Weak Damping - how to relate the amplitude and phase difference

[GayYoda]

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 can't work out how to simplify it further

 

[haruspex]

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 can't 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?

 

[vela]

You dropped $\phi$ as well.

 

[haruspex]

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.

 

[vela]

Ah, makes sense. I misunderstood what the OP was doing.