（School of Astronautics，Harbin Institute of Technology，Harbin 150001，China）

Analysis on Nonlinear Dynamic Properties of Dual-Rotor System

Hongliang Li?and Yushu Chen

（School of Astronautics，Harbin Institute of Technology，Harbin 150001，China）

In order to clarify the effects of support structure on a dual-rotor machine，a dynamic model is established which takes into consideration the contact force of ball bearing and the cubic stiffness of elastic support.Bearing clearance，Hertz contact between the ball and race and the varying compliance effect are included in the model of ball bearing.The system response is obtained through numerical integration method，and the vibration due to the periodic change of bearing stiffness is investigated.The motions of periodic，quasiperiodic and even chaotic are found when bearing clearance is used as control parameter to simulate the response of rotor system.The results reveal two typical routes to chaos：quasi-periodic bifurcation and intermittent bifurcation.Large cubic stiffness of elastic support may cause jump and hysteresis phenomena in resonance curve when rotors run at the critical-speed region.The modeling results acquired by numerical simulation will contribute to understanding and controlling of the nonlinear behaviors of the dual-rotor system.

dual-rotor system；ball bearing；elastic support；nonlinear dynamics；bifurcation；chaos

1 Introduction

As one of main factors affecting flight safety of aircraft，the vibration of aeroengine has become a research focus in the power engineering field.Due to the complicated mechanical structures and working conditions，many vibration sources exist in the aeroengine，such as rotor imbalance，misalignment supporting，airflow excitation，rolling bearing excitation and so on.Among all these sources，the rotor system contributes most to the whole-engine vibration which is comprised of rotors，rolling bearings and elastic supports.Therefore，the dynamic analysis of dual-rotor system plays an important role in enhancing the stability and reliability of aeroengines.

Since dual-rotor system is very common in aeroengine，much attention has been paid to its dynamics.Hibner［1］put forward the application of transfer matrix method to an equivalent engine system to compute the critical speed and nonlinearly damped response.Glasgow［2］analyzed the stability of a dual rotor system with component mode synthesis technique and achieved a significant reduction in the size of the problem.Gunter［3］studied the dynamics of a two-spool aircraft engine with squeeze film dampers.Yan［4］proposed a sub-structure transfer matrix procedure for the dynamic analysis of complex multiple-rotor systems. Huang［5］developed an impedance coupling procedure for predicting the frequency response of a composite rotor system with interconnections.

Besides，theoretical and experimental studies were also performed on the dynamics of counter-rotating rotors.Gupta［6-7］carried out an experiment on the dynamic response of a counter-rotating dual-rotor machine with an intershaft rolling bearing.An extended transfer matrix process with complex variable was theoretically formulated as well.Ferraris［8-9］presented a finite element study of a twin-spindle aircraft engine whose rotors spin in the same or opposite direction.The eigenmodes and mass unbalance response were investigated in their research.

As nonlinear problem continuously appeared in engineering，more and more attention is being attracted to nonlinear dynamics of dual-rotor system.Han［10］introduced a rigid dual-rotor model and analyzed the nonlinear vibrations in the system when external force，bearing clearance and rotating speed were used as control parameters to simulate the system response. Deng［11］discussed the influences of bearing parameters on the motion stability of a dual-rotor unit supported on rolling bearings.Hu［12］developed a five degree-offreedom bearing model for an aeroengine spindle dualrotor system analysis.However，the rotor unbalance was neglected in these works［10-12］.Luo［13］studied the nonlinear characteristics of an unbalanced dual-rotor test rig mounted on rolling bearings but without elastic supports.Yuan［14］set up a new model of a dual rotor-stator coupling system with rub-impact and numerical results showed different motion styles such as periodic，quasi-periodic and even chaotic motions.Chen［15］investigated the complex dynamic behaviors of a counter-rotating dual-rotor system when the two rotors rubbed with each other.Feng［16］conducted a misalignment analysis for support bearings in an aeroengine system.But the rolling bearing was simplified into linear spring and damping，and the nonlinear nature of bearing force was ignored in the researches［14-16］.

Based on the above studies，a dynamic model is established for an unbalanced dual-rotor system in this paper.Bearing clearance，Hertz contact between the ball and race and the varying compliance vibration are considered in both support and intershaft bearing model.The elastic support is simplified into a nonlinear spring with cubic stiffness.The obtained results are discussed via the vibration response，power spectrums，rotor axis orbits，bifurcation diagrams and Poincare maps，and some observations on the system nonlinear dynamic behaviors are extracted.

