**Wheel torque** can be calculated function of **engine torque** if the parameters and status of the transmission are known. In this tutorial, we are going to calculate the wheel torque and force for a given:

- engine torque
- 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**.

Image: Vehicle rear-wheel drive (RWD) powertrain diagram

As depicted in the image above, the engine is the source of torque. 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 engine torque *T _{e} [Nm]* is equal with the clutch/torque converter torque

*T*.

_{c}[Nm]\[T_c = T_e \tag{1}\]

Further, the **engine torque** is transmitted through the gearbox, where is multiplied with the gear ratio of the engaged gear *i _{x} [-]* and outputs the

**gearbox torque**

*T*.

_{g}[Nm]\[T_g = i_x \cdot T_e \tag{2}\]

The propeller shaft is transmitting the torque to the rear axle, where is multiplied with the final drive gear ratio *i _{0} [-]*. This gives the

**torque at the differential**

*T*.

_{d}[Nm]\[T_d = i_0 \cdot T_g \tag{3}\]

If the vehicle is driven on a straight line, the torque at the differential is equally split between the left wheel *T _{lw} [Nm]* and the right wheel

*T*.

_{rw}[Nm]\[T_{lw} = T_{rw} = \frac{T_d}{2} \tag{4}\]

Image: Vehicle rear-wheel drive (RWD) powertrain schematic

The sum of the left and right wheel torque gives the **torque at the differential**.

\[T_{lw} + T_{rw} =T_d \tag{5}\]

Replacing (2) in (3) in (4) gives the **mathematical expression of the wheel torque function of the engine torque**, for a given gearbox ratio *i _{x}* and a final drive ratio

*i*.

_{0}\[\bbox[#FFFF9D]{T_w = \frac{i_x \cdot i_0 \cdot T_e}{2} }\tag{6}\]

If we consider n_{w} [-] as the number of driving wheels, then the wheel torque formula will have the general form of:

\[\bbox[#FFFF9D]{T_w = \frac{i_x \cdot i_0 \cdot T_e}{n_{w}} }\tag{6.1}\]

If the vehicle is rear wheel drive (RWD) or front wheel drive (FWD) then n_{w} = 2, if the vehicle is four wheel drive (4WD) or all wheel drive (AWD) the n_{w} = 4. If the vehicle is a motorcycle then n_{w} = 1.

The **formula of the wheel torque** (6) applies to a vehicle which is driven on a straight line, where the left wheel torque is equal with the right wheel torque.

\[T_{lw} = T_{rw} = T_w \tag{7}\]

From mechanics (static), we know that the **torque** is the product between a **force** and its **lever arm length**. In our case, the wheel torque is applied in the wheel hub (center) and the lever arm is the wheel radius *r _{w} [m]*. For this example we assume that both wheel have the same radius

*r*.

_{w}\[T_{lw} = F_{lw} \cdot r_w \tag{8}\]

The same principle applies to the **right wheel torque**.

\[T_{rw} = F_{rw} \cdot r_w \tag{9}\]

Assuming that both left and right wheel torque and radius are equal, we can write a generic expression of the wheel force *F _{w} [N]*, function of wheel torque

*T*and wheel radius

_{w}[Nm]*r*.

_{w}[m]\[T_{w} = F_{w} \cdot r_w \tag{10}\]

From (10) we can extract the formula of the **wheel force** function of the **wheel torque** and **wheel radius**.

\[\bbox[#FFFF9D]{F_{w} = \frac{T_w}{r_w}} \tag{11}\]

Replacing (6) in (10) will give the mathematical expression of the **wheel force** function of **engine torque**, gearbox **gear ratio**, **final drive ratio** and **wheel radius**.

\[\bbox[#FFFF9D]{F_{w} = \frac{i_x \cdot i_0 \cdot T_e}{2 \cdot r_w}} \tag{12}\]

For a different number of driving wheels n_{w} [-], the **general formula for wheel force** becomes:

\[\bbox[#FFFF9D]{F_{w} = \frac{i_x \cdot i_0 \cdot T_e}{n_{w} \cdot r_w}} \tag{12.1}\]

**Example 1**. Calculate the **wheel torque** and **force** for a rear wheel drive vehicle (RWD) with the following parameters:

- engine torque, T
_{e}= 150 Nm - 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 (6).

\[T_w = \frac{i_x \cdot i_0 \cdot T_e}{2} = \frac{4.171 \cdot 3.460 \cdot 150}{2} = 1082.3745 \text{ Nm}\]

**Step 3**. Calculate the **wheel force** using equation (11).

\[F_{w} = \frac{T_w}{r_w} = \frac{1082.3745}{0.33965} = 3186.7349 \text{ N} \]

**Example 2**. For a given gearbox, with multiple gears (gear ratios), we can calculate the **wheel torque** and **force** for each gear. Let’s calculate the **wheel torque** and force for a vehicle with the following parameters:

- engine torque, T
_{e}= 150 Nm - wheel radius, r
_{w}= 0.33965 m

The gearbox is automatic (ZF6HP26), with the following gear ratios and final drive ratio.

