1. Chapter 10 Gear Mechanisms
§10－1 Applications and Types of Gear Mechanisms
§10－2 Fundamentals of Engagement of Tooth Profiles
§10－3 The Involute and Its Properties
§10－4 Terminology and Definition of Gears
§10－5 Gearing of Involute Spur Gears
§10－6 Introduction to Corrected Gear
§10－7 Helical Gears for Parallel Shafts
§10－8 Worm Gearing
§10－9 Bevel Gears

2. §10－1 Applications and Types of Gear Mechanisms
一、Introduction
1) Gear mechanisms are widely used in all kinds of machines to transmit motion and power between rotating shafts.
2) Circular gears have constant transmission ratio whereas, for non-circular gears, the ratio varies as the gears rotate.
3) Depending upon the relative shafts positions, circular gear mechanisms can be divided in­to planar gear mechanisms and spatial gear mechanisms.
4) In this chapter, only circular gears are considered.
二、Types of Gear Mechanisms

3. interse-cting shafts parallel shafts spur gear helical gear
double-helical gear
interse-cting shafts
bevel gear
helical bevel gear
spiral bevel gear
crossedshafts
Spiral
worm and worm wheel

4. §10－2 Fundamentals of Engagement of Tooth Profiles
Conjugate Profiles—— Meshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. (i12=ω1/ω2) O2 O1 2 1 C1 C2
一、Fundamental Law of Gearing
vk1
vk2
The driving pinion rotates clockwise with angular velocity ω1 while the driven gear rotates counterclockwise with angular velocity ω2 . The common normal n-n intersects the center line O1O2 at point P. the point P is the instant center of velocity of the gears . ω2 ω1 P n K
vk1k2 n
v12 ＝O1P ω1= O2 P ω2
i12 ＝ω1/ω2＝O2 P /O1P
The transmission ratio:
fundamental law of gearing:
The transmission ratio of two meshing gears is inversely proportional to the ratio of two line segments cut from the center line by the common normal of the tooth profiles through the contact point.

5. point P ——the pitch point.
As the center distance O1O2 is constant, the position of the point P must be fixed if a constant transmission ratio i12 is required. O2 O1 2 1 C1 C2
This implies that, wherever the teeth contact, the common normal n-n of the tooth profiles through the contact
point must intersect the center
line at a fixed point P, if a
constant transmission ratio i12 is required. r1 r2 ω2 ω1 P n K
pitch circle
pitch circle——The loci of P on the motion planes of both gears are called the circles.

6. 二、Conjugate Profiles
Meshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. For circular gears, the conjugate profiles are those that provide the desired constant transmission ratio. Generally speaking, for any specific tooth profile, we can find its conjugate profile. Theoretically. there is an infinity of pairs of conjugate profiles to produce any specific transmission ratio. Nevertheless, only a few curves have been used as conjugate profiles in practice. Among them, involutes are used most widely since gears using involutes as teeth profiles, or involute gears as they are called, can be manufactured and assembled easily.

7. §10－3 The Involute and Its Properties 一、 Generation of InvoluteB K
一、 Generation of Involute
The involute——is the curve generated by any point on a string which is unwrapped from a fixed cylinder.
Generating line
Base circle O rk A θk rb
二、Properties of the Involute
1） AB = BK;
2）The normal of an involute at any point is tangent to its base circle.
3）The tangent point B of the generating line with the base circle is the curvature center of the involute at the point K. The length of the segment BK is the radius of curvature of the involute at the point K.
4）The shape of an involute depends only on the radius of its base circle.

8. rk= rb/cosαk θk= invαk= tanαk-αk
5）No involute exists inside its base circle. t B K O A rk θk rb αk vk K O2 B2 A1 B1 O1 A2 αk rb O3 B3 ∞
三、Equation of the Involute
cosαk= rb/rk
tanαk= BK/rb=AB/rb
=rb(θk+αk) /rb
=θk+αk
rk= rb/cosαk
θk= invαk= tanαk-αk

9. 三、Gearing of Involute Profiles
1. The transmission ratio will remain constant. O2 O1 ω2 ω1
The common normal N1N2 to the meshing involute profiles through their contact point K must be the common tangent to their base circles. The position of this common tangent remains unchanged as both gears rotate, as does the common normal to the involute profiles. This results in a fixed pitch point P. Therefore, according to the fundamental law of gearing mentioned, the transmission ratio will remain constant.
rb2 N2 N1 K/ C1 C2 K P
i12=ω1/ω2= O2P/ O1P = constant

