AN INVESTIGATION OF MISALIGNMENT TO CONTACT PATH OF FACE-GEAR DRIVES

He Guo Qi1,2 Yan Hong Zhi1 Hu Wei1 Shu Tao Liang2

1 .School of Mechanical and Electrical Engineering Central South University HuNan ChangSha 410000 2. School of Mechanical Engineering HuNan University of Technology HuNan ZhuZhou 412000

tjhgq@163.com

Abstract—Described the methods and theories of the formation of point contacts of Face gear meshing process; Establish the point contact meshing coordinate system with Misalignment; Determine the meshing region with deformation when under load; Analysis the contact point position with different installation errors; Investigate the method to adjust the contact point to the Optimum position by axial displacement, which can provides a guide To improve the quality of face-gear drive.

Keywords- Face-gear drive; Point contact; Fixing Error; Contact path

I. INTRODUCTION

Face gear drives are a particular kind of cross axis transmission made of a pinion with involutes teeth meshing with a face gear, the structure of a tooth of a face gear generated by a shaper which is Similar to the pinion .For many years, People have done a lot of work to Improve the quality ,reliability and reduce the weight of air gear transmission, due to simple structure, transmission coincidence degree, power diversion effects, vibration, and many other advantages ,face gear drive technologystands out from many species transmission technology. According to foreign reports on the literature, face gear transmission used in the power split of Helicopter drive system decreased the weight by 40% than conventional devices, and with low noise and little vibration. With the further research of face gear transmission, Face gear rotation velocity and the load capacity increase significantly, thus further broadens the scope of its application . Due to the unique convenience of face-gear in stream-convergence transmission, face-gear transmission showed a potential advantage in aviation field, especially in the application of new combat helicopters .

II. THEORY OF POINT CONTACT FOR FACE GEAR TRANSMISSION

A. Theory of point contact formation for face gear drive According to the point contact theory, when processing

of face gear, imagine the face-gear Σ2 and the pinionΣ1 are all processing by the shaper ΣS . Installation

of the tool gear cutting tool position to imitate an imaginary gear and spur gear in meshing in Figure 1, Center distance b

is determined by and the shaper and pinion tooth number difference Δ , It’s value is:

� � 211 NNmrrb sopps ���� (1) The shaper's tooth surface , pinion tooth surface and

the Face gear tooth surface is mutual engagement, Formed after three meshing in to two intersect lines, The intersection is the pinion and face-gear meshing point of the instantaneous, These points at different time in the tooth surface formed the contact track. These points are converted to a fixed coordinate system is the curve formed by the meshing.

Figure 1. Imaginary cylindrical gears and shaper within the meshing

��

��

Figure 2. Face gear meshing between the instantaneous axis of the diagram

Face gear, shaper and pinion axis are in the same plane, figure 2 shows the plane between conic sections, shaper and pinion pitch line Is1 is parallel to the axis line, face-gear and pinion pitch line is I21,face-gear and shaper pitch line is

2011 Second International Conference on Digital Manufacturing & Automation

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DOI 10.1109/ICDMA.2011.53

189

Is2,through the point contact shows that the formation of three section lines intersect at one point p. Subscript S2, S1and 21, respectively representative the meshing of ΣS, Σ2 ,ΣS

andΣ1, Σ1 andΣ1. �s s is the angle of rotation of the shaper angle between the axis and IS2.It is determined by the following equation

� � ����� sincossincoscot 44

���

�����

sss N

Nm 2

Therefore, the point contact can achieve by changing the number of teeth of cutting tools.

III. THE CONTACT TRACK WITH MISALIGNMENT OF FACE-GEAR DRIVE

A. The coordinate system of point contact with misalignment

The basic parameters of face-gear are as follows: Cutter gear module, Pressure angle of pitch circle�0, tooth number of the shaper Ns, tooth number of the pinion N1,tooth number of the face-gear N2, transmission Shaft angle�. Suppose the coordinate system fixed on the pinion and face-gear are S1 x1 y1 z1 and S4 x4 y4 z4 , where z1and z4 are consistent with the axis of pinion and face-gear. as is shown in Figure 3,coordinate system S2

x2 y2 z2 is the initial position of S1 coincides with the fixed coordinate system, O2=O1 z2 =z1,as it Shown in Figure 4,coordinate system S3 x3 y3 z3 is the initial position of S4 coincides with the fixed coordinate system, O3= O4 z3 =z4.

