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International Journal of Automotive Technology Vol 10 No 4 pp 459 467 2009 DOI 10 1007 s12239 009 0052 y Copyright 2009 KSAE 1229 9138 2009 047 07 459 VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSION H W LEE 1 S H PARK 1 M W PARK 2 and N G PARK 1 1 School of Mechanical Engineering Pusan National University Busan 609 735 Korea 2 Wind Energy Biz Division Hyosung Corporation Nae dong Changwon si Gyeongnam 641 712 Korea Received 1 August 2008 Revised 5 January 2009 ABSTRACT A mathematical model of an automotive transmission is developed that considers the flexibility of the shafts bearings and gear teeth and gyroscopic effects of geared rotors The transverse torsional and axial motions are strongly coupled due to helical gearing The excitation forces acting on the automotive transmission are classified into first second and third grades based on the magnitudes of the forces that are determined by the perturbation method The excitation forces are caused by the mass imbalances among gears misalignment of shafts clearance and non linear deformation of bearings transmission errors and the periodic variation of the gear mesh stiffness A bench test on loading conditions is carried out for the third speed of the automotive transmission The experimental results of vibration characteristics are compared with those from theoretical analysis The results show good agreement i e within a tolerance of 3 3 KEY WORDS Automotive transmission Transmission error Critical speed Perturbation Loading condition bench test Helical gearing Vibration 1 INTRODUCTION Recently consumer preferences for automobiles have been focused on both performance and quality In particular consumers look for improved driving comfort safety power performance stability of the steering system and fuel economy The transmission which is a major component of the automobile is developed to satisfying the more stringent requirements of high capacity high endurance compact size and low vibration noise An automotive trans mission consists of shafts helical gears bearings gear rotor systems a case etc Lim and Singh 1991 undertook modal analysis of automotive transmissions by considering the mount and the case By changing the structure of the case and the layout of the power shaft Rondo 1990 designed an automotive transmission with less gear noise Honda 1990 studied the modal characteristics of axial vibration in a gear chain Lida et al 1985 found that the dynamic behavior of a gear shaft system when the bending and torsional effects in the spur gears are coupled together differs from the result that is obtained when the system is regarded as a simple non coupled model Schwibinger and Nordmann 1988 found that this coupling of bending and torsion in spur gears affects the stability of gear shaft systems Choy et al 1991 developed a dynamic model which couples the bending and torsional effects of a 3 speed spur gear rotor system that is forced by mass im balance he then calculated the transient and steady state responses Choy and Ruan 1993 modeled the reduction gear box with a 1 speed spur gear pair He employed the transfer matrix method for the parts of the gear rotor bearing and finite element method FEM for the case He then compared the calculated results of the vibrational spectra with experimental data Kahraman et al 1992 described the critical speed of a 1 speed gear chain by considering the coupled effect of the bending and torsional vibration and used FEM to solve for the forced response of the mass imbalance and the transmission error Kahraman 1994 calculated the forced response to the steady state error in the transmission of a 3 speed helical geared reducer Honda et al 1990 developed a thin 1 speed spur gear chain to investigate the vibrational effect of a gear shaft and compared the theoretical result of the noise level with experimental result Lee et al 2007 formulated the tooth profile modification curve by considering the errors in the manufacture of the profile and elastic de formation of the gear teeth in an automotive transmission He performed a comparative analysis of the calculated and measured responses to the excitations that arise from errors in transmission his aim was to verify the applicability to automotive transmissions This paper develops a mathematical model for analyzing the vibration characteristics of an automotive transmission that is composed of a multi helical gear system The model accounts for the shaft and bearing flexibilities gyroscopic effects and coupling of the forces that arise from the transverse torsional and axial motions due to gearing The excitation forces acting upon the automotive transmission are classified into first second and third grades based on the magnitudes of the forces that are determined by the Corresponding author e mail shpark01 pusan ac kr 460 H W LEE S H PARK M W PARK and N G PARK perturbation method The excitation forces are caused by the mass imbalances