Model Motion of Circling Airplane

Open Live Script

Start with an airplane moving along a circular track with a radius of 10 km at a horizontal speed of 100 m/s and descending at a rate of 1 m/sec. To create circular motion, specify a radially-inward acceleration and constrain the acceleration to lie in the horizontal plane. The acceleration of a body moving in a circle is $\frac{v^{2}}{r}$ . The rate of descent is constant. Set the initial orientation axes matrix of the platform to the identity matrix.

Set up the initial conditions

alt = 10000;
radcirc = 10000; % 10 km
phi = 60;
initPos = [cosd(phi)*radcirc;sind(phi)*radcirc;alt];
vs = 100.0;
vx = vs*sind(phi);
vy = -vs*cosd(phi);
vz = -1;
initVel = [vx,vy,vz]';
airplane = phased.Platform('MotionModel','Acceleration', ...
    'AccelerationSource','Input port','InitialPosition',initPos, ...
    'InitialVelocity',initVel,'OrientationAxesOutputPort',true, ...
    'InitialOrientationAxes',eye(3));
accelmag = vs^2/radcirc;
initPos1 = [cosd(phi)*radcirc;sind(phi)*radcirc;0];
unitvec = initPos1/radcirc;
accel = -accelmag*unitvec;

Compute the trajectory

Compute the trajectory for 20000 integration steps at T = 0.1 s intervals

N = 20000;
tstep = .10;
posmat = zeros(3,N);
for n = 1:N
    [pos,vel,oax] = airplane(tstep,accel);   
    velcirc2 = vel(1)^2 + vel(2)^2;
    vmag = sqrt(velcirc2);
    pos1 = [pos(1),pos(2),0]';
    radcirc = sqrt(pos1'*pos1);
    unitvec = pos1/radcirc;
    accelmag = velcirc2/radcirc;
    accel = -accelmag*unitvec;
    posmat(:,n) = pos;
end

Display the final orientation of the local coordinate system.

disp(oax)

    0.1271    0.9919    0.0001
   -0.9919    0.1271    0.0003
    0.0003   -0.0001    1.0000

Plot the trajectory

plot3(posmat(1,:)/1000,posmat(2,:)/1000,posmat(3,:)/1000,'b.')
xlabel('X (km)')
ylabel('Y (km)')
zlabel('Z (km)')
axis equal
grid

Figure contains an axes object. The axes object with xlabel X (km), ylabel Y (km) contains a line object which displays its values using only markers.