Question

A single-plate clutch has a friction disc with inner and outer radii of 20 mm and 40 mm, respectively. The friction lining in the disc is made in such a way that the coefficient of friction μ varies radially as μ = 0.01r, where r is in the mm. The clutch needs to transmit a friction torque of 18.85 kN-mm. As per uniform pressure theory, the pressure (in MPa) on the disc is ________.

Answer 1

Concept:

Total frictional force acting in a clutch:

\(W=2\pi\int_{r_i}^{r_o}pr\ dr\)

Total torque carrying capacity in a clutch:

\(T=2\pi μ\int_{r_i}^{r_o}pr^2\ dr\)

Assuming uniform pressure theory (p = constant)

\(W=\pi p(r_o^2\;-\;r_i^2)\)

In case of a new clutch, the intensity of pressure is approximately uniform, but in an old clutch, the uniform wear theory is more approximate.

Calculation:

Given:

W = p(2 πr)dr

Torque dT = μWr

dT = μP.2πr dr.r

\(T = \mathop \smallint \limits_{{r_i}}^{{r_0}} 0.01r.P.2\pi {r^2}dr\)

\(T = 0.01 \times 2\pi \times P\left[ {\frac{{r_0^4 - r_i^4}}{4}} \right]\)

\(18.85 \times {10^3} = \frac{{0.01 \times 2\pi \times P}}{4}\left( {{{40}^4} - {{20}^4}} \right)\)

P = 0.5 N/mm² = 0.5 MPa