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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 243758, 15 pages
Research Article

Nonlinear Vibration of a Multirotor System Connected by a Flexible Coupling Subjected to the Holonomic Constraint of Dynamic Angular Misalignment

Department of Mechanics, Xi'an University of Science and Technology, Xi'an 710054, China

Received 1 September 2011; Revised 28 November 2011; Accepted 13 December 2011

Academic Editor: Alex Elias-Zuniga

Copyright © 2012 M. 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.


This paper proposes a mathematical model of the multirotor system with a flexible coupling on spring supports on Lagrange's approach, which has taken into account the effects of dynamic angular misalignment and mass unbalance. Then its nonlinear dynamic behaviors of the system are discussed based on the method of multiple scales and numerical technique, respectively. The results show that the responses of the system in lateral directions contain a similar component to that of the mass unbalanced system on both the vibrating frequency and amplitude and involve the typical nonlinear components such as the ones from some combined frequencies; the results also reveal that the numerical agreements on the above-mentioned methods are perfect for the transient responses.

1. Introduction

The misalignment of rotor system is one of the most common mechanical faults in rotating machinery, which can cause excessive vibrations, produce bothering noises, and create damaging forces on rotors, bearings, and couplings. For large-scale rotating system, there are several reasons that lead to the misalignments such as the various deformations of rotors, uncoaxiality of bearings and couplings. According to the geometric relationship between the two rotors, the types of misalignment may be divided into the parallel and angle ones or their combination. In practical engineering, generally the misalignment appears in some main form, for example, the ground subsidence or looseness and improper assembly of the rotor system can induce the faults of angle misalignment. In fact, even a multirotor system is well aligned in static equilibrium state; it still presents the dynamic misalignment or offset in a rotor system which couples by a flexible coupling.

During the past decades, the dynamics of effects on the rotor misalignments has been studied in a variety of aspects, that is, firstly Gibbons [1] established a model to predict the additional forces due to the misalignment of the gear coupling, Dewell and Mitchell [2] analyzed the lateral vibration frequencies for a misaligned rotor system which were based on the experiment, and Li and Yu [3] discussed the coupled lateral and torsional vibration of a rotor-bearing system with the misalignment by a gear coupling. In paper [4], Al-Hussain and Redmond studied the effect of parallel misalignment on the lateral and torsional responses of two-rotor system, and in [5], Al-Hussain considered the effect of angular misalignment on the stability of rotors connected by a flexible mechanical coupling. Prabhakar et al. [6] dealt with the transient responses of a rotor system with a rotor misalignment by using the finite element method. Recently, Hili et al. [7] established a linear dynamic model of a multirotor system which couples a Cardan joint by using the finite element method, in which both of the faults misalignment and unbalance were considered. Lees [8] studied the misalignment of a rigidly coupled rotor which mounted on idealized linear bearings with a linear time-varying stiffness. Li [9] developed a mathematical model of a multirotor system by taking into account the effects of parallel misalignment and unbalance of rotors, in which the misalignment was dealt with as a holonomic constraint, and Slim et al. [10] paid attention to the dynamic behaviors of misaligned rotor system mounted on two journal bearings. Patel and Darpe [11] investigated vibrating responses of misaligned rotors modeled by using Timoshenko beam elements with six degrees of freedom, and in paper [12], they studied the frequency spectra of the system supported on rolling element bearings by experiment. In 2010, Redmond [13] developed a dynamic modal of misaligned shafts, which included both the static angular and parallel misalignment.

For the multirotor system, flexible couplings are commonly used to transmit torque from one rotor to another when the adjacent shafts are slightly misaligned. And yet it is required coaxially during the installation for avoiding the additional forces on bearings, for example, the force is not permitted to be produced for a flexible coupling under static conditions; however, it is aligned or coaxial in this state, the coupling still appears a dynamic misalignment at work [14], that is, a dynamic angular deformation is generated on the flexible coupling duo to the dynamic load. In the above discussions, while much work has been done on the dynamics of the system that an initial or static misalignment exists, the little attention has been paid to the dynamic one. Accordingly, the purpose of the present paper is to explain the vibration mechanism and dynamic characteristics of rotor system with a flexible coupling after considering the effect on the dynamic angular misalignment.