2 Physical Model and Governing Equation

The physical model includes a single-disk inner rotor and a hollow-shaft outer rotor that are connected by an intershaft ball bearing as shown in Fig.1.The inner rotor is supported on bearings at both ends but the outer rotor is supported at the left end.All of the support bearings are mounted into the pedestals with elastic supports.These rotors rotate in the same direction，and the speed of the outer rotor is usually higher than that of the inner rotor in practical work.

Fig.1 Schematic of the dual-rotor system

2.1 Reaction Force of Elastic Support

Elastic supports are widely used to adjust the critical speeds of dual-rotor system without modifying the rigidity of rotor.When rotors run through the loworder critical-speed region，elastic supports afford almost all the deformation and the bending deflection of rotor is rather small.Thus，the strain energy distribution of the system is improved，and the unbalanced vibration can be suppressed by the internal material damping of elastic support as well.

Squirrel cage［17］and rubber ring［18］are common supports whose deformation resistivity can be treated equivalent in each radial direction because of the axisymmetric structure.Therefore，the elastic support is modeled as an isotropic spring with nonlinear stiffness as shown in Fig.2.

Fig.2 Simplified model of elastic support

The reaction force of elastic support can be expressed as［19］

whereris the radial displacement of the bearing pedestal；kis the linear stiffness of the elastic support；αis the cubic stiffness of the elastic support；o（r3）is the high-order infinitesimal amount.

Sinceris much less than 1，o（r3）is omitted and the reaction force inx-ycoordinates can be written as

wherexandyrepresent the displacements of the bearing pedestal.

2.2 Contact Force of Ball Bearing

As shown in Fig.3，the bearing considered has equispaced elements rolling on the surfaces of the inner and outer races.It is assumed to be perfect rolling of elements on the races so that two points of the element touching the inner and outer races have different linear velocities.

Fig.3 Model of ball bearing

whereviis the velocity of the point touching the inner race；vois the velocity of the point touching the outer race；riis the radius of the inner race；rois the radius of the outer race；ωiis the rotating angular velocity of the inner ring；ωois the rotating angular velocity of the outer ring.

The linear velocity of the cage can be expressed as

Thus，the angular velocity of the cage is

With the periodic change of contact positions between the elements and races，stiffness of bearing vary periodically in actual work which is a parametric excitation to the rotor system.Thus，the so-called varying compliance vibration is inherent and its angular frequency is given byωvc＝ωcage×Nb．whereNbrepresents the number of the elements.

For support bearing，its outer ring is fixed rigidly to the pedestal and its inner ring rotates with the shaft，thereforeωo＝0，ωi＝ωrotorand the VC frequency can be written as

whereBNdepends on the dimensions of the bearing.

The angle location of thejth element is assumed as

Thus，the normal contact deformation between thejth element and the races is given by

wherexandyare the displacements of inner ring center relative to outer ring；γ0is the bearing clearance.

Then，the restoring force generated from contact between thejth element and the races can be obtained

whereCbis the Hertz contact stiffness related to the shape and material of contact objects［20］；ndepends on the bearing types：for ball bearing，n＝3／2［21］，for roller bearing，n＝10／9［22］，+indicates that only whenδj＞0，thejth element is loaded giving rise to a restoring forceFj，otherwise，the restoring forceFjis set to zero.

The total restoring force is the sum of restoring force from each of rolling elements and its components in thexandydirections are

2.3 Dynamic Equation

Regardingxandyas the translational displacements alongx-axis andy-axis，θxandθyas the rotational displacements aroundx-axis andy-axis，the system dynamic equation can be written as

wheremIandmIIare masses of the inner and outer rotors；are the equivalent diametric moments of inertia of the inner and outer rotors；are the equivalent polar moments of inertia of the inner and outer rotors；m1，m2andm4represent the masses of bearing pedestal 1，2 and 4 respectively；c1，c2andc4represent the dampings of support 1，2 and 4 respectively；are the reaction force components in thexandydirections of support 1；are the reaction force components inxandydirections of support 2；are the reaction force components in thexandydirections of support 4；represent the contacting force components in thexandydirections of bearing 1；represent the contacting force components in thexandydirections of bearing 2；represent the contact force components in thexandydirections of bearing 3；mean the contact force components in thexandydirections of bearing 4；l1，l2，l3，l4andl5are the dimensions of the dual-rotor system as shown in Fig.1；eIandeIIare eccentricities of the inner and outer rotors；gmeans the acceleration of gravity；ωIandωIIare the rotating speeds of the inner and outer rotors；φIandφIIare the initial eccentric angles.

3 Simulation Results and Analysis

Because of the strong nonlinear characteristics of the system dynamic equation，the vibration response will be quite complicated.It is difficult to obtain the exact solutions analytically，therefore the fourth-order Runge-Kutta method is used to solve the equations.The initial parameters of system used for computation are as follows：

Table 1 shows the bearing parameters adopted in the simulation.

