1.1 Toothed gearing : fundamentals
A "Gear" is a machine element used to transmit motion and power between rotating shafts by means of progressive engagement of protruding parts called as tooth.

1.1.1 General classification of gears :
There can be many types of gears however a general classification can be done based on their axes, tooth shape, tooth profile and sometimes also as per some special features.
General Classification of Gears

1.1.2 Principles of transmission and conjugate action :
Gear tooth mesh against each other to transmit rotary motion from the driver to the driven shaft. If the gear tooth profile is designed in such a way that there exist a constant angular velocity ratio then such profile are called as conjugate curves and the gears are called as conjugate gears. Mathematically it can be simply expressed as below
ω1ω2=constant−−− (Eqn.4.1) $\frac{{\omega}_{1}}{{\omega}_{2}}={\textstyle \text{constant}}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.1)}}$
Where,
ω1 = Angular velocity of the driver component, and
ω2 = Angular velocity of the driven component.
Now, in order to have a constant velocity ratio, the profiles of the mating tooth must be such that the law of gearing is satisfied. This law states that: “To have a constant angular velocity ratio, the tooth curves must be so shaped that the common normal to the tooth profiles at the point of contact should always pass through the pitch point, irrespective of the position of the point of contact during the course of action”.

1.1.3 Condition for constant velocity ratio of gears law of gearing :
Tooth profile 1 drives tooth profile 2 by the point of contact K, N1N2 is the common normal of the two profiles
N1 is the foot of the perpendicular from O1 to N1 N2 and
N2 is the foot of the perpendicular from O2 to N1 N2
Principle Of Conjugate Action
Principle of conjugate action (Fig. 4.2)
It is evident that the two profiles have different velocities V1 and V2 at point K , but their velocity along N1 N2 should be equal in magnitude and direction for the contact to exist. If such is the condition then component of the velocities along the common normal N1 N2, should be equal.
therefore, V1cosα=V2cosβ$\phantom{\rule{1em}{0ex}}{V}_{1}cos\alpha ={V}_{2}cos\beta $
(ω1×o1K)cosα=(ω2×o2K)cosβ$({\omega}_{1}\times {o}_{1}K)cos\alpha =({\omega}_{2}\times {o}_{2}K)cos\beta $
ω1×O1K×O1N1O1K=ω2×O2K×O2N2O2${\omega}_{1}\times {O}_{1}K\times \frac{{O}_{1}{N}_{1}}{{O}_{1}K}={\omega}_{2}\times {O}_{2}K\times \frac{{O}_{2}{N}_{2}}{{O}_{2}}$, therefore
ω1×O1N1=ω2×O2N2$\phantom{\rule{1em}{0ex}}{\omega}_{1}\times {O}_{1}{N}_{1}={\omega}_{2}\times {O}_{2}{N}_{2}$
ω1ω2=o2N2o1N1−−− (Eqn.4.2) $\frac{{\omega}_{1}}{{\omega}_{2}}=\frac{{o}_{2}{N}_{2}}{{o}_{1}{N}_{1}}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.2)}}$
Now,
∆O1N1P∼∆O2N2P
therefore,
ω1ω2=R2R1−−− (Eqn.4.3) $\frac{{\omega}_{1}}{{\omega}_{2}}=\frac{{R}_{2}}{{R}_{1}}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.3)}}$

1.2 Conjugate curve
Two types of curves known to be conjugate are Involute and Cycloidal. We will study in more details about these two curves.

