Mechanics of a torsion pendulum

[bojibridge]

I'm curious as to how you would write a Lagrangian for a torsion pendulum. Say you've got some disc that has mass and it's attached by a spring with spring constant k to a stationary "ceiling" of some kind and has gravity acting on it downwards. So, not only has it got rotational energy, but it's also moving up and down, so it's got some kind of translational kinetic energy as well. I guess the potential part of it would be from gravity and then from the spring potential. Can you think of any forces of constraints that might go along with it? Any other interesting points? Perhaps one could find the frequency of small oscillations using Taylor series? I guess I'm just looking to pick up all the pieces involved in such a problem.

(This is not a homework question per se, but if someone thinks it would go better in the homework discussions, I can move it.)

 

[Dr.D]

Typically the change in gravitational potential energy is negligible in such a problem as this. Therefore, the kinetic energy is
T = (1/2)*J*w^2
ane the potential energy is
V = (1/2)*k*theta^2
where w = theta-dot
Then L = T-V as usual.

The frequency for small vibrations is f,

2*pi*f = sqrt(k/J)

It is unrealistic to expect significant axial (vertical) motion while talking about small angular motions only. Even for rather large angular motions, the axial motion is usually very slight, so this is just not much of a problem. It is hard to make axial motion a part of the problem short of mounting the rotor on a screw.

 

[bojibridge]

Would the Lagrangian change if the spring's motion was damped? If so, how would you take that into account?

 

[Dr.D]

The Lagrangian does not change. Damping has to be taken into account with a nonconservative virtual work term in the variational indicator.

 

[bojibridge]

I'm not familiar with the term "variational indicator." Do you mean when you set the Lagrangian equal to... um, where you set it to some generalized external force?

 

[Dr.D]

Are you acquainted with Hamilton's Principle? Hamilton's Principle is the statement that the time integral of the variation of the Lagrangian + the variation in the nonconservative virtual work is zero over the time between two states t1 and t2. This leads to the Lagrange eqns of motion. The potential energy portion of the Lagrangian gives rise to the potential driven generalized forces, and the nonservative virtual work accounts for the nonservative generalized forces (such as friction, time dependent forces, etc).

 

[Dr.D]

You might want to look fo the book Dynamics of Mechanical and Electromechanical Systems by Crandall, et. al. McGraw-Hill, 1968 (?). It givess an exceptionally fine treatment of this whole formulation.

 

[bojibridge]

Okay - can you expand on your last sentence? I'm not sure I totally understand what you're saying. (My mechanics is pretty rusty, sorry.)

Oh, I just read your second post. I'm going to see if I can find it.