how to solve matrix differential equation

hi,

I' m trying to solve this differential equation for education purposes but I'm having difficulty to solve it.

does someone know a simple way to solve it?

? ∗ ?̈ = ?? − (?? + ??) ∗ ?

I attached a pdf where you can find all the information about the program.

thank you very much.

Answer 1

[David Wilson]

Here's my attempt:

M = [ 
    97227/21875, 0, 82269/109375, 0, 67311/43750, 0, -97227/218750, 0; ...
 0, 97227/21875, 0, 82269/109375, 0, 67311/43750, 0, -97227/218750;
 82269/109375, 0, 89748/546875, 0, 97227/218750, 0, -67311/546875, 0;
 0, 82269/109375, 0, 89748/546875, 0, 97227/218750, 0, -67311/546875;
 67311/43750, 0, 97227/218750, 0, 97227/21875, 0, -82269/109375, 0;
 0, 67311/43750, 0, 97227/218750, 0, 97227/21875, 0, -82269/109375;
 -97227/218750, 0, -67311/546875, 0, -82269/109375, 0, 89748/546875, 0;
 0, -97227/218750, 0, -67311/546875, 0, -82269/109375, 0, 89748/546875]

Qv = [ 0; -20651534751534355/422212465065984; 0; -4130306950306871/422212465065984; 0;
-20651534751534355/422212465065984; 0; 4130306950306871/422212465065984]; 

I couldn't be bothered using your matrices, so I just used some random, symmetric ones. You'll need to type in yours ...

Kt = randn(8,8); Kt = Kt'*Kt; % force symmetric 
Kl = randn(8,8); Kl = Kl'*Kl; % force symmetric 

L = 5; % ??? 
e0=[0; 0; 1; 0; L; 0; 0; 1];

Now I'm going to

(1) Convert the 2nd order ODE to Cauchy form, and

(2) Generate the appropriate Mass matrix, (although you can easily invert this since the condition number is reasonable.)

Once all set up, just throw to ode45 to solve it. Once done, extract the first 8 elements & plot them.

%-------------------------------------------------
K = (Kt+Kl); 
Ma = eye(16); % augmented mass matrix. 
Ma(9:end, 9:end) = M; % Bottom right corner is "M" 
z0 = [e0; zeros(8,1)]; 
odefun = @(t,z) [z(9:16,1); ... 
                 Qv - K*z(1:8,1)]; 

tspan = [0,5]; % who knows what this should be ??? 

optns = odeset('Mass',Ma); % Include Mass matrix & keep on LHS. 
[t,z] = ode45(odefun, tspan, z0, optns ); 

plot(t,z(:,[1:8])); 

[riccardo carrieri]

thank you very much for your quick reply, but the code don't work

this is the code:

M = eye(16); % augmented mass matrix. 
M(9:end, 9:end) = Ma
z0 = [e0; zeros(8,1)]; 
odefun = @(t,z) [z(9:16,1);inv(Ma)*(Qv - K*z(1:8,1))]; 
tspan = [0,2]; % who knows what this should be ??? 
% try explicit method 
options = odeset('Mass',M); 
[t,z] = ode45(odefun, tspan, z0,options); 
plot(t,z(:,[1:8])); 

this is the error:

Error using odearguments (line 113)

Inputs must be floats, namely single or double.

Error in ode45 (line 115)

odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in prova_del_code (line 135)

[t,z] = ode45(odefun, tspan, z0,options);

do you know why?

[David Wilson]

First up, you shouldn't use the inv(Ma) as well as the "Mass" matrix. Keep the mass matrix on the LHS, and drop the inv(Ma) part.

If you are going to use the inv(Ma), then drop the mass matrix part, however this is not generally recommended nor efficient.

You error seems to suggest that one (or more) ofthe arguments to ode45is the wrong type. Check each of the arguments.