A 5-DOF Model for Aeroengine Spindle Dual-rotor System Analysis

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1 Chinese Journal of Aeronautics 24 (2011) Contents lists available at ScienceDect Chinese Journal of Aeronautics journal homepage: A 5-DOF Model f Aeroengine Spindle Dual-rot System Analysis a b a, HU Qinghua, DEG Sier, TEG Hongfei * a School of Mechanical Engineering, Dalian University of Technology, Dalian , China b School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang , China Received 18 May 2010; revised 22 June 2010; accepted 26 September 2010 Abstract This paper develops a five degrees of freedom (5-DOF) model f aeroengine spindle dual-rot system dynamic analysis. In this system, the dual rots are suppted on two angular contact ball bearings and two deep groove ball bearings, one of the latter-mentioned bearings wks as the inter-shaft bearing. Driven by respective mots, the dual rots have different co-rotating speeds. The proposed model mathematically fmulates the nonlinear displacements, elastic deflections and contact fces of bearings with consideration of 5-DOF and coupling of dual rots. The nonlinear equations of motions of dual rots with 5-DOF are solved using Runge-Kutta-Fehlberg algithm. In der to investigate the effect of the introduced 5-DOF and nonlinear dynamic bearing model, we compare the proposed model with two models: the 3-DOF model of this system only considering three translational degrees of freedom (Gupta,1993, rotational freedom is neglected); the 5-DOF model where the deep groove ball bearings are simplified as linear elastic spring (Guskov, 2007). The simulation results verify Gupta s prediction (1993) and show that the rotational freedom of rots and nonlinear dynamic model of bearings have great effect on the system dynamic simulation. The quantitative results are given as well. Keywds: aeroengine; dual-rot; ball bearing; nonlinear dynamics; five degrees of freedom 1. Introduction1 The spindle dual-rot system is very common in aerospace field. Because of complicated dynamics and coupling between dual rots, the system features complicate nonlinear dynamics, which is crucial to stability of the aeroengine and plane. A spindle dual-rot system is investigated in this paper. In this system, the dual rots have co-rotating speeds and are connected by the intershaft bearing a deep groove ball bearing. The intershaft bearing inner race is mounted on the low-pressure rot while the *Cresponding auth. Tel.: address: tenghf@dlut.edu.cn Foundation items: ational atural Science Foundation of China ( , ); ational Key Technology Research and Development Program (JPPT ) Elsevier Ltd. Open access under CC BY-C-D license. doi: /S (11) outer race is mounted on the high-pressure rot. The high-pressure rot has higher rotational speed. Imptant progress has been made on studies of dynamics of dual-rot system. Early in 1975, Hibner [1] put fward the application of the transfer matrix method in der to compute the critical speeds and [2] nonlinearly damped response. In 1986, Li, et al. analyzed the dynamics of the dual-rot system which was connected by intershaft squeeze film damper. In 1993, Gupta, et al. [3] studied through experiment the critical speeds, mode shape and unbalanced response of the dual-rot system whose rots were connected by a deep groove ball bearing and developed theetically an extend transfer matrix procedure in complex variable to obtain dynamic response. The experimental results were in reasonable agreement with theetical results in the case of two degrees of freedom (2-DOF) model. In 1996, Ferraris, et al. [4] made theetical research on dynamic behavi of non-symmetric coaxial

2 o.2 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) co- counter-rotating rots. In his research, a 2-DOF model was built using finite element, and the natural frequency and mass unbalance responses were investigated. However, the displacements of rots at the bearing points were assumed the same and calculation was simplified in his model (The equations can be solved by hand calculations). In 2007, Guskov, et al. [5] presented a numerical 4-DOF model to study unbalanced response of a dual-rot test rig system where two coaxial shafts were suppted on deep groove ball bearings and connected by an intershaft bearing. Basing on the above wks, we numerically investigated a spindle dual-rot system with five degrees of freedom (5-DOF) in this study. This wk differs from the previous wk [5] in two main aspects. Fstly, the bearing type is different. The rots in Ref.