2.

Flow diagram of the controlled system.

The RFOC in Figure 2 is shown as Figure 3. The given speed $ω_{t}$ subtracts the feedback speed $ω_{1}$ , and then use the fuzzy PID method to obtain the electromagnetic torque $T_{e}^{*}$ which can calculate the stator current on the q axis $i_{s q}^{*}$ . The stator current on the d axis $i_{s d}^{*}$ can be obtained from the formulation $i_{s d}^{*} = ϕ_{r d}^{*} / L_{m}$ , where $ϕ_{r d}^{*}$ is the given flux linkage. Hence, $U_{s d}^{*}$ and $U_{s q}^{*}$ can be obtained. The synchronization electric angle $θ$ can be expressed as follows:

θ = \int (ω + ω_{s}) d t

(13)

where

ω_{s}

is the slip angular velocity,

ω_{s} = L_{m} i_{s q}^{*} / ϕ_{r d} T_{r}

. Through the coordinate transformation,

U_{s α}^{*}

and

U_{s β}^{*}

can be obtained. Finally, the speed of the motor

ω

and

i_{a b c}

with the SVPWM technology can be acquired.

Figure 3.

RFOC: rotor flux oriented control.

Design of the fuzzy PID method

The fuzzy system can be structured by the following steps.

Step 1: define a fuzzy set $A_{i}^{l_{i}} (l_{i} = 1, 2 \dots, p_{i})$ with the number $p_{i}$ of the variation $x_{i} (i = 1, 2)$ .

Step 2: use the fuzzy rules with the number of $\prod_{i = 1}^{2} p_{i}$ to structure the fuzzy system $\hat{f} (x | θ_{f})$ .

R^{(j)} : if x_{1} is A_{1}^{l_{1}} and x_{2} is A_{1}^{l_{2}} then \hat{f} is E^{l_{1} l_{2}}

(14)

where

l_{i} = 1, 2

(

i = 1, 2

). With the product inference engine, singleton obfuscator and center average obfuscator, the output of the fuzzy system can be expressed as

\hat{f} (x | θ_{f}) = \frac{\sum_{l_{1} = 1}^{p_{1}} \sum_{l_{2} = 2}^{p_{2}} {\bar{y}}_{f}^{l_{1} l_{2}} [\prod_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i})]}{\sum_{l_{1} = 1}^{p_{1}} \sum_{l_{2} = 2}^{p_{2}} [\prod_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i})]}

(15)

where

μ_{A_{i}^{l_{i}}} (x_{i})

is the member function of

x_{i}

. Consider

{\bar{y}}_{f}^{l_{1} l_{2}}

as a free parameter in the set

θ_{f} \in R^{\prod_{i = 1}^{2} p_{i}}

and then introduce a column vector

ξ (x)

, equation (15) can be derived as

\hat{f} (x | θ_{f}) = θ_{f} T ξ (x)

(16)

where

ξ (x)

is a column vector with the dimension number of

\prod_{i = 1}^{2} p_{i}

, in which the

l_{1}

th and

l_{2}

th elements are

ξ_{l_{1} l_{2}} (x) = \frac{\prod_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i})}{\sum_{l_{1} = 1}^{p_{1}} \sum_{l_{2} = 2}^{p_{2}} [\prod_{i = 1}^{2} μ_{A_{i}^{l_{i}}} (x_{i})]}

(17)

Two input variables and three output variables are given in this part. The input variables are respectively the error (e) and error change (ec). The output variables are respectively k_p, k_i and k_d. Each of the five variables is divided into seven situations which are respectively NB, NM, NS, Z, PS, PM and PB. NB and PB are the member function of the model Z (zmf). The others are the member function of the triangle model (trimf). The variables are shown in Figure 4.

Figure 4.

The subjection function of variables. (a) Error (e); (b) error change (ec); (c) parameter k_p; (d) parameter k_i and (e) parameter k_d.

According to equation (13), 49 rules are established with the fuzzy control regulation in this article and then the following equations of the PID parameters can be obtained.

