THE DESIGN AND MESHING EFFICIENCY ANALYSIS OF HELICAL SPUR GEARREDUCER WITH SINGLE GEAR PAIR FOR ELECTRIC SCOOTERS

Long-Chang Hsieh1, Tzu-Hsia Chen2 and Hsiu-Chen Tang31Department of Power Mechanical Engineering, National Formosa University, Taiwan, R.O.C.2Department of Mechanical Design Engineering, National Formosa University, Taiwan, R.O.C.

3Institute of Mechanical & Electro-Mechanical Engineering, National Formosa University, Taiwan, R.O.C.E-mail: jerry4421.hsieh@gmail.com, son.summer@msa.hinet.net, john91487@hotmail.com

ICETI-2014 J1054_SCINo. 15-CSME-24, E.I.C. Accession 3799

ABSTRACTTo achieve reduced costs and energy conservation, this paper proposes non-standard helical spur gear re-ducer with one gear pair (having reduction ratio 19.25) to be the gear reducer for electric scooter. Thispaper also focuses on the meshing efficiency analysis of non-standard helical spur gear pair. According toBuckingham’s research, the theoretical meshing efficiency formula of non-standard helical spur gear pair isderived. Three design cases of non-standard helical spur gear pair (4,77) are proposed as examples for ana-lyzing their meshing efficiencies at widely rotation speed range. The theoretical meshing efficiencies for thehelical spur gear pair (4, 77) are between 96.47–99.26%. Its best meshing efficiency occurs at 800–1000 rpmof pinion. The meshing efficiencies of these three design cases are almost same, and their differences areless than 0.5%. Considering the root strength of pinion, Cases II and III are better than Case I.

Keywords: friction coefficient, helical spur gear pair, meshing efficiency, sliding velocity.

ANALYSE ET EFFICACITÉ DE LA CONCEPTION D’UN RÉDUCTEUR PLANÉTAIRE ÀENGRENAGE HÉLICOÏDAL COMPORTANT UNE PAIRE D’ENGRENAGE UNIQUE POUR

MOBYLETTES ÉLECTRIQUES

RÉSUMÉDans un but de diminution des coûts et de conservation d’énergie, cet article présente un réducteur plané-taire hélicoïdal non-standard comportant une paire d’engrenage unique (ayant une moyenne de réduction de19,25) comme étant le réducteur planétaire pour une mobylette électrique. En outre, cet article met aussil’accent sur l’analyse de l’efficacité de la réduction de la paire d’engrenage hélicoïdal non-standard. En uti-lisant la formule théorique de Buckingam, nous avons établi la formule relative à l’efficacité du réducteurplanétaire hélicoïdal non-standard. Trois cas de paires d’engrenage hélicoïdal non-standard (4,77) sont pro-posés comme exemple pour l’analyse de l’efficacité de réduction à une plage étendue de vitesse de rotation.Les gains d’engrènement théorique pour l’engrenage hélicoïdal sont entre 96,47 et 99,26%. Le meilleur en-grènement se situe à 800 et 1000 rpm du pignon. Les gains d’efficacité de ces trois cas sont presque pareils,et leur différence est de moins de 0,5%. Pour de raisons de puissance du pignon, les cas II et cas III sontmeilleur que le cas I.

Mots-clés : coefficient de frottement; paire d’engrenage hélicoïdal; efficacité d’engrènement; vitesse glis-sante.

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Fig. 1. Spur gear pair.

1. INTRODUCTION

Gear reducers are widely used in various power systems, one of their applications is in the power system ofthe electric scooter. The purpose of gear reducer is to get large output torque than electric motor (or engine).In general, the reduction ratio of an ordinary spur gear pair is limited to 4–7. For the gear transmissionfor electric scooter, the reduction ratio is required to be 19.3, it is necessary to have two ordinary spur gearpairs. To achieve reduced costs and energy conservation, this paper proposes non-standard helical spur gearreducer with one gear pair (having reduction ratio 19.25) to substitute the gear reducer with two ordinarygear pairs.

The analysis of gear meshing efficiency involves involute theorem, sliding velocity of gears, calculationof friction coefficient, and other related issues. In the past, many researchers focused on involute theorem[1–3], sliding velocity of gears [4], frictional loss of gear meshing [5, 15], calculation of friction coefficient[6–10, 14, 15], and gear meshing efficiency [10–13].

Based on the equation of friction loss proposed by Chen [15], we derive the theoretical meshing efficiencyformula of non-standard helical spur gear pair. Then, based on the involute theorem, we derive the relativesliding velocity equations of non-standard helical spur gear pair for approach and recess. In this paper, wealso use same modified equation proposed by Chen [15] to calculate the average friction coefficients forapproach and recess. Three design cases of non-standard helical spur gear pair (4,77) are proposed forcalculating their meshing efficiencies.

