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Mathematical Problems in Engineering
VolumeΒ 2012, Article IDΒ 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: AlexΒ Elias-Zuniga

Copyright Β© 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 π‘š1 and π‘š2denote the lumped masses, respectively; π‘˜1 and π‘˜2 represent the stiffness of the bearings; π‘˜π‘‘ 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 π‘œπ‘₯𝑦𝑧 is set up in the static equilibrium position of rotor system as shown in Figure 1. Let 𝑂1(π‘₯1,𝑦1,0) and 𝑂2(π‘₯2,𝑦2,𝑧2) be the coordinates of the mass centers of disc 1 and disc 2, respectively, then they yieldπ‘₯1=π‘₯+𝑒1𝑦cos(Ω𝑑),1=𝑦+𝑒1𝑧sin(Ω𝑑),1π‘₯=0,(2.1)2=π‘₯+𝑙sinπœƒcos(Ω𝑑)+𝑒2𝑦cosπœƒcos(Ω𝑑+𝛾),2=𝑦+𝑙sinπœƒsin(Ω𝑑)+𝑒2𝑧cosπœƒsin(Ω𝑑+𝛾),2=𝑙+𝑙cosπœƒ,(2.2) where π‘₯ and 𝑦 are the displacements of disc 1 at the geometric center 𝐢1; πœƒ is the angle between adjacent rotors; Ξ© is the rotating speed; 𝛾 is the initial phase angle; 𝑙is the rotor length; 𝑒1, 𝑒2 are the mass unbalances. If the generalized coordinates π‘₯, 𝑦, and πœƒ are introduced, then (2.2) describes a nonstationary holonomic constraint because the displacement varies with time 𝑑. Generally, the offset β„Ž 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=2𝑖=112π‘šπ‘–ξ€·Μ‡π‘₯2𝑖+̇𝑦2𝑖+̇𝑧2𝑖,(2.3) and the potential energy isπ‘ˆ=2𝑖=112π‘˜π‘–ξ€·π‘₯2𝑖+𝑦2𝑖+12π‘˜π‘‘πœƒ2.(2.4) For simplifying the representation furthermore, let π‘š1=π‘š2=π‘š,π‘˜1=π‘˜2=π‘˜, and 𝑒1=𝑒, then substituting (2.1) and (2.2) into (2.3) and (2.4), and based on Lagrange’s equationπ‘‘ξ‚΅π‘‘π‘‘πœ•π‘‡πœ•Μ‡π‘žπ‘—ξ‚Άβˆ’πœ•π‘‡πœ•π‘žπ‘—+πœ•π‘ˆπœ•π‘žπ‘—=0,π‘žπ‘—=π‘₯π‘¦πœƒπ‘‡,(2.5) the motion equation in the generalized coordinates becomes