2. Motion Equations

The proposed model of the multirotor system connected by a flexible coupling is shown in Figure 1, in which 𝑚1 and 𝑚2denote the lumped masses, respectively; 𝑘1 and 𝑘2 represent the stiffness of the bearings; 𝑘𝑡 is the equivalent angular stiffness for the flexible couple. For simplification, the following assumptions upon the system of concern will be used hereinafter: (1) the two rotors are rigid; (2) the bearings are isotropic on dynamic performances; (3) the misaligned angle between the adjacent rotors is small.

Figure 1: Schematic diagram of angular-misaligned rotor system.

The coordinate system 𝑜𝑥𝑦𝑧 is set up in the static equilibrium position of rotor system as shown in Figure 1. Let 𝑂1(𝑥1,𝑦1,0) and 𝑂2(𝑥2,𝑦2,𝑧2) be the coordinates of the mass centers of disc 1 and disc 2, respectively, then they yield𝑥1=𝑥+𝑒1𝑦cos(Ω𝑡),1=𝑦+𝑒1𝑧sin(Ω𝑡),1𝑥=0,(2.1)2=𝑥+𝑙sin𝜃cos(Ω𝑡)+𝑒2𝑦cos𝜃cos(Ω𝑡+𝛾),2=𝑦+𝑙sin𝜃sin(Ω𝑡)+𝑒2𝑧cos𝜃sin(Ω𝑡+𝛾),2=𝑙+𝑙cos𝜃,(2.2) where 𝑥 and 𝑦 are the displacements of disc 1 at the geometric center 𝐶1; 𝜃 is the angle between adjacent rotors; Ω is the rotating speed; 𝛾 is the initial phase angle; 𝑙is the rotor length; 𝑒1, 𝑒2 are the mass unbalances. If the generalized coordinates 𝑥, 𝑦, and 𝜃 are introduced, then (2.2) describes a nonstationary holonomic constraint because the displacement varies with time 𝑡. Generally, the offset of disc 2 is larger than 𝑒2; therefore, the terms related to 𝑒2 are ignored in the paper.

