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Aeronautical Engineering - Wind Engineering

Presentazione Wind Engineering

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Titolo presentazione sottotitolo Milano, XX mese 20XX Berizzi Niccolò 10685173 Crotta Alessandro 10621786 Dall'Ora Matteo 10687723 W ind Engineering – A.Y. 2021/2022 Bridge Aerodynamics and Aeroelasticity Abstract • Sectional Model 1. Static Response 2. Stability Analysis 3. Buffeting Response • Full Bridge Model (Modal Approach) 1. Static Response 2. Stability Analysis 2 Titolo presentazione sottotitolo Milano, XX mese 20XX Sectional Model 1. Static Response WIND DECK SECT ION ST RUCTURAL MODEL Sectional Model: Static Response Coordinates : y, z, ϑ (3DOF) 4 Sectional Model: Static Response DECK SECT ION AERODYNAMIC MODEL Static T est → ������ = ������ (AoA ) → ������ ��� = ������ → � = ������ = 0 Τ � � (NO turbulence ) 5 Sectional Model: Static Response AERODYNAMIC COEFFICIENTS • (3 ° order polynomial fitting for CL, CM) • (2 ° order polynomial fitting for CD) STALL REGION EMPIRICAL DATA BORDERS KL>0 KL0 KL0 KM0) • Positive static tors . ϑ (M>0) • Static vertical z changes sign due to initial downforce (L0) • Linearized and non -linear approaches diverging at high ϑ 9 Titolo presentazione sottotitolo Milano, XX mese 20XX Sectional Model 2. Stability Analysis Dynamic Test + Turbulence → ������ = ������ + ψ (AoA ) DECK SECTION AERODYNAMICS (Using linearized QST ) → → (linearization ) Sectional Model: Dynamic Response 11 Sectional Model: Dynamic Response OVERALL FORMULAT ION OF FORCES (Using linearized QST ) • Static contribution • Steady and Quasi -Steady self excited contribution • Buffeting contribution 12 Sectional Model: Dynamic Response SYSTEM STABILITY ANALYSIS • No buffeting effects → u,w = 0 • Mean speed U varying from 0 to 80 m/s • Evaluate the eigenv alues of the dynamic system ( Matlab function: polyeig ) • Check if there are wind speeds for which at least one of the eigenvalues has positive real part: FLUTTER POINTS DECK SECTIONAL SYSTEM IN MATRIX FORM • Static equation has been taken out, perturbative coordinates are considered • Coefficients CD0, CL0, CM0 and slopes KD0, KL0, KM0 are evaluated at static equilibrium position 13 Sectional Model: Dynamic Response • Eigenvalues positions varying with increasing incoming wind speed • A couple of complex conjugate eigenvalues cross the imaginary axis at the speed at a speed of RESULTS U = 66 .3 Τ ������ � FLUTTER SPEED 14 Sectional Model: Dynamic Response At the FLUTTER SPEED U = 66 .3 Τ ������ � the damping ratio associated to one mode becomes negative, i.e. has positive real part: flutter instability 15 Sectional Model: Dynamic Response Red and yellow mode frequencies are getting closer when approaching the flutter speed 16 Titolo presentazione sottotitolo Milano, XX mese 20XX Sectional Model 3. Buffeting Response 3.1 Frequency Domain (PSD) 3.2 Time Domain Sectional Model: Buffeting Response • The buffeting response is the deck section ’s response to the incoming turbulence components u and w • Two strategies: time domain and frequency domain analysis TURBULENCE u w 18 Sectional Model: Buffeting Response TURBULENCE CHARACTERIZATION: Terrain Roughness Different surface types (open country, water surface , buildings, hills ) Conservative choice : z0 = 0.03 m (between cat . I and II) 3 km (Izmit Bay Osman Gazi Bridge) 19 TURBULENCE CHARACTERIZATION: Intensity and Integral Length Scales Turbulence Intensities: �� = 1 ln ������ ������0 �� = ������������ 2 Deck Clearance: 64 m (Izmit Bay Osman Gazi Bridge) (source: https://en.wikipedia.org/wiki/Os man_ Ga zi_Bri dge ) Integral Length Scales: � �� = 300 ( � 200 )[0.67 +0.05 ln �0 ] � �� = 0 .1 � �� Standard Dev iations: �������= �� ������ ������� = �� ������ Sectional Model: Buffeting Response Variable Value Iu 0.1305 Iw 0.0652 xLu 170.739 m xLw 17.0739 m Determination of Von Karman PSD uu and PSD ww 20 Sectional Model: Buffeting Response Longitudinal turbulence component PSD ������ �� = ������������2 � ∗ 4∗������∗������������������ ������ 1+ 70 .8∗ ������∗������������������ ������ 2 11 6 Vertical turbulence component PSD VON KARMAN PSDs (plotted for U = 45 m/s) 21 Sectional Model: Buffeting Response - Frequency Domain Compute the input PSD matrix Compute the transfer function matrix H(s) Compute the PSD of the displacements Compute RMS of the displacements for different mean U ������ � = � ∗ � ������ (������ �2 + ������ � + ������ )−1������ ������ ������ ������������ = ������ − ������� ∗ ������ ������������ ∗ ������ (������� )� ������ ������������ = ������ �� 0 0 ������ �� 22 Sectional Model: Buffeting Response - Frequency Domain Computed for ������ = 45 Τ � � 23 • The frequency content of the input signals (Von Karman PSDs ) ranges from 0 to 1 Hz • The natural frequencies of the 3 d.o.f . of the system fall in this range • The frequency content of the displacements ’ output signals , hence , shows a peak close to the respective natural frequency RESULTS Sectional Model: Buffeting Response - Frequency Domain The peaks are in correspondence of the flutter wind speed ������ � = 1 .6285 ������ ������ � = 1. 