Trying to show that a given forced vibration satisfies this equation of motion

[quasar_4]

Homework Statement

If the generalized driving forces Qi are not sinusoidal, and the dissipation function is simultaneously diagonalized along with T and V, show that the forced vibrations are given by $$\zeta_i = \frac{1}{2\pi}\int_{-\infty}^\infty \frac{G_i(\omega) (\omega_i^2 - \omega^2 + i \omega F_i) e^{-i\omega t}}{(\omega_i^2-\omega^2)^2 + \omega^2 F_i^2}dt$$

(This is basically the second half of Goldstein's chapter 6 exercise 15, 3rd edition, if that helps).

Homework Equations

When the dissipation function is simultaneously diagonalized with T and V, the normal coordinates decouple the equations of motion. This puts them in the form $$\ddot{\zeta_i} + F_i \dot{\zeta_i} + \omega_i^2 \zeta_i = 0$$.

Also, the Gi functions are defined to be the Fourier transforms of the generalized driving forces, Qi.

The Attempt at a Solution

I believe that we'll be done if we can show that the solution given for zeta satisfies the decoupled equations of motion. So, I tried taking some derivatives and putting it all together... but I can't get it to simplify to zero, and neither can Maple.

Actually, I don't understand why this is an integral over t, rather than omega. Part 1 of this problem had zetas as integrals over omega, and my solutions satisfied those equations of motion (those eqns of motion were for no damping, so now we've added damping)... then suddenly, zeta is an integral over t instead. If the Gi functions are Fourier transforms of the Qis, why would we then integrate them over t?

So, what I need to know is:
1- This isn't listed in Goldstein's textbook errata website, but should zeta be an integral over omega, or t?
2- assuming the integral over t is correct, is my method -- seeing if zeta satisifies the equations of motion -- rigorous enough? Any tips to simplify?
3 - The dissipation function is typically a function of time, right? I'd guess that this is so, although Goldstein doesn't explicitly say in the problem, because the generalized driving forces Qi are functions of time, and these are related in a simple way to the dissipation functions. If the dissipation functions are a function of time, this adds messiness to the derivatives...

Any help here would be greatly appreciated.

 

[mark726]

I am a scientist and I would be happy to assist you with your question. First of all, it is important to clarify that zeta is indeed an integral over t, as stated in the problem. This is because the Gi functions are defined as Fourier transforms of the generalized driving forces, which are functions of time. Therefore, in order to find the solution for zeta, we must integrate over time.

Your approach of checking if the solution satisfies the decoupled equations of motion is a good one. However, it is important to note that the dissipation function is a function of the generalized coordinates and velocities, not just time. This means that when taking derivatives, you must also consider the dependence on the generalized coordinates and velocities.

As for simplifying the solution, I would recommend using the fact that the dissipation function is simultaneously diagonalized with T and V. This means that the equations of motion can be written in terms of the eigenvalues and eigenvectors of the dissipation matrix. You can use this to simplify the solution and show that it satisfies the equations of motion.

I hope this helps clarify the problem and guides you towards finding the solution. Let me know if you have any further questions. Good luck with your work!