The kinetic energy of the system can be written as𝑇=2𝑖=112𝑚𝑖̇𝑥2𝑖+̇𝑦2𝑖+̇𝑧2𝑖,(2.3) and the potential energy is𝑈=2𝑖=112𝑘𝑖𝑥2𝑖+𝑦2𝑖+12𝑘𝑡𝜃2.(2.4) For simplifying the representation furthermore, let 𝑚1=𝑚2=𝑚,𝑘1=𝑘2=𝑘, and 𝑒1=𝑒, then substituting (2.1) and (2.2) into (2.3) and (2.4), and based on Lagrange’s equation𝑑𝑑𝑡𝜕𝑇𝜕̇𝑞𝑗𝜕𝑇𝜕𝑞𝑗+𝜕𝑈𝜕𝑞𝑗=0,𝑞𝑗=𝑥𝑦𝜃𝑇,(2.5) the motion equation in the generalized coordinates becomes ̈̇𝜃2𝑚̈𝑥+𝑚𝑙𝜃cos𝜃cosΩ𝑡𝑚𝑙2̇sin𝜃cosΩ𝑡2𝑚𝑙Ω𝜃cos𝜃sinΩ𝑡𝑚𝑙Ω2sin𝜃cosΩ𝑡+2𝑘𝑥+𝑘𝑙sin𝜃cosΩ𝑡=𝑚𝑒Ω2̈̇𝜃cosΩ𝑡,2𝑚̈𝑦+𝑚𝑙𝜃cos𝜃sinΩ𝑡𝑚𝑙2̇sin𝜃sinΩ𝑡+2𝑚𝑙Ω𝜃cos𝜃cosΩ𝑡𝑚𝑙Ω2sin𝜃sinΩ𝑡+2𝑘𝑦+𝑘𝑙sin𝜃sinΩ𝑡=𝑚𝑒Ω2sinΩ𝑡,𝑚𝑙2̈𝜃+𝑚𝑙̈𝑥cos𝜃cosΩ𝑡+𝑚𝑙̈𝑦cos𝜃sinΩ𝑡𝑚𝑙2Ω2sin𝜃cos𝜃+𝑘𝑙𝑥cos𝜃cosΩ𝑡+𝑘𝑙𝑦cos𝜃sinΩ𝑡+𝑘𝑙2sin𝜃cos𝜃+𝑘𝑡𝜃=0.(2.6) The above equations are second-order ordinary differential ones with variable coefficients, which is obviously the characteristic of strong nonlinearity. Based on the theory of differential equations, it is difficult to solve. By applying the inverse operation, (2.6) can be expressed as1̈𝑥𝑚𝑙1+sin2𝜃𝑚𝑙2̇𝜃2sin𝜃cosΩ𝑡+2𝑚𝑙2Ω̇𝜃cos𝜃sinΩ𝑡𝑚𝑙2Ω̇𝜃cos3𝜃sinΩ𝑡+𝑚𝑙2Ω2sin3𝜃cosΩ𝑡𝑘𝑙2sin3𝜃cosΩ𝑡𝑘𝑙𝑥1+sin2𝜃+𝑘𝑡𝜃cos𝜃cosΩ𝑡+𝑚𝑒Ω21𝑙𝑐𝑜𝑠Ω𝑡=0,̈𝑦𝑚𝑙1+sin2𝜃𝑚𝑙2̇𝜃2sin𝜃sinΩ𝑡2𝑚𝑙2Ω̇𝜃cos𝜃cosΩ𝑡+𝑚𝑙2Ω̇𝜃cos3𝜃cosΩ𝑡+𝑚𝑙2Ω2sin3𝜃sinΩ𝑡𝑘𝑙2sin3𝜃sinΩ𝑡𝑘𝑙𝑦1+sin2𝜃+𝑘𝑡𝜃cos𝜃sinΩ𝑡+𝑚𝑒Ω2̈1𝑙sinΩ𝑡=0,𝜃+𝑚𝑙21+sin2𝜃𝑚𝑙2̇𝜃2sin𝜃cos𝜃𝑚𝑙2Ω2sin𝜃cos𝜃+𝑘𝑙2sin𝜃cos𝜃+2𝑘𝑡𝜃+𝑚𝑒Ω2𝑙cos𝜃=0.(2.7) Let 𝑋=𝑥/𝑟, 𝑌=𝑦/𝑟 be the nondimensional displacements, in which 𝑟 is the radius of rotor; 𝐸=𝑒/𝑟, 𝐿=𝑙/𝑟 are the nondimensional mass eccentricity and rotor length, respectively; 𝜏=Ω𝑡 is the nondimensional time; 𝜔0=𝑘/𝑚Ω2, 𝜔𝑡=𝑘𝑡/𝑚𝑙2Ω2 are the nondimensional angular frequencies and denote 𝑑𝑥/𝑑𝑡=𝑥(𝑑𝑋/𝑑𝜏)=𝑋…; accordingly, (2.7) can be cast into the following nondimensional form:𝑋11+sin2𝜃𝐿(𝜃)2sin𝜃cos𝜏+2𝐿𝜃cos𝜃sin𝜏𝐿𝜃cos3𝜔𝜃sin𝜏201𝐿sin3𝜃cos𝜏𝜔20𝑋1+𝑠𝑖𝑛2𝜃+𝜔2𝑡𝑌𝐿𝜃𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜏+𝐸𝑐𝑜𝑠𝜏=0,11+sin2𝜃𝐿(𝜃)2sin𝜃sin𝜏2𝐿𝜃cos𝜃cos𝜏+𝐿𝜃cos3𝜔𝜃cos𝜏201𝐿sin3𝜃sin𝜏𝜔20𝑌1+sin2𝜃+𝜔2𝑡𝜃𝐿𝜃cos𝜃sin𝜏+𝐸sin𝜏=0,+11+sin2𝜃𝜃2𝜔sin𝜃cos𝜃+201sin𝜃cos𝜃+2𝜔2𝑡𝐸𝜃+𝐿cos𝜃=0.(2.8)

Equation (2.8) is a parametrically excited system with three degrees of freedom on the theory of nonlinear vibration, in which 𝜔0 and 𝜔𝑡 are the frequency ratios, respectively; εis the nondimensional mass eccentricity 𝐿 is the nondimensional length. In the above equation, the third one is decoupled with the first two equations, that is, the generalized coordinate θ is totally independent of 𝑋 and Y thereby it can be solved in the first place.

