Research Article Influence of Bearing Stiffness on the Nonlinear Dynamics of a Shaft-Final Drive System

Shock and Vibration Volume 16, Article ID 5469, 14 pages http://d.doi.org/1.1155/16/5469 Research Article Influence of Bearing Stiffness on the Nonlinear Dynamics of a Shaft-Final Drive System Xu Jinli, Wan Lei, and Luo Wenin School of Mechanical and Electronic Engineering, Wuhan University of Technology, Hubei 47, China Correspondence should be addressed to Wan Lei; wanlei1815@16.com Received 18 February 16; Revised May 16; Accepted 6 May 16 Academic Editor: Tai Thai Copyright 16 Xu Jinli et al. 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. The bearing stiffness has a considerable influence on the nonlinear coupling vibration characteristics of the shaft-final drive system. A 14-DOF nonlinear coupled vibration model was established by employing the lumped mass method so as to identify the coupling effects of the bearing stiffness to the vibration response of the shaft-final drive system. The engine s torque ripple, the alternating load from the universal joint (U-joint), and the time-varying mesh parameters of hypoid gear of the shaft-final drive system were also considered for accurate quantitative analysis. The numerical analysis of the vibration response of the coupled system was performed and the eperimental measurements were carried out for the validation test. Results show that, at the given driving speed, improving the bearing stiffness can reduce the vibration response of the given coupled system; however, when the bearing stiffness increases to a critical value, the effects of bearing stiffness on the vibration reduction become insignificant; when the driving speed changes, the resonance regions of the coupled system vary with the bearing stiffness. The results are helpful to determine the proper bearing stiffness and the optimum control strategy for the shaft-final drive system. It is hoped that the optimal shaft-final drive system can provide good vibration characteristics to achieve the energy saving and noise reduction for the vehicle application. 1. Introduction Itisacommoncaseinvehiclesthatpoweristransmitted through shafts and gears which are not in a direct line; this layout is even more comple with all-wheel drive vehicles. Some precision components are inherently nonlinear; for eample, the hypoid gear pair inside the gearbo is often coupled with bearing systems. On account of the increasing demand for high NVH performance, the research in the field of parameter matching on the transmission system becomes very important. As one of the major parameters in a shaftfinaldrivesystem,thebearingstiffnesssignificantlyaffects the performance of a high-speed pinion shaft and leads to the coupled vibration of the rear ale assembly [1, ]. In the transmission system, the dynamic mesh force ecited by the transmission error is believed to be an important vibration source. Therefore, some considerable research on the gear system has been carried on in recent years. Lim and Singh [] proposed and established the basic theory and coupling mechanism of vibration transmission through rolling element bearing. Donley et al. [4] studied the three-dimensional stress of a gear pair with finite element method. Bibel et al. [5] considered the contact stress analysis of spiral bevel gears with the invariable mesh point and lineof-action. Gosselin et al. [6] derived a general formula and applied it to the load distribution and transmission error of spiral bevel gear. Theodossiades et al. [7, 8] gave a model method to analyze the dynamic characteristic using timevarying stiffness and backlash nonlinearity. Wang et al. [9, 1] carried out the analysis and comparison from the gear pair to the gear system with the bearing. Cheng and Lim [11, 1] developed an analytical framework for analyzing both linear and nonlinear pure vibration characteristics of geared rotor system. Tao [1] further developed the coupled multibody dynamics and vibration model with the practical cases in his work. Yang and Lim [14] proposed a new capability to analyze the dynamic interaction of gear mesh characteristics and time-varying bearing stiffness. The tapered roller bearing (TRB) is widely used to compensate the installation errors along the aial and radial directions and the displacement by the movement of transmission system; the bearing stiffness in rotor-bearing system

