1、DIN 3971 x, = Tooth thickness alteration factor Other quantities as in Fig. 7 Figure 8. Standard basic rack tooth profile with positive tooth thickness alteration DIN3971 Page9 Y 9 II Datum. line ha=ho DIN 867 Datum line Figure 11. Standard basic rack tooth profile with negative addendum modificatio
2、n Page 10 DIN 3971 N o te : The addendum modification is stated for the tip distance R of the standard basic rack tooth profile. Depending on the manufacturing method it either in- creases steadily towards the centre of the crown gear or is constant at all tip distances. In the second case it is the
3、refore simpler to speak of the addendum modification of the complete basic crown gear versus the reference cone of the bevel gear. Because the reference cones of the gears have to coincide with the pitch cones of the gear pair (see Section 3.4) a bevel gear with positive addendum modification is alw
4、ays mated with a bevel gear having equally large negative addendum modification (gear train at reference centre distance): xhl + rh2 = 0 (19) 3.5 Parameters in the crown gear reference plane 3.5.1 Crown gear reference circle, crown gear radius RP The crown gear reference circle is determined by the
5、number of crown gear teeth zp and the module mpt exist- ing on its circumference. Its radius (crown gear radius) RP is given by Straight teeth 3.5.2 Crown gear angular pitch r p The crown gear angular pitch T is the centre angle at the crown gear centre which is enclosed by two crown gear radii runn
6、ing to adjacent right-hand or left-hand flanks, see Fig. 12. It is given by 3.5.3 Tooth thickness half angle #p (on a straight bevel gear) The tooth thickness half angle pp of a straight bevel gear is the centre angle at the crown gear centre which is en- closed by the tooth centre lineand an adjace
7、nt right-hand or left-hand flank of the standard basic rack tooth profile, see Fig. 12. It is given by SP *p=2.Rp Spiral teeth (left-hand) (22) Helical teeth (left-hand) Spiral teeth (right-hand) Figure 12. Straight, helical and spiral teeth on the crown gear DIN 3971 Page 11 3.5.4 Tooth space half
8、angle lp (on a straight bevel gear) The tooth space half angle Qp of a straight bevel gear is the centre angle at the crown gear centre which is en- closed by the tooth space centre line and an adjacent right-hand or left-hand flank of the standard basic rack tooth profile, see Fig. 12. It is given
9、by 23) 3.5.5 Datum tooth traces, reference tooth trace5 The datum tooth traces (reference tooth traces) are the linesof intersection of the crown gear tooth flanks (bevel gear tooth flanks) with the crown gear reference plane (with the reference cone). Depending on the form and disposition of these
10、lines, see Fig. 12, the following are distinguished: 3.5.5.1 Straight teeth, With straight teeth the datum tooth traces are straight lines passing through the crown gear centre. With straight bevel gears the prolongations of the (straight) reference tooth traces pass through the reference cone apex.
11、 3.5.5.2 Helical teeth, With helical teeth the datum tooth traces are straight lines touching a circle concentric with the crown gear axis and lying within the tooth system. With helical bevel gears the reference tooth traces are helices on the reference cone envelope surfaces. 3.5.5.3 S p i r a I t
12、e e t h, spiral bevel gears With spiral teeth the datum tooth traces are curves having various forms in the crown gear reference plane, depending on the manufacturing method used.Typical forms include the circle, involute, cycloids and sinoids. With spiral bevel gears the disposition of the referenc
13、e tooth traces is derivable from the datum tooth traces. The parameters for the datum tooth traces and reference tooth traces result from the manufacturing method. straight bevel gears helical bevel gears 3.5.6 Helix angle (spiral angle)/?, The helix angle/? is the acute angle formed at a tip dist-
14、ance R between a tangent to the datum tooth trace and the line through the tangent contact point to the crown gear centre, see Fig. 12. In the case of spiral bevel gears the helix angle is also termed the spiral angle. On a bevel gear the helix angle/? is the acute angle between a tangent to the ref
15、erence tooth trace and the reference cone envelope line through the tangent contact point. On straight bevel gears the helix angle /?equal to zero. The helix angles of a bevel gear may alter with the tip distance R. In thiscase the characteristic helix angles and the disposition of the tooth traces
16、shall be stated, for example the helix anglePP on the crown gear reference circle, or the helix angle/?, at the mean cone distance, or the helix angle/?, at the outer cone distance. right-hand and left-hand teeth A tooth system is right-hand (left-hand) if, on viewing the upright tooth from the apex
17、 of the cone, the tangent to the tooth trace is disposed to the right (left) at the refer- ence point observed. 35.7 Facewidth b The facewidth b is the portion of a reference cone envelope line lying between the outer and inner end faces of the teeth, see Figs 1 to 3 and Fig. 12. 3.6 Tip surface and