[5] were suppted on deep groove ball bearings, while the rots in our wk are suppted on two angular contact ball bearings and two deep groove ball bearings. Therefe, besides translational degrees of freedom along xy and rotational freedoms around x, y, we also consider the translational degree of freedom along z. Secondly, Ref.[5] simplified the bearing models as linear elastic spring elements so it igned the influence of nonlinear stiffness and the rotational freedoms of rots. In contrast, we fmulate the bearing nonlinear displacements, elastic deflections and contact fces mathematically considering 5-DOF of the dual rots. After contact fces of all the bearings are obtained, the equations of dual rots motion are proposed accding to rot dynamics. The 5-DOF model of the dual-rot system, which includes nonlinear fmulations of bearings, has not been repted in the previous wk. In der to indicate the effect of our introduced 5-DOF and nonlinear dynamic bearing model on the system dynamics, not only the proposed 5-DOF model but also the other two models of the system are simulated simultaneously f comparison. One is 3-DOF model only considering three translational degrees of freedom, and the other is 5-DOF model where the deep groove ball bearings are simplified as linear elastic spring elements as Ref.[5] (5-DOF-Sim f sht). 2. Mathematical Model of System Fig.1 shows the schematic diagram of the dual-rot system. The left hand of higher pressure rot a is suppted by two same angular contact Bearings 1, 2 and the right hand of lower pressure rot b is suppted by two different deep groove ball Bearings 3, 4, where Bearing 4 is the intershaft bearing. The dual rots have different co-rotating speeds. The coupling between the two rots through the intershaft bearing is also considered in the mathematical model. The mathematical model follows two steps. Fst of all, the displacements, elastic deflections and contact fces of the bearings are calculated considering 5-DOF of the rots. Then the motion equations of the dual rots are fmulated based on rot dynamics. Therefe, all the contact fces of bearings should be calculated fstly. F angular contact Bearings 1 and 2, calculations of the elastic deflections and contact fces only need consideration of freedoms of rot a, thus many studies about fmulations of angular contact bearing associated with one rot as Refs.[6]- [15] can provide reference f our wk. In this wk, the earliest rept about fmulations of angular contact ball bearing under one rot proposed by Aini [15] is adopted among all previous wk. The calculation is listed out in Section 2.1. The calculations of deep groove ball Bearings 3 and 4 differ from current research on some aspects. Although the current calculation [5] of intershaft bearing considered 4-DOF of rots, the bearing is simplified as linear elastic spring element and disregarded nonlinearity. Furtherme, the nonlinear calculation of deep groove ball bearing is involved in Refs.[16]-[19], but it only considers two translational degrees of freedom of the single rot. And last but not least, Zhou and Chen [20] calculated the nonlinear contact fce of intershaft deep groove ball bearing considering 2-DOF of rots. In our wk, not only the rotational degree of freedom is considered, but also the nonlinear displacements, deflections and contact fces are fmulated mathematically. Additionally, fmulations of Bearing 3 differ from that of Bearing 4 because Bear- Fig.1 Schematic diagram of dual-rot system.

3 226 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) o.2 ing 3 is only affected by rot b, while Bearing 4 is affected by dual rots since its inner ring is mounted on rot b and outer ring is mounted on rot a. The detailed fmulations of deep groove ball Bearings 3 and 4 are given in Section 2.2. After the contact fces of the bearings are obtained, the 5-DOF model of the dual-rot system is proposed in Section Elastic deflections and contact fces of angular contact Bearings 1,2 As is mentioned above, many wks about calculation of angular contact ball bearing associated with single rot [6, 14-15, 21-23] can be used f calculation of Bearings 1 and 2. Among them the earliest one, i.e., the one in Ref.