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

Abstract

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; 𝑘_𝑡 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 yield𝑥𝑦₁=𝑥+𝑒₁cos(Ω𝑡),₁=𝑦𝑧+𝑒₁sin(Ω𝑡),₁𝑥=0,(2.1)₂𝑦=𝑥+𝑙sin𝜃cos(Ω𝑡)+𝑒₂cos𝜃cos(Ω𝑡+𝛾),₂=𝑦+𝑙sin𝜃𝑧sin(Ω𝑡)+𝑒₂cos𝜃sin(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 ℎ 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=112𝑚^𝑖𝑥̇^2𝑖+𝑦̇^2𝑖+𝑧̇^2𝑖,(2.3) and the potential energy is𝑈=²_𝑖=112𝑘^𝑖𝑥^2𝑖+𝑦^2𝑖+12𝑘^𝑡𝜃².(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𝜕𝜕𝑞𝑇_𝑗̇−𝜕𝜕𝑞𝑇_𝑗+𝜕𝑈𝜕𝑞_𝑗=0,𝑞_𝑗=𝑥𝑦𝜃^𝑇,(2.5) the motion equation in the generalized coordinates becomes 2𝑚𝑥̈++2𝑚𝑘𝑙𝑥𝜃̈c+os𝑘𝜃𝑙scions𝜃Ωc𝑡o−sΩ𝑚𝑡𝑙=𝜃̇²𝑚si𝑒nΩ𝜃₂cosΩ𝑡−2𝑚𝑙Ω𝜃̇cos𝜃sinΩ𝑡−𝑚𝑙Ω²sin𝜃cosΩ𝑡2𝑚𝑦̈+𝑚𝑙𝜃̈cos𝜃sinΩ𝑡−𝑚𝑙𝜃̇₂cosΩ𝑡,+2𝑘𝑦+𝑘𝑙sin𝜃sinΩ𝑡=𝑚si𝑒nΩ𝜃₂sinΩ𝑡+2𝑚𝑙Ω𝜃̇cos𝜃cosΩ𝑡−𝑚𝑙Ω²sin𝜃sinΩ𝑡𝑚𝑙₂sinΩ𝑡,𝜃̈++𝑘𝑚𝑙𝑙𝑦𝑥c̈cooss𝜃𝜃scinosΩΩ𝑡𝑡++𝑘𝑚𝑙₂𝑙𝑦̈cos𝜃sinΩ𝑡−𝑚𝑙²Ω²sin𝜃cos𝜃+𝑘𝑙𝑥cos𝜃cosΩ𝑡sin𝜃cos𝜃+𝑘_𝑡𝜃=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 as𝑥̈−1𝑚𝑙1+sin²𝜃𝑚𝑙²+𝑚𝜃̇²𝑙₂sin𝜃cosΩ𝑡+2𝑚𝑙²Ω𝜃̇cos𝜃sinΩ𝑡−𝑚𝑙²Ω𝜃̇cos³𝜃sinΩ𝑡+𝑚𝑒ΩΩ²₂sin³𝜃cosΩ𝑡−𝑘𝑙²sin³𝜃cosΩ𝑡−𝑘𝑙𝑥1+sin²𝜃+𝑘_𝑡𝜃cos𝜃cosΩ𝑡𝑦̈−1𝑙𝑐𝑜𝑠Ω𝑡=0,𝑚𝑙1+sin²𝜃𝑚𝑙²+𝑚𝜃𝑙̇²₂sin𝜃sinΩ𝑡−2𝑚𝑙²Ω𝜃̇cos𝜃cosΩ𝑡+𝑚𝑙²Ω𝜃̇cos³𝜃cosΩ𝑡+𝑚𝑒ΩΩ²₂sin³𝜃sinΩ𝑡−𝑘𝑙²sin³𝜃sinΩ𝑡−𝑘𝑙𝑦1+sin²𝜃+𝑘_𝑡𝜃cos𝜃sinΩ𝑡𝜃̈+1𝑙sinΩ𝑡=0,𝑚𝑙²1+sin²𝜃𝑚𝑙²+𝜃̇²𝑚s𝑒inΩ𝜃₂cos𝜃−𝑚𝑙²Ω²sin𝜃cos𝜃+𝑘𝑙²sin𝜃cos𝜃+2𝑘_𝑡𝜃𝑙cos𝜃=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, 