Thinking in the Frequency Domain (1.0)
We have now solved the damped harmonic oscillator equation with (and without) forcing:
x + 2 x_ + !02 x = Fm(t) :
(1)
Like any other linear inhomogeneous ODE, the solution to this equation is a sum of the homogeneous
(transient) and particular (steady state) solutions:
x(t) = xh (t) + xp(t):
(2)
We have developed explicit expressions for xh (t) in general and for xp (t) for forcing functions that
satisfy the Dirichlet Conditions. All of these results, however, have been presented in the time
domain. Oscillatory phenomena are commonly studied in the frequency domain, and the purpose
of this note is to discuss the characteristics of these solutions in the frequency domain.
A. Absence of Forcing: Motion Represented in the Time Domain
In the absence of forcing, we showed in Homework 7 that the solutions to equation (1) break
into three cases depending on the relative size of !0 and . We applied the initial conditions
x(0) = x0 ; x_ (0) = v(0) = v0 . For the underdamped case ( 2 < !02):
x(t) = Ae,t cos(w1 t + )
h
i1=2
A = !1 x20 !12 + (v0 + x0)2
1
v
0 + x0
,
1
= tan
,! x ;
1
q
0
(3)
(4)
(5)
where we dene !1 = !02 , 2 . For the the critically damped case ( 2 = !02 ):
x(t) = (A + Bt)e,t
A = x0
B = v0 + x0 :
(6)
(7)
(8)
Finally, for the overdamped case ( 2 > !02 ):
q
x(t) = e,t (A cosh !2t + B sinh !2t)
A = x0
B = v0 +! x0 ;
2
(9)
(10)
(11)
where we dene !2 = 2 , !02 . In each case, we have replaced the arbitrary constants (A; ) or
(A; B ) with the initial conditions (x0 ; v0 ) which still have not been specied.
Now consider two common cases in which the oscillator either starts from rest but is displaced
from equilibrium (x0 6= 0; v0 = 0) or is imparted a non-zero initial velocity at it equilibrium position
(x0 = 0; v0 6= 0). Figures 1 and 2 show examples of these cases. To make the plots we need to
specify !0 ; ; x0 ; and v0 . In each case, !0 = 2 rad/s, f0 = 1 Hz, x0 = 1 m unless it's set to zero,
v0 = 10 m/s unless it's set to zero, and = 0:1!0 for underdamped, = !0 for critically damped
(of course), and = 10!0 for overdamped.
overdamped
1
0.75
0.5
0.25
0.5
1
1.5
2
2.5
3
t
-0.25
-0.5
underdamped
-0.75
critically damped
-1
Figure 1: Comparison between the underdamped, overdamped, and critically damped oscillators in the time domain
in which v0 = 0 and x0 = 1 m. In this case, !0 = 2 rad/s, = 0:1!0 for the underdamped oscillator, = !0 for the
critically damped oscillator, and = 10!0 for the overdamped oscillator.
underdamped
1.5
critically damped
1
0.5
0.5
1
1.5
2
2.5
3
t
-0.5
-1
-1.5
overdamped
Figure 2: Comparison between the underdamped, overdamped, and critically damped oscillators in the time domain
in which x0 = 0 and v0 = 10 m/s. In this case, !0 = 2 rad/s, = 0:1!0 for the underdamped oscillator, = !0 for
the critically damped oscillator, and = 10!0 for the overdamped oscillator. Note that the overdamped oscillator
hardly responds to an impulse.
B. Transformation to the Frequency Domain: The Fourier Transform
Recall that a function f (t) can be expressed as a Fourier Series if it satises the Dirichlet
Conditions, in which case:
f (t) =
1
X
n=,1
cn ei!n t
(12)
Z T
cn = 21T
f (t)e,i!n t dt;
,T
(13)
(14)
where !n = n=T . The complex coecients cn are said to form a \spectrum" (which usually
denotes a function of frequency) because n is an index that represents frequency. Because n is an
integer index, this is said to be a discrete spectrum. Thus, the Fourier Series can be seen as a
representation of the function f (t) in the frequency domain.
