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Transcript of SDEE: Lecture 3
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Structural Dynamics& Earthquake Engineering
Lectures #3 and 4: Fourier Analysis + FrequencyResponse Function for SDoF oscillators
Dr Alessandro Palmeri
Civil and Building Engineering @ Loughborough University
Tuesday, 11th February 2014
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Intended Learning Outcomes
At the end of this unit (which includes the tutorial nextweek), you should be able to:
Derive analytically the frequency response function(FRF) for a SDoF systemUse the Fourier Analysis to study the dynamicresponse of SDoF oscillators in the frequency domain
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Recap of Last-Week Key Learning Points
Unforced Undamped SDoF OscillatorEquation of motion (forces):
m u(t) + k u(t) = 0 (1)
Equation of motion (accelerations):
u(t) + ω20 u(t) = 0 (2)
Natural circular frequency of vibration:
ω0 =
√km
(3)
Time history of the dynamic response u(t) for giveninitial displacement u(0) = u0 and initial velocityu(0) = v0:
u(t) = u0 cos(ω0 t) +v0
ω0sin(ω0 t) (4)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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) + ω20 u(t) = 0 (6)
Viscous damping ratio and reduced (or damped) naturalcircular frequency:
ζ0 =c
2 mω0< 1 (7)
ω0 =√
1− ζ20 ω0 (8)
Time history for given initial conditions:
u(t) = e−ζ0 ω0 t[u0 cos(ω0 t) +
v0 + ζ0 ω0 u0
ω0sin(ω0 t)
](9)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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 particularintegral for the forcing term (up(t)) and the general solutionof the related homogenous equation (uh(t)):
u(t) = uh(t) + up(t) (11)
General solution (which includes two integration constantsC1 and C2):
uh(t) = e−ζ0 ω0 t[C1 cos(ω0 t) + C2 sin(ω0 t)
](12)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Recap of Last-Week Key Learning Points
Harmonically Forced SDoF Oscillator (2/2)
Particular integral:
up(t) = ustD(β) sin(ωf + ϕp) (13)
Static displacement and frequency ratio:
ust =F0
k(14)
β =ωf
ω0(15)
Dynamic amplification factor and phase lag:
D(β) =1√
(1− β2)2
+ (2 ζ0 β)2(16)
tan(ϕp) =2 ζ0 β
1− β2 (17)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Recap of Last-Week Key Learning Points
Dynamic Amplification Factor
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.5
1.0
5.0
10.0
50.0
Β
D
Ζ0 =0.50
Ζ0 =0.20
Ζ0 =0.10
Ζ0 =0.05
Ζ0= 0
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Recap of Last-Week Key Learning Points
Phase lag (= Phase of the steady-state response − Phaseof 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
jP
Ζ0 =0.50
Ζ0 =0.20
Ζ0 =0.10
Ζ0 =0.05
Ζ0= 0
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
Jean Baptiste Joseph Fourier(21 Mar 1768 – 16 May 1830)
Fourier was a Frenchmathematician andphysicis, born in Auxerre,and he is best known forinitiating the investigationof Fourier series and theirapplications to problemsof heat transfer andvibrations
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
We have obtained a closed-form solution for the dynamicresponse of SDoF oscillators subjected to harmonicexcitation
How can we extend such solution to a more general case?
Since the dynamic system is linear, the superpositionprinciple holds
The Fourier series allows us decomposing a periodic signalinto the sum of a (possibly infinite) set of simple harmonicfunctions
We can therefore: i) decompose the forcing function in itssimple harmonic components; ii) calculate the dynamicresponse for each of them; and then iii) superimpose allthese contributions to get the overall dynamic response
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
We have obtained a closed-form solution for the dynamicresponse of SDoF oscillators subjected to harmonicexcitation
How can we extend such solution to a more general case?
Since the dynamic system is linear, the superpositionprinciple holds
The Fourier series allows us decomposing a periodic signalinto the sum of a (possibly infinite) set of simple harmonicfunctions
We can therefore: i) decompose the forcing function in itssimple harmonic components; ii) calculate the dynamicresponse for each of them; and then iii) superimpose allthese contributions to get the overall dynamic response
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
We have obtained a closed-form solution for the dynamicresponse of SDoF oscillators subjected to harmonicexcitation
How can we extend such solution to a more general case?
