Weak Damping - how to relate the amplitude and phase difference

  • #1
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Homework Statement


Derive the relationship bewteen x_{max}, A_{+}, A_{-} and \phi

Homework Equations


x(t) = e^{\gamma t}(A_{+}e^{i \omega_d t} + A_{-}e^{-i \omega_d t})
x(t) = x_{max} e^{\gamma t} cos(\omega_d t + \phi)

The Attempt at a Solution


I know the e^{\gamma t} cancels and for the imaginary parts to cancel, A_{+} = A_{-} but then i get x_{max}(cos(\omega_d t) cos(\phi) - sin(\omega_d t)sin(\phi)) = 2A_{+} cos(\omega_d) and i cant work out how to simplify it further
 

Answers and Replies

  • #2
haruspex
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Homework Statement


Derive the relationship bewteen ##x_{max}##, ##A_{+}##, ##A_{-}## and ##\phi##

Homework Equations


##x(t) = e^{\gamma t}(A_{+}e^{i \omega_d t} + A_{-}e^{-i \omega_d t})##
##x(t) = x_{max} e^{\gamma t} cos(\omega_d t + \phi)##

The Attempt at a Solution


I know the ##e^{\gamma t}## cancels and for the imaginary parts to cancel, ##A_{+} = A_{-}## but then i get ##x_{max}(cos(\omega_d t) cos(\phi) - sin(\omega_d t)sin(\phi)) = 2A_{+} cos(\omega_d)## and i cant work out how to simplify it further
First, fixing up the LaTeX by inserting double hash as necessary.

You seem to have dropped a factor t.
After reinstating that, consider that this is supposed to be an identity valid for all t.
How can you use that?
 
Last edited:
  • #3
vela
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You dropped ##\phi## as well.
 
  • #4
haruspex
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You dropped ##\phi## as well.
That's not how I read it. One side has Φ, the other does not, and the value is to be deduced.
 
  • #5
vela
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Ah, makes sense. I misunderstood what the OP was doing.
 

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