major_2006

6
The University of Queensland School 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 ofyour 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 ˆ e r + r ˙ θ ˆ e θ 1

Upload: jackfin

Post on 17-Jan-2016

4 views

Category:

Documents


0 download

DESCRIPTION

c++ program test

TRANSCRIPT

Page 1: major_2006

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

1

Page 2: major_2006

r =(

r − rθ2)

er +1

r

d

dt

(

r2θ)

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.

2

Page 3: major_2006

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)

3

Page 4: major_2006

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

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.

4

Page 5: major_2006

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

5

Page 6: major_2006

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

6