Since the dynamic system is linear, the superpositionprinciple holds
The Fourier series allows us decomposing a periodic signalinto the sum of a (possibly infinite) set of simple harmonicfunctions
We can therefore: i) decompose the forcing function in itssimple harmonic components; ii) calculate the dynamicresponse for each of them; and then iii) superimpose allthese contributions to get the overall dynamic response
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
We have obtained a closed-form solution for the dynamicresponse of SDoF oscillators subjected to harmonicexcitation
How can we extend such solution to a more general case?
Since the dynamic system is linear, the superpositionprinciple holds
The Fourier series allows us decomposing a periodic signalinto the sum of a (possibly infinite) set of simple harmonicfunctions
We can therefore: i) decompose the forcing function in itssimple harmonic components; ii) calculate the dynamicresponse for each of them; and then iii) superimpose allthese contributions to get the overall dynamic response
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
We have obtained a closed-form solution for the dynamicresponse of SDoF oscillators subjected to harmonicexcitation
How can we extend such solution to a more general case?
Since the dynamic system is linear, the superpositionprinciple holds
The Fourier series allows us decomposing a periodic signalinto the sum of a (possibly infinite) set of simple harmonicfunctions
We can therefore: i) decompose the forcing function in itssimple harmonic components; ii) calculate the dynamicresponse for each of them; and then iii) superimpose allthese contributions to get the overall dynamic response
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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 =√
a2j + b2
j (for j ≥ 1) (20)
tan(Φj ) =aj
bj(for j ≥ 1) (21)
in which:
aj =2Tp
∫ Tp
0f (t) cos(Ωj t) dt (for j ≥ 0) (22)
bj =2Tp
∫ Tp
0f (t) sin(Ωj t) dt (for j ≥ 1) (23)
Ωj = j2πTp
(24)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
Approximating a square wave of unitary amplitude and periodTp = 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
forc
e
kN
n = 3
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 5
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 15
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
Approximating a square wave of unitary amplitude and periodTp = 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
forc
e
kN
n = 3
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 5
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 15
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
Approximating a square wave of unitary amplitude and periodTp = 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
forc
e
kN
n = 3
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 5
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 15
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
Approximating a square wave of unitary amplitude and periodTp = 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
forc
e
kN
n = 3
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 5
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
n = 15
0 1 2 3 4
-1.0
-0.5
0.0
0.5
1.0
time s
forc
e
kN
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Series
The same approach can beadopted for a non-periodicsignal, e.g. the so-calledFriedlander waveform, which isoften used to describe the timehistory of overpressure due toblast:
p(t) =
p0 , if t < 0p0 + ∆p e−t/τ
(1− t
τ
), if t ≥ 0
(25)
where p0 is the atmospheric pressure, ∆p is the maximumoverpressure caused by the blast, and τ defines the timescale ofthe waveform
Zero padding is however required, which consists of extendingthe signal with zeros
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
pre
ssure
kP
a
n = 20
-3 -2 -1 0 1 2 3-20
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 40
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 80
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
pre
ssure
kP
a
n = 20
-3 -2 -1 0 1 2 3-20
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 40
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 80
