How to use series expansion to simplify

[tadarah]

Hi, I'm new to this forum, so if there is anything wrong in this post please forgive me,

I'm not sure my post will be shown correctly, so I attached a doc file.

The question is

A lightly damped harmonic osillator, γ<<ω0, is driven at frequency ω.

1,
Find the frequency of the driving force such that the steady-state oscillation amplitude is one half of that at the resonance. How many solutions are there?

I found that two solutions, ω = ω0±√3γ (two solutions ± )

then

2,
Find the phase shifts between the driving force and the displacement at these frequencies to the first order in the ratio γ/ω0.
Give your answer as Φ= tan-1[a+bγ/ω0], where a, b are numbers; use series expansion for the messy analytic expressions

I got two solions,
I got
Φ1 = tan-1[2γ(ω0+√3γ) / ω02 – (ω0+√3γ)2]
and
Φ2 = tan-1[2γ(ω0-√3γ) / ω02 – (ω0-√3γ 2]

but I couldn't find the way to use series expansion to simplify my solutions,

Please someone help me!

 

[Mark Harder]

I don't know what the messy function is, so I can't answer the question specifically. Do you know about Taylor series expansions? A first-order approximation would be the sum truncated at the term that's first order in the derivative and the displacement from the initial point. There is a TS expansion for multi-variable functions, and a corresponding expansion for complex-valued functions.