ΜˆΜ‡πœƒ2π‘šΜˆπ‘₯+π‘šπ‘™πœƒcosπœƒcosΞ©π‘‘βˆ’π‘šπ‘™2Μ‡sinπœƒcosΞ©π‘‘βˆ’2π‘šπ‘™Ξ©πœƒcosπœƒsinΞ©π‘‘βˆ’π‘šπ‘™Ξ©2sinπœƒcosΩ𝑑+2π‘˜π‘₯+π‘˜π‘™sinπœƒcosΩ𝑑=π‘šπ‘’Ξ©2ΜˆΜ‡πœƒcosΩ𝑑,2π‘šΜˆπ‘¦+π‘šπ‘™πœƒcosπœƒsinΞ©π‘‘βˆ’π‘šπ‘™2Μ‡sinπœƒsinΩ𝑑+2π‘šπ‘™Ξ©πœƒcosπœƒcosΞ©π‘‘βˆ’π‘šπ‘™Ξ©2sinπœƒsinΩ𝑑+2π‘˜π‘¦+π‘˜π‘™sinπœƒsinΩ𝑑=π‘šπ‘’Ξ©2sinΩ𝑑,π‘šπ‘™2Μˆπœƒ+π‘šπ‘™Μˆπ‘₯cosπœƒcosΩ𝑑+π‘šπ‘™Μˆπ‘¦cosπœƒsinΞ©π‘‘βˆ’π‘šπ‘™2Ξ©2sinπœƒcosπœƒ+π‘˜π‘™π‘₯cosπœƒcosΩ𝑑+π‘˜π‘™π‘¦cosπœƒsinΩ𝑑+π‘˜π‘™2sinπœƒcosπœƒ+π‘˜π‘‘πœƒ=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 as1̈π‘₯βˆ’ξ€·π‘šπ‘™1+sin2πœƒξ€Έξ‚ƒπ‘šπ‘™2Μ‡πœƒ2sinπœƒcosΩ𝑑+2π‘šπ‘™2Ξ©Μ‡πœƒcosπœƒsinΞ©π‘‘βˆ’π‘šπ‘™2Ξ©Μ‡πœƒcos3πœƒsinΩ𝑑+π‘šπ‘™2Ξ©2sin3πœƒcosΞ©π‘‘βˆ’π‘˜π‘™2sin3ξ€·πœƒcosΞ©π‘‘βˆ’π‘˜π‘™π‘₯1+sin2πœƒξ€Έ+π‘˜π‘‘πœƒcosπœƒcosΩ𝑑+π‘šπ‘’Ξ©2ξ‚„1π‘™π‘π‘œπ‘ Ξ©π‘‘=0,Μˆπ‘¦βˆ’ξ€·π‘šπ‘™1+sin2πœƒξ€Έξ‚ƒπ‘šπ‘™2Μ‡πœƒ2sinπœƒsinΞ©π‘‘βˆ’2π‘šπ‘™2Ξ©Μ‡πœƒcosπœƒcosΩ𝑑+π‘šπ‘™2Ξ©Μ‡πœƒcos3πœƒcosΩ𝑑+π‘šπ‘™2Ξ©2sin3πœƒsinΞ©π‘‘βˆ’π‘˜π‘™2sin3ξ€·πœƒsinΞ©π‘‘βˆ’π‘˜π‘™π‘¦1+sin2πœƒξ€Έ+π‘˜π‘‘πœƒcosπœƒsinΩ𝑑+π‘šπ‘’Ξ©2ξ‚„Μˆ1𝑙sinΩ𝑑=0,πœƒ+π‘šπ‘™2ξ€·1+sin2πœƒξ€Έξ‚ƒπ‘šπ‘™2Μ‡πœƒ2sinπœƒcosπœƒβˆ’π‘šπ‘™2Ξ©2sinπœƒcosπœƒ+π‘˜π‘™2sinπœƒcosπœƒ+2π‘˜π‘‘πœƒ+π‘šπ‘’Ξ©2𝑙cosπœƒ=0.(2.7) Let 𝑋=π‘₯/π‘Ÿ, π‘Œ=𝑦/π‘Ÿ be the nondimensional displacements, in which π‘Ÿ is the radius of rotor; 𝐸=𝑒/π‘Ÿ, 𝐿=𝑙/π‘Ÿ are the nondimensional mass eccentricity and rotor length, respectively; 𝜏=Ω𝑑 is the nondimensional time; πœ”0=βˆšπ‘˜/π‘šΞ©2, πœ”π‘‘=ξ”π‘˜π‘‘/π‘šπ‘™2Ξ©2 are the nondimensional angular frequencies and denote 𝑑π‘₯/𝑑𝑑=π‘₯ξ…ž(𝑑𝑋/π‘‘πœ)=π‘‹ξ…žβ€¦; accordingly, (2.7) can be cast into the following nondimensional form:π‘‹ξ…žξ…žβˆ’1ξ€·1+sin2πœƒξ€Έξ‚ƒπΏ(πœƒξ…ž)2sinπœƒcos𝜏+2πΏπœƒξ…žcosπœƒsinπœβˆ’πΏπœƒξ…žcos3ξ€·πœ”πœƒsinπœβˆ’20ξ€Έβˆ’1𝐿sin3πœƒcosπœβˆ’πœ”20𝑋1+𝑠𝑖𝑛2πœƒξ€Έ+πœ”2π‘‘ξ‚„π‘ŒπΏπœƒπ‘π‘œπ‘ πœƒπ‘π‘œπ‘ πœ+πΈπ‘π‘œπ‘ πœ=0,ξ…žξ…žβˆ’1ξ€·1+sin2πœƒξ€Έξ‚ƒπΏ(πœƒξ…ž)2sinπœƒsinπœβˆ’2πΏπœƒξ…žcosπœƒcos𝜏+πΏπœƒξ…žcos3βˆ’ξ€·πœ”πœƒcos𝜏20ξ€Έβˆ’1𝐿sin3πœƒsinπœβˆ’πœ”20π‘Œξ€·1+sin2πœƒξ€Έ+πœ”2π‘‘ξ‚„πœƒπΏπœƒcosπœƒsin𝜏+𝐸sin𝜏=0,ξ…žξ…ž+1ξ€·1+sin2πœƒξ€Έξ‚ƒξ€·πœƒξ…žξ€Έ2ξ€·πœ”sinπœƒcosπœƒ+20ξ€Έβˆ’1sinπœƒcosπœƒ+2πœ”2π‘‘πΈπœƒ+𝐿cosπœƒ=0.(2.8)

