乘用車二軸式五檔變速器總成設(shè)計CAD圖紙+說明書,乘用車二軸式,五檔,變速器,總成,設(shè)計,CAD,圖紙,說明書 International Journal of Automotive Technology, Vol. 10, No. 4, pp. 459467 (2009)DOI 10.1007/s122390090052yCopyright 2009 KSAE12299138/2009/04707459VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSIONH. W. LEE1), S. H. PARK1)*, M. W. PARK2) and N. G. PARK1)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)ABSTRACTA 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 stronglycoupled due to helical gearing. The excitation forces acting on the automotive transmission are classified into first, second, andthird grades, based on the magnitudes of the forces that are determined by the perturbation method. The excitation forces arecaused 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 forthe third speed of the automotive transmission. The experimental results of vibration characteristics are compared with thosefrom 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, Vibration1. INTRODUCTIONRecently, 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 reductiongear box with a 1-speed spur-gear pair. He employed thetransfer-matrix method for the parts of the gear-rotor-bearing and finite element method (FEM) for the case. Hethen compared the calculated results of the vibrationalspectra with experimental data. Kahraman et al. (1992)described the critical speed of a 1-speed gear-chain byconsidering the coupled effect of the bending and torsionalvibration, and used FEM to solve for the forced response ofthe mass imbalance and the transmission error. Kahraman(1994) calculated the forced response to the steady-stateerror in the transmission of a 3-speed, helical-gearedreducer. Honda et al. (1990) developed a thin, 1-speed,spur-gear chain to investigate the vibrational effect of agear-shaft and compared the theoretical result of the noiselevel with experimental result. Lee et al. (2007) formulatedthe tooth-profile modification curve by considering theerrors 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 andmeasured responses to the excitations that arise from errorsin transmission; his aim was to verify the applicability toautomotive transmissions.This paper develops a mathematical model for analyzingthe vibration characteristics of an automotive transmissionthat is composed of a multi-helical gear system. The modelaccounts for the shaft and bearing flexibilities, gyroscopiceffects, and coupling of the forces that arise from thetransverse, torsional, and axial motions due to gearing. Theexcitation forces acting upon the automotive transmissionare classified into first, second, and third grades, based onthe magnitudes of the forces that are determined by the*Corresponding author. e-mail: shpark01pusan.ac.kr460H. W. LEE, S. H. PARK, M. W. PARK and N. G. PARKperturbation method. The excitation forces are caused bythe mass imbalances of gears, misalignment of shafts,clearance and non-linear deformation of bearings, trans-mission errors, and periodic variation of the gear meshstiffness. A bench test on loading conditions is carried outin the case of the third speed of the automotive trans-mission and the test results are compared with results froman analysis of vibration characteristics.2. A MATHEMATICAL MODEL OF AN AUTOMOTIVE TRANSMISSIONThe mathematical model of an FR-type manual trans-mission is shown in Figure 1. The model included 74 axiselements, 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 ofrotation, 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 P5represents the forth, first, second, third, and fifth-step gearpairs, respectively. P6 is the reverse gear-pair of G11 andG13, while P7 is the reverse gear-pair of G5 and G11. D1 isthe sleeve hub for the third and fourth speeds, D2 is thesleeve hub for the first and second speeds, and D3 is asleeve hub for the fifth speed and the reverse drive. Also,B1, B2, B3, and B6 denote ball bearings, B4 and B5 arecylindrical roller bearings, and B7-B13 are needle rollerbearings.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 TransmissionSystemThe gear-rotor system in an automotive transmission iscomposed of helical gear chains, shafts, rotors, and bear-ings. The model accounts for the flexibility of shafts andbearings, gyroscopic effects, and coupling of forces thatarise from transverse and torsional motions, which are dueto gearing. Gear mesh stiffness is considered with respectto the elastic deformation of the mating gear-teeth. As forthe rotor, which is a rigid body, gyroscopic effects areconsidered. 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 effectof the distributed mass and the elastic effect.The mathematical model of an automotive transmissionsystem is developed by assembling the models of theelements in the transmission by the substructure-synthesismethod. The equations of motion for an automotivetransmission 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, withthe corresponding lateral (x and y) and torsional (z)rotational vectors being as follows.