Four-bar mechanisms and cam-follower systems explained

A mechanism turns one kind of motion into another: rotation into swinging, rotation into straight-line movement, or a steady turn into a precisely timed lift. Linkages and cams are the two oldest ways of doing it, and they are still inside car engines, wiper systems, packaging machines, folding furniture and excavators.

Links, joints and degrees of freedom

A linkage is made of rigid links connected by joints: pin (revolute) joints that allow rotation and sliding (prismatic) joints that allow straight-line motion. One link is fixed: the frame or ground. The number of independent inputs needed to drive the mechanism is its degree of freedom (DOF), given for planar linkages by the Gruebler–Kutzbach equation:

$$F = 3\,(n - 1) - 2\,j_1 - j_2$$

where $n$ is the number of links (including the ground), $j_1$ the number of full joints (pins and sliders) and $j_2$ the number of half joints (rolling with slipping, such as a cam contact).

A result of 0 means a rigid structure; a negative result means it is over-constrained; 2 or more means you need more than one input (or the mechanism will flop around).

The four-bar linkage

Four links, four pins: the simplest mechanism with one degree of freedom, and the most widely used. The link fixed to the frame is the ground, the link driven by the motor is the input (often a crank), the link opposite the ground is the coupler, and the remaining link is the output (often a rocker).

The Grashof condition

Whether any link can turn all the way round depends only on the lengths. Call the shortest link $s$, the longest $l$ and the other two $p$ and $q$.

$$s + l \le p + q$$

If this holds (a Grashof linkage), at least one link can rotate fully. Which kind of mechanism you get then depends on which link is fixed:

If $s + l > p + q$, no link can rotate fully: every link only rocks (a triple rocker).

The slider-crank

Replace the rocker of a four-bar with a slider moving in a straight line and you have the slider-crank, the mechanism in every piston engine, compressor and reciprocating pump. The crank radius $r$ sets the stroke ($2r$), and the connecting rod length $l$ shapes the motion. The piston’s distance from the crank centre is:

$$x = r\cos\theta + \sqrt{l^2 - r^2 \sin^2\theta}$$

Where four-bar linkages are used

Windscreen wipers (crank–rocker), excavator and loader arms, car suspensions (double wishbone), toggle clamps, folding tables and chairs, bicycle suspension, door closers, oil-well pump jacks and the pantograph on trains.

Cams and followers

A cam is a shaped piece that pushes a follower through a planned motion as it turns. Where a linkage gives whatever motion its lengths produce, a cam can give almost any motion you design into its profile, including dwells where the follower stays still.

Cam and follower types

Describing the motion

A cam’s motion is drawn as a displacement diagram: follower lift against cam angle, usually split into rise, dwell, return and dwell. The shape of each rise matters:

• Constant velocity: simple, but the sudden start and stop gives infinite acceleration in theory, causing shocks and noise.
• Simple harmonic: smooth velocity, but acceleration still jumps at the ends of a dwell.
• Cycloidal: acceleration starts and ends at zero; the smoothest of the three and the usual choice for high-speed cams.

For a translating roller follower, keep the pressure angle (the angle between the follower’s direction and the contact force) below about 30°, otherwise the follower jams sideways in its guide. A larger base circle reduces the pressure angle at the cost of a bigger cam.

Where cams are used

Engine camshafts opening the valves, packaging and printing machines, automatic lathes, sewing machines, door locks and toys. Many cam functions are now done by servo motors, but cams are still the cheapest and most reliable way to repeat the same motion millions of times at speed.

Designing mechanisms in CAD

A CAD assembly with correctly defined mates is a working model of the mechanism: you can drive the crank and watch the coupler, check for collisions and measure the output angle at every position. Motion studies add velocities, accelerations and forces, and they show early whether a linkage reaches a toggle (dead) position where it locks or loses control.