KINEMATIC ANALYSIS AND DESIGN OF BALL … · The design program using Visual C++ and Auto Lisp to determine the design parameters and to generate the drawing sheet. Keywords: ball

KINEMATIC ANALYSIS AND DESIGN OF BALL REDUCER WITH WAVED GROOVE W. D. KIM Korea Institute of Machinery and Materials, 171, Jang, Yoosung, Daejon, KOREA; e-mail: [email protected] B. C. CHOI Liwoo Co., Limited, 973-6, Yangduk, Hoiwon, Masan, KOREA, e-mail: [email protected] SUMMARY A new conceptual ball reducer with waved groove was proposed and investigated. The kinematic analysis of the reducer carried out. The main components of the reducer were an eccentric input shaft and bearings, followers with balls, a flange, and a outer ring with waved groove. The force of each component was estimated, and the main design parameters were investigated. The design program using Visual C++ and Auto Lisp to determine the design parameters and to generate the drawing sheet.

Keywords: ball reducer, kinematic analysis, waved groove, design parameter

1 INTRODUCTION Various transmitting units reducing the rotational speed are widely used for engineering purpose. The trans-mitting unit mainly uses worm gears or epicyclic trains, and examples of the transmitting unit are disclosed in Japanese Patent No. Sho 60-91043 [1]. However, since the aforementioned transmitting unit uses relative move-ment of meshed gears, larger power loss, much noise and vibration is generated. Beyond that, the transmitting unit is mainly used for heavy load. Furthermore, the structure of the transmitting unit is so complicated that it is hard to install the unit, and an output shaft is reversibly rotated when an input shaft is rotated.

In order to overcome these problems, U.S. Patent No. 4736654 that uses bearings allowed [2]. According to this, an eccentric sleeve and bearings are rotated to push transmitting rods, when an input shaft is rotated. In addition, rolling members that are linearly moved in a radial direction on the outer sides moved along the waves of an outer ring. The rotational speed of an output shaft mounted to the output flange is reduced.

However, the structure of the transmitting unit has a disadvantage in that it is complicated and many parts are required to assemble the transmitting unit. As the clearance of each parts are cumulated in assembling

process, it causes of the large noise, vibration, and the shorten life.

To overcome these problems, a new conceptual ball reducer with waved grooves is presented. The kinematic analysis is carried out and the main design parameters are determined. 2 OPERATING PRINCIPLE Figures 1 and 2 show the operating principle, and components of this reducer. The main components of the reducer were an eccentric input shaft and bearings, followers with balls, a flange, and a outer ring with waved grooves. The principle of reducing is as followings.

1) The outer raceway of inner bearing assembled in the eccentric shaft takes a role as a cam.

2) Followers moved along the holes of the flange according to the rotation of the eccentric shaft.

3) The balls installed in tips of followers were rolled with the waved grooves of outer ring.

4) When the eccentric shaft was rotated as one revolution and the outer ring was fixed, the flange was rotated as the amount of one over number of followers.

1/4 rev. 2/4 rev. 3/4 rev. 4/4 rev.

Figure 1: Reducing principle

Figure 2: Components of ball reducer

3 KINEMATIC ANALYSIS The wave curve is shown figure 3. The co-ordinate of wave is determined by the geometric constraints.

The trace of the ball center ( )(θTc ) is the function of the eccentricity (e), the radius of inner bearing (R), and the length of follower (h). The co-ordinate of the trace of the ball groove in outer ring is (X, Y). The radius of ball is r.

})cos({ hRNweTc ++⋅⋅= θ (1) θθ sin)( ⋅= Tcx (2) θθ cos)( ⋅= Tcy (3)

kdydx

xXyY 1

)()( −=−=

−− (4)

222 )()( ryYxX =−+− (5) where,

θθθθθθ

sin)sin(coscos)sin(sin⋅⋅⋅⋅−⋅⋅⋅⋅⋅+⋅−=

NwNweTcNwNweTck (6)

Figure 3: Geometry of wave

The mechanism of the follower and input bearing is similar to cam-flat faced follower [3]. Fig. 4 shows the force diagram of each element. When the friction between the balls and groove is neglect, the next equations are induced.

αsin1 iPbaR = (7)

αsin2 iPb

baR += (8)

}sin)2({cos αµαb

baPF ii+−= (9)

where, iP is the contact load between the ball and

groove, 1R , 2R are the reaction forces of follower, iF is the reaction force of inner bearing, µ is the coefficient of friction between the follower and flange, α is the contact angle, and c is the radius of follower.

Fig. 4 : Force diagram

4 LOAD ANALYSIS When the capacity of the reducer is H (kW), the reduction ratio is Nr and the rotational speed is n (rpm), the input and output torque ( iT , oT ) are calculated as :

nHTi

⋅= 974000 (kg-mm) (10)

NrTT io ⋅= (kg-mm) (11) If the reaction force of input bearing at each location ( iF ) can be assumed as ii FF θsin= when F is the maximum reaction force, the relationship between the contact load and the output torque can be obtained by the next equation.