Shock and Vibration Node 1 Shaft 1 K 1 T K c Shaft Right-half ale θ 1 T 1 T 1 θ 1 c 1 θ T θ c K T 4 K c Pinion K 4 θ 9 c 4 Drive shaft θ 4 T 5 θ 5 θ 7 T 9 T 7 T 6 θ 6 Gear z o Left-half ale θ 8 K c y T 8 Figure 1: Mathematical model of a shaft-final drive system. can significantly affect the dynamic characteristics including natural frequencies and dynamic response [15]. In recent years, a theoretical analysis of lateral response and the optimization algorithm via the enhanced genetic algorithm waspresentedin[16].therunningerrorconditionsofthe misalignment rotor systems connected by the gear coupling were indicated in [17]. In [18], a nonlinear dynamic model ofaspurgearpairwithlinearmodelsofshafts,bearings,and disks was developed. Considering the rotor dynamic balance, the fleible rotor bearing system was studied by numerical analysis elsewhere [19 1]. These researches mentioned above mainly focused on the internal incentives of gear or rotor system; the interactional relationships between transmission system and the influence of bearing stiffness on a shaft-final drive system were still not well defined []; few studies have been done on fluctuations of the system with transmission error in which the bearing stiffness changes with the speed in operation. In this investigation, both nonuniformity of multiple U- joints and time-varying mesh parameters of hypoid gear are considered, a 14-DOF nonlinear model is established, and the numerical optimisation method is proposed to analyze the influence of bearing stiffness on the nonlinear dynamics of the shaft-final drive. Then the transmission system of a rear wheel drive vehicle is taken as an eample to obtain the reasonable region of bearing stiffness; eperiments are carried out to demonstrate the effectiveness of this method by simultaneously optimising system stiffness.. Mathematical Model.1. Nonlinear Dynamic Model. The shaft-final drive system model is shown in Figure 1. It mainly consists of the drive shaft,gears,andrearaleparts. Foraneffectivedynamicmodel,thelumpedmassmethod is adopted to deal with the components and parts. It allocates the moment of inertia of the shaft to the U-joints and pinion, the driven gear and differential can be counted as an assembly with a concentrated mass, the fleible shaft can be described by torsion springs and dampers, and the paired mounting bearings of the reducer can be treated as a whole and simulated by a spring-damper system. To simplify the calculation, the friction and lubrication and some inertia of the small components such as the needle bearings in the U- joints are neglected []... Universal Joint Geometry. On the basis of U-joints kinematics, there is tan θ = tan θ cos α 1, (1) where θ and θ are the input and output angle of U-joint, respectively, α 1 is the aial angle, and the angular speed and angular acceleration of main drive shaft are defined; thus, θ cos α = 1 (1 + cos α1 )/ cos θ (1 cos α1 )/ θ, θ = θ A Bcos θ B sin θ (A B cos θ ) ( θ ), A= 1+cos α 1 cos α 1, B = 1 cos α 1 cos α 1. Similarly, θ 4 and θ 5 also share the aforementioned relationship... Mesh Force Analysis. In order to make quantitative analysis, the uniform load of the pinion is replaced by the ()

Shock and Vibration F n F r α n F a z F n F r α n β1 F a F t β 1 δ 1 β 1 y t z y δ 1 F a F δ r 1 z δ 1 y Figure : Force analysis of the pinion. concentrated force acting on midpoint of the face width; the simplified mechanical model is shown in Figure. In process of the pinion mesh transmission, the normal force at the contact point and its components along -, y-, and z-aes are of the following form: F n =k m (t) f( n )+c m n, F = F n cos α n sin β 1 cos δ 1 +F n sin α n sin δ 1, F y = F n cos α n sin β 1 cos δ 1 F n sin α n cos δ 1, F z = F n cos α n cos β 1. The gear pair mesh stiffness function k m (t) is k m (t) =k m + N i=1 () k Ai cos (iω h t+φ k ). (4) The gear backlash function f can be epressed by the nonlinear combination: n b, n b, { f( n )=, b n b, { { n +b, n b. The relative displacement of the normal direction at the contact point n is (5) Generally, with e =, the coefficients in (6) can be epressed as follows: a 1 = cos α n sin β 1 cos δ 1 sin α n sin δ 1, a = cos α n sin β 1 sin δ 1 + sin α n cos δ 1, a = cos α n cos β 1.. Governing Differential Equations.1. Dynamic Differential Equations. Basedonthekineticsof the shaft and the mesh force analysis of the hypoid gear, the differential equations of the shaft-final drive system are as follows: J 1 θ 1 +c 1 ( θ 1 θ )+K 1 (θ 1 θ )=T 1, J θ +c ( θ θ 1 )+K (θ θ 1 )= T, J θ +c ( θ θ 4 )+K (θ θ 4 )=T, J 4 θ 4 +c 4 ( θ 4 θ )+K 4 (θ 4 θ )= T 4, J 5 θ 5 F z R 5 =T 5, J 6 θ 6 +F z R 6 = T 6, J 7 θ 7 +c 7 ( θ 7 θ 8 )+K 7 (θ 7 θ 8 )=T 7, (8) n =a 1 ( p g )+a (y p y g ) +a (z p z g +R p θ p R g θ g ) e(t). (6) J 8 θ 8 +c 8 ( θ 8 θ 7 )+K 8 (θ 8 θ 7 )= T 8, m p p +c p p +K p p =F, The synthetic error e(t), in consideration of manufacturing factors, can be described as e (t) =e + N i=1 e Ai cos (iω n t+φ e ). (7) m yp y p +c py y p +K py y p =F y, m zp z p +c pz z p +K pz z p =F z, m g g +c g g +K g g = F,