1.2.1 Involute Curve
An involute of a circle is a plane curve generated by a point or a tangent , which rolls on the circle without slipping.
Involute Profile
Generation Of involute curve
 Draw gear base circle.
 Divide the base circle into a number of equal parts. Say P1, P2, P3 etc.
 Measure arc length for P1, P2, P3 etc.
 Draw Tangent at the points P1, P2, P3 etc. Then along the tangents mark he arc length, name these new points as A1, A2, A3, A4 etc.
 The curve joining thepoints starting from A1, will be an involute curve.
In a gear, the involute profile of the tooth starts from the base circle and continues up to the addendum circle with continuously increasing radius.
Characteristic of involute curve :
Let O1 and O2 be the fixed centers of the two base circles as shown in the figure. The tooth in contact have involute profile. The contact point be K. If the tooth is dragged from the start of the contact to the end of the contact, it can be seen that the contact point moves in a straight line along the common normal N1N2 passing through the pitch point P. This is true for all teeth and entire meshing (start to end). Hence we say that the involute teeth satisfy the fundamental condition of constant velocity or the Law of gearing.
Charectaristic Of Involute Curve

Cycloidal curve :
To generate a cycloidal curve roll the circle C1 and C2 without slipping on the outside of the pitch circle . Roll the circles D1 and D2 on the inside of the pitch circle. The profile created by the point is a cycloidal profile.
Cycloidal profile
Advantages of involute profile over cycloidal profile
 Since the rack in an involute system has straight sides and since the generating cutters usually have rack profiles, these cutters can be easily manufactured. A hob cutter for the cycloidal gear is not as easily made. Consequently, involute gears can be produced more accurately and at a lesser cost.
 While the cycloidal tooth profile has double curvature, an involute tooth has single curvature which facilitates ease of manufacture.
 For effective conjugate action, i.e. for maintaining a constant velocity ratio, it is imperative that the pitch circles of cycloidal gears must be exactly tangent. In other words, for a mating pair of these gears there is only one theoretically correct center distance for which these will transmit motion maintaining a constant angular velocity ratio. In case of involute gearing system, the center distance can be changed without affecting the angular velocity ratio. This advantage of the involute system is of prime importance as most of the modern gears are corrected ones having extended centers.
 Also, even in case of a gearing system having standard center distance, it is practically not possible to accurately achieve or maintain that distance due to mounting inaccuracies, misalignment and a number of other diverse factors involved.
 In involute gearing as the path of contact is a straight line and the pressure angle is constant, there is a constant force acting on the axes. In cycloidal gears, the pressure angle continuously changes. This results in separating force of variable magnitude which in turn gives rise to unquiet operation, jerky running and consequently shorter life.
 The number of cutters required is less in case of involute gearing system to produce complete sets of inter changeable gears of any particular pitch.
Advantages of cycloidal profile
 Cycloidal gears do not have interference and problems.
 A cycloidal tooth is stronger than a standard involute tooth. This is because it has spreading flanks where as an involute tooth has radial flanks. Consequently, there is more material at the root portion of a cycloidal tooth as compared to an involute tooth.
 Cycloidal teeth have less sliding action and hence wear less due to rubbing.
 Since there is no problem of interference, pinions having number of teeth as low as 6 or 7 are possible in the cycloidal system. It can even be 3 or 4 in certain special cases. This is the reason why these gears are extensively used in clocks, watches and similar instruments where a low number of teeth combined with adequate strength is necessary.

1.3 Spur gear
Spur Gear
 Tip circle: is the outer most circle in the gear. It is also called as addendum circle.
 Root circle: is the inner most circle of the gear tooth. It is also known as the dedendum circle.
 Base circle: This is the circle from which the involute tooth profile is developed.
 Addendum: It is the radial distance between the pitch circle and the tip circle.
 Dedendum: It is the radial distance between the pitch circle and the root circle.
 Land: The top land and the bottom land are the surfaces at the top of the tooth and the bottom of the tooth space respectively.
 Working depth: This is the distance of engagement of two mating teeth and is equal to the sum of the addendums of the mating teeth of the two gears in case of standard system.
 Whole depth: This is the height of a tooth and is equal to the addendum plus dedendum.
 Clearance: This is the radial distance between the top land of a tooth and the bottom land ofthe mating tooth space.
 Face width: This is the width of the gear and is the distance from one end of a tooth to the other end.
 Face of tooth:This is the surface of the tooth between the pitch cylinder and the outside cylinder.
 Flank of tooth: This is the surface of the tooth between the pitch cylinder and the root cylinder.
 Module: : It is defined as the ratio of the pitch diameter to the number of teeth of a gear.
 Chordal addendum: This is the height bounded by the top of the tooth and the chord corresponding to the arc of the pitch circle representing the circular tooth thickness.
 Chordal tooth thickness: This is the chord referred to above.
 Diametral pitch: This is a term used in gear technology in the FPS system. It is defined as the ratio of the number of teeth to the pitch diameter in inch. It is equalto the number of gear teeth per inch of pitch diameter.