(2)The equation of motion shown in Equation (1) includesthe effects of inertia, M, gyroscopic forces, G, andstiffness, K.Based on the finite element modularization principle, weconsider the functions of individual parts of the auto-motive-transmission configuration. Thus, a vibrational sub-model is established for each part of the configuration ofthe gear-rotor element (Figure 2).M w + G w + K w = 0 w = xyzxyzFigure 1. Mathematical model of an FR-type manual trans-mission.VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSION461(i) In the case of a shaft of rotation, nodes are specified atthe 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, thediameter of the shaft at the location of the disk is extendedby half the thickness of the disk (Krmer, 1993).(iv) In the case of a shaft where a bearing that is fixed to thecase is assembled, a node is specified at the central point ofthe bearing (Figure 2d).(v) In the case of a shaft where the idling gear and theneedle bearing are attached, dual, collocated nodes areassigned to the gear and the central point of the needlebearing (Figure 2e).2.2. Components of a Vibrational Model2.2.1. Model of vibration of the gear-chainAn automotive transmission is composed of a very com-plex multi-helical gear system.The modeling of the vibration of the teeth-contact regionis as follows.(1) Calculate the equivalent mesh stiffness by consideringthe elastic deformation of the mating gear-teeth.(2) Neglect the frictional component of the distributedtransmitted force that is spread over the faces of themating gear-teeth; the distributed force can be definedby the average concentrated force at the pitch point andthe average coupling force. Neglecting the couplingforce, consider the lead crowning of the gear-toothsurface. We can define the transmitted force of matinggear-teeth as the average concentrated force at the pitchpoint, as shown in Figure 3.(3) Consider the elastic deformation of only a gear toothand not the body of the gear.(4) As shown in Figure 4, decompose the mating gear-teethinto two separate, compressed, linear springs, P-G1 andP-G2. Here, the orientation of the springs is perpen-dicular to the teeth-contact line, .ABFigure 2. Modular method for an automotive transmission.Figure 3. Model of a helical-gear pair.462H. W. LEE, S. H. PARK, M. W. PARK and N. G. PARK(5) The equivalent spring coefficients, K1 and K2, can becalculated by the method used by Choi (1987), whichconsiders bending and shear deformations by regardinga gear tooth as a cantilever beam and which derivesgear contact deformations from Hertzian contact-theory.The mathematical model of a helical-gear pair is shownin Figure 5. Let the center of the drive-gear be the origin ofthe 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 driveand the driven gear. As the rotational direction of the drive-gear is counterclockwise in Figure 5, the angle of the lineof 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).D1 and D2 are the proportional matrices, calculated fromthe linear correlation of the rigid body motion between thedisplacement of the tooth-contact and center of the gears.The tooth-stiffness coefficient, Kth, of the gear pair is cal-culated by a program developed by Park (1987). Theelement-stiffness matrix between two nodes can be calcu-lated by Equation (4). This equation expresses the potentialenergy, which is described with reference to the gene-ralized displacement vector at the center of both matinggears, assuming a lumped-parameter system.2.2.2. Model of vibration of the shaftThe model of vibration of the shaft is developed using thefinite element method. The model incorporates a momen-tum effect of the distribution mass and elastic effect.In Figure 6, N1 and N2 are the shape functions at nodes 1and 2, respectively. The generalized displacement vector atan 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 sectionalarea, I is the identity matrix, and J is the inertia matrix.t = cos cossin cossin = 2- - + Vth = 12- -q1q2TK11K12K21K22q1q2K11 = KthD1TttTD1K12 = KthD1TttTD2K21 = KthD2TttTD1K22 = KthD2TttTD2q = N1( )q1t ( ) + N2( )q2t ( )Ts = 12- -q1q2TM11M12M21M22q1q2Mij = 0LNiTMsNj()dz, i=1,2, j=1,2,Ms = A I 00JFigure 4. Model of a helical-gear pair.Figure 5. Schematic of the mathematical model of ahelical-gear pair.Figure 6. Vibrational model of a shaft element.VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSION463The potential energy between nodes 1 and 2 is obtainedas: (8)In Equation (8), the matrices, K11, K12, K21, and K22, arewritten as: (9)2.2.3. Vibrational model of bearingsA six degrees-of-freedom, roller-bearing stiffness-matrix,Kb, is used for each bearing support location.Note that the generated bearing stiffness-matrix not onlycouples the displacements in all three directions, it alsocouples the moments that arise from the bending of therotor.3. ANALYSIS OF THE VIBRATION CHARACTERISTICS AND EXPERIMENTAL VERIFICATION OF AN AUTOMOTIVE TRANSMISSION3.1. Excitation Frequency of an Automotive TransmissionAn automotive transmission is excited by the excitationsource, which is classified as: mass imbalance; errors in theassembly of bearings and shafts; tooth profiles and leaderrors of gears; clearance and non-linear deformation ofroller bearings and gear backlash; and the periodic vari-ation of gear-tooth stiffness.As with Figure 7, for deriving the excitation