Analysis of Rocket Flights

Section 4, Team 4

Student 1, Student 2, Student 3, Student 4

Temperature Predictions

Decrease in temperature upon ascent

Rise in temperature upon descent

Further increase after landing

Thermistor 3 (middle of rocket body, on surface)

Thermistor 2(on fin)

Altimeters

Expect decrease in pressure with increase in altitude, and vice versa

Used barometric equation to find altitude

Calibrated sensors in lab using vacuum chamber

))325.101

)((1(104544.1)( 1902.05 kPaP

fth −××=

Altimeter vs. Models for Flight

Flight Modeling (2-D)

CG

CPmg

D

T

y

z

rWind

T

CG

CPmg

rWind

D

y

z

θ

Euler’s Integration

Method for numerical integrationIterative

For given a(t) and initial conditions for x and v:

v(t+Δt)=v(t)+a(t)*t

x(t+Δt)=x(t)+v(t)*t

IMU Analysis: Mudd IIIC (Large) Rocket

Rotation from local to global axes

Euler integration of rotation matrix

az

ax

ay

Ay

Az

Ax

y

az

ayax

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Z

Y

X

XY

XZ

YZ

Z

Y

X

a

a

a

A

A

A

1

1

1

φφ

φφ

φφ

,cos1sin

)()( 22 ⎟

⎠

⎞⎜⎝

⎛ −++=+ BBItRttR

σσ

σσδ

,

0

0

0

tB

XY

XZ

YZ

δωω

ωωωω

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−=

222ZYXt ωωωδσ ++=

Processing Algorithm (Matlab)

Raw RDAS Data

(counts)

Local Acceleration

(m/s2)

Calibrations

Global Acceleration

(m/s2)

Global Velocity (m/s) and Position

(m)

Local Rotation Rate (radians/sec)

Rotation Matrix (radians/sec)

Calibrations

Euler integration

Euler integration Local Rotation Angle

(degrees)

Filtered Global Acceleration

(m/s2)

Acceleration Filtering (optional)

Principle Axis Rotation: Plot vs. Video

1-D Model Comparison (launch 2)

day 1 2nd

launch (13-20 mph

winds)

IMU data R-DAS pressure altimeter

% difference

from altimeter

Student model

% difference

from model

Rocksim model

% difference

from model

Apogee height

160.3390 meters

154.53 m +3.759% 147.3-182.3 m

(lies within range)

165.3 m -3.001%

Apogee time

6.38 sec 4.645-6.145 sec

+3.824% 5.763-6.293 sec

+1.382% 5.888 sec +8.356%

Max z vel

54.35 m/s N/A N/A 52.46-58.88 m/s

(lies within range)

57.24 m/s -5.049%

Max z accel

206.4 m/s2 N/A N/A 180.8-198.4 m/s2

+4.032% 202.5 m/s2 +1.926%

Acceleration Filtering (Before)

Acceleration Filtering (After)

Acceleration Filtering (Descent Plot)

E80 teams

wind

z

Bad Data

Mudd IIIA IMU rocket Failure to

eject parachute

Flat spin crash after apogee

WHY?

http://www.tribuneindia.com/2002/20020715/world.htm

Principle Axis Rotation Plot vs. Video, Round 2

Acceleration Data…

Very pronounced 0.2149 Hz oscillations

Possible causes: camera interference, camera overpowering

Band-stop filter might be able to retrieve data

Vibration Analysis

Tap tests on hollow tube are inaccurate

Mass spring damper system

Theoretical Analysis

Spring-Mass-Damper Model

Rocket can be modeled as a single degree of freedom spring-mass-damper system.

Effective mass, m

Spring constant, k

Damping Half-Power Bandwidth

kgMM rocketeff 813.035

17==

mNx

Fk /38536==

1.≤ς

221

2

ςςωω−

=Δ r

Predicted Resonance Frequency

Hzm

kfr 65.34

2

1==

π

ς

Analysis

No control variables!Treat Sensor 10 as input.

