## Contents

## Geometry definition

Beam with main axis z, and cross section is symmetric to axis x and y

clear all;close all;clc; geometry.L = 100; % beam length (m) geometry.E = 2.1e11; % Young Modulus (Pa) geometry.nu = 0.3; % Poisson ratio geometry.rho = 7850; % density (kg/m^3) % case of a cylinder % D = geometry.L/20; % beam diameter (m) % --> beam is symmetrical around axes y and z % d = D/100; % internal diameter if an annulus is centered at the origin; % Iy = pi.*D.^4./64; % Ix = pi.*D.^4./64; % geometry.I = Iy; % affectation of quadratic moment % number of discretisation points for the beam % geometry.y = linspace(0,geometry.L ,100); % V = pi.*D.^2*geometry.L; % geometry.m = geometry.rho.*V./geometry.L;

## Rectangular beam

B = geometry.L/100; H = geometry.L/100; Iy = B*H.^3/12; % quadratic moment Ix = H*B.^3/12; % quadratic moment geometry.I = Iy; % affectation of quadratic moment % number of discretisation points for the beam geometry.y = linspace(0,geometry.L ,100); V = B*H*geometry.L; geometry.m = geometry.rho.*V./geometry.L; % in kg/m

## Number of modes

```
Nmodes =4; % number of mode wanted
```

## modal analysis: Case 1

BC = 1; % pinned-pinned [phi,wn] = eigenModes(geometry,BC,Nmodes); figure for ii=1:Nmodes, subplot(Nmodes,1,ii) box on;grid on plot(geometry.y,phi(ii,:)); ylabel(['\phi_',num2str(ii)]) title(['w_',num2str(ii),' = ',num2str(wn(ii),3),' rad/s']); end set(gcf,'color','w') xlabel('y (m)');

Equation solved, inaccuracy possible. The vector of function values is near zero, as measured by the default value of the function tolerance. However, the last step was ineffective.

## modal analysis: Case 2

BC = 2;% clamped-free [phi,wn] = eigenModes(geometry,BC,Nmodes); figure for ii=1:Nmodes, subplot(Nmodes,1,ii) box on;grid on plot(geometry.y,phi(ii,:)); ylabel(['\phi_',num2str(ii)]) title(['w_',num2str(ii),' = ',num2str(wn(ii),3),' rad/s']); end set(gcf,'color','w') xlabel('y (m)');

Equation solved, fsolve stalled. fsolve stopped because the relative size of the current step is less than the default value of the step size tolerance squared and the vector of function values is near zero as measured by the default value of the function tolerance.

## modal analysis: Case 3

BC = 3; % clamped-clamped [phi,wn] = eigenModes(geometry,BC,Nmodes); figure for ii=1:Nmodes, subplot(Nmodes,1,ii) box on;grid on plot(geometry.y,phi(ii,:)); ylabel(['\phi_',num2str(ii)]) title(['w_',num2str(ii),' = ',num2str(wn(ii),3),' rad/s']); end set(gcf,'color','w') xlabel('y (m)');

Equation solved, inaccuracy possible. The vector of function values is near zero, as measured by the default value of the function tolerance. However, the last step was ineffective.

## modal analysis: Case 4

BC = 4;% clamped-pinned [phi,wn] = eigenModes(geometry,BC,Nmodes); figure for ii=1:Nmodes, subplot(Nmodes,1,ii) box on;grid on plot(geometry.y,phi(ii,:)); ylabel(['\phi_',num2str(ii)]) title(['w_',num2str(ii),' = ',num2str(wn(ii),3),' rad/s']); end set(gcf,'color','w') xlabel('y (m)');

Equation solved, inaccuracy possible. The vector of function values is near zero, as measured by the default value of the function tolerance. However, the last step was ineffective.