Alessandro Palmeri, profile picture
Uploaded byAlessandro Palmeri
PDF, PPTX16,223 views

SDEE: Lectures 3 and 4

This document summarizes key points from lectures on Fourier analysis and frequency response functions for single degree of freedom oscillators. It recaps concepts like natural frequency, damping ratio, and dynamic amplification factor. It introduces Fourier series as a way to decompose periodic signals into harmonic components. The Fourier transform is presented as a way to study the frequency content of both periodic and non-periodic signals. Examples are given to illustrate the effects of varying parameters in the time and frequency domains.

Embed presentation

Download as PDF, PPTX
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Structural Dynamics & Earthquake Engineering Lectures #3 and 4: Fourier Analysis + Frequency Response Function for SDoF oscillators Dr Alessandro Palmeri Civil and Building Engineering @ Loughborough University Friday, 6th October 2017
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap of Last-Week Key Learning Points Harmonically Forced SDoF Oscillator (2/2) Particular integral: up(t) = ust D(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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 Ζ0 0.50 Ζ0 0.20 Ζ0 0.10 Ζ0 0.05 Ζ0 0
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 ω α (39)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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?
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 HΩ0 2 H H H
Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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)

More Related Content

PPT
Vdocument.in vibration chapter05
bysujann5
 
PDF
Multiple Degree of Freedom (MDOF) Systems
byMohammad Tawfik
 
DOCX
Concreto Armado I
byerasmo palmadera
 
PPTX
Thirsty concrete (2)
byKarthi Kavya
 
PDF
Module4 s dynamics- rajesh sir
bySHAMJITH KM
 
PPTX
Chapter 2 seviceability and durability
byAnkit Patel
 
PPT
Earth retaining 2
byMarlizaAshiqin
 
PDF
Chapter 03
bySantistebanCampos
 
Vdocument.in vibration chapter05
bysujann5
 
Multiple Degree of Freedom (MDOF) Systems
byMohammad Tawfik
 
Concreto Armado I
byerasmo palmadera
 
Thirsty concrete (2)
byKarthi Kavya
 
Module4 s dynamics- rajesh sir
bySHAMJITH KM
 
Chapter 2 seviceability and durability
byAnkit Patel
 
Earth retaining 2
byMarlizaAshiqin
 
Chapter 03
bySantistebanCampos
 

What's hot

PDF
Ground Response Analysis - Geotechnical Earthquake engineering
byCristian Soriano-Camelo
 
PDF
psad 1.pdf
byRobin Arthur Flores
 
PPSX
Geotechnical Engineering-II [Lec #14: Timoshenko & Goodier Method]
byMuhammad Irfan
 
PPTX
Moving load
byAbdul Hai
 
PPTX
Self compacting concrete (scc)
byy Sahaja
 
PDF
TERZAGHI’S BEARING CAPACITY THEORY
bySANJEEV Wazir
 
PDF
Single Degree of Freedom Systems
byMohammad Tawfik
 
DOCX
Accelerated Bridge Construction( ABC)
byvarun km
 
PDF
Module 6, Spring 2020.pdf
byMohammad Javed
 
PDF
Portal and Cantilever Beam Method
bymisschand
 
PPTX
Constitutive modelling of soil (Duncan-chang Model)
byMostafa Abedi
 
PDF
Compressive strength of concrete cubs
byMuhammad Saleem
 
PDF
Lecture notes 05
byYakup Kocaman
 
DOCX
Truss problems
byPralhad Kore
 
PDF
Module4 s dynamics- rajesh sir
bySHAMJITH KM
 
PPTX
Lec 11 forces on vanes and turbo machinery 02 -fm 2
byCivil Zone
 
PPTX
Gravity Dam (numerical problem ) BY SITARAM SAINI
bySitaramSaini11
 
PPTX
Shear force and bending moment Solved Numerical
byRoyal University of Bhutan
 
PPSX
Geotechnical Engineering-II [Lec #26: Slope Stability]
byMuhammad Irfan
 
PDF
Concrete technology slides all.pdf
byRoshani Shahi
 
Ground Response Analysis - Geotechnical Earthquake engineering
byCristian Soriano-Camelo
 
psad 1.pdf
byRobin Arthur Flores
 
Geotechnical Engineering-II [Lec #14: Timoshenko & Goodier Method]
byMuhammad Irfan
 