Equation (2.8) is a parametrically excited system with three degrees of freedom on the theory of nonlinear vibration, in which πœ”0 and πœ”π‘‘ are the frequency ratios, respectively; Ξ΅is 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 ΞΈ 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 πœƒ=𝛼+πœ‘(𝑑) in which 𝛼 is the constant angle and πœ‘(𝑑) is its perturbation. After substituting πœƒ 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 obtainedπ‘‹ξ…žξ…žβˆ’1ξ€·1+sin2𝛼𝐿(πœ‘ξ…ž)2sin𝛼cos𝜏+πΏπœ‘ξ…žξ€·1+sin2π›Όξ€Έξ€·πœ”cos𝛼sinπœβˆ’20ξ€Έβˆ’1𝐿sin3𝛼cosπœβˆ’πœ”20ξ€·1+sin2𝛼𝑋+πœ”2π‘‘ξ‚„βˆ’πœ‘πΏπ›Όcos𝛼cos𝜏+𝐸cosπœξ€·1+sin2𝛼𝐿(πœ‘ξ…ž)2cos𝛼cos𝜏+πΏπœ‘ξ…žξ€·3cos2ξ€Έπ›Όβˆ’2sin𝛼sinπœβˆ’2πœ”20𝑋cos𝛼sinπ›Όβˆ’πœ”2𝑑𝐿(𝛼sinπ›Όβˆ’cos𝛼)cos𝜏+3𝐿1βˆ’πœ”20ξ€Έsin2ξ‚„+𝛼cos𝛼cosπœπœ‘sin2𝛼1+sin2𝛼2𝐿(πœ‘ξ…ž)2sin𝛼cos𝜏+πΏπœ‘ξ…žξ€·1+sin2𝛼cos𝛼sin𝜏+πœ”2𝑑𝛼𝐿cos𝛼cosπœβˆ’πœ”20ξ€·1+sin2𝛼𝑋+𝐿1βˆ’πœ”20ξ€Έsin3ξ‚„π‘Œπ›Όcos𝜏+𝐸cos𝜏=0,ξ…žξ…žβˆ’1ξ€·1+sin2𝛼𝐿(πœ‘ξ…ž)2sin𝛼sinπœβˆ’πΏπœ‘ξ…žξ€·1+sin2π›Όξ€Έξ€·πœ”cos𝛼cosπœβˆ’20ξ€Έβˆ’1𝐿sin3𝛼sinπœβˆ’πœ”20ξ€·1+sin2π›Όξ€Έπ‘Œ+πœ”2π‘‘ξ‚„βˆ’πœ‘πΏπ›Όcos𝛼sin𝜏+𝐸sinπœξ€·1+sin2𝛼𝐿(πœ‘ξ…ž)2cos𝛼sin𝜏+πΏπœ‘ξ…žξ€·2βˆ’3cos2𝛼sin𝛼cosπœβˆ’2πœ”20π‘Œcos𝛼sinπ›Όβˆ’πœ”2𝑑𝐿(𝛼sinπ›Όβˆ’cos𝛼)sin𝜏+3𝐿1βˆ’πœ”20ξ€Έsin2ξ‚„+𝛼cos𝛼sinπœπœ‘sin2𝛼1+sin2𝛼2𝐿(πœ‘ξ…ž)2sin𝛼sinπœβˆ’πΏπœ‘ξ…žξ€·1+sin2𝛼cos𝛼cos𝜏+πœ”2𝑑𝛼𝐿cos𝛼sinπœβˆ’πœ”20ξ€·1+sin2π›Όξ€Έξ€·π‘Œ+𝐿1βˆ’πœ”20ξ€Έsin3ξ‚„πœ‘π›Όsin𝜏+𝐸sin𝜏=0,ξ…žξ…ž+1ξ€·1+sin2π›Όξ€Έξ‚ƒξ€·πœ‘ξ…žξ€Έ2sin𝛼cos𝛼+2πœ”2π‘‘ξ€·πœ”π›Ό+20ξ€ΈπΈβˆ’1sin𝛼cos𝛼+𝐿+πœ‘cos𝛼1+sin2𝛼(πœ‘ξ…ž)2cos2𝛼+2πœ”2𝑑+ξ€·πœ”20ξ€ΈπΈβˆ’1cos2π›Όβˆ’πΏβˆ’sinπ›Όπœ‘sin2𝛼1+sin2𝛼2(πœ‘ξ…ž)2sin𝛼cos𝛼+2πœ”2π‘‘ξ€·πœ”π›Ό+20ξ€Έ+πΈβˆ’1sin𝛼cos𝛼𝐿cos𝛼=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 toπ‘‹ξ…žξ…žβˆ’πΏ(πœ‘ξ…ž)2πœ‘cosπœβˆ’πΏπœ‘ξ…žsin𝜏+πœ”20π‘‹βˆ’πœ”2π‘‘π‘ŒπΏπœ‘cos𝜏=𝐸cos𝜏,ξ…žξ…žβˆ’πΏ(πœ‘ξ…ž)2πœ‘sin𝜏+πΏπœ‘ξ…žcos𝜏+πœ”20π‘Œβˆ’πœ”2π‘‘πœ‘πΏπœ‘sin𝜏=𝐸sin𝜏,ξ…žξ…ž+(πœ‘ξ…ž)2ξ€·πœ”πœ‘+20+2πœ”2π‘‘ξ€ΈπΈβˆ’1πœ‘+𝐿=0.(3.1) The third of (3.1) is uncoupled with the first two; therefore, it can be simplified asπœ™ξ…žξ…ž+(πœ™ξ…ž)2πœ™+πœ›20πœ™=0,(3.2) where πœ™=πœ‘+𝐸/πΏπœ›2, πœ›20=πœ”20+2πœ”2π‘‘βˆ’1.