∑∑==

==Nb

iiiiii

Nb

iio RFPRT

11))(sin(tan)sin( θαα (12)

Where, Nb is the number of balls, and Ri is the distance from the center of reducer to the contact point. It is founded out that F and iF can be calculated from eq. (12). Therefore, the output torque is known, the remained load terms are determined.

5 CONTACT ANALYSIS The contact phenomena are generated between each element of ball reducer. Particularly, the contact between ball and groove is more severe than other elements, because this contact load transmits the power of reducer directly. It is predicted that the fatigue failure is originated in this contact area. Therefore, the maximum contact stress and the contact area are investigated to overcome the apprehension of the contact fatigue failure.

In the equation proposed by B. J. Hamrock [4], the maximum contact stress ( maxσ ) and the diameters of contact area ( xD , yD ) are calculated as :

3/12

)'

6(2E

PRD x πβχ= (13)

3/1)'

6(2EPRD x π

β= (14)

yx DDP

πσ 6

max = (15)

Where, R is the equivalent radius, E' is the equivalent Young’s modulus, and βχ, are the contact constants. The contact load is affected by the eccentricity of input shaft. Fig. 5 shows the effect of the eccentricity on the maximum contact stress. Fig. 6 shows the effect of the eccentricity on the elliptic contact diameter.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.190

100

110

120

130

140

150

10:1 (No. of ball:10)

20:1 (No, of ball:10)

30:1 (No. of ball:15)

Max.

Con

tact

Stre

ss (

kg/m

m2)

Eccentricity (mm) Fig. 5 : Effect of the eccentricity on the maximum

contact stress

0.20.30.40.50.60.70.80.91.01.1

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

10:1 (No. of ball:10)

20:1 (No. of ball:10)

30:1 (No. of ball:15)

Ell

ipti

c Conta

ct D

iame

ter

(mm)

Eccentricity (mm) Fig. 6 : Effect of the eccentricity on the elliptic contact

diameter 6 DESIGN PROGRAM The design program using Visual C++ and Auto Lisp to determine the design parameters and to generate the drawing sheet. Fig. 7 shows the flowchart of design program. The main input parameters are the capacity of reducer, the rotational speed, the reduction ratio, the number of wave, the number of balls, the size of ball, the eccentricity of input shaft, the PCD of balls, and the length and diameter of followers. The output presents the co-ordinate of waved groove, the drawing sheet, the predicted life, and the recommended bearing.

Fig. 8 shows the main menu of design program, and Fig. 9 shows the example of drawing sheet using Auto Lisp.

.

Input DataCapacity, Ball Number & Dia

Pitch Circle Dia,Eccentricity,

Inner Bearing TypeShaft Bearing Type

Flange Bearing Type

Input DataCapacity, Ball Number & Dia

Pitch Circle Dia,Eccentricity,

Inner Bearing TypeShaft Bearing Type

Flange Bearing Type

Design ParametersCalculation Program

Design ParametersCalculation Program

BearingData

ErrorCheck

DrawingData

DynamicAnalysis

Data

Wave Coord.for CNC

Manufacture

Tool Designfor CNC

Manufacture

ElementManufacture

ElementManufacture

DrawingProgram

by Auto Lisp

DrawingProgram

by Auto Lisp

DynamicAnalysis(DADS)

DynamicAnalysis(DADS)

Tool Dia. ForCNC

Manufacture

Tool Dia. ForCNC

Manufacture

HERBDrive

HERBDrive

Fig. 7 : Flowchart of design program

Fig. 8 : Main menu of design program

Fig. 9 : Example of drawing sheet

Fig. 13 : Test equipment of ball reducer

7 PROTOTYPE OF BALL REDUCER The prototype ball reducer was manufactured, assembled and experimented to realize the ball reducer with waved groove. Fig. 10 shows the outer ring with waved groove. Fig. 11 shows the inner part assembly. Fig. 12 shows the completed ball reducer. The fatigue life test was carried out to guarantee the durability. Fig. 13 shows the test equipment.

Fig. 10 : Outer ring Fig. 11 : Inner assembly

Fig. 12 : Completed ball reducer

8 CONCLUSIONS A new conceptual ball reducer with waved groove was proposed and investigated. The kinematic analysis of the reducer carried out. The force of each component was estimated, and the main design parameters were investigated. The design program using Visual C++ and Auto Lisp to determine the design parameters and to generate the drawing sheet. 9 REFERENCES [1] Reducer, Japanese Patent, Sho 60-91043. 1986. [2] Transmitting Unit, US Patent, 4736654, 1988. [3] H. A. Rothbart, CAMS Design, Dynamics and Accuracy, 1956. [4] Rolling Contact Fatigue Testing of Bearing Steels, ASTM STP 771, p.369, 1981. 10 ACKNOWLEDGEMENT This work was supported by grant No. 2000-1-30400-005-3 from the Basic Research Program of the Korea Science & engineering Foundation.