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

Mechanism and Machine Theory 45 (2010) 1583–1598

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Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t

Torsional vibration model based optimization of gearbox geometric design parameters 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

Article history: Received 23 November 2009 Received in revised form 8 June 2010 Accepted 22 June 2010 Available online 24 July 2010 Keywords: Optimization Gearbox Rattle noise Torsional vibration Automotive transmission

a b s t r a c t The optimization of gearbox geometric design parameters to reduce rattle noise in an automotive transmission based on a torsional vibration model approach is studied. Rattle noise is calculated and simulated based on the design parameters of a 5-speed gearbox, and all pinion gears and wheel gears are helical. The effect of the design parameters on rattle noise is analyzed. The observed rattle noise profiles are obtained depending on the design parameters. During the optimization, a four-degree-of-freedom torsional vibration model of the pinion gear–wheel gear system is obtained and the minimum singular value of the transfer matrix is considered as the objective functions and design parameters are optimized under several constraints that include bending stress, contact stress and a constant distance between gear centers. Therefore, by optimizing the geometric parameters of the gearbox such as, the module, number of teeth, axial clearance, and backlash, it is possible to obtain a light-weight-gearbox structure and minimize the rattling noise. It is concluded that the optimized geometric design parameters lower the rattle noise by 10% compared to the calculated rattle noise for sample gearbox. All optimized geometric design parameters also satisfy all constraints. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The optimization of gearbox geometric design parameters to reduce rattle noise in automotive transmissions based on a torsional vibration model approach is studied. Gear motion causes rattling and clattering noise, and noise level is considered to be a comfort factor in the automotive industry. Therefore, reducing rattling and clattering noise in the gearbox is important in the automotive transmission for a comfortable car design. Gears are widely used in automotive transmissions to transmit mechanical power from one shaft to another. The purpose of the gears 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 in neutral, and as clattering when the gear is engaged under power or in overrun [9–14]. Rattling and clattering are caused by torsional vibration of loose parts, i.e. parts, such as idler gears, synchronizer rings and 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 cars spend so much time idling under no load or very light loads [1].

⁎ 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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M. Bozca / Mechanism and Machine Theory 45 (2010) 1583–1598

The parameters that are responsible for rattle and clatter are classified as geometric parameters and operational parameter [9–14]. The geometrical parameters include the module m, number of teeth z, helix angle β, axial clearance sα ˆ˙ 1, and excitation frequency and backlash sv as shown in Fig. 3. The operational parameters include the angular acceleration ω ωan [9–14]. Transmission fluid as an engineering design parameter also has a considerable influence on rattling and clattering noises. The important factors include the type of oil, the additives used and the viscosity and the level of oil in the transmission which together act on a gear pair as drag torque, resulting in a reduction in rattling and clattering noises, especially at low speeds and when cold [9–14]. The analysis suggests that in order to reduce the gear rattle noise, rather than to increase the oil level in a gearbox and thus decreasing the mechanical efficiency, it would be opportune to guarantee the oil presence between the meshing teeth by means of a 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 the different kinds of nonlinearities in gear system, such as clearance, spatial variations and sliding friction, the effect of friction is the least understood. Certain unique characteristics of gear tooth sliding make it a potentially dominant factor. For instance, due to the reversal in direction during meshing action, friction is associated with a large oscillatory component, which causes a higher bandwidth in the system response. Furthermore, it becomes more significant at high values of torque and lower speeds, due to the tribological characteristics 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 with higher transmitted power levels. Optimization of the gear's micro-geometry i.e. trying to balance load-induced teeth deflections with profile corrections that has generally lead to less noisy transmission effects. This is not a suitable solution for an overall working range; therefore profile corrections must be determined statistically to take into account manufacturing deviations which will 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, and synchronizers, as shown in Fig. 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 in the 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 are idler gears and run in the needle bearings on the output shaft. The synchronizer of the 1st and 2nd wheel gears is connected to the output 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 on the 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 and 4th 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 automotive transmission, which is shown in Figs. 1 and 2. The rattle noise level of a complete automotive transmission is calculated as follows [9–13]: n

LpComp = 10log ∑

i=1

0;1L 10 p;i :

ð1Þ

The rattle noise Lp, is calculated as follows by correlating the computed noise value and the measured noise level [9–13]: 0;1L Lp = 10log kIm + 10 basic