4 Shock and Vibration m yg y g +c gy y g +K gy y g = F y, m zg z g +c gz z g +K gz z g = F z. Tosimplifythemodel,theintermediatesupportbearing assembly can be treated as a K -c elastic system; the K 1 and c 1 are the equivalent values of the two components of intermediate support bearing assembly and intermediate shaft. (9).. Nondimensionalization Parameters. The difference of parametermagnitudeisobvious;forinstance,themagnitude of stiffness is generally 1 7 to 1 9, but the vibration response is about 1 6, and some parameters in (9) are variable and nonlinear, so it is necessary for better solving to convert the dimension equations to nondimensionalization. The displacement is normalized by half of the gear backlash b and the frequency is normalized by the natural frequency of the gear pair ω n ; the conversion parameters are of the following form: m i = J i Ri, k i = ς i = K i (m i ω n R i ), c i (m i ω n R i ), i = R iθ i b, θ i i = R i (bω n ), i = R iθ i (bω n ), n = n b, g n = (bω n ), n = g (bω n ), p(g) = p(g) b, p(g) p(g) = (bω n ), p(g) = p(g) (bω n ), K p(g)j k p(g)j = (m p(g)j ωn ), ς p(g)j = c p(g)j (m p(g)j ω n ), k mpj (t) = K m (t) (m jp ωn ), ς mpj = c m (m jp ω n ), N k Ai k m (t) =1+ cos (iω k h t+φ k ), m ς m = i=1 c m (m e ω n ), f( n ) 1, f( n )>1, { f( n )=f( n )=, 1 f ( n ) 1, { { f( n )+1, f( n )< 1, (i=1,,...,8; j=,y,z). (1) Substituting (1) into (9), the 14-DOF differential equations are transformed to the no-dimensional form, and the further simplification is possible. Combining (1), (), and (1), the parameter canbealteredas = R θ b = R b arctan tan b /R, cos α = R θ = R 1 θ bω n bω = R, n R 1 M θ = R bω n M T = T θ =T θ M. = R R 1 M B sin θ M R b R 1 ( ), (11) Similarly, the parameter 4 shares the aforementioned relationship, so and 4 can be epressed by and 5 separately: M i =A Bcos θ i, P i =Bsin θ i, (i =,5). (1) The relationship between the driven gear and differential is given by θ 6 =θ 7, T 6 =T 7. (1)

Shock and Vibration 5 Combining (1) (1), (9) can be simplified to the following 11-DOF differential equations: T 1 +ς 1 ( 1 )+k 1 ( 1 ) 1 bωn m =, 1R 1 (m R 1 + m R R 1 M ) m P R b M R1 ( ) +(m R 1 ς + m ς R R 1 M ) m R 1 ς 1 m ς R 4 R R 5 M 5 M +k m R 1 k m R 1 1 + k m R [R bm arctan tan (b /R ) cos α R 4 arctan tan (b 5/R 5 ) ]=, cos α (m 5 R 5 + m 4R 4 R 5 M5 ) 5 + m 4ς 4 R 4 ( R 4 M 5 R 5 M 5 R R 1 M )+a k 5 (t) m 5 R 5 f( n ) + k 4m 4 R 4 [R bm 4 arctan tan (b 5/R 5 ) 5 cos α R arctan tan (b /R ) ] m 4P 5 R 4 b cos α M5 R5 ( 5 ) +a m 5 ς 5 n =, ( m 6R 6 +m 7R 7 ) R 6 +m 7 R 7 [ς 7 ( R 7 6 R 6 8 ) 6 5 5 m 8 +k 7 ( R 7 R 6 8 )] a m 6 R 6 [k 6 f( n ) ξ 6 n ] 6 =, 8 +m 8 ς 8 ( 8 R 7 6 )+m 8 k 8 ( 8 R 7 R 6 ) 6 + T 8 bωn R =, 8 R 6 p +ς p p +k p p +a 1 ς mp n +a 1 k mp (t) f( n ) =, y p +ς py y p +k py y p +a ς mpy n +a k mpy (t) f( n ) =, z p +ς pz z p +k pz z p +a ς mpz n +a k mpz (t) f( n ) =, g +ς g g +k g g a 1 ς mg n a 1 k mg (t) f( n ) =, y g +ς gy y g +k gy y g a ς mgy n a k mgy (t) f( n ) =, z g +ς gz z g +k gz z g a ς mgz n a k mgz (t) f( n ) =. 4. Numerical Analysis (14) To solve the vibration response of the system, the state space variables {q} are introduced: {q} = {q 1 q q q 4 q 5 q 6 q 7 q 8 q 9 q 1 q 11 q 1 q 1 q 14 q 15 q 16 q 17 q 18 q 19 q q 1 q } T. (15) Select the state space variables: q 1 = 1, q = 1, q =, q 4 =, q 5 = 5, q 6 = 5, q 7 = 6, q 8 = 6, q 9 = 8, q 1 = 8, q 11 = p, q 1 = p, q 1 =y p, q 14 = y p, q 15 =z p, q 16 = z p, q 17 = g, q 18 = g,

6 Shock and Vibration q 19 =y g, q = y g, q 1 =z g, q = z g. Combining (15) and (16), (14) can be altered as q 1 =q, T q = 1 bωn m ς 1 (q q 4 ) k 1 (q 1 q ), 1R 1 q =q 4, q 4 =[ m P M R b R1 q 4 (m R R 1 ς +m ς R 1 M )q 4 +m R 1 ς q + m ς R 4 R R 5 M 5 M q 6 k m R 1 q +k m R 1 q 1 k m R arctan tan (bq /R ) bm cos α + k m R 4 R arctan tan (bq 5/R 5 ) ](m bm cos α R 1 + m R 1 R 1 M ) ; q 5 =q 6, q 6 =[ m 4P 5 M5 R 4 b R5 q 6 m 4ς 4 R 4 ( R 4 q M 5 R 5 M 6 5 R R 1 M q 4 ) a k 5 (t) m 5 R 5 f( n ) (16) +a m 6 ξ 6 R 6 [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)]] (m 6 R 6 + m 7R 1 7 ), R 6 q 9 =q 1, T q 1 = 8 bωn m ς 8 (q 1 R 7 q 8R 8 R 8 ) k 8 (q 9 R 7 6 R 6 q 7 ), q 11 =q 1, q 1 = ς p q 1 k p q 11 a 1 k mp (t) f( n ) a 1 ς mp [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)], q 1 =q 