1.3.1 Relationship between gear parameters
Dimensions for standard gearing
Descriptions 
Pinion 
Gear 
Number of teeth 
Z1

Z2

Pitch circle diameter 
d1=Z1.m${d}_{1}={Z}_{1}.m$

d2=Z2.m${d}_{2}={Z}_{2}.m$

Tip circle diameter 
da1=d1+2m${d}_{a1}={d}_{1}+2m$

da2=d2+2m${d}_{a2}={d}_{2}+2m$

Root circle diameter 
df1=d1−2×1.25m${d}_{f1}={d}_{1}2\times 1.25m$

df2=d2−2×1.25m${d}_{f2}={d}_{2}2\times 1.25m$

Base circle diameter 
db1=d1Cosα${d}_{b1}={d}_{1}Cos\alpha $

db2=d2Cosα${d}_{b2}={d}_{2}Cos\alpha $

Tooth thickness on pitch circle 
S=θ=P2=(πm)2$S=\theta =\frac{P}{2}=\frac{(\pi m)}{2}$

S=θ=P2=(πm)2$S=\theta =\frac{P}{2}=\frac{(\pi m)}{2}$

Center distance 
a=(d1+d2)2=m(Z1+Z2)2$a=\frac{({d}_{1}+{d}_{2})}{2}=m\frac{({Z}_{1}+{Z}_{2})}{2}$

