2. Structural
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Dr Alessandro
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Intended Learning Outcomes
At the end of this unit (which includes the tutorial next
week), you should be able to:
Derive analytically the frequency response function
(FRF) for a SDoF system
Use the Fourier Analysis to study the dynamic
response of SDoF oscillators in the frequency domain
3. Structural
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Recap of Last-Week Key Learning Points
Unforced Undamped SDoF Oscillator
Equation of motion (forces):
m ¨u(t) + k u(t) = 0 (1)
Equation of motion (accelerations):
¨u(t) + ω2
0 u(t) = 0 (2)
Natural circular frequency of vibration:
ω0 =
k
m
(3)
Time history of the dynamic response u(t) for given
initial displacement u(0) = u0 and initial velocity
˙u(0) = v0:
u(t) = u0 cos(ω0 t) +
v0
ω0
sin(ω0 t) (4)
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Recap of Last-Week Key Learning Points
Unforced Damped SDoF Oscillator
Equation of motion (forces):
m ¨u(t) + c ˙u(t) + k u(t) = 0 (5)
Equation of motion (accelerations):
¨u(t) + 2 ζ0 ω0 ˙u(t) + ω2
0 u(t) = 0 (6)
Viscous damping ratio and reduced (or damped) natural
circular frequency:
ζ0 =
c
2 m ω0
< 1 (7)
ω0 = 1 − ζ2
0 ω0 (8)
Time history for given initial conditions:
u(t) = e−ζ0 ω0 t
u0 cos(ω0 t) +
v0 + ζ0 ω0 u0
ω0
sin(ω0 t) (9)
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Recap of Last-Week Key Learning Points
Harmonically Forced SDoF Oscillator (1/2)
Equation of motion (forces):
m ¨u(t) + c ˙u(t) + k u(t) = F0 sin(ωf t) (10)
The dynamic response is the superposition of any particular
integral for the forcing term (up(t)) and the general solution
of the related homogenous equation (uh(t)):
u(t) = uh(t) + up(t) (11)
General solution (which includes two integration constants
C1 and C2):
uh(t) = e−ζ0 ω0 t
C1 cos(ω0 t) + C2 sin(ω0 t) (12)
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Recap of Last-Week Key Learning Points
Phase lag (= Phase of the steady-state response − Phase
of the forcing harmonic)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
Π
4
Π
2
Π
3 Π
4
0
Π
4
Π
2
Π
3 Π
4
Β
P
Ζ0 0.50
Ζ0 0.20
Ζ0 0.10
Ζ0 0.05
Ζ0 0
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Fourier Series
Jean Baptiste Joseph Fourier
(21 Mar 1768 – 16 May 1830)
Fourier was a French
mathematician and
physicis, born in Auxerre,
and he is best known for
initiating the investigation
of Fourier series and their
applications to problems
of heat transfer and
vibrations
10. Structural
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Dr Alessandro
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Fourier Series
We have obtained a closed-form solution for the dynamic
response of SDoF oscillators subjected to harmonic
excitation
How can we extend such solution to a more general case?
