Numerically Integrating the differential equation below?

Bilal Arshed (view profile)

on 18 Nov 2019 at 16:48
Latest activity Edited by James Tursa

James Tursa (view profile)

on 18 Nov 2019 at 23:07
i know some boundary conditions such as, t0= 0 and v0=7000m/s (<===newly updated v0) and vf=0.... but do not know what, tf, i equal to.

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Bilal Arshed

Bilal Arshed (view profile)

on 18 Nov 2019 at 21:23
I don’t really know how I can implement ode45 to this problem. I have never used it before. Any guidance on it is much appreciated!
I know the basic syntax but since t_span requires tf I do not know what I can do if I no tf.
James Tursa

James Tursa (view profile)

on 18 Nov 2019 at 22:18
I'm not really sure what is being asked for here. The expressions for dr/dt and dtheta/dt are just going to be based on the integrals of your double_dot equations. What are you being asked for beyond that? The post doesn't ask for numerical solutions. Is that something you added?
Bilal Arshed

Bilal Arshed (view profile)

on 18 Nov 2019 at 22:23
so i need to numerically integrate both double dot equations in order to get them in the from which equals dr/dt and dtheta/dt so then i can integrate them again and have [r,theta] which i will use to plot the path taken by the reentry vehicle.
By numerically integrating it, i will have something to plot.
Also i have said v0=0 that is wrong, v0=7000ms^-1 (an arbitary choosen value)

darova (view profile)

on 18 Nov 2019 at 21:40

• Any guidance on it is much appreciated!
• I do not know what I can do if I no tf.
Try tf = 5. This should work

James Tursa

James Tursa (view profile)

on 18 Nov 2019 at 22:30
Please post the actual code as text that we can copy and run. We can't copy and run pictures of code.
Bilal Arshed

Bilal Arshed (view profile)

on 18 Nov 2019 at 22:30
i don't think so, that usually works just fine.
Bilal Arshed

Bilal Arshed (view profile)

on 18 Nov 2019 at 22:32
% { you can just use placeholder values for global variables like L and D.
function dv = myeqn(t, v)
mu_e = 3.986e14;
r=6378e3;
m=5000;
global L;
global D;
gamma=40; % not sure what to start at
v0=7000;
t0=0;
tend=5;
[t,v] = ode45(@myeqn, [t0,tend], v0)
dv = ((mu_e)./(r^2)) + ((v^2)./r) + (1./m)*(D*sin(gamma)+L*cos(gamma)); % radial accel equation

James Tursa (view profile)

on 18 Nov 2019 at 22:46
Edited by James Tursa

James Tursa (view profile)

on 18 Nov 2019 at 23:04

You've got two 2nd order DE's, so that means you have a 4th order system (2x2=4) and thus your state vector will contain four elements. You could define them as follows:
y = your 4x1 state vector with the following definitions
y(1) = r_r
y(2) = v_r
y(3) = r_t
y(4) = v_t
Then your derivative function outline would be:
function dy = myeqn(t, y)
% put some constants here or pass them in, e.g. mu etc.
dy = zeros(size(y));
dy(1) = y(2); % derivative of r_r is v_r
dy(2) = ___; % you fill this in from your r_r doubledot equation
dy(3) = y(4); % derivative of r_t is v_t
dy(4) = ___; % you fill this in from your r_t doubledot equation
end
For the dy(2) and dy(4) code, you will need to calculate your gamma value from the y vector. You could either hardcode the other stuff (D, L, m, etc.) or pass them in as input arguments. To start with, you will need to define initial values for all four states, not just v0. Also, I would have expected to see a factor somewhere on the r_t double dot equation based on the atmospheric density (a function of altitude), but I don't see it ... and this seems suspicious to me.

Bilal Arshed

Bilal Arshed (view profile)

on 18 Nov 2019 at 22:54
thanks, it is pretty late here so i will have a look into this tommorow morning. if all goes well i will accept the answer but i might have question when i come to atmepting it.
Thank you for helping, much appreciated!
P.S what is the y vector?
Is this the vector created by combining the tangent and radial velocities?
James Tursa

James Tursa (view profile)

on 18 Nov 2019 at 23:06
The y-vector is exactly what I have stated. It contains four elements as defined above. The radial position, radial velocity, tangential position, and tangential velocity.