Converting 2nd order ODEs to 2 1st order ODEs

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Converting 2nd order ODEs to 2 1st order ODEs

• Example

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Converting 2nd order ODEs to 2 1st order ODEs

• Example
• Define the state vector

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Converting 2nd order ODEs to 2 1st order ODEs

• Example
• Define the state vector
• Write state equations

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Converting 2nd order ODEs to 2 1st order ODEs

• Example
• Define the state vector
• Write state equations
• With initial conditions

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Solving Equations of Motions for Particles

• Example
• Particle in 3-D subject to thrust, gravity and
  drag
• State vector is 6x1

• 6 State Equations

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MATLAB Example

• A ball is thrown upward against gravitational
  attraction and air resistance
• with an initial velocity of 30 meters/second. The
  air resistance opposes the
• velocity and is proportional to the square of the
  velocity. The acceleration is
• a - g cv2 sign(v)
• where g 9.81 meter/sec2 and c 0.001 1/meter.
  Solve for the position and
• velocity of the particle as a function of time
  through a six second time interval.

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Figure SA2.2.1 (p. 91)Schematic of ejection
seat test device.
Example
The seat is moving at 600 knots when it is
ejected with a specified acceleration a-bt2 for
800 msec. A chute is released to slow the chair
down until the chair reaches a speed of 100
knots. Then a parachute is deployed. Assume the
parachute drag is 10 times the drag of the first
(drogue) chute.
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Ejection Seat
T 0 pilot pulls handle, T0.15 secs seat
clears cockpit T0.5-0.8 secs seat/man
separator fires T 200-400 seconds main
parachute deploys
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if t lt 0.8 this is phase I
accelerationa-btt xdot(3)
-accelerationsin(pi/9) sin(20 deg) sin
(pi/9 rads) xdot(4) accelerationcos(pi/9)
cos(20 deg) cos (pi/9 rads) else phase II
or III if speed gt v_crit this is phase II
xdot(3)-cspeedxdot(1)
xdot(4)-cspeedxdot(2)-g else we are in
phase III if x(2) gt 0 the ejection
seat is in the air xdot(3)-10cspeed
xdot(1) xdot(4)-10cspeedxdot(2)-
g else the ejection seat has hit the
ground xdot(3)0
xdot(4)0 xdot(1)0
xdot(2)0 end end end xdotxdot'
need to transpose the vector to get xdot
to be a column vector

• function xdotejectionSeat(t, x)
• This function computes the rate of change
• of position and velocity for the ejection seat
• x(1) position (y)
• x(2) velocity (ydot)
• k11.1508122 conversion from knots to miles
  per hour
• k21.4666249 conversion from miles per hour
  to ft/sec.
• kk1k2 conversion from knots to
  feet/second
• v_critk100 speed at which Phase III starts
• c0.003 c0.003 1/ft
• g32.2 g32.2 ft/sec/sec
• a16g a16g
• b9g b 9g