Linkages¶
Initial draft of course notes regarding linkages - this is a work in progress.
A linkage is a system of rigid parts connected by pivoting or sliding joints which conveys and transforms motion. These notes are intended to address some practical concerns involved in creating laser-cut linkage parts to use in your projects. The full history and theory of linkages is a much broader topic and beyond the scope of this page.
Four-Bar Linkages¶
Our most common form of linkage is a four-bar linkage which includes four rigid elements connected by four pivot joints with parallel axis. The links move in parallel planes with different types of trajectories depending on the link lengths. The elements are often long slender bars but very often one of the links is formed by the structure or chassis of a machine.
With these constraints, the resulting mechanism has only one degree of freedom, i.e., moving one joint a small displacement will move all the joints by a ratio which varies with the specific pose. There are complications, since in some singular poses the ratio may become zero or infinity, and some linkages can move between different configurations as the pose crosses a singularity.
The principal design problem is choosing the link lengths and nominal configuration which produces the motion of interest. In many cases, a particular link is identified as the input link and another as the output, so the desired motion is not just a path but a transfer function which implies the relative velocity of input and output.
Common Linkage Examples¶
Practical Problems¶
We can make links on the laser-cutter from flat stock, but there are some practical problems to consider.
Self-collision. A schematic diagram of a linkage does not always capture that the parts have volume and need to move past each other and not collide with the pivot pins. In practice, links need to be considered as moving in parallel planes and the full range of travel evaluated for striking other links or the pivot components.
Pivots and bearings. The connections between links need to move smoothly with some precision. Sometimes a clevis pin inserted through a wood hole is sufficient, but that can be improved by adding bushing or ball bearings. In either case, a pin which connects two parallel links is subject to a moment caused by the unbalanced bearing forces. With thin links close together this moment may be supported by the bearings, but sometimes links need to structured as two parallel plates to contact the shaft at two points for better moment transfer.
Actuator connection. A gearmotor-driven linkage system involves coupling shaft torque into a link. This usually involves a metal hub which can transfer the high shear forces of a small shaft to a larger radius. Servo-driven linkages can instead use a tie rod to form a secondary linkage. This avoids the high shear stresses but does not allow continuous rotation.