sdee: lecture 3

StructuralDynamics

& EarthquakeEngineering

Dr AlessandroPalmeri

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

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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

Fourier Series

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:

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)

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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)

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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)

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Harmonically Forced SDoF Oscillator (2/2)

Particular integral:

up(t) = ustD(β) sin(ωf + ϕp) (13)

Static displacement and frequency ratio:

ust =F0

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)

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Dynamic Amplification Factor

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Ζ0 =0.50

Ζ0 =0.20

Ζ0 =0.10

Ζ0 =0.05

Ζ0= 0

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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

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 Frenchmathematician andphysicis, born in Auxerre,and he is best known forinitiating the investigationof Fourier series and theirapplications to problemsof heat transfer andvibrations

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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

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Fourier Series

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Fourier Series

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Fourier Series

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Fourier Series

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Fourier Series

If the forcing function f (t) is periodic with period Tp:

f (t) = F0 +n∑

Fj sin(Ωj t + Φj ) = f (t + Tp) (18)

where:F0 =

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

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Fourier Series

Approximating a square wave of unitary amplitude and periodTp = 2 s with an increasing number n of harmonic terms

0 1 2 3 4

time s

0 1 2 3 4

time s

0 1 2 3 4

time s

n = 15

0 1 2 3 4

time s

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Fourier Series

0 1 2 3 4

time s

0 1 2 3 4

time s

0 1 2 3 4

time s

n = 15

0 1 2 3 4

time s

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Fourier Series

0 1 2 3 4

time s

0 1 2 3 4

time s

0 1 2 3 4

time s

n = 15

0 1 2 3 4

time s

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Fourier Series

0 1 2 3 4

time s

0 1 2 3 4

time s

0 1 2 3 4

time s

n = 15

0 1 2 3 4

time s

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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

), if t ≥ 0

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

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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

time ds

n = 20

-3 -2 -1 0 1 2 3-20

time ds

n = 40

-3 -2 -1 0 1 2 3

time ds

n = 80

-3 -2 -1 0 1 2 3

time ds

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Fourier Series

n = 10

-3 -2 -1 0 1 2 3-20

time ds

n = 20

-3 -2 -1 0 1 2 3-20

time ds

n = 40

-3 -2 -1 0 1 2 3

time ds

n = 80

-3 -2 -1 0 1 2 3

time ds

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Fourier Series

n = 10

-3 -2 -1 0 1 2 3-20

time ds

n = 20

-3 -2 -1 0 1 2 3-20

time ds

n = 40

-3 -2 -1 0 1 2 3

time ds

n = 80

-3 -2 -1 0 1 2 3

time ds

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Fourier Series

n = 10

-3 -2 -1 0 1 2 3-20

time ds

n = 20

-3 -2 -1 0 1 2 3-20

time ds

n = 40

-3 -2 -1 0 1 2 3

time ds

n = 80

-3 -2 -1 0 1 2 3

time ds

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Fourier Series

Once the forcing signal is expressed as:

f (t) = F0 +n∑

Fj sin(Ωj t + Φj ) (18)

The dynamic response can be evaluated as:

u(t) = uh(t) +F0

n∑j=1

uj (t) (26)

where:uj =

kD(βj ) sin(Ωj t + Φj + ϕj ) (27)

in which:βj =

ω0(28)

tan(ϕj ) =2 ζ0 βj

1− β2j

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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

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Fourier Transform

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Fourier Transform

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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)

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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 followingmathematical definitions:

F (ω) = F〈f (t)〉 =

∫ +∞

−∞f (t) e−ı ω t dt (31)

f (t) = F−1〈F (ω)〉 =1

∫ +∞

−∞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)

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Fourier Transform

F (ω) = F〈f (t)〉 =

∫ +∞

−∞f (t) e−ı ω t dt (31)

f (t) = F−1〈F (ω)〉 =1

∫ +∞

−∞F (ω) eı ω t dω (32)

