Torsional Vibration Model Based Optimization of Gearbox Geometric Design Parameters to Reduce Rattle Noise in an Automotive Transmission

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Torsional vibration model based optimization of gearbox geometric designparameters to reduce rattle noise in an automotive transmission

Mehmet Bozca

Yildiz Technical University, Mechanical Engineering Faculty, Machine Design Division, 34349, Yildiz, Istanbul, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:

Received 23 November 2009Received in revised form 8 June 2010Accepted 22 June 2010Available online 24 July 2010

The optimization of gearbox geometricdesign parameters to reducerattle noise in an automotivetransmission based on a torsional vibration model approach is studied. Rattle noise is calculatedand simulated based on the design parameters of a 5-speed gearbox, and all pinion gears andwheel gears are helical. The effect of the design parameters on rattle noise is analyzed. Theobserved rattle noise proles are obtained depending on the design parameters. During theoptimization, a four-degree-of-freedom torsional vibration model of the pinion gearwheel gearsystem is obtained and the minimum singular value of the transfer matrix is considered as theobjective functions and design parameters are optimized under several constraints that includebending stress, contact stress and a constant distance between gear centers. Therefore, byoptimizing the geometric parameters of the gearbox such as, the module, number of teeth, axialclearance, and backlash, it is possible to obtain a light-weight-gearbox structure and minimize therattling noise. It is concluded that the optimized geometric design parameters lower the rattlenoise by 10% compared to thecalculated rattle noise for sample gearbox.All optimized geometric

design parameters also satisfy all constraints. 2010 Elsevier Ltd. All rights reserved.

Keywords:

OptimizationGearboxRattle noiseTorsional vibrationAutomotive transmission

1. Introduction

The optimization of gearbox geometric design parameters to reduce rattle noise in automotive transmissions based on atorsional vibration model approach is studied.

Gear motion causes rattling and clattering noise, and noise level is considered to be a comfort factor in the automotiveindustry. Therefore, reducing rattling and clattering noise in the gearbox is important in the automotive transmission for acomfortable car design.

Gears are widely used in automotive transmissions to transmit mechanical power from one shaft to another. The purpose of thegears is to couple two shafts together such that the rotation of the output, or driven, shaft is a function of the rotation of the input,

or driving, shaft[4].Rattling and clattering noise in automotive transmissions are caused by torsional vibration that is transmitted from the

internal combustion engine to the transmission input shaft. This noise is known as rattling when the transmission is inneutral, and as clattering when the gear is engaged under power or in overrun[914].

Rattling and clattering are caused by torsional vibration of loose parts, i.e. parts, such as idler gears, synchronizer ringsand sliding sleeves, which are not under load and therefore can move within their functional clearances [9,14].

Gear rattling noise is one of the major problems facing the industry, and the car industry in particular, because carsspend so much time idling under no load or very light loads[1].

Mechanism and Machine Theory 45 (2010) 15831598

Tel.: +90 212 383 27 93.E-mail address:[email protected].

0094-114X/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2010.06.014

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The parameters that are responsible for rattle and clatter are classied as geometric parameters and operationalparameter [914]. The geometrical parameters include the module m, number of teeth z, helix angle , axial clearance sand backlashsvas shown inFig. 3. The operational parameters include the angular acceleration 1,and excitation frequencyan [914].

Transmissionuid as an engineering design parameter also has a considerable inuence on rattling and clattering noises. Theimportant factors include the type of oil, the additives used and the viscosity and the level of oil in the transmission which togetheract on a gear pair as drag torque, resulting in a reduction in rattling and clattering noises, especially at low speeds and when cold[914].

The analysis suggests that in order to reduce the gear rattle noise, rather than to increase the oil level in a gearbox and thusdecreasing the mechanical efciency, it would be opportune to guarantee the oil presence between the meshing teeth by means ofa suitable lubrication device feeding the lubricant only in the meshing zone[15].

Sliding friction between meshing teeth is one of the primary excitations for noise and vibration in geared systems. Among thedifferent kinds of nonlinearitiesin gear system,suchas clearance, spatial variations and slidingfriction, the effect of frictionis the leastunderstood. Certain uniquecharacteristics of gear tooth sliding make it a potentially dominant factor. For instance, dueto the reversalin direction during meshing action, friction is associated with a large oscillatory component, which causes a higher bandwidth in thesystem response. Furthermore, it becomes more signicant at high values of torque and lower speeds, due to the tribologicalcharacteristics as well as due to higher force transmissibility in the sliding direction[17].