2. POWER SYSTEM

The electric scooter has the following specifications: (1) Max. speed: 12.5 km/hr, (2) Load: 150 Kg,(3) Wheel diameter: 320 mm (0.32 m), and (4) Motor: 1280 W, 4000 rpm. For such an electric scooter, weobtain the following relationship:

VWheel = π×0.32m× 4000 rpmRr

× 601000

= 12.5 km/hr (1)

According to this equation, the reduction ratio of gear transmission for electric scooter is required to beRr = 19.3.

3. GEAR REDUCER

For the gear transmission for scooter, its reduction ratio is required to be 19.3. Considering the cost ofproduction, this paper proposes a helical spur gear reducer with one gear pair having reduction ratio 19.25.Figure 1 shows its corresponding kinematic skeleton, the teeth number of gear 2 is 4 (Z2 = 4) and the teethnumber of gear 3 is 77 (Z3 = 77) and its reduction ratio is 19.25 which is almost near 19.3.

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Table 1. Gear data of the helical spur gear reducer.Case I Case II Case III

Pinion 2 Gear 3 Pinion 2 Gear 3 Pinion 2 Gear 3Teeth Number of Gear 4 77 4 77 4 77Normal Module (mm) 1.25 1.25 1.25 1.25 1.25 1.25Helical Angle 32.5◦ (RH) 32.5◦

(LH)32.5◦ (RH) 32.5◦

(LH)32.5◦ (RH) 32.5◦

(LH)Shift coefficient (mm) 0.72 0.06 0.92 –0.57 0.92 0.03Operate Pitch Diameter(mm)

6.0346 116.1654 5.9852 115.21481 5.9852 114.1226

Base Diameter (mm) 5.4432 104.7817 5.4432 104.7817 5.4432 104.7817Out Diameter (mm) 10.0126 116.7726 10.2475 115.1976 10.2475 116.6976Root Diameter (mm) 4.6035 111.1476 5.1035 109.5726 5.1035 111.0726Gear machining Precision

HobbingPrecisionShaping

PrecisionHobbing

PrecisionShaping

PrecisionHobbing

PrecisionShaping

Surface roughness (µm) 0.8 0.8 0.8 0.8 0.8 0.8Center Distance (mm) 61.1 60.6 61.3Normal Pressure Angleφn

20 deg. 20 deg. 20 deg.

Normal Backlash (mm) 0.10174 0.10355 0.10650Contact ratio (Approach/Recess)

0.84083(0.16265 / 0.67818) 0.71953(–0.00485 / 0.72438) 0.74591(0.04046 / 0.70544)

For standard straight spur gear, the minimum number of teeth is 18, i.e. Z ≥ 18. So, the spur gear reducerwith few teeth must be the helical spur gear train and gear 2 must have large shift coefficient. Table 1 showsgear data of the three design examples.

4. THE MESHING EFFICIENCY EQUATION OF HELICAL SPUR GEAR PAIR

4.1. Meshing Efficiency of Standard Helical Spur Gear PairAccording to Chen’s paper [15], for the driving gear rotating at ω1, the friction loss per minute can beexpressed as:

Wf =WnRb1ω1(1+1/Rr)

βA1 +βR1

[fa

2β

2A1 +

fr

2β

2R1

](2)

where Wn is the normal force, Rb1 is the base radius of driving gear, Rr is the reduction ratio (Rr =Rb2/Rb1 =Z2/Z1), βA1 and βR1 are the angles of approach and recess of driving gear, fa is the average friction coefficientfor approach, and fr is the average friction coefficient for recess.

Figure 2 shows the force diagram of standard helical spur gear, where β is the helical angle at XY plane, φn

is the normal pressure angle, φt is the pressure angle at YZ plane, Fn is the normal force of helical spur gear,Ft is the tangent force of helical spur gear at XY plane, and Wn is the normal force at XY plane. Accordingto Fig. 3, we can obtain the following equations:

Ft = Fn cosφn cosβ (3)

Ft =Wn cosφt (4)

tanφn = tanφt cosβ (5)

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Fig. 2. The force diagram of standard helical spur gear.