The method of multiple scales (MMSs) is introduced for the approximation solution of the nonlinear vibration in this work. For the small angle πœ™, let 𝑇=πœ”πœ and expand πœ™ and πœ” [15] asπœ™(𝑇,πœ€)=πœ€πœ™1(𝑇)+πœ€3πœ™3(𝑇),(3.3)πœ”=πœ›0+πœ€2πœ”2,(3.4) where Ξ΅ stands for a small nondimensional parameter characterizing the amplitude of the motion. The term πœ€2πœ™2(𝜏) is missing from (3.3) because the nonlinearity appears at 𝑂(πœ€3). The term πœ€πœ”1 is absent from (3.4) because the frequency is independent of the sign of Ξ΅. Substituting (3.3) and (3.4) into (3.2) and equating coefficients of like powers of Ξ΅, after donating π‘‘πœ™/𝑑𝑇=πœ™βˆ—, it yieldsπœ›20ξ€·πœ™1βˆ—βˆ—+πœ™1ξ€Έπœ›=0,(3.5)20ξ€·πœ™3βˆ—βˆ—+πœ™3ξ€Έ=βˆ’2πœ›0πœ”2πœ™1βˆ—βˆ—βˆ’πœ›20ξ€·πœ™βˆ—1ξ€Έ2πœ™1.(3.6) The solution of (3.5) is of the formπœ™1=π‘Žcos(𝑇+𝛽),(3.7) where π‘Ž and 𝛽 are constants; hence, (3.6) becomesπœ™3βˆ—βˆ—+πœ™3=ξ‚΅πœ”2πœ›0βˆ’14π‘Ž2ξ‚Ά1π‘Žcos(𝑇+𝛽)+4π‘Ž3cos(3𝑇+3𝛽).(3.8) Eliminating the secular term in (3.8) gives πœ”2=(1/4)πœ›0π‘Ž2, then disregarding the solution of the homogeneous equation, its solution reduces toπœ™3π‘Ž=βˆ’332cos(3𝑇+3𝛽).(3.9) From the equation πœ™=πœ‘+𝐸/πΏπœ›2, it follows thatπΈπœ‘=βˆ’πΏπœ›20πœ€+πœ€π‘Žcosπœ“βˆ’3π‘Ž332cos3πœ“,(3.10) in which πœ“=πœ”πœ+𝛽, πœ”=πœ›0(1+πœ€2π‘Ž2/4). If imposing the initial conditions πœ‘|𝜏=0=0and πœ‘βˆ—|𝜏=0=0, then the constants πœ€π‘Ž and 𝛽 are obtainedπœ€πœ€π‘Žcosπ›½βˆ’3π‘Ž3𝐸32cos3𝛽=πΏπœ›20,ξ‚€3sin𝛽1+πœ€322π‘Ž2βˆ’38πœ€2π‘Ž2cos2𝛽=0.(3.11) When 𝛽=0, πœ€π‘Ž yieldsπœ€πœ€π‘Žβˆ’3π‘Ž3=𝐸32πΏπœ›20.(3.12) In practice, 𝐸/πΏπœ›20 is small, thus the solution becomes 𝛽=0,πœ€π‘Žβ‰ˆπΈ/πΏπœ›20. 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 forπœ€π‘Ž and 𝐸/πΏπœ›2 with increasing πœ”0 or πœ”π‘‘.

