How i can simulate a mass-spring-damper system with variable mass?

What is the proper form to simulate a mass-spring-damper system with variable mass?

I've analized 3 alternatives, based on Matlab Help

Option 1: mass change in the function

 m = 1.2;
 k = 15;
 c = 1.2;
 f = 1;
 tspan = 0:0.01:100-0.01;
 y0 = [0 0];
 opciones0=odeset('RelTol',1e-12,'AbsTol',1e-12);

 fun_1 = @(t,z) [z(2); (f*sin(2*pi*t) - k*z(1)-c*z(2))/(m*(1+0.01*sin(2*pi*t/10)))];
 [t1,z1]=ode15s(@(t,z) fun_1(t,z),tspan,y0,opciones0);

Option 2: mass change in odeset

 fun_2 = @(t,z) [z(2); (f*sin(2*pi*t) - k*z(1)-c*z(2))/m]; 
 fun_m = @(t,z) m*(1+0.01*sin(2*pi*t/10))*eye(2);

 opciones2=odeset('RelTol',1e-12,'AbsTol',1e-12,'Mass',fun_m,'MStateDependence','none','MassSingular','no');
 [t2,z2]=ode15s(@(t,z) fun_2(t,z),tspan,y0,opciones2);

Option 3: mass change in the function and odeset

 fun_3 = @(t,z) [z(2); (f*sin(2*pi*t) - k*z(1)-c*z(2))/(m*(1+0.01*sin(2*pi*t/10)))];
 fun_m = @(t,z) m*(1+0.01*sin(2*pi*t/10))*eye(2);
 opciones3=odeset('RelTol',1e-12,'AbsTol',1e-12,'Mass',fun_m,'MStateDependence','none','MassSingular','no');
 [t3,z3]=ode15s(@(t,z) fun_3(t,z),tspan,y0,opciones3);

But when i see the spectrums, with the 3 alternatives I get different results, and i don't know which is the correct. (the first graph is with constant mass)

Answer 1

[Ced]

You should be able to include the changing mass either in the function, or then through odeset, but I am not sure if you did it correctly.

Case 1: In function --> looks good to me

Case 2: Is a bit confusing to me. You have m in both fun_2 and in the settings. Also, afaik, matlab takes the odeset option to set M(t)*dy = f(....). Why would you multiply the first dimension with the mass as well? Is that intended?

Case 3: Well, as you already stated, you include the mass twice.

Case 4: For odeset, I think it should look something like

 fun_4 = @(t,z) [z(2); (f*sin(2*pi*t) - k*z(1)-c*z(2))]; 
 fun_m = @(t,z) diag([1, m*(1+0.01*sin(2*pi*t/10)]);

 opciones4=odeset('RelTol',1e-12,'AbsTol',1e-12,'Mass', ...
 fun_m,'MStateDependence','none','MassSingular','no');

 [t4,z4]=ode15s(@(t,z) fun_4(t,z),tspan,y0,opciones4);

[Manuel]

Thanks for you anser! Now I understood.

Case 2 and 3 are wrong so you mention.

Now I compared the results of cases 1 and 4, in the time and frequency domain they give the same results. So, when the mass matrix is time dependent and it's no singular, it's no necessary to use the 'Mass' option in odeset and solve like the case 1?

[Ced]

Correct. At least that's the theory. There might be cases were your mass matrix is not singular, but badly conditioned. In these cases, it might still be better to use that option.