Simuveliq

Planar mechanism simulator

Build linkages with pins, sliders, cams and gears, add a motor and watch them move. The tool computes the degrees of freedom (Grübler), the Grashof condition, the transmission angle, the time ratio and the position, velocity and acceleration of every point over a cycle, plus the input torque.

What you can do

Frequently asked questions

What is the Grashof condition?

In a four-bar linkage, if s + l ≤ p + q (shortest plus longest does not exceed the other two), at least one link can rotate fully.

How are the degrees of freedom counted?

With Grübler's equation: M = 3(n − 1) − 2j₁ − j₂.

Guide: analysis of planar mechanisms

A mechanism turns an input motion (usually a motor's rotation) into an output motion: the rocker of a windshield wiper, the piston of an engine, the ram of a shaper. Analyzing it means answering how many inputs it needs, whether the crank can make full turns and where each point is and how fast it moves. This guide covers those questions with the four-bar linkage.

Degrees of freedom (Grübler–Kutzbach)

M = 3(n − 1) − 2j1 − j2

n is the number of links (counting ground), j1 the number of lower pairs (pins and sliders, which remove 2 degrees of freedom) and j2 the number of higher pairs (cams, tooth contacts, which remove 1). M is the number of independent inputs the mechanism needs. If k links share one pin, it counts as k − 1 joints.

Grashof condition

In a four-bar linkage, with s the shortest link, l the longest and p, q the other two: if s + l ≤ p + q, at least one link can make full turns (Grashof). If, in addition, the shortest is the input crank, the linkage is a crank–rocker: the input turns fully and the output oscillates. If s + l > p + q, no link makes full turns.

Transmission angle

It is the angle between the coupler and the output link. The further it is from 90°, the worse force is transmitted; as a rule of thumb it should stay between about 40° and 140°.

Worked example

A four-bar with ground AB = 100 mm, crank AC = 40 mm, coupler CD = 120 mm and rocker BD = 80 mm, with the crank at θ2 = 60° turning at 60 rpm (ω2 = 2π rad/s). It is the linkage the tool opens with.

  1. Degrees of freedom: n = 4 and j1 = 4: M = 3·3 − 2·4 = 1, so one motor is enough.
  2. Grashof: s + l = 40 + 120 = 160 ≤ p + q = 100 + 80 = 180, and the shortest link is the crank: crank–rocker.
  3. Position: C = (40 cos 60°, 40 sin 60°) = (20, 34.64) mm. D is 120 mm from C and 80 mm from B (intersection of two circles): D ≈ (133.9, 72.47) mm. From there θ3 ≈ 18.38° and θ4 ≈ 64.94°.
  4. Velocities: differentiating the vector loop gives ω3 = ω2·r2·sin(θ4 − θ2)/(r3·sin(θ3 − θ4)) ≈ −0.249 rad/s and ω4 = ω2·r2·sin(θ2 − θ3)/(r4·sin(θ4 − θ3)) ≈ 2.874 rad/s. The speed of C is ω2·40 ≈ 251.3 mm/s.
  5. Transmission angle: by the law of cosines at the extreme positions, it ranges from ≈ 26.4° (θ2 = 0°) to ≈ 86.4° (θ2 = 180°). It drops below 40°, so there are positions where force is poorly transmitted.

The tool gives the same values at that position and shows them over the whole cycle.

How to use the simulator

  1. Draw the links, ground pivots, sliders, gears or cams, or load an example: four-bar, slider–crank, quick return, six-bar, cam with roller follower, planetary train…
  2. Define the inputs (motor or linear actuator) with their speed and direction; the simulator checks that their number matches the degrees of freedom.
  3. Animate the mechanism, draw paths and read the positions, velocities and accelerations of links and points, the input torque and the joint forces (with masses, inertias and gravity if you define them).

Common mistakes

Further reading

R. L. Norton, Design of Machinery (McGraw-Hill); J. J. Uicker, G. R. Pennock and J. E. Shigley, Theory of Machines and Mechanisms (Oxford University Press). Simuveliq is not affiliated with these authors or publishers.

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