Solving Ordinary Differential Equations

Problem - Solve the system of equations and determinate x(t) and y(t):

Equation 1: mv * A * L* y'' + 2 * mv * A * g * y = - mv * A * B * sin (wf * t)

Equation 2: ( M+ mv * A * L ) * x'' + mv * A * B * y'' + M * ws^2 * x= - (M + mv * A * L) * sin (wf * t)

Constants: mv=1; A=7.5*10^-4; B=20*10^-2; L=50*10^-2; g=9.81; wf=5.3; ws=5.3; M=(0.373+0.306/3+0.306/3)*10^-3;

It´s a engineering problem, there are motion equations and x and y are, respectively, the horizontal and vertical displacement. Anyone can help me solve this system? Already have the Symbolic Math Toolbox.

Answer 1

[Walter Roberson]

In Maple it is

dsolve([mv*A*L*(diff(y(t), t, t))+2*mv*A*g*y(t) = -mv*A*B*sin*wf*t, (M+mv*A*L)*(diff(x(t), t, t))+mv*A*B*(diff(y(t), t, t))+M*ws^2*x(t) = -(M+mv*A*L)*sin*wf*t])

and you should be able to use a quite similar syntax with the symbolic toolbox.

You have the second derivative of x and the second derivative of y, so you would need at least 4 equations to resolve all the constants of integrations (the boundary conditions). Maple's answer leaves four constants of integration undefined.

[Igor]

Thanks for the answer Walter Roberson.

I have some problems in MATHLAB because never now what window I introduce the code. I have a Editor with this code to solve that system (adapted from the MATHLAB Help):

function xprime = odetest(t,x)

% x(1) = x

% x(2) = y

% x(3) = w

% x(4) = z

mv=1;

A=7.5*10^-4;

B=20*10^-2;

L=50*10^-2;

g=9.81;

wf=5.3;

ws=5.3;

M=(0.373+0.306/3+0.306/3)*10^-3;

xprime(1) = x(3);

%x'=w

xprime(2) = x(4);

%y'=z

xprime(4) = (- mv * A * B * sin (wf * t) - 2 * mv * A * g * x(2))/(mv * A * L);

xprime(3) = (- (M + mv * A * L) * sin (wf * t) - mv * A * B * xprime(4) - M * ws^2 * x(1))/(( M + mv * A * L ));

xprime = xprime (:);

ti=0;

tf=20;

xi = [0 0 0];

[t,s] = ode45(@odetest,[ti,tf],xi);

x = s(:,1);

y = s(:,2);

Now I open the TOOLBOX MuPAD (for engenieering problems). But I don´t unsderstand how can I can get some results (the results are x(t) and y(t)) or if I really need that TOOLBOX. Can you help me with that?

Thanks

[Walter Roberson]

At the MATLAB command line, try

S = dsolve('mv*A*L*(diff(y(t), t, t))+2*mv*A*g*y(t) = -mv*A*B*sin*wf*t', '(M+mv*A*L)*(diff(x(t), t, t))+mv*A*B*(diff(y(t), t, t))+M*ws^2*x(t) = -(M+mv*A*L)*sin*wf*t');

Snum = subs(S);

The first line tries to solve the pair of differential equations completely symbolically, and the second line substitutes in the numeric values for the various constants.

[Igor]

Thank's for the help ;)