18、 root surface, tip cone and root cone The tip surface and root surface are generally surfaces which are conical with regard to the crown gear axis or cone axis; they are designated as the tip cone and root cone. The envelope surfaces on the basic crown gear determine the variation of the tooth depth
19、 along the facewidth. Since the tooth depths of the basic crown gear transfer directly to the bevel gear, their parameters coincide with those of the bevel gear in Section 4.3.1. 3.6.1 Tip angle , addendum angle 8, The angle enclosed by the gear axis and a tip cone envelope line is the tip angle 6,
20、see Fig. 1. The addendum angle 8, is the angle in an axial section between the tip cone envelope line and the reference cone envelope line (the crown gear reference plane in the case of a crown gear), see Figs 1 and 2. 3.6.2 Root angle 6f. dedendum angle Of The angle enclosed by the gear axis and a
21、root cone enve- lope line is the root angle df, see Fig. 1. The dedendum angle of is the angle in an axial section between the root cone envelope and the reference cone envelope line (the crown gear reference plane in the case of a crown gear), see Figs 1 and 2. 3.6.3 Bevel gears with tapering tooth
22、 depth Simple geometrical relationships result when the tip cone apex and the root cone apex coincide with the reference cone apex in the crown gear reference plane. In this case 8,=6+0, 6,= 6- Of, and the tooth depths are proportional to the tip distance R, see Fig. 1. In this case the bottom clear
23、ance is not constant. To obtain a constant bottom clearance over the facewidth it is necessary for the tipcone apex to lie within the refer- ence cone when the root cone apex and reference cone apex coincide. This usually leads to the following being adopted ha, = 6, +Of2 (.e. Sal = Sf2) (26) Sa, =6
24、, + Sfl (.e. Oa =Ofl), i 27) see Fig. 2. N o te : This design allows the use of larger tool radi- using without the risk of meshing interference at the inner end of the tooth. The versions according to equations (24) to (27) are to be regarded as limiting cases of the geometrical design between whic
25、h other versions are feasible, e.g. “inclined root cone line“ in which the root cone apex does not coincide with the reference cone apex. Page 12 DIN 3971 For bevel gears with tapering tooth depth it is usual to adopt as the standard basic rack tooth profile only the cylindrical section through the
26、basic crown gear, and hence the transverse module as the datum length. 3.6.4 Bevel gears with constant tooth depth Simple geometrical relationships also exist when the tip angle and root angle of a gear are equal. In this case the tooth depth is constant over the entire facewidth; this means that it
27、 is the same at each tip distance. The tip angle and root angle may be the same as the refer- ence cone angle, see Fig. 3. In this case the tip surface and root surface of the basic crown gear are circular surfaces parallel to the crown gear reference plane. In the case of bevel gears with constant
28、tooth depth the normal section through a tooth at the mean diameter of the basic crown gear is usually taken as the standard basic rack tooth profile, and hence the normal module as the datum length. 4 Additional definitions All definitions in this Section.relate to the bevel gear devoid of deviatio
29、ns. The equations thus apply to the nominal dimensions or the limiting dimensions of the toothsystem. and parameters on a bevel gear 4.1 Parameters far the tooth profile 4.1.1 Pressure anglea, The pressure anglea, at any flank point Y is formed by two tangentserected in this point to the spherical s
30、urface surrounding the apex, one of these tangents intersecting the gear axis whilst the other is tangent to the tooth flank. With helical and spiral bevel gears a distinction has to be made between the pressure angleafi on the spherical surface of radiusR and the pressure anglea, in a nor- mal sect
31、ion. On a straight bevel gear ayt and a, coincide. 4.1.2 Transverse pressure angle a The transverse pressure angle a is that pressure angle whose vertex lies on the reference cone. Assuming that the standard basic rack tooth profile is given in the normal section then the transverse pressure anglea,
32、 is equal to the pressure angleap. If the standard basic rack tooth profile is given in the cylindrical section then the transverse pressure angle a, is equal to the pres- sure angle ap. 4.2 Parameters for the flank spacing 4.2.1 Transverse tooth thickness s, The transverse tooth thicknessst is the
33、length of the refer- ence circle arc at the tip distance R between the two flanks of the tooth. The nominal dimension of the tooth thick- ness derives from the tooth thickness on the standard basic rack tooth profile and the addendum modification. If the standard basic rack tooth profile is given in