Optimization of the gear's macro-geometry i.e. the use of high contact ratio gears that has lead to minor noise emissions withhigher transmitted power levels. Optimization of the gear's micro-geometry i.e. trying to balance load-induced teeth deectionswith prole corrections that has generally lead to less noisy transmission effects. This is not a suitable solution for an overallworking range; therefore prole corrections must be determined statistically to take into account manufacturing deviations whichwill overlap their effect[16].

1.1. Gearbox mechanism

The gearbox mechanism includes pinion gears, wheel gears, an input shaft, an output shaft, a lay shaft, a bearing support, andsynchronizers, as shown inFig. 1.

1.1.1. Pinion gears and gear wheels

All pinion gears and wheel gears are helical, and all gears are made of 16MnCr5.

1.1.2. Input shaft

The constant pinion gear and rear wheel gear are engaged on the input shaft. The rear wheel gear is the idler gear and runs inthe needle bearing on the input shaft. The synchronizer of the rear wheel gear is connected to the input shaft.

1.1.3. Output shaft

The 1st, 2nd and 3rd wheel gears and 4th pinion gear are engaged on the output shaft. The 1st, 2nd and 3rd wheel gears areidler gears and run in the needle bearings on the output shaft. The synchronizer of the 1st and 2nd wheel gears is connected to theoutput shaft.

1.1.4. Lay shaft

The 1st, 2nd, and 3rd pinion gears, the rear pinion gear and the 4th wheel gear, and the constant wheel gears are engaged onthe lay shaft. The 4th wheel gear is the idler gear and runs in the needle bearing on the lay shaft. The synchronizer of the 3rd and4th wheel gears is connected to the lay shaft.

Fig. 1.5-Speed gearbox for automotive transmission.

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1.1.5. Synchronizer

All synchronizers run in the needle bearings to maximize smoothness.

2. Calculation of rattle noise

Rattle noise is calculated and simulated based on the gearbox design parameters for a 5-speed gearbox for an automotivetransmission, which is shown inFigs. 1 and 2.

The rattle noise level of a complete automotive transmission is calculated as follows [913]:

LpComp= 10logn

i = 1100;1Lp;i

: 1

The rattle noise Lp, is calculated as follows by correlating the computed noise value and the measured noise level [913]:

Lp= 10log kIm+ 100;1Lbasic

2

where k is the calibration factor [] and Im is the average impact intensity [N]. The average impact intensity Im, is written as follows[913]:

Im= m2 1rb1CIm 3

where m2 is a loose part [kg], 1 is the angular acceleration [rad/s2], rb1 is the pitch circle radius [mm] and CIm is the related average

impact intensity []. The average impact intensityCIm, is written as follows[913]:

CIm= ffiffiffiffiffiffiffi

Csvp

1; 462 0; 714CfaCsa

0; 016Cfa+ 0; 12Csv

! 4

whereCsvis the non-dimensional circumferential backlash []. The non-dimensional circumferential backlashCsv[] is dened asfollows[913]:

Csv= sv

2an

rb1 15

wheresvis backlash [mm], an is the excitation frequency [rad/s] and Csa is the non-dimensional axial clearance []. The non-dimensional axial clearanceCsa[] is dened as follows[913]:

Csa= sa

2antan

rb1 16

where sa is the axial clearance [mm],is the helix angle [], Cfa is the related axial friction force [] and Lbasicis the basic noise level[dB].

Fig. 2.View of 5-speed gearbox for automotive transmission.

1585M. Bozca / Mechanism and Machine Theory 45 (2010) 15831598
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3. Calculating the load capacity of helical gears

Determining the service life and/or determining the strength of gears is crucial for gear manufacturers. Gear strength is denedby the bending and contact strengths[22].

3.1. Tooth bending stress

The tooth bending stress is calculated as follows, Fig. 4[2,3,5,7]. According to the ISO 6336, shear stresses due to lateral forceswere not taken into account when determining the loading capacity of the gear [3,22,23]. A tooth-root bending fatigue fractureusually starts at the 30 tangent of the root [3,8,22,24].

The real tooth-root stressFis calculated as follows[2,3,5,7]:

F= Ftbmn

YFYSYYKAKVKFKF 7

whereFtis the nominal tangential load [N],bis the face width [mm],mnis the normal module [mm],YFis the form factor [],YSisthe stress correction factor [], Y is the contact ratio factor [], KAis the application factor [], KVis the internal dynamic factor [],

KFis the face load factor for tooth-root stress [

] andKFis the transverse load factor for tooth-root stress [

].The permissible bending stressFpis calculated as follows[2,3,5,7]:

Fp= F limYSTYNYYRYX 8

Fig. 3.Design parameters for a gearbox.