10. As shown in Fig., the transmission ratio:
2. The direction and magnitude of the reaction force does not change O2 O1 ω2 ω1
N1N2—— trajectory of contact（line of action）
α’ ——pressure angle
rb2 N2 N1
The reaction force is exerted along the line of action if there is no friction. As the position of the line of action stays unchanged during motion for an involute gear pair, the direction and magnitude of the reaction force does not change. α’ K/ C1 C2 K P
3. the separability of the center distance in involute gearing
△ O1N1P∽△O2N2P
As shown in Fig., the transmission ratio:
i12=ω1/ω2= O2P/ O1P = rb2 /rb1
A change in centre distance does not therefore affect the constant transmission ratio of an involute gear pair. This property is called the separability of the center distance in involute gearing .

11. §10－4 Terminology and Definition of Gearspi p
Addendum circle： da、ra ra ei e si pn ha
Dedendum circle： df、rf s r pb rb h
Tooth thickness： si rf hf
Spacewidth： ei
Circular pitch： pi= si + ei
Reference circle: Between the addendum circle and the dedendum circle, there is an important circle which is called the reference circle. Parameters on the reference circle are standardized and denoted without subscripts, such as d, s, e and p.
Addendum：ha
Base circle： db、rb
Dedendum：hf
Normal pitch： pn = pb
Tooth depth：h= ha+hf

12. m=p/π (as πd = zp，then d = zp /)
二、Basic Parameters
Number of teeth： z
Module：m The module m of a gear is introduced on the reference circle as a basic parameter, which is defined as:
m=p/π (as πd = zp，then d = zp /)
d=mz
(3.25) (3.75)
Second (6.5) (11)
Series (30)
Modules of involute cylindrical gears（GB1357－87）
First Series

13. Sizes of the teeth and gear are proportional to the module m.

14. Coefficient of addendum: ha*，be standardized: ha*＝1
Pressure angle：α
The pressure angle α is taken as a basic parameter to determine the base circle. The pressure angle α is also standardized. It is most commonly 20°.
Coefficient of addendum: ha*，be standardized: ha*＝1
Coefficient of bottom clearance : c*， be standardized:c*＝0.25
z、 m、α、ha*、c* are the fundamental parameters which determine the size and shape of a standard involute gear.
三、Parameters of Gear
Standard gear：
1）m ,α, ha* , c* are standardized
2）e = s

15. 三、Parameters of Gear
3） d = mz
4） da＝d + 2ha ＝（z＋2 ha*）m
5） df＝d － 2hf ＝（z－2 ha*－2c*）m
6） s＝e＝p / 2 =m / 2
7） ha=ha*m
8） hf=(ha* +c*)m

16. 四、The Rack and Internal Gears
A rack can be regarded as a special form of gear with an infinite number of teeth and its center at infinity. The radii of all circles be­come infinite and all circles become straight lines, such as the reference line, tip line and root line.
Characteristics
1) The involute tooth profile becomes a straight line too and the pressure angle remains the same at all points on the tooth profile. pb B α ha hf p e s
2) The pitch remains unchanged on the refer­ence line, tip line or any other line, i. e. pi= p =πm

17. 2. Internal Gears Characteristics: df > d > da da＝ d - 2ha
The teeth are distributed on the internal surface of a hollow cylinder. The tooth of an internal gear takes the shape of the tooth space of the corresponding external gear, while the tooth space of an internal gear
takes the shape of tooth of the
corresponding external gear. O B s e p
df > d > da
da＝ d - 2ha
df ＝d + 2hf rf r pb h ha hf rb ra N α
3) To ensure that the profile of the tooth on the top is an involute curve, da>db .

18. §10－5 Gearing of Involute Spur Gears
一、Proper Meshing Conditions for Involute Gears O1
rb1 r1 O1 ω1
rb1 ω1 r1
pb1 P N1 N2 B2 B1
pb2
pb1
rb2 r2 O2 ω2
rb2 r2 O2 ω2
pb2
pb1= pb2
pb1> pb2

19. m1cosα1=m2cosα2 m1 = m2 = m α1=α2 =α
To maintain the proper meshing of two pairs of profiles at the same time, the normal distances of the teeth on both gears must be the same.
rb2 r2 O2 ω2
rb1 r1 O1 ω1
pb2
pb1 P N1 N2 B2 B1
pb1= pb2
m1cosα1=m2cosα2
m1 = m2 = m
α1=α2 =α
The proper meshing condition for involute gears: the modules and pressure angles of two meshing gears should be the same.

