- #1

- 6

- 0

## Main Question or Discussion Point

I'd like to understand how the double rod pendulum works and how it changes when a point mass is added to one of the rods. Especially I'd like to understand whether it's possible to match the two rods with the point mass so that the movement is stable under sinusoidal applied angular acceleration.

The problem set is this: Assume two rods of masses ##M_1##/##M_2## and lengths ##L_1##/##L_2## attached together by a frictionless hinge. The mass is distributed evenly on the two rods. The motion is restricted to 2D by putting the rods on a frictionless table. The other end of the first rod is attached to a perfectly secured lossless attachment point. The question is: If one then applies a sinusoidal angular acceleration ##\alpha(t) = \alpha_0 \sin(\omega t)## at the attachment point to the first rod, what is the condition under which the second rod tends to follow the first rod? By following I mean that the rods either keep aligned on the same line or the rods both move periodically with the second rod only moving over a larger range of angles, but without flipping? Can the point mass be used on either or both of the rods to match the rods to the applied acceleration in this regard? If yes, where should the point mass(es) be added? And what is the logic how the double pendulum changes when the point mass is added to: 1) tip 2) tail 3) center of the first or the second rod?

How does the situation change when the applied angular acceleration is not sinusoidal, but a short burst of acceleration? Can the rods be forced to follow each other in this case?

My own intuition says that the moments of inertia and the centers of masses of the two rods are key parameters here, but I cannot understand what is the logic behind the movement, i.e what part of the movement slows down when mass is added to any of the three points on the two rods.

The problem set is this: Assume two rods of masses ##M_1##/##M_2## and lengths ##L_1##/##L_2## attached together by a frictionless hinge. The mass is distributed evenly on the two rods. The motion is restricted to 2D by putting the rods on a frictionless table. The other end of the first rod is attached to a perfectly secured lossless attachment point. The question is: If one then applies a sinusoidal angular acceleration ##\alpha(t) = \alpha_0 \sin(\omega t)## at the attachment point to the first rod, what is the condition under which the second rod tends to follow the first rod? By following I mean that the rods either keep aligned on the same line or the rods both move periodically with the second rod only moving over a larger range of angles, but without flipping? Can the point mass be used on either or both of the rods to match the rods to the applied acceleration in this regard? If yes, where should the point mass(es) be added? And what is the logic how the double pendulum changes when the point mass is added to: 1) tip 2) tail 3) center of the first or the second rod?

How does the situation change when the applied angular acceleration is not sinusoidal, but a short burst of acceleration? Can the rods be forced to follow each other in this case?

My own intuition says that the moments of inertia and the centers of masses of the two rods are key parameters here, but I cannot understand what is the logic behind the movement, i.e what part of the movement slows down when mass is added to any of the three points on the two rods.