# Demonstrate 2: Constrained Polynomials and Beam Measurements

This data set represents uncertain measurements of the deflection of a doubly clamped beam. The aim is to fit a curve through the data which fulfille the theoritical known constraints existing at both ends of the beam.

(c) 2013 Paul O'Leary and Matthew Harker

## Contents

The constraints on a doubly clamped beam are

## Prepare MATLAB for The Computation

close all; clear all; % % Set some defaults % FontSize = 12; set(0,'DefaultaxesFontName','Times'); set(0,'DefaultaxesFontSize',FontSize); set(0,'DefaulttextFontName','Times'); set(0,'DefaulttextFontSize',FontSize); set(0,'DefaultfigurePaperType','A4'); set(0,'DefaultTextInterpreter', 'latex'); %

## Load a Dataset Containing Perturbed Measurements and Constraints

The data points are contained in x and y.

load fitBeamData; %

## Define the Required Constraints

c1 = [0,x(1),0]; c2 = [1,x(1),0]; c3 = [0,x(end),0]; c4 = [1,x(end),0]; % Cs = [c1; c2; c3; c4]; %

## Perform the Constrained Approximation

Call `polyfitconstrained` to compute the constrained fit.

d = 5; [pc, Lp, S] = polyfitconstrained( x, y, d, Cs ); % % Evaluate at a lagre number of points % nrPts = 500; xc = linspace( min(x), max(x), nrPts)'; yc = polyval( pc, xc ); %

## Plot the Results of the Approximation

fig1 = figure; plot(x,y,'ko'); xlabel('x'); ylabel('y'); grid on; hold on; plot( xc, yc, 'b', 'LineWidth', 1.5); title( 'Constrained Polynomial Aproximation' ); % % Add the positions of the constraints to the plot % [n,m] = size(cs); csText = []; for k = 1:n xk = cs(k,2); vk = polyvander( xk, d ); yk = vk * pc; % plot( xk, yk, 'ko', 'markerFaceColor', 'r'); end;

## Zoom in on one End of the Beam

Zooming in on the end of the beam show that the fit fulfils both the zero value and zero derivative constraints.

```
figure(fig1);
axis([0.8,1,-0.01,0.01]);
%
```

## Present the Covariance of the Aolution as an Image

This is the covariance matrix of the approximation for the constrained y. Note: the zero covariance at the points where the value constraint is zero.

fig2 = figure; imagesc( x, x, S.Ly ); axis image; xlabel('Support'); ylabel('Support'); title('Covariance of the Approximation'); colorbar; hold on;