7674 ������ ������ ������� = 0 .7404 ������ At U = 45 Τ� � ������ � ��� = 66 .3 Τ ������ � 24 Sectional Model: Buffeting Response - Time Domain Discretize incoming turbulence PSDs Generate time history from the discrete PSDs Compute the turbulence aerodynamic forces acting on the system Apply the input forces to the dynamic system, and study the output spectrum ( FFT ) 25 � ���� = � � � (�) ������ (�) Sectional Model: Buffeting Response - Time Domain DISCRETIZATION OF T URBULENCE PSDs � �� = ������ �� ∆ � � �� = ������ �� ∆ � ∆ � = ൗ1 1000 ������� Frequency Resolution Stems 26 Sectional Model: Buffeting Response - Time Domain INPUT T IME HIST ORY GENERATION FROM DISCRETIZED PSDs � �� = 2� �� � �� = 2� �� u � = σ � �� sin (2������������� + �) w � = σ � �� sin (2������������� + �) � and � are random phases Amplitudes (associated to each frequency) Time signals (sum of sinusoids ) 27 Sectional Model: Buffeting Response - Time Domain RESULTS: FORCED RESPONSE • State Space model built from the second order system • Numerical integration in time of the State Space model (lsim ) • Time histories of the displacements are obtained 28 Sectional Model: Buffeting Response - Time Domain FFT s 29 • Frequency content of the output signals appears to be comparable to the one obtained through the PSD method (same U mean of 45 m/s) • The natural frequencies of the 3 d.o.f . of the system fall in this range • The frequency content still shows peaks close to the d.o.f.’s respective natural frequency Titolo presentazione sottotitolo Milano, XX mese 20XX Full Bridge Model 1. Static Response 2. Flutter Analysis Full bridge Model Generate the structural modal matrices : � �, ������ �, � � Evaluate the static displacements of each section and the associated aerodynamic coefficients Compute the aerodynamic matrices ������ ������, � ������ Look for the flutter points of the overall system, by gradually increasing the wind speed 31 ST RUCTURAL MAT RICES and MODES Structural matrices are generated from the provided data: Full bridge Model ������ = ������ (1) ������ (2) ������ (3) … ������ (14 ) ������ � = ������������������� (������ � ) ������ � = ������������������� (� � ) ������ � = ������������������� (�� ) Eigenvectors matrix PHI: 32 Eigenvectors ordered as follows: (Force vectors have been built accordingly to this choice ) Full bridge Model: Static Condition with U ST AT IC DISPLACEMENTS EVALUAT ION 33 ������ ������ = ������ ������ ������ The static displacements expressed in generalized coordinates are: Static displacements were evaluated for the following range of wind speeds: � = 0 ,… ,80 Τ ������ � The non -linear system of equations solved numerically to find ������������ is: W ith ������ ������ Full bridge Model: Static Condition with U 34 Full bridge Model: Stability Analysis AERODYNAMIC MAT RICES ������ � ሷ������ + ������ � + ������ ������� ������ ������ ሶ������ + ������ � + ������ ������� ������ ������ ������ = ������ ������������ and ������ ������ have been computed using aerodynamic coefficients and slopes evaluated in the static equilibrium solution related to a specific mean wind speed U 35 OVERALL DYNAMIC SYST EM (ST RUCTURE + AERO) Following the computation , we moved all our matrices in the modal coordinates space , through the eigenvectors matrix ������ to complete the overall system. The static equation was taken out, hence , perturbation coordinates are considered w.r.t the static condition Full bridge Model: Stability Analysis FLUTTER ANALYSIS Following the same procedure of sectional model flutter analysis but with a larger system: • Compute the eigenv alues of the dynamic system for increasing wind speeds • Check if there are wind speeds for which at least one of the eigenvalues has positiv e real part (polyeig routine): FLUTTER POINTS ��� [ ������ �������2 + ������ � + ������ ������� ������ ������ ������ + ������ � + ������ ������� ������ ������ ] = ������ 36 � = 0 ,… ,100 Τ ������ � RESULTS for • ROOT LOCUS • f-U DIAGRAM • x-U DIAGRAM Full bridge Model: Stability Analysis FLUTTER POINTS 37 The eigenvalue crossing the imaginary axis first ( at lower speed) is the one we’re actually most interested into . SYMMETRIC MODE SHAPES coupling (at lower speed) ANTISYMMETRIC MODE SHAPES coupling (at higher speed) Full bridge Model: Stability Analysis The FLUTTER SPEED has increased with respect to the one found in the previous section , showing that the simpler deck sectional model provides conservative results ������ ������ �������������� = 66 .3 Τ ������ � ������ ������ ��������������� = 76 .5 Τ ������ � 38 Full bridge Model: Stability Analysis 39