Mechanism Design Explorer

The degrees of freedom (DOF) of a planar mechanism tells you how many independent inputs are needed to fully define its position.

DOF = 3(n - 1) - 2j₁ - j₂

Here n is the number of links (including the ground), j₁ is the number of full joints (revolute or prismatic), and j₂ is the number of half joints (rolling or sliding contact). A four-bar linkage has 4 links and 4 revolute joints, giving DOF = 3(3) - 2(4) = 1.

Grashof's law determines whether at least one link in a four-bar mechanism can make a full 360-degree rotation.

s + l ≤ p + q

Where s is the shortest link, l is the longest, and p, q are the remaining two. When satisfied, the shortest link can fully rotate. The mechanism type (crank-rocker, double-crank, or double-rocker) depends on which link is shortest relative to the ground link.

The transmission angle (μ) is the angle between the coupler link and the output (follower) link. It measures how effectively force is transmitted through the mechanism.

An ideal transmission angle is 90 degrees, where force transfer is most efficient. As μ approaches 0 or 180 degrees, the mechanism reaches a toggle position where force transfer drops to zero.

In practice, transmission angles should stay between 40 and 140 degrees for reliable operation. The explorer shows a quality bar to help you evaluate the current angle.

Harmonic motion uses y = (h/2)(1 - cos(πt)), giving smooth displacement but non-zero acceleration at the start and end of the rise. This produces finite jerk but can cause vibration at high speeds.

Cycloidal motion uses y = h(t - sin(2πt)/(2π)), which starts and ends with zero velocity and zero acceleration. This is the smoothest standard profile and is preferred for high-speed applications.

Polynomial (3-4-5) motion uses y = h(10t³ - 15t⁴ + 6t⁵), giving zero velocity and acceleration at endpoints, similar to cycloidal but with a different acceleration shape. Used when specific boundary conditions must be met.