In order to describe the misalignment error of face gear transmission , Introduce auxiliary coordinate system Sq and Sd , �E indicates the installation of the offset distance, as is Shown in figure 5 ,By adjusting the amount of �E, B and Bcot� to determine the origin of the position Oq relative to the origin position O2.as is Shown in Figure 6 ,radial error △ q is expressed by the relative position of S3 to Sd, Axial angle error �f is expressed by the relative position of Sd to Sq,where �f =180º-�+△ �.

Figure 3. The fixed and rotation cylindrical gear coordinate system

Figure 4. The fixed and rotation Face gear coordinate system

Figure 5. The conversion coordinate system with offset error

Figure 6. The conversion coordinate system with Axial angle error and Radial error

B. Face gear meshing track solution of equation with misalignment

1) Determination of pinion tooth surface �1The shaper is involutes cylindrical gear, the face-gear

tooth surface is generated by conjugate method. If omitted the shear stress generated gear surface elastic deformation in processing, Face gear tooth surface is the envelope of the shaper. so cylindrical gear tooth surface equation and the equation of the tool gear tooth surface is basically the same, the coordinate parameters Os xs ys rbs �s0 �s

changes in to O x y rb �0 � The geometric meaning of each parameter the same. Tooth surface equations of the right side of the tooth profile is

190

� �� � � �� �� � � �� �

���

�

�

���

�

���

��

��������������

�� ,sin,cos,cos,sin

, b

b

rr

r 3

In which, � �10 ,��� � � �21,��� �

� �00 ���� ��� k � �22 1 ��� bp rr prk �tan�� The geometric meaning of each parameter are the same as the shaper.

The tooth surface normal vector, T

bkrn ��

������

�

� sincos 4

In which, 221 brk��

2) Face gear meshing track equation with misalignment Face-gear equation In the coordinate system S2 is:

� �� � � � � � � �!"

!#$

%�

�

ss

sssssss

sss

urMur

uf

�&&�

&�

,,,

0,,

2

)2(2

5

It normal line can transform from the normal line of shaper

� � � � � � � �s

ssssss nMn �&&� %� 2

)2(2 , 6

Transition the face gear tooth surface equations and the normal line from coordinate system S2 to the Sd

� � � � � �� � � � � �!"

!#$

%�

%�

sssd

dsssd

d

nMn

rMr

&�&

&�&

,

,2

22

)(2

222

)(2 7

Pinion tooth surface and its normal line in coordinate system Sd

� � � �� �� � � �� �

���

�

�

���

�

�

�����������

�u

rr

r b

bd

��&���&��&���&

0/

10/

1

0/

10/

1)(

1 sincoscossin

8

� �� �

���

�

�

���

�

�

������

�0

sincos

0/

1

0/

1)(

1 ��&��&

dn 9

In which, � is parameters of the involutes, the range is � �bbabbf rrrrrr 2222 , �� , fr is addendum circle radius, rag

is tooth vertex circle radius, u is the parameters of pinion in the direction of tooth length, �0 is determined by abduction angle of the basic circle. Its value is

01

0 2�'� inv

N�� 10

Vector equations of mesh track can be obtained as

� � � �� � � �!"

!#$

�

�

/2

)(2

/1

)(1

/2

)(2

/1

)(1

,,,

,,,,

&&�&�

&&�&�

ss

ddss

dd

nn

rur 11

/2& is the input parameter, we obtained other parameters and

get meshing point of different locations, it’s the mesh track with misalignment.

Through the above computation, we can obtained a series of mesh points on the face-gear, when these point is

between the addendum and the transition curve, we completely determine the mesh track with misalignment.

IV. THE EFFECTS OF MISALIGNMENT TO THE MESH TRACK ON TOOTH SURFACE

The existence of error makes the face gear meshing process in the contact area deviated from the ideal position of the contact line, concentrated load is loaded on the top and edges of face-gear. Face-gear misalignment can be divided into offset error, radial error and axial angle error. Under normal circumstances, these three errors are independent misalignment. Therefore, this paper considers only one misalignment.

The parameters of face-gear are: Modulus m=2mm ,pinion teeth number N=27,face-gear teeth number N=150, Tooth width B=26mm,Axis angle �=90º,Shaper tooth number N=30.

A. Effects of offset error to mesh track As Shown in figure 7,the offset error is expressed by

�E, assume the offset error is ±0.2mm ,the other two error is zero, trajectory obtained by simulation as shown in Figure 8,meshing track are expressed by 1,2,3.the corresponding offset errors are dE=-0.2mm dE=0mmdE=0.2mm,radial error and axial angle error are zero.