K_{P} = \frac{[\sum_{j = 1}^{49} μ_{K_{P j}} (E, E C) \times K_{P j}]}{\sum_{j = 1}^{49} μ_{K_{P j}} (E, E C)}

(18)

K_{I} = \frac{[\sum_{j = 1}^{49} μ_{K_{I j}} (E, E C) \times K_{I j}]}{\sum_{j = 1}^{49} μ_{K_{I j}} (E, E C)}

(19)

K_{D} = \frac{[\sum_{j = 1}^{49} μ_{K_{D j}} (E, E C) \times K_{D j}]}{\sum_{j = 1}^{49} μ_{K_{D j}} (E, E C)}

(20)

With equations (18) to (20), the surfaces of k_p, k_i and k_d can be obtained in Figure 5.

Figure 5.

The surfaces of the parameters with k_p, k_i and k_d. (a) Surface of k_p, (b) surface of k_i, and (c) surface of k_d. The horizontal axis x is e and the vertical axis y is ec in all the three diagrams.

The stability analysis of the vibrating system

In this chapter, the rotate speed of the motor 3 and the phase difference between motors 1 and 3 are controlled to reach the target value. With the Lyapunov stability theory, the validations are given.

The stability analysis of the speed

In this part, the speed of motor is chosen as a tracking variable. Then, it can be expressed as $ω = \dot{φ}$ . The dynamical equation of motor can be deduced as

J \dot{ω} + f ω = K_{T} u + W

(20)

where u means

i_{q s}^{*}

and

W = - T_{L}

which is the uncertain item. Assuming the target speed is

ω_{t}

and then the tracking error is E which can be expressed as

E = ω_{t} - ω

(21)

According to equation (20), the control law can be designed as

u = J / K_{T} [- \hat{f} (x | θ_{f}) + {\dot{ω}}_{t} + K^{T} E + f ω / J]

(22)

where

\hat{f} (x | θ_{f}) = θ_{f}^{T} ξ (x)

K = {[k_{p}, k_{i}]}^{T}

. So, the adaptive law can be obtained.

{\dot{θ}}_{f} = - γ E^{T} P b ξ (x)

(23)

To guarantee weight coefficient $θ_{f}$ has the boundary and an optimal weight coefficient $θ_{f}^{*}$ of the fuzzy system belongs to the convex set $Ω_{f}$ . The optimal weight coefficient $θ_{f}^{*}$ can be expressed as

θ_{f}^{*} = \arg \min_{θ_{f} \in Ω_{f}} [\sup | \hat{f} (x | θ_{f}) - f (x)]

(24)

Take equation (22) into equation (20), the dynamic equation of the fuzzy system can be obtained.

\dot{E} = Λ E + b [\hat{f} (x | θ_{f}) - f (x)]

(25)

where

Λ = (\begin{array}{l} 0 & 1 \\ - k_{p} & - k_{i} \end{array})

b = (\begin{array}{l} 0 \\ 1 \end{array})

Take equations (23) and (24) into equation (25), the approximation error equation of the system is deduced as

\dot{E} = Λ E + b [{(θ_{f} - θ_{f}^{*})}^{T} ξ (x) + Γ]

(26)

where

Γ = \hat{f} (x | θ_{f}^{*}) - f (x)

To realize minimum of the tracking error E and parameter error $θ_{f} - θ_{f}^{*}$ , the Lyapunov function can be defined as

V = E^{T} P E / 2 + {(θ_{f} - θ_{f}^{*})}^{T} (θ_{f} - θ_{f}^{*}) / (2 ζ)

(27)

where

ζ

is the positive constant and P is a positive definite matrix which meets the Lyapunov equation expressed as

Λ^{T} P + P A = - Q

(28)

where Q is a positive definite matrix with two rows and two columns. Set

V_{1} = E^{T} P E / 2

and

V_{2} = {(θ_{f} - θ_{f}^{*})}^{T} (θ_{f} - θ_{f}^{*}) / (2 ζ)

. Thus,

{\dot{V}}_{1} = - E^{T} Q E / 2 + {(θ_{f} - θ_{f}^{*})}^{T} E^{T} P b ξ (x) + E^{T} P b Γ

and

{\dot{V}}_{2} = {(θ_{f} - θ_{f}^{*})}^{T} {\dot{θ}}_{f} / ζ

. Because

\dot{V} = {\dot{V}}_{1} + {\dot{V}}_{2}

\dot{V} = - E^{T} Q E / 2 + E^{T} P b Γ

can be obtained. Only if choose the suitable

Γ

to reach the condition

\dot{V} \leq 0

, the controlling system is stable.