14, q 14 = ς py q 14 k py q 1 a k mpy (t) f( n ) a ς mpy [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)], q 15 =q 16, q 16 = ς pz q 16 k pz q 15 a k mpz (t) f( n ) a ς mpz [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)], q 17 =q 18, q 18 = ς g q 18 k g q 17 +a 1 k mg (t) f( n ) +a 1 ς mg [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 a m 5 ς 5 [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)] k 4m 4 R 4 (R bm 4 arctan tan (bq 5/R 5 ) 5 cos α R arctan tan (bq /R ) )] (m cos α 5 R 5 + m 4R 1 4 R 5 M5 ) ; q 7 =q 8, q 8 = [ m 7 ς 7 R 7 ( R 7 q R 8 q 1 ) m 7 k 7 R 7 ( R 7 q 6 R 7 6 q 9 )+a k 6 m 6 R 6 f( n ) q +q 6 q 8 )a e (t)], q 19 =q, q = ς gy q k gy q 19 +a k mgy (t) f( n ) +a ς mgy [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)], q 1 =q, q = ς gz q k gz q 1 +a k mgz (t) f( n ) +a ς mgz [(q 1 q 18 )a 1 +(q 14 q )a +(q 16 q +q 6 q 8 )a e (t)]. (17)

Shock and Vibration 7 Parameters Table 1: Parameters of the drive shaft. Shear modulus G (GPa) Length of the shaft L (mm) Outside diameter D (mm) Inside diameter d (mm) Shaft 1 79.4 58 6.5 59.9 Shaft 79.4 879 6.5 59.9 1 (solid Left (right) 8.8 719.5 circular half-shaft section) Table : Parameters of the pinion and gear. Pinion Gear Number of teeth 8 41 Transverse module 4.161 Average pressure angle (ALPHA) 1.5 Pitch diameter (D ) (mm) 17.6 Offset distance (mm) 5 (lower) Mean spiral angle (BETA) 47.97 8.7 Pitch cone angle (DELTA-B) 1.75 76.5 Tooth width (BF) (mm). 8 Base circle diameter (D b ) D b =D cos(alpha) Table : Values of the torsional stiffness and damping. Torsional stiffness (N mm/rad) Value Torsional damping (N mm s/rad) Value K 1,K 51945.46 c 1,c 11.86 K,K 4.57 c,c 4 11.86 K 7,K 8 65.118 c 7,c 8 47.416 The vibration characteristics of the system can be obtained by solving(17)withrunge-kuttamethod. 4.1. Parameter Setting. The main parameters of the drive shaft and gear pair are listed in Tables 1 and. Based on the initial parameters, the torsional stiffness anddampingoftheshaft-finaldrivesystemarecalculatedin Table. The average torsional stiffness and damping of the gear pair are as follows: k m =k mb ( r b1 +r b r ), b1 r b c m =ξ m k mb ( 1 m p + 1 m g ), (18) where r b1, r b are the pitch circle radius of the pinion and gear, m p, m g arethemassesofthepinionandgear,k mb is the gear mesh stiffness with a linear value: 1.84 1 N/mm, and ξ m is Table 4: Values of the equivalent moment of inertia and equivalent mass. Equivalent moment of inertia (kg mm ) Value Equivalent mass (kg) Value J 1 95.4 m p 1.1 J 16.616 m py 1.1 J,...,J 4 1979.6 m pz 1.1 J 5 955. m g 5.115 J 6 171.1 m gy 5.115 J 7 19.17 m gz 5.115 J 8 14675.6 thedampingratiowithavalueof.1here.thentheequivalent polarmomentofinertiaofthefirsttransmissionshaftatnode 1 of the U-joint can be described as J 1 = 1 (J szz1 +J z1 )+J cdc1. (19) J szz1 is the first universal joint cross-shaft s equivalent polar moment of inertia about the first transmission ais, J z1 is the first transmission shaft s equivalent polar moment of inertia about its geometric centre, and J cdc1 is the first joint shaftfork sequivalentpolarmomentofinertiaaboutthefirst transmission ais. Similarly, the equivalent moment of inertia and equivalent mass can be calculated by the D CAD model, and the values are listed in Table 4. In the gear system, there are two parallel TRBs for supporting the pinion and the other two TRBs for supporting the gear. The equivalent method is an effective way to deal with the bearing stiffness; the equivalent stiffness and damping of the bearings along -, y-, and z-aes are obtained as K pj = K pj1 + K pj, c pj = c pj1 + c pj, K gj = K gj1 + K gj, c gj = c gj1 + c gj, () where K ij1, K ij, c ij1, c ij (i = p,g; j =,y,z) are the four different bearing stiffnesses and dampings. 4.. Results at the Given Speed. Considering the drive shaft at a speed of rpm, and the input torque T 1 =. 1 5 N mm, the influence of bearing stiffness on the coupled fleural-torsional vibration of the shaft-final drive system can be analyzed. A bearing stiffness coefficient a is introduced in this paper. Figure shows the output dynamic response of the shaft and driven gear. The speed fluctuation value first increases and then decreases, but the fluctuation value of

