if the quality factor of the oscillator is large if implies that
Answers
Answer:
Explanation:
In driven oscillator it can be explained by the following differential equation
x¨+2βx˙+ω20x=Acos(ωt)
where the 2β is coefficient of friction, the ω0 is the frequency of simple harmonic oscillator, and Acos(ωt) is the driven force divided by the mass of oscillating object.
The particular solution xp of the equation is
xptanδ=A(ω20−ω2)2+4ω2β2−−−−−−−−−−−−−−−−√cos(ωt−δ)=2ωβω20−ω2
Now, in classical mechanics of particles and systems(Stephen T. Thornton, Jerry B. Marrion) it finds the amplitude's maximum by
ddωA(ω20−ω2)2+4ω2β2−−−−−−−−−−−−−−−−√∣∣∣∣∣ω=ωR=0∴ωR=ω20−2β2−−−−−−−√(ω20−2β2>0)
and defines Q factor in driven oscillator by
Q≡ωR2β
Here I have some questions about calculating Q factor in lightly damped driven oscillator.
Q=ωR2β≃ω0Δω
Δω is the width of 12√(amplitude maximum).
I searched Q factor in google, but there are so much confusion on understanding the condition "lightly damped". One says it means ω0>>2β, and the other says ω0>>β. Which is right?
In google, they calculate this very absurdly. They assume that ω≃ω0 and change the part of amplitude denominator by
(ω20−ω2)=(ω0+ω)(ω0−ω)≃2ω0(ω0−ω)
I don't understand this absurd approximation. Why (ω0+ω)≃2ω0 is possible and (ω0−ω)≃0 is not? Also, how can we assume ω≃ω0?
I want to know how to derive the Q≃ω0Δω