The stability analysis of the phase

In this chapter, the phase $φ_{3}$ is given as an initial value and the phase $n φ_{1}$ is considered as a target value. Then, the tracking error can be expressed as

E = n φ_{1} - φ_{3}

(29)

Similarly, the stability analysis of phase can be certified with the method in section ‘The stability analysis of the speed’.

Results and discussion

In this part, according to the former theory $p φ_{1} - q φ_{2}$ =constant, the result that the three ERs cannot realize the state of multifrequency self-synchronization in the dynamical model of Figure 1 is proved with the Matlab/Simulink first. And then, the multifrequency-controlled synchronization is accomplished with the fuzzy PID method. The target of $p φ_{1} - q φ_{2}$ =0 is realized with different parameters n₁ and n₂ which are arbitrary real numbers. Finally, the experimental verifications are given to certify the correctness and effectiveness. The parameters of the motors and system are respectively shown in Tables 1 and 2.

Table 1.

Parameters of the three inductor motors.

Parameters	Motor 1	Motor 2	Motor 3
Rated power P (kW)	0.2	0.2	0.2
Pole pairs n_p	3	3	3
Rated frequency f₀ (Hz)	50	50	50
Rated voltage U (V)	220	220	220
Rated speed (r/min)	910	910	910
Stator resistance R_s (Ω)	40.4	40.5	40.6
Rotor resistance referred R_r (Ω)	12	11.531	12.813
Stator inductance L_s (H)	1.21275	1.213	1.21275
Rotor inductance referred L_r (H)	1.222	1.225	1.222
Mutual inductance L_m (H)	1.116	1.116	1.116
Rated flux linkage $λ_{d r}^{*}$ (Wb)	0.98	0.98	0.98
Damping coefficients f_1,2 (Nms/rad)	0.005	0.005	0.005

Table 2.

Parameters of the vibration system.

Parameters	Value
m (kg)	275
J_p (kg.m²)	43.5
k_x (N/m)	129,332
k_y (N/m)	105,334
k_ψ (Nm/rad)	30,715
f_x (Ns/m)	615.5
f_y (Ns/m)	618
f_ψ (Nsm/rad)	180.2
l₁ (m)	0.45
l₂ (m)	0.3
l₃ (m)	0.45
θ (°)	30
β (°)	0
m₀ (kg)	6
r (m)	0.05

Numerical simulation of the multifrequency self-synchronization

In the theory of self-synchronization, when the speed and phase are both synchronous, the vibrating system is deemed to reach the synchronous state. Otherwise, it does not. As shown in Figure 6(a), the speed of motor 1 is set as 60 rad/s. The speed of motors 2 and 3 are respectively 60 rad/s and 90 rad/s. The initial phase angle $α_{0} = 0$ , and $m_{i} = 4 kg (i = 1, 2, 3)$ . The speeds of three motors are stable when the time reaches at about second one and then float with the given speeds slightly. (b) The phase differences among the three motors. $φ_{2} - φ_{1}$ is the phase difference between the motors 1 and 2. It is about $170 °$ , which is consistent with the theory of the self-synchronization. It is because that motors 1 and 2 do not realize the self-synchronization but vibrational synchronous motion. This phenomenon is a natural model of the vibrating system and has strong robustness. And motor 3 rotates in a periodical time. So, the influence of motor 3 on motors 1 and 2 can be neglected. According to the theory of non-liner dynamics in section ‘Responses of the vibrating system’, when the equations $φ_{3} - 1.5 φ_{1}$ =constant and $φ_{3} - 1.5 φ_{2}$ =constant are realized, the vibrating system reaches the synchronous state. However, they change with a monotone increasing tendency. So, the three ERs cannot reach the synchronous state. (c), (d) and (e) are respectively the responses of the x, y and ψ directions. A special phenomenon which is different from the base frequency situation needs to be pointed out. In the situation of the base frequency, the crest or trough values of the response curves are equal when the vibrating system is at the stable state. But in the situation of the multifrequency, the crest or trough values are on longer equal and appear an approximate sinusoidal variation. It is because that the parameters n₁ and n₂ have influences on the peak. These results in (c), (d) and (e) are consistent with equation (12) in the theory section.

