Shock and Vibration

Volume 2016, Article ID 8641754, 22 pages

http://dx.doi.org/10.1155/2016/8641754

## Investigation for Synchronization of a Rotor-Pendulum System considering the Multi-DOF Vibration

School of Mechanical Engineering, Southwest Petroleum University, Chengdu 610500, China

Received 30 May 2015; Accepted 28 September 2015

Academic Editor: Tai Thai

Copyright © 2016 Yongjun Hou and Pan Fang. 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

This work is a continuation for our published literature for vibration synchronization. A new mechanism, two rotors coupled with a pendulum rod in a multi-DOF vibration system, is proposed to implement coupling synchronization, and the dynamics equation of mechanism is derived by Lagrange equation. In addition, the coupling relationship between the vibrobody and the pendulum rod is ascertained with the Laplace transformation method, based on the dimensionless equation of the dynamics system. The Poincare method is employed to study the synchronization state between the two unbalanced rotors, which is converted into that of existence and the stability of solutions for synchronization-balance equations. The obtained results are supported by computer simulations. It is demonstrated that the values of the spring stiffness coefficient, the length of the pendulum, and the angular installation of the pendulum are important parameters with respect to the synchronous behavior in the rotor-pendulum system.

#### 1. Introduction

The word “synchronization” is often encountered in both science and daily life. Our surroundings are full of synchronization phenomenon, which is considered as an adjustment of rhythms of oscillating objects due to their internal weak couplings [1]. For example, violinists play in unison, insects in a population emit acoustic or light pulses with a common rate, birds in a flock flap their wings simultaneously, and the heart of a rapidly galloping horse contracts once per locomotory cycle [2]. Synchronization phenomena in large populations of interacting elements are the subjects of intense research efforts in physical, biological, chemical, and social system; however, the most representatives are synchronization of complex systems [3–5], coupled with pendula or mechanical rotors in recent years. For the synchronization of pendula, in the particular case of Huygens’ clocks system, the remarkable feature reported by Huygens in 1665 is that pendulum clocks synchronize in antiphase. Nowadays the synchronized limit behavior of Huygens’ clocks, synphase and antiphase synchronization of the pendula, is studied considering the different values of spring stiffness [6, 7]. Meanwhile, the synchronization of derivatizations of Huygens’ clocks, including two coupled double pendula [8], pendulum coupled by an elastic force [9], and pendula connected by linear springs [10], has been attracting many scholars’ attention. For the synchronization of rotors, Blekhman [1] proposed the Poincare method for the synchronization state and stability and by now this method is widely used in engineering. Based on Blekhman’s method, many scientists have been developing the other methods to analyze the synchronization of the rotors. Wen et al. [11] developed the average method to investigate synchronization and stability of multiple rotors in after-resonance. Zhang et al. [12, 13] described the average method of modified small parameters, which immensely simplify the process for solving the problems of synchronization of the rotors. Then, Fang et al. [14] employed the average method of modified small parameters to explore the synchronization of two homodromy rotors installed on a double vibrobody in a coupling vibration system. Sperling et al. [15] presented analytical and numerical investigation of a two-plane automatic balancing device, for equilibration of rigid-rotor unbalance. Balthazar [16, 17] examined self-synchronization of four nonideal exciters in nonlinear vibration system via numerical simulations. Djanan et al. [18] explored the condition, for which three motors working on a same plate can enter into synchronization with the phase difference depending on the physical characteristics of the motors and the plate.

The abovementioned researches are mainly synchronization of the pendula or the rotors; however, the synchronization of the rotors coupled with pendula is less reported. Recently, we have proposed synchronization of two homodromy rotors coupled with a pendulum rod in an postresonant system [19], but the influence of the multi-DOF vibration of vibrobody on the synchronous state of the rotors is less considered. This paper is a continuation of our published literature by means of the Poincare method, building on the original work of Blekhman. Here, we consider the model of two homodromy rotors coupled with a rigid pendulum rod through a torsion spring, and the vibrobody is connected to a fixed support by means of springs. It is demonstrated that the values of the spring stiffness coefficient, the length of the pendulum, and the angular installation of the pendulum are important parameters with respect to the synchronous behavior in the rotor-pendulum system.

This paper is organized as follows. Section 2 describes the strategy and considered model. In Section 3, we employ the Laplace transform method to calculate the value of the coupling coefficients. In Section 4, we derive the synchronization equation and the synchronization criterion of the system. In Section 5, we compare and analyze the values of the stable phase difference with theoretical computations and the computer simulations. Finally, we summarize our results in Section 6.

#### 2. Strategy and Model

##### 2.1. Strategy

Consider the dynamic equation of a rotation system:where , is a small parameter, is the rotational inertia of the th induction motors, is the mechanical damping of the motors, is the damping ratio of the system in the -direction, and is natural frequency of the system in -direction. and are mechanical velocity and phase angular of the th unbalanced rotor, respectively.

Based on (1), the following sequence of analysis for vibration system employing synchronizing rotors can be formulated:(1)Steady forced vibrations with are determined by from the supporting body or supporting system of bodies (i.e., from the second formula of (1) considering ) when rotors are uniformly rotating with initial phase ; that is, (2)Equation (3) mentioned above may correspond only to such values of constants , which satisfy Here and below the angle brackets show the average within ; that is, where symbol represents a function related to time [1].(3)If a certain set of constants , which satisfy (4), real parts of all roots of the th order algebraic equation are negative, then at sufficiently small this set of constants is indeed correlated with the unique, analytical relative to , asymptotically stale periodic solution of (1). This solution changes into the fundamental solution (see (4)) at . If the real part of at least one root of (6) is positive, then the corresponding solution is unstable. With purely imaginary zero roots, an additional analysis is required in general case [1].

##### 2.2. Model

A simplified rotor-pendulum system depicted in Figure 1(a) is considered. This model consists of a rigid vibrobody of mass [Kg], a rigid pendulum rod, and two unbalanced rotors. The rigid vibrobody is elastically supported via linear springs with stiffness [N/m], [N/m], and [N/rad] and linear viscous dampers with damping constants [Ns/m], [Ns/m], and [Nm/(rad/s)], respectively. Unbalanced rotor is modelled by a point mass [Kg] (for ) attached at the end of a massless rod of length [m]. One of the unbalanced rotors in the system is directly mounted on the vibrobody, and the other is fixed at the end of a pendulum rod, which is connected with the vibro-platform by a linear torque spring with stiffness [N/rad] and a linear viscous damper with damping constant [Nm/(rad/s)]. Note that the two rotors are driven separately in the same direction by two identical induction motors. The rotation angle of rotor is denoted by (for ) in [rad]; the oscillating angle of the pendulum rod is denoted by in [rad]; the installation angle of the pendulum rod is expressed by in [rad]; represents the electromagnetic torque of the induction motors in [N/m]; signifies resistance torque in bearings; and , , and are the vibration displacement of the vibrobody in [m], [m], and [rad], respectively.