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PHY354H1
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Erich Poppitz
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Lecture

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Physics

PHY354H1

Erich Poppitz

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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