Mechanisms and machine elements
Gears, linkages, cams, belts and chains: the mechanical building blocks that turn a motor’s rotation into the motion a machine needs.
A motor does one thing: it spins. Yet machines push, lift, grip, cut and walk. The theory of machines explains how we get from one to the other, using a small set of building blocks: linkages, gears, cams, belts and chains. This post introduces each of them in plain words.
Motion is what makes a machine
A machine without motion is just a piece of metal. Motion defines what a machine can do, and the first question for any design is: how many independent ways can it move?
Degrees of freedom
A free object in space can move in six ways, its degrees of freedom (DOF):
- 3 translations: forward–backward, left–right, up–down
- 3 rotations: roll, pitch and yaw, about the X, Y and Z axes
Joints take freedoms away. A hinge (revolute joint) leaves only one rotation; a slider (prismatic joint) leaves only one translation. For a flat (planar) mechanism, the Grübler–Kutzbach formula counts what is left:
$$ M = 3(n - 1) - 2j_1 - j_2 $$
where \(n\) is the number of links (including the fixed frame), \(j_1\) the number of one-freedom joints (hinges, sliders) and \(j_2\) the number of two-freedom joints (such as a cam contact).
Example: the four-bar linkage. \(n = 4\) links, \(j_1 = 4\) hinges, \(j_2 = 0\): \(M = 3(4-1) - 2(4) = 1\). One motor is enough to drive the whole mechanism.
Linkages: shaping the motion
A linkage is a set of rigid bars connected by joints. The four-bar linkage is the most common: one bar is fixed, one is driven (the crank), and a point on the connecting bar (the coupler) traces a curve that can be straight-ish, oval or figure-eight shaped. Wipers, folding chairs, excavator arms and walking toys all use it.
Whether a link can turn all the way round is decided by the Grashof condition. With \(s\) the shortest link, \(l\) the longest and \(p, q\) the other two:
$$ s + l \le p + q \quad\Longrightarrow\quad \text{at least one link can rotate fully} $$
The slider-crank is the other classic: it turns rotation into back-and-forth motion (or the reverse). It is the heart of every piston engine, pump and many cutting machines.
Gears: trading speed for force
Two meshing gears change speed and torque. With \(N_1\) teeth on the driving gear and \(N_2\) on the driven gear, the gear ratio is
$$ i = \frac{N_2}{N_1} = \frac{\omega_1}{\omega_2} \approx \frac{T_2}{T_1} $$
Example. A motor with a 12-tooth gear drives a 48-tooth gear: \(i = 4\). The output turns 4 times slower but with (almost) 4 times the torque. Robots use this all the time, since small motors are fast but weak.
| Gear type | Typical use |
|---|---|
| Spur gears | Simple, parallel shafts |
| Bevel gears | Turning motion through 90° |
| Worm gear | Very large reduction; usually cannot be driven backwards, so it holds a load without power |
| Planetary gearbox | Compact, high reduction, common in robot joints |
| Harmonic drive | Very high reduction with almost no backlash, used in precise robot arms |
Cams: motion by shape
A cam is a shaped disc that pushes a follower as it turns. The cam's profile is the motion program: rise, hold, fall. Cams time the valves in car engines and drive packaging machines, where the same motion repeats millions of times.
Belts and chains: power over a distance
When the motor cannot sit right next to the joint, belts and chains carry the motion:
- Timing belts (toothed) do not slip, so they are used where position matters: 3D printers, CNC machines, robot axes.
- V-belts are quiet and absorb shocks, but can slip.
- Chains carry high loads, as on bicycles and conveyors.
The speed ratio follows the pulley or sprocket sizes, just like gears.
From mechanism to automated machine
To turn a mechanism into an automated machine, engineers usually go through these steps:
$$ \text{Structure and constraints} \;\rightarrow\; \text{Workspace} \;\rightarrow\; \text{Degrees of freedom} \;\rightarrow\; \text{Motion equations} \;\rightarrow\; \text{Control} $$
- Define the parts and how they are connected.
- Specify the workspace: where the machine has to reach.
- Count the degrees of freedom, which tells you how many motors you need.
- Derive the equations of motion (kinematics and dynamics).
- Use those equations in the control software.
See it in action
For real examples of these mechanisms, watch Mechanisms of machines on YouTube.