In the case of the initial angular misalignment, it can be assumed that 𝜃=𝛼+𝜑(𝑡) in which 𝛼 is the constant angle and 𝜑(𝑡) is its perturbation. After substituting 𝜃 into (2.8) and expanding the relevant terms in the Taylor series, after neglecting the higher-order ones the governing equations that is of the initial or static angular misalignment between two rotors are obtained𝑋11+sin2𝛼𝐿(𝜑)2sin𝛼cos𝜏+𝐿𝜑1+sin2𝛼𝜔cos𝛼sin𝜏201𝐿sin3𝛼cos𝜏𝜔201+sin2𝛼𝑋+𝜔2𝑡𝜑𝐿𝛼cos𝛼cos𝜏+𝐸cos𝜏1+sin2𝛼𝐿(𝜑)2cos𝛼cos𝜏+𝐿𝜑3cos2𝛼2sin𝛼sin𝜏2𝜔20𝑋cos𝛼sin𝛼𝜔2𝑡𝐿(𝛼sin𝛼cos𝛼)cos𝜏+3𝐿1𝜔20sin2+𝛼cos𝛼cos𝜏𝜑sin2𝛼1+sin2𝛼2𝐿(𝜑)2sin𝛼cos𝜏+𝐿𝜑1+sin2𝛼cos𝛼sin𝜏+𝜔2𝑡𝛼𝐿cos𝛼cos𝜏𝜔201+sin2𝛼𝑋+𝐿1𝜔20sin3𝑌𝛼cos𝜏+𝐸cos𝜏=0,11+sin2𝛼𝐿(𝜑)2sin𝛼sin𝜏𝐿𝜑1+sin2𝛼𝜔cos𝛼cos𝜏201𝐿sin3𝛼sin𝜏𝜔201+sin2𝛼𝑌+𝜔2𝑡𝜑𝐿𝛼cos𝛼sin𝜏+𝐸sin𝜏1+sin2𝛼𝐿(𝜑)2cos𝛼sin𝜏+𝐿𝜑23cos2𝛼sin𝛼cos𝜏2𝜔20𝑌cos𝛼sin𝛼𝜔2𝑡𝐿(𝛼sin𝛼cos𝛼)sin𝜏+3𝐿1𝜔20sin2+𝛼cos𝛼sin𝜏𝜑sin2𝛼1+sin2𝛼2𝐿(𝜑)2sin𝛼sin𝜏𝐿𝜑1+sin2𝛼cos𝛼cos𝜏+𝜔2𝑡𝛼𝐿cos𝛼sin𝜏𝜔201+sin2𝛼𝑌+𝐿1𝜔20sin3𝜑𝛼sin𝜏+𝐸sin𝜏=0,+11+sin2𝛼𝜑2sin𝛼cos𝛼+2𝜔2𝑡𝜔𝛼+20𝐸1sin𝛼cos𝛼+𝐿+𝜑cos𝛼1+sin2𝛼(𝜑)2cos2𝛼+2𝜔2𝑡+𝜔20𝐸1cos2𝛼𝐿sin𝛼𝜑sin2𝛼1+sin2𝛼2(𝜑)2sin𝛼cos𝛼+2𝜔2𝑡𝜔𝛼+20+𝐸1sin𝛼cos𝛼𝐿cos𝛼=0.(2.9)

3. The Approximation Solution

For discussing the vibration mechanism and characteristics of the system conveniently, let 𝛼=0, namely, the initial or static angular misalignment of a flexible coupling vanishes, and only the dynamic angular misalignment presents in the system, then (2.9) reduces to𝑋𝐿(𝜑)2𝜑cos𝜏𝐿𝜑sin𝜏+𝜔20𝑋𝜔2𝑡𝑌𝐿𝜑cos𝜏=𝐸cos𝜏,𝐿(𝜑)2𝜑sin𝜏+𝐿𝜑cos𝜏+𝜔20𝑌𝜔2𝑡𝜑𝐿𝜑sin𝜏=𝐸sin𝜏,+(𝜑)2𝜔𝜑+20+2𝜔2𝑡𝐸1𝜑+𝐿=0.(3.1) The third of (3.1) is uncoupled with the first two; therefore, it can be simplified as𝜙+(𝜙)2𝜙+𝜛20𝜙=0,(3.2) where 𝜙=𝜑+𝐸/𝐿𝜛2, 𝜛20=𝜔20+2𝜔2𝑡1.