[15], is adopted. Fig.2 shows the geometry of the jth ball/race contact, where d is the pitch diameter, O i and O o are the centers of curvature of the inner and outer race grooves and move with them respectively. Assuming the outer race is fixed, O o may be considered as a fixed igin. ow take an auxiliary plane containing an axis parallel to the z axis at O o and a radial axis defining the pitch angle of the ball center with respect to the x axis. This plane also contains the line O i O o =A, the initial distance between the centers of curvature of the inner and outer races. Fig.3 Deflections of jth ball /race contact Elastic deflections and contact fces of deep groove ball Bearings 3,4 As is explained above, Bearing 3 is a suppt f rot b, while Bearing 4 is not only a suppt f rot b but also an intershaft bearing f the system. Thus, calculation of Bearing 3 needs only consider 5-DOF of rot b, but that of Bearing 4 must consider rot a and rot b simultaneously. The elastic deflections and contact fces of Bearings 3 and 4 are fmulated respectively as follows. The outer ring of bearing 3 is fixed and the inner ring is mounted on rot b. Fig.4 shows the deflection of Bearing 3 caused by rotational freedom. Fig.4 Deflection of deep groove ball Bearing 3 caused by rotational freedom. Fig.2 Geometry of jth ball/race contact. Fig.3 shows the relevant axes with the igin at O o. Thus, starting at O i (zero load), O i will move to (O i ) 1 under the preload with the contact angle p and preload deflection p. Under the external load axial component, O i will move to (O i ) 2 ; under the radial load component, O i will move to its final position (O i ) 3. The final contact angle is i. The calculation of deflections and contact fces between race and contacting ball is fmulated in Appendix A. As is shown in Fig.4, point P A is the initial contact point between the inner ring and the ball without consideration of rotational freedom of rot b. After rot b has the rotation angle around y axis, point P A moves to point P B causing the displacements dx and dz. The displacements dx and dz can be obtained as dx lsin cos b lcos sin b lsin dz lcos lcosb cos lsinbsin Since angle b is very small, cos b 1 and (1)

4 o.2 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) dx L sin, dz d sin (2) b3 b 3 b Only radial deflection dx is considered in this wk while the axial deflection dz is neglected. In a similar way the displacements caused by rotation angle b can be obtained. As the translational displacement of rot x b and y b is considered comprehensively, the relative displacements of the inner race relative to outer race in Bearing 3 can be calculated as X3o x b Lb3sin b (3) Y3o y b Lb3sin b And also the relative displacements are related to the ball location angles. The location angle of jth ball in Bearing 3 is obtained as 3j 3ct2 j/ 3 ( j 1,2,, 3) (4) where 3c is the rotational speed of the cage of Bearing 3 and 3 is the number of rollers of Bearing 3. ext, the deflection between the jth ball and the inner ring can be obtained considering the initial clearance 3 as X cos Y sin (5) 3j 3o 3j 3o 3j 3 Then the contact fce between the jth ball and the inner ring can be calculated using the deflection obtained above and contact stiffness. The sum contact fces of all the balls are the suppting fce of the Bearing 3 acting on rot a. The calculation is detailed in Appendix A. Calculation of elastic deflections and contact fces of intershaft Bearing 4 should consider 5-DOF of dual rots since its outer race and inner race are mounted on rot a and rot b respectively. The relative displacements between the inner and outer race can be obtained after the displacements are calculated. Then the contact fces are calculated using the deflections and contact stiffness. It should be noted that we assume the fce from Bearing 4 acting on rot b is equal to that on rot a in this wk. That is, the inertia, centrifugal fce and gyroscopic effect of Bearing 4 are neglected in this study. With 5-DOF of rot b considered, the displacements of the inner ring can be obtained as X4o x b Lb4sin b (6) Y4o y b Lb4sin b With 5-DOF of rot a considered, the displacements of the outer ring can be obtained as X4o x a La4sin a Y4o y a La4sin a (7) where x a, y a, a, a, x b, y b, b, b, are translational displacements and rotation angles of rot a and rot b respectively. Other parameters are shown in omenclature. Fig.5(a) shows that the location angle of the jth ball varies with time and is related to rotational speeds of inner and outer races (rot a and rot b). It can be obtained as 4j 4ct2 j/ 4 ( j 1,2,, 4) (8) where 4c is the rotational speed of the cage of Bearing 4 and 4 is the number of rollers of Bearing 4. As is shown in Fig.5(b), the following can be obtained: d4b d4a 4c (9) d d 4 where a and b are angular velocities of rot a and rot b, d 4 and d 4 are diameters of inner and outer race of Bearing 4 respectively. Based on the displacements of the inner and outer races and the initial clearance 4, the deflections between the jth ball and the races can be calculated as 4i [ xb Lb4 sin b ( xa La4 sin a)]cos 4 j [ yb La4sin b ( ya La4sin a)]sin4 j 4 (10) 4 Fig.5 Deep groove ball Bearing 4. The suppting fces of Bearing 4 can be calculated through the deflections and the contact stiffness obtained above. The calculation is detailed in Appendix A Proposed mathematical model of dual-rot system We propose the 5-DOF model of the dual-rot system based on the afementioned calculation of elastic