Fig. 4.Bending stress at the tooth root.

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whereFlim is the nominal stress number (bending) [N/mm2],YSTis the stress correction factor [],YNis the life factor for the

tooth-root stress [],Yis the relative notch sensitivity factor [],YRis the relative surface factor [] andYXis the size factor thatrepresents the tooth-root strength [].

The safety factor for bending stress SFis calculated as follows[2,3,5,7]:

SF= Fp

F9

3.2. Tooth contact stress

The tooth contact stressHCis calculated as follows, as seen inFig. 5[2,3,5,6].The real contact stressHis calculated as follows[2,3,5,6]:

H=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFt

bmn

u+ 1u

s ZHZEZZ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKA KVKHKH

q 10

whered1 is the reference diameter of the pinion [mm], u is gear ratio [], ZH is the zone factor [], ZEis the elasticity factorffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN=mm2

q ,Z is the contact ratio factor [],Z is the helix angle factor [], KH is the face load factor for contact stress [] and KH

is the transverse load factor for contact stress [].

The permissible contact stressHpis calculated as follows[2,3,5,6]:

Hp = H limZNZLZVZRZWZX 11

whereHlim is the allowable stress numbers (contact) [N/mm2],ZNis the life factor for contact stress [],ZL is the lubrication

factor [],ZVis the velocity factor [],ZR is the roughness factor [],ZWis the work hardening factor [] andZXis the size factor forcontact stress [].

The safety factor for contact stress SHis calculated as follows[2,3,5,6]:

SH=HpH

12

4. Calculating the torsional vibration

The torsional vibration transmitted from the combustion engine to the transmission excites idler components such as the idlergears, synchronizer rings and sliding sleeves, to vibrate within their functional clearances[11].

A dynamic gear vibration model is a useful tool to study the vibration response of a gearedsystem with various gear parametersand operating conditions[27].

Fig. 5.Contact stress at the tooth ank.



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Torsional vibration models of gears system are classied according to their rigidity and elasticity such as purely torsional multi-body models, rigid multi-body models, exible multibody models and semi-rigid-elastic multi-body models[2530].

Torsional vibration models of gears system are also classied according to time-invariant and time-variant such as linear time-invariant (LTI) models with stiffness, linear time-varying (LTV) models with stiffness, time-varying models with backlash, as wellas time-invariant average stiffness and time-invariant models in both backlash and stiffness simultaneously[31].

4.1. Equation of motion of gears system

A four-degree-of-freedom model of the pinion gearwheel gear system is considered for simplicity. The pinion gear body andwheel gear body are assumed to be rigid. The teeth are assumed to be elastic and parallel springdamper combinations areassumed to exist between the teeth and the gear body. A four-degree-of-freedom model is shown in Fig. 6.

The equations of motion of the pinion gearwheel gear system are written in terms of the four-degree-of-freedom model asfollows[2528]:Sp,p,Swand w.

Eq.(13)is written for tooth i of a pinion gear as follows:

JpthS::

p= Der2dp pSp+ Ker

2dppSp+ Tp 13

whereJpth is the moment of inertiaof tooth i of the pinion gear [kg.mm2], rdp is the pitch circle radius of the pinion gear [mm], p is

the rotational position of the pinion gear body [rad], Sp is the rotational position of tooth i of the pinion gear [rad], p is the

rotational velocity of the pinion gear body [rad/s], Spis the rotational velocity of the tooth i of the pinion gear [rad/s],S::

pis the

rotational acceleration of the toothi of the pinion gear [rad/s2], andTpis the contact torque applied to tooth i [N.mm].Eq.(14)is written for a pinion gear body as follows:

Jpb::

b= 0 14

where Jpbis the moment of inertia of the pinion gear body [kg.mm2],

::bis the rotational acceleration of the pinion gear body

[rad/s2].Eq.(15)is written for tooth i of the wheel gear as follows:

JwthS

::w= Der

2dw wSw+ Ker

2dwwSw+ Tw 15

whereJwth is the moment of inertia of tooth i of the wheel gear [kg.mm2], rdw is the pitch circle radius of the wheel gear [mm], w is

the rotational position of the wheel gear body [rad], Swis the rotational position of tooth i of the wheel gear [rad], w is therotational velocity of the wheel gear body [rad/s], Swis the rotational velocity of the toothi of the wheel gear [rad/s], S

::w is the

rotational acceleration of tooth i of the wheel gear [rad/s2], andTwis the contact torque applied to tooth i [N.mm].Eq.(16)is written for the wheel gear body as follows:

Jwb

::

w= 0 16

where Jwb

is the moment of inertia of the wheel gear body [kg.mm2], ::

wis the rotational acceleration of the wheel gear body[rad/s2].