8 Shock and Vibration Fluctuation value 1 7 1.4 1. 1.8.6.4..6.8 1 1. 1.4 1.6 1.8 Gear output speed Shaft output speed Stiffness coefficient a Figure : Influence of bearing stiffness on torsional vibration of the system. driven gear is more sensitive to the different bearing stiffness coefficients. When the coefficient a is.7, the fluctuation maimum occurs; it is because different bearing stiffness can change the system natural frequency, and the system resonance region appears at a certain ecitation frequency corresponding to a =.7. In cases where other parameters remain constant, the shaft and gear speed fluctuations are significantly reduced as the bearing stiffness coefficient a>.7; when it increases to a critical value, the reduction trend becomes less obvious. Figures 4(a) and 4(b) show the bending vibration of the hypoid gear when the coefficient increases from.6 to.. By comparison under the same conditions, a maimum displacement response occurs along z-ais, and the minimal fluctuation occurs along y-ais, the displacement response along -ais and z-ais has little difference, and the response of pinion is stronger than gear. Considering the influence of different bearing stiffness on bending vibration system, the displacement response follows a similar trend to the torsional vibration in Figure, and the displacement fluctuation increases to a peak value and then moves downward to the steady state. 4.. Results at the Different Speeds. For further analysis of the vibration characteristics, the gear transmission errors under different speeds are considered. Figure 5 shows the vibration response when the stiffness coefficient varies from.7 to 1.5 andthedriveshaftinputspeedchangesfrom8rpmto 5 rpm, in which a speed variation coefficient n V is used to describe the drive ecitation. Figure 5(a) shows the displacement fluctuation of the transmission error under different stiffness coefficients and different input regions, partial enlargements are shown in Figures 5(b) and 5(c), and the parameters in the resonance region of the system vary with the stiffness, but the resonance regions mainly occur from 1.45 to 1.65 and. to.. In the regions of 1.45 n V 1.65, themaimumfluctuation occurs when a is 1., and the dimensionless value is 4.84 1 5, but the fluctuation region is not obvious compared with other speed regions (see Figure 5(b)). In the regions of. n V., the minimum fluctuation occurs when a = 1. and then follows by.8 and 1., but when a is 1.5, the fluctuation increases significantly (see Figure 5(c)). Overall, the transmission error has a nonlinear relationship with the stiffness coefficient, considering the different input speeds; when a is.8, 1., and 1., the fluctuations are relatively smaller; meanwhile the influence on transmission error fluctuation is not obvious with the other stiffness coefficients. The bearing stiffness eerts a significant influence on the vibration characteristics of the shaft-final drive system. When the bearing stiffness increases at rpm, the speed fluctuation first increases and then decreases; when it increases to a critical value, the reduction trend is hardly changeable; then the resonance regions of the coupled system vary with the bearing stiffness under different speeds. Obviously, the system has a comprehensive minimum fluctuation at a = 1.. 5. Dynamic Simulation To investigate the dynamic response of the system under different bearing stiffness, a rigid-fleible coupling model is established in the ADAMS software, and then by introducing variables DV 1, DV, DV, and DV 4, where DV 1and DV arethedampingandbearingstiffnessofthepinion and DV anddv4 are the damping and bearing stiffness of the gear, the model adopts a Bushing connection to simulatethebearing,andtherotationalstiffnessanddamping values are set to ; then the measure functions of the virtual prototype model are established. The simulation is based on thefollowing:theinputspeedisrpm,theloadtorqueis. 1 5 N mm, and the speed fluctuation coefficient which comes from the engine eperiments of the vehicle is.5. As the bearing stiffness increases, the fluctuations in a certainty scope on output speed and torque of the system are nonlinear and complicated, and the output of drive shaft and driven gear could well describe the system performance, especially the influence of different bearing stiffness on coupling vibration of the shaft-final drive assembly. For the quantitative analysis, Trial 1 Trial 14 represent fourteen sets of data successively in Figure 6, corresponding to.6 a 1.9; it means that Trial 1 represents a =.6,andTrial14represents a = 1.9. Figures 6(a) and 6(b) show the output speed and torque fluctuations of drive shaft. From Trial 5 to Trial 7 (1. a 1.), the output speed fluctuations change slowly and the values are relatively small, but the output torque is not identical: the fluctuations are obvious in regions of.6 a.8 and 1.6 a 1.9, the minimum value occurs at a =1.5,and it is relatively smooth and steady at 1. a 1.4. Similarly, Figure 7 shows the fluctuation curves of driven gear output speed and torque. Fourteen sets of data, Trial 1 Trial14,arealsoselectedasthevariablescorrespondingto.6 a 1.9. From Trial 5 to Trial 7, the output speed and torque fluctuations are obvious in the regions of.6 a.8 or 1.6 a 1.9. Figures 7(a) and 7(b) show the similar fluctuation trend under different bearing stiffness and change slowly at 1. a 1.5. The simulation results in Figures 6 and 7 show that the dynamics of shaft-final drive system is affected by the bearing

