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University of Northern Colorado Greeley, Colorado

ANALYTICAL MODEL OF LIQUID SLOSH-VERIFICATION EXPERIMENT (SPLASHSAT)

Authors Names Nathan Clayburn

Casey Kuhns

Faculty Advisers Dr. Cynthia Galovich Dr. Mathew Semak Dr. Robert Walch

School of Physics and Astronomy

Contact Information clay8798@bears.unco.edu kuhnscasey@gmail.com

March, 2009

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Table of Contents Cover Page Signature PageTable of Contents.................................................................................................................2 1 Acronym List ....................................................................................................................3 STATEMENT OF THE PROBLEM 2 Abstract .............................................................................................................................4 REVIEW OF RELATED RESOUCES 3 Introduction.......................................................................................................................6 4 Traditional Modeling Methods .........................................................................................6 5 Traditional Passive Methods.............................................................................................7 METHODOLOGY AND RESEARCH DESIGN 6 Test Objectives..................................................................................................................7

6.1 Goal................................................................................................................................7 6.2 Hypothesis and Expected Results ..................................................................................7 6.3 Uniqueness.....................................................................................................................8

7 Test Description................................................................................................................8 7.1 Mathematical Model ......................................................................................................8 7.2 Data Collection ............................................................................................................11 7.3 Data Analysis ...............................................................................................................12

8 References.......................................................................................................................12 9 Experiment......................................................................................................................13

9.1 Experiment Description and Background....................................................................13 9.2 Electrical System .........................................................................................................14 9.3 Software System ..........................................................................................................18

Cost Estimate 10 Budget ...........................................................................................................................19

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1 Acronym List

UNCO University of Northern Colorado SPLASHSAT Spacecraft liquid attenuation simulation hypothesis SAT IRS Infrared Sensor

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STATEMENT OF THE PROBLEM

2 Abstract

The presence of liquid onboard spacecraft has important implications that must be

addressed. The following experiment is designed to study the dynamics of onboard

liquids so that better liquid management may become possible.

Due to the acceleration of their containers, onboard liquids manifest reactive

forces on their containers that can have adverse effects on the performance of the vehicle.

Loss of rotational kinetic energy due to the motions of onboard liquids can lead to

increased wobble that can affect the stability of, and present severe control problems for,

the craft.

Due to the non-linear dynamics of sloshing liquids their analysis can be complex

and computer intensive. A simpler analytical model is presented to describe liquid slosh.

This simplified model, although not comprehensive, may yield practical results. An

experiment to verify the validity of such a model will be conducted. By comparing the

predictions made by the analytical model and actual slosh data, the model's validity can

be assessed.

The research will be conducted onboard a Terrier-Orion sounding rocket launched

from Wallops Flight Facility in Virginia. Wallops will provide the rocket and launch

operations. This flight is being funded by a grant from NASA and significant cost

sharing by Wallops and the Colorado and Virginia Space Grant programs.

In our experiment our cylinder of liquid will be constrained to one axis of motion:

the z axis. Given that the resultant force in the x-y plane will be zero, we can focus our

study to the z direction. The liquid canister will be able to move freely along the z axis.

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The liquid inside is considered to be a “black box”. We will not be concerned with the

detailed dynamics of the liquid (all of its internal degrees of freedom), but only the

effects of the liquid's bulk motion on its container. In this sense the liquid will be treated

as a mass whose motion inside its container influences the motion of the entire system.

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REVIEW OF RELATED RESOUCES

3 Introduction

When considering spacecraft attitude controls one must take into consideration

the motion of liquids aboard the spacecraft. The motion of these liquids exerts a torque on

their tank’s wall and as a result the spacecraft must adjust accordingly. Although models

exist that predict the behavior of liquids onboard a spacecraft, the physical phenomena is

poorly understood. (Diagnosis of Water Motion in the Sloshsat FLEVO tank).

4 Traditional Modeling Methods

Numerous analytical models have been used to describe the motion of fluids. The

most accurate description of liquid motions requires use of the Navier-Stokes equations.

(Robust Nonlinear Attitude Control with Disturbance Compensation). These formulas,

however, are not practical for control implementations as they are highly dependent on

boundary conditions and are computationally expensive.

