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# SDEE: Lecture 3

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### SDEE: Lecture 3

1. 1. Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Structural Dynamics & Earthquake Engineering Lectures #3 and 4: Fourier Analysis + Frequency Response Function for SDoF oscillators Fast Fourier Transform Duhamel’s integral Dr Alessandro Palmeri Civil and Building Engineering @ Loughborough University Tuesday, 11th February 2014
2. 2. Intended Learning Outcomes Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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. 3. Recap of Last-Week Key Learning Points Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Unforced Undamped SDoF Oscillator Equation of motion (forces): ¨ m u (t) + k u(t) = 0 (1) Equation of motion (accelerations): 2 ¨ u (t) + ω0 u(t) = 0 (2) Natural circular frequency of vibration: k (3) m Time history of the dynamic response u(t) for given initial displacement u(0) = u0 and initial velocity ˙ u(0) = v0 : v0 u(t) = u0 cos(ω0 t) + sin(ω0 t) (4) ω0 ω0 =
4. 4. Recap of Last-Week Key Learning Points Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Unforced Damped SDoF Oscillator Equation of motion (forces): ¨ ˙ m u (t) + c u(t) + k u(t) = 0 (5) Equation of motion (accelerations): 2 ¨ ˙ u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = 0 (6) Viscous damping ratio and reduced (or damped) natural circular frequency: c <1 (7) ζ0 = 2 m ω0 ω0 = 2 1 − ζ0 ω0 (8) Time history for given initial conditions: u(t) = e−ζ0 ω0 t u0 cos(ω 0 t) + v0 + ζ0 ω0 u0 sin(ω 0 t) ω0 (9)
5. 5. Recap of Last-Week Key Learning Points Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Harmonically Forced SDoF Oscillator (1/2) Equation of motion (forces): ¨ ˙ m u (t) + c u(t) + k u(t) = F0 sin(ωf t) (10) Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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 C 1 and C 2 ): uh (t) = e−ζ0 ω0 t C 1 cos(ω 0 t) + C 2 sin(ω 0 t) (12)
6. 6. Recap of Last-Week Key Learning Points Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Harmonically Forced SDoF Oscillator (2/2) Particular integral: up (t) = ust D(β) sin(ωf + ϕp ) (13) Static displacement and frequency ratio: Fourier Series F0 k ωf β= ω0 Fourier Transform Fast Fourier Transform Duhamel’s integral (14) ust = (15) Dynamic ampliﬁcation factor and phase lag: 1 D(β) = (1 − 2 β2) tan(ϕp ) = (16) + (2 ζ0 β) 2 ζ0 β 1 − β2 2 (17)
7. 7. Recap of Last-Week Key Learning Points Structural Dynamics & Earthquake Engineering Dynamic Ampliﬁcation Factor 50.0 Dr Alessandro Palmeri Ζ0 0 Ζ0 0.05 Recap Fourier Series Fourier Transform 10.0 Ζ0 0.10 5.0 Ζ0 0.20 Ζ0 0.50 Fast Fourier Transform Duhamel’s integral 1.0 0.5 0.0 0.5 1.0 1.5 Β 2.0 2.5 3.0
8. 8. Recap of Last-Week Key Learning Points Structural Dynamics & Earthquake Engineering Phase lag (= Phase of the steady-state response − Phase of the forcing harmonic) Dr Alessandro Palmeri Π Π Recap Fourier Series 3Π 3Π 4 4 Fast Fourier Transform P Fourier Transform Π Π Ζ0 0 2 Duhamel’s integral 2 Ζ0 0.05 Ζ0 0.10 Π Π 4 4 Ζ0 0.20 Ζ0 0.50 0 0 0.0 0.5 1.0 1.5 Β 2.0 2.5 3.0
9. 9. Fourier Series Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Jean Baptiste Joseph Fourier (21 Mar 1768 – 16 May 1830)
10. 10. Fourier Series Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral We have obtained a closed-form solution for the dynamic response of SDoF oscillators subjected to harmonic excitation
