Mass Spring System ode45 with multiple damping values

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Logan Parrish
Logan Parrish 2022-2-22
评论： Sulaymon Eshkabilov 2022-2-22
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

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采纳的回答

Sulaymon Eshkabilov
Ran in:
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)')

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