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
pre
ssure
kP
a
n = 20
-3 -2 -1 0 1 2 3-20
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 40
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 80
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
pre
ssure
kP
a
n = 20
-3 -2 -1 0 1 2 3-20
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 40
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
n = 80
-3 -2 -1 0 1 2 3
0
20
40
60
80
100
time ds
pre
ssure
kP
a
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
kD(βj ) sin(Ωj t + Φj + ϕj ) (27)
in which:βj =
Ωj
ω0(28)
tan(ϕj ) =2 ζ0 βj
1− β2j
(29)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The Fourier Transform (FT) can be thought as anextension of the Fourier series, that results when theperiod of the represented function approaches infinity
The FT is a linear operator, often denoted with thesymbol F , which transforms a mathematical function oftime, f (t), into a new function, denoted byF (ω) = F〈f (t)〉, whose argument is the circularfrequency ω (with units of radians per second)
The FT can be inverted, in the sense that, given thefrequency-domain function F (ω), one can determinethe frequency-domanin counterpart, f (t) = F−1〈F (ω)〉,and the operator F−1 is called Inverse FT (IFT)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The Fourier Transform (FT) can be thought as anextension of the Fourier series, that results when theperiod of the represented function approaches infinity
The FT is a linear operator, often denoted with thesymbol F , which transforms a mathematical function oftime, f (t), into a new function, denoted byF (ω) = F〈f (t)〉, whose argument is the circularfrequency ω (with units of radians per second)
The FT can be inverted, in the sense that, given thefrequency-domain function F (ω), one can determinethe frequency-domanin counterpart, f (t) = F−1〈F (ω)〉,and the operator F−1 is called Inverse FT (IFT)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The Fourier Transform (FT) can be thought as anextension of the Fourier series, that results when theperiod of the represented function approaches infinity
The FT is a linear operator, often denoted with thesymbol F , which transforms a mathematical function oftime, f (t), into a new function, denoted byF (ω) = F〈f (t)〉, whose argument is the circularfrequency ω (with units of radians per second)
The FT can be inverted, in the sense that, given thefrequency-domain function F (ω), one can determinethe frequency-domanin counterpart, f (t) = F−1〈F (ω)〉,and the operator F−1 is called Inverse FT (IFT)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
In Structural Dynamics, the time-domain signal f (t) isoften a real-valued function of the time t , while itsFourier transform is a complex-valued function of thecircular frequency ω, that is:
F (ω) = FR(ω) + ıFI(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)| =√
F 2R(ω) + F 2
I (ω) is the absolute value (ormodulus) of F (ω)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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 followingmathematical definitions:
F (ω) = F〈f (t)〉 =
∫ +∞
−∞f (t) e−ı ω t dt (31)
f (t) = F−1〈F (ω)〉 =1
2π
∫ +∞
−∞F (ω) eı ω t dω (32)
Note that, according to the Euler’s formula, the followingrelationship exists between the complex exponential function andthe trigonometric functions:
eı θ = cos(θ) + ı sin(θ) (33)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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 followingmathematical definitions:
F (ω) = F〈f (t)〉 =
∫ +∞
−∞f (t) e−ı ω t dt (31)
f (t) = F−1〈F (ω)〉 =1
2π
∫ +∞
−∞F (ω) eı ω t dω (32)
Note that, according to the Euler’s formula, the followingrelationship exists between the complex exponential function andthe trigonometric functions:
eı θ = cos(θ) + ı sin(θ) (33)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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 followingmathematical definitions:
F (ω) = F〈f (t)〉 =
∫ +∞
−∞f (t) e−ı ω t dt (31)
f (t) = F−1〈F (ω)〉 =1
2π
∫ +∞
−∞F (ω) eı ω t dω (32)
Note that, according to the Euler’s formula, the followingrelationship exists between the complex exponential function andthe trigonometric functions:
eı θ = cos(θ) + ı sin(θ) (33)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The main reason why the FT is widely used in StructuralDynamics, is because it allows highlighting the distributionof the energy of a given signal f (t) in the frequency domain
The energy E is always proportional to the square of thesignal, e.g.:
Potential energy in a SDoF oscillator: V (t) = 12 k u2(t)
Kinetic energy in a SDoF oscillator: T (t) = 12 m u2(t)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The main reason why the FT is widely used in StructuralDynamics, is because it allows highlighting the distributionof the energy of a given signal f (t) in the frequency domain
The energy E is always proportional to the square of thesignal, e.g.:
Potential energy in a SDoF oscillator: V (t) = 12 k u2(t)
Kinetic energy in a SDoF oscillator: T (t) = 12 m u2(t)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
According to the Parseval’s theorem, the cumulative energyE contained in a waveform f (t) summed across all of time tis equal to the cumulative energy of the waveform’s FT F (ω)summed across all of its frequency components ω:
E =12α
∫ +∞
−∞f (t)2 dt =
12π
α
∫ +∞
0|F (ω)|2 dω (34)
where α is the constant appearing in the definition of theenergy (e.g. α = k for the potential energy and α = m forthe kinetic energy)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Example: For illustration purposes, let us consider thefollowing signal in the time domain:
f (t) = F0 e−(t/τ)2cos(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−τ
2 (ω2+Ω2)/4 cosh(
12
Ω τ2 ω
)(36)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
t s
f
F0 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
ÈFÈ
F0
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
t s
f2
F02 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
ÈF2HΠ
F02L
Τ= 5 s
Τ= 3 s
Τ= 1 s
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
t s
f
F0 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
ÈFÈ
F0
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
t s
f2
F02 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
ÈF2HΠ
F02L
Τ= 5 s
Τ= 3 s
Τ= 1 s
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
t s
f
F0 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
ÈFÈ
F0
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
t s
f2
F02 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
ÈF2HΠ
F02L
Τ= 5 s
Τ= 3 s
Τ= 1 s
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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
t s
f
F0 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
ÈFÈ
F0
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
t s
f2
F02 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
ÈF2HΠ
F02L
Τ= 5 s
Τ= 3 s
Τ= 1 s
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Effects of changing the circular frequency Ω = 1,3 rad/s (whileτ = 1 s)
f (t)
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
t s
f
F0
W= 5 rad.s
W= 1 rads
|F (ω)|
0 2 4 6 8 10 12
0
1
2
3
4
Ω s rad
ÈFÈ
F0
W= 5 rad.s
W= 1 rads
f (t)2
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
t s
f2
F02
W= 5 rad.s
W= 1 rads
|F (ω)|2/π
0 2 4 6 8 10 12
0
1
2
3
4
5
6
Ω s rad
ÈF2HΠ
F02L W= 5 rad.s
W= 1 rads
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Effects of changing the circular frequency Ω = 1,3 rad/s (whileτ = 1 s)
f (t)
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
t s
f
F0
W= 5 rad.s
W= 1 rads
|F (ω)|
0 2 4 6 8 10 12
0
1
2
3
4
Ω s rad
ÈFÈ
F0
W= 5 rad.s
W= 1 rads
f (t)2
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
t s
f2
F02
W= 5 rad.s
W= 1 rads
|F (ω)|2/π
0 2 4 6 8 10 12
0
1
2
3
4
5
6
Ω s rad
ÈF2HΠ
F02L W= 5 rad.s
W= 1 rads
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Effects of changing the circular frequency Ω = 1,3 rad/s (whileτ = 1 s)
f (t)
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
t s
f
F0
W= 5 rad.s
W= 1 rads
|F (ω)|
0 2 4 6 8 10 12
0
1
2
3
4
Ω s rad
ÈFÈ
F0
W= 5 rad.s
W= 1 rads
f (t)2
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
t s
f2
F02
W= 5 rad.s
W= 1 rads
|F (ω)|2/π
0 2 4 6 8 10 12
0
1
2
3
4
5
6
Ω s rad
ÈF2HΠ
F02L W= 5 rad.s
W= 1 rads
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Effects of changing the circular frequency Ω = 1,3 rad/s (whileτ = 1 s)