of gears misalignment of shafts clearance and non linear deformation of bearings trans mission errors and periodic variation of the gear mesh stiffness A bench test on loading conditions is carried out in the case of the third speed of the automotive trans mission and the test results are compared with results from an analysis of vibration characteristics 2 A MATHEMATICAL MODEL OF AN AUTOMOTIVE TRANSMISSION The mathematical model of an FR type manual trans mission is shown in Figure 1 The model included 74 axis elements 3 disk elements 13 gear elements seven helical gear pairs and 13 bearing elements In Figure 3 the pre fixes S G B P and D respectively denote the axes of rotation gears bearings helical gear pairs and disks Further S1 S2 S3and S4 are the input output counter and reverse idle gear shaft respectively Also G1 G2 G3 G4 G5 G6 are the forth step third step second step fifth step gears respectively G7 G12 are the counter shaft gears and G13 is the reverse idle gear P1 P2 P3 P4 and P5 represents the forth first second third and fifth step gear pairs respectively P6 is the reverse gear pair of G11 and G13 while P7 is the reverse gear pair of G5 and G11 D1 is the sleeve hub for the third and fourth speeds D2 is the sleeve hub for the first and second speeds and D3 is a sleeve hub for the fifth speed and the reverse drive Also B1 B2 B3 and B6 denote ball bearings B4 and B5 are cylindrical roller bearings and B7 B13 are needle roller bearings The first speed second speed and third speed trans mission delivery paths are S1 G1 G7 G10 G4 D2 S2 S1 G1 G7 G9 G3 D2 S2 and S1 G1 G7 G8 G2 D1 S2 respec tively In addition the fourth speed fifth speed and reverse drive transmission delivery paths are S1 G1 D1 S2 S1 G1 G7 G12 G6 D3 S2 and S1 G1 G7 G11 G13 G5 D3 S2 respectively 2 1 Equation of Motion in an Automotive Transmission System The gear rotor system in an automotive transmission is composed of helical gear chains shafts rotors and bear ings The model accounts for the flexibility of shafts and bearings gyroscopic effects and coupling of forces that arise from transverse and torsional motions which are due to gearing Gear mesh stiffness is considered with respect to the elastic deformation of the mating gear teeth As for the rotor which is a rigid body gyroscopic effects are considered It is assumed that the bearing element is an all linear spring and that the shaft of rotation is an Euler beam this enables the consideration of both the momentum effect of the distributed mass and the elastic effect The mathematical model of an automotive transmission system is developed by assembling the models of the elements in the transmission by the substructure synthesis method The equations of motion for an automotive transmission can be written in matrix form as 1 In Equation 2 the generalized displacement vector w consists of the three displacement vectors x y and with the corresponding lateral x and y and torsional z rotational vectors being as follows 2 The equation of motion shown in Equation 1 includes the effects of inertia M gyroscopic forces G and stiffness K Based on the finite element modularization principle we consider the functions of individual parts of the auto motive transmission configuration Thus a vibrational sub model is established for each part of the configuration of the gear rotor element Figure 2 g47 g89 g2g13g2 g41 g89 g2g13g2 g45 g89 g2g31g2 g18 g89 g2g31g2 g90 g91 g92 g90 g91 g92 Figure 1 Mathematical model of an FR type manual trans mission VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSION 461 i In the case of a shaft of rotation nodes are specified at the positions where the diameter of the shaft changes Figure 2a ii In the case of a disk a node is specified at the center point Figure 2b iii In the case of a shaft where a disk is assembled the diameter of the shaft at the location of the disk is extended by half the thickness of the disk Kr mer 1993 iv In the case of a shaft where a bearing that is fixed to the case is assembled a node is specified at the central point of the bearing Figure 2d v In the case of a shaft where the idling gear and the needle bearing are attached dual collocated nodes are assigned to the gear and the central point of the needle bearing Figure 2e 2 2 Components of a Vibrational Model 2 2 1 Model of vibration of the gear chain An automotive transmission is composed of a very com plex multi helical gear system The modeling of the vibration of the teeth contact region is as follows 1 Calculate the equivalent mesh stiffness by considering the elastic deformation of the mating gear teeth 2 Neglect the frictional component of the distributed transmitted force that is spread over the faces of the mating gear teeth the distributed force can be defined by the average concentrated force at the pitch point and the average coupling force Neglecting the coupling force consider the lead crowning of the gear tooth surface We can define the transmitted force of mating gear teeth as the