Fig. 6.4-Degree-of-freedom model.

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Keis the stiffness coefcient [N/mm] and is assumed to be time-invariant. The stiffness coefcientsKeresulting from the toothsurface contact are written as follows[29,32]:

Ke = ES0412

17

whereEis Young's modulus [N/mm2],is the Poisson's ratio [] andS0is the thickness of gear [mm] which is written as follows[3,5]:

S0= m n= 2 18

Deare the viscous damping coefcients [N.s/mm] and are assumed to be time-invariant. Assuming viscous damping, Rayleighdamping is written as follows[20,21].

De = Ke 19

where is the damping ratio [].The gear system equation of motion is written in matrix form as follows [2528]

J::

+D:

+K =T 20

whereJ is the moment of inertia matrix, D is the viscous damping matrix, Kis the stiffness matrix,Tis the vector of applied torques,::

is the rotational acceleration vector, :

is the rotational velocity vector, and is the rotational position vector.The moment of inertia matrix J is written as follows:

J=

Jpth

0 0 0

0 Jpb

0 0

0 0 Jwth 00 0 0 J

wb

26664

37775

4 x 4

21

The viscous damping matrix D is written as follows:

D=

Der2dp Der

2dp 0 0

0 0 0 00 0 Der

2dw Der

2dw

0 0 0 0

26666643777775

4 x 4

22

The stiffness matrixKis written as follows:

K=

Ker2dp Ker

2dp 0 0

0 0 0 0

0 0 Ker2dw Ker

2dw

0 0 0 0

2666664

3777775

4 x 4

: 23

The vector of applied torque vector Tis written as follows:

T=

Tp0

Tw0

2664

3775

4 x 1

24

The rotational acceleration vector::

is written as follows:

::

=

S::

p

::

p

S::

w

::

w

4 x 1

25



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The rotational velocity vector :

is written as follows:

:=

S:p

:p

S:

w

:

w

26664

37775

4 x 1

26

The rotational position vector is written as follows:

=

SppSww

2664

3775

4 x 1

27

5. Singular value decomposition (SVD)

Some basic properties of SVD are revisited below[1821].LetAFmxn whereFis the eld (the eld may be real or complex C). There exist unitary matrices

U= u1

; u2

; :::; um

Fmxm

and V= v1

; v2

; :::; vn

Fmxm

28

such that

A= UV* 29

where

= 1 0

0 0

;1= Diagf1;2; :::pg 30

1 2 3:::p 0; p= minfm; ng: 31

In the above equations,iis theithsingular value ofA, and the vectorsu iandviare, respectively, theith left and right singularvectors dened by the following eigenvalue problems

Avi= iu i or A*ui= ivi 32

where a superposed asterisk denotes conjugated transpose. The following notations for singular values are adopted:

A= maxA= 1A= the largest singular value of A 33

A= minA= pA= the smallest singular value of A 34

Suppose thatA and are arbitrary square matrices. Then, the following equations are valid:

1:maxA1

= 1minA

ifA is invertible: 35

2: iAiA1for any i: 36

3:i+j1A+ iA+ jfor anyi and j: 37

5.1. Structural singular value

Assuming zero initial conditions, one gets the following harmonic response of a structure by taking the Laplace transform of thetransfer matrix(20)as follows[20,21]:

Js2s+ Dss+ Ks= Ts 38

s= Js2 +Ds+ K1T: 39

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The complex Laplace transform variables is substituted bys =j, whereis the excitation frequency, andjis the imaginaryunit. Then, Eq.(39)is written in the frequency domain as follows[16,17]:

=J2

+Dj+ K1T 40

Note that singular values of the transfer matrix (J2+ Dj+ K)1 in Eq.(40)are called structural singular values in thisarticle, and are function of.

6. Optimization of gearbox design parameters

Constrained optimization is a very useful tool for light-weight-structure design of machine elements with constraints such asstress, deformation and vibration.

In optimization, the goal is usually to minimize the cost of a structure while satisfying the design specications[21]. Byoptimizing the responsible parameters, it is possible to obtain a light-weight-gearbox structure and minimize the rattling noise[13].

LetF(X) denote the objective function to be minimized, where Xis the design parameter (variable) vector to be determined.Then, to nd the constrained minimum ofF(X), the following optimization problem is solved[20,21]:

minFX 41

subject to :LB X UB and G X 0 42

Fig. 7.Flow chart to optimize gearbox design parameters.