𝜔_𝑡=𝑘_𝑡/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:𝑋^−11+sin²𝜃𝐿(𝜃^−𝜔)²₂₀sin𝜃cos𝜏+2𝐿𝜃^cos𝜃sin𝜏−𝐿𝜃^cos³𝜃sin𝜏−𝜔²⁰−1𝐿sin³𝜃cos𝜏𝑌𝑋1+𝑠𝑖𝑛²𝜃+𝜔^2𝑡𝐿𝜃𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜏+𝐸𝑐𝑜𝑠𝜏=0,_−11+sin²𝜃𝐿(𝜃^−𝜔)²sin𝜃sin𝜏−2𝐿𝜃^cos𝜃cos𝜏+𝐿𝜃^cos³𝜃cos𝜏₂₀𝜃−1𝐿sin³𝜃sin𝜏−𝜔²⁰𝑌1+sin²𝜃+𝜔^2𝑡𝐿𝜃cos𝜃sin𝜏+𝐸sin𝜏=0,_+11+sin²𝜃𝜃^²sin𝜃cos𝜃+𝜔²⁰−1sin𝜃cos𝜃+2𝜔^2𝑡𝜃+𝐸𝐿cos𝜃=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 𝜔_𝑡 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 obtained𝑋^−11+sin²𝛼𝐿(𝜑^−𝜔₂₀)²sin𝛼cos𝜏+𝐿𝜑^1+sin²𝛼cos𝛼sin𝜏−𝜔²⁰−1𝐿sin³𝛼cos𝜏−𝜑1+sin²𝛼𝑋+𝜔^2𝑡𝐿𝛼cos𝛼cos𝜏+𝐸cos𝜏1+sin²𝛼𝐿(𝜑^−𝜔_2𝑡)²cos𝛼cos𝜏+𝐿𝜑^3cos²𝛼−2sin𝛼sin𝜏−2𝜔²⁰𝑋cos𝛼sin𝛼+𝜑sin2𝛼𝐿(𝛼sin𝛼−cos𝛼)cos𝜏+3𝐿1−𝜔²⁰sin²𝛼cos𝛼cos𝜏1+sin²𝛼²−𝐿𝜔(𝜑^₂₀)²sin𝛼cos𝜏+𝐿𝜑^1+sin²𝛼cos𝛼sin𝜏+𝜔^2𝑡𝛼𝐿cos𝛼cos𝜏𝑌1+sin²𝛼𝑋+𝐿1−𝜔²⁰sin³𝛼cos𝜏+𝐸cos𝜏=0,_−11+sin²𝛼𝐿(𝜑^−𝜔₂₀)²sin𝛼sin𝜏−𝐿𝜑^1+sin²𝛼cos𝛼cos𝜏−𝜔²⁰−1𝐿sin³𝛼sin𝜏−𝜑1+sin²𝛼𝑌+𝜔^2𝑡𝐿𝛼cos𝛼sin𝜏+𝐸sin𝜏1+sin²𝛼𝐿(𝜑^−𝜔_2𝑡)²cos𝛼sin𝜏+𝐿𝜑^2−3cos²𝛼sin𝛼cos𝜏−2𝜔²⁰𝑌cos𝛼sin𝛼+𝜑sin2𝛼𝐿(𝛼sin𝛼−cos𝛼)sin𝜏+3𝐿1−𝜔²⁰sin²𝛼cos𝛼sin𝜏1+sin²𝛼²−𝐿𝜔(𝜑^₂₀)²sin𝛼sin𝜏−𝐿𝜑^1+sin²𝛼cos𝛼cos𝜏+𝜔^2𝑡𝛼𝐿cos𝛼sin𝜏𝜑1+sin²𝛼𝑌+𝐿1−𝜔²⁰sin³𝛼sin𝜏+𝐸sin𝜏=0,_+11+sin²𝛼𝜑^+𝜑²sin𝛼cos𝛼+2𝜔^2𝑡𝛼+𝜔²⁰−1sin𝛼cos𝛼+𝐸𝐿cos𝛼1+sin²𝛼(𝜑^)²cos2𝛼+2𝜔^2𝑡+𝜔²⁰−1cos2𝛼−𝐸−𝜑sin2𝛼𝐿sin𝛼1+sin²𝛼²(𝜑+^)𝐸²sin𝛼cos𝛼+2𝜔^2𝑡𝛼+𝜔²⁰−1sin𝛼cos𝛼𝐿cos𝛼=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 to𝑋𝑌^−𝐿(𝜑^)²𝜑cos𝜏−𝐿𝜑^sin𝜏+𝜔²⁰𝑋−𝜔^2𝑡𝐿𝜑cos𝜏=𝐸cos𝜏,_−𝐿(𝜑𝜑^)²𝜑sin𝜏+𝐿𝜑^cos𝜏+𝜔²⁰𝑌−𝜔^2𝑡𝐿𝜑sin𝜏=𝐸sin𝜏,_+(𝜑^)²𝜑+𝜔²⁰+2𝜔^2𝑡−1𝜑+𝐸𝐿=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+2𝜔^2𝑡−1.