ð2Þ

where k is the calibration factor [–] and Im is the average impact intensity [N]. The average impact intensity Im, is written as follows [9–13]: ˆ˙ 1 rb1 CIm Im = m2ω

ð3Þ

ˆ˙ 1 is the angular acceleration [rad/s2], rb1 is the pitch circle radius [mm] and CIm is the related average where m2 is a loose part [kg], ω impact intensity [–]. The average impact intensity CIm, is written as follows [9–13]: CIm

pffiffiffiffiffiffiffi = Csv 1; 462−

0; 714Cfa Csa −0; 016Cfa + 0; 12Csv

! ð4Þ

where Csv is the non-dimensional circumferential backlash [–]. The non-dimensional circumferential backlash Csv [–] is defined as follows [9–13]: Csv =

sv ω2an ˆ˙ 1 rb1ω

ð5Þ

where sv is backlash [mm], ωan is the excitation frequency [rad/s] and Csa is the non-dimensional axial clearance [–]. The nondimensional axial clearance Csa [–] is defined as follows [9–13]: Csa =

sa ω2an tanβ ˆ˙ 1 rb1ω

ð6Þ

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

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

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Fig. 3. Design parameters for a gearbox.

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 defined by 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 forces were not taken into account when determining the loading capacity of the gear [3,22,23]. A tooth-root bending fatigue fracture usually starts at the 30° tangent of the root [3,8,22,24]. The real tooth-root stress σF is calculated as follows [2,3,5,7]: σF =

Ft Y YY Y K K K K bmn F S ε β A V Fβ Fα

ð7Þ

where Ft is the nominal tangential load [N], b is the face width [mm], mn is the normal module [mm], YF is the form factor [–], YS is the stress correction factor [–], Yε is the contact ratio factor [–], KA is the application factor [–], KV is the internal dynamic factor [–], KFβ is the face load factor for tooth-root stress [–] and KFα is the transverse load factor for tooth-root stress [–]. The permissible bending stress σFp is calculated as follows [2,3,5,7]: σ Fp = σ F lim YST YN Yδ YR YX

ð8Þ

Fig. 4. Bending stress at the tooth root.


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where σFlim is the nominal stress number (bending) [N/mm2], YST is the stress correction factor [–], YN is the life factor for the tooth-root stress [–], Yδ is the relative notch sensitivity factor [–], YR is the relative surface factor [–] and YX is the size factor that represents the tooth-root strength [–]. The safety factor for bending stress SF is calculated as follows [2,3,5,7]: SF =

σ Fp σF

ð9Þ

3.2. Tooth contact stress The tooth contact stress σHC is calculated as follows, as seen in Fig. 5 [2,3,5,6]. The real contact stress σH is calculated as follows [2,3,5,6]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ft u + 1 Z H Z E Z ε Z β K A K V K Hβ K Hα σH = bm n u

ð10Þ

where d1 is the reference diameter of the pinion [mm], u is gear ratio [–], Z H is the zone factor [–], Z E is the elasticity factor qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N=mm2 , Zε is the contact ratio factor [–], Zβ is the helix angle factor [–], KHβ is the face load factor for contact stress [–] and KHα is the transverse load factor for contact stress [–]. The permissible contact stress σHp is calculated as follows [2,3,5,6]: σ Hp = σ H lim Z N Z L Z V Z R Z W Z X

ð11Þ

where σ H lim is the allowable stress numbers (contact) [N/mm2], Z N is the life factor for contact stress [–], Z L is the lubrication factor [–], Z V is the velocity factor [–], Z R is the roughness factor [–], Z W is the work hardening factor [–] and Z X is the size factor for contact stress [–]. The safety factor for contact stress SH is calculated as follows [2,3,5,6]: SH =

σ Hp σH

ð12Þ

4. Calculating the torsional vibration The torsional vibration transmitted from the combustion engine to the transmission excites idler components such as the idler gears, 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 geared system with various gear parameters and operating conditions [27].

Fig. 5. Contact stress at the tooth flank.