3.1 Varying Compliance Vibration of Ball Bearing

When the speeds of the two rotors are very low and both rotor eccentricities are sufficiently small，the unbalanced excitation is quite weak and the varying compliance vibration can be observed clearly because of the periodic variety of bearing stiffness.Figs.4 and 5 show the vibration response and axis orbits of the inner and outer rotors when they run atωI＝100 r／min，ωII＝120 r／min.Power spectrum is employed to detect the frequency components of the response.

Table 1 Parameters of ball bearing

Fig.4 Vibration response and axis orbit of the inner rotor

Fig.5 Vibration response and axis orbit of the outer rotor

The varying compliance frequency of bearing 1（bearing 2）and its harmonicsare detected in the vibration response of inner rotor，and a closed curve is observed for the axis orbit that mean the inner rotor is at a periodic motion as shown in Fig.4. But the VC frequencies of bearing 1（bearing 2），bearing 3 and bearing 4 are all determined in the outer rotor response，and the rotor axis orbit is nonclosed with complex net structure as illustrated in Fig.5.These results reveal that quasi-periodic motion emerges in the response due to mutual irreducibility of the VC frequencies

Fig.6 shows the three-dimensional spectrum waterfall plots of the inner and outer rotors using rotating speed as the control parameter with speed ratioare normalized frequencies， hereN1is the ratio of response frequency tois the ratio of response frequency toWith the increase of rotating speed，appear and increase gradually in the inner rotor response，but little change occurs in the outer rotor response.Therefore，the inner rotor will transit to quasi-periodic motion from original periodic vibration and the outer rotor remains in the quasi-periodic state.

The study of Han［10］reveals that when the dualrotor system is subjected to several bearing excitations，super harmonic components may appear in vibration response and the motion is nonperiodic due to mutual irreducibility of exciting frequencies.Apparently，the results of this paper accord with the conclusion.

Fig.6 Three-dimensional spectrum waterfall plots of the inner and outer rotors

3.2 Effect of Bearing Clearance on Unbalanced Vibration

Rotor unbalance is an unavoidable effect in engineering due to the manufacture error，installation error，thermal deformation and other factors.Even good balancing is made，and it cannot be completely eliminated.Particularly with the increase of rotating speed，the unbalanced excitation is significantly strengthened and the varying compliance vibration is so weak that it is submerged in system response.Thus，the unbalanced vibration plays a dominant role in highspeed rotor system.

In dual-rotor system，there are two unbalanced excitations from both the two rotors.It is assumed thatTIandTIIare rotor exciting periods，andTeqis the equivalent exciting period，that is the least common multiple ofTIandTII．Teqis used as the sample period to obtain Poincare map and bifurcation diagram of the system.

Fig.7 shows bifurcation diagrams of the system using bearing clearance as the control parameter when the rotors run atωI＝10 000 r／min，ωII＝12 000 r／min.As shown in Fig.7（a），a complete process from periodic，quasi-periodic state to complicated chaotic motion is illustrated with the increase of clearance of bearing 1.When the clearance is small and less than 2 μm，the system response is synchronous with periodic motion and its attractor is a single point in Poincare map as displayed in Fig.8（a）. As the clearance exceeds 2 μm，quasi-periodic motion appears and the points of the attractor are separated into a closed curve in Poincare map shown in Fig.8（b）.As the clearance continues to increase，the scattered points in Poincare map reveal that the system leaves quasi-periodic motion and evolves into chaotic motion as displayed in Figs.8（c）and 8（d）.

Fig.7 Bifurcation diagrams of the system with bearing clearance as the control parameter

As shown in Fig.7（b），the system response varies with clearance of bearing 2 and exhibits the process from periodic motion to chaotic motion through quasi-periodic bifurcation.When the clearance is small and less than 5 μm，the system is at periodic motion with one isolated point in Poincare map as shown in Fig.9（a）.As the clearance increases，quasi-periodic motion appears and a closed curve is observed in Poincare map as shown in Fig.9（b），but this state does not last for a long range of clearance.When the clearance reaches 20 μm，the system returns to periodic state and the points in Poincare map converge to one single point as shown in Fig.9（c）.Along with the increase of clearance，the closed form of the attractor suggests that the system enters quasi-periodic motion again and keeps it for a wide range of clearance until chaotic motion emerges in the system response as illustrated in Figs.9（d）-9（h）.

