Performance analysis of generated hypoid gear based on measured tooth flank form data

Mechanism and Machine Theory 72 (2014) 1–16

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Performance analysis of generated hypoid gear based on measured tooth flank form data Ryohei Takeda a, Masaharu Komori a,⁎, Tatsuya Nishino a, Yukihiko Kimura a, Takayuki Nishino b, Kenji Okuda b, Shinji Yamamoto b a b

Kyoto University, Department of Mechanical Engineering and Science, Kyoto daigaku-Katsura, Nishikyo-ku, Kyoto-shi, Kyoto 615-8530, Japan Mazda Motor Corporation, 3-1, Shinchi, Fuchu-cho, Aki-gun, Hiroshima 730-8670, Japan

a r t i c l e

i n f o

Article history: Received 18 February 2013 Received in revised form 19 September 2013 Accepted 22 September 2013 Available online 24 October 2013 Keywords: Vibration of rotating body Simulation Shape measurement Generating motion Hypoid gear Noise control

a b s t r a c t A method for predicting and analyzing the tooth contact pattern, vibration, and strength of a generated hypoid gear is needed to achieve a low-noise design and adequate quality control. However, it is not easy to analyze the performance of a generated hypoid gear because the tooth flank form is complicated and has a significant influence on the overall performance. In order to solve this problem, in this research, a method for analyzing one of the gear dynamics excitations and contact condition of a generated hypoid gear that considers the measured tooth flank form is proposed. The contact pattern and transmission error are measured experimentally and are compared with the analysis results. It is confirmed that the result from the proposed analysis method agrees with the experimental result. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Hypoid gears have advantages over spiral bevel gears due to their strength and smooth rotation, and thus, they are widely used in rear-wheel drive and four-wheel drive vehicles. Recently, engine and road noise in vehicles has been improved, and therefore, better noise and vibration quality are demanded for hypoid gears. Moreover, the load carrying capability must be increased in order to transmit higher torques. To meet these requirements, performance analysis technology is necessary. For hypoid gears, a lot of research has been conducted on theoretical tooth geometry [1–5]. Since CNC-controlled bevel and hypoid gear cutting machines and grinding machines were developed, much research related to CNC control has been conducted [6–9]. As for the analysis of real tooth flank form, Kin [10] studied spur gear adding the measured data to theoretical involute surface and interpolating them. And Zhang et al. [11] made unloaded tooth contact analysis of non-generated hypoid gear based on the measured tooth flank form data by CMM using similar approach with Kin. Most of it deals with the final drives of non-generated face mill hypoid gears for automobiles, of which the manufacturing method is rather simple. On the other hand, recently, cases that require a generated hypoid gear wheel instead of a non-generated one have increased. For example, these days, multipurpose four-wheel drive vehicles are very popular. In the case of a four-wheel drive system based on a front-wheel drive vehicle, a hypoid gear with a low ratio is used in the transfer gearbox. And in many applications the gear ratio becomes less than 2.5, and in that case, it is difficult to use non-generated cutting such as Formate® or Helixform® on such wheels. In that case, a generated hypoid gear must be used. In generated hypoid gears, the wheel tooth flank form becomes very complicated, and not much research has been reported on them. There is a report [12], in which the influence of misalignment to path of contact, minimal separation along the potential contact line and tooth contact pressure distribution are studied on generated spiral bevel

⁎ Corresponding author. Tel./fax: +81 75 383 3587. E-mail address: [email protected] (M. Komori). 0094-114X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2013.09.008