Generally for the above nonlinear equations, there are six real solutions, in which the root 𝛽=0,πœ€π‘Žβ‰ˆπΈ/πΏπœ›2is only satisfied for small oscillation in engineering, that is, the systematic parameters are considered as πœ”π‘œ=1.2, πœ”π‘‘=0.02, 𝐿=20.0, and 𝐸=0.05, accordinglyπœ›0=ξ”πœ”20+2πœ”2π‘‘βˆ’1=0.664, and after solving (3.11) on Maple, the six real roots of (3.12) are𝐸𝛽=0,πœ€π‘Ž=0.00567β‰ˆπΏπœ›2,𝐸𝛽=0,πœ€π‘Ž=βˆ’5.65969,𝛽=0,πœ€π‘Ž=5.65402,𝛽=πœ‹,πœ€π‘Ž=βˆ’0.00567β‰ˆβˆ’πΏπœ›2,𝛽=πœ‹,πœ€π‘Ž=5.65969,𝛽=πœ‹,πœ€π‘Ž=βˆ’5.65402.(3.13)

The proves that𝛽=0,πœ€π‘Žβ‰ˆπΈ/πΏπœ›2 is an available approximate solution, 𝛽=0, andπœ€π‘Ž=βˆ’5.696,5.65402 are not almost changed with increasing the parameters πœ”π‘œ and πœ”π‘‘, which are not suitable for the engineering. Thereby, the former will be paid more attention to the following analysis, then expression (3.10) leads toπΈπœ‘=πΏπœ›201(cos(πœ”πœ)βˆ’1)βˆ’ξƒ©πΈ32πΏπœ›20ξƒͺ3cos(3πœ”πœ).(3.14) Substituting expression (3.14) into (3.1), it yieldsπ‘‹ξ…žξ…ž+πœ”20πœ”π‘‹=𝐸20+πœ”2π‘‘βˆ’1πœ›20𝐸cos(𝜏)+πœ›20ξ€·πœ”2𝑑𝐸cos(πœ”πœ)cos(𝜏)βˆ’πœ”sin(πœ”πœ)sin(𝜏)+𝑂2ξ€Έ,π‘Œξ…žξ…ž+πœ”20πœ”π‘Œ=𝐸20+πœ”2π‘‘βˆ’1πœ›20𝐸sin𝜏+πœ›20ξ€·πœ”2𝑑𝐸cos(πœ”πœ)sin(𝜏)+πœ”sin(πœ”πœ)cos(𝜏)+𝑂2ξ€Έ,(3.15) and its general solution gives𝑋=𝐢1cosπœ”0𝜏+𝐢2sinπœ”0πœ”πœ+𝐸20+πœ”2π‘‘βˆ’1πœ›20ξ€·πœ”20ξ€Έ+πΈβˆ’1cos𝜏2πœ›20ξƒ©πœ”2𝑑+πœ”πœ”20βˆ’(πœ”+1)2πœ”cos(πœ”+1)𝜏+2π‘‘βˆ’πœ”πœ”20βˆ’(πœ”βˆ’1)2ξƒͺ𝐸cos(πœ”βˆ’1)𝜏+𝑂2ξ€Έ,π‘Œ=𝐢3cosπœ”0𝜏+𝐢4sinπœ”0πœ”πœ+𝐸20+πœ”2π‘‘βˆ’1πœ›20ξ€·πœ”20ξ€Έ+πΈβˆ’1sin𝜏2πœ›20ξƒ©πœ”2𝑑+πœ”πœ”20βˆ’(πœ”+1)2πœ”sin(πœ”+1)πœβˆ’2π‘‘βˆ’πœ”πœ”20βˆ’(πœ”βˆ’1)2ξƒͺ𝐸sin(πœ”βˆ’1)𝜏+𝑂2ξ€Έ,(3.16) where ξ”πœ”=πœ”20+2πœ”2π‘‘βˆ’1(1+(𝐸/2πΏπœ›2)2), and 𝐢1,𝐢2,𝐢3,and𝐢4are the constants.