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Torsional vibration models of gears system are classified according to their rigidity and elasticity such as purely torsional multibody models, rigid multi-body models, flexible multibody models and semi-rigid-elastic multi-body models [25–30]. Torsional vibration models of gears system are also classified according to time-invariant and time-variant such as linear timeinvariant (LTI) models with stiffness, linear time-varying (LTV) models with stiffness, time-varying models with backlash, as well as 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 gear–wheel gear system is considered for simplicity. The pinion gear body and wheel gear body are assumed to be rigid. The teeth are assumed to be elastic and parallel spring–damper combinations are assumed 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 gear–wheel gear system are written in terms of the four-degree-of-freedom model as follows [25–28]: Sp, ϕp, Sw and ϕw. Eq. (13) is written for tooth i of a pinion gear as follows: :: 2 2 Jpth S p = De rdp ðϕ˙ p −S˙ p Þ + K e rdp ðϕ p −S p Þ + T p

ð13Þ

where Jpth is the moment of inertia of tooth i of the pinion gear [kg.mm2], r dp is the pitch circle radius of the pinion gear [mm], ϕp is the rotational position of the pinion gear body [rad], Sp is the rotational position of tooth i of the pinion gear [rad],ϕ˙ p is the :: rotational velocity of the pinion gear body [rad/s], S˙ p is the rotational velocity of the tooth i of the pinion gear [rad/s], S p is the 2

rotational acceleration of the tooth i of the pinion gear [rad/s ], and Tp is the contact torque applied to tooth i [N.mm]. Eq. (14) is written for a pinion gear body as follows: :: Jpb ϕ b = 0

ð14Þ

:: where Jpb is the moment of inertia of the pinion gear body [kg.mm2], ϕ b is the rotational acceleration of the pinion gear body [rad/s2]. Eq. (15) is written for tooth i of the wheel gear as follows: :: 2 2 J wth S w = De rdw ðϕ˙ w −S˙ w Þ + Ke rdw ðϕw −Sw Þ + Tw

ð15Þ

where Jwth 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], ϕw is the rotational position of the wheel gear body [rad], Sw is the rotational position of tooth i of the wheel gear [rad], ϕ::˙ w is the rotational velocity of the wheel gear body [rad/s], Ṡw is the rotational velocity of the tooth i of the wheel gear [rad/s], S w is the rotational acceleration of tooth i of the wheel gear [rad/s2], and Tw is the contact torque applied to tooth i [N.mm]. Eq. (16) is written for the wheel gear body as follows: :: J wb ϕw = 0

ð16Þ

:: where Jwb is the moment of inertia of the wheel gear body [kg.mm2], ϕw is the rotational acceleration of the wheel gear body [rad/s2].

Fig. 6. 4-Degree-of-freedom model.


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Ke is the stiffness coefficient [N/mm] and is assumed to be time-invariant. The stiffness coefficients Ke resulting from the tooth surface contact are written as follows [29,32]: πES0 4ð1−υ2 Þ

Ke =

ð17Þ

where E is Young's modulus [N/mm2], υ is the Poisson's ratio [–] and S 0 is the thickness of gear [mm] which is written as follows [3,5]: S0 = πm n = 2

ð18Þ

De are the viscous damping coefficients [N.s/mm] and are assumed to be time-invariant. Assuming viscous damping, Rayleigh damping is written as follows [20,21]. De = ξKe

ð19Þ

where ξ is the damping ratio [–]. The gear system equation of motion is written in matrix form as follows [25–28] :: : JΩ + DΩ + KΩ = T

ð20Þ

where J is the moment of inertia matrix, :D is the viscous damping matrix, K is the stiffness matrix, T is the vector of applied torques, :: Ω is the rotational acceleration vector, Ω is the rotational velocity vector, and Ω is the rotational position vector. The moment of inertia matrix J is written as follows: 2

J pth 6 0 6 J=6 4 0 0

0 J pb 0 0

0 0 J wth 0

0 0 0

3 7 7 7 5

J wb

ð21Þ

4x4

The viscous damping matrix D is written as follows: 2 6 6 6 D=6 6 4

2 −De rdp

2 De rdp

0

0

0

0

0

0

2 −De rdw

0

0

0

0

3

7 7 7 7 2 7 De rdw 5 0 4x4 0

ð22Þ

The stiffness matrix K is written as follows: 2 6 6 6 K=6 6 4

2

2

−Ke rdp

Ke rdp

0

0

0

0

0

0

−Ke rdw

0

0

0

0

3

7 7 7 : 7 2 Ke rdw 7 5 0 4x4 0

2

ð23Þ

The vector of applied torque vector T is written as follows: 2

3 Tp 6 0 7 7 T=6 4 Tw 5 0 4x1

ð24Þ

:: The rotational acceleration vector Ω is written as follows:

½

:: S ::p :: ϕp Ω = :: Sw :: ϕw

4x1

ð25Þ

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: The rotational velocity vector Ω is written as follows: 2 : 3 Sp 6 : 7 : 6 ϕp 7 Ω=6 : 7 4S 5 :w ϕw 4 x 1

ð26Þ

The rotational position vector Ω is written as follows: 2

3 Sp 6 ϕp 7 7 Ω=6 4 Sw 5 ϕw 4 x 1

ð27Þ

5. Singular value decomposition (SVD) Some basic properties of SVD are revisited below [18–21]. Let A ∈ F mxn where F is the field (the field may be real ℜ or complex C). There exist unitary matrices U = ½u1 ; u2 ; :::; u m ∈ F such that

mxm

and V = ½v1 ; v2 ; :::; vn ∈ F

mxm

ð28Þ

ð29Þ

A = U∑V* where ∑=

∑1 0

0 ; ∑1 = Diagfμ 1 ; μ 2 ; :::μ p g 0

μ 1 ≥ μ 2 ≥ μ 3 :::≥ μ p ≥ 0; p = minfm; ng:

ð30Þ ð31Þ

In the above equations, μ i is the ith singular value of A, and the vectors u i and v i are, respectively, the ith left and right singular vectors defined by the following eigenvalue problems Avi = μ i u i

or A*ui = μ i vi

ð32Þ

where a superposed asterisk denotes conjugated transpose. The following notations for singular values are adopted: μ ðAÞ = μ max ðAÞ = μ 1 ðAÞ = the largest singular value of A

ð33Þ

μ ðAÞ = μ min ðAÞ = μ p ðAÞ = the smallest singular value of A

ð34Þ

Suppose that A and Δ are arbitrary square matrices. Then, the following equations are valid: 1: μ max ðA

−1

Þ=

1 if A is invertible: μ min ðAÞ

ð35Þ

2: μ i ðA ΔÞ ≤ μ i ðAÞμ 1 ðΔÞ for any i:

ð36Þ

3: μ i+j−1 ðA + ΔÞ≤ μ i ðAÞ + μ j ðΔÞ for any i and j:

ð37Þ

5.1. Structural singular value Assuming zero initial conditions, one gets the following harmonic response of a structure by taking the Laplace transform of the transfer matrix (20) as follows [20,21]: 2

Js ΘðsÞ + DsΘðsÞ + KΘðsÞ = TðsÞ

2

−1

ΘðsÞ = ðJs + Ds + KÞ

T:

ð38Þ

ð39Þ


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The complex Laplace transform variable s is substituted by s = jω, where ω is the excitation frequency, and j is the imaginary unit. Then, Eq. (39) is written in the frequency domain as follows [16,17]: 2

−1

Ω = ð−Jω + Djω + KÞ

ð40Þ

T

Note that singular values of the transfer matrix (−J ω2 + Djω + K)− 1 in Eq. (40) are called structural singular values in this article, and are function of ω. 6. Optimization of gearbox design parameters Constrained optimization is a very useful tool for light-weight-structure design of machine elements with constraints such as stress, deformation and vibration. In optimization, the goal is usually to minimize the cost of a structure while satisfying the design specifications [21]. By optimizing the responsible parameters, it is possible to obtain a light-weight-gearbox structure and minimize the rattling noise [13]. Let F(X) denote the objective function to be minimized, where X is the design parameter (variable) vector to be determined. Then, to find the constrained minimum of F(X), the following optimization problem is solved [20,21]: minFðXÞ

