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Archived How to use series expansion to simplify

  1. Oct 4, 2008 #1
    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,

    Please someone help me!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
  3. Feb 6, 2016 #2

    Mark Harder

    User Avatar
    Gold Member

    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.
     
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