20. There are two requirements in designing a gear pair.
二、Center Distance and Working Pressure Angle of a Gear Pair O1
There are two requirements in designing a gear pair. a ω1
ra1 r1
rb1
1) The backlash should be zero to prevent shock between the gears.
rf1 N1
To obtain zero backlash of a gear pair:
Zero backlash P
C=C*m C
s’1= e’2 s’2= e’1 N2
2) The bottom clearance should take the standard value
rf2 r2
c=c*m
2. Standard(reference) center distance ω2
Standard mounting
working center distance a’=r’1+ r’2 O2
reference center distance a = r1+ r2
If two gears are mounted with the reference center distance, then：

21. a’cosα’= a cosα 3. Center distance a and working pressure angle α’
rb2 ω2
ra2 O1 ω1
rb1
ra1 r1 r2 P N1 N2 a α’
rb2 O2 ω2 O1 ω1
rb1 a’ α’ P N1 N2 r2
r’2 r1
r’1
r1’ =r1
α’ =α
r2’ =r2
3. Center distance a and working pressure angle α’
α’ >α
α’>α
r’2 >r2
r1’ >r1
1) Standard mounting(a’ = a)
The reference circles coincide with their pitch circles.
r’1=r1 r’2=r α’=α c=c*m
2)Nonstandard mounting(a’ >a)
The reference circles do not coincide with their pitch circles.
r’1> r1 r’2> r α’>α c’>c*m
rb1＋rb2 = (r1’+r2’)cosα’ rb1＋rb2 =（r1＋r2）cosα
a’cosα’= a cosα

22. Meshing of a rack and pinionω1
ra1
1） Standard mounting r1
rf1 1
The pitch line of the rack coincides with its reference line :
r1’ = r1 ，α’＝α N1
α’=α v2 2 N2 P
2） Nonstandard mounting
The pitch line of the rack does not coincides with its reference line :
r1’ = r1 ，α’＝α
As mentioned above, α’＝α, and r '＝r are characteristics of rack and pinion gearing and differ from those of two spur gears.

23. 1. Mating Process of a Pair of Gears
三、Mating Process of a Pair of Gears and Continuous Transmission Condition O1
1. Mating Process of a Pair of Gears ω2 ω1
rb1
B2 ——meshing begins at point B2
ra1 N1
B1 ——meshing ends at point B1 P B2 B1 N2
B1B2 ——the actual line of action
ra2
rb2
N1N2 ——the theoretical line of action
N1、N 2 ——meshing limit points O2

24. 2. Continuous Transmission Condition
In order to get a continuous motion transmission, the second
pair of teeth must have meshed before the first pair moves out of contact.
The condition of continuous motion
transmission is： B1B2≥pb O1 N2 N1 K O2 ω2 ω1 B1 B2
Contact ratio:  = B1B2/pb pb
B1B2
Theoretically, if ＝1, a pair of gears can transmit continuously. Considering the manufacture tolerance, the contact ratio  should be larger than 1. Actually, the contact ratio should be equal to or larger than the permissible contact ratio[].
  []

25. Equations of Contact Ratio
rb1
rb2 O2 P
εα＝ B1B2/pb ＝(PB1+P B2) /πmcosα
αa2
αa1
PB1＝B1 N1-PN1＝rb1tanαa1- rb1tanα’
＝z1mcosα(tanαa1-tanα’ )/2
ra1 B2
ra2 B1
PB2＝z2mcosα(tanαa2-tanα’ )/2
εα＝[z1(tanαa1-tanα’)
　　+z2(tanαa2-tanα’)]/2π
The value of the contact ratio indicates the average number of tooth pairs in contact during a cycle to share the load. The higher the contact ratio, the greater the average number of tooth pairs to share the load and the higher the capacity of the gear set to transmit the power.

26. =1.46
1.46 pb B1 B2 pb C D
Two pairs
One pair
0.46 pb
0.54 pb

27. §10－6 Introduction to Corrected Gear
一、Standard gears have some disadvantages
Standard gears enjoy interchangeability and are widely used in many kinds of machines. However, they also have some disadvantages.
1）It is not fit that a’≠a. When a’<a , the pair of gears can not be installed at all. When a’>a, the backlash will increase and the contact ratio will decrease.
2）The curvature radius of the tooth profile and the tooth thickness of the pinion on the dedendum circle are less than those of the gear. The strength of the pinion is much lower than that of the gear, and contact time of the pinion is more than that of the gear.
Base
circle
Reference
3）When z< zmin，undercutting will occur.
Cutter interference——In a generating process, it is sometimes found that the top of the cutter enters the profile of the gear and some part of the involute profile near the root portion is removed.