a=(d1+d2)2=m(Z1+Z2)2$a=\frac{({d}_{1}+{d}_{2})}{2}=m\frac{({Z}_{1}+{Z}_{2})}{2}$


1.3.2 Other gear parameters
3.2. Other gear parameters
Arc of action:
This is the arc on the pitch circle through which a tooth travels from the beginning of contact with the mating gear tooth to the point where the contact ends. Since the two pitch cylinders are in rolling contact the lengths of the arcs of action of the two gears are the same. That is,
Arc of action =r1θ1=r2θ2−−− (Eqn.4.4) $\text{Arc of action}}={r}_{1}{\theta}_{1}={r}_{2}{\theta}_{2}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.4)}$
where r1 and r2 are the pitch circle radii, and θ1, and θ2, are the angles subtended by the two arcs at their respective centers.
Arc of approach:
Is the arc through which the tooth moves from the initial contact up to the pitch point P .
Arc of recess:
Is the arc through which the tooth travels from the pitch point to the end of contact.
Relation Between Gear Parameter
Line of action:
This is the line along which the point of contact of the two mating tooth profiles moves. This is also known as the path of contact.
Line of Action
Length of action:
That portion of the line of action on which the point of contact moves during the course of action is known as the length of action. The length of action can also be subdivided as length of approach and length of recess.
Length of Action
Pressure angle :
If a tangent is drawn to the involute profile of a tooth at any point on the curve and if a radial line is drawn through this point of tangency, connecting this point with the centre of the gear, then the acute angle included between this tangent and the radial is defined as the pressure angle at that point.
Pressure Angle
In an involute profile the pressure angle remains constant throughout the course of meshing which is not case with cycloidal profile.
Effect of Increasing pressure angle
 The limiting number of teeth to avoid under cutting is lowered. That is to say if the pressure angle is increased, pinions with comparatively lesser number of teeth can be generated without undercutting.
 The shape of the tooth becomes more pointed or peaked.
 Tooth flank becomes more curved.
 The relative sliding velocity is reduced.
 The contact ratio and overlap are reduced.
 The tooth pressure and axial pressure increase.
 Tooth loadcarrying capacity increases.
Length of arc of contact
Arc of contact is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth. Arc of contact is the sum of arc of approach and arc of recess. Hence,
Length of arc of approach=Length of Path of approachCos α,and$\text{Length of arc of approach}}=\frac{{\textstyle \text{Length of Path of approach}}}{{\textstyle \text{Cos \alpha}}}\phantom{\rule{2em}{0ex}},{\textstyle \text{and}$
Length of arc of recess=Length of path of recessCos α$\text{Length of arc of recess}}=\frac{{\textstyle \text{Length of path of recess}}}{{\textstyle \text{Cos \alpha}}$
Therefore, the length of arc of contact is
Length of arc of contact=Length of path of contactCos α$\text{Length of arc of contact}}=\frac{{\textstyle \text{Length of path of contact}}}{{\textstyle \text{Cos \alpha}}$
Contact ratio:
The contact ratio or the number of pairs of teeth in contact is defined as the ratio of the length of the arc of contact to the circular pith. Mathematically,
Contact ratio or the number of pairs of teeth in contact=Length of the arc of contactCircular pitch,π.m$\text{Contact ratio or the number of pairs of teeth in contact}}=\frac{{\textstyle \text{Length of the arc of contact}}}{{\textstyle \text{Circular pitch,\pi .m}}$
Where, m=Module
Contact ratio (Fig. 4.10)
Backlash in gears
Backlash is the play between a mating pair of gear teeth. It is the amount by which the width of a tooth space exceeds the thickness of the meshing tooth measured on the pitch circle. If the backlash is measured on the line of action, it is termed as the normal or linear backlash. Proper amount of backlash ensures smooth running of the gear set. Excessive backlash may also cause noise and impact loads in case of reversible drives.
Backlash in Gear

1.4 Interference and undercutting of gear tooth
When the gear tooth proportions are such that, the start or the end of the contact happens outside the path of contact during meshing we may see interference. This is because the involute curves begin at the base circle and extend out wards to form the gear tooth profiles. Obviously, there is no involute inside the base circle. The line of action of the two intermeshing gears is tangent to the two base circles. The extreme points on this line represent the length of action and are called as interference points. Any meshing outside of the involute portion will result in nonconjugate action. In case the mating teeth are of such proportion that the beginning or the end of contact or both, occur outside of theinterference points on the path of contact, then the involute portion of one member will mate with the noninvolute portion of the other member. At this point the flank of the tooth of the driver is forced into contact with the tip of the tooth of the driven gear. This phenomenon is termed as “interference”.
Interference in gear tooth is undesirable for several reasons. Due to interference, the tip of one tooth of the gear of the pair will tend to diginto operations of the flank of the tooth of the other member of the pair. Moreover, removal of portions of the involute profile adjacent to the base circle may result in serious reduction in the length of action. All these factors weaken the toothand are detrimental to proper tooth action.
interference and undercutting