force of anautomotive transmission, we will consider a simple spring-mass system that includes a time-varying-stiffness systemand a nonlinear section.The equation of motion is given by:(10)In Equation (10), represents an infinitesimal value, x is aresponse signal, m is the mass parameter of the system, K0is also a system parameter (assuming the system is time-invariant), and f0(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. K1(t) and K2(t) respectivelydenote the first and the higher-order, time-variant, harmonicterms of the gear-tooth stiffness coefficients. x2+x3 refersto the non-linear effect of the gear backlash and the bearingclearance. In the right-hand-side of the equation, f1(t)represents the combined effect of all the second- or higher-order harmonic excitation sources. The first excitationsource physically indwells the system. The frequency of thisforced vibration is related to the speed of the shaft in termsof 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 excitingforce.Let x(t) be as follows.(12)By substituting (11) and (12) into (10), and combiningfirst-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 secondor higher order, f1(t), arising from errors in the assembly ofbearings and shafts and tooth errors of gears, can be writtenas:(16)In Eqation (16), i and i respectively denote the speed ofthe shaft and the tooth-passing frequency.Letting(17)while substituting (14), (16), and (17) into (15) yieldsseveral excitation frequencies that correspond to the forceterms. VS = 12- -q1q2TK11K12K21K22q1q2Kij = 0LMiTK0Mj()dz, i=1,2, j=1,2Kb = KxxKxyKxzKxxKxy0KyyKyzKyxKyy0KzzKzxKzy0symmetricKxxKxy0Kyy00mx + K0 + K1t ( ) + 2K2t ( )x + x2x3+()= f0t ( ) + f1t ( )f0t ( ) = i1=Nf0ieji1( )ti1( )x t ( ) = x0t ( ) + x1t ( ) + 2x2t ( ) + O 3()mx 0 + K0 x0 = i1=Nf0ieji1( )tx0t ( ) = i1=Nx0ieji1( )tmx 1 + K0 x1 = K1t ( )x0 + f1x( )x02 + x03f1t ( ) = i1=3 k2=f0iejkit + i1=2 k2=f0iejkitK1t ( ) = K1ejitFigure 7. Simple spring-mass system.464H. W. LEE, S. H. PARK, M. W. PARK and N. G. PARKThe term yields:(18)From the term of , we get:(19)The second excitation frequency, , i = 1, 2, N2, thenincludes the frequencies in Eqs. (18)(19).The rewriting of (10) in terms of O(2) gives:(20) In Equation (20), K2(t) can be expressed as:(21)From Equation (14) and Equation (21), the term, ,in Equation (21) is derived as:, k = 2,3, , i = 1, 2, N1(22)Similarly, the term, , yields the following., j = 1, 2, N2(23)Therefore, the third excitation frequency, , i = 1, 2, N3, includes the frequencies in Equations (22)(23). Thesecalculations yield the possible excitation frequencies in anautomotive transmission, as listed in Table 1.3.2. Vibrational Experiment on Automotive TransmissionA test rig of the automotive transmission for a rear-wheel-drive car is shown in Figure 8. The test-rig uses the T/MDYNAMO TESTER(K.H.I Company), which consists of a120 kw motor, a dynamometer, a controller, two gearboxes,and two torque meters. Figure 9 shows the configuration ofthe experimental device for an automotive transmission.The controller of the testing machine adds to the largesttorque by each speed of the automotive transmission andincreases the rotational speed up to 2500 rpm. A triple-axisaccelerometer is attached to the upside of the automotivetransmission case, and a tachometer is installed in theinput-motor portion. Vibration signals and rotational speedsare measured by the accelerometer and the tachometer. Themeasuring equipment used is the 3560 PULSE frequencyanalyzer (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 5002500 rpmwith increments of 10 rpm. The order is on the horizontalaxis, while the rotational speed and magnitude are on theleft and right sides of the vertical axis, respectively. If thehorizontal area appears light, it implies a high order of themeasured vibration signals, namely, 18.15X, 23X, 36.3X,46X, 54.45X, 56.7X, and 69X. In relation to the force thatis the excitation source, it is important to compare the firstforcing frequencies of the tooth-passing frequencies of theK1t ( )x0ii1( ), i = 1,2,N1f1t ( )ki, kj, k = 2,3, , i = 1,2,3, j = 1,2i2( )mx 2 + K0 x2 = K1t ( )x1K2t ( )x0K2t ( ) = k2=K2ejkitK2t ( )x0kii1( )K1t ( )x1kij2( )i3( )Figure 8. Test rig of the automotive transmission.Figure 9. Configuration of the experimental device for anautomotive transmission.Figure 10. Waterfall diagram of the automotive trans-mission (third speed).VIBRATIONAL CHARACTERISTICS OF AUTOMOTIVE TRANSMISSION465input (1=23X) and output (2=18.15X). The secondforcing frequency, which is an integral multiple of thetooth-passing frequency (21, 31, 22, 32), and thirdforcing frequency, 56.7X (=1+22212), which iscaused by the backlash of gears and the nonlinear charac-teristics of the bearing gap, are also critical excitationsources (Table 1). The bode plot of every order is shown inFigure 11.3.3. Comparison of the Analytical and the ExperimentalResults of the Automotive TransmissionIn the case of the third speed, analytical results of thevibration/noise characteristics were compared with theexperimental results. Figure 12 shows the Campbell dia-gram of the analytical results in the case of the third speedof a vibrational model of