5 228 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) o.2 deflections and contact fces of bearings. Compared with the previous wks [3, 5], the system model is extended to 5-DOF. The following assumptions are made as Ref.[5]. (1) All the balls are massless, i.e. the centrifugal fces and gyroscopic effects are all neglected. (2) The intershaft Bearing 4 is massless, so its suppting fce acting on rot a is equal to that on rot b but opposite in dection. (3) All balls are positioned equi-pitched around the inner ring and there is no interaction between them. (4) The initial clearances of Bearings 1 and 2 are 0, and the initial clearances of Bearings 3 and 4 are shown in Table 1. (5) The shafts are rigid and there is no elastic deflection happening. The equations of the dual rot motions f 5-DOF can be derived as follows M x W cos cos W cos cos W cos M gf (11a) a a 1j 1j 1j 2j 2j 2j 4j 4j a ar j1 j1 j M y W cos sin W cos sin W sin (11b) a a 1j 1j 1j 2j 2j 2j 4j 4j j1 j1 j1 1 2 M z F W sin W sin (11c) a a aa 1j 1j 2j 2j j1 j I I L W cos sin L W cos sin L W sin (11d) ax a az a a a1 1j 1j 1j a2 2j 2j 2j a4 4 j 4j j1 j1 j I I L W cos cos L W cos cos L W cos (11e) ay a az a a a1 1j 1j 1j a2 2j 2j 2 j a4 4 j 4 j j1 j1 j1 3 4 where, Eqs.11(a)-(c) describe the translational motions of rot a along x-axis, y-axis, z-axis; Eqs.11(d)-(e) state the rotary motions of rot a around x-axis and y-axis; Eqs.11(f)-(g) are description of the translational motions of rot b along x-axis and y-axis; Eqs.11(h)- (i) state the rotary motions of rot b around x-axis and y-axis respectively. M a, M b are masses of rot a and rot b respectively; I ax, I ay, I bx, I by are moments of inertia of rot a and rot b around x-axis, y-axis; W 1j, W 2j, W 3j, W 4j are loads of jth ball of Bearing 1, 2, 3 and 4 respectively; 1j, 2j are contact angles of jth ball of Bearing 1 and 2 respectively; L a1, L a2, L a4, L b3, L b4 are distances between the bearings and the centers of gravity of rots as shown in Fig.1; F ar, F aa are fce effects on rot a in the radial and axial dections; F br is fce effect on rot b in the radial dection. Appendix A details the calculation. 3. umerical Simulation Results The numerical simulation is aimed at illustrating the effects of additional consideration of the 5-DOF and the nonlinear dynamic bearing model. Comparison M x W cos W cos M g F (11f) b b 3j 3j 4j 4j b br j1 j1 3 4 M y W sin W sin (11g) b b 3j 3j 4j 4j j1 j1 4 3 I I L W sin L W sin (11h) bx b bz b b b4 4j 4j b3 3j 3j j1 j1 4 I I L W cos L W cos (11i) 3 by 2 bz b b b4 4j 4j b3 3j 3j j1 j1 between the simulation results of 5-DOF and 3-DOF is f the purpose of illustrating the effect of additional consideration of rotational degrees of freedom. Comparison between the simulation results of 5-DOF model and 5-DOF-Sim aims to indicate the effect of introduced nonlinear dynamic bearing model. ote that the proposed model is compared with the 5-DOF-Sim model rather than the 4-DOF model in Ref.[5]. In Ref.[5], only deep groove ball bearings are adopted and the axial translational DOF is not concerned. However, two angular contact ball bearings are used in our system and the axial translational degree of freedom should be considered additionally in this wk. Three models mentioned above are all solved by Runge-Kutta-Fehlberg algithm and the responses are obtained. The integral initial values f 5-DOF model and 5-DOF-Sim model are expressed as X 5-DOF 0 = 5-DOF-Sim X 0 ={110 6, 110, 110, 110 6,110,1 10,110,110,110,0,0,0,0,0,0,0,0,0} and integral initial values f 3-DOF model are ={1 3-DOF X ,110,110,110 6,110,0,0,0,0,0}. Table 1 shows the parameter values adopted in the simulation.