Since the dynamic system is linear, the superposition
principle holds
The Fourier series allows us decomposing a periodic signal
into the sum of a (possibly infinite) set of simple harmonic
functions
We can therefore: i) decompose the forcing function in its
simple harmonic components; ii) calculate the dynamic
response for each of them; and then iii) superimpose all
these contributions to get the overall dynamic response
11. Structural
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Fourier Series
If the forcing function f(t) is periodic with period Tp:
f(t) = F0 +
n
j=1
Fj sin(Ωj t + Φj ) = f(t + Tp) (18)
where:
F0 =
a0
2
(19)
Fj = a2
j + b2
j (for j ≥ 1) (20)
tan(Φj ) =
aj
bj
(for j ≥ 1) (21)
in which:
aj =
2
Tp
Tp
0
f(t) cos(Ωj t) dt (for j ≥ 0) (22)
bj =
2
Tp
Tp
0
f(t) sin(Ωj t) dt (for j ≥ 1) (23)
Ωj = j
2 π
Tp
(24)
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Fourier Series
Approximating a square wave of unitary amplitude and period
Tp = 2 s with an increasing number n of harmonic terms
n = 1
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
time s
forcekN
n = 3
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
time s
forcekN
n = 5
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
time s
forcekN
n = 15
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
time s
forcekN
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Fourier Series
The same approach can be
adopted for a non-periodic
signal, e.g. the so-called
Friedlander waveform, which is
often used to describe the time
history of overpressure due to
blast:
p(t) =
p0 , if t < 0
p0 + ∆p e−t/τ
1 − t
τ , if t ≥ 0
(25)
where p0 is the atmospheric pressure, ∆p is the maximum
overpressure caused by the blast, and τ defines the timescale of
the waveform
Zero padding is however required, which consists of extending
the signal with zeros
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Fourier Series
Approximating a Friedlander waveform (p0 = 0, ∆p = 100kPa,
τ = 0.01 s) with an increasing number n of harmonic terms
n = 10
3 2 1 0 1 2 3
20
0
20
40
60
80
100
time ds
pressurekPa
n = 20
3 2 1 0 1 2 3
20
0
20
40
60
80
100
time ds
pressurekPa
n = 40
3 2 1 0 1 2 3
0
20
40
60
80
100
time ds
pressurekPa
n = 80
3 2 1 0 1 2 3
0
20
40
60
80
100
time ds
pressurekPa
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Fourier Series
Once the forcing signal is expressed as:
f(t) = F0 +
n
j=1
Fj sin(Ωj t + Φj ) (18)
The dynamic response can be evaluated as:
u(t) = uh(t) +
F0
k
+
n
j=1
uj (t) (26)
where:
uj =
Fj
k
D(βj ) sin(Ωj t + Φj + ϕj ) (27)
in which:
βj =
Ωj
ω0
(28)
tan(ϕj ) =
2 ζ0 βj
1 − β2
j
(29)
16. Structural
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Dr Alessandro
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Fourier Transform
The Fourier Transform (FT) can be thought as an
extension of the Fourier series, that results when the
period of the represented function approaches infinity
The FT is a linear operator, often denoted with the
symbol F, which transforms a mathematical function of
time, f(t), into a new function, denoted by
F(ω) = F f(t) , whose argument is the circular
frequency ω (with units of radians per second)
The FT can be inverted, in the sense that, given the
frequency-domain function F(ω), one can determine
the frequency-domanin counterpart, f(t) = F−1 F(ω) ,
and the operator F−1 is called Inverse FT (IFT)
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Fourier Transform
In Structural Dynamics, the time-domain signal f(t) is
often a real-valued function of the time t, while its
Fourier transform is a complex-valued function of the
circular frequency ω, that is:
F(ω) = FR(ω) + ı FI(ω) (30)
where:
ı =
√
−1 is the imaginary unit
FR(ω) = F(ω) is the real part of F(ω)
FI(ω) = F(ω) is the imaginary part of F(ω)
|F(ω)| = F2
R(ω) + F2
I (ω) is the absolute value (or
modulus) of F(ω)
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Dr Alessandro
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Fourier Transform
There are several ways of defining the FT and the IFT
(depending on the applications)
In this module, we will always use the following
mathematical definitions:
F(ω) = F f(t) =
+∞
−∞
f(t) e−ı ω t
dt (31)
f(t) = F−1
F(ω) =
1
2 π
+∞
−∞
F(ω) eı ω t
dω (32)
Note that, according to the Euler’s formula, the following
relationship exists between the complex exponential function and
the trigonometric functions:
eı θ
= cos(θ) + ı sin(θ) (33)
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Fourier Transform
The main reason why the FT is widely used in Structural
Dynamics, is because it allows highlighting the distribution
of the energy of a given signal f(t) in the frequency domain
The energy E is always proportional to the square of the
signal, e.g.:
Potential energy in a SDoF oscillator: V(t) = 1
2 k u2(t)
Kinetic energy in a SDoF oscillator: T(t) = 1
2 m ˙u2(t)
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Fourier Transform
According to the Parseval’s theorem, the cumulative energy
E contained in a waveform f(t) summed across all of time t
is equal to the cumulative energy of the waveform’s FT F(ω)
summed across all of its frequency components ω:
E =
1
2
α
+∞
−∞
f(t)2
dt =
1
2 π
α
+∞
0
|F(ω)|2
dω (34)
where α is the constant appearing in the definition of the
energy (e.g. α = k for the potential energy and α = m for
the kinetic energy)
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Fourier Transform
Example: For illustration purposes, let us consider the
following signal in the time domain:
f(t) = F0 e−(t/τ)2
cos(Ω t) (35)
consisting of an exponentially modulated (with time scale τ)
cosine wave (with amplitude F0 and circular frequency Ω),
whose FT in the frequency domain is known in closed form:
F(ω) = F f(t) =
√
π τ F0 e−τ2 (ω2+Ω2)/4
cosh
1
2
Ω τ2
ω
(36)
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Dr Alessandro
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Fourier Transform
Effects of changing the time scale τ = 1, 3, 5 s (while Ω = 1 rad/s)
f(t)
10 5 0 5 10
0.5
0.0
0.5
1.0
s
fF0
Τ 5 s
Τ 3 s
Τ 1 s
|F(ω)|
0 2 4 6 8 10 12
0
1
2
3
4
Ω s rad
FF0
Τ 5 s
Τ 3 s
Τ 1 s
f(t)2
10 5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
s
f2
F0
2
Τ 5 s
Τ 3 s
Τ 1 s
|F(ω)|2
/π
0 2 4 6 8 10 12
0
1
2
3
4
5
6
Ω s rad
F2
ΠF0
2
Τ 5 s
Τ 3 s
Τ 1 s
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Dr Alessandro
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Fourier Transform
Effects of changing the circular frequency Ω = 1, 3 rad/s (while
τ = 1 s)
f(t)
10 5 0 5 10
1.0
0.5
0.0
0.5
1.0
s
fF0
5 rad.s
1 rad s
|F(ω)|
0 2 4 6 8 10 12
0
1
2
3
4
Ω s rad
FF0
5 rad.s
1 rad s
f(t)2
10 5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
s
f2
F0
2
5 rad.s
1 rad s
|F(ω)|2
/π
0 2 4 6 8 10 12
0
1
2
3
4
5
6
Ω s rad
F2
ΠF0
2
5 rad.s
1 rad s
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Dr Alessandro
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Fast Fourier Transform
The FT is a very powerful tool, but we can use it mainly if we have a
simple mathematical expression of the signal f(t) in the time domain
Very often the signal f(t) is known at a number n discrete time instants
within the time interval [0, tf]
In other words, we usually have an array of the values fr = f(tr ), where:
tr = (r − 1) ∆t is the rth
time instant
r = 1, 2 · · · , n is the index in the time domain
∆t = tf/(n − 1) is the sampling time (or time step)
νs = ∆t−1
is the sampling frequency (i.e. the number of points
available per each second of the record)
Can we still use the frequency domain for the dynamic analysis of linear
structures?
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Dr Alessandro
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Fast Fourier Transform
The answer is yes...And the Fast Fourier Transform (FFT) can be used to
transform in the frequency domain the discrete signal fr
The FFT (which is implemented in any numerical computing language,
including MATLAB and Mathematica) is indeed an efficient algorithm to
compute the Discrete Fourier Transform (DFT), of great importance to a
wide variety of applications (including Structural Dynamics)
The DFT is defined as follows:
Fs = DFT fr =
n
r=1
fr e2 π ı(r−1)(s−1)/n
(42)
where n is the size of both the real-valued arrays fr in the time domain
and of the complex-valued array Fs in the frequency domain (i.e.
r = 1, 2, · · · , n and s = 1, 2, · · · , n), while ı =
√
−1 is once again the
imaginary unit
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Dr Alessandro
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Fast Fourier Transform
It can be proved mathematically that, for ω < 2 π νN = π/∆t, the
array Fs, computed as the DFT of the discrete signal fr , gives a
numerical approximation of the analytical FT of the continuous
signal f(t).
