Rocketry Analysis

7/27/2019 Rocketry Analysis

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Rocketry Theory, Part 2:More Numerical Analysis and MATLAB

Tim BretlUniversity of Illinois at Urbana-Champaign

AE100SDOctober 7, 2010
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Deriving Euler Integration


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The equation of rocket motion

We derived the following:

mRvR=TmRg kv2R

Some terms may be functions of time:

the rocket velocity vR

the rocket mass mR

the thrust T

Some terms will always be assumed constant:

the acceleration of gravity g

the drag constant k= 12airARCD
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Solving the rocket equation in simple cases

Neglect drag and assume T and mRare constant:

mRvR=TmRg

Rearrange:

vR= T

mR g

Integrate:

vR(t) =vR(0) +

t0

T

mR g

ds

= 0 +TmR

g t=

T

mR g

t
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Solving the rocket equation in less simple cases (1)

Restore drag but still assume T and mRare constant:

mRvR=TmRg kv2RRearrange:

vR= TmR

g kmR

v2R

Integrate:

vR(t

) =vR(0) +

t0 TmR

g

k

mRvR

2 dsThat doesnt look right!
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Solving the rocket equation in less simple cases (2)

Its actually still possible to integrate this:

vR= TmR

g kmR

v2R

But we need to use some tricks from calculus:

vRv=0

TmR g

k

mRv21

dv= ts=0

ds

The result can be found in a table of integrals:

mRtanh1 k

T

mRg

vRk(TmRg)

=t

vR(t) =

TmRgk

tanh

tk(TmRg)

mR
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A different approach

This is an ordinary differential equation:

vR= T

mR g k

mRv2R

It has a standard form:x=f(t, x)

Equations of this form are, in general, easy to solvenumerically

using a tool like MATLAB.
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The basis for numerical integration

t0 t

x(t0)

x(t)

t t0

x(t) x(t0) + x(t0) (t t0)

x(t0) (t

t0)

error
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Where it comes from

The following is a good approximation when t t0:x(t) x(t0) + x(t0) (t t0)

This is the linear part of a Taylors series expansion:

x(t) =x(t0) + x(t0) (t t0) +12x(t0) (t t0)2 +

It also corresponds to our notion of what a derivativeis:

x(t0) = limtt0

x(t) x(t0)t t0
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How we use it

Given x(0) and a fixed time step t, iterate:

x(t+ t) x(t) +f(t, x(t))tThis algorithm is calledEuler integration.
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Application to the rocket equation

Assume T= 20, mR= 1, g= 10, and k= 1:

vR= T

mR g k

mRv2R

= 10 v2RChoose the time step t= 0.1 and iterate:

vR(0.1) =vR(0) +

10 vR(0)2

t

= 0 + (10 0) 0.1= 1

vR(0.2) =vR(0.1) +

10 vR(0.1)2

t

= 1 + (10 1) 0.1= 1.9
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Compare to the exact expression

We have found:

vR(0.1) = 1vR(0.2) = 1.9

The exact solution was:

vR(t) =

TmRgk

tanhtk(TmRg)

mR

=

10 tanht

10

Plug in to find the exact values:

vR(0.1) = 0.968

vR(0.2) = 1.770
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Implementing Euler Integration Using MATLAB(see example.m)
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% Clean up the workspace.

clear;

clc;

What it means:

Forget all previously defined variables Erase the command window

In other words,give me a new sheet of paper.

Why:

Dont accidentally reference an old variable (doing so now willcause an error, so it will be easy to debug)

Dont get confused looking at the results of old scripts
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% Define constants.

T = 2 0 ;

m R = 1 ;

g = 1 0 ;k = 1 ;

% Choose a time step.

dt = 0.1;

% Choose a time horizon.

tmax = 1;

What it means:

Assign a value of 20 to the variable T, etc.