Shock and Vibration 9 Fluctuation value 1 8 5 4.5 4.5.5 1.5 1.5.5 1 1.5 Stiffness coefficient a X-direction of gear Y-direction of gear Z-direction of gear (a) Fluctuation value 1 8.5.5 1.5 1.5.5 1 1.5 X-direction of pinion Y-direction of pinion Z-direction of pinion Stiffness coefficient a (b) Figure 4: Influence of bearing stiffness on bending vibration of the hypoid gear. Fluctuation value.16.14.1.1.8.6.4..5 1 1.5.5 Speed variation coefficient n Fluctuation value 1 5 5 4.5 4.5.5 1.5 1.5 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 Speed variation coefficient n 1 a = 1.5 a = 1. a =.8 (a) Fluctuation value a =.7 a = 1. a =.9 a = 1.4 1 a = 1.5 a = 1. a =.8.16.14.1.1.8.6.4. 1.9 1.95.5.1.15..5. Speed variation coefficient n (b) a =.7 a = 1. a =.9 a = 1.4 1 a = 1.5 a = 1. a =.8 (c) a =.7 a = 1. a =.9 a = 1.4 Figure 5: Gear transmission error n.

1 Shock and Vibration Fluctuation value (rad/s) 65. 6. 55. 5. 1. 4.5 7.5 1.75 14. Stiffness coefficient a Drive shaft output speed (a) Fluctuation value (N mm),.,1..,9.,8.,7.,6.,5.,4.,. 1. 4.5 7.5 1.75 14. Drive shaft output torque Stiffness coefficient a (b) Figure 6: Output speed and torque of the drive shaft. Fluctuation value (rad/s),5.,4.,.,.,1.. 1,99. 1,98. 1,97. 1,96. 1. 4.5 7.5 1.75 14. Gear output speed Stiffness coefficient a (a) Fluctuation value (N mm),55.,5.,45.,4. 1. 4.5 7.5 1.75 14. Stiffness coefficient a Gear output torque Figure 7: Output speed and torque fluctuations of the driven gear. (b) stiffness. Considering the output fluctuations of drive shaft and driven gear, and the influence of gear transmission error, the fluctuation values of drive shaft are lower than that of driven gear. Compared with the initial speed and torque, the fluctuations are relatively smaller. From the fluctuation trend, there eists a reasonable region of bearing stiffness coefficient 1. a 1.5 in which the dynamic fluctuations of the shaftfinal drive system keep a smaller and steady state. In fact, the speed and torque fluctuations are closely related to engine output dynamics and transmission ratio. When running in the rd gear, the power matching parameters in detail are as shown in Table 5. The curves of seven different colours from a1 to a7 which represent the output fluctuations under different speeds are shown in Figure 8, corresponding to seven groups of bearing stiffness.theselectedstiffnessreflectsthekeydynamicsof this shaft-final drive system as it originates from the base points:.6,.7,.8, 1., 1., 1., and 1.5 in turn. Figures 8(a) and 8(b) are the output speed fluctuation curves of drive shaft anddrivengear. In Figure 8(a), the output fluctuations of drive shaft increase as the input speed increases at 1. a 1.5, but the fluctuations increase sharply above 57 rpm from.6 to.8. Figure 8(b) shows that the output fluctuations are not obvious under different bearing stiffness at the low input speeds, and Table 5: Drive shaft input speed/torque matching parameters. Engine output speed Engine output torque 1 (N mm) Output speed in third gear Output torque in third gear 1 (N mm) 5 5 4 45 5 55 99 14 18 11 19 16 14 1785.7 14.9 5 857.1 14. 571.4 98.6 18.6 145.6 151. 154 15.6 148.4 145.6 it is minimal from 18 to 4 rpm at a =.8.However, thecurvesshowanobviousdifferenceabove5rpm,the difference is minimal at a =1.andthenfollowsby1.and 1.5, and the fluctuations of driven gear have a sudden change at.6 a.8. 6. Eperimental Verification From the above theoretical analysis, the influence of bearing stiffness on the nonlinear dynamics of a shaft-final drive

