Vibration of a Cantilever beam

here is my main code:

clear all

clc

Ne=6;

l=1; %length

t=0.02; %thickness

b=0.02; %width

modulus=2e11; %(E)

area=b*t;

imoment=(b*((t)^3))/12;

Le=l/Ne; %length of element

Rho=7850; %density

%Element stiffness matrix

K1=(modulus*imoment/(Le^3))*[12,6*Le,-12,6*Le; ...

6*Le,4*Le*Le,-6*Le,2*Le*Le; ...

-12,-6*Le,12,-6*Le; ...

6*Le,2*Le*Le,-6*Le,4*Le*Le];

Kglobal=zeros(2*(Ne+1),2*(Ne+1));

M1=[156 22*Le 54 -13*Le;...

22*Le 4*Le*Le 13*Le -3*Le*Le;...

54 13*Le 156 -22*Le;...

-13*Le -3*Le*Le -22*Le 4*Le*Le]*(Rho*Le*b*t)/420;

Mglobal=zeros(2*(Ne+1),2*(Ne+1));

for ii=1:Ne

Kglobal(2*ii-1:2*(ii+1),2*ii-1:2*(ii+1))=Kglobal(2*ii-1:2*(ii+1),2*ii-1:2*(ii+1))+K1;

Mglobal(2*ii-1:2*(ii+1),2*ii-1:2*(ii+1))=Mglobal(2*ii-1:2*(ii+1),2*ii-1:2*(ii+1))+M1;

end

K=Kglobal;

K(1:2,:)=[];

K(:,1:2)=[];

M=Mglobal;

M(1:2,:)=[];

M(:,1:2)=[];

C=0.05*Kglobal;

C(1:2,:)=[];

C(:,1:2)=[];

K

M

C

u=(2*Ne)+1;

%dt=1/(max_freq*20);

dt=0.001;

T=300;

%Displacement initials

y0=zeros(2*(2*(Ne+1))-4,1);

% y0(1:2,:)=[];

% y0(u:u+1,:)=[];

y0(end-1,1)=0.5;

% t_array = a(1,:); % This is t array from xls file

F=xlsread('l&d.xlsx','O1:O300'); % This is F array from xls file

%% ode solver

for i = 1:length(F)

[tsol ysol]=ode15s(@(t, y) beam_function(t, y, F(i), K, M, C, u),[1:dt:T],y0);

end

ysol(:,u:end)=[];%eliminate velocity nodes 5001*12

size(ysol)

ii=1:2:(u-2); %eliminate angular displacement nodes

ysol(:,ii+1)=[]; %eliminating angular dis components

size(ysol)

%% plotting

plot(tsol,ysol(:,Ne))

here is my function

function [dy]=beam_function(t, y, F, K, M, C, u)

dy=[y(u:end);

M\(F-K*y(1:u-1)-C*y(u:end))]

Right now the force seems to be applied on the complete beam at once.

I want force (F) to be applied only on the last node, like the case of a cantilever beam. Can someone tell me a way how i can do that?

Answer 1

[Umang Pandey]

Hi Bhanu,

To apply the force only on the last node of the beam, like in a cantilever beam scenario, you need to adjust the way the force is incorporated into your system of equations within the beam_function. Given that the force F is read from an Excel file and is intended to be applied at discrete time steps, you should construct a force vector that applies this force only to the last node.

Here's how you can modify your beam_function to apply the force only to the last node:

function [dy]=beam_function(t, y, F, K, M, C, u)
    % Initialize force vector of size u-1 (since you've removed the first two DOFs)
    F_vec = zeros(u-1, 1);
    
    % Apply the force to the second last entry, which corresponds to the last node's displacement
    % This is because the last entry would correspond to the rotation at the last node
    F_vec(end-1, 1) = F;
    
    % Dynamics equation
    dy=[y(u:end);
        M\(F_vec-K*y(1:u-1)-C*y(u:end))];
end

This adjustment ensures that the external force F (read from the Excel file at each time step) is applied only to the displacement of the last node, accurately simulating a cantilever beam subjected to a point load at its free end.

Best,

Umang