This simulation brings the building blocks of mechanical engineering to life: gear trains, crank–slider mechanisms, four-bar linkages, epicyclic (planetary) gears, chain drives and cam-and-follower pairs. Each one is a kinematic chain — rigid links joined so that motion in one part rigidly determines motion in the others. The same principles drive everything from a bicycle and a car engine to automatic gearboxes and the camshafts that open engine valves. It is interesting because complex, useful motion emerges from a handful of simple geometric constraints.
ω₂/ω₁ = -N₁/N₂ — for a meshing gear pair, output speed equals input speed scaled by the inverse ratio of tooth counts N, with the sign showing reversed rotation. For epicyclic gearing the sun-to-carrier reduction is 1 + N_ring/N_sun, and a crank of radius r gives a piston displacement x = r·cosθ + √(L² − r²sin²θ) where L is the connecting-rod length.
The four-bar linkage is the simplest closed-loop mechanism, yet Grashof's 1883 criterion still tells engineers at a glance whether a bar can fully rotate. James Watt invented his "parallel motion" four-bar in 1784 to guide a piston in a straight line and was prouder of it than of any other invention.
Explore fundamental mechanical systems: gear trains, crank-slider mechanisms, chain drives and four-bar linkages. Adjust gear ratios and watch how torque and motion transfer through each mechanism.
Kinematic constraints connect rigid bodies. Gear teeth enforce velocity ratios, cranks convert rotation to linear motion, chains transmit power across distance.
Click to start/stop each mechanism. Adjust gear ratios, crank length and linkage geometry. Watch how changes propagate through the system.
The four-bar linkage is the simplest closed-loop mechanism and appears everywhere from car suspensions to robotic arms. It was first analysed by Grashof in 1883.
This simulation animates six fundamental machine elements from mechanical engineering: gear trains, the crank–slider used in piston engines, four-bar linkages, epicyclic (planetary) gears, chain drives and cam-and-follower pairs. Each is a kinematic chain — rigid links joined so that the motion of one part fully determines the motion of the others. You spin the input shaft at a chosen speed and watch motion propagate through the whole machine, with coupler traces, angular-velocity arcs and live readouts for input and output angular speed, gear ratio and efficiency updating every frame.
These six mechanisms are the building blocks of nearly all machinery. Gear trains and chain drives transmit rotation while trading speed for torque; the crank–slider converts rotation into the straight-line motion of a piston (and vice versa) at the heart of every internal-combustion engine; four-bar linkages guide complex paths in everything from car suspensions to robotic arms; epicyclic gears form the core of automatic gearboxes and bicycle hub gears; and cams translate a shaft's rotation into precisely timed lift, opening the valves in your engine. What makes them fascinating is how richly useful motion emerges from just a handful of simple geometric constraints.
What is a gear ratio and why does it matter?
A gear ratio is the ratio of the tooth counts (or radii) of two meshing gears, and it sets the trade-off between speed and torque. A pair where the driver has twice as many teeth as the driven gear doubles the output speed but halves the torque, following ω₂/ω₁ = −N₁/N₂; the minus sign shows the gears spin in opposite directions.
How does a crank–slider mechanism work?
A rotating crank is joined by a connecting rod to a slider (the piston) that can only move in a straight line. As the crank turns, the rod pushes and pulls the piston back and forth, converting rotary motion into reciprocating motion. The piston's position follows x = r·cosθ + √(L² − r²sin²θ), where r is the crank radius and L the rod length — this is the principle behind every piston engine and pump.
What is a four-bar linkage?
A four-bar linkage is the simplest closed-loop mechanism: four rigid bars connected by four pivots, with one bar fixed as the ground. Driving one bar makes the others trace precise, repeatable paths, which is why four-bars appear in car suspensions, windscreen wipers, folding chairs and robotic arms.
An epicyclic gearset has a central sun gear, several planet gears riding on a carrier, and an outer ring gear. By holding one member fixed and driving another, you can get large speed reductions in a compact, coaxial package — the sun-to-carrier reduction with a fixed ring is 1 + N_ring/N_sun. This is why planetary gears are used in automatic transmissions, bicycle hub gears and electric-drive reducers.
A cam is a specially shaped rotating disc; as it turns, its profile pushes a follower up and down in a precisely timed pattern called the lift curve. This converts steady rotation into a custom reciprocating motion, which is exactly how an engine's camshaft opens and closes its valves at the right moment in each cycle.
Where two external gears mesh, their teeth push against each other at the contact point, so one must turn clockwise while the other turns anticlockwise. That is why the gear-ratio equation carries a minus sign. Adding an idler gear in between reverses the direction again, letting engineers make the output spin the same way as the input when needed.
Grashof's 1883 criterion is a quick test that tells engineers whether a four-bar linkage can fully rotate. It states that if the sum of the shortest and longest bar lengths is less than or equal to the sum of the other two, at least one bar can make a complete revolution — vital for knowing whether a crank will spin freely or merely rock back and forth.
The coupler trace is the path drawn by a point on the moving connecting link as the mechanism runs. These curves can be surprisingly complex — figure-eights, ovals and even near-straight lines — and engineers exploit them to make a single linkage produce a specific desired motion without motors or controls.
Every real mechanism loses a little energy to friction at the teeth, pins and sliding surfaces, plus losses to lubricant and slight deformation. Gear and chain drives are very efficient (often 97–99% per stage), while sliding pairs like cams and pistons lose a bit more, which is why the readouts show realistic values just below 100%.
In 1784 James Watt devised his "parallel motion" four-bar linkage to guide the piston of his steam engine along an almost perfectly straight path without a rigid slide. He considered it one of his most ingenious inventions, and it is a famous early example of using linkage geometry to approximate straight-line motion.