In other words:
Fs ≈ F(ωs) (43)
where:
ωs = (s − 1) ∆ω is the sth
circular frequency where the DFT
is computed
∆ω = 2 π/(n ∆t) is the discretisation step on the frequency
axis
νN = νs/2 is the Nyquist’s frequency, and only signals with
the frequency content below the Nyquist’s frequency can be
represented
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Fast Fourier Transform
Comparing FT (red solid lines) with FFT (blue dots)
(∆t = 0.7 s; n = 58; ∆ω = 0.155 rad/s; νN = 0.714 Hz)
f(t)
0 10 20 30 40
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
s
ffmax
|F(ω)|
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
Ω s rad
Ffmax
FR(ω)
0 1 2 3 4
2
1
0
1
2
Ω s rad
FRfmax
FI(ω)
0 1 2 3 4
2
1
0
1
2
Ω s rad
FIfmax
30. Structural
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Dr Alessandro
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Fast Fourier Transform
Working with discrete signal can be tricky... A typical example is the phenomenon
of aliasing
In signal processing, it refers to: i) different signals becoming indistinguishable
when sampled; ii) the distortion that results when the signal reconstructed from
samples is different from the original continuous signal
In the figure above, the red harmonic function of frequency νred = 0.9 Hz is
completely overlooked as the sampling rate is νs = 1 Hz (black dots), and
therefore the Nyquist’s frequency is νN = 0.5 Hz < νred
The reconstruction will then identify (incorrectly) the blue harmonic function of
frequency νblue = 0.1 Hz < νN
31. Structural
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Dr Alessandro
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Frequency Response Function
The equation of motion for a SDoF oscillator in the time domain
reads:
¨u(t) + 2 ζ0 ω0 ˙u(t) + ω2
0 u(t) =
1
m
f(t) (44)
By applying the FT operator to both sides of Eq. (44), one obtains:
F ¨u(t) + 2 ζ0 ω0 ˙u(t) + ω2
0 u(t) = F
1
m
f(t)
∴ F ¨u(t) + 2 ζ0 ω0 F ˙u(t) + ω2
0 F u(t) =
1
m
F f(t)
∴ (ı ω)2
U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω2
0 U(ω) =
1
m
F(ω)
∴ −ω2
+ 2 ı ζ0 ω0 ω + ω2
0 U(ω) =
1
m
F(ω)
(45)
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Frequency Response Function
The equation of motion in the frequency domain (the last of Eqs.
(45)) has been posed in the form:
−ω2
+ 2 ı ζ0 ω0 ω + ω2
0 U(ω) =
1
m
F(ω) (46)
where F(ω) = F f(t) and U(ω) = F u(t) are the FTs of
dynamic load and dynamic response, respectively
We can rewrite the above equation as:
U(ω) = H(ω)
F(ω)
m
(47)
in which the complex-valued function H(ω) is called Frequency
Response Function (FRF) (or Transfer Function), and is defined
as:
H(ω) = ω2
0 − ω2
+ 2 ı ζ0 ω0 ω
−1
(48)
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Frequency Response Function
Procedure for the dynamic analysis of SDoF oscillators in the frequency domain:
1 Compute the DFT of the dynamic force at discrete frequencies ωs (see Eqs.
(42) and (43)):
F(ωs) ≈ DFT fr =
n
r=1
fr e2 π ı(r−1)(s−1)/n
(49)
2 Define analytically the complex-valued FRF of the oscillator:
H(ω) = ω2
0 − ω2
+ 2 ı ζ0 ω0 ω
−1
(48)
3 Compute the dynamic response in the frequency domain (see Eq. (47)):
U(ωs) = H(ωs)
F(ωs)
m
(50)
4 Compute the dynamic response in the time domain at discrete time instants
tr through the Inverse DFT (IDFT):
u(tr ) ≈ IDFT Us =
1
n
n
s=1
Us e2 π ı(r−1)(s−1)/n
(51)