34、 the cylindrical section then normal tooth thickness s, mp *I 2 St = - + 2 - mp - (x, + xh * tan ap). (28) If the standard basic rack tooth profile is given in the normal section then the transverse tooth thickness st is calculated from the normal tooth thickness s, as follows Sn st =cosp ( 29) wher
35、e mp *I 2 ,=-+2-m*(x+.tanap). (30) For the tip distances according to Sections 2.4.1 to 2.4.3 it is necessary to distinguish the following: the tooth thickness se on the back cone, the tooth thickness si on the inner cone distance and the tooth thickness s, on the mean cone distance. The tooth thick
36、ness on cones other than the reference cone can only be calculated in conjunction with the manufacturing method concerned, or approximately by way of the equivalent cylindrical gear teeth (see Section 4.4.3). With the tooth thickness allowances A, (upper allowance A, lower allowance Ad) referred to
37、the reference circle the tooth thickness limits are found as (31) N o te : It should be noted in this connection that the allowancesgenerally have negative signs so that the tooth thickness limiting sizes are smaller than the nominal di- mension. 4.2.2 Spacewidth e, The spacewidth et is the length o
38、f the reference circle arc at the tip distance R within a tooth space. (32) Corresponding to Section 4.2.1, the following are to be distinguished: the spacewidth e, on the back cone, the spacewidth ei on the inner cone distance and the space- width e, on the mean cone distance. With the tooth thickn
39、ess allowance A, (upper allowance A, lower allowance Ag) referred to the reference circle the spacewidth limits are found as (331 N o te : It should be noted in this connection that the allowancesgenerally have negative signs so that the space- width limiting dimensions are larger than the nominal d
40、imension. 4.2.3 Normal chordal tooth thickness The normal chordal tooth thickness is the chord of the reference circle arc at the tip distance R between the two flanks of a tooth. It is found as e, = Pt - St %(Ci) = et - A, - St st = d * sin - d (34) If this equation is calculated first with the upp
41、er tooth thickness limiting dimension ste and then with the lower tooth thickness limitingdimension sti this yields the limit- ing dimensions for the normal chordal tooth thickness and hence the normal chordal tooth thickness allowances. These allowances can generally be made equal to the tooth thic
42、kness allowances. It is therefore usually suffi- cient to calculate the nominal dimension of the normal chordal tooth thickness and to add the tooth thickness allowances to this. DIN 3971 Page 13 Only with small numbers of teeth may it be necessary to calculate the normal chordal tooth thickness all
43、owances. For this purpose it is usually sufficient to convert with the aid of the allowance factor A: which gives the ratio of a normal chordal tooth thickness allowance Ayto the tooth thickness allowance A,. The allowance factor is i A- A-=?=co S- st a AS d (35) 4.2.4 Height ha above the chord The
44、height h7, above the chord S, is stated at tip distance R in the middle of the tooth perpendicular to the refer- ence cone envelope line. Starting from the tip circle of the complementary cone it is found as - d ha =ha + -. (1 - cos;) * cos8 2 (36) For a straight bevel gear the values for the transv
45、erse sec- tion% and quantities on this virtual tooth system are designated by the subscript v. The quantities mainly involved are as follows: 4.4.3.1 Reference diameter d, The length of the cone envelope line between the com- plementary cone apex and the reference circle of the bevel gear at the tip
46、 distance R is half the reference diameter d, of the virtual equivalent cylindrical gear tooth system. d d,=- cos6 (41) 4.4.3.2 N u m be r of teeth z, The number of teeth L, is that number of teeth which, multiplied by the reference circle pitch pt, yields the circumference of the reference circle o
47、f the virtual equiv- alent cylindrical gear tooth system. d;n z z,=-=- pt cos8 (42) N o t e : In the case of helical and spiral bevel gears it is necessary to distinguish between the number of teeth 2, arising in rhe plane of the developed back cone, and the number of teeth z, ruling for a normal se
48、ction of the virtual equivalent cylindrical gear tooth system. The following can be taken as an approximation: z 2, = cos 6 * cos3 (43) 4.4.3.3 Ti p d i am et e r d, The length of the envelope line between the complemen- tary cone apex and the tip circle of the bevel gear is half the tip diameter d,
49、 of the virtual equivalent cylindrical gear tooth system. d,=d,+ 2 *ha (44) 4.4.3.4 R o o t d i a m e t e r dd The length of the envelope line between the complemen- tary cone apex and the root circle of the bevel gear is half the root diameter dd of the virtual equivalent cylindrical gear tooth system. dd=d, - 2 *hi (45) 4.4.3.5 Ba se d i a m e t e r dvb For approximate representation in drawings and for the manufacture of templates and patterns the virtual equiv- alent cylindrical gear tooth system is usually regarded as an involute tooth system with parameters acco