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gear with mismatched surface. But these studies are for the theoretical pinion surface and not using the actual tooth flank form. And tooth contact pattern and transmission error are not studied. Therefore, in this research, a method for analyzing the gear dynamics excitation and contacting condition of a generated face mill hypoid gear is developed. During manufacturing of a generated hypoid gear, a lapping process is used after gear tooth cutting and heat treatment. If gear tooth grinding is used, lapping is typically performed afterward. The gear dynamics excitation of a hypoid gear is largely affected by any small waviness of the tooth surface. For that reason, to accurately analyze the dynamic performance of a hypoid gear, detailed information on the tooth flank form must be considered. In the case where lapping is used, the tooth flank form of the hypoid gear after lapping becomes different from that after gear tooth cutting based on gear tooth cutting theory. Therefore, it is difficult to accurately analyze an actual gear set using tooth cutting theory. Also, analysis of the tooth flank form after heat treatment and lapping is required. There are few studies of analysis of hypoid gear using measured tooth flank form [11,13]. In the study [11], pinion tooth form measurement is made on machine setting base, but it is not suitable to analyze the tooth flank form after heat treatment or lapping accurately. And those studies use tooth flank form measurement data of 5 × 9 grid points, but they are not sufficient to get detailed information of the tooth flank form after heat treatment or lapping. The authors have presented an analysis method for the generated face mill hypoid gear tooth geometry based on conjugate tooth flank theory and developed a tooth flank scanning measurement method in the previous report [14]. This was then used to obtain detailed information on the tooth flank form of a generated face mill hypoid gear. In this research, a performance analysis method for a generated face mill hypoid gear that utilizes tooth flank form data obtained from the scanning measurement and takes the small waviness of the tooth flank form into account is developed. By comparing the result of the analysis with that of the experiment, the effectiveness of the proposed method is confirmed. 2. Introduction of theoretical and conjugate tooth flank form and composite deviation 2.1. Theoretical tooth flank form The purpose of this research is to predict the gear dynamics excitation and strength of generated face mill hypoid gears by measuring the tooth flank form and analyzing the tooth contact condition using the measured data. To achieve this, it is desirable to use a reference surface with an ideal contact condition for the gear performance. The gear set with tooth flank forms conjugate to each other has tooth contact on the full tooth flank form and has no transmission error. Then, the conjugate tooth flank form can be described as the ideal tooth contact condition. For this reason, in this research, the conjugate tooth flank form is used as the reference surface. Specifically, the reference surface of the wheel is defined as the surface derived from the envelope of the gear cutter and that of the mating pinion at the conjugate surface of the wheel [15]. 2.2. Calculation of theoretical tooth flank form for generated face mill hypoid gear Recently developed CNC-controlled hypoid pinion cutting machines do not have the mechanical devices (such as eccentric angle and cradle) that old mechanical machines have, but the relative motion of the cutter and workpiece duplicates that of old mechanical machines. Therefore, this research is done based on the Gleason No. 106 generator shown in Fig. 1.

Cradle angle Yc1

Cutter spindle rotation angle Xc1 Work offset

Eccentric angle

Zc1

Swivel angle

Sliding base

Machine center to crossing point Machine root angle Fig. 1. Hypoid gear generator.

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Fig. 2 shows the analysis model, which is a simplified model of the construction of the generator. The cutter is attached to a tilting mechanism, where the “Cutter spindle rotation angle” adjusts the cutter axis tilt angle θi. This assembly is mounted on a swivel mechanism, where the “Swivel angle” adjusts the direction of the tilting axis θj. This assembly is mounted on a table to adjust the eccentricity of the cutter axis from the machine center, where the “Eccentric angle” indicates the radius of orbital motion of the cutter axis Sr. This assembly is mounted on a rotating table or “Cradle,” which rotates around the machine center and creates the orbital motion of the cutter. The work gear spindle is located on a vertical slide, which adjusts the vertical position of the spindle (“Blank offset EM”). This is mounted on a work gear axial adjustment mechanism (“Machine center to back ΔA”). It is set on the rotating table, which adjusts the root angle of the work gear (“Machine root angle γ1”), and on a feed slide (“Sliding base”), which adjusts the in-feed position of the workpiece (ΔB). In this model, the following coordinate systems are used to define the relationship between the cutter and work gear: Of − Xf,Yf,Zf Cutter coordinate system Ob − Xb,Yb,Zb Swivel angle coordinate system Oc1 − Xc1,Yc1,Zc1 Cradle coordinate system O1 − X1,Y1,Z1 Work gear coordinate system Oh − Xh,Yh,Zh Adjusted work gear coordinate system The cutter rotates on the cutter axis Zr, and by rotating the cradle axis Zc1 and work gear axis Zh at a constant speed ratio, the tooth flanks are generated. As mentioned in the previous report [14], for a point on the cutting edge and its contact point on the tooth flank to be cut, the direction of their relative velocities is perpendicular to their common normal vector, and from this condition, the contact point of the cutting edge and tooth flank can be obtained. From this point, the coordinates of the tooth flank point in the work gear coordinate system can be obtained as follows. ! Point M on the cutting edge is determined by the rotation angle αc of cutter axis zf, the distance along the generating cone ! ! sp and rotation angle of cradle α1 which moves the cutter axis around the cradle axis Oc1. The normal vector N of point M can be described by αc and α1. α1 and rotation angle of work gear α3 have the relation shown by Eq. (1). I f ¼ α 1 =α 3 :

ð1Þ

! ! Then M and N are described as Eqs. (2) and (3).  ! !
M ¼ M Sp ; α C ; α 3

ð2Þ

! ! N ¼ N ðα C ; α 3 Þ:

ð3Þ

Fig. 2. Coordinate systems of hypoid generator and analysis model [16].

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! ! The generated work tooth and the cutting edge contact at point M . They have common normal vector N and move in the ! different direction. And at the contact point M , normal component of velocity of 2 surfaces must be equal. From this condition, Eq. (4) is obtained.