Figure 3 shows the example of a discrete spectrum that we considered in an earlier hand-out,
for the box-car function f (t) = 0 on (0; T ) and = 1 on (,T; 0].
0.5
0.4
0.3
0.2
0.1
0
-20
-10
0
10
20
-20
-10
0
10
20
100
50
0
-50
-100
n
Figure 3: Plot of the Discrete Fourier \Spectrum" of a box-car function. At top, the modulus of the complex
coecient cn is plotted versus n and at bottom the phase is plotted. Note that each n denes a frequency !n = n=T .
Here we have set T = 1. Plots such as this one can be seen as representing amplitude versus frequency; hence, the
word spectrum.
The Fourier Series is dened for a function over a nite interval (,T; T ). The spacing between
the frequencies !n = n=T is inversely proportional to T . As T increases, the spacing decreases
until, in the limit as T ! 1, the spectrum is no longer discrete, but becomes continuous. In this
limit, the analogue of the discrete Fourier coecients cn is the continuous complex function F (!):
1
f (t)e,i!t dt;
,1Z
1
F ,1 [F (!)] = f (t) = 21
F (!)ei!t d!:
,1
F [f (t)] = F (!) =
Z
(15)
(16)
Equation (15) denes the forward Fourier Transform in which a function of time is converted
to a function of frequency and equation (16) denes the inverse Fourier Transform in which
a function of frequency is transformed into a function of time. F and F ,1 are the forward and
inverse Fourier operators, respectively. It is easy to show that they are both linear operators. That
is:
F [af (t) + bg(t)] = aF [f (t)] + bF [g(t)];
F , [aF (!) + bG(!)] = aF , [F (!)] + bF , [G(!)]:
1
1
1
(17)
(18)
Note that there is a factor of 1=(2) in front of the integral in equation (16). Forward and
inverse Fourier Transforms must be dened such that the round-trip from the time domain, to
the frequency domain, and back to the time domain involves a factor of 1=(2). Textbooks and
disciplines dier as to the location of this factor. Some, like Boas' book, place it in equation (15),
p
others place a factor of 1= 2 in both equations (15) and (16). We put it in equation (16) because
we will only be doing forward transforms, so in this way we can ignore it. The other aspect of this
that diers between textbooks is the signs of i!t in the argument of the exponentials. Again, all
that matters is that the signs dier between equations (15) and (16). You have to choose either
one to be positive and then the other is negative. Physicists usually use signs like in equations (15)
and (16), whereas engineers typically use the opposite sign convention. As you mature as scientists
and engineers you will need to keep the conventions clear to avoid mistakes. Consistency is the
key!
Time and frequency are known as conjugate quantities, they are related to one another
through a Fourier Transform. Position (or displacement) and wavenumber are also conjugate
quantities. In quantum mechanics, frequency and energy are linearly related, as are wavenumber
and momentum. Hence, time and energy are conjugate quantities as are position and momentum.
We will see later on in this write-up that the characteristics of a function in one domain has
denite implications for characteristics in the conjugate domain. In particular, a function that
is narrow in the time domain is broad in the frequency domain. We will see this because as you
Forward Fourier Transform
F(ω)
f(t)
time (or space)
domain
frequency
(or wavenumber)
domain
f(x)
F(k)
Inverse Fourier Transform
Figure 4: Time and the frequency are conjugate quantities because a function of time, f (t), is converted to function
of frequency, F (!), by a Fourier Transform. Similarly the spatial and wavenumber domains are conjugate domains.
increase damping you limit a function to have signicant amplitude in a narrower time ban, but this
broadens the peak in the frequency domain. This inverse relation between the width of functions in
conjugate domains has great implications in quantum mechanics. Because position and momentum
(or time and energy) are conjugate quantities, knowing the position (or time) of a particle very well
necessitates that the momentum (or energy) is poorly known. This is the conceptual underpinning
of the Heisenberg Uncertainty Principle. It all comes down to the properties of Fourier Transforms
and being able to think in the frequency and time domains!