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Fourier Transform

F (ω) = F〈f (t)〉 =

∫ +∞

−∞f (t) e−ı ω t dt (31)

f (t) = F−1〈F (ω)〉 =1

∫ +∞

−∞F (ω) eı ω t dω (32)

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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)

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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)

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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 =

∫ +∞

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)

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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(

Ω τ2 ω

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Fourier Transform

Effects of changing the time scale τ = 1,3,5 s (while Ω = 1 rad/s)

-10 -5 0 5 10

F0 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

Τ= 5 s

Τ= 3 s

Τ= 1 s

f (t)2

-10 -5 0 5 10

F02 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

Τ= 5 s

Τ= 3 s

Τ= 1 s

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Fourier Transform

-10 -5 0 5 10

F0 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

Τ= 5 s

Τ= 3 s

Τ= 1 s

f (t)2

-10 -5 0 5 10

F02 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

Τ= 5 s

Τ= 3 s

Τ= 1 s

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Fourier Transform

-10 -5 0 5 10

F0 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

Τ= 5 s

Τ= 3 s

Τ= 1 s

f (t)2

-10 -5 0 5 10

F02 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

Τ= 5 s

Τ= 3 s

Τ= 1 s

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Fourier Transform

-10 -5 0 5 10

F0 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

Τ= 5 s

Τ= 3 s

Τ= 1 s

f (t)2

-10 -5 0 5 10

F02 Τ= 5 s

Τ= 3 s

Τ= 1 s

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

Τ= 5 s

Τ= 3 s

Τ= 1 s

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Duhamel’sintegral

Fourier Transform

Effects of changing the circular frequency Ω = 1,3 rad/s (whileτ = 1 s)

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

W= 5 rad.s

W= 1 rads

f (t)2

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

F02L W= 5 rad.s

W= 1 rads

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FourierTransform

Duhamel’sintegral

Fourier Transform

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

W= 5 rad.s

W= 1 rads

f (t)2

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

F02L W= 5 rad.s

W= 1 rads

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Fourier Transform

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

W= 5 rad.s

W= 1 rads

f (t)2

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

F02L W= 5 rad.s

W= 1 rads

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Fourier Transform

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|

0 2 4 6 8 10 12

Ω s rad

W= 5 rad.s

W= 1 rads

f (t)2

-10 -5 0 5 10

W= 5 rad.s

W= 1 rads

|F (ω)|2/π

0 2 4 6 8 10 12

Ω s rad

ÈF2HΠ

F02L W= 5 rad.s

W= 1 rads

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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

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Fourier Transform

Linearity:

F〈a1 f1(t) + a2 f2(t)〉 = a1 F1(ω) + a2 F2(ω) (37)

Time scaling:

F〈f (α t)〉 =1|α|

F(ωα

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Fourier Transform

Linearity:

F〈a1 f1(t) + a2 f2(t)〉 = a1 F1(ω) + a2 F2(ω) (37)

Time scaling:

F〈f (α t)〉 =1|α|

F(ωα

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Fourier Transform

Moreover (very importantly):

Derivation rule:

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)

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Fourier Transform

Moreover (very importantly):

Derivation rule:

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)

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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

Fourier Series

FourierTransform

Duhamel’sintegral

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

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FourierTransform

Duhamel’sintegral

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∑

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

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FourierTransform

Duhamel’sintegral

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∑

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

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FourierTransform

Duhamel’sintegral

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

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Duhamel’sintegral

Comparing FT (red solid lines) with FFT (blue dots)(∆t = 0.7 s; n = 58; ∆ω = 0.155 rad/s; νN = 0.714 Hz)

ææææææææææææææææææææææ

0 10 20 30 40

|F (ω)|

ææ æ æ æ æ æ æ æ æ æ æ æ æ

0 1 2 3 4

Ω s rad

FR(ω)

0 1 2 3 4

Ω s rad

FI(ω)

æ ææ

ææ æ æ æ æ æ æ æ æ æ æ æ æ æ

0 1 2 3 4

Ω s rad

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0 10 20 30 40

|F (ω)|

0 1 2 3 4

Ω s rad

FR(ω)

