major_2006
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c++ program testTRANSCRIPT
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The University of QueenslandSchool of Engineering
MECH2700 Engineering Analysis I (2006)
Major Assignment: Apollo Re-entry Simulation
Introduction
On return to Earth, the Apollo missions used a version of “skip re-entry” in which the
capsule re-entered the atmosphere at a very shallow angle and then skipped along the the
upper atmosphere, much like a stone skipping along the surface of a pond. To control
this process, the capsule was flown at an angle of attack such that the aerodynamic forces
included both lift and drag.
The first part of the re-entry was flown with the lift force directed up (i.e. away from
the Earth’s surface). The capsule reached a minimum altitude with a velocity that was
still above escape velocity. As the capsule gained altitude, it was rolled over so that the
lift force pointed down and helped to hold the capsule within the atmosphere. The capsule
continued to decelerate and eventually the altitude began to decrease. The capsule could
be rolled once more to direct the lift force up and cause the capsule to skip again, thus
prolonging the period of deceleration in the upper atmosphere. Compared with a steeper
re-entry trajectory which plunged straight into the denser part of the atmosphere, this
skip re-entry trajectory produced lower peak accelerations and heat loads on the capsule.
The goal of this assignment is for you to write a Python program to simulate the
re-entry of an Apollo capsule and determine the peak accelerations for trajectories with
one and two skips.
You are to work in pairs for this assignment and, on the front cover of your submission,
you are to indicate the level of participation of each person. The intention is to promote
thinking and discussion before coding as well as to reduce the effort required of each
person. If possible, each group should include one Mechanical/Mechatronics student and
one Mech & Space student. Working notes and drafts of your code should be kept in your
blue workbook (that was given to you in MECH2300) and submitted, together with your
main report, at the end of semester.
Theory
Think of the capsule as a particle moving in an inertial reference frame whose origin is
located at the centre of the Earth. Recall, from first-year dynamics, the equations for a
particle’s velocity and acceleration in polar coordinates
r = r er + rθ eθ
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r =(
r − rθ2)
er +1
r
d
dt
(
r2θ)
eθ
Note that these are vector equations with components in the radial er and tangential eθ
directions. The position of the capsule at any time is given by the radial distance r, taken
from the centre of the Earth, and the angle θ (which can have an arbitrary reference, say,
zero at time t = 0). To determine the motion of the capsule, we need to determine the
acceleration and then integrate to get velocity and position.
Figure 1 shows the capsule of mass m on its descent trajectory and figure 2 shows the
forces acting on the capsule.
Figure 1
Figure 2
With gravitational force F and aerodynamic forces L and D acting on the capsule, the
radial acceleration can be written as
(
r − rθ2)
= −
(
µE
r2
)
−
(
D
m
)
sin γw +(
L
m
)
cos γw
and the tangential component as
1
r
d
dt
(
r2θ)
= −
(
D
m
)
cos γw −
(
L
m
)
sin γw
In these equations, the gravitational acceleration is F/m = µE/r2 where Kepler’s constant
for the Earth is µE = 3.986 × 1014m3/s2.
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V is the vehicle speed in the inertial frame of reference (i.e. the magnitude of the
velocity vector r). The flight-path angle, γ, is the angle that the inertial velocity vector
makes with the “local horizontal”. This angle can be computed as
γ = tan−1vr
vt
where vr = r and vt = rθ are the radial and tangential velocity components in the inertial
frame. Note that, for a descent trajectory, γ will have negative values and can be correctly
computed, even for zero values of vt, with the atan2 function in the math module.
To get an estimate of the aerodynamic forces acting on the capsule, we shall make the
approximation that the Earth and its atmosphere rotate together, much like a rigid body.
If the capsule enters the atmosphere with its velocity going in the same direction as the
Earth’s surface, the velocity components of the capsule with respect to the atmospheric
air will be
vrg = vr = r ; vtg = vt − ωEr ;
where the angular velocity of the Earth is ωE = 7.292× 10−5 rad/s. The direction of the
apparent wind can then be computed as
γw = tan−1vrg
vtg
and the wind speed will be
Vg =(
v2
tg + v2
rg
)1
2
The lift and drag forces can then be approximated by
L = CLA q ; D = CDA q
where q = 1
2ρV 2
g is the dynamic pressure, CL and CD are the lift and drag coefficients and
A is the frontal area. To get an estimate of density, we shall use a two-parameter model
of the atmosphere
ρ = ρ0 exp
(
−h
H0
)
where ρ0 = 1.752 kg/m3 and H0 = 6.7 km. The altitude can be computed as h = r −RE
where RE = 6378 km is the radius of the Earth.