Moving load
byAbdul Hai
 
Self compacting concrete (scc)
byy Sahaja
 
TERZAGHI’S BEARING CAPACITY THEORY
bySANJEEV Wazir
 
Single Degree of Freedom Systems
byMohammad Tawfik
 
Accelerated Bridge Construction( ABC)
byvarun km
 
Module 6, Spring 2020.pdf
byMohammad Javed
 
Portal and Cantilever Beam Method
bymisschand
 
Constitutive modelling of soil (Duncan-chang Model)
byMostafa Abedi
 
Compressive strength of concrete cubs
byMuhammad Saleem
 
Lecture notes 05
byYakup Kocaman
 
Truss problems
byPralhad Kore
 
Module4 s dynamics- rajesh sir
bySHAMJITH KM
 
Lec 11 forces on vanes and turbo machinery 02 -fm 2
byCivil Zone
 
Gravity Dam (numerical problem ) BY SITARAM SAINI
bySitaramSaini11
 
Shear force and bending moment Solved Numerical
byRoyal University of Bhutan
 
Geotechnical Engineering-II [Lec #26: Slope Stability]
byMuhammad Irfan
 
Concrete technology slides all.pdf
byRoshani Shahi
 

Similar to SDEE: Lectures 3 and 4

PDF
CVD020 - Lecture Week 2
byAlessandro Palmeri
 
PDF
Master Thesis on Rotating Cryostats and FFT, DRAFT VERSION
byKaarle Kulvik
 
PDF
Eece 301 note set 14 fourier transform
bySandilya Sridhara
 
PDF
Circuit Network Analysis - [Chapter3] Fourier Analysis
bySimen Li
 
PDF
MEG_453-Lecture_02_Vibration_Systems_Basic_Concepts.pdf
bynazzybunny
 
PPT
Nt lecture skm-iiit-bh
byIIIT Bhubaneswar
 
PPT
DISP-2003_L01B_20Feb03.ppt
bygopal556359
 
PPTX
Fourier_series_Lec 1(how to find FS coeffiecients).pptx
byjunaidzaheer2311
 
PPTX
SP_SNS_C2.pptx
byIffahSkmd
 
PDF
DTS 01discrete-fourier-transform (1).pdf
byvikalankit84710
 
PPTX
Eq Decipher
byNAXUTL, StratGlobal, Nature Acoustical
 
PDF
Find the compact trigonometric Fourier series for the periodic signal.pdf
byarihantelectronics
 
PDF
rcg-ch4a.pdf
byKaviprathi
 
PDF
4. a Find the compact trigonometric Fourier series for the periodic s.pdf
byjovankarenhookeott88
 
PDF
Chapter 3
byFawad Masood
 
PDF
Fourier Transform ppt and material for mathematics subject
byprateekjain415709
 
PPT
fourier transforms
byUmang Gupta
 
PPT
Lecture6 Signal and Systems
bybabak danyal
 
DOCX
Running Head Fourier Transform Time-Frequency Analysis. .docx
bycharisellington63520
 
PDF
3.Frequency Domain Representation of Signals and Systems
byINDIAN NAVY
 
CVD020 - Lecture Week 2
byAlessandro Palmeri
 
Master Thesis on Rotating Cryostats and FFT, DRAFT VERSION
byKaarle Kulvik
 
Eece 301 note set 14 fourier transform
bySandilya Sridhara
 
Circuit Network Analysis - [Chapter3] Fourier Analysis
bySimen Li
 
MEG_453-Lecture_02_Vibration_Systems_Basic_Concepts.pdf
bynazzybunny
 
Nt lecture skm-iiit-bh
byIIIT Bhubaneswar
 
DISP-2003_L01B_20Feb03.ppt
bygopal556359
 
Fourier_series_Lec 1(how to find FS coeffiecients).pptx
byjunaidzaheer2311
 
SP_SNS_C2.pptx
byIffahSkmd
 
DTS 01discrete-fourier-transform (1).pdf
byvikalankit84710
 
Eq Decipher
byNAXUTL, StratGlobal, Nature Acoustical
 
Find the compact trigonometric Fourier series for the periodic signal.pdf
byarihantelectronics
 
rcg-ch4a.pdf
byKaviprathi
 
4. a Find the compact trigonometric Fourier series for the periodic s.pdf
byjovankarenhookeott88
 