You can now write Tinstead of 20 whenever you want

Why:

Makes code easier to read (matches formula on paper)

Changing the value ofT is easy


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% Define an array of times.

t = 0:dt:tmax;

What it means:

Create a list of numbers

0 t 1 t 2 t N t

such that N t tmax and (N+ 1) t>tmax Its anarray, or arow vector, or amatrix of size 1 N

Why:

Define it b/c its a compact way to store a lot of numbers

Define it this way b/c t(1) = 0 t, t(2) = 1 t, etc., istedious and error-prone
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% Define the initial condition.

vR(1) = 0;

% Iterate to use Euler integration.

for i=2:length(t)vR(i) = vR(i-1) + ((T/mR)-g-((k/mR)*vR(i-1)2))*dt;

end

What it means (Euler integration):

Given vR(0) = 0 and a fixed time step t, iterate:

vR(t+ t) vR(t) +T

mR g k

mRv2R

t

NOTE:() means a different thing for us vs. MATLAB!Why:

The use offorallows you torepeat exactly the sameprocessmany times but write it only once

Less tedious, fewer errors, easier to changeanddebug
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These things are equivalent:

% Define the initial condition.vR(1) = 0;


for i=2:length(t)

vR(i) = vR(i-1) + ((T/mR)-g-((k/mR)*vR(i-1)2))*dt;

end

% Define the initial condition.

vR(1) = 0;


vR(2) = vR(1) + ((T/mR)-g-((k/mR)*vR(1)2))*dt;vR(3) = vR(2) + ((T/mR)-g-((k/mR)*vR(2)2))*dt;

vR(4) = vR(3) + ((T/mR)-g-((k/mR)*vR(3)2))*dt;

% etc.
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% Compute the exact values for comparison.

vR Exact = sqrt((T-mR*g)/k)*tanh(t*sqrt(k*(T-mR*g))/mR);

What it means:

Evaluate

vR(t) =

TmRg

k tanh

tk(T mRg)

mR

at every value of t in our list t, and put the result in a new

list called vR Exact We call this anelement-wiseoperation

Both t and vR Exact will have the same size
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% Display the results as text.

[t' vR' vR Exact']

What it means:

Take the row vectors t, vR, and vR Exact, transpose them to

get column vectors, and insert these side-by-side in a matrix If each row vector is 1 N, the matrix is N 3

Why:

Column vectors are easier to view in the command window

Putting data side-by-side makes it easier to compare

Eyeballing the data is one way to debug
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% Plot the results.

figure(1);

clf;

plot(t,vR,'r.-',t,vR Exact,'ko-');

xlabel('time');ylabel('velocity');

legend('Euler','Exact');

axis([0,tmax,0,5]);

grid on;
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% Plot the results.

figure(1);

clf;

plot(t,vR,'r.-',t,vR Exact,'ko-');

What it means:

Switch to the figure named 1 if it exists, otherwise create it

Erase everything in this figure

Plot the approximate value ofvR(t) as a red line, with eachdata point as a red dot

Plot the exact value ofvR(t) as a black line, with each datapoint as a black circle
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xlabel('time');

ylabel('velocity');

legend('Euler','Exact');

axis([0,tmax,0,5]);

grid on;

What it means:

Label the xaxis time

Label the yaxis velocity

Add a legend that says the red line is approximate (from Eulerintegration) and the black line is exact

Make the xaxis run from 0 to tmax

Make the yaxis run from 0 to 5

Draw a grid, making it easier to interpret the results
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Dropping Assumptions

Making thrust a function of time (1)
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Time (s)

Thrust (N)266110

0.1 1.7 1.8Figure: A typical thrust profile for motors like ours. It is piecewise-linear,with a totalburn timeof 1.8 seconds.

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Time(s)

Thrust (N)266110

0.1 1.7 1.8

T =

0 +26600.10

(t 0) if 0 t


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T =

0 + 26600.10 (t0) if 0

t


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Making mass a function of time

mR=

mrocket+mpropellant

1 t

1.8

if t


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Making drag point in the right direction (1)

The drag term is

D= k

mRv2

R

where

k=1

2AIRARCD,

i.e., its a force that points down. This is only correct if the rocketis going up! In general

D=

k

mRv2R ifvR 0

kmR

v2R ifvR


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Making drag point in the right direction (2)

D=

kmR

v2R ifvR

0

kmR

v2R ifvR= 0)

D = -(k/mR)*(vR(i-1)2)

elseD = (k/mR)*(vR(i-1)2)

end

D= kmRv2R vR

|vR|

D = -(k/mR)*(vR(i-1)2)*(vR(i-1)/abs(vR(i-1)))

Other assumptions


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Other assumptions

We have made other assumptions along the way:

gravity is constant with altitude

air density is constant with altitude

motion is entirely vertical

no wind (at least, none that affects vertical motion)

etc.

Effects are negligible, so we will continue to ignore them.


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Uncertain Parameters and Real Data

The exact value ofCD is unknown
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D

The drag term is

D= k

mRv2R

vR

|vR|

where

k=1

2AIRARCD.