The method of multiple scales (MMSs) is introduced for the approximation solution of the nonlinear vibration in this work. For the small angle 𝜙, let 𝑇=𝜔𝜏 and expand 𝜙 and 𝜔 [15] as𝜙(𝑇,𝜀)=𝜀𝜙1(𝑇)+𝜀3𝜙3(𝑇),(3.3)𝜔=𝜛0+𝜀2𝜔2,(3.4) where ε stands for a small nondimensional parameter characterizing the amplitude of the motion. The term 𝜀2𝜙2(𝜏) is missing from (3.3) because the nonlinearity appears at 𝑂(𝜀3). The term 𝜀𝜔1 is absent from (3.4) because the frequency is independent of the sign of ε. Substituting (3.3) and (3.4) into (3.2) and equating coefficients of like powers of ε, after donating 𝑑𝜙/𝑑𝑇=𝜙, it yields𝜛20𝜙1+𝜙1𝜛=0,(3.5)20𝜙3+𝜙3=2𝜛0𝜔2𝜙1𝜛20𝜙12𝜙1.(3.6) The solution of (3.5) is of the form𝜙1=𝑎cos(𝑇+𝛽),(3.7) where 𝑎 and 𝛽 are constants; hence, (3.6) becomes𝜙3+𝜙3=𝜔2𝜛014𝑎21𝑎cos(𝑇+𝛽)+4𝑎3cos(3𝑇+3𝛽).(3.8) Eliminating the secular term in (3.8) gives 𝜔2=(1/4)𝜛0𝑎2, then disregarding the solution of the homogeneous equation, its solution reduces to𝜙3𝑎=332cos(3𝑇+3𝛽).(3.9) From the equation 𝜙=𝜑+𝐸/𝐿𝜛2, it follows that𝐸𝜑=𝐿𝜛20𝜀+𝜀𝑎cos𝜓3𝑎332cos3𝜓,(3.10) in which 𝜓=𝜔𝜏+𝛽, 𝜔=𝜛0(1+𝜀2𝑎2/4). If imposing the initial conditions 𝜑|𝜏=0=0and 𝜑|𝜏=0=0, then the constants 𝜀𝑎 and 𝛽 are obtained𝜀𝜀𝑎cos𝛽3𝑎3𝐸32cos3𝛽=𝐿𝜛20,3sin𝛽1+𝜀322𝑎238𝜀2𝑎2cos2𝛽=0.(3.11) When 𝛽=0, 𝜀𝑎 yields𝜀𝜀𝑎3𝑎3=𝐸32𝐿𝜛20.(3.12) In practice, 𝐸/𝐿𝜛20 is small, thus the solution becomes 𝛽=0,𝜀𝑎𝐸/𝐿𝜛20. For example, if the parameters 𝐿=20.0 and 𝐸=0.03, 0.05, the errors between them can be neglected at all. The numerical results are shown in Figure 2, which indicate a good agreements under the concerned parameters.

Figure 2: Numerical results for𝜀𝑎 and 𝐸/𝐿𝜛2 with increasing 𝜔0 or 𝜔𝑡.

Generally for the above nonlinear equations, there are six real solutions, in which the root 𝛽=0,𝜀𝑎𝐸/𝐿𝜛2is only satisfied for small oscillation in engineering, that is, the systematic parameters are considered as 𝜔𝑜=1.2, 𝜔𝑡=0.02, 𝐿=20.0, and 𝐸=0.05, accordingly𝜛0=𝜔20+2𝜔2𝑡1=0.664, and after solving (3.11) on Maple, the six real roots of (3.12) are𝐸𝛽=0,𝜀𝑎=0.00567𝐿𝜛2,𝐸𝛽=0,𝜀𝑎=5.65969,𝛽=0,𝜀𝑎=5.65402,𝛽=𝜋,𝜀𝑎=0.00567𝐿𝜛2,𝛽=𝜋,𝜀𝑎=5.65969,𝛽=𝜋,𝜀𝑎=5.65402.(3.13)