1.4.1 Ways to reduce undercutting
There are several practical ways of tackling the problem of interference and undercutting. They are listed as below:
Increasing Pressure angle
Since
Base Circle diameter = Pitch Circle diameter Cosα$\text{Base Circle diameter = Pitch Circle diameter}}Cos\alpha $
Where α , is the pressure angle, Increasing the pressure angle results in smaller base circle diameter. This effectively means more of the tooth profile will be involute thereby reducing interference.
Tooth stubbing
Tooth Stubbing
Another practical way of eliminating interference is to limit the addendum of the driven gear so that it passes through the interference point as such that the whole depth of the tooth consists of the normal depth minus the shaded portion, making it a stubtooth.
Increasing center distance
Increasing slightly the centre distance between the meshing gears would also eliminate interference
Minimum number of tooth to avoid interference
Interference may only be avoided, if the point of contact between the two teeth is always on the involute profiles of both the teeth. In other words, interference may only be prevented, if the addendum circles of the two mating gears cut the common tangent to the base circles between the points of tangency.In order to avoid interference addendum circles for the two mating gears must cut the common tangent to the base circles between the points of tangency. The limiting condition reaches, when the addendum circles of pinion and wheel pass through points of tangency. Say N and M respectively.
Let,
t = Number of teeth on the pinion,
T = Number of teeth on the wheel,
m = Module of the teeth,
r = pitch circle radius of pinion = m.t/2
G = Gear ratio = T/t = R/r
Φ = Pressure angle or angle of obliquity.
From triangle O1 NP.
(O1N)2=(O1P)2+(PN)2−2×O1P×PNcosO1PN$({O}_{1}N{)}^{2}=({O}_{1}P{)}^{2}+(PN{)}^{2}2\times {O}_{1}P\times PNcos{O}_{1}PN$
r2+R2sin2ϕ−2r.Rsinϕcos(90o+ϕ)${r}^{2}+{R}^{2}si{n}^{2}\varphi 2r.Rsin\varphi cos({90}^{o}+\varphi )$…(becausePN=O2Psinϕ=Rsinϕ)$\dots (because\phantom{\rule{1em}{0ex}}PN={O}_{2}Psin\varphi =Rsin\varphi )$
=r2+R2sin2ϕ+2r.Rsin2ϕ$={r}^{2}+{R}^{2}si{n}^{2}\varphi +2r.Rsi{n}^{2}\varphi $
=r2[1+R2sin2ϕr2+2Rsin2ϕr]=r2[1+Rr(Rr+2)sin2ϕ]−−− (Eqn.4.5) $={r}^{2}[1+\frac{{R}^{2}si{n}^{2}\varphi}{{r}^{2}}+\frac{2Rsi{n}^{2}\varphi}{r}]={r}^{2}[1+\frac{R}{r}(\frac{R}{r}+2)si{n}^{2}\varphi ]\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.5)}}$
Therefore, Limiting radius of the pinion addendum circle,
O1N=r1+Rr(Rr+2)sin2ϕ−−−−−−−−−−−−−−−−−√=m.t21+Tt[Tt+2]sin2ϕ−−−−−−−−−−−−−−−−√−−− (Eqn.4.6) ${O}_{1}N=r\sqrt{1+\frac{R}{r}(\frac{R}{r}+2)si{n}^{2}\varphi}=\frac{m.t}{2}\sqrt{1+\frac{T}{t}[\frac{T}{t}+2]si{n}^{2}\varphi}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.6)}}$
Let,
APm = Addendum of the pinion, where AP is a fraction by which the standard addendum of one module for the pinion should be multipled in order to avoid interference.
We know that the addendum of the pinion
=O1N−O1P$={O}_{1}N{O}_{1}P$
Therefore,
AP.m=m.t21+Tt(Tt+2)sin2ϕ−−−−−−−−−−−−−−−−−√−m.t2…(becauseO1P=r=m.t/2)${A}_{P}.m=\frac{m.t}{2}\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)si{n}^{2}\varphi}\frac{m.t}{2}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\dots (because\phantom{\rule{2em}{0ex}}{O}_{1}P=r=m.t/2)$
=m.t2[1+Tt(Tt+2)sin2ϕ−−−−−−−−−−−−−−−−−√−1]$=\frac{m.t}{2}[\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)si{n}^{2}\varphi}1]$
AP=t2[1+Tt(Tt+2)sin2ϕ−−−−−−−−−−−−−−−−−√−1]−−− (Eqn.4.7) ${A}_{P}=\frac{t}{2}[\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)si{n}^{2}\varphi}1]\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.7)}}$
Therefore,
t=2AP1+Tt(Tt+2)sin2ϕ√−1=2AP1+G(G+2)sin2ϕ√−1−−− (Eqn.4.8) $t=\frac{2{A}_{P}}{\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)si{n}^{2}\varphi}1}=\frac{2{A}_{P}}{\sqrt{1+G(G+2)si{n}^{2}\varphi}1}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.8)}}$
This equation gives the minimum number of teeth required on the pinion in order to avoid interference.
The minimum number of teeth on the pinion which will mesh with any gear(also rack) without interface are given in the following table.
13.4 Minimum number of teeth on the pinion
S. No. 
System of gear teeth 
Minimum number of teeth on the pinion 
1. 
1412o${14\frac{1}{2}}^{o}$ Composite

