SPEED AND TORQUE CALCULATION IN GEARS

Before entering the design criteria of gears, let's explain the relationship between the number of teeth (Z) and speed (N) and torque (T).

Gears are designed to transmit torque and rotational motion. Since a gear pair touches each other in a tangential position to the segment circle, the force (F) and linear velocity (V) generated at this point are the same for both gears, but the torque and speed values ​​differ from each other in the ratio of radii.

Ri: Section circle radius
Di: Section circle diameter
F: Force transmitted
V: Environmental speed
Ti: Transmitted Torque
S: Thread pitch
Zi: Number of teeth

Since we multiply or divide both sides of an equation by the same number, the value of the equation will not be distorted.

The number of teeth in the environment must be integers and the tooth pitch of the gears that will work together must also be equal. Thus;

SAMPLE:

Using the relationships above, let's calculate the torque and speed to be taken from a two-stage gearbox driven by a motor at P = 10 KW and N = 3000 rpm.

Reducer gears in 1st stage Z1 = 10 teeth, Z2 = 40 teeth
In the 2nd stage Z3 = 16 teeth, Z4 = 50 teeth

Let's first calculate the input torque:

P = T1 * N1 / 9550 (See important formulas)

=> T1 = 9550 * P / N1 => T1 = 9550 * 10/3000 => T1 = 31.8 N-m

T1 / T2 = Z1 / Z2 => T2 = T1 * Z2 / Z1 => T2 = 31.8 * 40/10 => T2 = 177.12 N-m

Since Z2 and Z3 Gears are on the same shaft, T2 = T3 = 127.2N-m

T3 / T4 = Z3 / Z4 => T4 = T3 * Z4 / Z3 => T4 = 127.2 * 50/16 => T4 = 397.5 N-m

Torque on output shaft T = 397.5N-m

Transfer Account

N1 / N2 = Z2 / Z1 => N2 = N1 * Z1 / Z2 => N2 = 3000 * 10/40 => N2 = 750rpm

Since Z2 and Z3 Gears are on the same shaft, N2 = N3 = 750rpm

N3 / N4 = Z4 / Z3 => N4 = N3 * Z3 / Z4 => N4 = 750 * 16/50 => N4 = 240rpm

Revolution at output shaft N = 240 rpm

Reduction ratio of reducer

i = Input revolution / Output revolution

i = 3000/240 => i = 12.5 / 1

PRACTICAL SOLUTION

The example above has been solved in a long way for easy understanding.
In practice, this problem should be solved as follows.

Discount ratio = First stage reduction * Second stage reduction

i = i1 * i2

i = (Z2 / Z1) * (Z4 / Z3)

i = (40/10) * (50/16)

i = 12.5

OUTPUT = NInput / i

OUTPUT = 3000 / 12.5 = 240 rpm

EXIT = TInput * i

P = Tinput * Niris / 9550 (See important formulas)

=> Tinput = 9550 * P / Nentry => Tinput = 9550 * 10/3000 => Tinput = 31.8 N-m

EXIT = 31.8 * 12.5

OUTPUT = 397.5 N-m

AUTOMOBILE TRANSMISSION BOX

Automobile gearboxes (transmission) operate on the principle of the RPM-Torque relationship described above.

The gearbox mechanism works simply like this. Gears with different numbers of teeth on the same shaft are shifted by the gear lever mechanism, one gear pair is separated from each other and matched with another gear pair.
Thus, while the speed of the gear pairs increases depending on the tooth ratios, the torque they transmit decreases or vice versa.

The animation on the right shows how the gear-gearbox works.