11. 11. Fourier Series Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri We have obtained a closed-form solution for the dynamic response of SDoF oscillators subjected to harmonic excitation Recap How can we extend such solution to a more general case? Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral
12. 12. Fourier Series Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri We have obtained a closed-form solution for the dynamic response of SDoF oscillators subjected to harmonic excitation Recap How can we extend such solution to a more general case? Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Since the dynamic system is linear, the superposition principle holds
13. 13. Fourier Series Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri We have obtained a closed-form solution for the dynamic response of SDoF oscillators subjected to harmonic excitation Recap How can we extend such solution to a more general case? Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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
14. 14. Fourier Series Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri We have obtained a closed-form solution for the dynamic response of SDoF oscillators subjected to harmonic excitation Recap How can we extend such solution to a more general case? Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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
15. 15. Fourier Series Structural Dynamics & Earthquake Engineering If the forcing function f (t) is periodic with period Tp : n f (t) = F0 + Dr Alessandro Palmeri Recap Fj sin(Ωj t + Φj ) = f (t + Tp ) where: F0 = Fourier Series Fourier Transform Fj = Fast Fourier Transform Duhamel’s integral (18) j=1 aj2 + bj2 (for j ≥ 1) tan(Φj ) = in which: aj = bj = 2 Tp 2 Tp a0 2 aj bj (for j ≥ 1) (19) (20) (21) Tp f (t) cos(Ωj t) dt (for j ≥ 0) (22) f (t) sin(Ωj t) dt (for j ≥ 1) (23) 0 Tp 0 Ωj = j 2π Tp (24)
16. 16. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a square wave of unitary amplitude and period Tp = 2 s with an increasing number n of harmonic terms n=1 Dr Alessandro Palmeri 1.0 Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 0.5 force kN Recap 0.0 0.5 1.0 0 1 2 time s 3 4
17. 17. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a square wave of unitary amplitude and period Tp = 2 s with an increasing number n of harmonic terms n=1 Dr Alessandro Palmeri n=3 Fourier Transform Fast Fourier Transform Duhamel’s integral 0.5 force kN Fourier Series 1.0 0.5 force kN 1.0 Recap 0.0 0.5 0.0 0.5 1.0 1.0 0 1 2 time s 3 4 0 1 2 time s 3 4
18. 18. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a square wave of unitary amplitude and period Tp = 2 s with an increasing number n of harmonic terms n=1 Dr Alessandro Palmeri n=3 Fourier Transform 0.5 force kN Fourier Series 1.0 0.5 force kN 1.0 Recap 0.0 0.5 0.5 1.0 Fast Fourier Transform 1.0 0 1 2 time s 3 4 3 4 n=5 Duhamel’s integral 1.0 0.5 force kN 0.0 0.0 0.5 1.0 0 1 2 time s 0 1 2 time s 3 4
19. 19. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a square wave of unitary amplitude and period Tp = 2 s with an increasing number n of harmonic terms n=1 Dr Alessandro Palmeri n=3 Fourier Transform 0.5 force kN Fourier Series 1.0 0.5 force kN 1.0 Recap 0.0 0.5 0.5 1.0 Fast Fourier Transform 0.0 1.0 0 1 2 time s 3 4 0 1 n=5 Duhamel’s integral 3 4 3 4 n = 15 1.0 0.5 0.5 force kN 1.0 force kN 2 time s 0.0 0.5 0.0 0.5 1.0 1.0 0 1 2 time s 3 4 0 1 2 time s
20. 20. Fourier Series Structural Dynamics & Earthquake Engineering 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: Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral p(t) = p0 , p0 + ∆p e−t/τ 1 − t τ if t < 0 , 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