The proves that𝛽=0,𝜀𝑎𝐸/𝐿𝜛2 is an available approximate solution, 𝛽=0, and𝜀𝑎=5.696,5.65402 are not almost changed with increasing the parameters 𝜔𝑜 and 𝜔𝑡, which are not suitable for the engineering. Thereby, the former will be paid more attention to the following analysis, then expression (3.10) leads to𝐸𝜑=𝐿𝜛201(cos(𝜔𝜏)1)𝐸32𝐿𝜛203cos(3𝜔𝜏).(3.14) Substituting expression (3.14) into (3.1), it yields𝑋+𝜔20𝜔𝑋=𝐸20+𝜔2𝑡1𝜛20𝐸cos(𝜏)+𝜛20𝜔2𝑡𝐸cos(𝜔𝜏)cos(𝜏)𝜔sin(𝜔𝜏)sin(𝜏)+𝑂2,𝑌+𝜔20𝜔𝑌=𝐸20+𝜔2𝑡1𝜛20𝐸sin𝜏+𝜛20𝜔2𝑡𝐸cos(𝜔𝜏)sin(𝜏)+𝜔sin(𝜔𝜏)cos(𝜏)+𝑂2,(3.15) and its general solution gives𝑋=𝐶1cos𝜔0𝜏+𝐶2sin𝜔0𝜔𝜏+𝐸20+𝜔2𝑡1𝜛20𝜔20+𝐸1cos𝜏2𝜛20𝜔2𝑡+𝜔𝜔20(𝜔+1)2𝜔cos(𝜔+1)𝜏+2𝑡𝜔𝜔20(𝜔1)2𝐸cos(𝜔1)𝜏+𝑂2,𝑌=𝐶3cos𝜔0𝜏+𝐶4sin𝜔0𝜔𝜏+𝐸20+𝜔2𝑡1𝜛20𝜔20+𝐸1sin𝜏2𝜛20𝜔2𝑡+𝜔𝜔20(𝜔+1)2𝜔sin(𝜔+1)𝜏2𝑡𝜔𝜔20(𝜔1)2𝐸sin(𝜔1)𝜏+𝑂2,(3.16) where 𝜔=𝜔20+2𝜔2𝑡1(1+(𝐸/2𝐿𝜛2)2), and 𝐶1,𝐶2,𝐶3,and𝐶4are the constants.

If the initial conditions𝑋(0)=𝑋(0)=0 and 𝑌(0)=𝑌(0)=0 are introduced, the constants 𝐶1,𝐶2,𝐶3, and 𝐶4 can be determined easily; therefore, the responses can be shown to reduce to𝑋=(𝐴+𝐵+𝐷)cos𝜔0𝜏+𝐴cos𝜏+𝐵cos(𝜔+1)𝜏+𝐷cos(𝜔1)𝜏,𝑌=𝐷(𝜔1)𝐵(𝜔+1)𝐴𝜔0sin𝜔0𝜏+𝐴sin𝜏+𝐵sin(𝜔+1)𝜏𝐷sin(𝜔1)𝜏,(3.17) in which 𝐴=𝐸((𝜔20+𝜔2𝑡1)/𝜛20(𝜔201)),𝐵=𝐸(𝜔2𝑡+𝜔)/2𝜛20(𝜔20(𝜔+1)2), and 𝐷=𝐸(𝜔2𝑡𝜔)/2𝜛20(𝜔20(𝜔1)2).

The above solution shows that the response consists of the following components: (a) the free vibration at natural frequency 𝜔0 of the derived linear system. If the damping is concerned in 𝑥 and 𝑦 direction, its amplitude will be decreased exponentially on the theory of vibration; (b) forced vibration excited by the mass unbalance at the frequency of rotating speed. When 𝜔0=1, that is, the rotating speed coincides with the lateral natural frequency, the bending resonance of rotor system occurs; (c) the angular motions due to the flexible coupling, in which the frequencies 𝜔1and 𝜔+1 are combined by 𝜔 and the rotating speed. It is a typical nonlinear oscillation because the combination resonances exist at𝜔0=𝜔1and𝜔0=𝜔+1.

Certainly, it should be noted that the above solution only reveals some dynamic behaviors of the rotor system connected to a flexible coupling, and some other characteristics such as the subharmonic resonances may be emerged if suitable conditions are satisfied.