! ωc


!  ! ! !  N ¼ 0:  M −! ω 1  M− O 1

ð4Þ

Then substituting Eqs. (2) and (3) into Eq. (4) and solving it, sp is determined by αc and α3. Sp ¼ Sp ðα C ; α 3 Þ:

ð5Þ

! sp can be eliminated from Eq. (2) and M is describe as Eq. (6). ! ! M ¼ M ðα C ; α 3 Þ:

ð6Þ

! Then by coordinate transform, contact point in the work gear coordinate P is also described as Eq. (7). ! ! P ¼ P ðα C ; α 3 Þ:

ð7Þ

By obtaining all the required points, the tooth flank form can be determined. 2.3. Calculation method for surface grid points of generated face mill hypoid gear wheel For hypoid gear performance analysis, it is required to divide the surface in the profile and lengthwise directions and to describe the tooth flank form using grid points. To calculate the theoretical tooth flank form of the hypoid gear wheel by numerical analysis as described in the previous section, a point on the cutting edge is taken, and the line generated by the point is calculated. In the case of a non-generated face mill hypoid gear, on which much research has been performed, the tip of the cutter sweeps the tooth root line, and the straight-sided cutter shape is transformed to the flank form. As shown in Fig. 3, all the lines generated by the points on the cutting edge are parallel to the root line. When the cutting edge rotates to a certain phase angle, the contacting points of the cutting edge and tooth flank lie on a straight line perpendicular to the generating direction. For this reason, the grid points of the tooth flank form for a non-generated face mill hypoid gear wheel can be described by the points on the cutting edge and the rotational phase angles. In other words, the tooth profile (from root to tip) can be divided by the points on the cutting edge, and the tooth trace (from toe to heel) can be divided by the cutter rotation angle. Using this method, the tooth flank form of a non-generated mill hypoid gear wheel can be described easily. And this approach was reported by Kubo et al. [5]. On the other hand, a wheel tooth flank of a generated face mill hypoid gear is cut by the generating motion, and the position of the cutter axis changes in relation to the tooth flank. Then, as shown in Fig. 4, the shapes of the paths of the points on the cutting edge are not straight lines but are curves, and an instantaneous contact point has different normal directions at different

Fig. 3. Position of tooth surface generated by each point of cutter edge in the case of a non-generated face mill hypoid gear.

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positions. As a result, it becomes complicated and impractical to describe the tooth flank of a generated face mill hypoid gear wheel using a point on the cutting edge and the cutter rotation angle. For this reason, in this research, a method for describing the tooth flank using the radial position R and the axial position Z is proposed, as shown in Fig. 4, and this method is suitable to describe the generated face mill hypoid gear wheel using the grid points.

2.4. Calculation method for conjugate pinion tooth form Now, the wheel tooth surface of the generated face mill hypoid gear is defined as the envelope of the cutting edge controlled by the cutting machine setting. The pinion tooth surface is defined as the surface conjugate to the wheel. Fig. 5 shows the wheel and pinion coordinate systems and the contact between their surfaces. E αg αp ∑

Pinion offset Wheel rotation angle Pinion rotation angle Shaft angle ! O pg Offset vector ! ! r g; r p Point vectors of the points on the tooth flank of the wheel and pinion ! ! V g; V p Velocity vectors of the points on the tooth flank of the wheel and pinion ! ! ω g ; ω p Angular velocity vectors of the wheel and pinion Og − Xg, Yg, Zg Wheel coordinate system G (the 3rd axis is the wheel axis, and the 2nd axis is the offset direction) Op − Xp, Yp, Zp Pinion coordinate system P (the 3rd axis is the pinion axis, and the 2nd axis is the offset direction) Based on the rotation angle of the wheel, the coordinates and normal vectors of the points on the tooth flank are changed. The angle position of a certain point, of which the coordinates and nominal vector are known, is established as the reference angle. When the tooth flank of the wheel and its mating pinion have conjugate action, the contact point satisfies following equation:
! ! ! Vp − Vg  Vg ¼ 0

ð8Þ

In such case following equation can be obtained. −xg0  Nyg0 ð cos ∑ ¼ 1=γÞ þ yg0  Nxg0 ð cos ∑ ¼ 1=γÞ þ Nzg0  sin ∑  E
 þ −E cos ∑  N yg0 þ zg0  Nyg0  sin ∑−yg0  Nyg0  sin ∑ cos α g
 þ E cos ∑  Nyg0 þ zg0  N xg0  sin∑−xg0  N zg0  sin ∑  sinα g ¼ 0:

ð9Þ

Point C Point B Z

Tooth tip

Point A Generation by point C

Cutter Generation by point B Generation by point A Tooth root

R Fig. 4. Position of tooth surface generated by each point of cutter edge in the case of a generated face mill hypoid gear.

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ωg

The coordinate system of gear Og

Gear

Zg

- vg

αg

Σ

axis

rg Xg Yg

Q

v p- v g

Op g E Yp

vp

rp

v = ω g× r g g v p = ω p× r p

Xp Op The coordinate system of pinion

Zp

ωp

αp Pinio

n axi

s

Fig. 5. Definition of gear and pinion coordinates [17].

Z

Y X

Fig. 6. Structure of the measuring machine for the hypoid gear tooth flank form.