21. 21. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a Friedlander waveform (p0 = 0, ∆p = 100kPa, τ = 0.01 s) with an increasing number n of harmonic terms n = 10 Dr Alessandro Palmeri 100 Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral pressure kPa 80 Recap 60 40 20 0 20 3 2 1 0 time ds 1 2 3
22. 22. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a Friedlander waveform (p0 = 0, ∆p = 100kPa, τ = 0.01 s) with an increasing number n of harmonic terms n = 10 Dr Alessandro Palmeri n = 20 Fast Fourier Transform Duhamel’s integral 80 pressure kPa Fourier Transform pressure kPa Fourier Series 100 80 Recap 100 60 40 20 0 20 60 40 20 0 3 2 1 0 time ds 1 2 3 20 3 2 1 0 time ds 1 2 3
23. 23. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a Friedlander waveform (p0 = 0, ∆p = 100kPa, τ = 0.01 s) with an increasing number n of harmonic terms n = 10 Dr Alessandro Palmeri n = 20 80 pressure kPa Fourier Transform pressure kPa Fourier Series 100 80 Recap 100 60 40 20 0 20 Fast Fourier Transform 2 1 0 time ds 1 2 3 1 2 3 100 80 pressure kPa 40 20 0 3 n = 40 Duhamel’s integral 60 60 40 20 0 3 2 1 0 time ds 20 3 2 1 0 time ds 1 2 3
24. 24. Fourier Series Structural Dynamics & Earthquake Engineering Approximating a Friedlander waveform (p0 = 0, ∆p = 100kPa, τ = 0.01 s) with an increasing number n of harmonic terms n = 10 Dr Alessandro Palmeri n = 20 pressure kPa 80 60 40 20 0 20 Fast Fourier Transform 60 40 20 0 3 2 1 0 time ds 1 2 20 3 3 2 100 0 time ds 1 2 3 1 2 3 100 80 pressure kPa 1 n = 80 n = 40 Duhamel’s integral 80 pressure kPa Fourier Transform pressure kPa Fourier Series 100 80 Recap 100 60 40 20 0 60 40 20 0 3 2 1 0 time ds 1 2 3 3 2 1 0 time ds
25. 25. Fourier Series Structural Dynamics & Earthquake Engineering Once the forcing signal is expressed as: n f (t) = F0 + Dr Alessandro Palmeri Recap Fj sin(Ωj t + Φj ) The dynamic response can be evaluated as: Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral (18) j=1 u(t) = uh (t) + where: uj = in which: F0 + k n uj (t) Fj D(βj ) sin(Ωj t + Φj + ϕj ) k Ωj ω0 2 ζ 0 βj tan(ϕj ) = 1 − βj2 βj = (26) j=1 (27) (28) (29)
26. 26. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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
27. 27. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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)
28. 28. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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)
29. 29. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap 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: Fourier Series F (ω) = FR (ω) + ı FI (ω) Fourier Transform Fast Fourier Transform Duhamel’s integral 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) 2 |F (ω)| = FR (ω) + FI2 (ω) is the absolute value (or modulus) of F (ω) (30)
30. 30. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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:
31. 31. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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 2π +∞ −∞ F (ω) eı ω t dω (32)
32. 32. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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 2π +∞ 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)
33. 33. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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
34. 34. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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) = Kinetic energy in a SDoF oscillator: T (t) = 1 2 1 2 k u 2 (t) ˙ m u 2 (t)
