Springs and clutches: storing energy and controlling drive

Springs and clutches look unrelated, but they work together in many machines. A spring stores energy by deflecting and gives it back, keeps parts pressed together, and absorbs shock. A clutch connects and disconnects a drive while it is running, and in most clutches it is a spring that provides the clamping force.

Springs

Types of spring

Spring rate

Within its working range, a spring’s deflection is proportional to the force (Hooke’s law):

$$F = k\,x, \qquad k = \frac{G\,d^4}{8\,D^3\,n}$$

The rate depends on the wire diameter to the fourth power, so a small change in wire size changes the spring a lot. Springs in series (one after the other) have a combined rate $1/k = 1/k_1 + 1/k_2$; springs in parallel (side by side) add: $k = k_1 + k_2$.

Spring stress

The wire of a helical spring is mainly twisted, so the stress is shear:

$$\tau = K_w\,\frac{8\,F\,D}{\pi\,d^3}, \qquad K_w = \frac{4C - 1}{4C - 4} + \frac{0.615}{C}, \qquad C = \frac{D}{d}$$

$K_w$ is the Wahl factor, which adds the extra stress on the inside of the coil caused by its curvature. The spring index $C$ is usually kept between about 4 and 12: below 4 the spring is hard to coil, above 12 it tangles and buckles.

Spring materials

Music wire and other cold-drawn carbon spring steels are the cheapest and strongest for small springs. Oil-tempered and chrome–silicon wires are used for larger springs and higher temperatures, such as engine valve springs. Stainless steel (302, 17-7PH) is used where corrosion matters, and phosphor bronze or beryllium copper where the spring also conducts electricity.

Clutches

A clutch lets a driving shaft (such as an engine or motor) connect to and disconnect from a driven shaft while it is turning.

How a single-plate clutch works

When engaged, springs (usually a single diaphragm spring) push the pressure plate towards the flywheel, clamping the friction disc between them. Friction on both faces of the disc carries the engine torque to the disc, whose hub is splined to the gearbox input shaft. Pressing the pedal pushes the release bearing against the spring fingers, the pressure plate moves back, and the disc spins freely.

Clutch torque capacity

Using the uniform-wear assumption, which suits a clutch that has bedded in:

$$T = \mu\,F\,R_m\,N, \qquad R_m = \frac{R_o + R_i}{2}$$

The clutch is usually sized to carry 1.2 to 1.5 times the engine’s maximum torque, so it doesn’t slip in normal use but still slips before something more expensive breaks.