f (t)
-10 -5 0 5 10
-1.0
-0.5
0.0
0.5
1.0
t s
f
F0
W= 5 rad.s
W= 1 rads
|F (ω)|
0 2 4 6 8 10 12
0
1
2
3
4
Ω s rad
ÈFÈ
F0
W= 5 rad.s
W= 1 rads
f (t)2
-10 -5 0 5 10
0.0
0.2
0.4
0.6
0.8
1.0
t s
f2
F02
W= 5 rad.s
W= 1 rads
|F (ω)|2/π
0 2 4 6 8 10 12
0
1
2
3
4
5
6
Ω s rad
ÈF2HΠ
F02L W= 5 rad.s
W= 1 rads
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The FT enjoys a number of important properties, including:
Linearity:
F〈a1 f1(t) + a2 f2(t)〉 = a1 F1(ω) + a2 F2(ω) (37)
Time shift:F〈f (t − τ)〉 = e−ı ω τ F (ω) (38)
Time scaling:
F〈f (α t)〉 =1|α|
F(ωα
)(39)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The FT enjoys a number of important properties, including:
Linearity:
F〈a1 f1(t) + a2 f2(t)〉 = a1 F1(ω) + a2 F2(ω) (37)
Time shift:F〈f (t − τ)〉 = e−ı ω τ F (ω) (38)
Time scaling:
F〈f (α t)〉 =1|α|
F(ωα
)(39)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
The FT enjoys a number of important properties, including:
Linearity:
F〈a1 f1(t) + a2 f2(t)〉 = a1 F1(ω) + a2 F2(ω) (37)
Time shift:F〈f (t − τ)〉 = e−ı ω τ F (ω) (38)
Time scaling:
F〈f (α t)〉 =1|α|
F(ωα
)(39)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Moreover (very importantly):
Derivation rule:
F⟨
dn
dtn f (t)⟩
= (ı ω)n F (ω) (40)
Convolution rule:
F〈f ∗ g(t)〉 =
∫ +∞
−∞f (t) g(t − τ) dτ
=
∫ +∞
−∞f (t − τ) g(t) dτ
= F (ω) G(ω)
(41)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fourier Transform
Moreover (very importantly):
Derivation rule:
F⟨
dn
dtn f (t)⟩
= (ı ω)n F (ω) (40)
Convolution rule:
F〈f ∗ g(t)〉 =
∫ +∞
−∞f (t) g(t − τ) dτ
=
∫ +∞
−∞f (t − τ) g(t) dτ
= F (ω) G(ω)
(41)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fast Fourier Transform
The FT is a very powerful tool, but we can use it mainly if we have asimple 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 instantswithin 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 r th 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 pointsavailable per each second of the record)
Can we still use the frequency domain for the dynamic analysis of linearstructures?
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fast Fourier Transform
The FT is a very powerful tool, but we can use it mainly if we have asimple 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 instantswithin 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 r th 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 pointsavailable per each second of the record)
Can we still use the frequency domain for the dynamic analysis of linearstructures?
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fast Fourier Transform
The answer is yes...
And the Fast Fourier Transform (FFT) can be used totransform in the frequency domain the discrete signal fr
The FFT (which is implements in any numerical computing language,including MATLAB and Mathematica) is indeed an efficient algorithm tocompute the Discrete Fourier Transform (DFT), of great importance to awide variety of applications (including Structural Dynamics)
The DFT is defined as follows:
Fs = DFT〈fr 〉 =n∑
r=1
fr e2π ı(r−1)(s−1)/n (42)
where n is the size of both the real-valued arrays fr in the time domainand 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
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fast Fourier Transform
The answer is yes...And the Fast Fourier Transform (FFT) can be used totransform in the frequency domain the discrete signal fr
The FFT (which is implements in any numerical computing language,including MATLAB and Mathematica) is indeed an efficient algorithm tocompute the Discrete Fourier Transform (DFT), of great importance to awide variety of applications (including Structural Dynamics)
The DFT is defined as follows:
Fs = DFT〈fr 〉 =n∑
r=1
fr e2π ı(r−1)(s−1)/n (42)
where n is the size of both the real-valued arrays fr in the time domainand 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
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fast Fourier Transform
It can be proved mathematically that, for ω < 2π νN = π/∆t , thearray Fs, computed as the DFT of the discrete signal fr , gives anumerical approximation of the analytical FT of the continuoussignal f (t).