35. 35. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series 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: Fourier Transform Fast Fourier Transform Duhamel’s integral E= 1 α 2 +∞ −∞ f (t)2 dt = 1 α 2π +∞ |F (ω)|2 dω (34) 0 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)
36. 36. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Example: For illustration purposes, let us consider the following signal in the time domain: 2 f (t) = F0 e−(t/τ ) 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) = √ 2 2 2 π τ F0 e−τ (ω +Ω )/4 cosh 1 Ω τ2 ω 2 (36)
37. 37. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the time scale τ = 1, 3, 5 s (while Ω = 1 rad/s) f (t) Dr Alessandro Palmeri 1.0 Τ 1s Τ 3s Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral f F0 0.5 Τ 5s 0.0 0.5 10 5 0 5 s 10
38. 38. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the time scale τ = 1, 3, 5 s (while Ω = 1 rad/s) |F (ω)| f (t) Dr Alessandro Palmeri 1.0 4 Τ 1s Τ 1s Τ 3s Fourier Series 0.0 Τ 3s 2 Τ 5s F0 3 Τ 5s F Recap f F0 0.5 1 Fourier Transform Fast Fourier Transform Duhamel’s integral 0.5 0 10 5 0 5 s 10 0 2 4 6 Ω s rad 8 10 12
39. 39. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the time scale τ = 1, 3, 5 s (while Ω = 1 rad/s) |F (ω)| f (t) Dr Alessandro Palmeri 1.0 4 Τ 1s Τ 1s Τ 3s Fourier Series 0.0 Τ 3s 2 Τ 5s F0 3 Τ 5s F Recap f F0 0.5 1 0.5 Fourier Transform 0 10 5 0 5 10 s Fast Fourier Transform f (t)2 Duhamel’s integral 1.0 Τ 1s 2 f 2 F0 0.8 Τ 3s 0.6 Τ 5s 0.4 0.2 0.0 10 5 0 5 s 10 0 2 4 6 Ω s rad 8 10 12
40. 40. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the time scale τ = 1, 3, 5 s (while Ω = 1 rad/s) |F (ω)| f (t) Dr Alessandro Palmeri 1.0 4 Τ 1s Τ 1s Τ 3s Fourier Series 0.0 Τ 3s 2 Τ 5s F0 3 Τ 5s F Recap f F0 0.5 1 0.5 Fourier Transform 0 10 5 0 5 0 10 2 4 s Fast Fourier Transform 8 10 12 |F (ω)|2 /π f (t)2 Duhamel’s integral 6 Ω s rad 1.0 6 Τ 1s 5 0.4 2 Π F0 F2 2 f 2 F0 Τ 5s Τ 1s Τ 3s Τ 5s Τ 3s 0.6 4 3 0.8 0.2 2 1 0 0.0 10 5 0 5 s 10 0 2 4 6 Ω s rad 8 10 12
41. 41. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the circular frequency Ω = 1, 3 rad/s (while τ = 1 s) f (t) Dr Alessandro Palmeri 1.0 1 rad s Fourier Transform f F0 Recap Fourier Series 5 rad.s 0.5 0.0 0.5 1.0 Fast Fourier Transform Duhamel’s integral 10 5 0 5 s 10
42. 42. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the circular frequency Ω = 1, 3 rad/s (while τ = 1 s) |F (ω)| f (t) Dr Alessandro Palmeri 1.0 4 1 rad s 1 rad s Fourier Transform F0 0.0 0.5 Duhamel’s integral 2 1 0 1.0 Fast Fourier Transform 5 rad.s 3 F f F0 Recap Fourier Series 5 rad.s 0.5 10 5 0 5 s 10 0 2 4 6 Ω s rad 8 10 12
43. 43. Fourier Transform Structural Dynamics & Earthquake Engineering Effects of changing the circular frequency Ω = 1, 3 rad/s (while τ = 1 s) |F (ω)| f (t) Dr Alessandro Palmeri 1.0 4 1 rad s 1 rad s F0 0.0 0.5 Fourier Transform 0 10 Fast Fourier Transform 2 1 1.0 5 0 5 10 s f (t)2 Duhamel’s integral 1.0 1 rad s 0.8 2 f 2 F0 5 rad.s 3 F f F0 Recap Fourier Series 5 rad.s 0.5 5 rad.s 0.6 0.4 0.2 0.0 10 5 0 5 s 10 0 2 4 6 Ω s rad 8 10 12
45. 45. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral The FT enjoys a number of important properties, including: Linearity: F a1 f1 (t) + a2 f2 (t) = a1 F1 (ω) + a2 F2 (ω) (37)
46. 46. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series The FT enjoys a number of important properties, including: Linearity: F a1 f1 (t) + a2 f2 (t) = a1 F1 (ω) + a2 F2 (ω) (37) Fourier Transform Fast Fourier Transform Duhamel’s integral Time shift: F f (t − τ ) = e−ı ω τ F (ω) (38)