6 o.2 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) Table 1 Data used in simulation Parameter Mass/kg M a =15, M b =13.47 Value Angle/( o ) 0 =15 External fce/ F ar =40, F aa =200, F br =0 Moment of inertia/(k m 2 ) Distance/m I ax =I ay = , I bx =I by =0.627, I az = , I bz = L a1 =0.45, L a2 =0.15, L a4 =0.6, L b3 =0.34, L b4 =0.34 Ball number 1 = 2 =16, 3 =20, 4 =24 Stiffness/(10 9 m - ) K 3 =7.063, K 4 =8.235 Race diameter/m d 1 =d 2 = , d 1 =d 2 = , d 3 =0.112, d 3 =0.146, d 4 =0.115, d 4 =0.145 Ball diameter/m D=0.02 Groove curvature radius/m r 1 =r 2 = , r 1 =r 2 = Clearance/m, 5 Preload/ P a =100 Elastic modulus/gpa E=207 Poisson ratio The running envonment of the algithm is AMD Athlon 64 CPU with main frequency of 2 GHz, memy of 1 GB. The programming languages are MAT- LAB and C Effects of additional considered rotational freedom In der to verify the effects of the additional consideration of rotational freedoms of the rots, the simulation results of 5-DOF model are compared with 3-DOF model. In both models, no bearing is simplified as linear elastic spring element. The rotational speed of rot bn 2 is r/min and the speed of rot a n 1 varies from r/min to r/min in the simulation. Fig.6 shows the obtained amplitudes of the dual rots. Great differences between the simulation results of 5-DOF and 3-DOF are shown in Fig.6. The change range of amplitudes of 5-DOF model is larger and its amplitude values are bigger than those of 3-DOF. Table 2 presents the statistical results. Fig.6 Responding amplitudes of system of 3-DOF and 5-DOF models. Table 2 shows that the amplitudes of the rots in 5-DOF model are greater than those of 3-DOF model. The amplitudes of rot a and rot b increase by 22.76% and 22.33% respectively. Besides, the standard deviations of amplitudes of rot a and rot b in 5-DOF model are 2.83 times and 2.45 times as large as those in 3-DOF model respectively. Furtherme, 5-DOF model is me time-consuming in simulation than 3-DOF model because of increment of the parameters and additional calculation of nonlinear contact fces after the rotational freedom of the rot is considered. In der to further study the effect between the dual rots, let rotational speed of rot b n 2 =3 000 r/min and the rotational speed of rot a n 1 change from r/min to r/min, the amplitudes of the dual rots are obtained as Fig.7(a). Comparing with the results in Fig.6(b), we analyze the different effects of rot a on rot b while rot b has different rotational speeds. The comparison of the results in Fig.6(b) and Fig.7(a) matches up to Gupta analysis in Ref.[3]. In Fig.6(b) and Fig.7(a), rot a runs at the same rotational speed while rot b runs at different speeds of r/min and r/min. This comparison shows the dual rots have different amplitudes. That is rot a has different effects on rot b when rot b has different rotational speeds. We can conclude that the effects between the dual rots depend on speeds of both rots. Gupta investigated a 2-DOF model of dual-rot system and the results indicate that the effect of rot a on rot b is independent of the speed of rot b in Ref.[3] f the reason that gyroscopic moments and rotary inertia are neglected in his simulations. However, in our proposed model, the involved rotational freedom of the rots cresponds to the consideration of rotary inertia. Fig.8 shows time domain responses of displacements of rot b at n 2 = r/min and n 1 = r/min, Table 2 Statistical amplitudes of 3-DOF model and 5-DOF model Model Amplitude of rot a /m Amplitude of rot b /m Time/s Average Deviation Maximum Average Deviation Maximum 3-DOF DOF otethe time index is the time it takes to obtain the response at a parameter set in the above running envonment (sic passim).

7 230 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) o.2 freedoms are considered additionally. Therefe, it is necessary to investigate the effects of the intershaft bearing on dynamics of the 5-DOF model. Let n 1 = r/min, n 2 =9 600 r/min respectively and the initial clearance of Bearing 4 changes from 5 m to 15 m. Fig.9 shows the simulation results of 3-DOF model and 5-DOF model. Fig.7 Responding amplitudes of system of 5-DOF and 5-DOF-Sim models. Fig.9 Responding amplitudes of system of 3-DOF and 5-DOF models (n 1 = r/min, n 2 =9 600 r/min). Fig.8 Time domain responses of rots at different speeds of rot a. From Fig.9(a), we find that the amplitudes of the dual rots of 3-DOF model have different changing trends. The amplitude of rot a has a decreasing trend while the amplitude of rot b has increasing trend with increasing of. When the value of reaches 12 m, the amplitudes of the dual rots become stationary as is shown in Fig.9(a), where stationary means change ranges of the amplitudes of dual rots are within 1 m. Different from 3-DOF model, the amplitude of rot a in the 5-DOF model oscillates in a range. The amplitude of rot b has a larger change range and its change is me complicated. When the value of reaches m, the dual rots have a stationary range as is in 3-DOF model during the simulation. The frequency spectrums of 3-DOF and 5-DOF models obtained as 4 =13 m, n 1 = r/min, n 2 = 0.8n 1 =9 600 r/min are shown in Fig r/min and r/min respectively. The results match up to Gupta s analysis too. In Fig.8(a), A 2 < A 1 < A 3 and in Fig.8(b), A 1 >A 2 >A 3, where A 1, A 2 and A 3 are maximum amplitudes at n 1 = r/min, r/min and r/min respectively. There is no definite change low f amplitude variations as is shown in Fig.6(b). In addition, speed of rot a has greater effect on y b -phase than it on x b -phase. As is mentioned above, there is complicated coupling between the rots and the bearings. Thus, the effects of intershaft on system dynamics differ when the bearing types are different and when, the rotational