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)𝜔==𝜀𝜙𝜛₁₀(𝑇)+𝜀³𝜙₃(𝑇),(3.3)+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 yields𝜛𝜛²⁰𝜙^1∗∗+𝜙₁=0,(3.5)₂₀𝜙^3∗∗+𝜙₃=−2𝜛₀𝜔₂𝜙^1∗∗−𝜛²⁰𝜙^∗1²𝜙₁.(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) becomes𝜙^3∗∗+𝜙₃=𝜔²𝜛₀−14𝑎²𝑎cos(𝑇+𝛽)+14𝑎³cos(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 to𝜙₃=−𝑎³32cos(3𝑇+3𝛽).(3.9) From the equation 𝜙=𝜑+𝐸/𝐿𝜛², it follows that𝜑=−𝐸𝐿𝜛²⁰+𝜀𝑎cos𝜓−𝜀³𝑎³32cos3𝜓,(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 obtained𝜀𝑎cos𝛽−𝜀³𝑎³32cos3𝛽=𝐿𝐸𝜛²⁰sin𝛽1+3,32𝜀²𝑎²−38𝜀²𝑎²cos²𝛽=0.(3.11) When 𝛽=0, 𝜀𝑎 yields𝜀𝑎−𝜀³𝑎³32=𝐿𝐸𝜛²⁰.(3.12) In practice, 𝐸/𝐿𝜛²⁰ is small, thus the solution becomes 𝛽=0,𝜀𝑎≈𝐸/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.

(a) 𝜔0=𝟏.𝟐

(b) 𝜔𝑡=𝟎.𝟎𝟐

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,𝜀𝑎≈𝐸/𝐿𝜛²is only satisfied for small oscillation in engineering, that is, the systematic parameters are considered as 𝜔_𝑜=1.2, 𝜔_𝑡=0.02, 𝐿=20.0, and 𝐸=0.05, accordingly𝜛₀=𝜔²⁰+2𝜔^2𝑡−1=0.664, and after solving (3.11) on Maple, the six real roots of (3.12) are𝛽=0,𝜀𝑎=0.00567≈𝐸𝐿𝜛²𝛽=𝜋𝛽,𝛽==0𝜀0,𝑎,=−𝜀𝜀𝑎0𝑎.=0=0−5556.6.7655≈49062−9,,𝐸,𝐿𝜛²𝛽𝛽==𝜋𝜋,,𝜀𝜀𝑎𝑎==−55.6.655946092,.,(3.13)