In other words:
Fs ≈ F (ωs) (43)
where:
ωs = (s − 1) ∆ω is the sth circular frequency where the DFTis computed
∆ω = 2π/(n ∆t) is the discretisation step on the frequencyaxis
νN = νs/2 is the Nyquist’s frequency, and only signals withthe frequency content below the Nyquist’s frequency can berepresented
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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)
ææææææææææææææææææææææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææææææææææææææææææææææ
0 10 20 30 40
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
t s
f
f max
|F (ω)|
æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
Ω s rad
ÈFÈ
f max
FR(ω)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FR
f max
FI(ω)
æ ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FI
f max
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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)
ææææææææææææææææææææææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææææææææææææææææææææææ
0 10 20 30 40
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
t s
f
f max
|F (ω)|
æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
Ω s rad
ÈFÈ
f max
FR(ω)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FR
f max
FI(ω)
æ ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FI
f max
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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)
ææææææææææææææææææææææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææææææææææææææææææææææ
0 10 20 30 40
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
t s
f
f max
|F (ω)|
æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
Ω s rad
ÈFÈ
f max
FR(ω)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FR
f max
FI(ω)
æ ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FI
f max
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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)
ææææææææææææææææææææææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææææææææææææææææææææææ
0 10 20 30 40
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
t s
f
f max
|F (ω)|
æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
Ω s rad
ÈFÈ
f max
FR(ω)
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FR
f max
FI(ω)
æ ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ æ æ æ æ æ æ æ æ æ æ æ æ æ
0 1 2 3 4
-2
-1
0
1
2
Ω s rad
FI
f max
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Fast Fourier Transform
Working with discrete signal can be tricky... A typical example is the phenomenonof aliasing
In signal processing, it refers to: i) different signals becoming indistinguishablewhen sampled; ii) the distortion that results when the signal reconstructed fromsamples is different from the original continuous signal
In the figure above, the red harmonic function of frequency νred = 0.9 Hz iscompletely overlooked as the sampling rate is νs = 1 Hz (black dots), andtherefore 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
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Frequency Response Function
The equation of motion for a SDoF oscillator in the time domainreads:
u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =
1m
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⟨
1m
f (t)⟩
∴ F〈u(t)〉+ 2 ζ0 ω0 F〈u(t)〉+ ω20 F〈u(t)〉 =
1mF〈f (t)〉
∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω20 U(ω) =
1m
F (ω)
∴(−ω2 + 2 ı ζ0 ω0 ω + ω2
0)
U(ω) =1m
F (ω)
(45)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Frequency Response Function
The equation of motion for a SDoF oscillator in the time domainreads:
u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =
1m
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⟨
1m
f (t)⟩
∴ F〈u(t)〉+ 2 ζ0 ω0 F〈u(t)〉+ ω20 F〈u(t)〉 =
1mF〈f (t)〉
∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω20 U(ω) =
1m
F (ω)
∴(−ω2 + 2 ı ζ0 ω0 ω + ω2
0)
U(ω) =1m
F (ω)
(45)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Frequency Response Function
The equation of motion for a SDoF oscillator in the time domainreads:
u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =
1m
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⟨
1m
f (t)⟩
∴ F〈u(t)〉+ 2 ζ0 ω0 F〈u(t)〉+ ω20 F〈u(t)〉 =
1mF〈f (t)〉
∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω20 U(ω) =
1m
F (ω)
∴(−ω2 + 2 ı ζ0 ω0 ω + ω2
0)
U(ω) =1m
F (ω)
(45)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Frequency Response Function
The equation of motion for a SDoF oscillator in the time domainreads:
u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =
1m
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⟨
1m
f (t)⟩
∴ F〈u(t)〉+ 2 ζ0 ω0 F〈u(t)〉+ ω20 F〈u(t)〉 =
1mF〈f (t)〉
∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω20 U(ω) =
1m
F (ω)
∴(−ω2 + 2 ı ζ0 ω0 ω + ω2
0)
U(ω) =1m
F (ω)