28. 二、Manufacturing Methods of Involute Profiles
To improve the performance of gears, addendum modification is employed.
二、Manufacturing Methods of Involute Profiles
1. Cutting of Tooth Profiles
pinion-shaped shaper cutter
rack-shaped shaper cutter
The cutting motion is the reciprocation of the cutter while the feed is the movement of the cutter toward the blank. The blank should retreat a little as the cutter goes back to prevent scraping on the finished flank by the cutter.

29. Gear hobbing

30. 2. Cutting a Standard Gear with Standard Rack-shaped CutterO1
Gear blank
O1’
The reference line of the cutter should be tangent to
the reference circle of the gear
rb’
N1’
O1’’
N1’’
rb’’ ra
Involute
c*m r v rb
Reference circle N1 α
ha*m
Reference line B1 B2
e = s = p / 2
ha= ha* m; hf =(ha*+ c*)m; P
1）The addendum line of the cutter does not exceed the limit point N1’’ of the line of action, cutter interference will not occur.
2）Cutter interference will occur if the addendum line of the cutter passes the limit point N1’’ of the line of action.

31. 3. Minimum Teeth Number of Standard Gear Without Undercutting
To prevent cutter interference, the point B2 should not pass point Nl ,：PN1≥PB2
PN1=rsinα=mzsinα/2
PB2=ha*m/sinα=mzsinα/2
4. Methods to Avoid Undercutting
There are several methods to avoid undercutting：
1）Decrease the coefficient of addendum depth ha*
ha*  zmin 
ha*    the transmission characteristics will be influenced and the cutter will not be standard.

32. 2）Increase the pressure angle of cutter a
a   zmin 
a   rb   This procedure will reduce the active length and the contact ratio will reduce too, which will also lead to rougher, noisier gear operation and the cutter will not be standard.
3）Corrected gear O1
The method commonly used to eliminate undercutting is to cut the gears with profile-shifted, i.e., with unequal addendum and dedendum teeth. α
xminm xm N1
ha*m
Therefore, parameters m , a , ha* , c* , of the corrected gear remain the same as those of standard gears, but s≠e，the gear is called corrected gear (profile-shifted gear). Q α P xm
The cutter will be standard.

33. 5. Corrected gear O1
Modification distance（xm）—— In cutting the corrected gear, the rackshaped cutter is located a distance xm from the position used for cutting the standard gear.
x ——modification coefficient α
xminm xm N1
ha*m Q α P xm
Positive modification( x>0) ——The cutter is placed further away from the position for cutting a standard gear.
positive modification gear
Negative modification( x<0) ——The cutter is placed towards the axis of the blank negative modification gear

34. Reference line of cutter
三、Geometric Dimensions of Corrected Gears
1. Geometric dimensions are identical with that of the standard gear
d = mz
db = mzcos
p = m
Reference circle P N1 O1 α rb K J I
2. Geometric dimensions are not identical with that of the standard gear α xm
1）Tooth thickness and spacewidth
Base circle B2
Reference line of cutter
Pitch line of cutter xm K I J
πm/2

35. Negative modification gear x<0
2）Addendum and dedendum
Positive modification gear x>0
Standard gear x＝0
Negative modification gear x<0
Reference circle

36. 四、 Gearing of a Corrected Gear Pair
Proper meshing conditions and condition of continuous transmission
Proper meshing conditions： m1= m α1=α2
Condition of continuous transmission：   []
2. Centers distance of a pair corrected gear
1) Gearing equation without backlash
To keep zero backlash for a corrected gear pair, the following relations should hold, as in the case of standard gears, i.e., sl'= e2' , s2'= el' , therefore,
p'＝ s'1+ e'1 ＝ s'2+ e'2＝ s'1+ s'2

37. Analysis
(x1+x2)    a’  a
The two pitch circles will not overlay on the two reference circles
acos＝acos  a’  a
2） Shifting coefficient of centers distance y
Difference of the centers distance a’ with standard centers distance a ：
ym = a’- a
y——Shifting coefficient of centers distance

38. 3） Shifting coefficient of addendum depth y With no backlash：
Clearance be standard：
If two gears mating with no backlash and remaining standard clearance, therefore
a'=a'' y=x1+x2
Problem： (x1+x2) > y if x1+ x2≠0  a' > a''
Solution：No backlash can be assured, the depth of addendum circle is decreased.