8 o.2 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) Fig.10 Frequency spectra of 3-DOF and 5-DOF models. As is demonstrated in Fig.10, the frequency spectrum of 5-DOF model includes the same main frequencies as 3-DOF model, but its frequency division is me complicated because 5-DOF model introduces me exciting frequencies with additional rotational freedom. As is shown in Appendix A, the deflection calculations of both angular contact ball bearings and deep groove ball bearings have me frequency divisions, such as sin, sin and etc. Table 3 Statistical amplitudes of 5-DOF-Sim model 3.2. Effect of nonlinearity of deep groove ball bearings Besides the consideration of the rotational freedom, the other difference between the proposed model and previous wks [5] is the calculation of the bearings. In our wk, nonlinear fmulations of all the bearings with consideration of 5-DOF of the dual rots are not simplified as linear elastic spring in Ref.[5]. We compare simulation results of 5-DOF model with 5-DOF-Sim model to indicate the influence of the introduced nonlinearity of deep groove ball bearings. Fst of all, we investigate effects of nonlinearity of deep groove ball bearings on rot amplitudes. Fig.6(a) and Fig.7(b) show the rot amplitude simulation results of 5-DOF model and 5-DOF-Sim model respectively, and Table 2 and Table 3 show the statistical results. Through comparison, we have the following findings. Fstly, 5-DOF model and 5-DOF-Sim model have different rot amplitude variation curves. Secondly, the nonlinear dynamic deep groove ball bearing model and linear model have different effects on amplitudes of rots. The fmer decreases the amplitude of rot a and increases the amplitude of rot b. Spe- Model Amplitude of rot a /m Amplitude of rot b /m Time/s Average Deviation Maximum Average Deviation Maximum 5-DOF-Sim DOF cifically, the amplitude average, standard deviation and maximum of rot a decrease by 6.7%1.7% and 1.9%, while these values of rot b increase by 7.7% 48.4% and 13.9%. Thdly, the rot stationarity (standard deviation) in two models is different. In comparison with 5-DOF- Sim model, stationarity of rot a increases (the standard deviation decreases), while stationarity of rot b decreases (standard deviation increases). Fourthly, in 5-DOF model, the intershaft bearing has less effect on rot a than rot b. When it is reflected in values, the amplitude average, standard deviation and maximum values of rot b are larger than those of rot a. This is because rot a is suppted on two angular contact ball Bearings 1, 2 and the intershaft Bearing 4 while rot b is only suppted on deep groove ball Bearing 3 and the intershaft bearing. Then, the effect of intershaft bearing in 5-DOF model and 5-DOF-Sim model is investigated. Set n 1 = r/min, n 2 =9 600 r/min respectively, and assume that initial clearance of Bearing 4 changes from 5 m to 15 m, the simulation results of 5-DOF- Sim are shown in Fig.11. As is shown in Fig.11 and Fig.9(b), the amplitudes of dual rots in 5-DOF and 5-DOF-Sim models display the same growth tendency while the increases. However, the amplitudes reach maximums at different values, =11 m f 5-DOF model and 4 =13.25 m f 5-DOF-Sim model. In the range m m, the dual-rot amplitudes are stationary f 5-DOF model while they oscillate f 5-DOF-Sim model. Meover, f the whole simulation range of 5-DOF-Sim from 5 m to 15 m, there is no such stationary range as 5-DOF model. Fig.12 depicts x a -displacements and x b -displacements at different initial clearances of intershaft Bearing 4 in 5-DOF model. This simulation results are obtained at n 1 = r/min, n 2 = r/min. At three different clearance values mentioned above, both x a and x b have the largest amplitude at =15 m. However, as the increases, the equilibrium position of rot a does not have significant change (see Fig.12(a)) while the migration of equilibrium position of rot b is significant (see Fig.12(b)). The reason is Fig.11 Responding amplitudes of 5-DOF-Sim model (n 1 = r/min, n 2 =9 600 r/min).