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From Eq. (9), wheel rotation angle αg when point Q on the wheel is in contact with pinion surface is obtained. And pinion rotation angle αp is αp ¼ γ  αg :

ð10Þ

By coordinate transformation, the position vector and the normal vector of the contact point of pinion surface can be obtained on pinion coordinate system P. 3. Measurement method for tooth flank form of generated face mill hypoid gear For measuring the tooth flank form of a generated face mill hypoid gear, a CNC gear measuring machine with three orthogonal axes and a rotational axis (main spindle) that has been developed by the authors [14] is used, as shown in Fig. 6. In this measuring machine, mounting surface of the measuring gear is used as the datum surface of Z axis, and the measuring position is chosen such that the X component of the normal vector of the measured tooth flank becomes as small as possible, as shown in Fig. 7(a). Then, detecting the Y–Z directions is sufficient for the measurement, and only two axis probe is required. In the case of an ordinary coordinate measuring machine, the measurement is performed by touching each point, but the developed machine performs the measurement by scanning the line on the tooth flank; this makes it possible to measure the tooth flank form in detail. The measuring lines are set in both the profile direction and the tooth trace direction, as shown in Fig. 7(b). By this method, the measurement, which is similar to that of a cylindrical gear, is possible. In this research, measurement data for 29 profile lines and one tooth trace line are used. 4. Analysis method for tooth flank contact condition and transmission error under load In this section, an analysis method is developed for the tooth flank contact condition and transmission error of a generated face mill hypoid gear under load. When the tooth flanks of a gear pair are meshed and transmitting load, the tooth flank causes elastic deformation, and certain areas make contact. This instantaneous contact area moves according to the rotation of the gear pair; then, the envelope of all these contact areas creates a tooth contact pattern. In this research, a performance analysis program for a generated face mill hypoid gear that utilizes tooth flank measurement data based on the conjugate tooth flank principal (tooth flank deviation from conjugate reference surfaces) is developed. Additionally, the analysis program uses the conjugate tooth flank principal. In the case of conjugate tooth flanks, instantaneous tooth flank contact can be described by the contact line. Then, the load distribution over a certain contact area Ωℓ for a number of engaging pairs of tooth flanks ℓ (ℓ=1,…,nℓ, where nℓ is the number of pairs of tooth flanks engaging at the same time) of the pinion and wheel satisfies the following equation [5]:

∫Ωℓ K Bℓ ðη; ξÞdξ þ K ðηÞ  P ℓ ðηÞ ¼ δℓ ðη; ΔÞ  eRℓ ðηÞ

O

ð11Þ

Y

Gear rotation to target position θt

Gear tooth surface

Z Profile

Profile

Toe

Toe

Normal vector

Q

Heel

Heel

Tooth trace Probe headof sensor Yp =Constant

R

Movingdirection of probe head on tooth trace measurement

Tooth trace

Offset plane X

(a) Probe position and direction of normal vector

(b) Profile and tooth trace measurement

on gear tooth surface Fig. 7. Measurement method for tooth flank form of generated hypoid gear.

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Hypoid Gear dimensions; Wheel Machine setting; Gear Theoretical Surface ; Conjugate Pinion Surface; Wheel and Pinion Tooth Surface Measurement Composite Tooth Flank Form Deviation of Pinion and Wheel

Transmitting Torque; Position of rotational angle

Calculating the contact range on the geometrical contact line Dividing the contact range Interference value of each divided area Compliance of bending and shearing deflection Initial value of load distribution Compliance of contact deformation Convergence of load distribution

Y N Modification of contact area

Load distribution and torque Y N

Existence of tooth pairs simultaneously The amount of torque = ? = Input torque

Y N

Modification of transmission error

Y N

Existence of other position of rotation angle

End Fig. 8. Flow chart of calculation of tooth contact under load.

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Table 1 Gear dimension.

Number of teeth Module Face width Pinion offset Shaft angle Pitch angle

mm mm mm Degree Degree

Pinion

Gear

19

24 4.0 24.0

5.0 90 39.27

50.58

where ( δℓ ðη; ΔÞ ¼

n o Δ− epℓ ðηÞ þ egℓ ðηÞ ; for δℓ ðη; ΔÞN0 for δℓ ðη; ΔÞb0

0;

Ωℓ Contact area η Coordinate of deflection measuring point in the contact area ξ Coordinate of load point in the contact area P ℓ ðξÞ Contact pressure at point ξ KB(η,ξ) Compliance of tooth deflection by bending, shearing, and inclination of tooth root KC(η) Compliance of tooth contact deformation δℓ ðη; ΔÞ Tooth deflection of gear set at point η Δ Delayed angle of the driven gear from the theoretical position in relation to the drive gear epℓ ðηÞ Tooth flank form deviation of pinion (angle) egℓ ðηÞ Tooth flank form deviation of wheel (angle) eRℓ ðηÞ ¼ −yg ðηÞ  nxg ðηÞ þ xg ðηÞ  nyg ðηÞ Conversion factor of deviation from angle to distance at point η xg(η), yg(η) X and Y components of coordinate at point η nxg(η), nyg(η) X and Y components of normal vector at point η Compliance of tooth contact deformation Kc(η) is the function of local radius of curvature and is not influenced by the tooth flank form so much, then it is considered that the same model as the involute gear can be used. And Compliance of tooth deflection by bending, shearing, and inclination of tooth root is expressed by following equation. [18]


 K B jη ; jξ ¼ K bt jη ; jξ þ K br jη ; jξ

ð12Þ

where, Kbt(jη,jξ) Compliance of tooth deflection by bending and shearing Kbr(jη,jξ) Compliance of tooth deflection by inclination of tooth root.