47. 47. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series The FT enjoys a number of important properties, including: Linearity: F a1 f1 (t) + a2 f2 (t) = a1 F1 (ω) + a2 F2 (ω) (37) Fourier Transform Fast Fourier Transform Time shift: F f (t − τ ) = e−ı ω τ F (ω) Duhamel’s integral (38) Time scaling: F f (α t) = 1 ω F |α| α (39)
48. 48. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Moreover (very importantly): Derivation rule: Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral F dn f (t) dt n = (ı ω)n F (ω) (40)
49. 49. Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Moreover (very importantly): Derivation rule: Recap Fourier Series F Fourier Transform Fast Fourier Transform dn f (t) dt n = (ı ω)n F (ω) (40) Convolution rule: Duhamel’s integral +∞ F {f ∗ g}(t) = f (t) g(t − τ ) dτ −∞ +∞ f (t − τ ) g(t) dτ = −∞ = F (ω) G(ω) (41)
50. 50. Fast Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series 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 r th time instant Fast Fourier Transform r = 1, 2 · · · , n is the index in the time domain Duhamel’s integral ∆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)
51. 51. Fast Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series 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 r th time instant Fast Fourier Transform r = 1, 2 · · · , n is the index in the time domain Duhamel’s integral ∆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?
52. 52. Fast Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral The answer is yes...
53. 53. Fast Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series 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 implements 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) Fourier Transform Fast Fourier Transform The DFT is deﬁned as follows: n fr e2 π ı(r −1)(s−1)/n Fs = DFT fr = Duhamel’s integral (42) r =1 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
54. 54. Fast Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap 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: Fourier Series Fs ≈ F (ωs ) Fourier Transform Fast Fourier Transform Duhamel’s integral (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
55. 55. Fast Fourier Transform Structural Dynamics & Earthquake Engineering 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) Dr Alessandro Palmeri 1.0 0.8 Recap Fourier Transform fmax 0.6 0.4 f Fourier Series 0.2 0.0 0.2 0.4 Fast Fourier Transform Duhamel’s integral 0 10 20 s 30 40
56. 56. Fast Fourier Transform Structural Dynamics & Earthquake Engineering 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 (ω)| f (t) Dr Alessandro Palmeri 1.0 2.5 0.8 Recap 2.0 Fourier Transform fmax fmax 1.5 0.2 F Fourier Series 0.4 f 0.6 1.0 0.0 0.5 0.2 0.4 Fast Fourier Transform Duhamel’s integral 0.0 0 10 20 s 30 40 0 1 2 Ω s rad 3 4
57. 57. Fast Fourier Transform Structural Dynamics & Earthquake Engineering 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 (ω)| f (t) Dr Alessandro Palmeri 1.0 2.5 0.8 Recap 2.0 fmax fmax 1.5 0.2 F Fourier Series 0.4 f 0.6 1.0 0.0 Fourier Transform 0.5 0.2 0.4 Fast Fourier Transform 0.0 0 10 20 s 30 40 FR (ω) Duhamel’s integral 2 FR fmax 1 0 1 2 0 1 2 Ω s rad 3 4 0 1 2 Ω s rad 3 4
58. 58. Fast Fourier Transform Structural Dynamics & Earthquake Engineering 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 (ω)| f (t) Dr Alessandro Palmeri 1.0 2.5 0.8 Recap 2.0 fmax fmax 1.5 0.2 F Fourier Series 0.4 f 0.6 1.0 0.0 Fourier Transform 0.5 0.2 0.4 Fast Fourier Transform 0.0 0 10 20 s 30 40 0 1 FR (ω) Duhamel’s integral 3 4 3 4 FI (ω) 2 1 1 fmax 2 0 0 FI FR fmax 2 Ω s rad 1 1 2 2 0 1 2 Ω s rad 3 4 0 1 2 Ω s rad