9 232 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) o.2 Fig.12 Time domain responses of dual rots under different clearance of Bearing 4. as follows. Rot a is suppted on Bearing 1, Bearing 2 and intershaft Bearing 4, and so its vibration is mainly limited by Bearing 1 and Bearing 2. However, rot b is only suppted on Bearing 3 and intershaft Bearing 4. As increases, rot b has larger amplitude to obtain enough contact fce from the bearings and its equilibrium position migrates as its amplitude increases. The simulation time f 5-DOF model is 1.6 times of that of 5-DOF-Sim model since calculation of nonlinear dynamic bearing models is me complicated than linear dynamic models. 4. Conclusions (1) A 5-DOF model of aeroengine spindle dual-rot system is proposed f dynamic analysis. The proposed model is compared with several previous models to investigate the effect of the introduced 5-DOF and nonlinear dynamic bearing models. (2) The simulation results of the proposed 5-DOF model accd with Gupta s conclusions. In our proposed model, the involved rotational freedom of the rots cresponds to the consideration of the rotary inertia. Our simulation results show the effects of dual rots depend on the speeds. (3) The dual rots of 5-DOF model have bigger simulation amplitudes than those of 3-DOF model. The introduced rotational freedom of the rots also increases the simulation amplitudes deviation (lowered the stationarity). (4) The amplitudes of 5-DOF model and 3-DOF model show different changing trends as the initial clearance of intershaft bearing increases. In 3-DOF model, the amplitude of rot a has decreasing trend while the amplitude of rot b has increasing trend, and the dual rots have stationary amplitudes when is bigger than 12 m. In 5-DOF model, amplitudes of dual rots oscillate in a certain range and have stationary amplitudes when m m. (5) Although the main frequencies of the frequency spectrum of 5-DOF model are the same with 3-DOF model, the frequency division is me complicated. The exciting frequencies increase as the rotational freedom is considered additionally. (6) Compared with 5-DOF-Sim model, the rot a of 5-DOF model has smaller amplitude and higher stationarity (smaller deviation) while rot b has bigger amplitude and lower stationarity (bigger deviation). The rot amplitudes of both models tend to increase with the increase of while they get the maximums at different values. (7) Because of me complicated calculation of nonlinear bearing model, the simulation time of 5-DOF model is 1.6 times and 18 times of that of 5-DOF-Sim model and 3-DOF model respectively. (8) Although in the 5-DOF model, the rotational freedom of rot is considered and the nonlinearity of the bearing is fmulated considering the coupling of the rots, the centrifugal fce and gyroscopic effect of the balls are neglected. When the system runs at high speed, the neglected elements may play an imptant part in the system dynamics. This deserves further study. References [1] Hibner D H. Dynamic response of viscous-damped multi-shaft jet engines. Journal of Acraft 1975; 12(4): [2] Li Q, Yan L, Hamiltion J F. Investigation of the steady-state response of a dual-rot system with intershaft squeeze film damper. Transactions Journal of Engineering f Gas Turbines and Power 1986; 108(4): [3] Gupta K, Gupta K D, Athre K. Unbalance response of a dual rot system. They and experiment. Journal of Vibration and Acoustics, Transactions of the ASME 1993; 115(4): [4] Ferraris G, Maisonneuve V, Lalanne M. Prediction of the dynamic behavi of non-symmetric coaxial co- counter-rotating Rots. Journal of Sound and Vibration 1996; 154(4): [5] Guskov M, Sinou J J, Thouverez F, et al. Experimental and numerical investigation of a dual-shaft test rig with intershaft bearing. International Journal of Rotating Machinery 2007; doi: /2007/ [6] Akturk, Uneeb M, Gohar R. The effect of number of balls and preload on vibrations associated with ball bearings. Transactions of ASME Journal of Tribology 1997; 119(4): [7] Akturk. The effect of waviness on vibrations associated with ball bearings. Journal of Tribology 1999; 121(4): [8] Arslan H, Akturk. An investigation of rolling element vibrations caused by local defects. Journal of Tribology 2008; 130(4): [9] Cao Y, Altintas Y. A general method f the modeling of spindle-bearing systems. Journal of Mechanical Design, Transactions of the ASME 2004; 126(6):

