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1D bar, fixed at one ends, Forced at the other end

1D bar, fixed at one ends, Forced at the other end

1D bar, fixed at one ends, Forced at the other end

(OP)
Hi All,
I am having difficulty understanding a set of results I have obtained by solving the dynamic equations of a 1D bar, fixed on the left hand side, and excited by an external force of F0*sin(w*t)on the right hand side. I have integrated in time using an explicit Euler scheme. I suspect for a simple case, that should not cause instabilities.
I have also tried to work out the natural frequencies of the forced system analytically.
I was expecting the results to be similar to a free vibration results, when you have the amplitude waves that either damp over time (provided frequency not close to the natural frequency). But here I get waves that are not based around a straight line at (0,0) but the fluctuate around a line that has a gradient. Not sure if I am clear here, so I have attached my code which produces the results. The velocity results seem more reasonable.
I would appreciate it if you could run my code and guide me through this. I have spent quite a bit of time on it. I want to understand this fully. Are there ways to check my answers? How can I use the eigenfrequency and eigenvectors to check my results?
I would appreciate any guidance ASAP. Thank you very much.

clear all
clc

%%
% MATERIAL PROPERTIES
nn = 4;
E = 30e6;
A =1 ;
L0 = 100*(nn-1);
le0 = (L0/nn);
irhos0 = 0.00073*386.22;

k1 = [1 -1 ; -1 1]; k2 = k1; k3 = k1;
m1= [2 1;1 2]; m2 = m1; m3 = m1;
K = ((E*A)/le0)*[k1(1,1) k1(1,2) 0 0;
k1(2,1) k1(2,2)+k2(1,1) k2(1,2) 0;
0 k2(2,1) k2(2,2)+k3(1,1) k3(1,2);
0 0 k3(2,1) k3(2,2)];
M = ((irhos0*A*le0)/6)*[m1(1,1) m1(1,2) 0 0 ;
m1(2,1) m1(2,2)+m2(1,1) m2(1,2) 0;
0 m2(2,1) m2(2,2)+m3(1,1) m3(1,2);
0 0 m3(2,1) m3(2,2)];

F0 = 1000;
%%
% ANALYTICAL, CALCULATES THE NATURAL FREQUENCIES OF THE FORCED SYSTEM
% Let x(t) = Xsin(wt), xdot = wXcos(wt), xddot = -(w^2)X(sin(xt)
% Let f(t) = Fsin(wt)
% sub above in [M]{xddot}+[k]{x}={f}
% => ([k]-[m](w^2)){X}={F}
% => {X} = inv[[k]-[m](w^2)]*{F}
% => CHOOSE F function, here F = F0*w^2

% EIGENVALUES
X = zeros(nn,1); % AMPLITUDE OF VIBRATIONS
b = 1;
w1 = 0:3*180;
freq = w1/(2*pi);
Fex1 = 0; % EXTERNAL FORCE
F1 = zeros(nn,1);
for j=1:length(w1)
Fex1(j) = F0*(w1(j)^2);
F1(:,j) = [zeros(nn-1,1);Fex1(j)];
A = K-M*(w1(j)^2);
X(:,j) = inv(A)*F1(:,j);
end
figure
subplot(4,1,1)
plot(w1,abs(X(1,:)),'-r')
subplot(4,1,2)
plot(w1,abs(X(2,:)),'-r')
subplot(4,1,3)
plot(w1,abs(X(3,:)),'-r')
subplot(4,1,4)
plot(w1,abs(X(4,:)),'-r')
xlabel('\omega rads/sec')
ylabel('Displacement')
% close all


% Eigenvectors
% ??

%%
% FEA - EXPLICIT EULER TIME SCHEME
m = 5000; % NUMBER OF TIME STEPS
wn = sqrt(E/irhos0); %[E*A/L/rho]=
w = wn/100;
tp = (2*pi)/w;
tf = 2*tp;
dt = tf/m;

% [m][d2u/dt2]+[k][u] = [F]
% =>[m][dv/dt]+[k][u]=[F]
% => dv/dt = inv[m]*([F]-[k]{u}) EQ1
% du/dt = v EQ2
u = zeros(nn,1);
v = zeros(nn,1);
u(:,1) = 0;
v(:,1) = 0;
u(1,:) = 0;
v(1,:) = 0;
Fex2 = 0;
F2(:,1) = zeros(nn,1);
t = 0;
ww = 0;
for i = 1:m
t(i+1) = t(i)+dt;
Fex(i+1) = F0*sin(t(i)*w);
F2(:,i) = [zeros(nn-1,1);Fex(i)];
v(:,i+1)= v(:,i)+dt*inv(M)*(F2(:,i)-K*u(:,i));
u(:,i+1)= u(:,i)+dt*v(:,i);
t(i+1) = t(i)+dt;
end

% PLOT DISPLACEMENT WITH TIME
figure
subplot(4,1,1)
plot(t,(u(1:nn,:)))
subplot(4,1,2)
plot(t,(u(2,:)))
subplot(4,1,3)
plot(t,(u(3,:)))
subplot(4,1,4)
plot(t,(u(4,:)))

% PLOT OF VELOCITY WITH TIME
figure
subplot(4,1,1)
plot(t,(v(1,:)))
subplot(4,1,2)
plot(t,(v(2,:)))
subplot(4,1,3)
plot(t,(v(3,:)))
subplot(4,1,4)
plot(t,(v(4,:)))

% PLOT OF FORCE WITH TIME
figure
plot(t(1:end-1),Fex)


RE: 1D bar, fixed at one ends, Forced at the other end

(OP)
I just found out that I have not really implemented the boundary condition in the code. Have done that now and results seem more reasonable. Having said that, I would appreciate some guidance on how I can evaluate the results, and how to work out the eigenvectors of a forced system.

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