39. 3. Types of Corrected Gear Pairs
Types of corrected gear pairs can be divided into three types
by the sum of the shifting coefficients（x1+ x2）.
（1）Standard transmission（ x1+ x2＝0，and x1＝x2＝0）
z1 > zmin , z2 > zmin
（2）Zero transmission (height shifting gears transmission）
x1+ x2＝0，and x1＝-x2≠0
As x1+x2 = 0 and the above three equations 
a’ = a , ’=， y = 0，  y = 0
The pinion should be positive corrected gear( x1 >0)；the gear should be negative corrected gear（x2<0）.
Two gears should not be undercutting：
z1 + z2 ≥ 2zmin

40. （3）Angle shifting gear transmission ( x1+x2≠0 )
1）Positive transmission（ x1+x2 > 0 ）
As x1+x2 > 0 and the above three equations 
a’ > a , ’ > ， y > 0，  y > 0
As x1+x2 > 0  z1+z2 < 2 zmin
Since gears are positive corrected gear, the strengths of two gears increase. But the contact ratio decreases since the working pressure angles decrease.

41. 2）Negative transmission（ x1+x2 < 0 ）
As x1+x2 < 0 and the above three equations 
a’ < a , ’ < ， y < 0，  y > 0
AS x1+x2 < 0， therefore  z1+z2 > 2 zmin
This transmission is contrary to positive transmission. Since gears are negative corrected the strengths of the two gears decrease.But the contact ratio increases since the working pressure angle decrease.

42. §10－7 Helical Gears for Parallel Shafts
Properties:
Tooth profiles go into and out of contact along the whole facewidth at the same time；
Sudden loading and sudden unloading on teeth；
Vibration and noise are produced.
Spur gear
Properties:
The tooth surfaces of two engaging helical gears contact on a straight line inclined to the axes of the gears；
The length of the contact line changes gradually from zero to maximum and then from maximum to zero；
The loading and unloading of the teeth become gradual and smooth.
Helical gear

43. 一、 Basic Parameters of Helical Gears
There are two sets of parameters for a helical gear.One set is on the transverse plane and the other set on the normal plane.
The parameters on the normal plane are the standard values.
To make use of the formulae for spur gear, the parameter in the equations for spur gears should be replaced by those on the transverse plane of helical gears. Therefore, it is necessary to set up relationships between both sets of parameters.
（一） Basic Parameters of Helical
1. Helix angleβ
helix angle（β）——is the helix angle on the reference cylinder. β β
righthanded
lefthanded

44. 2. Normal module mn and transverse module mtβ pt pn β B πd
3. Normal pressure angle n and transverse pressure angle t
4. Coefficient of addendum（ h*an 、h*at）and coefficient of bottom clearance(c*n 、c*t)
ha=h*anmn = h*atmt
hf=(h*an+cn*)mn = (h*at+ct*)mt

45. (二） Sizes of helical gear
Reference diameter：
Center distance:
Modification coefficient：
二、Gearing of a pair of helical gears
1. Proper Meshing Conditions for Helical Gears or

46. 2. Contact Ratio for a Helical Gear PairB1 B2
Spur gear： B B1 B2
Helical gear： B1 B2 βb B B1 B2
ea --- transverse contact ratio △L L
The contact ratio of a helical gear pair is much higher than that of a spur gear pair.
eb --- is the face contact ratio or overlap ratio.

47. 三、 Virtual Number of Teeth for Helical Gears
Virtual gear——the tooth profile of the spur gear is equivalent to that of a helical gear on the normal plane. The spur gear is called the virtual gear of the helical gear. The number zv of teeth of the virtual gear is called the virtual number of teeth （zv）.
The minimum number of teeth of the standard helical gear without cutter interference：
zmin=zvmincos3β

48. 四、The main advantages and disadvantages of helical gearsFn Ft β
1. Main advantages： β
１）Better meshing properties.
２）A much higher total contact ratio.
３）Being more compact means of mechanical
power transmission.
2. Main disadvantages：
The helix angle results in a thrust load in addition to the usual tangential and separating loads. Fa=Ft tgb，b ，Fa
Herringbone gear：b = 25～35 
β＝8°～20°

49. §10－8 Worm Gearing
Worm gear drives are used to transmit motion and power between non­intersecting and non-parallel shafts, usually crossing at a right angle. ＝90
一、Worm Gearing and its Characteristics
1) Smooth silent operation as screw drives.
2) Greater speed reduction in a single step. This means compact designs.
3) If the lead angle of a worm is less than the friction angle, the back-driving is self-locking.
4) Lower efficiency due to the greater relative sliding speed . The friction loss may result in overheating and serious wear. There­fore, brass is usually used as the material for the worm wheel to reduce friction and wear.

