- #1
tadarah
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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!
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!