Chapter 3
byFawad Masood
 
Fourier Transform ppt and material for mathematics subject
byprateekjain415709
 
fourier transforms
byUmang Gupta
 
Lecture6 Signal and Systems
bybabak danyal
 
Running Head Fourier Transform Time-Frequency Analysis. .docx
bycharisellington63520
 
3.Frequency Domain Representation of Signals and Systems
byINDIAN NAVY
 

More from Alessandro Palmeri

PDF
CVD020 - Lecture Week 1
byAlessandro Palmeri
 
PDF
Complex strains (2nd year)
byAlessandro Palmeri
 
PDF
Complex stresses (2nd year)
byAlessandro Palmeri
 
PDF
Unsymmetrical bending (2nd year)
byAlessandro Palmeri
 
PDF
Using blurred images to assess damage in bridge structures?
byAlessandro Palmeri
 
PDF
Failure of slender and stocky columns (2nd year)
byAlessandro Palmeri
 
PDF
Shear stresses (2nd year)
byAlessandro Palmeri
 
PDF
SDEE: Lecture 6
byAlessandro Palmeri
 
PDF
Structural Mechanics: Shear stress in Beams (1st-Year)
byAlessandro Palmeri
 
PDF
SDEE: Lectures 1 and 2
byAlessandro Palmeri
 
PDF
SDEE: Lecture 1
byAlessandro Palmeri
 
PDF
Structural Mechanics: Deflections of Beams in Bending
byAlessandro Palmeri
 
PPTX
Toward an Improved Computational Strategy for Vibration-Proof Structures Equi...
byAlessandro Palmeri
 
PPTX
Spectrum-Compliant Accelerograms through Harmonic Wavelet Transform
byAlessandro Palmeri
 
PDF
Wind Actions According To EC1
byAlessandro Palmeri
 
PPTX
Toward overcoming the concept of effective stiffness and damping in the dynam...
byAlessandro Palmeri
 
CVD020 - Lecture Week 1
byAlessandro Palmeri
 
Complex strains (2nd year)
byAlessandro Palmeri
 
Complex stresses (2nd year)
byAlessandro Palmeri
 
Unsymmetrical bending (2nd year)
byAlessandro Palmeri
 
Using blurred images to assess damage in bridge structures?
byAlessandro Palmeri
 
Failure of slender and stocky columns (2nd year)
byAlessandro Palmeri
 
Shear stresses (2nd year)
byAlessandro Palmeri
 
SDEE: Lecture 6
byAlessandro Palmeri
 
Structural Mechanics: Shear stress in Beams (1st-Year)
byAlessandro Palmeri
 
SDEE: Lectures 1 and 2
byAlessandro Palmeri
 
SDEE: Lecture 1
byAlessandro Palmeri
 
Structural Mechanics: Deflections of Beams in Bending
byAlessandro Palmeri
 
Toward an Improved Computational Strategy for Vibration-Proof Structures Equi...
byAlessandro Palmeri
 
Spectrum-Compliant Accelerograms through Harmonic Wavelet Transform
byAlessandro Palmeri
 
Wind Actions According To EC1
byAlessandro Palmeri
 
Toward overcoming the concept of effective stiffness and damping in the dynam...
byAlessandro Palmeri
 

Recently uploaded

PPTX
ELIMINATION NEEDS Fundamentals of Nursing .pptx
bySonali Gupta
 
PPTX
How to Set Recurring Tasks in Odoo Projects
byCeline George
 
PPTX
Surface tension is the force acting per unit lenth of thec surface
bysavithakps95
 
PDF
Four Stars Of Destiny By General Manoj Mukund Naravane
byAman744562
 
PDF
A Study of W. H. Auden’s September 1, 1939
bypriyarathod315
 
PPTX
Methods & Applications of Enzyme Immobilization Technique.pptx
byAnupkumar Sharma
 
PPTX
WEEK 2 (2).pptx TLE COOKERY 10 QUARTER 4
byResynFayeCortez2
 
PPTX
Renal Physiology -Nephron.pptx
byDisha Soriya
 
PPTX
Greengnorance Toolkit Module 2 Energy saving
byKarl Donert
 
PDF
Intellectual Property Rights I Types (IPR)
byAmit Gangwal
 
PPTX
A Memory of Two Mondays by Arthur Miller
byDhatriParmar
 
PDF
Holm Community Heritage at St Nicholas Kirk - 2025 AGM Minutes (30.04.2025)
byAntoine Pietri
 
PDF
Power, Propaganda, and Fear: A Study of W. H. Auden’s Epitaph on a Tyrant
bypriyarathod315
 
