What you can learn here
Build your own mechanism
In the Builder you draw bars, plates, fixed pivots and sliders, pick a motor and the lab animates it. Before moving it, the lab counts its degrees of freedom with Grübler’s formula, M = 3(n − 1) − 2j₁ − j₂: with 1 DOF one motor is enough, with 0 you have a structure and with 2 you need another motor. It also checks the result numerically, because there are special geometries where the formula gets it wrong.
Four-bar linkages and Grashof’s rule
If the shortest bar plus the longest add up to less than the other two (s + l < p + q), at least one bar can make full turns. Which one turns depends on which bar is fixed: that gives the crank-rocker of a windscreen wiper, the double crank or the double rocker. The transmission angle tells you how well the coupler pushes; below about 40° the mechanism jams or vibrates.
Slider-crank and quick return
It is the heart of engines, compressors and pumps: it turns rotation into reciprocating motion. If the guide is offset, the forward and return strokes take different times, which is the principle of quick-return shapers and saws. The lab computes the stroke and the time ratio, and shows that the piston’s acceleration is not a pure sine wave.
How velocities and accelerations are calculated
Every mechanism is a closed vector loop. For the four-bar, the position comes from Freudenstein’s equation; in the Builder, the constraints form a system Φ(q) = 0 solved with Newton–Raphson from the previous position. Differentiating that system gives the velocities and accelerations of any point. All formulas are checked against finite differences in autotest mode.
Instant centers and polygons
At every instant, two links rotate relative to each other about an instant center of rotation. A mechanism with n links has n(n − 1)/2 centers, and Kennedy’s theorem says that the three centers of any three links lie on a straight line. The velocity polygon draws the velocities of all points from a single pole: the image of each bar is its shape rotated 90° and scaled by ω. The Builder draws the centers, the velocity and acceleration polygons, and the vectors at every point.
Forces and motor torque (inverse dynamics)
If you know the motion, you can work out the forces that produce it. Each bar gets a mass, a center of mass and an inertia; weight, springs, dampers and external loads are added, and Φqᵀλ = Qext − M·q̈ is solved. The multipliers λ are the pin forces and the torque the motor needs. With inertia switched off the calculation is pure statics. The lab checks that the motor power equals the rate of change of energy.
Synthesis: designing the mechanism backwards
Analysis starts from the mechanism and computes its motion; synthesis starts from the desired motion. With Freudenstein’s equation, K₁ cos ψ − K₂ cos φ + K₃ = cos(φ − ψ), a four-bar can “compute” a function at three precision positions (Chebyshev spacing keeps the error to a minimum). With the dyad method, the coupler passes through three positions (motion generation) or three points with prescribed timing (path generation). The result opens in the Builder for analysis.
Cams: motion laws
A disc cam forces the follower to rise, dwell and return according to a motion law. The harmonic law is smooth in position, but its acceleration jumps abruptly at the start and end, giving infinite jerk. The cycloidal and 3-4-5 polynomial laws start and end with zero acceleration, which is why they are used in fast machines such as engine camshafts.
Cams: pressure angle and undercutting
With a roller follower, the pressure angle is the angle between the contact force and the direction the follower moves. Above about 30°, the force pushes the stem sideways and the follower jams in its guide. The fix is a larger base circle or a slower rise. With a flat-faced follower the risk is different: if the profile’s radius of curvature reaches zero, the cam can’t be manufactured.
Gears: ratio, speed and torque
In a gear pair, speeds are inversely proportional to the tooth counts: ω₂ = ω₁·Z₁/Z₂, and torque rises in the same proportion as speed falls, minus the losses of each mesh. In a simple train, intermediate (idler) gears only change the direction of rotation, not the ratio. See also the gear ratio calculator.
Compound and planetary trains
With two gears on the same shaft, the ratios multiply: that is how a large reduction fits in a small space. In a planetary train, the sun, planets, ring and carrier obey Willis’s formula, (ωsun − ωcarrier)/(ωring − ωcarrier) = −Zring/Zsun. Depending on which member is held, you get a reduction, an overdrive or a reverse gear, as in automatic gearboxes.
Export your curves
In any mode, “Download CSV” saves position, velocity and acceleration against angle and time, ready to open in a spreadsheet or in MATLAB, Python or Octave. “Download PNG” saves the chart as an image for your report.
Keep learning with the angular velocity calculator, the torque and power calculator, the Robot Lab or the Mechanical Engineering path.