Spur Gear Torque Calculation Explained

This problem involves calculating the torque transmitted by a driven gear when it meshes with a driving gear (pinion). We are given the number of teeth and rotational speed for the pinion, along with the torque it transmits. We need to find the torque transmitted by the larger gear.

Gear Torque Transmission Principles

In a pair of meshing spur gears, the power transmitted is assumed to be constant (neglecting efficiency losses). Power ($P$) is the product of torque ($Torque$) and angular velocity ($\omega$).

The relationship is given by: $P = Torque \times \omega$

For two meshing gears (pinion and gear), the power is conserved:

$P_{pinion} = P_{gear}$
$Torque_{pinion} \times \omega_{pinion} = Torque_{gear} \times \omega_{gear}$

Angular velocity ($\omega$) is related to rotational speed in revolutions per minute ($N$) by the formula $\omega = \frac{2 \pi N}{60}$. Since $\frac{2 \pi}{60}$ is a constant factor for both gears, we can simplify the power conservation equation in terms of speed (rpm):

$Torque_{pinion} \times N_{pinion} = Torque_{gear} \times N_{gear}$

The ratio of rotational speeds is inversely proportional to the ratio of the number of teeth ($T$):

$\frac{N_{pinion}}{N_{gear}} = \frac{T_{gear}}{T_{pinion}}$

Substituting this speed ratio into the torque equation:

$Torque_{gear} = Torque_{pinion} \times \frac{N_{pinion}}{N_{gear}} = Torque_{pinion} \times \frac{T_{gear}}{T_{pinion}}$

This formula shows that the torque transmitted by the gear is equal to the torque transmitted by the pinion multiplied by the gear ratio (ratio of teeth).

Calculating Gear Torque Step-by-Step

Let's list the given parameters:

• Number of teeth on the pinion, $T_{pinion} = 40$
• Number of teeth on the gear, $T_{gear} = 120$
• Rotational speed of the pinion, $N_{pinion} = 1200$ rpm
• Torque transmitted by the pinion, $Torque_{pinion} = 20$ Nm

We need to find the torque transmitted by the gear, $Torque_{gear}$.

Using the formula derived above:

$Torque_{gear} = Torque_{pinion} \times \frac{T_{gear}}{T_{pinion}}$

Substitute the given values into the formula:

$Torque_{gear} = 20 \text{ Nm} \times \frac{120}{40}$

Calculate the ratio of teeth:

$\frac{120}{40} = 3$

Now, calculate the final torque:

$Torque_{gear} = 20 \text{ Nm} \times 3$

$Torque_{gear} = 60 \text{ Nm}$

Therefore, the torque transmitted by the gear is 60 Nm.