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Example.m
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%% Example
%
% The Critical flutter velocity is computed for 2 single-span suspension
% bridges: One with a main span of 1.2 km and the other one with the main span
% of 0.45 km. The coupling of the lateral, vertical and torsional motions of the bridge are
% accounted for the calculation of the critical velocity. The function
% flutterFD estimates the flutter velocity using [1]. For comparison, the
% Selberg's [2] and Rocard's [3] algorithm are also used (function VcrFlutter).
%
% References
%
% [1] Jain, A., Jones, N. P., & Scanlan, R. H. (1996).
% Coupled aeroelastic and aerodynamic response analysis of long-span bridges.
% Journal of Wind Engineering and Industrial Aerodynamics, 60, 69-80.
%
% [2] Selberg, A., & Hansen, E. H. (1966). Aerodynamic stability and related aspects of suspension bridges.
%
% [3] Rocard, Y. (1963). Instabilite des ponts suspendus dans le vent-experiences sur modele reduit. Nat. Phys. Lab. Paper, 10.
%
%% Case of a suspension bridge with a main-span of 1200 m
clearvars;close all;clc;
% Input the bridge parameters
% Modal parameters
load('modalParameters_case1.mat','wn','phi');
% load the mode shapes and eigen frequencies
% The modal parameters can be computed using another Matlab FileExchange
% submission available here:
% https://se.mathworks.com/matlabcentral/fileexchange/51815-calculation-of-the-modal-parameters-of-a-suspension-bridge
[Ndof,Nmodes,Nyy] = size(phi);
Bridge.wn= wn; % eigen frequencies (rad/s)
Bridge.phi= phi; % Mode shapes
Bridge.zetaStruct = 5e-3*ones(Ndof,Nmodes); % structural modal damping ratios, chosen as the same for every mode.
% Structural parameters
Bridge.L = 1200 ; % length of main span (m)
Bridge.B = 20 ; % deck width (m)
Bridge.D = 3; % Deck height (m)
Bridge.m =13e3 ; % lineic mass of the girder + the two main cables (kg/m)
Bridge.m_theta = 0.43e6; %kg.m^2/m torsional mass
% static coefficient for lift and overturning moment
Bridge.Cd = 1;% drag coefficient
Bridge.dCd = 0;% first derivative of the drag coefficient
Bridge.Cl = -0.3;% lift coefficient
Bridge.dCl = 3.0;% first derivative of the lift coefficient
Bridge.Cm = 0.01 ;% Overturning moment coefficient
Bridge.dCm = 0.5 ;% first derivative of the overturning moment coefficient
% Compute the critical flutter velocity
[Ucr,wCr,meanU] = flutterFD(Bridge,'Nfreq',2000,'Niter',300);
% Comparison with selberg's method (2 expressions) and Rocard's method:
fz = Bridge.wn(2,2)./(2*pi); % Mode HS1
ftheta= Bridge.wn(3,1)./(2*pi); % mode TS1
[Vcr] = VcrFlutter(Bridge.B,Bridge.m,Bridge.m_theta,fz,ftheta,'method','Selberg1');
fprintf([' Cricial flutter velocity with Selberg''s formula 1 is: ',num2str(Vcr,4),' m/s \n'])
[Vcr] = VcrFlutter(Bridge.B,Bridge.m,Bridge.m_theta,fz,ftheta,'method','Selberg2');
fprintf([' Cricial flutter velocity with Selberg''s formula 2 is: ',num2str(Vcr,4),' m/s \n'])
[Vcr] = VcrFlutter(Bridge.B,Bridge.m,Bridge.m_theta,fz,ftheta,'method','Rocard');
fprintf([' Cricial flutter velocity with Rocard''s formula is: ',num2str(Vcr,4),' m/s \n'])
% Display the variation of the eigen freqiencies with the mean wind
% velocity
clf,close all;
figure
title(' Variation of some of the eigen-frequencies with the mean wind velocity')
hold on; box on;
plot(meanU,squeeze(wCr(:,3,1))/(2*pi),'r')
plot(meanU,squeeze(wCr(:,2,2:3))/(2*pi),'k')
ylim([0,0.5])
grid on; box on;
grid minor
xlabel('U (m/s)');
ylabel('f (Hz)');
legend('TS1','VS1','VS2')
set(gcf,'color','w')
%% Case of the Lysefjord Bridge (main-span of 446 m)
clearvars;close all;
load('modalParameters_case2.mat');
[Ndof,Nmodes,Nyy] = size(phi);
Bridge.wn= wn; % eigen frequencies (rad/s)
Bridge.phi= phi; % Mode shapes
Bridge.zetaStruct = 5e-3*ones(Ndof,Nmodes); % structural modal damping ratios, chosen as the same for every mode.
% Structural parameters
Bridge.L = 446 ; % length of the main span (m)
Bridge.B = 12.3 ; % deck width (m)
Bridge.D = 2.76; % Deck height (m)
Bridge.m = 6166; % lineic mass of the girder + the two main cables (kg/m)
Bridge.m_theta = 59e3; %kg.m^2/m torsional mass
% static coefficient for lift and overturning moment
Bridge.Cd = 1;% drag coefficient
Bridge.dCd = 0;% first derivative of the drag coefficient
Bridge.Cl = 0.1;% lift coefficient
Bridge.dCl = 3.0;% first derivative of the lift coefficient
Bridge.Cm = 0.01 ;% Overturning moment coefficient
Bridge.dCm = 1 ;% first derivative of the overturning moment coefficient
% Compute the critical flutter velocity
[Ucr,wCr,meanU] = flutterFD(Bridge,'Nfreq',2000,'Niter',200,'Umin',50,'Umax',200);
% Comparison with selberg's method (2 expressions) and Rocard's method:
fz = wn(2,2)./(2*pi); % Mode HS1
ftheta= wn(3,1)./(2*pi); % mode TS1
[Vcr] = VcrFlutter(Bridge.B,Bridge.m,Bridge.m_theta,fz,ftheta,'method','Selberg1');
fprintf([' Cricial flutter velocity with Selberg''s formula 1 is: ',num2str(Vcr,4),' m/s \n'])
[Vcr] = VcrFlutter(Bridge.B,Bridge.m,Bridge.m_theta,fz,ftheta,'method','Selberg2');
fprintf([' Cricial flutter velocity with Selberg''s formula 2 is: ',num2str(Vcr,4),' m/s \n'])
[Vcr] = VcrFlutter(Bridge.B,Bridge.m,Bridge.m_theta,fz,ftheta,'method','Rocard');
fprintf([' Cricial flutter velocity with Rocard''s formula is: ',num2str(Vcr,4),' m/s \n'])
% Display the variation of the eigen freqiencies with the mean wind
% velocity
clf,close all;
figure
title(' Variation of some of the eigen-frequencies with the mean wind velocity')
hold on; box on;
plot(meanU,squeeze(wCr(:,3,1))/(2*pi),'r')
plot(meanU,squeeze(wCr(:,2,2:3))/(2*pi),'k')
ylim([0,1.5])
grid on;
grid minor
xlabel('U (m/s)');
ylabel('f (Hz)');
legend('TS1','VS1','VS2')
set(gcf,'color','w')
%%