I need to know the order of solution vector i-e how solution is stored in solution vector of coupled 2nd order differential equations

I have these two coupled differential equations

m_1 * z_1_dot_dot + (k_1+k_2 ) z_1 + b * z_1_dot = k_2 * z_2 + b * z_dot_2

m_2 * z_2_dot_dot + k_2 z_2 + b * z_2_dot = k_2 * z_1 + b * z_1_dot + f

( 2 Degree of freedom mass spring system with damping in between Attached image )

Now I have written a code for the solution of these two coupled diffferential equations but I dont know how to write correct initial conditions order (z1 , z_1_dot , z_2 , z_2_dot)

And also I dont know in what order the solution is stored in Solution vector i-e which columns contains the z1 displacement and velocity and which columns contain the displacement and velocity of second z2 (z_2 and z_2_dot)

Here is the Code

clc

clear

close all

% Use Symbolic toolbox

syms m1 m2 k1 k2 f b z1(t) z2(t) t Y;

dz1 = diff(z1);

d2z1 = diff(z1,2);

dz2 = diff(z2);

d2z2 = diff(z2,2);

% Write two Coupled Differential Equations

Eq1 = d2z1 == (k2*z2(t) + b*dz2 - (k1+k2)*z1(t) - b*dz1)/m1; % 1st Equation

Eq2 = d2z2 == (k2*z1(t) + b*dz1 + f - k2*z2(t) - b*dz2)/m2;

% Convert the equations to vector Field (Coupled first order linear differential equations)

[VF,Subs] = odeToVectorField(Eq1, Eq2);

ftotal = matlabFunction(VF,'Vars',{t,Y,m1,m2,k1,k2,f,b});

% Define the Known Parameters

m1 = 10;

m2 = 5;

k1 = 40;

k2 = 20;

b = 10;

f = 5;

% Use ode45 Command to solve the system of coupled equations

tspan = [0 20]; % Choose Appropriate Simulation Time

ic = [0 0 0.3 2]; % Choose Appropriate Initial Conditions (% This is where I need Help The conditions are m1 initial displacement "0.3" and velocity of "2" and for m2 both are zero How do I write in Correct Order)

[t,y] = ode45(@(t,y) ftotal(t,y,m1,m2,k1,k2,f,b), tspan, ic);

% Here t is time vector and y is the solution vector containing solution to

% differential equations

Please please Anyone help and also please make me understand to avoid this confusion in Future.

Any Type of help would be EXTREMELY APPRECIATED !!!!! ( I have given the code which works)

[Haseeb Hashim]

Anyone to help (I just need to know in what order initial conditions are put and the order of columns solution vector) !!!!!!!!!!

Answer 1

[Sam Chak]

@Haseeb Hashim, Aha, I see.

You will have to remove the semicolon at the end of this line

[VF,Subs] = odeToVectorField(Eq1, Eq2);

so that the output Subs can be displayed and you will know the sequence of the system states. Refer to

https://www.mathworks.com/help/symbolic/odetovectorfield.html#bvlo69z-2

For example,

syms f(t) g(t)
eqn1  = diff(g)   == g-f;
eqn2  = diff(f,2) == g+f;
eqns  = [eqn1 eqn2];
[V,S] = odeToVectorField(eqns)

V = 

S = 

[Haseeb Hashim]

Oh!! Thanks Now I get it. Thankyou soo much for help. Much Appreciated