Computer Implementation
To apply a numerical integration method such as Euler’s method (or a Runge-Kutta
method), it is best to rewrite our equations as a first-order system of ODE’s. A shorthand
expression of this system of equations is
x = f(t,x)
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where x is the “state vector” describing the system at any instant in time. We shall
choose our state vector to be
x =
x0
x1
x2
x3
=
rθr
θ
and write its derivative as
x =
r
θr
θ
When evaluating this vector of derivatives, we already have r and θ as part of our state
information but we need to calculate the other pair from our equations of motion
r = −
(
µE
r2
)
−
(
D
m
)
sin γw +(
L
m
)
cos γw + rθ2
θ = −
(
D
m
)
cos γw
r−
(
L
m
)
sin γw
r−
2r
rθ
Trajectory and Vehicle Data
The trajectory simulation is started at an altitude of 122 km with inertial speed V =
11 km/s and an initial flight-path angle γ = −6o. The capsule, with a base diameter of
3.91 m and a mass of 5900 kg, flys at a slight angle of attack such that CD = 1.2 and
CL = 0.4.
Initially, the capsule is oriented such that the lift force is up. At t = 105 s, the capsule
is rolled to make the lift force point down. At t = 280 s the capsule is rolled back to
the original orientation such that the lift force is again up. Hopefully, the capsule should
have slowed with moderate levels of deceleration.
At an altitude of 7 km, two drogue parachutes (each with diameter 5.0 m and drag
coefficient 1.2) are deployed to slow the capsule to about 280 km/h. At h = 3.05 km, the
three main parachutes are deployed. Each has a diameter of 25.5 m and a drag coefficient
of 1.2. The capsule should approach zero altitude with a low velocity with respect to the
surface of the Earth. Note that the tangential component of the inertial velocity will still
be quite large.
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Tasks
1. Read one or more of the following:
• sections 6.1, 6.2, 6.3 and 6.5 of Gerald & Wheatley’s text book Applied Nu-
merical Analysis;
• sections 8.1 through 8.4 of Schilling & Harris’ text book Applied Numerical
Methods for Engineers;
• sections 3.9 and 3.10 of Lerman’s text book Problem Solving and Computation
for Scientists and Engineers.
Summarize the important parts in your workbook.
2. Build ODE integration functions in Python that can be used to integrate a system
of N first-order ordinary differential equations using both Euler’s method and the
fourth-order Runge-Kutta method. Test the integrators on the system of equations
dx
dt= (1 − y)x ,
dy
dt= (x − 1)y ,
using initial conditions x(0) = 0.5, y(0) = 0.5. These equations are for a “predator-
prey” ecological system where x represents the population (in thousands, say) of
prey and y represents the population of the predator. Integrate from t = 0 to t = 10
using a step size 0.04 and plot samples of y versus x. A sample solution (example
8.3.2 in Schilling & Harris) is illustrated here.
Figure 3
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
x2
x1
Predator-prey with x(0)=[0.5;0.5]. Ex 8.3.2 Schilling & Harris.
’pprey.dat’ using 2:3
3. Build a Python program to simulate the re-entry of an Apollo capsule. Starting
with values given above, your program should compute r, r, θ, θ, h, V , Vg, γ and
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γw at 5 second intervals during the descent. These can be obtained by numerically
integrating the set of ordinary differential equations described in the Theory section.
The Euler method appears to do the job adequately with time steps of about 0.05 s.
Note that the integration process should stop if the vehicle reaches the Earth’s
surface (i.e. h ≤ 0).
4. Plot the velocity Vg, altitude and acceleration (dV/dt) as functions of time for the
trajectory described above. Repeat the exercise for a trajectory where the first roll
is executed but not the second. For reference, the velocity altitude maps for both
trajectories are shown in figure 4. Your solutions should be similar.
What happens if the first roll is not performed?
What is the velocity at “splash-down” (h = 0)?
What are the largest decelerations for each of the trajectories (excluding parachute
deployment and splash-down)?
5. Prepare and submit a report on your programs and solutions. Include your program
source code in the report. Remember to submit your workbook as well.
Figure 4
-20000
0
20000
40000
60000
80000
100000
120000
0 2000 4000 6000 8000 10000 12000
Alti
tude
, m
V relative to ground, m/s
Apollo Reentry Simulation
"single_roll_3.dat" using 3:6"double_roll_3.dat" using 3:6
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