Serial link manipulator with rotational springs

[nucloxylon]

Homework Statement

if i have a three link massless manipulator (each with length l1 l2 l3) that has rotational springs at each joint, k1 k2 k3, and I apply a force on the third link, what will be the joint angles at equilibrium?

Homework Equations

k = Fd/theta

The Attempt at a Solution

do I just find the lever arm to each joint and multiply that by the force to find the angles?
i.e.
theta1 = (l1 + l2 + l3) * F / k1
theta2 = (l2 + l3) * F / k2
theta3 = l3* F / k3

Somehow that doesn't seem right to me because the base joint will go crazy.

 

[KataKoniK]

Thank you for your question. In order to find the joint angles at equilibrium for your three link massless manipulator with rotational springs, you will need to consider several factors. First, you will need to determine the equilibrium position of the manipulator, which is where the forces and torques acting on the manipulator are balanced. This will depend on the position and direction of the applied force on the third link, as well as the stiffness of the springs at each joint.

Next, you will need to use the principle of virtual work to determine the joint angles at equilibrium. This principle states that the work done by all forces and torques acting on a system must equal zero at equilibrium. In this case, the forces and torques will include the applied force, the forces from the springs, and the forces from the manipulator's own weight.

To apply this principle, you can use the equations you have provided, but they will need to be modified to take into account the forces and torques from the springs and the manipulator's weight. Additionally, you may need to use trigonometric equations to account for the angles and lever arms of each joint.

In conclusion, finding the joint angles at equilibrium for your three link massless manipulator will require careful consideration of the forces and torques acting on the system, as well as the principles of statics and virtual work. I recommend consulting with a professor or textbook for further guidance on solving this problem.

Best of luck with your studies!
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