4. Numerical Analysis

Because (2.8) is a strongly nonlinear system, the numerical technique is preferable. Accordingly, the Runge-Kutta method is carried out to predict the dynamic characteristics in the present paper. As a nonautonomous system, traditionally (2.8) is discussed on the state space, and hence the three second-order equations can be converted easily to six first-order ones𝑋=𝜂,𝑌=𝜉,𝜃1=𝜗,𝜂=1+sin2𝜃𝐿𝜗2sin𝜃cos𝜏+2𝐿𝜗cos𝜃sin𝜏𝐿𝜗cos3𝜔𝜃sin𝜏201𝐿sin3𝜃cos𝜏𝜔20𝑋1+sin2𝜃+𝜔2𝑡,1𝐿𝜃cos𝜃cos𝜏+𝐸cos𝜏𝜉=1+sin2𝜃𝐿𝜗2sin𝜃sin𝜏2𝐿𝜗cos𝜃cos𝜏+𝐿𝜗cos3𝜔𝜃cos𝜏201𝐿sin3𝜃sin𝜏𝜔20𝑌1+sin2𝜃+𝜔2𝑡,𝜗𝐿𝜃cos𝜃sin𝜏+𝐸sin𝜏1=1+sin2𝜃𝜗2𝜔sin𝜃cos𝜃+201sin𝜃cos𝜃+2𝜔2𝑡𝐸𝜃+𝐿.cos𝜃(4.1)

For the following calculation, the values of systematic parameters are considered as 𝜔0=1.20, 𝜔𝑡=0.02, and 𝐿=20, and all the initial generalized displacements and velocities are set to zero.

Figures 3(a) and 4(a) depict the displacement responses of rotor system in 𝑥 and 𝑦 directions when𝐸=0.05, respectively, in which the curves are demonstrated by using MMS and the numerical technique. The results show that the agreements between their responses are pretty almost at each interval, and thereby it proves that the dynamic behaviors of the system are in the form of (3.17) under certain initial conditions. Figures 3(b) and 4(b) appear as the frequency spectra of displacement responses by the numerical method, in which there are mainly four components from the vibrating frequencies:𝑓1=(𝜔1)/2𝜋=0.05349,𝑓2=1/2𝜋=0.15915, 𝑓3=𝜔0/2𝜋=0.19099, and 𝑓4=(𝜔+1)/2𝜋=0.26482. Actually, it reveals that the synchronous motion 𝑓2 emerges in the responses, which are obviously from the mass unbalance, and 𝑓3 is the natural frequency of the derived linear system as shown in (3.15), 𝑓1 and 𝑓4 are the combinations between 𝜔/2𝜋 and the rotating speed Ω, which occur frequently for a nonlinear system. Figures 5 and 6 also show the relevant responses and the spectra when 𝐸=0.03. Figures 7 and 8 illustrate the responses and their spectra when𝐸=0.05, and 𝜔0=1.20, 𝜔𝑡=0.50, namely, the angular stiffness of flexible coupling increases these vibration characteristics display the multifrequency signatures in the rotor system connected a flexible coupling with a dynamic angular misalignment.

Figure 3: The nondimensional response and its frequency spectrum in 𝑥 direction when 𝐸=0.05: (a) response; (b) spectrum.
Figure 4: The nondimensional response and its frequency spectrum in 𝑦 direction when𝐸=0.05: (a) response; (b) spectrum.
Figure 5: The nondimensional response and its frequency spectrum in 𝑥 direction when𝐸=0.03: (a) response; (b) spectrum.
Figure 6: The nondimensional response and its frequency spectrum in 𝑦 direction when𝐸=0.03: (a) response; (b) spectrum.
Figure 7: The nondimensional response and its frequency spectrum in 𝑥 direction when 𝜔0=1.20, 𝜔𝑡=0.50, and 𝐸=0.05: (a) response; (b) spectrum.
Figure 8: The nondimensional response and its frequency spectrum in 𝑦 direction when 𝜔0=1.20, 𝜔𝑡=0.50, and 𝐸=0.05: (a) response; (b) spectrum.

5. Conclusions

The misalignment of rotor system is an important reason which causes mechanical vibration; a dynamic misalignment or offset of the rotors can be produced at work even the system is aligned under the static conditions. In this paper, firstly a mathematic model of the multirotor system with a flexible coupling is established after considering the effect on a dynamic angular misalignment based on Lagrange’s equation, and it shows that the system is of a parametric oscillation with strongly nonlinear characteristics. Then, the method of multiple scales and Runge-Kutta numerical technique are carried out, respectively. And the results indicate that the responses in lateral direction consist of some parts, that is, the synchronous vibration due to the mass unbalance, the free vibration at natural frequency of the derived linear system, and the angular motions caused by the flexible coupling at the combined frequencies.


The author acknowledges the financial support from the National Natural Science Foundation of China, China (Grant no. 11072190).


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