Concave Profile (Toe)

Wheel

Gear set A Convex Profile (Toe)

Pinion

Concave Profile (Toe)

Wheel

Gear set B Convex Profile (Toe)

Profile (Mean)

Profile (Mean)

Profile (Mean)

Profile (Mean)

Profile (Heel)

Profile (Heel)

Profile (Heel)

Profile (Heel)

Tooth trace

Tooth trace

Tooth trace

Fig. 9. Tooth flank form measurement results.

Tooth trace

Pinion

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Transfer of tooth contact pattern using adhesive transparent tape

Fig. 10. Recording method for actual tooth contact pattern.

Then hypoid gear tooth model is created and those compliances are calculated by FEA. And by the comparison with the FEA result, following approximation equations are created. Approximation equation of compliance of tooth deflection by bending and shearing is as follows.
 U R R K bt jη ; jξ ¼ Rt   V tr  RtF1  RtF2  tR1 tR2  cos ϕjη  cos ϕjξ  RtT1  RtT2  RtA1  RtA2 m RtR3 where, Rt U m Vtr RtF1 RtF2 RtR1 RtR2 RtR3 φjη φjξ RtT1, RtT2 RtA1, RtA2

ð13Þ

Tooth form factor for hypoid gear Inverse number for spring constant of directly below the tip load (finite width) Module Distance factor for load and calculating points Position factor for calculating point in tooth trace direction Position factor for load point in tooth trace direction Position factor for calculating point in profile direction Position factor for load point in profile direction Distance factor for load and calculating points in profile direction Normal angle for calculating point (pressure angle) Normal angle for load point (pressure angle) Tooth depth variation factor Tooth width variation factor

All of distance factors are non-dimensionalized by dividing by mean tooth depth Th. Approximation equation of compliance of tooth deflection by inclination of tooth root is as follows.
 R R K br jη ; jξ ¼ Rr  V rr  RrF1  RrF2  rR1 rR2  RrT1  RrT2  RrA1  RrA2 RrR3 where, Rr

ð14Þ

Tooth form factor for hypoid gear

(a) Gear set A

(b) Gear set B

Fig. 11. Measured actual tooth contact pattern (V = 0 μm, H = 0 μm, load = 20 Nm).

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a) Gear set A

11

b) Gear set B

Fig. 12. Simulated tooth contact pattern (V = 0 μm, H = 0 μm, load = 20 Nm).

The direction of the distributed force Pℓ(ξ)dξ at a certain point ξ on the tooth flank is the same as that of the normal vector at that point. In the plane perpendicular to the wheel axis, assuming the component of P ℓ ðξÞdξ orthogonal to the radial direction of the point of the force Pℓ(ξ) is P ℓ ðξÞdξ, the torque caused by the load distribution is described by the following equation [5]:

T¼

n Z X ℓ¼1



Ωℓ

P ℓ ðξÞ  r ℓ ðξÞdξ:

ð15Þ

By obtaining the delayed angle of the driven gear from the theoretical position in relation to the drive gear Δ, which satisfies Eqs. (11) and (15), the condition of elastic deformation for the mating teeth at a certain instance is also obtained. Additionally, the calculation is performed according to the following procedure. Assume that the drive tooth flank and driven tooth flank of a hypoid gear have a point contact; then, turn the drive gear to obtain a certain amount of relative phase angle. In this case, interference of the tooth flanks occurs for the drive and driven gears. In the calculation of the interference, the measured small waviness of the tooth flanks is taken into account. Then, the load distribution of the interference area is calculated using Eq. (11). For the compliance of the tooth deflection by bending, shearing, and inclination of the tooth root and that of the contact deformation of the teeth, the previously reported values are used [5]. Additionally, if there is waviness on the tooth flank, in some cases, the contact line is divided into several portions instead of one contact line. In this case, Eq. (11) is used for those partial contact lines. From this calculation, the load distribution caused by the contact of a pair of teeth at a certain instant is obtained. Then, the same calculations are performed for those pairs of teeth, which can be in contact simultaneously, and their load distributions are obtained. Then, the torque of the wheel is calculated using Eq. (15) with the load distributions of each pair of teeth. Although the obtained torque from this calculation must coincide with the given torque as a precondition of the analysis, if the assumed delayed angle of the driven gear from the theoretical position in relation to the drive gear Δ is too large, the calculated torque becomes too large, and vice versa. Then, by convergence calculation, the final delayed angle is determined with the given torque as the convergence condition and the delayed angle as the variable. By applying a similar process to each rotational step of the gear set, the delayed angles of the driven gear are obtained from the theoretical position relative to the drive gear for all rotational steps of the gear set. The variation in the rotational delayed angle of the driven gear in relation to the drive gear corresponds to a transmission error of the gear set. Finally, by substituting all of the obtained delayed angles into Eq. (11), the load distributions on the tooth surfaces can be calculated. And the flow chart of calculation of tooth contact under load is shown in Fig. 8.

