Mass Spring System ode45 with multiple damping values

I am working on this problem where I have to model multiple mass spring systems that all have different damping values. My model uses the equation of with the initial conditions of m = 5 kg, k = 0.5 N/m, and . I converted this ODE into a system of 1st order ODEs so I could use ode45. I need to model the above equation for when (overdamped) , (critcally damped), and (underdamped). I am confused on how I could be able to specify the different damping conditions with ode45, I was thinking to use it within a for loop for each value c but I am having errors occur within my ode45 function. Any help would be greatly appreciated.

% Constants

m = 5; % kg

k = 0.5; % N/m

c = [0 , sqrt(4*m*k)-1 , sqrt(4*m*k)+1, sqrt(4*m*k)]; % damping constants

A = 1.00002;

B = 3.16228;

w0 = sqrt(k/m);

t = 20;

% Function

% Let

% u1 = y ---> u1' = y' = u2

% u2 = y' ---> u2' = 1/m*(-k*u1-c*u2)

f = @(t,u) [u(2) ; 1/m*(-k*u(1)-c*u(2))];

fA = @(t) A*cos(w0*t)+B*sin(w0*t);

% IC

u0 = [1 1];

tspan = [0 t];

% Solve

for tspan = [0 t]

[t,u] = ode45(f,tspan,u0);

if abs(c^2) > 4*m*k

plot(t,u(:,1))

end

end

[KSSV]

You may refer this: https://in.mathworks.com/matlabcentral/fileexchange/33082-simplependulum

You can change C here in a loop and get the equation solved.

Answer 1

[Sulaymon Eshkabilov]

It is realtively easy to attain the issues of solving the exercises for all damping values and plot them all with appropriate legends:

% Constants 
m = 5; % kg
k = 0.5; % N/m
c = [0 , sqrt(4*m*k)-1 , sqrt(4*m*k)+1, sqrt(4*m*k)]; % damping constants
A = 1.00002;
B = 3.16228;
w0 = sqrt(k/m);
t = 20;
% IC
u0 = [1 1]; 
tspan = [0 t];
% Solve, simulate and plot 
figure('name', 'One case of damping: c')
for ii=1:numel(c)
    f = @(t,u) [u(2) ; 1/m*(-k*u(1)-c(ii)*u(2))];
    fA = @(t) A*cos(w0*t)+B*sin(w0*t);
     [t,u] = ode45(f,tspan,u0);
     if abs(c(ii)^2) > 4*m*k 
         plot(t,u(:,1))
     end
end
grid on; xlabel('time, [s]'), ylabel('Solution, y(t)')

%% OR you can plot all values of c
clearvars
% Constants 
m = 5; % kg
k = 0.5; % N/m
A = 1.00002;
B = 3.16228;
w0 = sqrt(k/m);
c = [0 , sqrt(4*m*k)-1 , sqrt(4*m*k)+1, sqrt(4*m*k)]; % damping constants
t = 20;
% IC
u0 = [1 1]; 
tspan = [0 t];
figure('name', 'All cases of damping: c')
for ii=1:numel(c)
    f = @(t,u) [u(2) ; 1/m*(-k*u(1)-c(ii)*u(2))];
    fA = @(t) A*cos(w0*t)+B*sin(w0*t);
    [t,u] = ode45(f,tspan,u0);
    plot(t,u(:,1)), hold all
    L{ii} = ['c = ', num2str(c(ii))];
    legend(L{:})
end
grid on; xlabel('time, [s]'), ylabel('Solution, y(t)')

[Logan Parrish]

Thanks for the help I was really lost on how to set up the loop for the systems.

[Sulaymon Eshkabilov]

Most welcome!