Create FRFs of other sensors to see relative peaks

Sensor 10

FRF plots

Removed DC offsetfdomain.m used to generate Fourier

CoefficientsRelative AmplitudesFirst set of data is not trustworthySecond set of data has more coherent

peaksUsed 1st second of data, short motor burn

time

0 20 40 60 80 1000

2

4

6

8

10FRF of Sensor 7

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4

5FRF of Sensor 6

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4

5FRF of Sensor 1

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4

5FRF of Sensor 15

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

2

4

6

8

10FRF of Sensor 12

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

2

4

6

8

10Input (sensor 10)

Frequency

Magnitude (Gain)

1st Set of Data Results

Peak around 60 or 70 HzOther peaks are inconsistentSensor 15 seems to be malfunctioningLocally, 3 sensors show local peaks

between 60-80No video

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1FRF of Sensor 1

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4

5FRF of Sensor 3

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4FRF of Sensor 6

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3FRF of Sensor 8

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1FRF of Sensor 13

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4Input (sensor 10)

Frequency

Magnitude (Gain)

2nd Set of Data Results

Consistent peaks at 64 HzPossible peaks around 30 Hz, but not

consistentSensors 1, 3, and 8 are 13 show peak

frequenciesSensor 13 farther away from the input

source

Noises

Only 64 Hz showed in every FRFOthers are jumbled by the noiseRunning averages smoothes out the data

too much. Too little data during the 1st second of input Ineffective way of removing noise

0 20 40 60 80 1000

1

2

3

4Input (sensor 10)

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1FRF of Sensor 1

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4

5FRF of Sensor 3

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3

4FRF of Sensor 6

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

1

2

3FRF of Sensor 8

Frequency

Magnitude (Gain)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1FRF of Sensor 13

Frequency

Magnitude (Gain)

Mode ShapesAbsolute magnitude of Fourier Coefficients

vs Relative Sensor DistancesSensor 10 was normalized as “0” point.

5 10 15 20 25 30 35-2

-1.5

-1

-0.5

0

0.5

1

Location

Absolute Magnitude

Fundamental Modal Shape - 64 Hz

Results from FRF

Not enough frequencies to test all 3 mode shapes

Does not deal well with noise, especially with highly aliased data

Problems with FFT

Using just FFT coefficients to calculate Frequency Response Functions assumes a clean periodic signal.

The rocket data is neither. A better estimator is Power Spectral Density (PSD).

Power Spectral Density

2)(

1lim)( ωωφ j

Tj TXX Χ=

)()(1

lim)( * ωωωφ jjT

j TTXY ΥΧ=

)()(

)(lim

)(

)( ωωω

ωφωφ

jHjj

jj

T

T

XX

XY =ΧΥ

=

Auto power spectral density

Cross power spectral density

Frequency Response Function0 10 20 30 40 50 60 70 80 90 100

-10

-5

0

5

10

15

20

25

30

Frequency (Hz)

Power/frequency (dB/Hz)

PSD of Sensor 4

PSD and Noise

dttntvtxT

P Txy )]()([)(*1

lim +⋅= ∫∞

∞−

xnxvxy φφφ +=

H(jωx(t) v(t) y(t)

n(t)

Assume n(t) is unrelated to v(t)

0

)()(

)()(1 ω

ωφωφ

ω jHjj

jHxx

xy ≅=

Hamming Window

Time Domain Frequency Domain

Averaging Overlap

Overlapping windowed segments by 50% minimizes attenuation of time domain signal near the end of segment

Frequency Response Function

Waterfall Analysis

freq (Hz)

time (.1 sec)

mag

nitu

de (

dB)

FRF of Sensor 5 over time

Conclusions

Thermistor behavior depends on locationEuler Integration Method not sufficient to

model whole flight pathSpring-Mass-Damper model can simplify

system to find theoretical resonance FFT method of finding FRF is not

consistent due to large noise componentPSD method gives much sharper peaks in

FRF

Interesting Precautions...

Check battery…sensors are sensitive!

Wait until last moment to turn on R-DAS and video camera…otherwise, ejection charge could go off early!

Don’t try to catch rocket…it may have a chute, but it’s still falling fast!

Extra: Altimeter Plots

Extra: Altimeter Plots

Extra: Why We Didn’t Do 2-D Model Comparison

Extra: Why We Didn’t Do 2-D Model Comparison

Acknowledgements

The professors and proctors who helped to make this beta-test a success.

All of our classmates for their infinite support and advice during this semester

Student A for a discussion on the causes of small rocket IMU corruption

Student B for his help with setting up the Single Degree of Freedom model

References

E80 The Next Generation Spring 2008, http://www.eng.hmc.edu/New E80/index.html.

R. Wang, http://fourier.eng.hmc.edu/e80/inertialnavigation/ Q. Yang, http://www.eng.hmc.edu/NewE80/PDFs/Lecture_PressureSensor

Thermistors.ppt H. Buchholdt, Structural Dynamics for Engineers (Telford, 1997), pp. 17-22. The Hanning Window, http://www.dliengineering.com/vibman/thehanning

window.htm