Finding resonant amplitude

homework-and-exercises oscillators resonance differential-equations

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Since you want to find the maximum R, you must differentiate R(ω) with respect to ω (that is dR(ω)/dω ) and then set the derivative equal to zero to find at which value of ω you have a maximum (be careful not to accept negative values or zero). Then insert that value of ω into the response amplitude equation to find your maximum.
Hope it helps!

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raptakem

Updated on August 01, 2022

Comments

raptakem 10 months

For a system of oscillations described by the differential equation:

$$ \cfrac{d^2x}{dt^2} -\epsilon \cfrac{dx}{dt} + x = \cos(\omega t)$$

We find that the response amplitude $R(\omega)$ to be:

$$R(\omega) = \cfrac{1}{\sqrt{(1-\omega^2)^2+\epsilon^2\omega^2}}$$

When first looking at this it looks like the simplest way to find the maximum response amplitude would be to set $\omega = 1$ and hence $R(1) = \cfrac{1}{\epsilon}$. However, I have been told that this is not actually the case and that the maximum amplitude is also dependent on the $\epsilon$ of the system. Why is this and how could one get to that point from the response amplitude equation?
- mastrok almost 7 years
  
  Complete the square in the denominator in $\omega^2$.
- Alfred Centauri almost 7 years
  
  Note that as $\omega \rightarrow 0$, the amplitude $R(\omega)$ goes to 1. Thus, if $\frac{1}{\epsilon} < 1$, it is not the case that the maximum is $\frac{1}{\epsilon}$. Also, the negative sign in the diff eq is troubling. This system has right half plane poles and thus is unstable, i.e., the modes grow without bound.