Designing a Damped Pendulum with a Forcing Function for Timekeeping

[Crosson]

This was part of the final in my ODE class last semester:

"Design a pendulum to be used as a clock. It should include friction and a forcing function. "

Of course, by "design" I mean write the equation of motion. My class did not come up with very good answers, I would like to see what you guys come up with.

This was my solution (which I will explain):

m\ddot{x} + b\dot{x}^2 + mglSin(x) = C\delta (x - x_m_a_x)

Where m is the mass, b is the dampening constant and L is the length of the pendulum. C is the constant that took most of the work, finding the magnitude of the delta function.

 

[saltydog]

This was part of the final in my ODE class last semester:

"Design a pendulum to be used as a clock. It should include friction and a forcing function. "

Of course, by "design" I mean write the equation of motion. My class did not come up with very good answers, I would like to see what you guys come up with.

This was my solution (which I will explain):

m\ddot{x} + b\dot{x}^2 + Lx = C\delta (x - x_m_a_x)

Where m is the mass, b is the dampening constant and L is the length of the pendulum. C is the constant that took most of the work, finding the magnitude of the delta function.

May I ask why you didn't use the standard (non-linear) form:

ml^2\theta^{''}+b\theta^{'}+mgl\sin(\theta)=A\cos(w_d t)

And then solve it numerically?

 

[dextercioby]

As for friction,well,for small velocities,u can assume it to be linear in the modulus of the velocity vector...And why do you have that "delta-Dirac" in the RHS,is this supposed to be an equation for the Green function...??


Daniel.

 

[Crosson]

First, I had a mistake in my solution above, so check that out.

Yes Salty Dog, that is what everyone else in my class did. I think that is a bad answer to a good question because:

1) Trial and error numerics should be a last resort.

2) If you already have a forcing function capable of oscillating at any frequency, then why would you build a clock? i.e. my classmates submitted an applied math solution to a non-problem: how to build a clock using a clock.

Okay Dexter, here is how I approached the problem:

"A necessary but insufficient condition for the pendulum to tick regularly is that the frictional force and the forcing function together contribute zero work (over some interval)".

This is not necessary if the length of the pendulum varies, but in my solution it does not. I hope it is clear that I am saying: all the energy the frictional force takes away from B to A to B has to be put back by the forcing function by the time we get back to B.

To avoid the complexity of the forcing function being out a phase, I have it depend on theta (same force in the same spot, regardless of time). Furthermore, I make the physically realistic assumption that a hammer delivers the required impulse over an infinitesmally brief interval, hence the dirac-delta as a forcing function.

That is the conceptual approach, all that remains is to do the math to calculate the coefficient of the dirac-delta.

 

[saltydog]

Alright, how about this:

How did you solve the homogeneous case and can you report that solution?

 

[Crosson]

The equation of motion is the solution to the question. The difficulty is to design an equation of motion whos solution will be a regularly periodic function, without being able to solve the differential equation.