Figure 6.

The multifrequency self-synchronization on the situation n₁=1 and n₂=1.5, $α_{0} = 0$ . (a) Speeds of the motors; (b) phase differences; (c) response of the x direction; (d) response of the y direction and (e) response of the ψ direction.

Numerical simulation of the multifrequency-controlled synchronization

According to section ‘Numerical simulation of the multifrequency self-synchronization’, the three ERs cannot reach the synchronous state. To realize the stable multifrequency synchronous state, the fuzzy PID method and the master–slave controlling strategy based on the phase ratio are introduced.

As shown in Figure 7, the same values of the parameters n₁ and n₂ are used to make a comparison with Figure 6(a) shows the speeds of three motors. The given speed $ω_{t}$ is 60 rad/s. Motors 2 and 3 trace motor 1 with phase ratio which are used as controlled variables. Comparing (a) with (c), the given speed of the master motor is stable at about 3 s in (a). And then the values of the phase ratio n₁ and n₂ are stable at about 5 s through the phase control in (c). Finally, the speeds of motors 2 and 3 are stable at about 8 s. This result is consistent with Figure 2 and certifies the effectiveness and robustness of the controlling system. (b) The values of the three motors’ load torques which are between –2 and 2. This result is the same as the conclusion in document²² and the load torque in this article is smaller than the rated torque. So, when the vibrating system is at the stable synchronous state, three motors can operate normally. (d) The phase differences of three motors. $φ_{3} - 1.5 φ_{1}$ =0 proves that $p φ_{1} - q φ_{2}$ =0 which is needed in the engineering can be realized. (e), (f) and (g) are respectively the responses of the x, y and ψ directions. Because the speed of motor 3 is faster than motors 1 and 2’s, the response curves have a modal superposition. This result leads the curves being not typically sinusoids but still regular and periodic variation ones.

Figure 7.

Multifrequency controlled synchronization on the situation n₁=1 and n₂=1.5, $α_{0} = 0$ . (a) Speeds of the motors; (b) load torques; (c) phase ratio; (d) phase differences; (e) response of the x direction; (f) response of the y direction and (g) response of the ψ direction.

To certify the arbitrariness of the parameters n₁ and n₂, they are changed in Figures 8 and 9. And then, the influence on the vibrating system by changing the parameters is discussed.

Figure 8.

Multifrequency controlled synchronization on the situation n₁=1 and n₂=1.2, $α_{0} = 0$ . (a) Speeds of the motors; (b) load torques; (c) phase ratio; (d) phase differences; (e) response of the x direction; (f) response of the y direction and (g) response of the ψ direction.

Figure 9.

Multifrequency controlled synchronization on the situation n₁=1.2 and n₂=1.5, $α_{0} = 0$ . (a) Speeds of the motors; (b) load torques; (c) phase ratio; (d) phase differences; (e) response of the x direction; (f) response of the y direction and (g) response of the ψ direction.

