i am not getting the exact data for finite element model updating

clear;

close all;

clc;

% __________ M3=40tons

% | | K3=50*10^6

% ---------- M2=45tons

% | | K2=50*10^6

% ---------- M1=50tons

% | | K1=50*10^6

% --- ---

set-up for stiffness matrix and other

m1=50000;

m2=50000;

m3=50000;

mass=diag([m1 m2 m3]);

fprintf('The mass matrix \n');

disp(mass)

k1=50*10^6;

k2=50*10^6;

k3=50*10^6;

stiffness=[k1+k2 -k2 0;-k2 k2+k3 -k3;0 -k3 k3];

fprintf('The stiffness matrix \n');

disp(stiffness)

Natural frequency(eigen value) and Mode shape(eigen vector) calculation

[v,d]=eig(stiffness,mass);

fprintf('lambda value \n');

disp(d);

w=[sqrt(d)];

w1=w(1,1);

w2=w(2,2);

w3=w(3,3);

W=[w1,w2,w3];

fprintf('The natural frequencies of this structure are as follows(HZ) \n');

disp(W)

fprintf('The mode shape of the structure \n');

disp(v)

fprintf('lambda value \n');

disp(d);

w=[sqrt(d)];

w1=w(1,1);

w2=w(2,2);

w3=w(3,3);

W=[w1,w2,w3];

fprintf('The natural frequencies of this structure are as follows(HZ) \n');

disp(W)

fprintf('The mode shape of the structure \n');

disp(v)

Individusal mode shapes vector

fprintf('individusal mode shapes \n');

v1=v(:,1)

v2=v(:,2)

v3=v(:,3)

Plotting of mode shape

% height=[0;3;6;9];

% for i=1:5

% subplot(1,3,i)

% plot([0; v(:,i)],height);

% xlabel('mode shape','FontSize',12);

% ylabel('Height of the structure','FontSize',12);

% title(['mode shape', num2str(i)],'FontSize',18);

% end

Sensitivity matrix

l1 = [1 0 0;0 0 0;0 0 0]; % DIFFERENTIATION OF STIFFNESS WRT K1

l2 = [1 -1 0;-1 1 0;0 0 0]; % DIFFERENTIATION OF STIFFNESS WRT K2

l3 = [0 0 0;0 1 -1;0 -1 1]; % DIFFERENTIATION OF STIFFNESS WRT K3

Mode shape decomposition

a111=0;

a112=(v2'*l1*v1)/(w1-w2);

a113=(v3'*l1*v1)/(w1-w3);

a121=0;

a122=(v2'*l2*v1)/(w1-w2);

a123=(v3'*l2*v1)/(w1-w3);

a131=0;

a132=(v2'*l3*v1)/(w1-w2);

a133=(v3'*l3*v1)/(w1-w3);

a211=(v1'*l1*v2)/(w2-w1);

a212=0;

a213=(v3'*l1*v2)/(w2-w3);

a221=(v1'*l2*v2)/(w2-w1);

a222=0;

a223=(v3'*l2*v2)/(w2-w3);

a231=(v1'*l3*v2)/(w2-w1);

a232=0;

a233=(v3'*l3*v2)/(w2-w3);

a311=(v1'*l1*v3)/(w3-w1);

a312=(v2'*l1*v3)/(w3-w2);

a313=0;

a321=(v1'*l2*v3)/(w3-w1);

a322=(v2'*l2*v3)/(w3-w2);

a323=0;

a331=(v1'*l3*v3)/(w3-w1);

a332=(v2'*l3*v3)/(w3-w2);

a333=0;

SENSITIVITY MATRIX FORMATION (CONSIDERING BOTH EIGEN VALUE AND MODE-SHAPE DECOMPOSITION)

s11 = v1'*l1*v1;

s12 = v1'*l2*v1;

s13 = v1'*l3*v1;

s21=a111*v1+a112*v2+a113*v3;

s22=a121*v1+a122*v2+a123*v3;

s23=a131*v1+a132*v2+a133*v3;

s31 = v2'*l1*v2;

s32 = v2'*l2*v2;

s33 = v2'*l3*v2;

s41=a211*v1+a212*v2+a213*v3;

s42=a221*v1+a222*v2+a223*v3;

s43=a231*v1+a232*v2+a233*v3;

s51 = v3'*l1*v3;

s52 = v3'*l2*v3;

s53 = v3'*l3*v3;

s61=a311*v1+a312*v2+a313*v3;

s62=a321*v1+a322*v2+a323*v3;

s63=a331*v1+a332*v2+a333*v3;

S = [s11 s12 s13;s21 s22 s23;s31 s32 s33;s41 s42 s43;s51 s52 s53;s61 s62 s63]

SENSITIVITY MATRIX FORMATION 2 (CONSIDERING ONLY EIGEN VALUE DECOMPOSITION)

% s11 = v1'*l1*v1;

% s12 = v1'*l2*v1;

% s13 = v1'*l3*v1;

% s21 = v2'*l1*v2;

% s22 = v2'*l2*v2;

% s23 = v2'*l3*v2;

% s31 = v3'*l1*v3;

% s32 = v3'*l2*v3;

% s33 = v3'*l3*v3;

% S=[s11 s12 s13;s21 s22 s23;s31 s32 s33];

UPDATING PARAMETER (CONSIDERING BOTH EIGEN VALUE AND MODE-SHAPE DECOMPOSITION)

K = [k1 k2 k3];

WA = [w1 v1' w2 v2' w3 v3'];

WM = [14.7257 -0.0015 -0.0027 -0.0038 35.4870 0.0029 0.0021 -0.0031 55.2413 -0.0031 0.0033 -0.0011];

K1 = K'+inv(S'*S)*S'*(WM'-WA')

UPDATING PARAMETER 2 (CONSIDERING ONLY EIGEN VALUE DECOMPOSITION)

% K = [k1 k2 k3];

% WA = [w1 w2 w3];

% WM = [13.3221 36.7237 51.7097];

% K1 = K'+inv(S*S')*S'*(WM'-WA')

[Steven Lord]

It's not clear what problem you're experiencing or what question you'd like help with.

[Susanta Barik]

I AM USING SENSITIVITY METHOD TO UPDATE FINITE ELEMENT DATA BUT WHAT THE RESULT I AM GETTING THAT IS NOT EXACT DATA . ALSO WHEN I AM COMPAIRING MY EXPERIMENTAL DATA WITH UPDATED DATA I AM GETTING NOT EXACT DATA

Answer 1

[Nipun]

Hi Susanta,

I understand that you are facing issues with finite element model updating and not getting the exact data. Here’s a concise approach to check and improve your FEM updating process:

1. Verify Model and Data Consistency: Ensure that the mesh, material properties, boundary conditions, and loads are accurately defined.

2. Check for Convergence: Ensure your solver settings are appropriate for achieving convergence.

3. Use Optimization Toolbox: Apply optimization techniques to update your model parameters.

Here’s a basic outline for using optimization in MATLAB:
% Define your objective function (difference between model and experimental data)
objectiveFunction = @(params) computeDifference(params, experimentalData);

% Set initial guess for parameters
initialParams = [E, density, poisson]; % Example parameters like Young's modulus, density, Poisson's ratio

% Define optimization options
options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');

% Run optimization
[optimizedParams, fval] = fmincon(objectiveFunction, initialParams, [], [], [], [], lb, ub, [], options);

disp(optimizedParams);

Replace `computeDifference` with your function to calculate the difference between model and experimental data, and define `experimentalData`, `initialParams`, `lb`, and `ub` appropriately.

Hope this helps.

Regards,

Nipun