12 
2. 
1412o${14\frac{1}{2}}^{o}$ Full depth involute

32 
3. 
20o${20}^{o}$ Full depth involute

18 
4. 
20o${20}^{o}$ Stub involute

14 

1.5 Systems of gear teeth
The following four systems of gear teeth are commonly used in practice

1412o${14\frac{1}{2}}^{o}$; Composite system

1412o${14\frac{1}{2}}^{o}$; Full depth involute system

20o${20}^{o}$; full depth involute system, and

20o${20}^{o}$; stub involute system
Standard proportions of gear systems :
The table below shows the standard proportions in module (m) for the four gear systems shown above
Standard proportions of gear systems
S. No. 
particulars 
1421o;
composite or full depth involute system 
20o;
full depth involute system 
20o;
stub involute system 
1. 
Addendum 
1 m 
1 m 
0.8 m 
2. 
Dedendum 
1.25 m 
1.25 m 
1 m 
3. 
Working Depth 
2 m 
2 m 
1.6m 
4. 
Minimum total depth 
2.25 m 
2.25 m 
1.80m 
5. 
Tooth thickness 
1.5708 m 
1.5708 m 
1.5708 m 
6. 
Minimum clearance 
0.25 m 
0.25 m 
0.2 m 
7. 
Fillet radius at root 
0.4 m 
0.4 m 
0.4 m 

1.6 Gear train
A gear train is combination of gears that is used for transmitting motion from one shaft to another. Two types of gear trains are generally used
 Ordinary gear train in which all gears rotate on fixed axes relative to a single frame of reference, and
 Epicyclic gear train in which at least one gear axis rotates relative to the frame in addition to the gear's rotation about its own axis.
An ordinary gear train can be simple or compound. In a simple train, each shaft carries one gear only.
Simple Train
As it can be seen that the gear mounted on shaft1 is driving the gear mounted on shaft2 through a small gear called as the idler gear. This idler gear changes the direction of the rotation of the second gear in shaft2 and also bridges the distance between the shafts, thereby reducing the usage of bigger gears.
In a compound train, the shafts can carry more than one gear. Compound gear train offer large speed ratio from a very small space. Most of the modern automobiles has compound gears trains in the gearbox.
Compound Train
In any train, the ultimate speed ratio “i”, is given by
i=Rotational speed of first driving gearRotational speed of last driven gear$i=\frac{{\textstyle \text{Rotational speed of first driving gear}}}{{\textstyle \text{Rotational speed of last driven gear}}}$

