Developable Mechanisms Explained
What is a mechanism?
A mechanism is a mechanical device used to transfer or transform motion, force, or energy. Mechanisms typically use links and pin joints to achieve their motion.
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One example of a mechanism is vise grips. This mechanism transfers energy from the input (your hand) to the output (the gripper teeth). Since energy is conserved between the input and output (neglecting friction losses), the output force may be much larger than the input force, but the output displacement is much smaller than the input displacement. Another example is an umbrella. The input displacement (sliding the piece up the shaft) causes an output motion (opening up the umbrella).
Mechanisms are all around us. There are many examples in automobiles, sports equipment, furniture, construction equipment, robotics, and almost anything that has moving parts.
What is a developable surface?
A developable surface is a surface with curvature in a single direction. The four types of developable surfaces are planes, cylinders, cones, and tangent-developed surfaces.
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For each point on a 3D developable surface there is a straight line along the surface that passes through that point. This straight line is called a ruling line.
A good way to visualize developable surfaces is with a piece of paper. Any shape you make by bending the piece of paper (as long as it doesn't stretch or tear) is a developable surface. For example, you can roll the paper up into a cylinder or into a cone, but you can't make it into a sphere. A sphere is a non-developable surface because it curves in two directions (latitude and longitude).
Developable surfaces are all around us. They are used in clothing and shoe design, ship building, aerospace, origami, and many engineering applications.
A developable mechanism is a mechanism that, at some point in its motion, conforms to a developable surface. In other words, it is a device that can move and that is shaped to match a curved surface in a certain position. A simple example of a developable mechanism is shown below.
In the left image, the mechanism is conformed to the developable surface (in this case, a cone). In the right image the mechanism has been deployed off the surface.
Groundbreaking research was recently published on developable mechanisms in the Science Robotics academic journal.
To allow motion, the mechanism's joint axes (where the pin joints are) must be along the ruling lines on the developable surface. Developable mechanisms are possible on all types of developable surfaces. Below is an animation showing developable mechanisms on cylindrical, conical, and tangent developed surfaces.
Advantages of Developable Mechanisms
Developable mechanisms are useful because they can be compactly stored and they can exhibit predictable motion. This means we can mathematically model and plan their movement.
Another advantage is that they can be contained within (or placed on) developable surfaces. Developable surface are very common in engineering applications, but currently those surfaces are static and don't move.
Developable mechanisms can give increased functionality to common surfaces and can be compactly stored when not in use.
Challenges of Developable Mechanisms
Traditional mechanism design assumes that the shape of the links does not matter. But for developable mechanisms, the shape of the links does matter because the links must conform to the developable surface.
As noted above, another challenge is where the joint axes can be placed. The joint axes must be coincident with the ruling lines on the developable surface.