C. Absence of Forcing: Motion Represented in the Frequency Domain
We wish to be able to understand how the motion represented in the time domain in Figures 1
and 2 appears in the frequency domain. In doing so, let's concentrate on underdamped oscillations
(equations (3) - (5)) which is our principle interest. Note that x(t) dened by equation (3) is zero
for t < 0, so the lower limit on the Fourier Transform should be zero. Instead of taking the Fourier
Transform of Ae,t cos(!1 t + ) let's replace the e,t cos(!1 t + ) with a product of exponentials:
e,t cos(!1 t + ) ! e,t ei(!1 t+) = eie(,+i!1 )t
= ei ei(i+!1 )t
= ei ei!^ t ;
(19)
(20)
(21)
where we have dened !^ = !1 + i . With this denition note that !^ , ! = (!1 , !) + i and
i(^! , !) = , + i(!1 , !). (!1; !; are all real, but !^ is complex.) Afterward we will reconstruct
the Fourier Tranform of the decaying cosine.
Now, doing the Fourier Transform:
F
h
Ae,t ei(!1 t+)
i
= Ae
i
1
Z
0
e
i!^ t
Z 1
,
i!t
i
e dt = Ae
ei(^!,!)t dt
0
1
i(^
! ,!)t = Aei i(^! 1, !) e
0
1
1
i
,
= Ae , + i(! , !) e t ei(!1 ,!)t 1
(22)
(23)
(24)
0
= Aei [ + i(! , !1 )],1 = Aei 2 ++ i((!!1 ,, !!))2 :
1
(25)
Recall the following:
e,t cos(!1 t + ) = 12 e,t ei(!1 +)t + 21 e,t e,i(!1 +)t
(26)
Because the Fourier Transform is a linear operator, from equations (17) (26) we know:
h
i
h
i
h
i
F e,t cos(!1 t + ) = 21 F e,t ei(!t +)t + 21 F e,t e,i(!t+)t :
(27)
The rst term on the RHS of equation (27) is given by equation (25) and the second is determinable
from equation (25) with the following substitutions: ! , and !1 ! ,!1. Doing so, we get:
h
i
F (!) = F Ae,t cos(!1 t + ) = 21 Aei 2 ++i((!!1 ,, !!))2 + 21 Ae,i 2 ,+ i((!!1 ++!!))2 (28)
1
1
= F1 (!) + F2 (!):
(29)
Consider the modulus and phase ( ) of these two components:
F1 (!) = jF1 (!)jei 1
F2 (!) = jF2 (!)jei 2
h
i,1=2
jF1 (!)j = (F1 (!)F1 (!))1=2 = 21 A 2 + (!1 , !)2
jF (!)j = (F (!)F (!)) = = 21 A + (! + !)
!
,
!
,
+
(!) = tan
2
2
1 2
2
h
2
1
2
i,1=2
(30)
(31)
(32)
(33)
(34)
= tan,1 ,(!1+ !) , :
(35)
2 (! )
Thus, the moduli of F1 (!) and F2 (!) peak at ! = !1 and ! = ,!1 , respectively. Perhaps it's a
1
1
1
little harder to see, but the two terms in equation (28) exhibit an important symmetry about the
origin ! = 0: F2 (,!) = F1 (!). This symmetry results from the realness of the function in the
time domain.
Figure 5 attempts to summarize much of the information contained in equation (28). For
simplicity of presentation, we let A = 1; = 0. This is non-physical, because by equation (5), an
underdamped oscillator is destined to have 6= 0 if 6= 0. However, all that does is add a positive
shift to the phase spectrum at positive frequencies and a negative shift to the phase spectrum at
negative frequencies and we can imagine this after we make the plot with = 0.
1
1
(a)
(c)
amplitude or modulus
0.8
0.8
decaying
cosine
(solid)
0.6
0.6
decaying
exponential
(dashed)
0.4
0.4
0.2
0.2
0
0
-10
-5
0
5
10
-10
frequency (rad/s)
0
5
10
5
10
(d)
(b)
1
1
0
0
-1
-1
-10
-5
frequency (rad/s)
-5
0
5
10
-10
frequency (rad/s)
-5
0
frequency (rad/s)
Figure 5: Plots of (a) the modulus, (b) the phase, (c) the real part, and (d) the imaginary part of the Fourier
Transforms of two functions: e,t cos !1 t and e,tei!1 t. We set !1 = 2; = !1 =10 so these spectra correspond
to the underdamped oscillator. For simplicity, we set A = 1 and = 0 even though this is physically unrealistic.