5. Comparison of analysis and experiment results for tooth contact and transmission error The tooth contact pattern and transmission error have a strong correlation with the noise, vibration, and strength of a hypoid gear, so they are used to ensure quality control. In this chapter, the tooth contact pattern and transmission error are analyzed using the developed analysis program and measured tooth flank form, and the results are compared with the experimental results.

a) Gear set A b) Gear set B (V= 20µm, H= 40µm, Load=20Nm) (V= 20µm, H= -220µm, Load=20Nm) Fig. 13. Simulated tooth contact pattern corresponding to measured actual tooth contact pattern.

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Experiment

Simulation

a) Change of alignment V

Experiment

Simulation

a) Change of alignment H Fig. 14. Tooth contact pattern of gear set A.

5.1. Target gears The dimensions of the target generated face mill hypoid gears are shown in Table 1. They are cut, heat treated, and lapped. Two pairs of gear sets (gear sets A and B) with the same dimensions are prepared. These gear sets were taken from the volume production line. Because of it, those gear sets had some alignment errors. Though it is the best for the verification of the analysis program to use gear sets with no alignment error or with the alignment error of which amounts are known, it was not possible to prepare such kind of gear sets, then authors used those volume production gear sets. However the method to analyze gear set with uncertain alignment error like volume production gears is also required. And a purpose of this research is to realize the performance analysis method for gear sets with uncertain alignment errors by estimate the alignment error. It is also the reason to choose volume production gears. The measurement results for the gear sets measured using the developed measuring machine

Experiment

Simulation

a) Change of alignment V

Experiment

b) Change of alignment H

Fig. 15. Tooth contact pattern of gear set B.

