Damped and Driven Oscillation of a Bridge

[Canuck156]

Sorry that I had to use an image file, I was having a lot of trouble using the Latex system.

http://www.flamingice.5gigs.com/Question.gif

Ok... We know the amplitude of the oscillations, and the force per person, so all we need to do is find Fmax, by finding other values and substituting into the equation for amplitude.

By using the equation for oscillation of an underdamped system I have found that b=m/3T, where T is the period of the undriven, undamped system, but I'm not sure how to find T in order to get a numerical value for b. I could find it if I knew k, the force constant of the system, but I have no idea how to find that.

So, I guess that my question is, how do I find the k of the system? If I know that, I think I'll be able to solve the rest of the problem.

Thanks!

 

[HallsofIvy]

Well, your quote says: "Take the damping constant to be such that the amplitude of the undriven oscillations would decay to 1/e of its original value in a time t= 6T where T is the period of the undriven, undamped system."

The "undriven, undamped" system has equation of the form my"+\omega_n = 0 and so has solution of the form y= A cos(\omega)+ B sin(\omega). What is \omega for that system? What is its period, T? What it 6T? What must k be so that the solution to my"- ky'+ \omega_ny= 0 reduces to 1/e of its original value in time 6T?

 

[Canuck156]

I can see what you mean (I think ). If I knew \omega_n I would be able to solve the problem. However, I'm not sure how I should go about finding \omega_n. In the physics course that I'm taking we are not really expected to be able to solve differential equations.

Thanks!

 

[Canuck156]

Anybody got any ideas?

 

[Canuck156]

I've solved it now... I was supposed to assume that since resonance was occurring, the natural frequency was approximately equal to the frequency of the driving force.