Effect of Transition Curve on Gear Bending Strength

  Abstract. Transition curve is very important to bending strength and life of gear. In  order to improve the bending strength of gear, the shape of three kinds of transition curve (ordinary fillet cutter tip, single circle arc cutter tip and double circle arc cutter tip) was analyzed respectively, the relationship of double circle arc cutter tip parameters was established, the relation model  of sensitive part of gear and stress concentration factor ( J and γ) was determined, the effect of different transition curves of gear on bending strength ( Jmin ) was explored. The results of analysis show that the bending strength of gear cut by double circle arc cutter tip increases by 10% comparing with that of gear cut by ordinary fillet cutter tip. It provides theories foundation for the design of high bending strength gear.

  Introduction

  Improving the bending strength of gear is an important goal of gear design. The selection of transition curve is the key to improving the bending strength of gear. The transition curve of gear is the most susceptible to bending failure [1] because there are physical mutation and stress concentration in the transition curve. Jize Wu, etc. [2] studied transition curves formed by the cutters with ordinary fillet cutter tip and single circle arc cutter tip and single circle arc transition curve. But research reports on transition curve formed by cutter with double circle arc cutter tip have not been found. In this paper, transition curves formed by three cutters with different cutter tip are studied respectively, and distribution of bending strength on the transition curve is analyzed, and the best transition curve is selected. The results are very significant to the design of high bending strength gear.

  Transition Curve of Gear

  The transition curve is cut out by the cutter tip of the cutter. The shape of the transition curve is directly dependent on processing methods and the shape of the cutter tip.

  Reference [3] listed seven shapes of the rack cutter. Here, three of them will be analyzed.

  The shape of the first cutter tip is ordinary fillet. As shown in Fig. 1(a), the arc of the cutter tip is tangent with the line of the profile and the line of the cutter tip. The relationship of its parameters is as in reference [2].

  The shape of the second cutter tip is single circle arc. As shown in Fig. 1(b), the arc center is located on the central line of the tooth. The arc of the cutter tip is tangent with the line of the tooth profile. The relationship of its parameters is as in reference [2].  

Fig. 1 Shape of three kinds of cutter tip

  The shape of the third cutter tip is double circle arc. As shown in Fig. 1(c), large radius arc of the cutter tip is tangent with the line of tooth profile, and it is also tangent with the small radius arc whose center is located on the central line of the tooth. The relationship of its parameters is as follows:

  Where

a1 ——distance from the center  Cr1 of the large radius arc to midline;

b1 ——distance from the center  Cr1 of the large radius arc to central line of cogging;

—— radius of the large arc;

——distance from the center  Cr2 of the small radius arc to midline;

b2 ——distance from the center  Cr2  of the small radius arc to central line of cogging;

rr2  —— radius of the small arc;

a2 ——pressure angle on tangency point between large circle arc and small circle arc. 

  Other parameters is as in reference [2].

  There has equation of transition curve in reference [2]. Put the parameters of three kinds of

cutters into the equation respectively and then coordinates of different points on transition curve will be obtained.

  Analysis of Tooth Root Stress

  The calculation of tooth root bending stress based on the broken—line section method [2] is as follows:

  Where

F ——the force on unit tooth width;

J ——geometry factor.

  Geometric factor reflects the differential properties of points on transition curve and the relationship  between  other  parameters  and  stress  [2].   J is  point  function  on  curve  if   other

parameters of gear are  fixed. J = J (g , rr ) .   J   about different points on transition curve will been obtained using analytic operation to the equation of transition curve.

  It can be seen know from above：The smaller   J ，the bigger of tooth root is. The point that has minimum J is the point that has maximum bending stress of points on the transition curve. The research on effects that different transition curve on J is equal to the research on effects that different transition curve on bending strength [4].

  Evaluation About Bending Strength of Tooth Root

  Bending Strength of Three Kinds of Transition Curve. The parameters of gears that will be processed are hypothesized as: Z1=30, Z2=30, m=1,a = 20° , f=1。The effect that parameters of

every kind of cutter tip on J should be analyzed. A series of J about transition curve formed by each cutter tip can be obtained.

  The first cutter: It will be standard rack cutter if c=0.25 , rr  = 0.38 . We can obtain Jmin about the transition curve:

Jmin =0.2857 , at this time, g= 23.3°。

  The second cutter: rr  = 0.4485 , c =0.2951 can be deduced by using of original parameters of gear. We can obtain Jmin about the transition curve: Jmin=0.2913 , at this time, g= 22°.

  The third cutter: Its shape is slightly more complex. It is composite curve of large circular arc and small circular arc. We should analyze the bending strength of transition curve formed by large

circular arc in the range a  to a ² , and analyze the bending strength of transition curve formed by small circular arc in the range a ²to  90²。

  As shown in Fig. 1(c), rr1 and rr2 are two independent variables. The cutter tip curve will be single circle arc if rr1  =0.4485.Therefore, there must be rr2  >0.4485.

  The tangent point between large circle arc and small circle arc will coincide with the tangent point between large circle arc and line of the profile. The cutter tip curve is still the single circle arc at this time. Therefore, there must be 0< rr2  <0.4485。

  The value of a2 and  c  will be determined if the value of rr1 and rr2 are determined. Then we can calculate the value of  J and Jmin。The changing trend of Jmin due to rr1  and rr2 is shown in Fig. 2.

Fig. 2 effects of and rr2 on J                                    Fig. 3 effects of  g on J

  Comparison of Tooth Root Bending Strength of Three Kinds of Transition Curve. The change curve that   J   of three kinds of transition curve with  g  are shown in Fig. 3.

  From Fig. 3, these can be found：

  (1)The bending strength of gear cut by cutter with single circle arc cutter tip is higher than that of gear cut by cutter with ordinary fillet cutter tip.

  It is noted that: The fillet radius of cutter with ordinary fillet cutter tip increase as top clearance c increases, and their centers are closer to centra1 line of tooth profile [5]．If the fillet center of cutter tip moved to centra1 line of tooth profile, the cutter tip is a complete arc. At this time，it is single circle arc cutter tip. So the cutter tip is ordinary fillet cutter tip is only c<0.2951.In short, the bending strength of gear cut by cutter with single circle arc cutter tip is higher than the bending strength of gear cut by cutter with ordinary fillet cutter tip.

  (2)The value of Jmin about transition curve formed by double circle arc cutter tip( rr1 =1.1 ,rr2 =0.16 )is larger than the value of Jmin about the first two transition curve. The tooth root bending strength is about 10% higher than that of the first transition cursve, and it is about 7.9% higher than the tooth root bending strength of the second transition curve.

  (3)On the third transition curve, there is a point that is corresponding to the tangent point between large circle arc and small circle arc. The curvature radius changes discontinuously on this point.

  There is an abrupt stress on this point. Actual situation needs further experimental verification.

  Different shapes of cutter tip will lead to different transition curve and different tooth root bending strength. It will bring different effects to the life and reliability of gear [1]. In the theory, the tooth root bending strength of gear cut by the cutter with double circle arc cutter tip is about 10% higher than the tooth root bending strength of gear cut by the cutter with ordinary fillet cutter tip. However, the shape of the transition curve is also dependent on processing methods in production practice.

  The transition curve will have saw tooth due to the discontinuous cutting in manufacturing process. It will reduce the tooth strength. These cases need to be deep studied.