For standard helical spur gear pair, the friction loss per minute of Eq. (2) shall be rewritten as

Wf H = FnRb1ω1(1+1/Rr)

βA1 +βR1

[fa

2β

2A1 +

fr

2β

2R1

](6)

Substituting Eqs. (3–5) into Eq. (6), the friction loss per minute of standard helical spur gear pair can beexpressed as

Wf H =cosφt

cosφn cosβWnRb1ω1

(1+1/Rr)

βA1 +βR1

[fa

2β

2A1 +

fr

2β

2R1

](7)

According to the definition of mechanical efficiency, the meshing efficiency for standard helical spur gearpair can be expressed as

ηm =WIn−Wf H

WIn(8)

For standard helical spur gear pair, the input power (WIn) is equal to WnRb1ω1, i.e. WIn =WnRb1ω1. Substi-tuting Eq. (7) into Eq. (8), the meshing efficiency of standard helical spur gear pair (ηmH(s)) can be expressedas

ηm H(s) = 1− (cosφt

cosφn cosβ)(1+1/Rr)

βA1 +βR1

[fa

2β

2A1 +

fr

2β

2R1

](9)

4.2. Meshing Efficiency of Non-standard Helical Spur Gear PairFor non-standard helical spur gear pair, the operation meshing pressure angle at YZ plane will be defined asφt(n), which can be expressed as

φt(n) = cos−1(

mn(N1 +N2)cosφt

2C× cosβ

)(10)

where C is the center distance of non-standard helical spur gear pair; N1 (N2) is the teeth number of gear 1(gear 2); mn is the normal module of gears. Therefore, the meshing efficiency for non-standard helical spurgear pair (ηmH(ns)) can be modified as

ηmH(ns) = 1−(

cosφt(n)

cosφn cosβ

)(1+1/Rr)

βA1 +βR1

[fa

2β

2A1 +

fr

2β

2R1

](11)

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5. THE SLIDING VELOCITY

In this section, we will derive the equations of angles of approach and recess, sliding velocities between twogears, friction coefficients, and meshing efficiency for helical spur gear pairs.

5.1. Angles of a Approach and RecessBased on Buckingham [10], the angles of approach and recess for non-standard helical spur pinion (drivinggear) can be expressed as

βA1 =

√(Ra2)2− (Rb2)2−R2 sinφt(n)

Rb1(12)

βR1 =

√(Ra1)2− (Rb1)2−R1 sinφt(n)

Rb1(13)

where Ra1, R1 and Rb1 are addendum, pitch, and base radius of driving gear. And, Ra2, R2 and Rb2 areaddendum, pitch, and base radius of driven gear. φt(n) is operation meshing pressure angle in the Y Z plane.

5.2. Sliding Velocity of Approach and RecessAccording to Buckingham [10], for non-standard helical spur gear pair, the average sliding velocity forapproach (VS(a)) and sliding velocity for recess (VS(r)) can be expressed as

VS(a) =Vt

2

[1

R1+

1R2

]×(√

(Ra2)2− (Rb2)2−R2 sinφt(n)

)(14)

VS(r) =Vt

2

[1

R1+

1R2

]×(√

(Ra1)2− (Rb1)2−R1 sinφt(n)

)(15)

where Vt is the velocity of pitch point and can be expressed as

Vt = ω1×R1 (16)

6. FRICTION COEFFICIENTS

Ffriction coefficient is not only a function of sliding velocity between meshing gears but also a function ofcontact pressure (P) of gear, dynamic viscosity (t) of oil, and surface roughness (S) of gear. Stribeck curve[14], shown in Fig. 3, was proposed to indicate the relationship among friction coefficient, load of gear,dynamic viscosity of oil, and sliding velocity between meshing gears. However, the relationship also lacksthe parameter of surface roughness (S) of gear.

By several experiments and verification, Buckingham [10] proposed the semi-empirical formula of fric-tion coefficients for gears in mesh. In this paper, the average friction coefficient for approach (fa) and averagefriction coefficient for recess (fr) are modified and expressed as

fa =4Ka

3

[0.05

e0.125VSa+0.002

√VS(a)

](17)

fr =2Ka

3

[0.05

e0.125VSr+0.002

√VS(r)

](18)

where Ka is the modified coefficient for average friction coefficient, which can be expressed as

Ka = KS×Kµ ×KL =

[1

1−S/45

]×[

1+µ/902

]×[

log(

10 Wn(max)

Wn

)](19)

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Fig. 3. Stribeck curve [14].

Table 2. Surface roughness and modified factor of surface roughness [14].Gear machining Surface roughness (S), µm KS =

11−S/45

Rough hobbing 3.2–1.6 1.0766–1.0369Precision hobbing 1.6–0.8 1.0369–1.0181Shaping 3.2–0.8 1.0766–1.0181Grinding 0.4–0.1 1.0090–1.0022

In Eq. (19), the first term is the modified factor of surface roughness (KS), the second term is the modifiedfactor of dynamic viscosity (Kµ ), and the third term is the modified factor of normal force (KL). Table 2shows the value of modified factor of surface roughness (KS) and Table 3 shows the value of modified factorof dynamic viscosity (Kµ ).