If the initial conditions𝑋(0)=π‘‹ξ…ž(0)=0 and π‘Œ(0)=π‘Œξ…ž(0)=0 are introduced, the constants 𝐢1,𝐢2,𝐢3, and 𝐢4 can be determined easily; therefore, the responses can be shown to reduce to𝑋=βˆ’(𝐴+𝐡+𝐷)cosπœ”0𝜏+𝐴cos𝜏+𝐡cos(πœ”+1)𝜏+𝐷cos(πœ”βˆ’1)𝜏,π‘Œ=𝐷(πœ”βˆ’1)βˆ’π΅(πœ”+1)βˆ’π΄πœ”0sinπœ”0𝜏+𝐴sin𝜏+𝐡sin(πœ”+1)πœβˆ’π·sin(πœ”βˆ’1)𝜏,(3.17) in which 𝐴=𝐸((πœ”20+πœ”2π‘‘βˆ’1)/πœ›20(πœ”20βˆ’1)),𝐡=𝐸(πœ”2𝑑+πœ”)/2πœ›20(πœ”20βˆ’(πœ”+1)2), and 𝐷=𝐸(πœ”2π‘‘βˆ’πœ”)/2πœ›20(πœ”20βˆ’(πœ”βˆ’1)2).

The above solution shows that the response consists of the following components: (a) the free vibration at natural frequency πœ”0 of the derived linear system. If the damping is concerned in π‘₯ 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 πœ”0=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 πœ”βˆ’1and πœ”+1 are combined by πœ” and the rotating speed. It is a typical nonlinear oscillation because the combination resonances exist atπœ”0=πœ”βˆ’1andπœ”0=πœ”+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′=πœ‚,π‘Œξ…ž=πœ‰,πœƒξ…ž1=πœ—,πœ‚β€²=ξ€·1+sin2πœƒξ€Έξ€ΊπΏπœ—2sinπœƒcos𝜏+2πΏπœ—cosπœƒsinπœβˆ’πΏπœ—cos3βˆ’ξ€·πœ”πœƒsin𝜏20ξ€Έβˆ’1𝐿sin3πœƒcosπœβˆ’πœ”20𝑋1+sin2πœƒξ€Έ+πœ”2𝑑,1πΏπœƒcosπœƒcos𝜏+𝐸cosπœπœ‰β€²=ξ€·1+sin2πœƒξ€Έξ€ΊπΏπœ—2sinπœƒsinπœβˆ’2πΏπœ—cosπœƒcos𝜏+πΏπœ—cos3βˆ’ξ€·πœ”πœƒcos𝜏20ξ€Έβˆ’1𝐿sin3πœƒsinπœβˆ’πœ”20π‘Œξ€·1+sin2πœƒξ€Έ+πœ”2𝑑,πœ—πΏπœƒcosπœƒsin𝜏+𝐸sinπœξ…ž1=βˆ’ξ€·1+sin2πœƒξ€Έξ‚ƒπœ—2ξ€·πœ”sinπœƒcosπœƒ+20ξ€Έβˆ’1sinπœƒcosπœƒ+2πœ”2π‘‘πΈπœƒ+𝐿.cosπœƒ(4.1)

For the following calculation, the values of systematic parameters are considered as πœ”0=1.20, πœ”π‘‘=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 π‘₯ 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=(πœ”βˆ’1)/2πœ‹=0.05349,𝑓2=1/2πœ‹=0.15915, 𝑓3=πœ”0/2πœ‹=0.19099, and 𝑓4=(πœ”+1)/2πœ‹=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 πœ”/2πœ‹ and the rotating speed Ξ©, 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 πœ”0=1.20, πœ”π‘‘=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 π‘₯ 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 π‘₯ 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 π‘₯ direction when πœ”0=1.20, πœ”π‘‘=0.50, and 𝐸=0.05: (a) response; (b) spectrum.
Figure 8: The nondimensional response and its frequency spectrum in 𝑦 direction when πœ”0=1.20, πœ”π‘‘=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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