1.6.1 Simple Gear Train
Consider the case as shown below:
Simple gear train (Fig. 4.16)
Let t1 , t2 be number of teeth on gears 1 and 2.
Let N1 , N2 be speed in rpm for gears 1 and 2. The velocity of P,
Vp=2πN1d160=2πN2d260−−− (Eqn.4.15) ${V}_{p}=\frac{2\pi {N}_{1}{d}_{1}}{60}=\frac{2\pi {N}_{2}{d}_{2}}{60}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.15)}}$
N1N2=d2d1=t2t1$\phantom{\rule{1em}{0ex}}\frac{{N}_{1}}{{N}_{2}}=\frac{{d}_{2}}{{d}_{1}}=\frac{{t}_{2}}{{t}_{1}}$
In simple gear train, there can be more than two gears also, like the case below
Gear train (Fig. 4.17)
Let N1, N2, N3, . . . be speed in rpm of gears and t1, t2, t3, . . . be number of teeth of respective gears , then the gear ratio can be given as
Gear Ratio=N1N4=N1N2×N2N3×N3N4$\text{Gear Ratio}}=\frac{{N}_{1}}{{N}_{4}}=\frac{{N}_{1}}{{N}_{2}}\times \frac{{N}_{2}}{{N}_{3}}\times \frac{{N}_{3}}{{N}_{4}$
N1N2=t2t1;N2N3=t3t2andN3N4=t4t3$\phantom{\rule{2em}{0ex}}\frac{{N}_{1}}{{N}_{2}}=\frac{{t}_{2}}{{t}_{1}};\phantom{\rule{1em}{0ex}}\frac{{N}_{2}}{{N}_{3}}=\frac{{t}_{3}}{{t}_{2}}\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}\frac{{N}_{3}}{{N}_{4}}=\frac{{t}_{4}}{{t}_{3}}$
N1N4=t2t1×t3t2×t4t3=t4t1−−− (Eqn.4.16) $\frac{{N}_{1}}{{N}_{4}}=\frac{{t}_{2}}{{t}_{1}}\times \frac{{t}_{3}}{{t}_{2}}\times \frac{{t}_{4}}{{t}_{3}}=\frac{{t}_{4}}{{t}_{1}}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.16)}}$
Thus we see that the intermediate gears have no effect on the gear ratio it only changes the gear rotation direction.

1.6.2 Compound gear train
Consider the case as shown below:
Compound Gear Train
Let N1, N2, N3, . . . be speed in rpm of gears 1, 2, 3, . . . , etc. and t1, t2, t3 , . . . , etc. be number of teeth of respective gears 1, 2, 3, . . . , etc.
GearRatio=N1N4=N1N2×N2N4=N1N2×N3N4$GearRatio=\frac{{N}_{1}}{{N}_{4}}=\frac{{N}_{1}}{{N}_{2}}\times \frac{{N}_{2}}{{N}_{4}}=\frac{{N}_{1}}{{N}_{2}}\times \frac{{N}_{3}}{{N}_{4}}$
=t2t1×t4t3−−− (Eqn.4.17) $=\frac{{t}_{2}}{{t}_{1}}\times \frac{{t}_{4}}{{t}_{3}}\phantom{\rule{2em}{0ex}}{\textstyle \text{(Eqn.4.17)}}$
Therefore, unlike simple gear train the gear ratio is contributed by all the gears.

