How to solve two coupled differential equations using ode45.

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I need help to solve two paired differential equations. After hours of unsuccessful attemps, I am asking for help from you. Any suggestion will be highly appriciated.
The equations are -
m*x''(t) + U*x(t) + γ*x'(t) + kv*V(t) - ζ = 0
V'(t) - kc*x'(t) + τ*V(t) = 0
initial conditions are x(0) = 1, X'(0) = 0, V(0) =0
x and V are the variables. All others are constant.
Thanks in advance.

Respuesta aceptada

Alan Stevens
Alan Stevens el 19 de Oct. de 2020
The following should help:
% tspan = [0 tend];
% IC = [1 0 0]; % initial conditions
% [t, X] = ode45(@fn, tspan, IC);
% x = X(:,1);
% z = X(:,2);
% V = X(:,3);
%
% function dXdt = fn(t,X)
% x = X(1); z = X(2); V = X(3);
% constants = ... list them with their values
% dXdt = [z;
% -(y*z+U*x+kv*V)/m;
% kc*z-tau*V];
% end
  22 comentarios
Alan Stevens
Alan Stevens el 19 de En. de 2022
I'd be inclined to manipulate the equations to get the following (I'm assuming you know time zero values for y1, y2, dy2/dt and V2):
% Let dy2/dt = v2
% Define the following functions
% fna = @(t,y1,y2,v2,VR) (F0*cos(omega*t)+theta*VR+keq*(y1-y2)-c*v2)/m;
% fnb = @(y1,v2) keq*v2/(knl1+3*knl2*y1^2+keq);
% Set up a rate of change function that will be called by an ODE solver
% function dYdt = rate(t,Y,fna,fnb,fnc)
% y1 = Y(1);
% y2 = Y(2);
% v2 = Y(3);
% VR = Y(4);
% dYdt = [fnb(y1,v2);
% v2;
% fna(t,y1,y2,v2,VR);
% (-R*theta*k1/(k1+kp)*(v2-fnb(y1,v2))-VR)/(R*Cp)];
% end
I can't check these numerically as you haven't supplied vaies for the constants, initial conditions or integration time.
If you intend to try this approach you must very carefully double check that I have manipulated the equations correctly!
haohaoxuexi1
haohaoxuexi1 el 19 de En. de 2022
dx(1)=(k1*x(3))/(knl_1+3*knl_3*x(1)^2+k1);
dx(2)=x(3);
dx(3)=(-c*x(3)-k1*(x(2)-x(1))+kAmp*cos(w*t)+theta_p*x(4))/m;
dx(4)=(-theta_p*(kspring/(kpiezo+kspring))*(x(3)-((k1*x(3))/(knl_1+3*knl_3*x(1)^2+k1)))*R_s-x(4))/R_s/C_p;
I changed the code to the above way and removed the mass matrix anymore, which follow your suggestions, and it worked with ode45.
I want to know if there is any logical mistake in my previous code? Why is my previous code is not working properly by treating the problem to be a DAE.
Also I noticed you wrote the fna fnb as separate function in your code instead of putting them all in dYdt function? Is there specific reason for you to do it this way? Like it can help calculating faster or it is just your habit?
Thank you for your help anyway.

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Más respuestas (1)

Alan Stevens
Alan Stevens el 20 de En. de 2022
"Why is my previous code is not working properly by treating the problem to be a DAE."
I don't know!
"Also I noticed you wrote the fna fnb as separate function in your code instead of putting them all in dYdt function? Is there specific reason for you to do it this way? Like it can help calculating faster or it is just your habit?"
You could put them inside the dYdt function here if you like. I tend to define them outside in case I need to call them separately outside of dYdt.

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