Simulation

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are shown in Fig. 9. The measurements are taken using the conjugate method, and the points of the tooth flank measurement curve for the wheel and pinion correspond to the contact points of the wheel and pinion tooth flanks. Even though the dimensions of gear sets A and B are the same, there are differences in their tooth flank forms. In the figure, three lines of profile measurements and one line of tooth trace measurement are shown for each tooth flank, but for the analysis, 29 lines of profile measurement data are used. 5.2. Comparison of measurement and analysis results for tooth contact pattern The tooth contact patterns are measured for gear sets A and B on a gear rolling tester. The gear rolling tester used has the ability to adjust the alignment error in the direction of the hypoid gear offset V and the direction of the pinion axis H for the observation of the tooth contact pattern. To record the tooth contact pattern, the gear set is mounted on the tester, the wheel teeth are painted with a marking compound, and the gear set is run under a given load. As a result, the marking compound on the contacting area of the wheel tooth flanks is rubbed off. Then, by attaching transparent adhesive tape to the tooth flank and removing it, the remaining compound on the tooth flank is transferred to the tape. Then, the tape is attached to a record sheet. The area without the compound corresponds to the contact area of the wheel tooth flank as shown in Fig. 10. Fig. 11 shows the measured tooth contact patterns of gears sets A and B on the rolling tester with the alignment error for V = 0 μm, H = 0 μm. From now on, these contact patterns are called the reference contact patterns of the gear sets. Also, the contact patterns shown in Fig. 11 are the results of the rolling tests with 20 Nm of torque on the wheel axis. The tooth contact patterns obtained from simulation are shown in Fig. 12. The simulation was performed with the same conditions of alignment (V = 0 μm, H = 0 μm) and torque as the measurement. Those tooth contact patterns in the measurement and simulation results do not show any edge contact on the tip, toe end, or heel end. In this regard, the measurement and simulation have common results. However, from detailed observation, it can be seen that the simulated contact patterns are located closer to the toe. Also, the shapes of the simulated contact patterns are different in length from those of the measured ones. The reason for the difference is considered to be the difference of the actual alignment errors V and H from that of simulation. In the simulation program, the calculation is made for the condition where the pinion and wheel are located at the theoretical positions. On the other hand, the actual hypoid gear is equipped with shims on the mounting surface to adjust their axial position, and its mounting surfaces are not necessarily at the theoretical position; as a result, even though the origin point of the rolling tester is shifted, tooth contact pattern can be corrected when gear set is assembled. In the case of hypoid gears, these alignment errors affect the position of tooth contact, and this shift of the origin point is possibly the reason for the difference in tooth contact patterns between the measurement and simulation. The developed simulation program is capable of adjusting the alignment errors V and H. Then, using this function, the combination of alignment errors V and H, with which the simulated tooth contact pattern becomes close to that of the measurement, is obtained. Fig. 13(a) shows the simulated tooth contact pattern of gear set A with an alignment error of V = 20 μm, H = 40 μm. In comparison with Fig. 11(a), the lengths of the tooth contact pattern in the profile and lengthwise directions are close to each other. Additionally, the contact pattern in Fig. 11(a) has an elongated shape on both the toe and heel sides, and this characteristic can also be seen in Fig. 13(a). As a result, it can be said that the simulated tooth contact pattern with alignment errors of V = 20 μm, H = 40 μm gives close agreement with the contact pattern of an actual gear set with alignment errors of V = 0 μm, H = 0 μm. The results for gear set B are shown in Fig. 13(b) in the same manner. In the case of gear set B, it is found that with alignment errors of V = 40 μm, H = − 200 μm, the simulated tooth contact pattern is in close agreement with the reference contact pattern of the actual gear set. These obtained differences in the values of V and H between the measurement and simulation are defined as the “amounts of alignment shift.” The amount of alignment shift H is different between gear sets A and B. However, the alignment error H is in the pinion axis direction, and it is common for each gear set to have some deviation in the alignment error H. Additionally, the deviation is normally adjusted by shims, which are attached behind the mounting surface of the pinion. From this condition, the difference in alignment error H between gear sets A and B is considered to be acceptable. Then, the tooth contact pattern for different alignment errors V and H is studied on both a gear rolling tester and a simulation program. Fig. 14(a) shows the measured and simulated results for the tooth contact pattern change with a different alignment error V, and (b) shows that for a different alignment error H. Fig. 15 shows the same study result for gear set B. Those simulations are made taking the alignment shift discussed earlier into account. From Fig. 14(a), it can be seen that the measured and simulated tooth contact patterns in gear set A are in good agreement with respect to size, shape, and position when the alignment error V is changed. From Fig. 14(b), it also can be seen that the measured and simulated results agree with each other when the alignment error H is changed. Fig. 15 shows that the simulated results also agree with that of the measurement for gear set B. The estimation of the amount of alignment shift in Fig. 13 was based on the condition of V = 0 μm, H = 0 μm, but it is shown that the amount of alignment shift is the same for the different alignment errors V and H. Because the amount of alignment shift of the simulated tooth contact pattern from that of the measurement is caused by the shift of the origin point of the gear rolling tester and deviation of the pinion tooth flank position from its mounting surface, it is considered that the amount of alignment shift for the same gear set does not change, even though the alignment errors V and H of the gear rolling tester are changed. The results shown in Figs. 14 and 15 indicate these characteristics. From this, the estimated amount of alignment shift is considered to be correct. Also, Figs. 14(a) and 15(a) show the characteristic when the alignment error V is increased and the tooth contact pattern moves toward the heel and tip directions and when V is decreased and the pattern moves toward the toe and bottom directions. On the other hand, when the alignment error H is increased, it moves toward the tip and slightly toward the toe, and when it is decreased, it moves toward the bottom and slightly toward the heel. These characteristics are well known, and

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the movement of the simulated tooth contact patterns shows the same characteristics. From this, it can be said that the influence of the alignment errors V and H is correctly evaluated by the developed simulation program. 5.3. Comparison of measured and simulated transmission errors In this section, the transmission errors of gear sets A and B are measured and compared with the simulated results. The transmission error is measured using a gear rolling tester, which is equipped with transmission measuring system made by Ono Sokki. In the system, rotary encoder of 36000p/r are attached both pinion and wheel axis and by analyzing phase of both axis, total transmission error of wheel against pinion is measured with the resolution of 0.2 μrad. Then 1st and 2nd order transmission errors are calculated by FFT analyzer. During the measurement of the transmission error, the tooth contact pattern is also checked. The experimental transmission error is measured by changing the alignment errors V and H. Fig. 16 shows the measurement results for the 1st and 2nd order transmission errors of the tooth mesh frequency. The results for when the alignment error H is changed are shown in (a), and the results for when the alignment error V is changed are shown in (b). In Fig. 16, the simulated results for the developed simulation program are also shown. In the same manner, the measured result and simulated result using gear set B are shown in Fig. 17. These simulations are made taking into account the amount of alignment shift, which was estimated earlier. In other words, the experiment and simulation were performed under the same assembly conditions and with the same tooth contact pattern. In Figs. 16 and 17, comparisons of the experiment and simulation results are made, and it was found that the characteristics of the increase and decrease in transmission error according to the variation in alignment errors V and H agree between the experiment and simulation. In the case that tooth contact pattern is moved toward tooth tip or tooth root extremely by the change of alignment, experimental and simulated results have some differences but when tooth contact pattern is in the active tooth area, the experimental and simulated results have almost the same quantitative values. The agreement between experiment and simulation is not only for the 1st order transmission error but also for the 2nd order transmission error. From these comparisons, it is confirmed that the developed simulation program is capable of analyzing the transmission error accurately. Because in this research actual tooth flank form is measured and elastic deformation of tooth flank is analyzed taking the measured tooth flank form into account, the accuracy of the simulation is realized like Figs. 16 and 17. And because no other method has been reported to achieve the same level of accuracy, this method is considered to be effective.

