! APMonitor Modeling Language
! http://www.apmonitor.com
! Endeavor shuttle launch simulation
! The shuttle rises to 380 km and has a velocity of 7.68 km/sec at the final state
Model endeavor
Parameters
! cross-sectional area of shuttle
! solid rocket booster (srb) => cross-sectional area 10.8 m^2 => diameter 3.71 m
! 1st stage => cross-sectional area 55.4 m^2 => diameter 8.4 m
! orbiter => cross-sectional area ~ 20 m^2
d0 = 100.0 ! 1st drag constant (N/(m/s)^2)
e0 = 1.0 ! 2nd drag constant (dimensionless)
g0 = 9.8 ! gravity at launch (m/s^2)
m0 = 2029203 ! mass of shuttle at launch
h0 = 1000 ! initial height from earth's center (m)
c = 1.5e7 ! impulse of rocket fuel (N/(kg/sec))
t_srb = 12.5e6 ! 2 solid rocket boosters (HMX) each have 12.5e6 N of thrust at lift-off
! they burn for 124 sec to height of 45.7 km
! they are released at 126 sec
t_1st = 5.25e6 ! 1st stage fuel tank (liq H2/O2) has 5.25e6 N of thrust at lift-off
! it fires for 480 sec when it is released
t_orb = 53.0e3 ! orbiter (MMH/N2O4) has 53e3 N of thrust at lift-off
! it fires for 1250 sec
n_srb = 2 ! number of solid rocket boosters
n_1st = 1 ! number of 1st stage thrusters
n_orb = 1 ! number of orbiter thrusters
End Parameters
Variables
t ! thrust force (N) 0 < t < t_max
m = m0 ! mass of shuttle and fuel (kg) m_shuttle < m < m_full
g = g0 ! gravitational force
h = h0 ! altitude from earth's center (m) h > h0
d = 1 ! drag force (N)
v = 0 ! velocity (m/s)
a = 0 ! acceleration (m/s^2)
End Variables
Equations
! thrust
t = n_srb * t_srb + n_1st * t_1st + n_orb * t_orb
! gravitational variation with height
g = g0 * (h0/h)^2
! velocity
$h = v
! acceleration
$v = a
! force balance
! inertial + gravitational + drag = thrust
m*a + m*g + d = t
! aerodynamic drag force
d = d0 * v^2 * exp(-e0*(h-h0)/h0)
! mass loss due to burn-off of fuel
c * $m = -t + 0*m
End Equations
End Model