7. THE MESHING EFFICIENCY ANALYSIS

7.1. Design case I (C = 61.1)7.1.1. Sliding velocityFor Case I shown in Table 1 (C = 61.1 mm), according Eq. (5), the pressure angle at the YZ plane (φt) canbe obtained as follows:

φt = tan−1(

tanφn

cosβ

)= tan−1

(tan20◦

cos32.5◦

)= 23.3429◦

According to Eq. (10), the operation meshing pressure angle at the YZ plane (φt(n)) can be obtained asfollows:

φt(n) = cos−1(

mn(N1 +N2)cosφt

2C× cosβ

)=

1.25× (4+77)× cos23.3429◦

2×61.1× cos32.5◦= 25.5773◦

According to Eqs. (12–13), the angles of approach and recess for non-standard helical spur pinion (drivinggear) can be obtained as follows:

βA1 =

√(Ra2)2− (Rb2)2−R2 sinφt(n)

Rb1=

√58.38632−52.39082−58.0827× sin25.5773◦

2.7216= 0.2555 rad

βR1 =

√(Ra1)2− (Rb1)2−R1 sinφt(n)

Rb1=

√5.00632−2.72162−3.0173× sin25.5773◦

2.7216= 1.0653 rad

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Table 3. Dynamic viscosities and modified factor of dynamic viscosity [14].

Oil product Dynamic viscosity (cst 40◦C) Kµ = 1+µ/902

Motor Oil 5W40 82.16 0.9564Motor Oil 10W40 91.5 1.0083Motor Oil 5W50 106.5 1.0917Motor Oil 10W50 128.5 1.2139Grease No.1 164.5 1.4139

Fig. 4. The average sliding velocities of Case I.

When pinion rotates at speed 100 rpm, according to Eqs. (14–16), the average sliding velocity for ap-proach (VS(a)) and sliding velocity for recess (VS(r)) can be obtained as

VS(a) =(3.0173)×100×2π

2×[

13.0173

+1

58.0827

]×(√

(58.3863)2− (52.3908)2−58.0827× sin25.5772◦)= 229.86 mm/min = 0.75 ft/min

VS(r) =(3.0173)×100×2π

2×[

13.0173

+1

58.0827

]×(√

5.00632−2.72162−3.0173× sin25.5722◦)= 958.17 mm/min = 3.14 ft/min

Based on the above process, the average sliding velocities for approach (VS(a)) are between 0.75–27.14 ft/min for rotation speed between 100–3600 rpm. Also, the average sliding velocities for recess(VS(r)) are between 3.14–113.17 ft/min for rotation speed between 100–3600 rpm. The results are shown inFig. 4.

7.1.2. Friction coefficientsFor the gear reducer shown in Fig. 2, the pinion (Z = 4) is machined by precision hobbing and has surfaceroughness S = 0.8 µm, and the gear (Z = 77) is machined by precision shaping and has surface roughnessS = 0.8 µm, then, according to Table 2, the modified factor of surface roughness is equal to 1.0181 (KS =1.0181). If the lubricant used in the gear reducer is “Grease No. 1” with dynamic viscosity µ = 164.5,then, according to Table 3, the modified factor of dynamic viscosity is equal to 1.4319 (Kµ = 1.4139). The

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Fig. 5. The 3D representation of the average friction coefficient of Case I.

Fig. 6. The average modified friction coefficients.

maximum input torque is 40 Nm, for the input torque between 10–40 Nm, the load modified coefficientKL = 1.6021–1.0. Then, the total modified coefficients (Ka) are between 1.4395–2.3355.

When pinion rotates at speed 100 rpm and input torque is 40 Nm, according to Eqs. (17–19), the averagefriction coefficients for approach (fa) and recess (fr) are obtained as

fa =4Ka

3

[0.05

e0.125VSa+0.002

√VS(a)

]=

4×1.43953

[0.05

e0.125×0.75 +0.002√

0.75]= 0.0907

fr =2Ka

3

[0.05

e0.125VSr+0.002

√VS(r)

]=

2×1.43953

[0.05

e0.125×3.14 +0.002√

3.14]= 0.0358

Then, based on the above reasoning, the average friction coefficients for approach (fa) and recess (fr) areobtained and shown in Figs. 5(a) and 5(b), respectively. For the input torque at 40 Nm, average modifiedfriction coefficients for approach (fa) are between 0.0907–0.0232 and average modified friction coefficientsfor recess (fr) are between 0.0358–0.0204 for rotation speed between 100–3600 rpm. The results are shownin Fig. 6.