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6. Conclusion For generated hypoid gears, improvements in noise and vibration quality and strength are required. To achieve this, a method of performance analysis that considers the tooth contact condition under load is necessary, especially for generated hypoid gears, where small waviness affects their performance and must be considered in the analysis. In this research, a performance analysis program for generated hypoid gears that is capable of utilizing the measured tooth flank form data was developed. The analysis method for the generated tooth flank form of the generated hypoid gear and the calculation method for its deformation under load were presented, and a tooth contact simulation program was created. The actual tooth contact patterns were measured by experiment and were compared with the simulated results; it was confirmed that they had good agreement with each other. It was also confirmed that the measured and simulated tooth contact patterns agree with each other even with alignment error in the offset and pinion axis directions. The transmission error of the generated hypoid gear was also measured by experiment and was compared with the simulated results, and it was confirmed that they had good agreement with each other for the 1st and 2nd order transmission errors of the tooth mesh frequency. From these results, it was confirmed that the developed performance analysis method for a generated hypoid gear is capable of accurate analysis. Acknowledgments The authors are grateful to Mr. Yosuke Miyata in Kyoto University, for his contribution to this research. References [1] F.L. Litvin, et al., “Methods of synthesis and analysis for hypoid gear-drives of formate and helixform”, part 1, 2, and 3, Trans. ASME J. Mech. Des. 103 (1981) 83–113. [2] F.L. Litvin, et al., Straddle design of spiral bevel and hypoid pinion and gears, Trans. ASME J. Mech. Des. 113 (1991) 422–426. [3] F.L. Litvin, et al., Minimization of deviations of gear real tooth surfaces determined by coordinate measurement, Transactions of the ASME International Transmission and Gearing Conference, 1992, pp. 193–200. [4] C. Gosselin, et al., Simulation and experimental measurement of transmission error of real hypoid gear under load, Trans. ASME J. Mech. Des. 122 (2000) 122. [5] A. Kubo, et al., On simulation method of performance of hypoid and spiral bevel gears (1st report, definition of reference for tooth form accuracy and method of simulation), Trans. JSME Ser. C 62 (599) (1996) 2833–2841. [6] H.J. Stadtfeld, Hand Book of Bevel and Hypoid Gears, Rochester Institute of Technology, 1993.

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[7] H.J. Stadtfeld, et al., The ultimate motion graph, Trans. ASME J. Mech. Des. 122 (2000) 317–322. [8] Q. Fan, et al., Higher-order tooth flank form error correction for face-milled spiral bevel and hypoid gears, Trans. ASME J. Mech. Des. 130 (2008) 072601-1-7. [9] Q. Fan, Enhanced algorithm of contact simulation for hypoid gear drives produced by face-milling and face hobbing processes, Trans. ASME J. Mech. Des. 129 (2007) 31–37. [10] V. Kin, Computerized analysis of gear meshing based on coordinate measurement data, ASME Int. Power Transmission and Gearing Conference, DE-Vol. 43-2, Scottsdale, AZ, 1992, pp. 543–550. [11] Y. Zhang, et al., Computerized analysis of meshing and contact of gear real tooth surfaces, Trans. ASME J. Mech. Des. 116 (1994) 677–682. [12] V. Simon, Influence of tooth errors and misalignments on tooth contact in spiral bevel gears, Mech. Mach. Theory 43 (2008) 1253–1267. [13] M. Kolivand, et al., An ease-off based method for loaded tooth contact analysis of hypoid gears having local and global surface deviations, J. Mech. Des. ASME 132 (2010) 071004. [14] R. Takeda, et al., Development of scanning measurement of tooth flank form of generated face mill hypoid gear pair with reference to the conjugate mating tooth flank form using 2 axes sensor, MAPAN - JSMI 26 (1) (2011) 55–67. [15] S. Kato, Proposal of reference surface of hypoid gear tooth measurement, JSME MD & T Division Conference MPT-100, 1997, pp. 325–330. [16] M. Sugimoto, et al., Tooth fillet profile of hypoid pinion generated with toprem blade cutter, Jsme Mpt'91-Hiroshima, 9d21991. 711–716. [17] Z. Wang, Basic Research on Simulation of Hypoid Gear Running Quality and Performance Analysis, 1998. (Dissertation). [18] Z. Wang, et al., Tooth root stress analysis of hypoid gears (1st report, introduction of predicting method for the tooth root stress), Trans. JSME Ser. C 66 (652) (2000) 4024–4032.