ð41Þ

subject to : LB ≤ X ≤ UB and Gð X Þ ≤ 0

ð42Þ

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

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where, LB and UB define, the sets of lower and upper bounds on the design parameter (variable) X. The iterations start with the initial design parameter vector X0 and a solution vector X is found that minimizes the objective function F(X)subject to the nonlinear inequalities G(X) ≤ 0 [20,21]. 7. Numerical example Constrained optimization approaches are applied to the 5-speed gearbox for automotive transmission. All programs are developed using the MATLAB program. In all optimization studies, the sequential quadratic programming (SQP) method is employed. To find the optimum design parameter, the initial design parameters of the 5-speed gearbox for automotive transmission such as m, z, β, b, sa, and sv are varied. Thirty-six design parameters are optimized simultaneously using the developed programs. During optimization, different initial value vectors are used to identify the global minimum solution of the objective function T (m, z, β, b, sa, sv). 7.1. Objectives function Singular values of the gear system transfer matrix are considered as objective functions, and the design parameters are optimized considering bending stress, contact stress and distance between gear center constraints. The flowchart of the design parameter optimization procedure is shown in Fig. 7. The following objective function is employed: F = μ min ðTÞ:

ð43Þ

The minimum singular values of the transfer matrix μ min(T) are defined as follows 2

μ min ðTÞ = μ min ð−J ω + Djω + KÞ

−1

ð44Þ

The minimum singular values of the transfer matrix are considered to be the objective functions to be minimized, where module m, the number of teeth z, helix angle β, axial clearance sa and backlash sv are the design parameters (variables) to be determined. Then, to find the constrained minimum of the transfer matrix T (m, z, β, b, sa, sv), the following optimization problem is solved. minTðm; z; β; b; sa ; sv Þ

ð45Þ

subject to : LB ≤ m; z; β; b; sa ; sv ≤ UB and Gð X Þ ≤ 0

ð46Þ

where, LB and UB define, the sets of lower and upper bounds on the design parameter (variables) vector such as m, z, β, b, sa, and sv. The iteration begins with the initial design parameter vector, which include, e.g., m0, z0, β0, b0, sa0 and sv0 and a solution vector with m, z, β, b, sa, and sv is found that minimizes the objective function T (m, z, β, b, sa, sv) subject to the nonlinear inequalities G (X) ≤ 0. Although the number of DOF of structures is very large in practice, the computational cost of the associated singular value problems is quite low for the objective function, because it is only necessary to compute the largest and smallest singular values (μ max and μ 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

Torque TL Gear ratio u Stress correction factor YST Form factor YF Stress correction factor YS Transverse contact ratio εα Overlap ratio εβ Application factor KA Internal dynamic factor KV Transverse load factor for tooth-root stress KFα Nominal stress number (bending) σFlim Life factor for tooth-root stress YN Relative notch sensitivity factor Yδ Relative surface factor YR Size factor relevant to tooth-root strength YX

[N.mm] [–] [–] [–] [–] [–] [–] [–] [–] [–] [N/mm2] [–] [–] [–] [–]

392.103 1.814 2 2.75 1.60 1 1 1.25 1.14 1.2 300 1 1 1 1

392.103 1.147 2 2.75 1.60 1 1 1.25 1.14 1.2 300 1 1 1 1

316.103 1.242 2 2.75 1.60 1 1 1.25 1.14 1.2 300 1 1 1 1

252.103 1.560 2 2.75 1.60 1 1 1.25 1.14 1.2 300 1 1 1 1

200.103 1 2 2.75 1.60 1 1 1.25 1.14 1.2 300 1 1 1 1

1148.103 2.84 2 2.75 1.60 1 1 1.25 1.14 1.2 300 1 1 1 1


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Table 2 Tooth contact stress parameters. Parameter

Unit

Reference diameter of pinion d1 Gear ratio u Zone factor ZH Elasticity factor ZE

[mm] [–] [–] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N=mm2

1st pinion

189.8

189.8

189.8

189.8

189.8

189.8

Transverse load factor for contact stress KHα Allowable stress numbers (contact) σHlim Life factor for contact stress ZN Velocity factor ZV Roughness factor ZR Work hardening factor ZW Size factor for contact stress ZX

[–] [N/mm2] [–] [–] [–] [–] [–]

1.2 800 1 1 1 1 1

1.2 800 1 1 1 1 1

1.2 800 1 1 1 1 1

1.2 800 1 1 1 1 1

1.2 800 1 1 1 1 1

1.2 800 1 1 1 1 1

61.116 1.814 1

2nd pinion 80.124 1.147 1

3rd pinion 76.716 1.242 1

4th pinion

Constant pinion

67.199 1.560 1

58.151 1 1

Rear pinion 41.319 2.84 1

7.2. Constraint functions Tooth bending stress, contact stress and the distance between gear centers are considered to be the constraint functions in the optimization. The tooth bending stress parameters, tooth contact stress parameters and torsional vibration parameters are shown in Tables 1–3 respectively. The following constraints are considered to be constraint functions 1: σF −σFp ≤ 0