PDF
Auden’s "In Memory of W.B. Yeats": The Role of Poetry in the Modern World
byKruti Vyas
 
PDF
The Sheep and the Goat: Beckett’s Subversion of Divine Justice in Waiting for...
bysejadchokiya
 
PPTX
Greengnorance Toolkit Module1 Climate Change
byKarl Donert
 
PPT
Card repertory , Boger card , Kishore card
byDr Dhwanika Dhagat
 
PPTX
PRE TERM LABOR ( PREMATURE LABOUR IN PREGNANCY)
byMuthu Kumar
 
PDF
BP801T BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory) Unit 2 Part 1
byRushi Mandali
 
PPTX
Drug acting on ANS.pptx (Drug acts on Sympathetic and parasympathetic NS)
byManarat International University
 
ELIMINATION NEEDS Fundamentals of Nursing .pptx
bySonali Gupta
 
How to Set Recurring Tasks in Odoo Projects
byCeline George
 
Surface tension is the force acting per unit lenth of thec surface
bysavithakps95
 
Four Stars Of Destiny By General Manoj Mukund Naravane
byAman744562
 
A Study of W. H. Auden’s September 1, 1939
bypriyarathod315
 
Methods & Applications of Enzyme Immobilization Technique.pptx
byAnupkumar Sharma
 
WEEK 2 (2).pptx TLE COOKERY 10 QUARTER 4
byResynFayeCortez2
 
Renal Physiology -Nephron.pptx
byDisha Soriya
 
Greengnorance Toolkit Module 2 Energy saving
byKarl Donert
 
Intellectual Property Rights I Types (IPR)
byAmit Gangwal
 
A Memory of Two Mondays by Arthur Miller
byDhatriParmar
 
Holm Community Heritage at St Nicholas Kirk - 2025 AGM Minutes (30.04.2025)
byAntoine Pietri
 
Power, Propaganda, and Fear: A Study of W. H. Auden’s Epitaph on a Tyrant
bypriyarathod315
 
Auden’s "In Memory of W.B. Yeats": The Role of Poetry in the Modern World
byKruti Vyas
 
The Sheep and the Goat: Beckett’s Subversion of Divine Justice in Waiting for...
bysejadchokiya
 
Greengnorance Toolkit Module1 Climate Change
byKarl Donert
 
Card repertory , Boger card , Kishore card
byDr Dhwanika Dhagat
 
PRE TERM LABOR ( PREMATURE LABOUR IN PREGNANCY)
byMuthu Kumar
 
BP801T BIOSTATISITCS AND RESEARCH METHODOLOGY (Theory) Unit 2 Part 1
byRushi Mandali
 
Drug acting on ANS.pptx (Drug acts on Sympathetic and parasympathetic NS)
byManarat International University
 

SDEE: Lectures 3 and 4

  • 1.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri StructuralDynamics & Earthquake Engineering Lectures #3 and 4: Fourier Analysis + Frequency Response Function for SDoF oscillators Dr Alessandro Palmeri Civil and Building Engineering @ Loughborough University Friday, 6th October 2017
  • 2.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri IntendedLearning 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 Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recapof 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)
  • 4.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recapof 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)
  • 5.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recapof 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)
  • 6.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recapof Last-Week Key Learning Points Harmonically Forced SDoF Oscillator (2/2) Particular integral: up(t) = ust D(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)
  • 7.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recapof 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 Ζ0 0.50 Ζ0 0.20 Ζ0 0.10 Ζ0 0.05 Ζ0 0
  • 8.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recapof 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
  • 9.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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 Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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 Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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)
  • 12.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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
  • 13.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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
  • 14.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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
  • 15.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierSeries 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 Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 17.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 18.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 19.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 20.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 21.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 22.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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
  • 23.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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
  • 24.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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 ω α (39)
  • 25.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FourierTransform 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)
  • 26.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FastFourier 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?
  • 27.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FastFourier 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
  • 28.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FastFourier 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
  • 29.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FastFourier 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 Dynamics & Earthquake Engineering Dr Alessandro Palmeri FastFourier 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 Dynamics & Earthquake Engineering Dr Alessandro Palmeri FrequencyResponse 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)
  • 32.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FrequencyResponse 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)
  • 33.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FrequencyResponse 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 HΩ0 2 H H H
  • 34.
    Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri FrequencyResponse 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)