1.6.3 Planetary gears
An epicyclic or planetary gear train consists of one or more rotating gears revolving around a central gear. The basic components of such a gear train are shown in figure.
Planetary gears (Fig. 4.19)
The train usually consists of a central gear or sun gear (S1), one or more gears surrounding the sun gear called planet gears (P1) , a member with one or more arms to which the planet gears are mounted called arm (A), or planet carrier or spider, and an annular gear or ring gear (R) which is concentric to the sun gear.
The aspects which characterize a planetary gear system are its compactness, coaxial arrangement of driving and driven shafts, large speed reduction possibilities visa vis its overall size, possibilities of a number of combinations of driving and driven inputs and outputs, large torque conversion possibilities, and different possibilities of orientation of drives. A unique feature of this type of gearing is that it permits some of the gear axes to rotate with respect to others.
Calculating reduction ratio
Planetary gear train
This can be explained well by taking an example. Let us take the below example
Planetary gear train (Fig. 4.21)
For the above type of gear train we can have three possible configuration:
 When the arm is fixed and the sun and ring gear rotates – In this case since the planet gears acts like a idler gear the ring gear and sun gear will rotate opposite to each other.
 When the ring gear is fixed and the sun gear and arm rotates – In this case both the input and output shaft will rotate in the same direction.
 When the sun gear is fixed and the ring gear and arm rotates – In this case also both the input and output shaft will rotate in the same direction.
Let,
NS, NA, NP = number of teeth on the sun wheel, the ring gear, and the planet gearsrespectively RS, RA, RL RP =rotation (revs) of the sun wheel, the ring gear, the planetary arm, and the planet gears respectively.ωS ,ωA ,ωL,ωP = angular velocity of the sun wheel, the ring gear, the planetary arm, and the plant gearsrespectively. If the ratio of rotation of the follower member (RA) has a ratio of R relative to the driving member (RB) rotation i.e. RA = R. RB. Then RB = RA /R
Calculation of Ratios
Configuration 1
The driver is the planetary arm ( L ).
The driven member is the Sun (S).
The Ring Gear is fixed (A).
Note NA = NS + 2 . NP
Action 
Rotation (CW = +ve) from action....................Nx = number of teeth 
L 
A 
P 
S 
Turn whole gear through 1 rev Clock Wise 
1 
1 
1 
1 
Fix arm L and rotate A back CCW 1 rev 
0 
1 
−NaNp$\frac{{N}_{a}}{{N}_{p}}$ 
NaNs$\frac{{N}_{a}}{{N}_{s}}$ 
Add the two motions above 
1 
0 
1−NaNP$1\frac{{N}_{a}}{{N}_{P}}$ 
1+NaNs$1+\frac{{N}_{a}}{{N}_{s}}$ 
Summary : Rotation of the planet shaft(L) 1 rev CW results in the rotation of the shaft (S) 1 + ( NA / NS) revs (CW).
Ratio = RS = RL ( 1 + ( NA / NS)
ratio r1 = RS / RL = ( 1 + ( NA / NS)
Configuration 2
The driver is the Sun ( S ).
The driven member is the Ring (A)
The planetary arm is fixed. (L)
Note NA = NS + 2 . NP
Action 
Rotation (CW = +ve) from action....................Nx = number of teeth 
S 
L 
P 
A 
Fix L and rotate Sun 1 rev CW 
1 
0 
NS/NP 
NS/NA 
Configuration 3
The driver is the planetary Arm ( L ).
The follower is the annulus ring (A)
The the sun gear (S) is fixed.
Note NA = NS + 2 . NP
Action 
Rotation (CW = +ve) from action....................Nx = number of teeth 
L 
A 
P 
S 
Turn whole gear through 1 rev Clock Wise 
1 
1 
1 
1 
Fix arm L and rotate S back CCW 1 rev 
0 
1 
NA/NP 
+NA/NS 
Add the two motions above 
1 
1 + (NS/NA) 
1 + (NS/NP) 
0 

1.7 Types of gear boxes
 Selective type gear boxes :
 Sliding mesh gear box
 Constant mesh gear box
 Synchromesh gear box
 Progressive type gear box
 Epicyclic type gear box.

Function of gear box
An automobile is able to provide varying speed and torque through its gear box. Various functions of a gear box are listed below :
 The gear box is necessary in the transmission system to maintain engine speed at the most economical value under all conditions of vehicle movement.
 An ideal gear box would provide an infinite range of gear ratios, so that the engine speed should be kept at or near that the maximum power is developed whatever the speed of the vehicle.
 To provide high torque at the time of starting, vehicle acceleration, climbing up a hill.
 Gear box provides a reverse gear for driving the vehicle in reverse direction.

Conventional automobile gear box
A conventional gear box of an automobile uses compound gear train. For different gear engagement, it may use sliding mesh arrangement, constant mesh arrangement or synchromesh arrangement.

1.7.1 Sliding mesh gear box:
It is simplest type of gear box out of the available gear boxes. In this type of gear box,
gears are changed by sliding one gear on the other. This gear box consists of three shafts;
main shaft, clutch shaft and a counter shaft.
Working of this gear box can be seen with the help of a three speed gearbox.
Sliding Mesh gearbox

1.7.2 Constant mesh gearbox:
Constant Mesh Gearbox

1.7.3 Synchromesh gearbox:
Syncro mesh Gear Transmission
