conservation of energy in an undamped driven harmonic oscillator

[inkliing]

This isn't homework. I'm reviewing physics after many years of neglect.

 

Since a simple harmonic oscillator is a conservative system with no energy losses, then a driven undamped harmonic oscillator, once the transient solution has died out, can't be receiving any energy from the driving mechanism, else the system's M.E. would increase without bound.

 

But when a simple harmonic oscillator is driven at the system's natural frequency, the amplitude of the position, and therefore the system's M.E. increases without bound, which can't happen unless the system is receiving energy from the driving mechanism.

 

Specifically:

 

Let [latex]\omega_{\circ} = \sqrt{k/m}[/latex], driving function = [latex]f(t) = \frac{F_{\circ}}{m}\cos\omega t[/latex], therefore [latex]a + \omega_{\circ}^2x = \frac{F_{\circ}}{m}\cos\omega t[/latex], therefore, ignoring the transient solution, [latex]x = \frac{F_{\circ}}{m(\omega_{\circ}^2-\omega^2)}\cos\omega t[/latex], a steady state solution of clearly constant amplitude. Therefore the system's mechanical energy, [latex]E = U + K = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 =[/latex] a constant. This is true for all input frequencies [latex]\neq\omega_{\circ}[/latex]. Therefore, at all input frequencies [latex]\neq\omega_{\circ}[/latex], the driving function can't be inputing energy into the system, since the system's M.E. = a constant. That is, since there are no energy losses, if the driving mechanism were inputing a steady stream of energy, the system's M.E. would go to infinity as time goes to infinity.

 

Now let [latex]\omega = \omega_{\circ}[/latex]. Therefore [latex]a + \omega_{\circ}^2x = \frac{F_{\circ}}{m}\cos\omega_{\circ} t[/latex], therefore, ignoring the transient solution, [latex]x = \frac{F_{\circ}}{2m\omega_{\circ}}t\cos\omega_{\circ} t[/latex], a steady state solution with an amplitude which clearly [latex]\rightarrow\infty[/latex] as [latex]t\rightarrow\infty[/latex]. Therefore the system's mechanical energy, [latex]E = U + K = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 \rightarrow\infty[/latex] as [latex]t\rightarrow\infty[/latex]. Therefore, the driving function must be inputing energy into the system when [latex]\omega = \omega_{\circ}[/latex].

 

So which is it? Either the driving mechanism is supplying the system with energy or not.

 

If energy is being input, then why doesn't the system energy tend to infinity at all driving frequencies?

 

If energy is not being input, then why does the system energy tend to infinity when driven at the natural frequency?

 

Both must be explained.

[swansont]

 

But when a simple harmonic oscillator is driven at the system's natural frequency, the amplitude of the position, and therefore the system's M.E. increases without bound, which can't happen unless the system is receiving energy from the driving mechanism.

 

 

That can only be true if you have a driving mechanism that did no work. How is that a driving mechanism?

[Enthalpy]

[...] ignoring the transient solution, [...]

 

You have written an equation for an oscillator without damping, so it never attains a stabilized state, and you can't ignore the transient solution.

 

Especially, if you excite it at resonance, with a feeding system that doesn't weaken nor saturate (not feasible), then the energy will increase indefinitely. This corresponds to the energy provided by the excitation.