(45)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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) =1m
F (ω) (46)
where F (ω) = F〈f (t)〉 and U(ω) = F〈u(t)〉 are the FTs ofdynamic 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 FrequencyResponse Function (FRF) (or Transfer Function), and is definedas:
H(ω) =(ω2
0 − ω2 + 2 ı ζ0 ω0 ω)−1
(48)
StructuralDynamics
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Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Frequency Response Function
FRF for ζ0 = 0.05 – Note that: |H(ω)| = D(ω/ω0)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-10
-5
0
5
10
Ω Ω0
HΩ
02 IXH\
RXH\
ÈH È
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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=1fr e2π 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 instantstr through the Inverse DFT (IDFT):
u(tr ) ≈ IDFT〈Us〉 =1n
∑n
s=1Us e2π ı(r−1)(s−1)/n (51)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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=1fr e2π 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 instantstr through the Inverse DFT (IDFT):
u(tr ) ≈ IDFT〈Us〉 =1n
∑n
s=1Us e2π ı(r−1)(s−1)/n (51)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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=1fr e2π 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 instantstr through the Inverse DFT (IDFT):
u(tr ) ≈ IDFT〈Us〉 =1n
∑n
s=1Us e2π ı(r−1)(s−1)/n (51)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
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=1fr e2π 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 instantstr through the Inverse DFT (IDFT):
u(tr ) ≈ IDFT〈Us〉 =1n
∑n
s=1Us e2π ı(r−1)(s−1)/n (51)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Duhamel’s integral
If we now apply the convolution rule of Eq. (41) to thedynamic response in the frequency domain (see Eq. (47)),given as a product of two complex-valued function of thecircular frequency ω, the dynamic response for a genericexcitation f (t) can be derived in the time domain as follows:
u(t) = F−1〈U(ω)〉 =1mF−1〈H(ω) F (ω) 〉
=1mh ∗ f(t) =
1m
∫ +∞
∞h(t − τ) f (τ) dτ
(52)
where the function h(t) = 1m F
−1(ω) is the IFT of the FRF,and constitutes the Green’s function for the equation ofmotion of a SDoF oscillator
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Duhamel’s integral
This integral solution in the time domain is called Duhamel’sintegral, and furnishes a way for evaluating the dynamic responseto a generic forcing function:
u(t) =1m
∫ +∞
∞h(t − τ) f (τ) dτ (53)
It can be shown that:
h(t) = F−1〈H(ω)〉 =1ω0
e−ζ0 ω0 t sin(ω0 t) η(t) (54)
-2 -1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
t
Η
in which η(t) is the so-calledHeaviside’s Unit Step Function,so defined:
η(t) =
0 , if t < 012 , if t = 01 , if t > 0
(55)
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Duhamel’s integral
Green’s function for ζ0 = 0.05
-20 0 20 40 60-1.0
-0.5
0.0
0.5
1.0
Ω0 t
hΩ
02
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Duhamel’s integral
Final observation for today...
If you compare Eq. (9), which gives the time history forgiven initial conditions, with the Green’s function of Eq. (54),it appears that the function h(t) describes the free vibrationof a SDoF oscillator with initial conditions u(0) = 0 andu(0) = 1
Since such initial unit velocity could have been produced byan impulsive force applied at t = 0, the function h(t) is oftencalled Impulse Response Function (IRF), and indeed playsa fundamental role in the dynamic analysis of linearstructures
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Duhamel’s integral
Final observation for today...
If you compare Eq. (9), which gives the time history forgiven initial conditions, with the Green’s function of Eq. (54),it appears that the function h(t) describes the free vibrationof a SDoF oscillator with initial conditions u(0) = 0 andu(0) = 1
Since such initial unit velocity could have been produced byan impulsive force applied at t = 0, the function h(t) is oftencalled Impulse Response Function (IRF), and indeed playsa fundamental role in the dynamic analysis of linearstructures
StructuralDynamics
& EarthquakeEngineering
Dr AlessandroPalmeri
Recap
Fourier Series
FourierTransform
Fast FourierTransform
Duhamel’sintegral
Duhamel’s integral
Final observation for today...
If you compare Eq. (9), which gives the time history forgiven initial conditions, with the Green’s function of Eq. (54),it appears that the function h(t) describes the free vibrationof a SDoF oscillator with initial conditions u(0) = 0 andu(0) = 1
Since such initial unit velocity could have been produced byan impulsive force applied at t = 0, the function h(t) is oftencalled Impulse Response Function (IRF), and indeed playsa fundamental role in the dynamic analysis of linearstructures