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- 1. 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
- 2. 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
- 3. 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)
- 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)
- 5. 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)
- 6. 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)
- 7. 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
- 8. 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
- 9. 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
- 10. 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
- 11. 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)
- 12. 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
- 13. 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
- 14. 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
- 15. 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)
- 16. 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)
- 17. 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)
- 18. 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)
- 19. 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)
- 20. 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)
- 21. 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)
- 22. 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
- 23. 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
- 24. 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)
- 25. 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)
- 26. 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?
- 27. 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
- 28. 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
- 29. 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
- 30. 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
- 31. 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)
- 32. 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)
- 33. 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
- 34. 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)

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