Actually，more intuitive understanding can be provided by the rotor axis orbit though its resolution for motion state is not as high as Poincare map.When the system response is periodic，the rotor axis orbits are discovered as regular closed curves as shown in Figs.10（a）and 11（a）.As the system enters quasi-periodic motion，the rotor axis orbits become nonclosed andexhibit typical warped surfaces as shown in Figs.10（b）and 11（b）.Then as the chaotic motion occurs in the system response，the rotor axis orbits get disordered as shown in Figs.10（d）and 11（d）.The rotor axis orbits in transition stage between quasi-periodic motion and chaos are displayed in Figs.10（c）and 11（c）.

Fig.8 Poincare maps of the system with different clearance of bearing 1

Fig.9 Poincare maps of the system with different clearance of bearing 2

Fig.10 Axis orbits of the inner rotor with different clearance of bearing 2

Fig.11 Axis orbits of the outer rotor with different clearance of bearing 2

As illustrated in Fig.12（a），the system undergoes a special route from periodic motion to chaos with the increase of clearance of bearing 3.When the clearance is equal to 2 μm，the system response is periodic and its attractor is one single point in Poincare map as shown in Fig.13（a）.As the clearance exceeds 3.5 μm，the system enters into chaos directly as displayed in Figs.13（b）-13（d），and this route to chaos is called intermittent bifurcation.Furthermore，the same way to chaos is found when clearance of bearing 4 is used as the control parameter to simulate system response as illustrated in Fig.12（b）.Poincare maps of the system with different clearance of bearing 4 are displayed in Fig.14.

Fig.12 Bifurcation diagrams of the system with bearing clearance as the control parameter

Fig.13 Poincare maps of the system with different clearance of bearing 3

Figs.15 and 16 show the rotor axis orbits with different clearance of bearing 3.With the increase of the clearance，the rotor axis orbits become more and more disordered，and it is proved that the system response changes from periodic motion to chaos directly through intermittent bifurcation.

The study of Tiwari et al.［23］confirms the quasiperiodic route to chaos in single rotor system with the increase of bearing clearance.Then，both quasiperiodic and intermittent bifurcation are discovered as the route to chaos in rotor system when bearing clearance is used as the control parameter to obtain the system response by Chen［24］.Thus，the results of this paper keep consistent with the former studies.

3.3 Effect of Cubic Support Stiffness on Resonance Curve

As is known，the dual-rotor system has two kinds of critical speeds excited by its two rotors separately［25］.Thus the resonance curve is studied in both cases，and a dimensionless parameter is introduced to evaluate the magnitude of cubic support stiffness.This parameter is defined asβ＝αe2／（k+αe2），wherekis the linear stiffness of the elastic support；αis the cubic stiffness of the elastic support；eis the rotor eccentricity.

Fig.17 shows resonance curves of the system excited by the inner rotor with rotor eccentricitye＝50 μm.As the cubic support stiffness increases，the resonance curve leans to the right side gradually and excessive incline causes jump and hysteresis phenomena［26］when rotors run through the criticalspeed zone.Besides，since the resonance curve leans to high-speed zone，the unbalanced excitation is strengthened in resonance region and the rotor vibration peak value is increased at the same time which is also a negative impact on the steady operation of rotor system.

Fig.18 shows resonance curves of the system excited by the outer rotor.Jump and hysteresis behavior is observed in the second order critical-speed zone，but not the first order critical-speed zone.It is indicated that large vibration amplitude of rotor can also aggravate this unstable behavior.

Fig.15 Axis orbits of the inner rotor with different clearance of bearing 3

Fig.16 Axis orbits of the outer rotor with different clearance of bearing 3

Fig.17 Resonance curves of the system excited by the inner rotor

Fig.18 Resonance curves of the system excited by the outer rotor

4 Conclusions

A dynamic model is established for an unbalanced dual-rotor system in this paper.The motion equation is solved by numerical integration method and the varying compliance vibration of ball bearing is analyzed through vibration response and power spectrum.It is observed that the motion is nonperiodic due to mutual irreducibility of VC frequencies.With the increase of bearing clearance，the system finally goes into chaos. But the route to chaos changes in different bearings. For support bearings of the inner rotor，quasi-periodic bifurcation is discovered as the way to chaos and the threshold value is relatively high.For support bearing of the outer rotor and intershaft bearing，the system response varies from periodic motion to chaos directly by intermittent bifurcation.Cubic stiffness of elastic support may cause jump and hysteresis phenomena in resonance curve when rotors run at the critical-speed zone，and large rotor amplitude can also aggravate the unstable behavior.Thus high linearity should be taken as an important criterion in elastic support design.

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TB123

：

：1005-9113（2015）05-0046-09

10.11916／j.issn.1005-9113.2015.05.008

2014-12-20.

Sponsored by the National Natural Science Foundation of China（Grant No.11302058）.

?Corresponding author.E-mail：andypang141592＠163.com.