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where,LBandUBdene, the sets of lower and upper bounds on the design parameter (variable) X. The iterations start with theinitial design parameter vector X0 and a solution vector Xis found that minimizes the objective function F(X)subject to thenonlinear inequalitiesG(X) 0 [20,21].

7. Numerical example

Constrained optimization approaches are applied to the 5-speed gearbox for automotive transmission. All programs aredeveloped using the MATLAB program. In all optimization studies, the sequential quadratic programming (SQP) method isemployed.

To nd the optimum design parameter, the initial design parameters of the 5-speed gearbox for automotive transmission suchas m,z, , b, sa, and sv are varied. Thirty-six design parametersare optimized simultaneously using the developed programs. Duringoptimization, different initial value vectors are used to identify the global minimum solution of the objective functionT(m,z,,b,sa,sv).

7.1. Objectives function

Singular values of the gear system transfer matrix are considered as objective functions, and the design parameters areoptimized considering bending stress, contact stress and distance between gear center constraints. The owchart of the designparameter optimization procedure is shown inFig. 7.

The following objective function is employed:

F= minT: 43

The minimum singular values of the transfer matrixmin(T) are dened as follows

minT= minJ2

+Dj+ K1

44

The minimum singular values of the transfer matrix are considered to be the objective functions to be minimized, wheremodulem, the number of teethz, helix angle , axial clearancesaand backlash svare the design parameters (variables) to bedetermined. Then, to nd the constrained minimum of the transfer matrix T(m,z,,b,sa,sv), the following optimization problemis solved.

minTm;z;; b; sa; sv 45

subject to :LB m;z; ;b;sa;sv UBand G X 0 46

where, LB and UB dene, the sets of lower and upper bounds on the design parameter (variables) vector such as m,z, , b, sa, and sv.The iteration begins with the initial design parameter vector, which include, e.g., m0,z0,0,b0, sa0and sv0and a solution

vector with m, z, , b, sa, and sv is found that minimizes the objective function T(m, z, , b, sa, sv) subject to the nonlinearinequalitiesG(X)0.

Although the number of DOF of structures is very large in practice, the computational cost of the associated singular valueproblems is quite low for the objective function, because it is only necessary to compute the largest and smallest singular values(maxand min) that can be achieved by using selective eigenvalue solvers; the other singular values are not needed[20,21].

Table 1

Tooth bending stress parameters.

Parameter Unit 1st pinion 2nd pinion 3rd pinion 4th pinion Constant pinion Rear pinion

TorqueTL [N.mm] 392.103 392.103 316.103 252.103 200.103 1148.103

Gear ratiou [] 1.814 1.147 1.242 1.560 1 2.84Stress correction factor YST [] 2 2 2 2 2 2Form factorYF [] 2.75 2.75 2.75 2.75 2.75 2.75Stress correction factor YS [] 1.60 1.60 1.60 1.60 1.60 1.60Transverse contact ratio [] 1 1 1 1 1 1Overlap ratio [] 1 1 1 1 1 1Application factorKA [] 1.25 1.25 1.25 1.25 1.25 1.25Internal dynamic factorKV [] 1.14 1.14 1.14 1.14 1.14 1.14Transverse load factor for tooth-root stress KF [] 1.2 1.2 1.2 1.2 1.2 1.2Nominal stress number (bending) Flim [N/mm

2] 300 300 300 300 300 300Life factor for tooth-root stressYN [] 1 1 1 1 1 1Relative notch sensitivity factorY [] 1 1 1 1 1 1Relative surface factor YR [] 1 1 1 1 1 1

Size factor relevant to tooth-root strengthYX [] 1 1 1 1 1 1



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7.2. Constraint functions

Tooth bending stress, contact stress and the distance between gear centers are considered to be the constraint functions in theoptimization. The tooth bending stress parameters, tooth contact stress parameters and torsional vibration parameters are showninTables 13respectively.

The following constraints are considered to be constraint functions

1: FFp 0 47

whereFis the real tooth-root stress [N/mm2] andFpis the permissible bending stress [N/mm

2].

2: HHp 0 48

whereHis the real contact stress [N/mm2] andHpis the permissible contact stress [N/mm

2].

3: a1= a2= a3= a4= a5= aR = constant 49

where a1 is the center distance of the 1st speed, a2 is the center distance of 2nd speed, a3 is the center distance of the 3rd speed, a4is the center distance of the 4th speed, a5is the center distance of the 5th speed and aRis the center distance of rear speed.

