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.

For any questions, observations and queries regarding this article, use the comment form below.

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## AJ

Hey, could you please explain the purpose of the elementary area?

Thanks in advance.

## Alan Thomas

Robert, yes, it happens when the rear crankshaft seal breaks down. Seriously, you could but it is unlikely the friction material will like the environment. Lube effect could cause slipping withthe same clamping force; it could swell, causing drag when it should be open; it could become spongy and lead to judder on engagement or disengagement, and slight shunting on torque reversal, also acting as a torsional damper which could either be nice or awful. On the other hand, if the lining tolerates the oil, and the housing doesn’t leak is the only concern I can see is churning losses and extra weight.

## Robert

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