Modern automotive powertrain and drivetrain systems have at least one clutch as a component. An AWD or 4WD vehicle can have several clutches, depending on the architecture and type of the powertrain and drivetrain.

A clutch can be a standalone component, used to connect the internal combustion engine (ICE) to the transmission, or can be the sub-component or another main component: torque converter, automatic transmission, transfer case, limited-slip differential (LSD), etc.

Depending on the number of friction plates, a clutch can be:

**single plate clutch****multi-disc clutch**

Depending on the type of friction, we can have:

**dry clutches****wet (oil) clutches**

Usually, single plate clutches (except torque converter lock-up clutches) have dry friction and multi-disc clutches are with wet friction.

In operation, a clutch can have 4 operating states:

**open**, zero torque is transmitted between the input and the output shafts**slipping**, some amount of torque is transmitted between input and output shafts; the speed difference between input and output shaft is significant (e.g. 500 rpm)**micro-slipping**, almost all of the input torque is transmitted through the clutch; the speed difference between input and output shaft is very small, around 5-10 rpm**closed**(clamped, locked-up), there is no slip between input and output shaft, all the input torque is transmitted through the clutch

Regardless of the type, every clutch has a **torque capacity**. The torque capacity of the clutch is the amount of torque that can be transmitted by the clutch when it’s slipping or when it’s fully closed. The torque capacity of a clutch depends on a series of factors:

- total area of the friction surface
- friction coefficient
- normal force acting on the clutch
- number of friction elements

To calculate the torque capacity of the clutch we’ll have a look at the geometry of the clutch (friction) disc. Within the area of the friction surface we are going to represent an elementary area *dx*, at the distance *x* from the center.

where:

*F _{a} [N]* – the normal force pressing the clutch plate

*T*– the torque capacity of the clutch

_{c}[Nm]*r*– the inner radius of the friction surface

_{1}[m]*r*– the outter radius of the friction surface

_{2}[m]The pressure *p [Pa]* acting on the clutch is equal with the ratio between the normal force *F _{a}* and the area of the friction surface

*S [m*:

^{2}]Assuming that the area of the rivets is neglijable, the area of the friction surface is calculated as:

\[S = S_2 – S_1 = \pi r_2^2 – \pi r_1^2 = \pi (r_2^2 – r_1^2) \tag{2}\]Replacing (2) in (1), we get the expression for the clutch pressure:

\[p = \frac{F_a}{\pi (r_2^2 – r_1^2)} \tag{3}\]The elementary area *dA* is calculated as:

The elementary normal force *dN*, acting on the elementary area is calculated as:

Replacing (3) and (4) in (5), we get:

\[dN = \frac{2 F_a x dx}{r_2^2 – r_1^2} \tag{6}\]The elementary friction force *dF* is calculated as:

where *μ [-]* is the friction coefficient of the clutch disc.

Replacing (6) in (7), we get:

\[dF = \frac{2 \mu F_a x dx}{r_2^2 – r_1^2} \tag{8}\]The elementary friction torque *dT* is calculated as:

Replacing (8) in (9), we get:

\[dT =\frac{2 \mu F_a x^2 dx}{r_2^2 – r_1^2} \tag{10}\]Integrating equation (10) from *r _{1}* to

*r*, we get the mathematical expression of the torque capacity of the clutch:

_{2}T_c &= \int_{r_1}^{r_2} dT & \\

&= \frac{2 \mu F_a}{r_1^2 – r_2^2} \int_{r_1}^{r_2} x^2 dx \\

&= \frac{2}{3} \mu \frac{r_2^3 – r_1^3}{r_2^2 – r_1^2} F_a

\end{split} \end{equation*} \]

The resulting mathematical expression for the **torque capacity of a single plate clutch** is:

For a **multi-disc clutch** expression (11) becomes:

where *z [-]* is the number of friction plates (discs).

Assuming a mean radius *r _{m} [m]* of the clutch calculated as:

we can deduce a simplified expression, with an acceptable error, for the **torque capacity of the clutch**:

**Example 1**. Calculate the **torque capacity of single plate dry clutch**, which has: the normal force 250 N, the outer radius 0.3 m, the inner radius 0.2 m and the friction coefficient 0.4.

Replacing the given parameters in equation (11), we get:

\[T_c = \frac{2}{3} 0.4 \frac{0.3^3 – 0.2^3}{0.3^2 – 0.2^2} 250 = 25.3 \quad Nm\]**Example 2**. Calculate the **torque capacity of multi-disc wet clutch**, which has: 5 friction discs (plates), the normal force 250 N, the outer radius 0.3 m, the inner radius 0.2 m and the friction coefficient 0.07.

Replacing the given parameters in equation (12), we get:

\[T_c = 5 \frac{2}{3} 0.07 \frac{0.3^3 – 0.2^3}{0.3^2 – 0.2^2} 250 = 22.2 \quad Nm\]The same algorithm can be used to calculate the braking torque of a vehicle equipped with disc brake system.

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## Robert

Hi, Can a dry clutch be converted to a wet clutch.