7.3. Optimization results

The optimization results using objective function Fare presented inTable 4. Because of the limited space, only the importantresults are presented.

It is observed in solution 1 that the obtained optimum module changes between 3.3178 [mm] and 4.6740 [mm]. The optimumnumber of teeth varies between 14 [] and 17[]. In addition, the helix angle varies between 26.3342 [] and 26.0866 [] while facewidth varies between 25 [mm] and 28 [mm].

It is observed in solution 2 that the optimum module ranges from 2.9926 [mm] to 4.3290 [mm]. The numbers of teeth varybetween 14 [] and 19 []. Moreover, the optimum helix angle varies between 26 [] and 26.4142 [] and the optimum face widthvaries between 28 [mm] and 32 [mm].

The results from solution 3 show that the optimum module ranges between 3.2862 [mm] and 4.3290 [mm]; the optimumnumber of teeth vary between 14 [] and 19[]; the optimum helix angle is between 30.1647 [0] and 32[0]; and the optimum face

width varies between 30 [mm] and 32 [mm].The solution 4 results indicate that the optimum module varies between 3.1241 [mm] and 4.3290 [mm]; the optimumnumber of teeth is between 14 [] and 20 []; the optimum helix angle varies between 30.1411 [] and 32 []; and theoptimum face width ranges from 30 [mm] and 31.0004 [mm].

Although the results given above represent the optimum solution, the standard design parameter values used by gearmanufacturers do not necessarily reect these results because some of the solutions are impossible in practice.

Table 2

Tooth contact stress parameters.


Reference diameter of piniond1 [mm] 61.116 80.124 76.716 67.199 58.151 41.319Gear ratiou [] 1.814 1.147 1.242 1.560 1 2.84Zone factorZH [] 1 1 1 1 1 1

Elasticity factorZE ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN=mm2

q 189.8 189.8 189.8 189.8 189.8 189.8Transverse load factor for contact stressKH [] 1.2 1.2 1.2 1.2 1.2 1.2Allowable stress numbers (contact)Hlim [N/mm

2] 800 800 800 800 800 800Life factor for contact stress ZN [] 1 1 1 1 1 1Velocity factor ZV [] 1 1 1 1 1 1Roughness factor ZR [] 1 1 1 1 1 1Work hardening factor ZW [] 1 1 1 1 1 1Size factor for contact stress ZX [] 1 1 1 1 1 1

Table 3

Torsional vibration parameters.


Gear ratiou [] 1.814 1.147 1.242 1.560 1 2.84Young's modulusE [N/mm2] 21.104 21.104 21.104 21.104 21.104 21.104

Poisson's ratio [] 0.3 0.3 0.3 0.3 0.3 0.3

Damping ratio [] 0.1 0.1 0.1 0.1 0.1 0.1



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Table 4

Optimization results by using objective function F.

No 1

Lb=[2 2 2 2 2 2 14 14 14 14 14 14 20 20 20 20 20 20 20 20 20 20 20 20 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2]Ub=[6 6 6 6 6 6 24 24 24 24 24 24 32 32 32 32 32 32 28 28 28 28 28 28 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5]x0=[5 5 5 5 5 5 18 18 18 18 18 18 26 26 26 26 26 25 25 25 25 25 25 25 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4]

Solution

1st pinion 2nd pinion 3rd pinion 4th pinion Constant pinion Rear pinion

m 3.3178 4.3494 4.1650 3.6474 4.6740 4.3290z 17.0000 17.0000 17.0000 17.0000 17.0000 14.0000 26.0866 26.0000 26.0000 26.0000 26.0000 20.3342b 25.0026 25.0000 25.0000 25.0000 25.0000 28.0000sa 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000sv 0.4000 0.4000 0.4000 0.4000 0.4000 0.4000SF 1.0196 1.5550 1.7689 1.7012 3.5160 1.0000SH 1.2653 1.4276 1.5506 1.5949 2.0757 2.1206a 79.3590 79.3735 79.3720 79.3669 79.4575 80.0000Lp 72.9004 72.1098 71.9973 71.6494 72.2932 84.4584LpComp 85.3506 85.3916 85.3968 85.4120 85.5909 79.1996Lpaverage 84.3903CPU 4.4665SV 0.0134

No 2

Lb=[2 2 2 2 2 2 14 14 14 14 14 14 20 20 20 20 20 20 24 24 24 24 24 24 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2]Ub=[7 7 7 7 7 7 24 24 24 24 24 24 32 32 32 32 32 32 32 32 32 32 32 32 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5]x0=[6 6 6 6 6 6 20 20 20 20 20 20 26 26 26 26 26 26 28 28 28 28 28 28 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5]

