Euler's Formula

  • Thread starter krnhseya
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Homework Statement



Derive...

x(t)=(exp^(-[tex]\zeta[/tex][tex]\omega[/tex]t))*(a1(exp^(i[tex]\omega[/tex]sqrt(1-[tex]\zeta^2[/tex])*t)))+a2(exp^(-i[tex]\omega[/tex]sqrt(1-[tex]\zeta^2[/tex])*t))))

into

x(t)=(exp^(-[tex]\zeta[/tex][tex]\omega[/tex]t))*(A sin ([tex]\omega[/tex]*t + [tex]\varphi[/tex]))

Homework Equations



n/a

The Attempt at a Solution



I've managed to get x(t) = (exp^(-[tex]\zeta[/tex][tex]\omega[/tex]t))*((a1(cos[tex]\omega[/tex]*t) + i sin ([tex]\omega[/tex]*t))+(a2(cos[tex]\omega[/tex]*t) - i sin ([tex]\omega[/tex]*t))) then when i simplify things...sin terms cancel out and i end up geting...

exponential term * (a1+a2) * (2cos [tex]\omega[/tex]*t)
 

Answers and Replies

  • #2
Tom Mattson
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Hi,

Is the [itex]\zeta[/itex] a constant here?

I've managed to get x(t) = (exp^(-[tex]\zeta[/tex][tex]\omega[/tex]t))*((a1(cos[tex]\omega[/tex]*t) + i sin ([tex]\omega[/tex]*t))+(a2(cos[tex]\omega[/tex]*t) - i sin ([tex]\omega[/tex]*t))) then when i simplify things...sin terms cancel out and i end up geting...

exponential term * (a1+a2) * (2cos [tex]\omega[/tex]*t)
No, the sine terms do not cancel out. You have the following.

[tex]x(t)=e^{-\zeta\omega t}\left[(a_1+a_2)\cos\left(\omega\sqrt{1-\zeta^2}t\right)+(a_1-a_2)\sin\left(\omega\sqrt{1-\zeta^2}t\right)\right][/tex].

The only way the sine term will vanish is if [itex]a_1=a_2[/itex], and you did not say that that is the case in your problem statement.
 
  • #3
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Hi,

Is the [itex]\zeta[/itex] a constant here?



No, the sine terms do not cancel out. You have the following.

[tex]x(t)=e^{-\zeta\omega t}\left[(a_1+a_2)\cos\left(\omega\sqrt{1-\zeta^2}t\right)+(a_1-a_2)\sin\left(\omega\sqrt{1-\zeta^2}t\right)\right][/tex].

The only way the sine term will vanish is if [itex]a_1=a_2[/itex], and you did not say that that is the case in your problem statement.
Hello. It is a constant.
Yeah, I figured out that I made a mistake.

that's as far as i can go but i need to combine that into sin like i've posted above.

thanks for the reply!

And, there should be a bracket infront of "sqrt" so exponential part it outside the whole thing.

[edit] I think I got it...Thank you very much! :)
 
Last edited:

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