As shown in Figure 8, keep n₁=1 and change n₂ from 1.5 to 1.2. Similar to Figure 7, the vibrating system still reaches the stable synchronous state which can be obtained from (a) and (c). This conclusion proves that the parameter n₂ can be an arbitrary rather than a fixed value only if it satisfies the synchronization and stability conditions of the vibrating system. Comparing (b) in Figure 8 with (b) in Figure 7, the load torque increases with the parameter n₂. When n₂ equals to 1.5, the values of the load torque are from –2 to 2. When n₂ equals to 1.2, the values of the load torque are from –1.5 to 1.5. Comparing (d) in Figure 8 with (d) in Figure 7, there are two phenomena. One is that the curves of the phase difference in Figure 8 are smoother than the curves in Figure 7. The other is that the values of the phase difference in Figure 8 are larger than those in Figure 7. The first phenomenon can be illustrated that the smaller n₂ is, the easier the vibrating system can be controlled until n₂ equals to 1 which is the base frequency synchronization. The second is summarized that the complexity to realize the controlling of the zero phase difference is decided by the trajectory of the system which is obtained from the responses. The more complex the trajectory of the system is, the more difficult the zero phase difference can be realized.

Different from Figure 8, in Figure 9 keep n₂=1.5 and then change n₁ from 1 to 1.2. (a) and (c) are shown to prove that the stability of the system has nothing to do with the position of the two slave motors. Changing the position of the motors 2 and 3, the system can still reach the synchronization state. The results in (b) and (d) are consistent with the former conclusion which is obtained in Figure 8. The responses (e), (f) and (g) in Figure 9 appear the characters of three different frequency. Take the x response as an example. When the time is between 33.1 s and 33.3 s, the response presents the character of 1.2 times frequency. When the time is between 33.5 s and 33.7 s, the response presents the character of 1.5 times frequency.

Experimental verification

For combining the theory with the engineering, the experimental verification is given to certify the accuracy of the theory and the effectiveness in the engineering. Figure 10 shows the experiment of multifrequency self-synchronization. In this experiment, the frequencies of three motors which are driven by three inverters are respectively set as 30 Hz, 30 Hz and 45 Hz. There are three acceleration sensors on the frame. One is on the x direction. Another is on the y direction in the center of the frame. The other is on the y direction in the edge of the frame. When the time is about 15 s, the speeds of three motors are stable in (a). (b) The phase difference between motor 1 and motor 2. It is about 160°, which is consistent with the result of the base frequency self-synchronization. However, (c) presents the difference between the base frequency synchronization and the multifrequency synchronization. Obviously, the two groups of phase differences are not constants but monotone increasing curves. The phases are not synchronous. So, this phenomenon proves that the vibrating system cannot reach the synchronous state. And this result is consistent with (b) in Figure 6 of section ‘Numerical simulation of the multifrequency self-synchronization’. (d), (e) and (f) are respectively the responses of x, y₁ and y₂. Taking the x response as an example, (d) has the same variation tendency with (c) in Figure 6. This conclusion shows the same result between the numerical simulation and the experiment.

Figure 10.

The experiment of the multifrequency self-synchronization, where $η_{i} = 50 % (i = 1, 2, 3)$ , $α_{0} = 0$ , n₁=1 and n₂=1.5. (a) Speeds of the three motors. (b) The phase difference of motor 2 and motor 1. (c) The phase differences of motor 3 with motors 1 and 2. (d) Response of the x direction. (e) Response of the y₁ direction and (f) Response of the y₂ direction.