Solution


m 2.9926 3.9175 3.7509 3.2833 4.2065 4.3290z 19.0000 19.0000 19.0000 19.0000 19.0000 14.0000 26.1083 26.0000 26.0000 26.0000 26.0000 26.4142b 28.0031 28.0000 28.0000 28.0000 28.0000 32.0000sa 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000sv 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000SF 1.0625 1.5687 1.7842 1.7152 3.5440 1.0000

SH 1.3602 1.5109 1.6410 1.6879 2.1967 1.2453a 79.9999 79.9036 79.8912 79.8500 79.9226 80.0000Lp 73.8058 72.6827 72.5724 72.2320 72.8624 78.9505LpComp 81.7958 81.9502 81.9631 82.0006 82.4361 79.8542Lpaverage 81.6667CPU 3.3858SV 0.0012

No 3

Lb=[2 2 2 2 2 2 14 14 14 14 14 14 20 20 20 20 20 20 24 24 24 24 24 24 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2]Ub=[7 7 7 7 7 7 28 28 28 28 28 28 32 32 32 32 32 32 32 32 32 32 32 32 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6]x0=[6 6 6 6 6 6 20 20 20 20 20 20 32 32 32 32 32 30 30 30 30 30 30 32 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5]

Solution


m 4.0593 3.9201 3.7537 3.2867 4.2087 4.3290z 14.0000 19.0000 19. 0000 19.0000 19. 0000 14.0000 32.0000 32. 0000 32.0000 32.0000 32.0000 30.1647b 30.0000 30. 0000 30.0000 30.0000 30.0000 32.0000sa 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000v 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000SF 1.0368 1.3123 1.4927 1.4354 2.9643 1.0000SH 1.1542 1.3772 1.4958 1.5385 2.0024 1.1867a 79.9604 79.9561 79.9505 79.9317 79.9648 80.0000Lp 61.8075 63.4120 63.1788 62.4124 63.7788 80.1381LpComp 80.4780 80.4515 80.4560 80.4692 80.5366 76.9647Lpaverage 79.8926CPU 2.2650SV 0.0010



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The rattle noise values vary between 76 [dB] and 80 [dB] during optimization. Thus, by optimizing the designparameters, the rattle noise values are reduced to 76 [dB]. These rattle noise values are between 7% and 10% lower than thecalculated rattle noise values for the sample gearbox.

The safety factor for bending stress SF ranges between 1.00 and 3.54 during optimization. In addition, the safety factorfor contact stress SHvaries between 1.15 and 2.19 during optimization. Thus, all optimized design parameters satisfy allconstraints.

The CPU time varies between 2 [s] and 4 [s] using objective function Fduring optimization.

8. Comparison of rattle noise levels

The sample gearbox rattle noise level and optimized gearbox rattle noise levels using objective function F for the 1st,2nd, 3rd, 4th, 5th and rear speed are shown below.

A comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 1st speed is showninFig. 8. While the rattle noise of the sample gearbox for the 1st speed is 88.1524 [dB], the rattle noise of the optimizedgearbox for 1st speed is 80.4780 [dB].

A comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 2nd speed isshown inFig. 9. While the rattle noise of the sample gearbox for 2st speed is 88.1893 [dB], rattle noise of the optimizedgearbox for the 2nd speed is 80.4515 [dB].

A comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 3rd speed isshown inFig. 10. While the rattle noise of the sample gearbox for the 3rd speed is 86.3327 [dB], the rattle noise of theoptimized gearbox for the 3rd speed is 80.4560 [dB].

Table 4 (continued)

No 4

Lb=[2 2 2 2 2 2 14 14 14 14 14 14 20 20 20 20 20 20 24 24 24 24 24 24 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2]Ub=[7 7 7 7 7 7 28 28 28 28 28 28 32 32 32 32 32 32 32 32 32 32 32 32 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6]x0=[7 7 7 7 7 7 21 21 21 21 21 21 32 31 31 31 31 30 30 30 30 30 30 31 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6]

Solution


m 4.0607 3.7254 3.5675 3.1241 3.9993 4.3290z 14.0000 20.0000 20.0000 20.0000 20.0000 14.0000 32.0000 31.0000 31.0000 31.0000 31.0000 30.1411b 30.0000 30.0000 30.0000 30.0000 30.0000 31.0004sa 0.6000 0.6000 0.6000 0.6000 0.6000 0.6000sv 0.6000 0.6000 0.6000 0.6000 0.6000 0.6000SF 1.0371 1.0847 1.2339 1.1867 2.4500 1.0000SH 1.1542 1.3152 1.4284 1.4693 1.9122 1.2122a 79.9869 79.9841 79.9826 79.9776 79.9864 80.0000Lp 62.9285 75.4959 75.4104 75.1493 75.6364 80.8208LpComp 84.1556 83.5577 83.5709 83.6094 84.1882 81.5079Lpaverage 83.4316CPU 2.0074SV 0.0012

Fig. 8.Comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 1st speed.