11 Drive shaft output speed 4 5 5 15 1 5 15 Fluctuation value (rad/s) Fluctuation value (rad/s) Shock and Vibration 5 Input speed.6.7.8 1. 5 4 a = 1. a = 1. a = 1.5 Driven gear output speed 8 6 4 18 16 14 15 5 Input speed.6.7.8 1. 5 4 a = 1. a = 1. a = 1.5 (a) (b) Figure 8: Speed fluctuations of the drive shaft and driven gear. Photoelectric sensor The vertical direction The aial direction Figure 9: Layout and installation of two sensors. system is obvious, and the different bearing stiffness coefficients lead to the resonance parameters of the system changing. In the installation process, bearing preload adjustment is a simple and effective method to change the bearing stiffness, but it is still not enough for the improved system. To obtain suitable preload adjustment, the initial bearing pairs are replaced by other ones with the improved stiffness. Comparison test of the initial (a = 1.) and optimized bearing stiffness (a = 1.) of the drive system is designed to analyze the rear ale vibration behaviour and interior noise under different conditions. 6.1. Shaft-Final Drive System Vibration Behaviour. For obtaining better test results, the self-developed software of Portable vibration noise performance testing system is designed by Wuhan University of Technology (WUT) to detect the relationship between shaft speed and rear ale vibration in vehicle driving. Its hardware part consists of computer, sensors, DAQ Card, preamplifier, and so forth and the key structure components are shown in Figure 9. There are two different types of sensors: one is photoelectric sensor, which is used to detect the speed; its magnetic base is fied on flat chassis, and it is installed to vertical alignment of reflective tape pasted in the drive shaft keeping mm distance; the other is the vibration acceleration sensor, which is used to rear ale vibration behaviour. The two sensors are fied on the reducer along vertical and horizontal ais, the signal lines connect two channels of instrument bo, and the sensitivity values are 4.77 pc/s and 4.94 pc/s, respectively. In order to analyze the relative error between the simulation and eperiment, comparative analysis is used to research the system response. The speed fluctuations of driven gear which reflect the system output characteristics are obvious compared to that of drive shaft; it can be tested in the shaftfinal drive assembly with the Rear ale noise test platform in SAIC-GM-Wuling Automobile Co., Ltd. (SGMW). The statistical data are listed in Table 6. From the comparative analysis in Table 6, the results of dynamic simulation match with the eperiment data in the commonly used speed range of 15 rpm rpm, the relative errors of the initial and optimized driven gear speed fluctuations are less than %, the relative error maimum is.6% at a = 1. with the input speed of 15 rpm, and the fluctuations at a = 1. are less than that at a = 1.; it can be interpreted as the ecitation frequency is away from the system natural frequency at a = 1., which can provide good vibration characteristics. Overall, the relative errors of the

1 Shock and Vibration Amplitude (m/s ) 45 Output along vertical ais 4 5 5 15 1 5 1 4 5 6 Target-5th gear-wot and Coast down 5th gear-wot 5th gear-coast down (a1) Initial test Amplitude (m/s ) 45 Output along vertical ais 4 5 5 15 1 5 1 4 5 6 Target-5th gear-wot and Coast down 5th gear-wot 5th gear-coast down (a) Improved test Amplitude (m/s ) 45 Output along horizontal ais 4 5 5 15 1 5 1 4 5 6 Target-5th gear-wot and Coast down 5th gear-wot 5th gear-coast down (b1) Initial test Amplitude (m/s ) 45 Output along horizontal ais 4 5 5 15 1 5 1 4 5 6 Target-5th gear-wot and Coast down 5th gear-wot 5th gear-coast down (b) Improved test Figure 1: Test curves along vertical and horizontal ais. Table 6: Comparative analysis on the speed fluctuations of the driven gear. a = 1. a = 1. Input speed Simulation (rad/s) Eperiment data (rad/s) Relative error (%) 15 5.9 11.8.6 14.77 8.5.66 5 195.95 1975.7 1.11 1899.1 191.9 1.7 15 4.5 74.88.1 5.67 15.74.5 5 1947. 1966.8.98 189.6 1918.56 1.49 simulation and eperiment results follow the same trend; the dynamic simulation can meet the system test accuracy. When the field test layout is finished, the vibration responses under different conditions are obtained in the test, and the initial and improved vibration signals along vertical and horizontal ais are shown in Figure 1. In Figure 1, the vertical fluctuations are larger from rpm to 5 rpm and stronger than those along horizontal ais obviously. The dynamic behaviour is slightly different under different conditions, considerable fluctuation mainly occurs from 5 rpm to 4 rpm in Figure 1(a1), and it has an abrupt horizontal vibration in 5th gear of WOT in Figure 1(b1). Compared with the initial test, the improved vibration signals shown in Figures 1(a) and 1(b) are more stable, and the vertical fluctuation amplitudes are below the target;thistrendisobviousfrom5rpmto4rpm. 6.. Interior Noise in Vehicle Driving. LMS Test. Lab is widely adopted for data