# Homework Help: Archived How to use series expansion to simplify

1. Oct 4, 2008

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 expantion 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,

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

File size:
25.5 KB
Views:
97
2. Feb 6, 2016

### 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.