ODE45 to solve multiple degree of system free vibration

Hello,

I am trying to slove free vibration for four degree system using ode45 and function. However I am not sure why but the response of the system is a streight line. My system has four masses and four spring coefficients, with initial condtions for dispalcemnt for m1= 0.01, for m2= 0.01, for m3= 0.02, for m4= 0.01 and inital velocity 0 0 0 0 for each mass.

I am pasisng my code her. I would apprciate the feedback!

Function

function xdot = f_4degree(t,x)
global M K
A=[zeros(4,4) eye(4,4);-M\K zeros(4,4)]*x;
xdot = A.*x;

ODE45

global M K
m1=10; m2=10; m3=10; m4=10;
k1=50; k2=50; k3=50; k4=50;
M=[m1 0 0 0;0 m2 0 0;0 0 m3 0;0 0 0 m4];
K=[k1+k2 -k2 0 0;-k2 k2+k3 -k3 0;0 -k3 k3+k4 -k4;0 0 -k4 k4];
%ODE45
tspan=[0 20];
IC=[0.01 0.01 0.02 0.01 0 0 0 0];
[t,x]= ode45('f_4degree',tspan,IC);
figure(1)
plot(t,x(:,1));
legend('Repsonse')
ylabel('x(t)')
xlabel('time') 

[Rik]

I recovered the removed content from the Google cache (something which anyone can do). Editing away your question is very rude. Someone spent time reading your question, understanding your issue, figuring out the solution, and writing an answer. Now you repay that kindness by ensuring that the next person with a similar question can't benefit from this answer.

Answer 1

[Torsten]

Maybe you mean

m1=10; m2=10; m3=10; m4=10;
k1=50; k2=50; k3=50; k4=50;
M=[m1 0 0 0;0 m2 0 0;0 0 m3 0;0 0 0 m4];
K=[k1+k2 -k2 0 0;-k2 k2+k3 -k3 0;0 -k3 k3+k4 -k4;0 0 -k4 k4];
result = -M\K;
%ODE45
tspan=[0 20];
IC=[0.01 0.01 0.02 0.01 0 0 0 0];
[t,x]= ode45(@(t,x)f_4degree(t,x,result),tspan,IC);
figure(1)
plot(t,x(:,1));
legend('Repsonse')
ylabel('x(t)')
xlabel('time') 

function xdot = f_4degree(t,x,result)
  A=[zeros(4,4), eye(4,4);result, zeros(4,4)];
  xdot = A*x;
end

[Karolina Kugiel]

Thank you so much!

[Karolina Kugiel]

Is there any way how I could find response of each mass? Instead of total response?

[Torsten]

I don't have experience with the physical background of your problem. Can you describe mathematically what the output for the "response of each mass" would be ?

Answer 2

[Sam Chak]

Hi @Karolina Kugiel

ode45 solves the problem by integrating where . So I guess you looking for states .

m1=10; m2=10; m3=10; m4=10;
k1=50; k2=50; k3=50; k4=50;
M = [m1 0 0 0;
     0 m2 0 0;
     0 0 m3 0;
     0 0 0 m4];
K = [k1+k2 -k2 0 0;-k2 k2+k3 -k3 0;0 -k3 k3+k4 -k4;0 0 -k4 k4];
result = -M\K;
%ODE45
tspan=[0 20];
IC=[0.01 0.01 0.02 0.01 0 0 0 0];
[t,x]= ode45(@(t,x)f_4degree(t,x,result),tspan,IC);

figure(1)
subplot(221)
plot(t, x(:,1)), title('Response of m_{1}'), xlabel('t')
subplot(222)
plot(t, x(:,2)), title('Response of m_{2}'), xlabel('t')
subplot(223)
plot(t, x(:,3)), title('Response of m_{3}'), xlabel('t')
subplot(224)
plot(t, x(:,4)), title('Response of m_{4}'), xlabel('t')

% legend('Repsonse')
% ylabel('x(t)')
% xlabel('time') 
function xdot = f_4degree(t,x,result)
  A=[zeros(4,4), eye(4,4); result, zeros(4,4)];
  xdot = A*x;
end