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I'm studying a linearization of a differential equation. $x(t)$ and $r(t)$ are really small signals and G, K, B and M are constants. I understand everything until I reach

$$ \frac{d^2x(t)}{dt^2}=G+\frac{K}{M}\sqrt{\left| x(t)-r(t)-\bigg(\frac{MG}{K}\bigg)^2\right|} -\frac{B}{M}\bigg(\frac{dx(t)}{dt}-\frac{dr(t)}{dt}\bigg)$$

And I don't know how they pass from that to:

$$ \frac{d^2x(t)}{dt^2}=-\frac{K^2}{2M^2G} (x(t)-r(t)) -\frac{B}{M}\bigg(\frac{dx(t)}{dt}-\frac{dr(t)}{dt}\bigg)$$

They are obviously performing a linearization of the expression $$G+\frac{K}{M}\sqrt{\left| x(t)-r(t)-\bigg(\frac{MG}{K}\bigg)^2\right|}$$

to

$$\frac{K^2}{2M^2G} (x(t)-r(t)) $$

Can someone clarify what they are doing here? Thanks!

user2662833
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Granger Obliviate
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1 Answers1

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They're using $\sqrt{a^2-b}=a\sqrt{1-b/a^2}\approx a(1- b/(2a^2))$. The last step is the linearization and it is valid when $b/a^2$ is small.

The reason this applies in your situation is you are assuming that x(t)-r(t) is small, so $\sqrt{|x(t)-r(t)- \left(\frac{MG}{K}\right)^2|} = \sqrt{\left(\frac{MG}{K}\right)^2-(x(t)-r(t))}$.

Antti S.
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  • Aaaaah ok, I was looking for that! Do you know where I can find a list of common linearizations? It might be useful. Thank you very much – Granger Obliviate Oct 09 '18 at 22:38
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    I don't know of a list. But if you're familiar with calculus you can easily derive these, this is the tangent line approximation. In this case the function is the function $f(x)=x^{1/2}$ and you want the tangent line approximation at $x=1$, which is $f(1+x)\approx 1+x/2$, because the derivative of $f(x)$ is $\frac{1}{2} x^{-1/2}$ which is $1/2$ at $x=1$.

    So to your answer your question, a list of derivatives is basically a list of linearizations :)

     – Antti S. Oct 09 '18 at 22:41