Figure 11 is the experiment of multifrequency-controlled synchronization. It corresponds to Figure 8 of the numerical simulation. In this experiment, the speeds of three motors are set to 30 Hz, 30 Hz and 36 Hz by three inverters. Motor 1 is the master motor and there is no controlling method on it. Motors 2 and 3 are slave motors which are respectively controlled by two PLCs (programmable logic controller) through the pulses. The measurement and collection of the pulses are accomplished by the Hall sensor. The acceleration sensors on the frame are installed with the same method as the experiment above. (a) The speeds of three motors. When the time is at about 12 s, the speeds of three motors reach the synchronous state. The speed of motor 2 is same as motor 1’s and the speed of motor 3 is 1.2 times than motor 1’s. (b) The phase difference between motor 2 and motor 1. It is about –4°. The experiment always has some error. For example, the installation of the motors (motor 2 may have a few distance from the center line of the vibrating system), the pulse losing (few pulses are missed by the Hall sensor) and so on. So, this result in this article is not considered as $φ_{2} - φ_{1}$ =constant but $φ_{2} - φ_{1}$ =0. (c) The phase difference between motor 3 and motor 1. It floats around 12°. In the same way, the result in (c) can also be regarded as $φ_{3} - 1.2 φ_{1} = 0$ . (d), (e) and (f) are respectively the responses of x, y₁ and y₂. Similarly, take the response of the x direction as an example. Through contrasting the enlarged view of (d) in Figure 11 with (e) in Figure 8, they have the same waveform and variation tendency. So, the result of the numerical simulation is consistent with the experiments. Thus, the multifrequency-controlled synchronization can be well realized. Figure 12 represents the experiment result on the situation n₁=1.2 and n₂=1.5 and it corresponds to Figure 9 in section ‘Numerical simulation of the multifrequency-controlled synchronization’. This experiment is to certify the arbitrary of the parameters n₁ and n₂. With changing the parameters, the speeds of three motors also reach the synchronous state in (a). (b) and (c) are respectively the phase differences of the motors. According to the former conclusion, $φ_{2} - 1.2 φ_{1} = 0$ and $φ_{3} - 1.5 φ_{1} = 0$ are realized. So, the phases of three motors reach the synchronous state. (d) is the response of x direction. When the time is between 90.3 s and 90.5 s, the character of the 1.5 times frequency is shown. When the time is between 90.7 s and 90.85 s, the character of the 1.2 times frequency is shown. This result is consistent with (e) in Figure 9 through the enlarge drawing.

Figure 11.

The experiment of the multifrequency controlled synchronization, where $η_{i} = 30 % (i = 1, 2, 3)$ , $α_{0} = 0$ , n₁=1 and n₂=1.2. (a) Speeds of the three motors. (b) The phase difference of motor 2 and motor 1. (c) The phase difference of motor 3 and motor 1. (d) The response of the x direction. (e) The response of the y₁ direction and (f) The response of the y₂ direction.

Figure 12.

The experiment of the multifrequency controlled synchronization, where $η_{i} = 30 % (i = 1, 2, 3)$ , $α_{0} = 0$ , n₁=1.2 and n₂=1.5. (a) Speeds of the three motors. (b) The phase difference of motor 2 and motor 1. (c) The phase difference of motor 3 and motor 1. (d) The response of the x direction. (e) The response of the y₁ direction and (f) The response of the y₂ direction.

Conclusions

In this article, multifrequency-controlled synchronization of three ERs driven by induction motors in the same direction is investigated. The multifrequency self-synchronization with the dynamic model in Figure 1 cannot be realized. To realize the multifrequency synchronization, the fuzzy PID method based on the phase ratio is introduced and the result is very effective. Not only $p φ_{1} - q φ_{2}$ =constant can be realized but also $p φ_{1} - q φ_{2}$ =0. And the arbitrariness of the parameters n₁ and n₂ are certified. The influence of the parameters on the vibrating system is divided into two parts. One is that the difficulty to control the system is decided by the parameter n ( $n_{1} \in n, n_{2} \in n$ ). The other is that the complexity to realize the controlling of the zero phase difference is decided by the trajectory of the system which is obtained from the responses. Comparing the base frequency synchronization with multifrequency synchronization, the responses curves are not typically sinusoids but still regular and periodic variation ones in multifrequency synchronization. This result realizes the diversity of the motion trail of the vibrating system. Consequently, the multifrequency-controlled synchronization is in favor of promoting the effectiveness of the vibrating screen in the engineering.

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: This project is supported by the National Nature Science Foundations of China (51675090).

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Multifrequency-controlled synchronization of three eccentric rotors driven by induction motors in the same direction

Abstract

Keywords

Introduction

Dynamic model of the vibration system

The model of the mechanical system

The model of the electromagnetic system

Responses of the vibrating system

Design of the controlling system

Design of the electromechanical coupling system

Design of the fuzzy PID method

The stability analysis of the vibrating system

The stability analysis of the speed

The stability analysis of the phase

Results and discussion

Numerical simulation of the multifrequency self-synchronization

Numerical simulation of the multifrequency-controlled synchronization

Experimental verification

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References