# Weak Damping - how to relate the amplitude and phase difference

## 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

haruspex
Homework Helper
Gold Member

## 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:
vela
Staff Emeritus
Homework Helper
You dropped ##\phi## as well.

haruspex
Homework Helper
Gold Member
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
Staff Emeritus