50. 二、 Types of Worms Archimedes worm Involute helicoid worms
Cylindrical worms
Arc-contact worms
Types of Worms
Enveloping worms
spiroids

51. 三、Proper Meshing Conditions for Worm Drives
mid-plane：The transverse plane of a worm wheel passing through the axis of
the worm
The engagement between a worm and a worm wheel on the mid-plane corresponds to that of a rack and pinion
Proper Meshing Conditions：
The modules and pressure angles of the worm and worm wheel on the mid-plane should be equal to each other.
The directions of both helices should be the same.

52. 四、 Main Parameters and Dimensions for Worm Drives The number of teeth
The number of threads on the worm z1 : usually, z1＝1 ~ 10，
the recommended value of z1:
z1＝1、2、4、6。
The number of teeth on the worm gear z2 is determined according to the speed ratio and the selected value of z1. For power transmission, z2＝29 ~ 70.
The module
The series of modules for worms is somehow different from those for gears.
The profile angle of worm (pressure angle)
Archimedes worm ：a = 20º
In power transmission： a = 25º
In indexing devices： a =1 5º or 12º

53. 4. The lead angleγ1 of the worm
5. reference diameter
The mid-diameter d1 of worm：the mid-diameter d1 of
the worm is standardized.
The reference diameter d2 of worm wheel： d2 = mz2
6. The center distance a of the worm gear pair

54. §10－9 Bevel Gears 一、Introduction to Bevel Gears
1. Characteristics of Bevel Gears
Bevel gears are used to transmit motion and power between intersecting shafts. The teeth of a bevel gear are distributed on the frustum of a cone. The corresponding cylinders in cylindrical gears become cones, such as the reference cone, addendum cone and dedendum cone. The dimensions of teeth on different transverse planes are different. For convenience, parameters and dimensions at the large end are taken to be standard values.
The shaft angle of a bevel gear pair can be any required value. In most cases, the two shafts intersect at a right angle.

55. 2. Types and Applications or Bevel Gears
are most widely used as they are easy to design and manufacture.
Straight bevel gears：
Bevel
Gears
Helical bevel gears：
operate smoothly and easy to design .
Spiral bevel gears：
operate smoothly and have higher load capacity.

56. 二、Back Cone and Virtual Gear of a Bevel Gear
Crown gear ----d 2 = 90 ，the surface of the reference cone becomes a plane.
Back cone——the cone , the element of which crosses the large end of a bevel gear and is perpendicular to the element of the reference cone.
Virtual gear of the bevel gear： mv = m ； αv = α ；rv= r
The tooth profile of the virtual gear is almost the same as that of the bevel gear at the large end.
Virtual number of teeth zv ：The tooth number of the virtual gear r2
Crown gear
=90° δ2 2 ∑ P P P1 δ1 1 O2
rv1 O1 r1

57. Virtual number of teeth zv
The engagement of bevel gears
The engagement of spur gears r2
Crown gear
=90° δ2 2 ∑ P P P1 δ1 1 O2
rv1 O1 r1

58. 三、Parameters and Dimensions of Bevel Gears
Proper Meshing Conditions： m1=m2 , α1=α2
The contact ratio of the bevel gear set. The virtual number of teeth zv should not be less than the minimum number of teeth of the virtual gear. zmin=zvmincosδ
三、Parameters and Dimensions of Bevel Gears
The most dimensions of bevel gears are measured at the large end being standardized.
1. The reference diameter is
2. The transmission ratio of a gear pair is
（∑＝90°）

59. Transmission ratio： i12＝ω1 / ω2 ＝z2 /z1 ＝r2 / r1 ＝sinδ2 /sinδ1
R—Outer cone distance 2 ha hf O θf 1
δ—Reference cone angle
δa1 d1
δa—Addendum cone angle R
b—Face width δ1
d1 , d2—Reference diameter δ2 b
da—Addendum diameter
da2
δa2 d2
df—dedendum diameter
df 2
Transmission ratio：
i12＝ω1 / ω2
＝z2 /z1
＝r2 / r1 δ1 r1 r2 δ2 R
＝sinδ2 /sinδ1
∑=90°
When ∑＝90°，
δ2 +δ1 ＝90°
i12 ＝ tanδ2
＝cotδ1