Gear Train

A gear train is a combination of gears used to transmit motion from one shaft to another. The followings are  main type of gear trains:

• Simple Gear Train
• Compound Gear Train
• Reverted Gear Train
• Planetary or Epicyclic Gear Train

Speed Ratio and Train Value:

It is the ratio of the speed of driving gear to that of the driven gear. The reverse of the speed ratio is known as train value.

\[\boxed{Train\quad value = \frac{1}{Speed \quad ratio}}\]

1. Simple Gear Train

All the gear axes remain fixed relative to the frame and each gear on the separate shaft. 

Simple Gear Train

By gear rules: $ \frac{N_2}{N_1} = \frac{T_1}{T_2}$, $ \frac{N_3}{N_2} = \frac{T_2}{T_3}$, $ \frac{N_4}{N_3} = \frac{T_3}{T_4}$, $ \frac{N_5}{N_4} = \frac{T_4}{T_5}$

\[Train\quad value = \frac{N_5}{N_1} = \frac{N_5}{N_4 } * \frac{N_4}{N_3 }*\frac{N_3}{N_2 }*\frac{N_2}{N_1 } = \frac{T_4}{T_5 }*\frac{T_3}{T_4 }* \frac{T_2}{T_3 } * \frac{T_1}{T_2 }\]

\[Train\quad value = \frac{T_1}{T_5} = \frac{number \quad of \quad teeth \quad on \quad driving \quad gear}{number \quad of \quad teeth \quad on \quad driven \quad gear}\]

Note:

• Two external gears of a pair always moves in opposite direction. For example gear 3 is anti CW then gear 4 is CW in direction.
• All odd number gear(1,3,5) moves in one direction and all even number gear(2,4) moves in the opposite direction.
• The intermediate gears have no effect on the speed ratio or train value therefore they are known as idlers.

2. Compound Gear Train

When a series of gears are connected in such a way that two or more gears rotate about an axis with the same angular velocity. 

\[Train\quad value = \frac{N_4}{N_1} = \frac{T_1*T_3}{T_2*T_4}  = \frac{product \quad of \quad number \quad of \quad teeth \quad on \quad driving \quad gear}{product \quad of \quad number \quad of \quad teeth \quad on \quad driven \quad gear}\]

3. Reverted Gear Train

It is a compound gear trian where input gear axis and output gear axis coincides. 

Reverted Gear Train

\[Train\quad value = \frac{N_4}{N_1} = \frac{T_1*T_3}{T_2*T_4}  = \frac{product \quad of \quad number \quad of \quad teeth \quad on \quad driving \quad gear}{product \quad of \quad number \quad of \quad teeth \quad on \quad driven \quad gear}\]

also,

\[r_1 + r_2 = r_3 + r_4\]

\[m_{12}[T_1 + T_2] = m_{34}[T_3 + T_4]\]

m is the module of the gear

4. Planetary or Epicyclic Gear Train

Epicyclic Gear Train

• Epicyclic gear train is the one in which the axis of some gears have relative motion. 
• Large speed reductions are possible with epicyclic gears.
• Important application of epycyclic gears are in transmission, computing devices.

Relative Velocity Method

$N_{pa}$ : Angular vel. of P relative to arm a = Np - Na

$N_{sa}$ : Angular vel. of S relative to arm a = Ns - Na

\[\frac{N_{pa}}{N_{sa}} = -\frac{T_s}{T_p}\]

\[\boxed{\frac{N_p - N_a}{N_s - N_a} = -\frac{T_s}{T_p}}\]

Tabulation Method

5. Sun and Planet

When an annular wheel is added to the epicyclic gear train, the combination is referred as Sun and Planet gear train.

Sun and Planet Gear Train

In general, S, A and a are free to rotate independently of each other. Table can be prepared:

References:

• Theory of machines By SS Rattan

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