ð47Þ

where σF is the real tooth-root stress [N/mm2] and σFp is the permissible bending stress [N/mm2]. 2: σH −σHp ≤ 0

ð48Þ

where σH is the real contact stress [N/mm2] and σHp is the permissible contact stress [N/mm2]. ð49Þ

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

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, a4 is the center distance of the 4th speed, a5 is the center distance of the 5th speed and aR is the center distance of rear speed. 7.3. Optimization results The optimization results using objective function F are presented in Table 4. Because of the limited space, only the important results are presented. It is observed in solution 1 that the obtained optimum module changes between 3.3178 [mm] and 4.6740 [mm]. The optimum number of teeth varies between 14 [–] and 17 [–]. In addition, the helix angle varies between 26.3342 [°] and 26.0866 [°] while face width 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 vary between 14 [–] and 19 [–]. Moreover, the optimum helix angle varies between 26 [°] and 26.4142 [°] and the optimum face width varies 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 optimum number 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 optimum number of teeth is between 14 [–] and 20 [–]; the optimum helix angle varies between 30.1411 [°] and 32 [°]; and the optimum 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 gear manufacturers do not necessarily reflect these results because some of the solutions are impossible in practice. Table 3 Torsional vibration parameters. Parameter

Unit

1st pinion

2nd pinion

3rd pinion

4th pinion

Constant pinion

Rear pinion

Gear ratio u Young's modulusE Poisson's ratio υ Damping ratio ξ

[–] [N/mm2] [–] [–]

1.814 21.104 0.3 0.1

1.147 21.104 0.3 0.1

1.242 21.104 0.3 0.1

1.560 21.104 0.3 0.1

1 21.104 0.3 0.1

2.84 21.104 0.3 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

m z β b sa sv SF SH a Lp LpComp Lpaverage CPU SV

1st pinion

2nd pinion

3rd pinion

4th pinion

Constant pinion

Rear pinion

3.3178 17.0000 26.0866 25.0026 0.4000 0.4000 1.0196 1.2653 79.3590 72.9004 85.3506 84.3903 4.4665 0.0134

4.3494 17.0000 26.0000 25.0000 0.4000 0.4000 1.5550 1.4276 79.3735 72.1098 85.3916

4.1650 17.0000 26.0000 25.0000 0.4000 0.4000 1.7689 1.5506 79.3720 71.9973 85.3968

3.6474 17.0000 26.0000 25.0000 0.4000 0.4000 1.7012 1.5949 79.3669 71.6494 85.4120

4.6740 17.0000 26.0000 25.0000 0.4000 0.4000 3.5160 2.0757 79.4575 72.2932 85.5909

4.3290 14.0000 20.3342 28.0000 0.4000 0.4000 1.0000 2.1206 80.0000 84.4584 79.1996

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


1st pinion

2nd pinion

3rd pinion

4th pinion

Constant pinion

Rear pinion

2.9926 19.0000 26.1083 28.0031 0.5000 0.5000 1.0625 1.3602 79.9999 73.8058 81.7958 81.6667 3.3858 0.0012

3.9175 19.0000 26.0000 28.0000 0.5000 0.5000 1.5687 1.5109 79.9036 72.6827 81.9502

3.7509 19.0000 26.0000 28.0000 0.5000 0.5000 1.7842 1.6410 79.8912 72.5724 81.9631

3.2833 19.0000 26.0000 28.0000 0.5000 0.5000 1.7152 1.6879 79.8500 72.2320 82.0006

4.2065 19.0000 26.0000 28.0000 0.5000 0.5000 3.5440 2.1967 79.9226 72.8624 82.4361

4.3290 14.0000 26.4142 32.0000 0.5000 0.5000 1.0000 1.2453 80.0000 78.9505 79.8542

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 z β b sa v SF SH a Lp LpComp Lpaverage CPU SV