59. 59. Fast Fourier Transform Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral 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
60. 60. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri The equation of motion for a SDoF oscillator in the time domain reads: 1 2 ¨ ˙ u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = f (t) (44) m Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral By applying the FT operator to both sides of Eq. (44), one obtains: 2 ¨ ˙ F u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = F 1 f (t) m 1 F f (t) m 1 2 ∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω0 U(ω) = F (ω) m 1 2 ∴ −ω 2 + 2 ı ζ0 ω0 ω + ω0 U(ω) = F (ω) m 2 ¨ ˙ ∴ F u (t) + 2 ζ0 ω0 F u(t) + ω0 F u(t) = (45)
61. 61. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri The equation of motion for a SDoF oscillator in the time domain reads: 1 2 ¨ ˙ u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = f (t) (44) m Recap Fourier Series Fourier Transform Fast Fourier Transform By applying the FT operator to both sides of Eq. (44), one obtains: 2 ¨ ˙ F u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = F 1 f (t) m Duhamel’s integral 1 F (ω) m 1 U(ω) = F (ω) m 2 ∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω0 U(ω) = ∴ 2 −ω 2 + 2 ı ζ0 ω0 ω + ω0 (45)
62. 62. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri The equation of motion for a SDoF oscillator in the time domain reads: 1 2 ¨ ˙ u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = f (t) (44) m Recap Fourier Series Fourier Transform Fast Fourier Transform By applying the FT operator to both sides of Eq. (44), one obtains: 2 ¨ ˙ F u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = F 1 f (t) m Duhamel’s integral (45) ∴ 2 −ω 2 + 2 ı ζ0 ω0 ω + ω0 U(ω) = 1 F (ω) m
63. 63. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri The equation of motion for a SDoF oscillator in the time domain reads: 1 2 ¨ ˙ u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = f (t) (44) m Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral By applying the FT operator to both sides of Eq. (44), one obtains: 2 ¨ ˙ F u (t) + 2 ζ0 ω0 u(t) + ω0 u(t) = F 1 f (t) m 1 F f (t) m 1 2 ∴ (ı ω)2 U(ω) + 2 ζ0 ω0 (ı ω) U(ω) + ω0 U(ω) = F (ω) m 1 2 ∴ −ω 2 + 2 ı ζ0 ω0 ω + ω0 U(ω) = F (ω) m 2 ¨ ˙ ∴ F u (t) + 2 ζ0 ω0 F u(t) + ω0 F u(t) = (45)
64. 64. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral The equation of motion in the frequency domain (the last of Eqs. (45)) has been posed in the form: 2 −ω 2 + 2 ı ζ0 ω0 ω + ω0 U(ω) = 1 F (ω) m (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: −1 2 H(ω) = ω0 − ω 2 + 2 ı ζ0 ω0 ω (48)
65. 65. Frequency Response Function Structural Dynamics & Earthquake Engineering FRF for ζ0 = 0.05 – Note that: |H(ω)| = D(ω/ω0 ) 10 Dr Alessandro Palmeri H Recap H 5 Fourier Transform Fast Fourier Transform Duhamel’s integral H Ω2 0 Fourier Series H 0 5 10 0.0 0.5 1.0 1.5 Ω Ω0 2.0 2.5 3.0
66. 66. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 = Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral n r =1 fr e2 π ı(r −1)(s−1)/n (49)
67. 67. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 = Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral n r =1 fr e2 π ı(r −1)(s−1)/n (49) 2 Deﬁne analytically the complex-valued FRF of the oscillator: 2 H(ω) = ω0 − ω 2 + 2 ı ζ0 ω0 ω −1 (48)