0��q 0��� 2.0���E 0��E 2.0��E

Figure 7. Face-gear mesh track with offset error

B. Effects of radial error to mesh track As shown in figure8, the radial error is expressed by

�q. Assume the radial error is (0.1mm ,the other two error is zero, trajectory obtained by simulation as shown in figure 9,meshing track are expressed by 1,2,3.the corresponding radial errors are dq=-0.2mm dq=0mm dq=0.2mm,offset error and axial angle error are zero.

2.0���q 0��q 2.0��q 0��� 0��E

Figure 8. Face-gear mesh track with radial error

191

C. Effects of Axial angle error to mesh trackAs Shown in figure 9,the axial angle error is expressed

by ��=�+�f-180º.Assume the axial angle error is (0.05º ,the other two error is zero, trajectory obtained by simulation as shown in figure 10,Meshing track are expressed by 1,2,3.the corresponding axial angle errors are d�=-0.02 º d�=0ºd�=0.02 º,offset error and radial error are zero.

0��q 02.0���� 0��� 02.0��� 0��E

Figure 9. Face-gear mesh track with Axial angle error

The results can be obtained from the above analysis, when the offset error �E enlarged as positive, mesh track move in the direction of undercutting to pointing, otherwise in the direction of pointing to undercutting. When the radial error �q enlarged as positive, mesh track move in the direction of tooth root, otherwise to the addendum. When the axial angle error �� enlarged as positive, mesh track move in the direction of tooth root, otherwise to the addendum. The mesh track is most sensitive to axial angle error ��, followed by radial error �q, offset error �E is The least sensitive.

V. METHOD OF MESH TRACK ADJUSTMENT

As the processing, installation constraints, deformation and other factors, there is always a certain amount of installation error in face-gear drive, these installation errors are bound to affect the mesh track. In order to get rational mesh track, we need to control the installation error. In the actual installation, the axial angle error Δγ and offset error ΔE are difficult to measure and control, but we can measure and control the radial error Δq, by adjustment the thickness of gasket, and thus improve the quality of mesh track and transmission.

Figure10shows the mesh track by adjustment the axial displacement of face gear under the coupling installation error. Figure 10(a) shows the mesh track offset to the big end by 8.52mm under the coupling installation error, when Δq=-0.16mm,by adjust the radial error, the mesh trace back to the middle of tooth width. Figure 10(b) shows the mesh track offset to the left by 7.63mm under the coupling installation error, when Δq=0.16mm,by adjust the radial error, the mesh trace back to the middle of tooth width. After adjustment, the mesh track is coincides with the ideal track, so it can improve the quality of face-gear drive.

a

b

Figure 10. Face-gear mesh tracks by Adjustment

From the above it can concluded that we can control the mesh track by adjust the radial error, the method of adjustment are as follows:1.Measure the location of meshing track under initial installation;2.Calculated parameters of the three installation error;3.Calculate The adjustment amount Δq;4.Adjust the radial error in actual situation.

VI. CONCLUSION

This paper applied Principle of differential geometry and meshing, deduced the meshing equation of face-gear drive. Analysis of deformation of tooth surface points between its curvature and the installation error effect to the meshing area. Give a certain elastic deformation of tooth surface changes under the scope of the installation error. Meanwhile, in order to get a reasonable mesh track, we can control the mesh track by adjust the installation error. This will give certain guide to improve the quality of face-gear drive.

REFERENCES

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[2] F. L. Litvin. Face Gear Drive With Helical Involutes Pinion: Geometry, Generation by a Shaper and a Worm, Avoidance of Singularities and Stress Analysis[R] . NASA/CR-2005-213-443

[3] Litvin F L, Fuentes A, Howkins M. Design, generation and TCA ofnew type of asymmetric face-gear drive with modified geometry[J].Compute. Methods Appl. Mech. Engrg.,2001,190:5837-5865.

[4] Litvin F L, Fuentes A, Zanzi C, etal. Face-gear drive with spur involutes pinion: Geometry, generation by a worm, stress analysis [J]. Compute. Methods Appl. Mech.Engrg., 2002, 191: 2785- 2813.

[5] Guing and M, de Vaujany J P, Jacquin C Y. Quasi-static analysis of a face gear under torque [J]. Compute. Methods Appl.Mech.Engrg.,2005,194:4301-4318.

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