Transmission and Differential Gear Ratio Calculation
With the wide use of transmission gears and Differential gears in almost every automobile, the concept to calculate transmission and differential gear ratios is important. The concept has vital importance in the SSC JE exams.
Transmission Gear Ratio
Take a look at our types of gears article. Here you will understand that a vehicle’s gearbox most probably has helical gears. The design procedure for these types of gears is different from differential gears. The method to calculate the gear ratio we learned above remains the same.
A 3D view of a transmission gearbox with an assembly of helical gears is shown in Fig 3 below.
Fig 3: Transmission gear 3D
As the transmission box or gearbox directly corresponds to the speed of the vehicle, the definition for transmission gear ratio remains the same as the original gear ratio definition. And, the transmission gear ratio formula also remains the same.
But only the way we calculate the number of teeth changes with helical gears. Since the gear tooth forms a helix angle, we consider the formative number of teeth (\(Z_e\)) here. This is also known as the equivalent number of teeth.
\(Z_e = \frac{Z}{cos^3 \beta} \)
\(Z_e\) → Actual number of teeth on a helical gear
\(beta\) → Helix Angle
Again, the method of calculation of the number of teeth changes for differential gear assembly.
Differential Gear Ratio
Let us know the definition of differential gear ratio in the first place: “It is the ratio of the number of rotations by pinion for every rotation of ring gear.”
Fig 4: Differential Gear System Representation
This comes into the picture when the vehicle is taking a turn. So, how do we calculate the differential gear ratio? Before that, we need to know that the differential system mainly consists of bevel gears that rest at an angle to each other.
Hence, the differential gear ratio calculation depends on the design criteria of the bevel gears. Here, we can see an addition of a concept called equivalent gear ratio.
This ratio is considered in terms of the spur gear ratio.
\(i_e = i^2\)
where i = gear ratio and \(i_e\) = equivalent gear ratio
The method of calculating the number of teeth (regarding the original gear ratio formula) differs from the transmission gear ratio method. Consider these equations:
Virtual Number of Teeth on pinion: \(Z_{v1} = \frac{Z_1}{cos \delta} \)
Virtual Number of Teeth on Gear: \(Z_{v2} = \frac{Z_2}{cos \delta} \)
Here,
\(\delta \) → Angle between the Pitch Line and the Center of the Axis (Consider the figure below to understand it better)
Fig 3: Dimensions of Bevel Gear,
However, the method to arrive at the final gear ratio using the given gear ratio formula or equation remains the same.
Utilizing these concepts, we can now learn about calculating transmission gear ratios that are used in determining the RPM for each gear shift for the given vehicle.