The coefficient of drag CDis hard to predict exactly, and in many

cases must be estimated from experimental data. How?

A similar but simpler problem
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p p

Say the position xof an object as a function of time t is

x(t) = sin (ct) ,

where c is an unknown constant in the range

cmin c cmax.

Given experimental data

xmeas(t1), . . . , xmeas(tn)

collected at times t1, . . . , tn, how might we estimate c?

What makes for a good estimate of c?
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g

A guess at cgives us the prediction

x(t1) = sin (ct1) , . . . , x(tn) = sin (ctn) .

Define theprediction errorassociated with this guess by

(x(t1) xmeas(t1))2

+ + (x(tn) xmeas(tn))2

,

or in other words by

n

i=1

(x(ti)xmeas(ti))

2 .

A good guess is one that results in minimum prediction error.

Some notation
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Make the dependence of prediction error on cclear by writing

x(t, c) = sin(ct)

and

e(c) =

ni=1

(x(ti, c) xmeas(ti))2 .

An algorithm to find a good estimate of c
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1. Sample m guesses according to

ck=cmin+

k 1m 1

cmax

fork= 1, . . . ,m

2. Compute the prediction error

e(ck) =n

i=1

(x(ti, ck) xmeas(ti))2

associated with each guess3. Choose the estimate c=ck, where

k = arg minke(ck)

How to implement this algorithm in MATLAB (1)
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Sample m guesses according to

ck=cmin+

k 1m 1

cmax

fork= 1, . . . ,m

% Define the possible range of c values

cmin = 0;

cmax = 10;

% Define the number of guesses to make

m = 5 0 ;% Generate m guesses, equally spaced from cmin to cmax

c = linspace(cmin,cmax,m);

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Compute the prediction error

e(ck) =

ni=1

(x(ti, ck) xmeas(ti))2

associated with each guess

% Load the experimental data

% (first column is time, second column is position)

load sinusoid.data

t = sinusoid(:,1);

xmeas = sinusoid(:,2);

% Loop to compute the prediction errorfor k=1:m

x = sin(c(k)*t);

e(k) = sum((x-xmeas).2);

end

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Choose the estimate c=ck, where

k = arg minke(ck)

% Find the value of c giving minimum error

[emin,kmin] = min(e);

cest = c(kmin)

Example results (see parameter.m)
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0 0.2 0.4 0.6 0.8 12

1.5

1

0.5

0

0.5

1

1.5

2Results of Best Fit: c=5.95

time

position

predicted

measured

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90

100Error for Different Values of c

c

predictione

rror

An important note
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This approach only makes sense for certain assumptions about the

data, which we will not discuss.
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Homework

Problem 1
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The data in decay.data was generated by the process

x(t) =ebt sin(ct) ,

where c= 6 but b is known only to be between 0 and 5. Write aMATLAB script that estimates the value ofband produces two

sub-plots showing the results of the best fit and the error fordifferent values ofb, just as in the example parameter.m.

Name your script netidP1.m, where netid is your NetID (i.e.,

the start of your @illinois.edu

email address). Attach it to anemail and send it to Shreyas Narsipur ([email protected])by Monday, October 25.

Problem 2
http://localhost/var/www/apps/conversion/tmp/scratch_3/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/[email protected]://find/


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The data in rocket.data is from a rocket with parameters

Duration of burn tb= 1.8s

Time delay after burn is over before parachute opens td= 13s Mass of rocket mrocket= 1.8934kg

Mass of propellant mpropellant= 0.178kg

Acceleration of gravity g= 9.81m/s2

Air density AIR = 1.23kg/m3

Rocket diameter d= 3 inches

Thrust:

Time (s)

Thrust (N)266110

0.1 1.7 1.8

Problem 2, cont.


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Write a MATLAB script that estimates the value ofCD andproduces two sub-plots showing the results of the best fit and theerror for different values ofCD, just as in the previous problem.

Name your script netidP2.m, where netid is your NetID (i.e.,the start of your @illinois.edu email address). Attach it to anemail and send it to Shreyas Narsipur ([email protected])by Monday, October 25.

Collaboration
http://localhost/var/www/apps/conversion/tmp/scratch_3/[email protected]://localhost/var/www/apps/conversion/tmp/scratch_3/[email protected]://find/http://goback/


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You are encouraged to work together on these problems, but pleasestate explicitly (in your MATLAB scripts) who you worked with andwhat each person did. Each of you must submit a script separately.
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