Shock and Vibration

Volume 2019, Article ID 7481746, 17 pages

https://doi.org/10.1155/2019/7481746

## Phase Synchronization Control of Two Eccentric Rotors in the Vibration System with Asymmetric Structure Using Discrete-Time Sliding Mode Control

School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

Correspondence should be addressed to Lingxuan Li; moc.361@il_nauxgnil

Received 11 January 2019; Revised 25 April 2019; Accepted 14 May 2019; Published 10 June 2019

Academic Editor: José J. Rangel-Magdaleno

Copyright © 2019 Xiaozhe Chen and Lingxuan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Control synchronization of two eccentric rotors (ERs) in the vibration system with the asymmetric structure is studied to make the vibration system obtain the maximum excited resultant force and the driven power. Because this vibration system is essentially an underactuated system, a decoupling strategy for the control goal of the same phase motion between two ERs is proposed to reduce the order of state equation of the vibration system. According to the master-slave control scheme, the complex control objects are converted into the velocity control of the master motor and the phase control of the slave motor. Considering the self-adjusting of the vibration system as interference, controllers of the velocity and the phase difference are designed by applying the discrete-time sliding mode control, which is proved by Lyapunov theory. A vibration machine is designed for evaluating the performance of the proposed controllers. Two control schemes are presented: controlling one motor and controlling two motors, and two group experiments are achieved to investigate the dynamic coupling characteristic of the vibration system in the state of control synchronization. The experimental results show that control synchronization is an effective and feasible technology to remove the limitation of vibration synchronization.

#### 1. Introduction

Vibration machine is a new type of machine quickly developed in the twentieth century, which utilizes vibration principle to perform various processing tasks and has been widely used in various fields of industry [1]. They usually adopt an eccentric rotor (ER) as the excited source. By combining two ERs in different rotational directions and different installation positions, the vibration machine acquires the different resultant force and the motion trajectory to satisfy the processing requirements [2–4]. Hence, how to guarantee the synchronous motion between two ERs becomes a research topic [5, 6]. In the early design of vibration machine, the method of forced synchronization, such as gear and belt, is the only way to achieve the synchronous motion of two ERs, which is actually a passive control.

Since Blekhman studied self-synchronization of two ERs in the vibration system using the small parameter of Poincare-Lyapunov [5], more and more scholars are attracted to study self-synchronization from every respect [7–14]. Self-synchronization of the vibration system (also called vibration synchronization) is of great significance in engineering and technology fields because it replaces the technology of forced synchronization as the design technology of 2nd generation vibration machine. The technology of vibration synchronization utilizes the dynamic coupling characteristic of the vibration system to guarantee the synchronous motion of two ERs, which reflects the vibration system has the ability of the self-adjusting [2]. Hence, vibration synchronization is essentially semiactive control of applying the self-adjusting of the vibration system.

In some vibration machines, such as vibration mill, eccentric excitation caused by the asymmetric structure is needed to prevent the same amplitude at all positions of the system [1]. Although vibration synchronization utilizes the dynamic coupling characteristic of the vibration system to guarantee the synchronization of two ERs, the phase difference between two ERs is not usually equal to zero because of the stability limit of the system [3, 15, 16]. Because the stability of the vibration system depends on its structure parameters when two are ERs are operating with the asymmetric structure, the phase difference between two ERs is not equal to zero, which results in the smaller resultant force [3]. It is remarked that only when two ERs are operating in the same phase, their resultant force is the maximum value. To obtain the maximum excited resultant force and the driven power, the viewpoint of introducing control theory into the design of a vibration machine (also called control synchronization) is proposed [5]. Therefore, the technology of active control is the development trend of 3rd vibration machine.

Because the vibration system has the ability of the self-adjusting, the control synchronization of two ERs in the vibration system is different from that of two motors in other mechanical systems [17]. Now, there are several control strategies for multimotors in the general system, e.g., the master-slave control [18], the cross-coupling control [19], the virtual shaft control [20], and the ring coupling control [21]. In engineering, the master-slave control is the most direct and effective method compared with other methods. Based on the master-slave control strategy, several scholars studied control synchronization of two ERs in the vibration system with symmetric structure. Kong [17] applied slide mode control (SMC) to study control synchronization of the vibration system with symmetrical structure. Tomchina [22] adopted PI control to investigate control synchronization of the vibration system with one degree of freedom. Jia [23] proposed a fuzzy PID method to study multiple-frequency synchronization of the vibration system with symmetrical structure. Fradkov [24] applied PI control to study the multiple-frequency control synchronization for 3-rotor vibration unit with varying payload. Miklos [11] applied PI control to study control synchronization of a dual-rotors system. To some extent, these results promote the development of control synchronization of the vibration system with symmetric structure. However, some results still need to be proved by experiment, because control methods of continuous-time cannot be directly applied in a microprocessor to control practical systems.

Considering the engineering requirement, this paper studied control synchronization of two ERs in the vibration system with the asymmetric structure. For this type of nonlinear system, this paper uses discrete-time sliding mode control (DSMC) to carry out the experimental research [25, 26]. To easily observe the dynamic coupling characteristic of the vibration system in the state of control synchronization, DC motor is adopted as the driven source. In the next section, the electromechanical mathematic model is presented. Next, controllers are designed. Additionally, an experimental system is introduced. Later, experimental results are achieved. Finally, conclusions are presented.

#### 2. Dynamic Model of the Vibration System

The dynamic model, the vibration machine, the experimental system, and experimental flow are shown in Figure 1. Comparing Figures 1(a) and 1(b), the dynamic model of the vibration system mainly consists of a fixed base (Num 1), springs (Num 2), a rigid body (Num 3), and two ERs (Num 4). Additional details about the numbers are shown in Table 1. The springs connect the base with the rigid body. The mass center of the vibration system translations is and , and angular rotation is . Two ERs are direct-driven by DC motor in the clockwise direction, respectively. are the rotational centers of ERs, is the eccentric radius of two ERs, and denote ER rotates about its spin axis, . Two vibration motors (two eccentric mass blocks are installed on both ends of the motor shaft) are not fixed symmetrically based on the *Y*-axis passing through the mass center of the system, and they are fixed to the left of the *Y*-axis. So, the system of Figure 1 is a type of the asymmetric structure because of external excitation from two ERs. Because the spring is made of high-carbon steel and is cylindrical, the spring stiffness can be approximately linear when the system works at the far resonance [1]. Selecting the , , , , and as the generalized coordinates and using Lagrange’s equation, the electromechanical mathematic model of the vibration system is expressed as follows [16]:where , are the electromagnetic torques of two motors, are the circuit equations of two motors, are armature resistances, are electromotive force constants, are electromagnetic torque constants, are the input voltages of DC motor, are the input currents of DC motor, are the moments of inertia of ERs, are the damping coefficients of the axis of motor, and ; is the mass of the rigid body, and are the masses of ERs, , is the moment of inertia of the vibration system; is the anticlockwise direction of ER, is the clockwise direction of ER; are the distances between the rotational center and the center of mass , and are the angles between line and -axis, ; are stiffness of springs and are damping of springs; and denotes and .