1st pinion

2nd pinion

3rd pinion

4th pinion

Constant pinion

Rear pinion

4.0593 14.0000 32.0000 30.0000 0.5000 0.5000 1.0368 1.1542 79.9604 61.8075 80.4780 79.8926 2.2650 0.0010

3.9201 19.0000 32. 0000 30. 0000 0.5000 0.5000 1.3123 1.3772 79.9561 63.4120 80.4515

3.7537 19. 0000 32.0000 30.0000 0.5000 0.5000 1.4927 1.4958 79.9505 63.1788 80.4560

3.2867 19.0000 32.0000 30.0000 0.5000 0.5000 1.4354 1.5385 79.9317 62.4124 80.4692

4.2087 19. 0000 32.0000 30.0000 0.5000 0.5000 2.9643 2.0024 79.9648 63.7788 80.5366

4.3290 14.0000 30.1647 32.0000 0.5000 0.5000 1.0000 1.1867 80.0000 80.1381 76.9647


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Table 4 (continued) 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


1st pinion

2nd pinion

3rd pinion

4th pinion

Constant pinion

Rear pinion

4.0607 14.0000 32.0000 30.0000 0.6000 0.6000 1.0371 1.1542 79.9869 62.9285 84.1556 83.4316 2.0074 0.0012

3.7254 20.0000 31.0000 30.0000 0.6000 0.6000 1.0847 1.3152 79.9841 75.4959 83.5577

3.5675 20.0000 31.0000 30.0000 0.6000 0.6000 1.2339 1.4284 79.9826 75.4104 83.5709

3.1241 20.0000 31.0000 30.0000 0.6000 0.6000 1.1867 1.4693 79.9776 75.1493 83.6094

3.9993 20.0000 31.0000 30.0000 0.6000 0.6000 2.4500 1.9122 79.9864 75.6364 84.1882

4.3290 14.0000 30.1411 31.0004 0.6000 0.6000 1.0000 1.2122 80.0000 80.8208 81.5079

The rattle noise values vary between 76 [dB] and 80 [dB] during optimization. Thus, by optimizing the design parameters, the rattle noise values are reduced to 76 [dB]. These rattle noise values are between 7% and 10% lower than the calculated 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 factor for contact stress SH varies between 1.15 and 2.19 during optimization. Thus, all optimized design parameters satisfy all constraints. The CPU time varies between 2 [s] and 4 [s] using objective function F during 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 shown in Fig. 8. While the rattle noise of the sample gearbox for the 1st speed is 88.1524 [dB], the rattle noise of the optimized gearbox 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 is shown in Fig. 9. While the rattle noise of the sample gearbox for 2st speed is 88.1893 [dB], rattle noise of the optimized gearbox 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 is shown in Fig. 10. While the rattle noise of the sample gearbox for the 3rd speed is 86.3327 [dB], the rattle noise of the optimized gearbox for the 3rd speed is 80.4560 [dB].

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

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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.

A comparison between the rattle noise level of the sample gearbox and optimized gearbox for the 4th speed is shown in Fig. 11. While the rattle noise of the sample gearbox for the 4th speed is 86.3327 [dB], the rattle noise of the optimized gearbox 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. While the rattle noise of the sample gearbox for the 5th speed is 88.4915 [dB], the rattle noise of the optimized gearbox for the 5th 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 the rattle noise of the sample gearbox for the rear speed is 87.7589 [dB], the rattle noise of the optimized gearbox for the rear speed is 76.9647 [dB].

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


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

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

It is shown that the rattle noise values of the optimized gearbox are lower than the calculated rattle noise values for the sample 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 of teeth results in increased rattle noise. In addition, increased axial clearance results in increased rattle noise until the axial clearance reaches its maximum value, then the rattle noise decreases. Moreover, increased backlash causes decreased rattle noise until the backlash reaches its maximum value, then the rattle noise increases. Changing the helix angle resulted in different levels of rattle noise, while changing the face width resulted in a constant level of rattle noise. Furthermore, increasing the gearbox operational parameters, 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, and some backlash are necessary to allow room for an oil film 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 safety requirement. 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 for the sample gearbox. All optimized geometric design parameters also satisfy all constraints. Optimizing the geometric design parameters 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 clearance are optimized, the operational parameters, such as angular acceleration and excitation frequency are not optimized because these 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.

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

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