average concentrated force at the pitch point as shown in Figure 3 3 Consider the elastic deformation of only a gear tooth and not the body of the gear 4 As shown in Figure 4 decompose the mating gear teeth into two separate compressed linear springs P G1 and P G2 Here the orientation of the springs is perpen dicular to the teeth contact line g35g36 Figure 2 Modular method for an automotive transmission Figure 3 Model of a helical gear pair 462 H W LEE S H PARK M W PARK and N G PARK 5 The equivalent spring coefficients K1 and K2 can be calculated by the method used by Choi 1987 which considers bending and shear deformations by regarding a gear tooth as a cantilever beam and which derives gear contact deformations from Hertzian contact theory The mathematical model of a helical gear pair is shown in Figure 5 Let the center of the drive gear be the origin of the coordinates radial horizontal direction be the x axis and positive rotational direction be the z axis The direc tional vector of the tooth contact force t is defined as 3 In Equation 3 is the helix angle of the base circle while refers to the angle between the center of the drive and the driven gear As the rotational direction of the drive gear is counterclockwise in Figure 5 the angle of the line of action is expressed as In the above expression is the transverse running pre ssure angle The potential energy of a helical gear tooth is derived as 4 In Equation 4 D 1 and D 2 are the proportional matrices calculated from the linear correlation of the rigid body motion between the displacement of the tooth contact and center of the gears The tooth stiffness coefficient K th of the gear pair is cal culated by a program developed by Park 1987 The element stiffness matrix between two nodes can be calcu lated by Equation 4 This equation expresses the potential energy which is described with reference to the gene ralized displacement vector at the center of both mating gears assuming a lumped parameter system 2 2 2 Model of vibration of the shaft The model of vibration of the shaft is developed using the finite element method The model incorporates a momen tum effect of the distribution mass and elastic effect In Figure 6 N 1 and N 2 are the shape functions at nodes 1 and 2 respectively The generalized displacement vector at an arbitrary position is obtained as 5 The kinetic energy between nodes 1 and 2 is as follows 6 In Equation 6 7 In Equation 9 represents the density A is the sectional area I is the identity matrix and J is the inertia matrix g86 g2g31g2 g69g81g85 g69g81g85 g85g75g80 g69g81g85 g85g75g80 g2 g2g31g2 pi g20 g2g13g2 g56 g86g74 g2g31g2 g19 g20 g83 g19 g83 g20 g54 g45 g19g19 g45 g19g20 g45 g20g19 g45 g20g20 g83 g19 g83 g20 g45 g19g19 g2g31g2g45 g86g74 g38 g19 g54 g86 g86 g54 g38 g19 g45 g19g20 g2g31g2 g45g332 g86g74 g38 g19 g54 g86 g86 g54 g38 g20 g45 g20g19 g2g31g2 g45g332 g86g74 g38 g20 g54 g86 g86 g54 g38 g19 g45 g20g20 g2g31g2g45 g86g74 g38 g20 g54 g86 g86 g54 g38 g20 g83g2g31g2g48 g19 g83 g19 g86 g2g13g2g48 g20 g83 g20 g86 g54 g85 g2g31g2 g19 g20 g83 g19 g83 g20 g54 g47 g19g19 g47 g19g20 g47 g20g19 g47 g20g20 g83 g19 g83 g20 g47 g75g76 g2g31g2 g2 g18 g46 g48 g75 g54 g47 g85 g48 g76 g70g92g14g2g75g31g19g14g20g14g2g2g76g31g19g14g20g14 g47 g85 g2g31g2 g35g43 g18 g18 g44 Figure 4 Model of a helical gear pair Figure 5 Schematic of the mathematical model of a helical gear pair Figure 6 Vibrational model of a shaft element VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSION 463 The potential energy between nodes 1 and 2 is obtained as 8 In Equation 8 the matrices K 11 K 12 K 21 and K 22 are written as 9 2 2 3 Vibrational model of bearings A six degrees of freedom roller bearing stiffness matrix K b is used for each bearing support location Note that the generated bearing stiffness matrix not only couples the displacements in all three directions it also couples the moments that arise from the bending of the rotor 3 ANALYSIS OF THE VIBRATION CHARACTERISTICS AND EXPERIMENTAL VERIFICATION OF AN AUTOMOTIVE TRANSMISSION 3 1 Excitation Frequency of an Automotive Transmission An automotive transmission is excited by the excitation source which is classified as mass imbalance errors in the assembly of bearings and shafts tooth profiles and lead errors of gears clearance and non linear deformation of roller bearings and gear backlash and the periodic vari ation of gear tooth stiffness As with Figure 7 for deriving the excitation force of an automotive transmission we will consider a simple spring mass system that includes a time varying stiffness system and a nonlinear section The equation of motion is given by 10 In Equation 10 represents an infinitesimal value x is a response signal m is the mass parameter of the system K 0 is also a system parameter assuming the system is time invariant and f 0 t represents the main sinusoidal excita tion force of the automotive transmission system e g mass imbalance the centrifugal force of the rolling bear ing tooth errors of