acquisition and processing in vehicle NVH performance evaluation. In the test road, the repetitive tests ofthewideopenthrottle(wot)andcoastdownwith5th gear are finished with the test system, and then the initial and improved vertical vibration signals are shown in Figure 11. In Figure 11(a), the interior noise mostly ranges from 4.5 db to 67.5 db in WOT and 4 db to 57.5 db in Coast down; the peak occurs at 9 rpm and 5 rpm nearby; when the bearing stiffness is improved in Figure 11(b), the overall interior noise decreases obviously by comparison, especially in the peak region from 9 rpm to 5 rpm. When running at a low speed, the average decibel value is larger than the initial test, but the noise level is mostly below the target, which reflects the consistency to the theoretical analysis on the nonlinear dynamics of the shaft-final drive system. 7. Conclusions A 14-DOF coupled vibration model of a shaft-final drive system is established; the influence of nonlinear factors was considered. Based on the deduced dynamic differential equations, numerical analysis method is adopted to analyze the

Shock and Vibration 1 (Pa) (db) (A) 7. 1. 65. 6. 55. 5. 45. 4. 5.. 5... 18 5 5 75 5 5 7 Amplitude (Pa) (db) (A) 7. 1. 65. 6. 55. 5. 45. 4. 5.. 5.. 18 5 5 75 5 5 7. Amplitude F F F Target--5th gear WOT and Coast down Order 1.1 left (A)--5th gear WOT Order 1.1 left (A)--5th Coast down F F F Target--5th gear WOT and Coast down Order 1.1 left (A)--5th gear WOT Order 1.1 left (A)--5th Coast down (a) Initial test (b) Improved test Figure 11: Interior noise test under different conditions. dynamic response with the introduced stiffness coefficient. Results show that improving the bearing stiffness can reduce the vibration response at the given speed; when the bearing stiffness increases to a critical value, the reduction becomes insignificant, and the resonance regions of the coupled system vary with the bearing stiffness at the different driving speeds. Influence of different bearing stiffness on the nonlinear system was simulated to investigate the dynamic response at the given speed and different speeds. When the stiffness coefficient a is 1., compared with other coefficients, the output fluctuations of the drive shaft and gear can be decreased, and the system performance improves significantly. Then the eperiments on the system vibration behaviour and interior noise in vehicle driving were carried out, and the selfdeveloped software of Portable vibration noise performance testing system was designed for performance detection of the vibration system. By comparative analysis of the initial and improved system, the vibration behaviour and interior noise can be improved when a is 1.; meanwhile, the relative error analysis of the simulation and eperiment demonstrates the correction of numerical results. Nomenclature θ i : Angular displacement of transmission components (where i=1,...,1) i, y i, z i : Displacement of hypoid gear pair along -, y-, and z-aes (where i=p,g) c m : Average damping of hypoid gear α n : Normal pressure angle β 1 : Mean spiral angle δ 1 : n : Pitchconeangle Relative displacement of normal direction at the contact point a: Bearing stiffness coefficient (ratio of selected stiffness and initial bearing stiffness) n V : Speed variation coefficient (ratio of selected speed and given speed) k m : Mean mesh stiffness k Ai :Fluctuationamplitudeinithorder Ω h : Mesh frequency of hypoid gear φ k : Initial phase of dynamic mesh stiffness R i : Radius of pinion and gear at the contact point (where i=p,g) b: Half of gear backlash φ e : Initial phase of static transmission error e : Mean transmission error at the contact point e Ai : Transmission error in ith order at the contact J i : point Equivalent moment of inertia of transmission components (where i=1,...,8) m ji : Equivalent lumped mass of pinion and gear (where i=p,g; j=,y,z; m p =m yp =m zp, m g =m yg =m zg ) K i : Equivalent stiffness of shaft (where i=1,...,4) and half-shaft (where i=7,8) c i : Damping of shaft (where i=1,...,4)and half-shaft (where i=7,8) K ij : Equivalent stiffness along -, y-, and z-aes c ij : (where i=p,g; j=,y,z) Equivalent damping along -, y-, and z-aes (where i=p,g; j=,y,z). Competing Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work was supported and funded by the Cooperation and Echange Projects of Guangi Science and Technology Department (Grant no. 14611-6) and the Scientific and Technological Project (Grant no. 1F). Thanks are due to SAIC-GM-Wuling Automobile Co., Ltd., for assistance

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