0 1 2 3 4

Ω s rad

FI(ω)

æ ææ

0 1 2 3 4

Ω s rad

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Duhamel’sintegral

0 10 20 30 40

|F (ω)|

0 1 2 3 4

Ω s rad

FR(ω)

0 1 2 3 4

Ω s rad

FI(ω)

æ ææ

0 1 2 3 4

Ω s rad

StructuralDynamics

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FourierTransform

Duhamel’sintegral

0 10 20 30 40

|F (ω)|

0 1 2 3 4

Ω s rad

FR(ω)

0 1 2 3 4

Ω s rad

FI(ω)

æ ææ

0 1 2 3 4

Ω s rad

StructuralDynamics

Fourier Series

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Duhamel’sintegral

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

Fourier Series

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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) =

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⟨

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) =

F (ω)

∴(−ω2 + 2 ı ζ0 ω0 ω + ω2

U(ω) =1m

F (ω)

StructuralDynamics

Fourier Series

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Duhamel’sintegral

u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =

f (t) (44)

F⟨u(t) + 2 ζ0 ω0 u(t) + ω2

0 u(t)⟩

= F⟨

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) =

F (ω)

∴(−ω2 + 2 ı ζ0 ω0 ω + ω2

U(ω) =1m

F (ω)

StructuralDynamics

Fourier Series

FourierTransform

Duhamel’sintegral

u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =

f (t) (44)

F⟨u(t) + 2 ζ0 ω0 u(t) + ω2

0 u(t)⟩

= F⟨

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) =

F (ω)

∴(−ω2 + 2 ı ζ0 ω0 ω + ω2

U(ω) =1m

F (ω)

StructuralDynamics

Fourier Series

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Duhamel’sintegral

u(t) + 2 ζ0 ω0 u(t) + ω20 u(t) =

f (t) (44)

F⟨u(t) + 2 ζ0 ω0 u(t) + ω2

0 u(t)⟩

= F⟨

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) =

F (ω)

∴(−ω2 + 2 ı ζ0 ω0 ω + ω2

U(ω) =1m

F (ω)

StructuralDynamics

Fourier Series

FourierTransform

Duhamel’sintegral

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

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)

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

StructuralDynamics

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Duhamel’sintegral

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

Ω Ω0

02 IXH\

ÈH È

StructuralDynamics

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Duhamel’sintegral

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)

4 Compute the dynamic response in the time domain at discrete time instantstr through the Inverse DFT (IDFT):

u(tr ) ≈ IDFT〈Us〉 =1n

s=1Us e2π ı(r−1)(s−1)/n (51)

StructuralDynamics

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Duhamel’sintegral

r=1fr e2π ı(r−1)(s−1)/n (49)

H(ω) =(ω2

0 − ω2 + 2 ı ζ0 ω0 ω

)−1(48)

s=1Us e2π ı(r−1)(s−1)/n (51)

StructuralDynamics

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Duhamel’sintegral

r=1fr e2π ı(r−1)(s−1)/n (49)

H(ω) =(ω2

0 − ω2 + 2 ı ζ0 ω0 ω

)−1(48)

s=1Us e2π ı(r−1)(s−1)/n (51)

StructuralDynamics

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Duhamel’sintegral

r=1fr e2π ı(r−1)(s−1)/n (49)

H(ω) =(ω2

0 − ω2 + 2 ı ζ0 ω0 ω

)−1(48)

s=1Us e2π ı(r−1)(s−1)/n (51)

StructuralDynamics

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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) =

∫ +∞

∞h(t − τ) f (τ) dτ

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

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Duhamel’sintegral

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

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

StructuralDynamics

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Duhamel’sintegral

Green’s function for ζ0 = 0.05

-20 0 20 40 60-1.0

StructuralDynamics

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Duhamel’sintegral

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

Fourier Series

FourierTransform

Duhamel’sintegral

StructuralDynamics

Fourier Series

FourierTransform

Duhamel’sintegral

sdee: lecture 3

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