gears etc K 1 t and K 2 t respectively denote the first and the higher order time variant harmonic terms of the gear tooth stiffness coefficients x 2 x 3 refers to the non linear effect of the gear backlash and the bearing clearance In the right hand side of the equation f 1 t represents the combined effect of all the second or higher order harmonic excitation sources The first excitation source physically indwells the system The frequency of this forced vibration is related to the speed of the shaft in terms of the imbalance the centrifugal force of the rolling bear ing and tooth passing frequency Thus 11 In Equation 11 is the frequency of the first exciting force Let x t be as follows 12 By substituting 11 and 12 into 10 and combining first order terms we get 13 For Equation 13 14 Also by rewriting 10 in terms of O we get 15 In Equation 15 the harmonic exciting force of the second or higher order f 1 t arising from errors in the assembly of bearings and shafts and tooth errors of gears can be written as 16 In Eqation 16 i and i respectively denote the speed of the shaft and the tooth passing frequency Letting 17 while substituting 14 16 and 17 into 15 yields several excitation frequencies that correspond to the force terms g56 g53 g2g31g2 g19 g20 g83 g19 g83 g20 g54 g45 g19g19 g45 g19g20 g45 g20g19 g45 g20g20 g83 g19 g83 g20 g45 g75g76 g2g31g2 g2 g18 g46 g47 g75 g54 g45 g18 g47 g76 g70g92g14g2g75g31g19g14g20g14g2g2g76g31g19g14g20 g45 g68 g2g31g2 g45 g90g90 g45 g90g91 g45 g90g92 g45 g90 g90 g45 g90 g91 g18 g45 g91g91 g45 g91g92 g45 g91 g90 g45 g91 g91 g18 g45 g92g92 g45 g92 g90 g45 g92 g91 g18 g85g91g79g79g71g86g84g75g69 g45 g90 g90 g45 g90 g91 g18 g45 g91 g91 g18 g18 g79g90 g2g13g2 g45 g18 g2g13g2 g45 g19 g86 g2g13g2 g20 g45 g20 g86 g90g2g13g2 g90 g20 g90 g21 g13 g31g2g72 g18 g86 g2g13g2 g72 g19 g86 g72 g18 g86 g2g31g2 g2 g75 g19g31 g48 g72 g18g75 g71 g76 g75 g19 g86 g75 g19 g90g86 g2g31g2g90 g18 g86 g2g13g2 g90 g19 g86 g2g13g2 g20 g90 g20 g86 g2g13g2g49 g21 g79g90 g18 g2g13g2g45 g18 g90 g18 g2g31g2 g2 g75 g19g31 g48 g72 g18g75 g71 g76 g75 g19 g86 g90 g18 g86 g2g31g2 g2 g75 g19g31 g48 g90 g18g75 g71 g76 g75 g19 g86 g79g90 g19 g2g13g2g45 g18 g90 g19 g2g31g2 g45 g19 g332 g86 g90 g18 g2g13g2g72 g19 g90 g90 g18 g20 g2g13g2 g90 g18 g21 g72 g19 g86 g2g31g2 g2 g75 g19g31 g21 g2 g77 g20g31 g72 g18g75 g71 g76g77 g75 g86 g2g13g2 g2 g75 g19g31 g20 g2 g77 g20g31 g72 g18g75 g71 g76g77 g75 g86 g45 g19 g86 g2g31g2g45 g19 g71 g76 g75 g86 Figure 7 Simple spring mass system 464 H W LEE S H PARK M W PARK and N G PARK The term yields 18 From the term of we get 19 The second excitation frequency i 1 2 g718 N 2 then includes the frequencies in Eqs 18 19 The rewriting of 10 in terms of O 2 gives 20 In Equation 20 K 2 t can be expressed as 21 From Equation 14 and Equation 21 the term in Equation 21 is derived as k 2 3 g718 i 1 2 g718 N 1 22 Similarly the term yields the following j 1 2 g718 N 2 23 Therefore the third excitation frequency i 1 2 g718 N 3 includes the frequencies in Equations 22 23 These calculations yield the possible excitation frequencies in an automotive transmission as listed in Table 1 3 2 Vibrational Experiment on Automotive Transmission A test rig of the automotive transmission for a rear wheel drive car is shown in Figure 8 The test rig uses the T M DYNAMO TESTER K H I Company which consists of a 120 kw motor a dynamometer a controller two gearboxes and two torque meters Figure 9 shows the configuration of the experimental device for an automotive transmission The controller of the testing machine adds to the largest torque by each speed of the automotive transmission and increases the rotational speed up to 2500 rpm A triple axis accelerometer is attached to the upside of the automotive transmission case and a tachometer is installed in the input motor portion Vibration signals and rotational speeds are measured by the accelerometer and the tachometer The measuring equipment used is the 3560 PULSE frequency analyzer B K Co The signals measured by accelero meter and tachometer are analyzed For measuring the critical speed the vibration experi ment is carried out for each speed of the automotive trans mission The experimental results of the worst vibration noise in the case of the third speed are displayed as follows Figure 10 depicts the Waterfall diagram of lateral magni tude wherein the range of the input speed is 500 2500 rpm with increments of 10 rpm The order is on the horizontal axis while the rotational speed and magnitude are on the left and right sides of the vertical axis respectively If the horizontal area appears light it implies a high order of the measured vibration signals namely 18 15X 23X 36 3X 46X 54 45X 56 7X and 69X In relation to the force that is the excitation source it is important to compare the first forcing frequencies of the tooth passing frequencies of the g45 g19 g86 g90 g18 g75 g75 g19 g14g2g2g75g2g31g2g19g14g20g14 g48 g19 g72 g19 g86 g77 g75 g14g2g77 g76 g14g2g77g2g31g2g20g14g21g14g2 g14g2g