68. 68. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 = Recap Fourier Series Fourier Transform n r =1 fr e2 π ı(r −1)(s−1)/n (49) 2 Deﬁne analytically the complex-valued FRF of the oscillator: 2 H(ω) = ω0 − ω 2 + 2 ı ζ0 ω0 ω −1 (48) Fast Fourier Transform Duhamel’s integral 3 Compute the dynamic response in the frequency domain (see Eq. (47)): U(ωs ) = H(ωs ) F (ωs ) m (50)
69. 69. Frequency Response Function Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri 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 = Recap Fourier Series Fourier Transform n r =1 fr e2 π ı(r −1)(s−1)/n (49) 2 Deﬁne analytically the complex-valued FRF of the oscillator: 2 H(ω) = ω0 − ω 2 + 2 ı ζ0 ω0 ω −1 (48) Fast Fourier Transform Duhamel’s integral 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)
70. 70. Duhamel’s integral Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series If we now apply the convolution rule of Eq. (41) to the dynamic response in the frequency domain (see Eq. (47)), given as a product of two complex-valued function of the circular frequency ω, the dynamic response for a generic excitation f (t) can be derived in the time domain as follows: Fourier Transform Fast Fourier Transform Duhamel’s integral 1 −1 F H(ω) F (ω) m +∞ 1 1 = {h ∗ f }(t) = h(t − τ ) f (τ ) dτ m m ∞ u(t) = F −1 U(ω) = (52) 1 where the function h(t) = m F −1 (ω) is the IFT of the FRF, and constitutes the Green’s function for the equation of motion of a SDoF oscillator
71. 71. Duhamel’s integral Structural Dynamics & Earthquake Engineering This integral solution in the time domain is called Duhamel’s integral, and furnishes a way for evaluating the dynamic response to a generic forcing function: Dr Alessandro Palmeri u(t) = Recap +∞ 1 m h(t − τ ) f (τ ) dτ (53) ∞ Fourier Series Fourier Transform It can be shown that: h(t) = F −1 H(ω) = Fast Fourier Transform Duhamel’s integral 1.0 0.8 Η 0.6 0.4 0.2 0.0 2 1 0 t 1 2 1 −ζ0 ω0 t e sin(ω 0 t) η(t) ω0 (54) in which η(t) is the so-called Heaviside’s Unit Step Function, so deﬁned:  0 , if t < 0  η(t) = 1 , if t = 0 (55) 2  1 , if t > 0
72. 72. Duhamel’s integral Structural Dynamics & Earthquake Engineering Green’s function for ζ0 = 0.05 1.0 Dr Alessandro Palmeri Recap 0.5 Fourier Transform Fast Fourier Transform h Ω2 0 Fourier Series 0.0 Duhamel’s integral 0.5 1.0 20 0 20 Ω0 40 60
73. 73. Duhamel’s integral Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Final observation for today...
74. 74. Duhamel’s integral Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Final observation for today... If you compare Eq. (9), which gives the time history for given initial conditions, with the Green’s function of Eq. (54), it appears that the function h(t) describes the free vibration of a SDoF oscillator with initial conditions u(0) = 0 and ˙ u(0) = 1
75. 75. Duhamel’s integral Structural Dynamics & Earthquake Engineering Dr Alessandro Palmeri Recap Fourier Series Fourier Transform Fast Fourier Transform Duhamel’s integral Final observation for today... If you compare Eq. (9), which gives the time history for given initial conditions, with the Green’s function of Eq. (54), it appears that the function h(t) describes the free vibration of a SDoF oscillator with initial conditions u(0) = 0 and ˙ u(0) = 1 Since such initial unit velocity could have been produced by an impulsive force applied at t = 0, the function h(t) is often called Impulse Response Function (IRF), and indeed plays a fundamental role in the dynamic analysis of linear structures
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