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Lecture

L12_qualityfactor(1).pdf

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Department
Physics
Course
PHY354H1
Professor
Erich Poppitz
Semester
Winter

Description
Lecture 12: Varying the Damping Force, and adding sinusoidal driving forces What does mean? ▯ At some point we made the variable change: = b=2m in our damped equations. It was presented as a way to simplify the algebra, but it also has a physical meaning. ▯ In an underdamped (no driving) system, we have found that the solu- tion is: x(t) = e t[Acos(~t) + B sin~t)] ▯ We can ask the question, at what period of the oscillation has the amplitude A decayed to e▯1 of its original value? At t = 0: 0 x(0) = e A = A Remember the de▯nition of logarithmic decrement (where T is some multiple of the period). Since T is a multiple of the period: T = 2▯n=!~ ) !~T = 2▯n, so ▯ (t+T) x(t + T) = e x(t) e▯ t If we set t = 0 we get: x(T) = e▯ T x(0) The time T when the amplitude has fallen to x(0)=e is therefore given by: ▯1 ▯ T 1 e = e ) T = 1 ) = T So physically, is the reciprocal of the time it takes for the amplitude ▯1 to decay to e of its original value. Its a measure of how fast the mechanical energy dissipates to thermal energy from friction. 1 ▯ Notice that the units of are inverse seconds, the same units a0 ! . We can therefore de▯ne a non-dimensional ratio of the two quantities: ! Q = 0 Q is called the "Quality factor" of a system. ▯ Notice that Q is large for systems with lots of oscillations before de- caying fully (i.e. systems where is much smaller than0! ). We can write some of our previous quantities describing damped oscillations that involve (lik~;A(! d;▯(! d) in terms of Q. For example: 1. The angular velocity of an underdamped system: s q ~ = ! ▯ = ! 1 ▯ 1 0 0 Q 2 Notice that if Q is large, then~ ▯ ! 0nd the motion of the oscillator is given very nearly by: ▯! t=Q x(t) = e 0 [Acos(!0t) + B sin(0 t)] Assume we have completed n full oscillations (i.e. n periods so t = 2▯n=! 0, then ▯2▯n=Q ▯2▯n=Q ▯2▯n=Q x(t) ▯ e Acos(2▯n) = e A = e x(0) So the oscillation amplitude decays by a factor of 1=e in n = Q=2▯ full oscillations (or cycles). We can therefore get a quick estimate of how many oscillations occur before damping by 1=e of the amplitude occurs just by looking at Q. 2. The amplitude of the particular solution in a driven/damped sys- tem: 2 3 ▯ ▯ F0 4q 1 5 A(! d = m 2 2 2 2 (1) (!0▯ ! d + (2 ! )d ▯ ▯ 2 3 F0 1 = 4q 5 (2) m (!0▯ ! ) + (2! !0=d) 2 d 2 We are going to rewrite this a little by putting all the angular frequencies in ratio form (e.g. d =0 ). Notice that: !0 ! d !0▯ ! d ▯ = !d ! 0 !0!d Using this in equation (2) we get: ! 2 3 F ! =! A(! ) = 0 4q 0 d 5 d m! 2 !0 !d 2 4 0 (!d ▯ !0) + Q2 Using di▯erent Q values we can plot this as a function of ! d (each curve represents a di▯erent Q value). See python program Aphase dependence onQ.py. 10 A vs omega_d for different Q Q=1 Q=4 8 Q=8 Q=10 Q=20 6 A 4 2 0.0 0.5 1.0 1.5 2.0 omegad Notice that there is a maximum amplitude (i.e. a resonance) for all except the most heavily damped systems. Using our expression for the frequency where the maximum occurs, we can solve for the required Q value: q q 2 2 2 !max = !0▯ 2 = ! 0 1 ▯ 2=Q p So a max will occur when Q > 2 (i.e. when the argument of the square root is positive). 3 3. The phase of the particular solution in a driven/damped system: 2 ! d tan▯ = 2 2 ! 0 ! d
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