# Chapter 10 Gear Mechanisms

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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

§10－1 Applications and Types of Gear Mechanisms

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

§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.

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.

§10－3 The Involute and Its Properties 一、 Generation of Involute
B 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.

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

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

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 .

§10－4 Terminology and Definition of Gears
pi 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

m=p/π (as πd = zp，then d = zp /)

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

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

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

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 .

§10－5 Gearing of Involute Spur Gears

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.

There are two requirements in designing a gear pair.

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

Meshing of a rack and pinion
ω1 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.

1. Mating Process of a Pair of Gears

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[].   []

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.

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

§10－6 Introduction to Corrected Gear

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.

Gear hobbing

2. Cutting a Standard Gear with Standard Rack-shaped Cutter
O1 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.

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.

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.

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

Reference line of cutter

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

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

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

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.

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

（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.

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.

§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

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

2. Normal module mn and transverse module mt
β 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

(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

2. Contact Ratio for a Helical Gear Pair
B1 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.

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

Fn 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°

§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.

Cylindrical worms Arc-contact worms Types of Worms Enveloping worms spiroids

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.

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

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

§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.

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.

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

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

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°）

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