Note that the transform of the decaying cosine is doubly peaked, but the transform of the decaying exponential is
singly peaked. The realness of the decaying cosine necessitates the apparent symmetries around zero frequency. Also
note that decay broadens the the peak. In the absence of damping ( = 0), the peak would be innitely high and
innitessimally wide.
In Figure 5, you can see that with fair accuracy you can approximate F (!) in the neighborhood
of ! = !1 with F1 (!); that is one can approximate the Fourier Transform of the decaying cosinusoid
with the Fourier Transform of the decaying exponential. Thus, in the neighborhood of !1 : F (!) F1 (!), jF (!)j jF1 (!)j, and (!) 1 (!). The peak value of F (!) occurs when ! = !1 and is:
jF (! )j 2A
1
whereas the phase
jF (! )j 4A ;
1
2
2
2
(36)
! 0 at the peak. The spectrum is purely real at the peak. jF (!)j is
referred to as the amplitude spectrum and jF (!)j2 as the power spectrum. The function
[ 2 + (!1 , !)2 ],1 , the square of equation (32) found in the power spectrum, is so commonly
encountered it has a name: the Lorentzian, presumably named after the Dutch physicist Hendrik
Antoon Lorentz of the \Lorentz transformation" and the \Lorentz force", etc., but I'm unsure of
the origin.
It is important to recognized that damping acts to increase the width of the peak. Notice at
! = !1 + , by equation (32), the power (which is the square of the modulus of the spectrum) is:
2
h
i
jF (!1 + )j2 41 A2 2 + (!1 , (!1 + ))2 ,1 = 8A 2 = 21 jF (!1)j2 :
(37)
Equation (37) says that the value of the power of the spectrum at ! = !1 + is half the value at
the peak (! = !1 ). Thus, the full width of the peak at half its maximum power (FWHM) is equal
to 2 . (FWHM is typically read \full-width at half max".) The larger the damping, the larger ,
and the wider the peak! This is illustrated in Figure 6. This also provide a means for estimating
from the spectrum of a measured time series if it is unknown.
|F(ω)|
2
2
A
2
4β
2
A
2
8β
ω -β
1
ω
1
ω +β
ω
1
Figure 6: This plot illustrates that the full width at half the peak power of the spectrum is equal to 2 . This fact
is very useful in estimating for damped oscillating systems in which is unknown.
Let's call the FWHM of the power spectrum !. We see that we can get an expression for the
Quality Factor Q as follows:
! = 2 = 22!Q1 ! Q = !!1 !!0 ;
(38)
where the nal equality holds in the case of weak damping.
Recall that for a damped oscillator, the total energy of the oscillator will decrease with time.
At the maximum of each swing all energy is potential energy, and E = 21 kx2 . The maximum
of each swing is determined by the envelope function e,t , so E / x2 / e,2t . Over one cycle
E (t) ! E (t + 1 ), where 1 = 2=!1 is the period of the damped oscillator. So the change in energy
in one cycle is E = E (t + 1 ) , E (t). We see then that:
E = e,2(t+1 ) , e,2t = e,21 , 1 = e,4=!1 , 1
E
e,2t
ln 1 + E = ,4
E
!1
(39)
(40)
! (38)
,!1 ln 1 + E ,!1 E ;
=
=
(41)
2
4
E
4 E
where the nal equality uses the binonmial expansion, ln(1 + x) x, which holds if x << 1. In
this case, equation (41) will be approximately valid if damping is weak so that E << E . Thus,
if damping is weak the relative energy lost per cycle is approximately;
E 2 ! :
(42)
E
!0
This says that the energy lost per cycle is proportional to the relative width of the peak in the
frequency domain.