Gear # | Gear ratio symbol | Gear ratio |

1 | i_{1} | 4.171 |

2 | i_{2} | 2.340 |

3 | i_{3} | 1.521 |

4 | i_{4} | 1.143 |

5 | i_{5} | 0.867 |

6 | i_{6} | 0.691 |

Final drive | i_{0} | 3.460 |

To speed up calculations, we can use a Scilab script.

clc// Input dataTe = 150;ix = [4.171 2.340 1.521 1.143 0.867 0.691];i0 = 3.460;rw = 0.33965;nw = 2;// Wheel torque and force calculationTw = (ix .* i0 .* Te)/nw;Fw = Tw ./ rw;// Display resultsmprintf("\n%s\t\t%s\t\t%s\t\t%s\n","Gear","ix [-]","Tw [Nm]","Fw [N]")for i=1:length(ix) mprintf("%d\t\t%.3f\t\t%.2f\t\t%.2f\n",i,ix(i),Tw(i),Fw(i));end

Executing the above script will output the following results in the Scilab console:

Gearix [-]Tw [Nm]Fw [N]14.1711082.373186.7322.340607.231787.8131.521394.701162.0841.143296.61873.2850.867224.99662.4160.691179.31527.94

**Example 3**. For our third example we are going to use the full load torque curve of an engine and calculate the **wheel torque and force** (traction) in each gear. Calculate the wheel torque and force (traction) for a vehicle with the following parameters:

- engine torque, T
_{e}= 150 Nm - wheel radius, r
_{w}= 0.33965 m - the gear ratios of ZF6HP26 (see
**Example 2**)

The engine torque at full load is given by the following parameters:

N_{e} [rpm] | 800 | 1312 | 1800 | 2276 | 2800 | 3316 | 3806 | 4300 | 4770 | 5300 | 5800 | 6300 |

T_{e} [Nm] | 116 | 135 | 148 | 157 | 165 | 172 | 178 | 184 | 188 | 187 | 183 | 171 |

where N_{e} is **engine speed** and T_{e} is **engine torque**.

The graphical representation of the engine speed and torque points is depicted in the image below.

Image: Engine torque at full load function of engine speed

Since we need to perform a lot of calculations, we’ll use a Scilab script to calculate the **wheel torque** and force curves for each gear. The results are going to be plotted in a graphical window.

clc// Input dataNe = [800 1312 1800 2276 2800 3316 3806 4300 4770 5300 5800 6300];Te = [116 135 148 157 165 172 178 184 188 187 183 171];ix = [4.171 2.340 1.521 1.143 0.867 0.691];i0 = 3.460;rw = 0.33965;nw = 2;// Plot engine torquefigure(1)hf = gcf();hf.background = 8;plot(Ne,Te,"LineWidth",2)xgrid()xlabel("Engine speed [rpm]","FontSize",3)ylabel("Engine torque [Nm]","FontSize",3)title("x-engineer.org","Color","blue","FontSize",2)// Calculate wheel torque and forcefor i = 1:length(ix) for j = 1:length(Te) Tw(i,j) = (ix(i) .* i0 .* Te(j))/nw; Fw(i,j) = Tw(i,j) ./ rw; endend// Plot wheel torque and forcefigure(2)hf = gcf();hf.background = 8;plot(Ne,Tw,"LineWidth",2)xgrid()xlabel("Engine speed [rpm]","FontSize",3)ylabel("Wheel torque [Nm]","FontSize",3)title("x-engineer.org","Color","blue","FontSize",2)legend("1st gear","2nd gear","3rd gear","4th gear","5th gear","6th gear",2)figure(3)hf = gcf();hf.background = 8;plot(Ne,Fw,"LineWidth",2)xgrid()xlabel("Engine speed [rpm]","FontSize",3)ylabel("Wheel force [N]","FontSize",3)title("x-engineer.org","Color","blue","FontSize",2)legend("1st gear","2nd gear","3rd gear","4th gear","5th gear","6th gear",2)

Executing the script will output the following graphical windows.

Image: Wheel torque at full load function of engine speed and gear

Image: Wheel force at full load function of engine speed and gear

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

You can also check your results using the calculator below.

### Wheel torque calculator

T_{e} [Nm] | i_{x} [-] | i_{0} [-] | r_{w} [m] | n_{w} [-] |

T_{w} [Nm] = | ||||

F_{w} [N] = |

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