Table 4(continued)



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A comparison between the rattle noise level of the sample gearbox and optimized gearbox for the 4th speed is shown inFig. 11. While the rattle noise of the sample gearbox for the 4th speed is 86.3327 [dB], the rattle noise of the optimizedgearbox for the 4th speed is 80.4692 [dB].

The rattle noise level of the sample gearbox and the optimized gearbox for the 5th speed is compared in Fig. 12. Whilethe rattle noise of the sample gearbox for the 5th speed is 88.4915 [dB], the rattle noise of the optimized gearbox for the5th speed is 80.5366 [dB].

The rattle noise level of the sample gearbox and optimized gearbox for rear speed is compared in Fig. 13. While therattle noise of the sample gearbox for the rear speed is 87.7589 [dB], the rattle noise of the optimized gearbox for the rearspeed is 76.9647 [dB].

Fig. 11.Comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 4th speed.

Fig. 9.Comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the for 2nd speed.

Fig. 10.Comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 3rd speed.

http://localhost/var/www/apps/conversion/tmp/scratch_2/image%20of%20Fig.%E0%B1%B0http://localhost/var/www/apps/conversion/tmp/scratch_2/image%20of%20Fig.%E0%B9%80http://localhost/var/www/apps/conversion/tmp/scratch_2/image%20of%20Fig.%E0%B1%B1


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It is shown that the rattle noise values of the optimized gearbox are lower than the calculated rattle noise values for thesample gearbox for each speed.

9. Conclusion

It is concluded through simulation that increasing the geometric parameters of the gearbox, such as the module and number ofteeth results in increased rattle noise. In addition, increased axial clearance results in increased rattle noise until the axial clearancereaches its maximum value, then the rattle noise decreases. Moreover, increased backlash causes decreased rattle noise until thebacklash reaches its maximum value, then the rattle noise increases. Changing the helix angle resulted in different levels of rattlenoise, while changing the face width resulted in a constant level of rattle noise. Furthermore, increasing the gearbox operationalparameters, such as the angular acceleration and excitation frequency, caused increased gearbox rattle noise [33].

Some geometric design parameters, such as the module, and number of teeth must satisfy desired safety protocols, andsome backlash are necessary to allow room for an oil lm for all conditions of thermal expansion and contraction. Although,there is no relationship between face width and rattle noise, face width is necessary to satisfy the desired contact safetyrequirement. Therefore, by optimizing the geometric parameters of the gearbox, including the module, number of teeth,axial clearance, and backlash, it is possible to obtain a light-weight-gearbox structure and minimize the rattling noise.

Optimized geometric design parameters lower the rattle noise by 10% compared to the calculated rattle noise values forthe sample gearbox. All optimized geometric design parameters also satisfy all constraints. Optimizing the geometric designparameters not only reduces the rattle noise but also increases the desirable bending stress and contact stress level.

While geometric parameters, such as the module, number of teeth, helix angle, face width, backlash and axial clearanceare optimized, the operational parameters, such as angular acceleration and excitation frequency are not optimized becausethese operational parameters are given by the automotive manufacturer as input values.

Acknowledgements

This research study is supported by the Institute of Machine Elements (IMA), University of Stuttgart. The author thanks Prof.

Dr.-Ing. B. Bertsche, Dipl.-Ing. P. Fietkau, Dipl.-Ing. A. Baumann, Dipl.-Ing. W. Novak, Dipl.-Ing. S. Nebel and Dipl.-Ing. S.Sanzenbacher for their helpful co-operation.

Fig. 13.Comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the rear speed.

Fig. 12.Comparison between the rattle noise level of the sample gearbox and the optimized gearbox for the 5th speed.

http://localhost/var/www/apps/conversion/tmp/scratch_2/image%20of%20Fig.%E0%B1%B2http://localhost/var/www/apps/conversion/tmp/scratch_2/image%20of%20Fig.%E0%B1%B3


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Documents

Torsional Vibration Model Based Optimization of Gearbox Geometric Design Parameters to Reduce Rattle Noise in an Automotive Transmission