7.1.3. Meshing efficienciesFor Case I shown in Table 1, the angle of approach βA1 = 0.2555 rad and the angle of recess βR1 =1.0653 rad. When the pinion rotates at speed 100 rpm and the input torque is 40 Nm, according to Eq. (11),

462 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

Fig. 7. The meshing efficiency diagram of Case I.

Table 4. Meshing efficiencies of Case I of the non-standard helical spur gear pair (4,77).Nm

rpm 10 15 20 25 30 35 40100 0.966208 0.969922 0.972557 0.974601 0.976272 0.977684 0.978907300 0.979278 0.981555 0.983171 0.984425 0.985449 0.986315 0.987065900 0.988015 0.989332 0.990267 0.990992 0.991584 0.992085 0.992519

1000 0.988084 0.989393 0.990323 0.991044 0.991633 0.992131 0.9925621100 0.988038 0.989353 0.990286 0.99101 0.991601 0.992101 0.9925341200 0.987917 0.989245 0.990188 0.990918 0.991516 0.992021 0.9924581800 0.986566 0.988042 0.98909 0.989903 0.990567 0.991128 0.9916142400 0.985012 0.98666 0.987828 0.988735 0.989476 0.990102 0.9906453000 0.983507 0.98532 0.986606 0.987604 0.988419 0.989108 0.9897053600 0.982075 0.984045 0.985443 0.986527 0.987413 0.988162 0.988811

its meshing efficiency is obtained as

ηmH(ns) = 1−(

cosφt(n)

cosφn cosβ

)(1+1/Rr)

βA1 +βR1

[fa

2β

2A1 +

fr

2β

2R1

]= 1−

(cos25.5773◦

cos20◦× cos32.5◦

)× 1+1/19.25

0.2555+1.0653×(

0.09072×0.25552 +

0.03582×1.06532

)= 97.89%

Then, based on the above results, the meshing efficiencies of Case I are between 96.62–99.26, shown in Ta-ble 4. The best meshing efficiency occurs at pinion speed 1000 rpm. Figure 7 shows the meshing efficiencydiagram of Case I.

7.2. Case II (C = 60.6)For Case II shown in Table 1, based on the above process, the meshing efficiencies are between 96.13–99.20%. The best meshing efficiency occurs at pinion speed 800 rpm. Figure 8(a) shows the meshingefficiency diagram of Case II.

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Fig. 8. The meshing efficiency diagram of (a) Case II and (b) Case III.

7.3. Case III (C = 61.3)For Case III shown in Table 1, based on the above process, the meshing efficiencies are between 96.43–99.26%. The best meshing efficiency also occurs at pinion speed 900 rpm. Figure 8(b) shows the meshingefficiency diagram of Case III.

7.4. Comparison of the Three CasesObserving Figs. 7 and 8, Case I has the best meshing efficiencies, however, they are almost same. Theirdifferences of meshing efficiencies are less than 0.5%, which can be neglected. The root diameter of pinion(Z = 4) of Case I is 4.6035 mm and the root diameters of pinion (Z = 4) of Cases II and III both are5.1035 mm. Due to the root strength, Cases II and III are better than Case I. These two cases can be usedfor engineering design.

8. CONCLUSION

Due to shortage of energy, the meshing efficiency of gear train becomes an important factor for powersystem. This paper focuses on the meshing efficiency of non-standard helical spur gear pair. In this paper,non-standard helical spur gear reducer with one gear pair (having reduction ratio 19.25) has been proposed asthe gear reducer of electric scooter. According to Buckingham’s research, the theoretical meshing efficiencyformula of non-standard helical spur gear pair is derived. Three design cases of non-standard helical spurgear pair (4,77) are proposed as examples for analyzing their meshing efficiencies at widely rotation speedranged. The theoretical meshing efficiencies for the gear pair (4,77) are between 96.47–99.26%. Its bestmeshing efficiency occurs at 800–1000 rpm of pinion. The meshing efficiencies of these three design casesare almost same, and the differences are less than 0.5%. Due to the reason of root strength of pinion, CasesII and III are better than Case I. The results of this paper can be used as a reference for engineering to designthe non-standard helical spur gear reducers and will enhance the competitiveness of gear industry.

ACKNOWLEDGEMENTS

The authors are grateful to the National Science Council of the Republic of China for the support of thisresearch under NSC 102-2221-E-150-045. The authors are also grateful to Li Yuan Machinery Co., Ltd. forthe support of this research under 103AF13.

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