Vehicle and wheel speed can be calculated as a function of engine speed, if the parameters and status of the transmission are known. In this tutorial, we are going to calculate the vehicle and wheel speed for a given:

- engine speed
- gear ratio (of the engaged gear)
- final drive ratio (at the differential)
- (free static) wheel radius

Also, we are going to assume that there is no slip in the clutch or torque converter, the engine being mechanically linked to the wheels.

This method can be applied to any powertrain architecture (front-wheel drive or rear-wheel drive) but, for an easier understanding of the components, we are going to use a read-wheel drive (RWD) powertrain.

where:

*ω _{e} [rad/s]* – rotational speed of the engine

*ω*– rotational speed of the gearbox output shaft speed

_{g}[rad/s]*ω*– rotational speed of the differential crown wheel

_{d}[rad/s]*ω*– rotational speed of the right wheel

_{wr}[rad/s]*ω*– rotational speed of the left wheel

_{wl}[rad/s]*v*– linear speed of the left wheel

_{wl}[m/s]*v*– linear speed of the right wheel

_{wr}[m/s]*i*– gear ratio of the engaged gear

_{x}[-]*i*– gear ratio of the differential

_{0}[-]*r*– static radius of the wheel

_{w}[m]To have a simple calculation, we are going to assume that the vehicle is moving in a straight line, and also that both wheels have the same radius. This means that:

\[\omega_{wr}=\omega_{wl}=\omega_{w} \tag{1}\]where *ω _{w} [rad/s]* is the common wheel rotational speed.

Since both vehicle and wheel move together in a linear direction, the vehicle (linear) speed is equal to the linear speed of the wheel. So if we calculate the wheel linear speed, we also have the vehicle’s speed.

\[v_{wr}=v_{wl}=v_{w}=v_{v} \tag{2}\]Where *v _{w} [m/s]* is the common wheel linear speed and

*v*is the vehicle speed.

_{v}[m/s]Since the gearbox is connected to the engine through the clutch (on a manual transmissions) or torque converter (on an automatic transmissions), we consider that there is absolutely no slip in the clutch (fully closed) or in the torque converter (lock-up clutch closed). In this case, the clutch speed *ω _{c} [rad/s]* is equal with the engine speed

*ω*.

_{e}[rad/s]Contrary to the wheel torque calculation, the gear ratios will decrease the wheel speed. The speed of the gearbox output shaft is equal with the clutch speed divided by the gear ratio:

\[\omega_{g} = \frac{\omega_{c}}{i_{x}} \tag{4}\]The rotational speed of the differential crown wheel is also reduced, being equal gearbox output shaft speed divided by the differential gear ratio:

\[\omega_{d} = \frac{\omega_{g}}{i_{0}} \tag{5}\]The left and right wheel speeds are equal with the differential speed:

\[\omega_{wr}=\omega_{wl}=\omega_{d} \tag{6}\]Combining all above equations, gives the formula for wheel speed function of engine speed:

\[\omega_{w} = \frac{\omega_{e}}{i_{x} \cdot i_{0}} \tag{7}\]For engine speed, the conversion from *rpm* to *rad/s* is done as:

Where *N _{e}* is engine speed in

*[rpm]*.

If we want the wheel speed *N _{w}* in

*[rpm]*, from

*[rad/s]*, we need to apply the inverse conversion:

Also, the wheel’s linear speed is calculated function of rotational speed and radius as:

\[v_{w} = \omega_{w} \cdot r_{w} \tag{10}\]Combining equations (7), (8) and (10), gives the expression of the vehicle and wheel speed function of engine speed and gearbox and differential gear ratios:

\[v_{v} \text{ [m/s]} = v_{w} \text{ [m/s]} = \frac{N_{e} \cdot \pi \cdot r_{w}}{30 \cdot i_{x} \cdot i_{0}} \tag{11}\]If we want to have the speed in *[kph]*, the formula becomes:

**Example 1**. Calculate vehicle speed in *[kph]* for a vehicle with the following parameters:

- engine speed,
*N*_{e}= 2300 rpm - gearbox (1
^{st}) gear ratio,*i*_{x}= 4.171 - final drive ratio,
*i*_{0}= 3.460 - tire size marking
*225/55R17*

**Step 1**. Calculate the (free static) wheel radius from the tire size marking. The method for calculating the wheel radius is described in the article How to calculate wheel radius. The calculated wheel radius is *r _{w} = 0.33965 m*.

**Step 2**. Calculate the wheel torque using equation (12).

The same method can be applied for an electric vehicle, the engine speed being replaced by the motor speed.

You can also check your results using the calculator below.

N_{e} [rpm] | i_{x} [-] | i_{0} [-] | r_{w} [m] |

ω_{w} [rad/s] = | |||

N_{w} [rpm] = | |||

V_{v} [kph] = |

For more tutorials, click the links below.

## Starbuck

Hi, thanks for providing these equations. Just want to know, can they be applied to motorcycles as well ?

## Anthony Stark

Hi, yes it can be applied to motorcycles.