10 o.2 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) [10] Jang G H, Jeong S W. Analysis of a ball bearing with waviness considering the centrifugal fce and gyroscopic moment of the ball. Journal of Tribology 2003; 125(3): [11] Jang G H, Jeong S W. onlinear excitation model of ball bearing waviness in a rigid rot suppted by two me ball bearings considering five degrees of freedom. Journal of Tribology 2002; 124(1): [12] Bai C Q, Xu Q Y. Dynamic model of ball bearings with internal clearance and waviness. Journal of Sound and Vibration 2006; 294(1-2): [13] Bai C Q, Zhang H Y, Xu Q Y. Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rot-bearing system. onlinear Dynamics 2008; 53(3): [14] Alfares M A, Elsharkawy A A. Effects of axial preloading of angular contact ball bearings on the dynamics of a grinding machine spindle system. Journal of Materials Processing Technology 2003; 136(1-3): [15] Aini R, Rahnejat H, Gohar R. A five degrees of freedom analysis of vibrations in precision spindles. International Journal of Machine Tools and Manufacture 1990; 30(1): [16] Tiwari M, Gupta K. Effect of radial internal clearance of ball bearing on the dynamics of a balanced hizontal rot. Journal of Sound and Vibration 2000; 238(5): [17] Tiwari M, Gupta K, Prakash O. Dynamic response of unbalanced rot suppted on ball bearing. Journal of Sound and Vibration 2000; 238(5): [18] Chen G. onlinear dynamic response analysis of an unbalanced rot suppted on ball bearing. China Mechanical Engineering 2007; 18(23): [in Chinese] [19] Harsha S P. onlinear dynamic analysis of a high-speed rot suppted by rolling element bearings. Journal of Sound and Vibration 2006; 290(1): [20] Zhou H L, Chen G. Dynamic response analysis of dual rot-ball bearing-stat coupling system f aero-engine. Journal of Aerospace Power 2009; 24(6): [in Chinese] [21] Aini R, Rahnejat H, Gohar R. A five degrees of freedom analysis of vibrations in precision spindles. International Journal of Rotating Machinery 1990; 30(1): [22] Alfares M, Elsharkawy A. Effect of grinding fces on the vibration of grinding machine spindle system. International Journal of Machine Tools & Manufacture 2000; 40(14): [23] Harris T A. Rolling bearing analysis. 3rd ed. ew Yk: John Wiley and Sons, Biographies: HU Qinghua Bn in 1981, he received his B.S. degree from School of Mechanical Engineering, Dalian University of Technology, Dalian, China, in He is currently wking towards the Ph.D. degree from the same university. His main research interests include dynamic analysis, simulation of rot systems, evolutionary computation and optimization design of mechanical system. hutsinghua@yahoo.com.cn DEG Sier Bn in 1963, he received his Ph.D. degree from School of Mechanical Engineering, Dalian University of Technology, China, in He is currently a profess in School of Mechatronics Engineering, Henan University of Science and Technology, China. His main research interests include rolling bearing design and they, intelligent CAD, optimization and computer simulation. dse@mail.haust.edu.cn TEG Hongfei Bn in 1936, he received his B.S. degree from School of Mechanical Engineering, Dalian University of Technology, Dalian, China, in He is currently a profess in School of Mechanical Engineering, Dalian University of Technology. He is also a Pluralistic Profess with the Department of Computer Science and Engineering, Dalian University of Technology. His maj research interests include rapid response design, CAD and optimization, computational intelligence and complex engineering system. tenghf@dlut.edu.cn Appendix A Calculation of the parameters of the mathematical model in Section 2.3 W 1j K1j1j, W 2j K2j2j, W 3j K3 3j, W 4j K4 4j 2 2 1/2 1j {[ sin 0 1 m1/ 2( acos 1j asin 1j)] ( cos p 0cos p 1cos 1j 1sin 1j) } A Z d A X Y A 2 2 1/2 2j {[ sin 0 2 m2 / 2( a cos 2j asin 2j)] ( cos p 0cos p 2cos 2j 2sin 2j) } A Z d A X Y A ( x L sin )cos ( y L sin )sin 3j b b3 b 3j b b3 b 3j 3 [ x L sin ( x L sin )]cos [ y L sin ( y L sin )]sin 4j b b4 b a b4 a 4j b b4 b a b4 a 4j 4 1j 1ct2 j/ 1, 2j 2ct2 j/ 2, 3j 3ct2 j/ 3, 4j 4ct2 j/ 4 D d3b (1 cos ), 3c d d a 1c 2c p 2 dm 3 3 d d 4 b 4, 4c d4 d4 X1 xa La1 sin a, X2 xa La2 sin a, Y1 ya La1 sin a, Y2 ya La2 sin a a

11 234 HU Qinghua et al. / Chinese Journal of Aeronautics 24(2011) o.2 Z1 Z0 za, Z2 Z0 za, Z0 Asin( p 0) / cos p, A r r D Asin 0 Z1dm1/ 2( a cos 1j asin 1j) 1 j arctan Acos p 0cos p X1cos 1j Y1sin 1j 2 j Asin 0 Z2 dm2 / 2( a cos 1j asin 1j) arctan Acos p 0cos p X2cos 1j Y2sin 1j where A is distance between centers of curvature of inner and outer race grooves, D and d m are ball diameter and pitch diameter respectively. The nonlinear stiffness coefficient K 1j, K 2j of angular contact Bearings 1 and 2 is a function of contact angle, which can be calculated as follows. 1 2/3 K 1 1 K K 2/3 F( ), 4 K 2 2 E v f 1, 1 2 f F( ) 4 D f 1, f r D r, D, f, K 1 2 f f E v D f 1 Dcos d where * can be determined from the following equations: If 0<F()< ( F( )) ( F( )) ( F( )) ( F( )) If <F()< If F()> m ( F( )) ( F( )) ( F( )) ( F( )) ( F( )) ( F( )) ( F( )) ( F( )) The preload contact angle p, and the initial contact deflection 0 due to the elasticity of the Bearings 1, 2 f a given preload P a can be determined by solving the following equations simultaneously using iterative ewton-raphson scheme: 1(2) 0 sin p K P, a cos 0 0 A 1 cos p F deep groove ball Bearings 3 and 4, the stiffness coefficients K 3, K 4 are constant in the calculation. They are related to the material and shape of contacting objects. The values are /m and /m respectively.