Finally, note what happens to F (!) as you continuously turn o the damping; that is, as
! 0. The width ! ! 0, !1 ! !0, and the amplitude at the peak jF (!0 )j ! 1. This function,
which has innite amplitude at the peak (! , !0 ) and zero amplitude everywhere else, is called a
delta function: (! , !0). Thus, the Fourier Transform of cos !0t is doubly peaked, with innite
amplitude at ! = !0 and zero amplitude elsewhere. From equation (28), letting ! 0, we see
that:
F [cos ! t] = i(! , ! ) , i(! + ! ):
0
0
0
(43)
The phase of this function is zero everywhere, except at !0 where it is and at ,!0 where it
is ,. As before when we were considering the damped oscillator, because this is the Fourier
Transform of a real function (cos !t), the spectrum at positive and negative frequencies exhibits
complex conjugate symmetry. Note that cosine is innitely extended in the time domain, whereas
the delta function is innitessimally extended in the frequency domain. This is really the paradigm
of the inverse relationship between quantities expressed in conjugate domains. An innitely broad
function in the time domain, the cosine, transforms to an innitely narrow function in the frequency
domain, the delta function.
D. Forcing and Resonance
The discussion above concentrated on visualizing and understanding the homogeneous or transient solution in the frequency domain. In a previous hand-out we showed that if F (t) in equation
(1) is expanded in a Fourier Series:
1
1
X
X
a
0
F (t) = 2 + an cos !nt + bn sin !nt;
n=1
n=1
(44)
then the particular or steady-state solution can written as:
xp(t) =
An =
Bn =
n =
1
1
X
a0 + X
2m!02 n=1 An cos(!n t + n ) + n=1 Bn sin(!n t + n );
an h(!2 , !2 )2 + 4 2 !2 i,1=2 ;
n
mh 0 n
bn (!2 , !2 )2 + 4 2 !2 i,1=2 ;
m 0 n n
tan,1 !,22,!!n2 :
0
n
(45)
(46)
(47)
(48)
Consider a single cosinusoidal force with frequency !^ and amplitude a^,
F (t) = a^ cos !^ t;
(49)
then:
xp(t) = A^ cos(^! t + ^)
h
i,1=2
;
A^ = ma^ (!02 , !^ 2 )2 + 4 2 !^ 2
^ = tan,1 !,2 2, !!^^ 2 :
0
(50)
(51)
(52)
The response of the oscillator to this force will be a cosine at the forcing frequency, but the amplitude
and phase of the response depend on two factors: the dierence between the natural and forcing
frequencies (!0 and !^ ) and the damping . The amplitude of the response will maximize when
!0 = !^ . The phase at !0 is always ,=2. Figure 7 displays the response amplitude and phase, A^
and ^, as a function of forcing frequency !^ relative to natural frequency !0 .
Figure 7: To be lled in later..... See Figure 3.15 of Marion and Thornton until that time. His D is our A^ and his
is our ,^.
It is worthwhile noting that the peak in the response curves shown in Figure 7 does not occur
at either the natural frequency of the undamped oscillator, !0 , or the frequency of the damped
oscillator, !1 , but at a dierent frequency called the resonance frequency, !R . The extremal
points of A^(^!) will occur when:
dA^ = 0:
(53)
d!^
From equation (51):
dA^ = , a^ 2^!(2 2 + (^!2 , !02 ))
(54)
d!^
m (^!2 , !02 )2 + 4 2 !23=2
q
^
0 = ddA
!
!
^
=
0
or
!
^
=
!02 , 2 2
(55)
!^
Equation (55) shows that there are extremal points: !^ = 0 is the minimum and the maximum is
the resonance frequency:
q
!R = !02 , 2 2
(56)
Thus, the resonance frequency isqlowered as damping is increased. Compare this with the frequency
of the damped oscillator; !1 = !02 , 2 . Note that no resonace will occur if the argument of the
p
square root in equation (56) is imaginary. That is, resonance will only occur if < !0 = 2.
It is important to reconginze that Figure 7 is not the spectrum of the particular solution in this
example. Afterall, the particular solution is simply a cosine and we know that the spectrum of a
cosine is a sum of two delta functions (equation (43). Rather Figure 7 shows the response of an
oscillator as you vary the forcing frequency !^ .