The proves that𝛽=0,𝜀𝑎≈𝐸/𝐿𝜛² is an available approximate solution, 𝛽=0, and𝜀𝑎=−5.696,5.65402 are not almost changed with increasing the parameters 𝜔_𝑜 and 𝜔_𝑡, which are not suitable for the engineering. Thereby, the former will be paid more attention to the following analysis, then expression (3.10) leads to𝜑=𝐸𝐿𝜛²⁰(cos(𝜔𝜏)−1)−132𝐿𝐸𝜛²⁰³cos(3𝜔𝜏).(3.14) Substituting expression (3.14) into (3.1), it yields𝑋^+𝜔²⁰𝑋=𝐸𝜔²⁰+𝜔^2𝑡−1𝜛²⁰cos(𝜏)+𝐸𝜛²⁰𝑌_𝜔^2𝑡cos(𝜔𝜏)cos(𝜏)−𝜔sin(𝜔𝜏)sin(𝜏)+𝑂𝐸²,+𝜔²⁰𝑌=𝐸𝜔²⁰+𝜔^2𝑡−1𝜛²⁰sin𝜏+𝐸𝜛²⁰𝜔^2𝑡cos(𝜔𝜏)sin(𝜏)+𝜔sin(𝜔𝜏)cos(𝜏)+𝑂𝐸²,(3.15) and its general solution gives𝑋=𝐶₁cos𝜔₀𝜏+𝐶₂sin𝜔₀𝜏+𝐸𝜔²⁰+𝜔^2𝑡−1+𝐸𝜛²⁰𝜔²⁰−1cos𝜏2𝜛²⁰𝜔^2𝑡+𝜔𝜔²⁰−(𝜔+1)²cos(𝜔+1)𝜏+𝜔^2𝑡−𝜔𝜔²⁰−(𝜔−1)²𝑌=𝐶₃cos(𝜔−1)𝜏+𝑂𝐸²,cos𝜔₀𝜏+𝐶₄sin𝜔₀𝜏+𝐸𝜔²⁰+𝜔^2𝑡−1+𝐸𝜛²⁰𝜔²⁰−1sin𝜏2𝜛²⁰𝜔^2𝑡+𝜔𝜔²⁰−(𝜔+1)²sin(𝜔+1)𝜏−𝜔^2𝑡−𝜔𝜔²⁰−(𝜔−1)²sin(𝜔−1)𝜏+𝑂𝐸²,(3.16) where 𝜔=𝜔²⁰+2𝜔^2𝑡−1(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 to𝑋𝑌==−𝐷((𝐴𝜔+−𝐵1)+−𝐷𝐵)(c𝜔os+𝜔1₀)𝜏−+𝐴𝐴cos𝜏+𝐵cos(𝜔+1)𝜏+𝐷cos(𝜔−1)𝜏,𝜔₀sin𝜔₀𝜏+𝐴sin𝜏+𝐵sin(𝜔+1)𝜏−𝐷sin(𝜔−1)𝜏,(3.17) in which 𝐴=𝐸((𝜔²⁰+𝜔^2𝑡−1)/𝜛²⁰(𝜔²⁰−1)),𝐵=𝐸(𝜔^2𝑡+𝜔)/2𝜛²⁰(𝜔²⁰−(𝜔+1)²), and 𝐷=𝐸(𝜔^2𝑡−𝜔)/2𝜛²⁰(𝜔²⁰−(𝜔−1)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 𝜔−1and 𝜔+1 are combined by 𝜔 and the rotating speed. It is a typical nonlinear oscillation because the combination resonances exist at𝜔₀=𝜔−1and𝜔₀=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 ones𝑋𝜂′′==𝜂,1𝑌^=𝜉,𝜃^=𝜗,1+sin²𝜃𝐿𝜗²−𝜔sin𝜃cos𝜏+2𝐿𝜗cos𝜃sin𝜏−𝐿𝜗cos³𝜃sin𝜏₂₀𝜉′=1−1𝐿sin³𝜃cos𝜏−𝜔²⁰𝑋1+sin²𝜃+𝜔^2𝑡𝐿𝜃cos𝜃cos𝜏+𝐸cos𝜏,1+sin²𝜃𝐿𝜗²−𝜔sin𝜃sin𝜏−2𝐿𝜗cos𝜃cos𝜏+𝐿𝜗cos³𝜃cos𝜏₂₀𝜗−1𝐿sin³𝜃sin𝜏−𝜔²⁰𝑌1+sin²𝜃+𝜔^2𝑡𝐿𝜃cos𝜃sin𝜏+𝐸sin𝜏_,=−11+sin²𝜃𝜗²sin𝜃cos𝜃+𝜔²⁰−1sin𝜃cos𝜃+2𝜔^2𝑡𝜃+𝐸𝐿cos𝜃.(4.1)

For the following calculation, the values of systematic parameters are considered as 𝜔₀=1.20, 𝜔_𝑡=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=(𝜔−1)/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, 𝜔_𝑡=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.

(a)

(b)

Figure 3 

The nondimensional response and its frequency spectrum in 𝑥 direction when 𝐸=0.05: (a) response; (b) spectrum.

(a)

(b)

Figure 4 

The nondimensional response and its frequency spectrum in 𝑦 direction when𝐸=0.05: (a) response; (b) spectrum.

(a)

(b)

Figure 5 

The nondimensional response and its frequency spectrum in 𝑥 direction when𝐸=0.03: (a) response; (b) spectrum.

(a)

(b)

Figure 6 

The nondimensional response and its frequency spectrum in 𝑦 direction when𝐸=0.03: (a) response; (b) spectrum.

(a)

(b)

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.

(a)

(b)

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.

Acknowledgment

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