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Today’s Objectives:
Students will be able to:
a) Determine the Fourier Coefficients for a periodic signal
b) Find the steady-state response for a system forced with general periodic forcing
Fourier Series – (Lecture 13)
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Rarely is forcing actually harmonic
Fourier’s Theorem: Any periodic function can be expressed by a constant term plus an infinite series of sins and cosines with increasing frequency
Rose-Hulman Institute of TechnologyMechanical Engineering
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Let’s look at Fourier Series
Given a periodic function f(t) with a period T and f(t) piecewise continuous then f(t) can be expressed as
f t a a t a t b t b t
a a n t b n tnn
nn
( ) cos( ) cos( ) sin( ) sin( )
cos( ) sin( )
= + + + + + +
= + +=
∞
=
∞
∑ ∑0 1 0 2 0 1 0 2 0
0 01
01
2 2ω ω ω ω
ω ω
L L
where
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Fourier Coefficients
aT
f t dt
aT
f t n t dt
bT
f t n t dt
T
n
T
n
T
00
00
00
1
2
2
=
=
=
∫
∫
∫
( )
( ) cos( )
( ) sin( )
ω
ω
∑∑∞
=
∞
=
++=1
01
00n
nn
n )tnsin(b)tncos(aa)t(f ωω
Where
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Summary of Fourier Series
0i tdtn 0 n 0 n
n n 1f (t) X e c c cos(n t )
∞ ∞ω
=−∞ =
= = + ω + φ∑ ∑
0
Ti t
n0
1X f (t)e dtT
− ω= ∫
0 0
n n
n n
c X
c 2 XX
=
=
φ = ∠
Complex Form
where
and
aT
f t dt
aT
f t n t dt
bT
f t n t dt
T
n
T
n
T
00
00
00
1
2
2
=
=
=
∫
∫
∫
( )
( ) cos( )
( ) sin( )
ω
ω
Summary
Note: These integrals can be over any period
0 2
2
2
3T
T
T
T
T
∫ ∫ ∫−
or or /
/
etc.
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You will often see magnitude and phase plots of the spectra
Square wave – 10 terms of Fourier Series
Spectra
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Websites and tools are available
http://www.jhu.edu/~signals/fourier2/index.html
Rose-Hulman Institute of TechnologyMechanical Engineering
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Odd and Even Functions
Odd: f(x) is odd if f(-x) = -f(x)Examples: sin(x), sin(nωt)
Even: f(x) is even if f(-x) = f(x)Examples: cos(x), cos(nωt)
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Fourier Series of Even and Odd functions …Why do we care?
• Fourier series of even functions– Even functions cannot be expressed in terms of odd functions.
Therefore:
• Fourier series of odd functions– Odd functions have an average value of zero and cannot be
expressed in terms of even functions. Therefore:
• Helps us interpret Maple results
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Example
Periodic 015 0
50 3
⎪⎩
⎪⎨⎧
<<
<<=
t
t)t(fFind the Fourier Series of:
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System response to multiple inputs
Assuming the transfer function is: ( )12
1
2
2
++=
ssk/sH
nm ωζ
ω
f1(t) = 2sin(5t)
f2(t) = 1cos(15t)
f3(t) = 4cos(25t)
xss(t)System with
transfer functionH(s)
We know the steady state response is:
( ) ( )( ) ( )( )ωωω jHtsinjHtxxx ∠+= AmplitudeInput
So, all we need to do is apply superposition!
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System response to multiple inputs (cont.)
( ) ( )( ) ( )( )ωωω jHtsinjHtxxx ∠+= AmplitudeInput
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30 35
Frequency (rad/s)
Mag
nific
atio
n fa
ctor
= x
ss/(F
0/k)
If we plot the magnitude and phase of this after letting s = jω we get:
-180-160-140-120-100
-80-60-40-20
00 5 10 15 20 25 30 35 40
Frequency (rad/s)
Pha
se o
f H (d
egre
es)
0.532.5
1.11MF
-159425f3(t) = 4cos(25t)
-90115f2(t) = 1cos(15t)
-8.525f1(t) = 2sin(5t)
PhaseInput amp.ωι (rad/s)Term
So xss(t) = 2(1.11)sin(5t-8.5°)+1(2.5)cos(15t-90°)+4(0.53)cos(25t-159°_
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General Periodic Forcing Steady State Response
harmonic n at the evaluated phasefunction transfer =
harmonic n at the evaluated magnitudefunction transfer =
gainfrequency zeroor bias DC atscoefficienFourier
harmonic n
frequency lfundamenta 2 of Period
function periodicknown a
th0
th0
0
th0
0
)jn(TF
)jn(TF
b,an
T
)t(fT)t(f
nn
ω
ω
ω
πω
∠
===
==
==
))jn(Htnsin()jn(Hb))jn(Htncos()jn(TFa)j(Ha)t(yn
nn
nss 001
0001
00 0 ωωωωωω ∠++∠++= ∑∑∞
=
∞
=
∑∑∞
=
∞
=
++=1
01
00n
nn
n )tnsin(b)tncos(aa)t(f ωω
and
where
f(t) yss(t)TF(s)If a known periodic function f(t) is applied to a linear system represented by the transfer function H(s) we use the principle of superposition.
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How many terms do you need to keep?
Often we only have to keep a few because those terms whose frequencies lie outside the bandwidth can be neglected as a result of the filtering property of the system (look at the frequency response plots).
TF j
a bn b
( )
,
ω ω⎯→⎯
⎯→⎯
0
0
as increases (usually)
as n increases (always)0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30 35
Frequency (rad/s)
Mag
nific
atio
n fa
ctor
= x
ss/(F
0/k)
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Maple Example
Periodic 21 010
)(2
⎩⎨⎧
<<<<
=ttt
tfwhere:
Determine the steady state response of the system.
A vibrating system is found to be governed by the differential equation:
)(1002 tfxxx =++ &&&
See Maple worksheet
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What you need to modify in Maple
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Homework
• Start the paper• Input the Maple worksheet
• Bring working Maple worksheet to class on Monday