Additional models have been suggested including (single and multi) mass-spring-

damper, pendulum liquid slug, and CFD/FEA models. (Robust Nonlinear Attitude

Control with Disturbance Compensation). These models work very well when dealing

with small linear or angular motions and are considered acceptable for some aerospace

craft. For example, they work well for rockets whose fuel pools at the bottom after the

main engine is fired. However, these methods have their limitations and a model needs to

be developed in which the fuel can display a large range of movement.

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5 Traditional Passive Methods

A modeling system that accounts for both the motion of the spacecraft and the

liquid fuel simultaneously would be most ideal. This is very difficult as one can not

control or measure the position or orientation of the fuel aboard the spacecraft accurately.

It is only possible to measure the effects of the fuel slosh on the total system.

As a result, many passive ways have been developed to dissipate the energy of the

fuel sloshing: baffles, slosh absorbers, and breaking a large tank into a smaller one (A

Standing-wave type Sloshing Absorber to Control Transient Oscillations). However,

these methods add weight and therefore increase launch cost.

METHODOLOGY AND RESEARCH DESIGN

6 Test Objectives

6.1 Goal

The primary mission of the SplashSAT experiment is to determine the validity of

our analytical method. This method assumes the liquid acts as an elastic mass distribution

that influences the motion of its container. In order to validate our hypothesis we will

measure the motion of a fluid filled container onboard a sounding rocket. Comparison of

experimental data and mathematical modeling will allow us to check the accuracy of such

a model.

6.2 Hypothesis and Expected Results

We hypothesize that our mathematical model will accurately describe the motion of

the liquid filled container. The SplashSAT experiment is designed to collect data

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continuously during successive parabolic flights. These data will be analyzed, and it will

be determined how well the mathematical model developed predicts real flight data.

6.3 Uniqueness

The experiment is unique in its relevance. Current unmanned and future manned

missions will require a careful understanding of liquid slosh and its dynamics. Both the

experiment and the mathematical model were constructed and developed by students

anticipating the importance of liquid dynamics to the aerospace field in the coming years.

7 Test Description

7.1 Mathematical Model

Our mathematical model, which follows, can predict the modes of oscillation which

the undamped system can display. We begin with the following experimental apparatus

(Figure 1.1). We then represent this situation as a pair of coupled damped harmonic

oscillators where m1 represents the liquid's mass and m2 represents the mass of the tank.

The motion of the liquid is communicated to the tank by k', the constant describing the

strength of the coupling spring.

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From the diagram the force equations are as follows:

2221222

1112111

)()(

xgmxxkxkxmxgmxxkkxxm&&&

&&&

βα−−′+′−=

−−′+−=

−

− (1)

m1 and m2 are the masses of the liquid and container, respectively. k and k' are the spring

constants. k speaks to the container's connection to the craft and k' to the liquid's

elasticity. Also note that α and β are damping coefficients. α describes the frictional

damping affecting the liquid and β quantifies the damping due to the coupling with the

craft. Gravity will be incorporated in the derivation but g, the acceleration due to gravity,

will approach zero in a free fall situation as we will see later.

Liquid filled Tank

Springs

Springs

Figure 1.1

m1

m2

k’

k’

k

Figure 1.2

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Factoring out a k and k', respectively, results in the following:

2212

222

1121

111

)()(

)()(

xxxkk

gmxkxm

xxxkk

gmxkxm

&&&

&&&

β

α

−′++′−=

−−′++−=

−

(2)

By substituting kgmxU 11 += and k

gmxV 22 += we find:

VmmkgkVUkVkVm

UmmkgkUVkkUUm

&&&

&&&

β

α

−−′

+−′+′−=

−−′

+−′+−=

)()(

)()(

212

121

(3)

Next we assume the standard trial solution:

tieAA

VU ω

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

2

1 (4)

This solution leads to the following in matrix notation:

⎥⎦

⎤⎢⎣

⎡−

−′

+⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−′+′−−′+−

=⎥⎦

⎤⎢⎣

⎡−

11

)()()(

122

1

212

121

22

11 mmkgk

AA

iAAkAkAAkkA

AmAm

βωαω

ω (5)

Next we find the homogeneous equation:

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⎥⎦

⎤⎢⎣

⎡−

−′

+⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

−′−′−′′−′−−

=⎥⎦

⎤⎢⎣

⎡1

1)(

00

122

1

22

12

mmkgk

AA

ikkmkkikkm

βωωαωω

(6)

During a free fall situation, g=0. We can then find the determinant of the matrix in order

to form a constraint for the solutions:

( 0)2)( 222

12 =′−−′−−′−− kikmikkm βωωαωω (7)

For the undamped case, this equation can be analytically solved for ω revealing the

frequencies for normal mode oscillations. However, the damped situation cannot be

solved analytically for ω. Still, a computer can solve the damped case by approximating

roots of the characteristic equation above.

The only unknown variable is ω. All other variables represent measurable physical

quantities. Even α and β can be measured; from our trial solution we understand that:

t=1/α and t= 1/β are time scales for damping. These critical values are the time over

which an oscillation amplitude will decrease by a factor of 1/e.

7.2 Data Collection

Velocity: During the rocket flight a series of photogates will record the displacement and

velocity of the canister along the rail.

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Accelerometer: To record the acceleration of the rocket we will use our own

accelerometers.

7.3 Data Analysis

The data recorded will be quantitative in nature. The data collected by the

photogates will allow us to determine the position, velocity, and acceleration of the

container along the rail. The frequency of the system's oscillation will be obtained from

these results. These results will then be used in conjunction with the mathematical model

to determine the model’s accuracy.

8 References

El-Sayad, M., Hanna, S., and Ibrahim, R “Parametric Excitation of Nonlinear Elastic Systems involving Hydrodynamic Sloshing Impact,” Nonlinear Dynamics, Vol 18, 1999, pp 25-50. Vreeburg, J.P.B., “Diagnosis of water motion in the Sloshsat FLEVO tank”, National Aerospace Laboratory NLR, 2000.

Walchko, K., “Robust Nonlinear Attitude Control with Disturbance Compensation”, Graduate Thesis, University of Florida, 2003. Anderson J., Turan, O., and Semercigil, S., “A Standing-wave type Sloshing Absorber to Control Transient Oscillations,” Journal of Sound Vibration, Vol 232, No 5, 2000, pp 839-856. Sidi, M., Spacecraft Dynamics and Controls, Cambridge University Press, New York, 1997. Hughes. P., Spacecraft Attitude Dynamics, John Wiley & Sons, New York 1986.

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9 Experiment

9.1 Experiment Description and Background

The goal of the experiment is to determine the viability of our mathematical model.

A tank partially filled with liquid (water) will be constrained by rails so that it may only

move along one axis throughout the duration of the flight. The displacement along the

rails as the liquid filled tank moves will be recorded. This data will be later used for

analysis on the ground. We expect that the actual flight data will match the predictions of

our mathematical model. Figure 7.1 shows the experimental apparatus; Figure 7.2 shows

the experimental apparatus confined by mounting brackets that will be used by other

payloads.

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(Figure 7.1 Experimental Apparatus)

(Figure 7.2 Full Canister)

9.2 Electrical System

The electrical system is very simple in this experiment. There is only a sensor

board that is powered by a NiCd battery. This sensor board consists of an AVR micro

controller, a 3-axis accelerometer and connections for the photogates.

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(Figure 7.3 Block Diagram of Electronics)

Experiment (5V)

IR

IR

IR

IR

Atmega32

Accelerometers (3.3V)

Mechanical Restraints

Data Logger

SD Card

Power Battery

Kill

G-switch

Latch5 V Reg

3.3 V Reg Power

Switch (5V)

5 V Out

3.3 V Out

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(Figure 7.4 Detailed Schematics of Data Acquisition Connections)

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(Figure 7.5 Schematics for the Power Circuit)

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9.3 Software Systems

When the G-switch is enabled data collection will began. Upon activation the data

logger will collect data until the battery runs out.

(Figure 7.6 Software Flow Chart)

Activation of G-Switch

Data Logger Initializes

Read Accelerometer

Read Encoder

Perform Velocity Calculation

Write to Data Logger

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COST ESTIMATE

10 Budget

The money for the SplashSAT experiment will be provided in part by COSGC funding

and UNC travel funds.

Part Amount ($) Manufacture

3-Axis Accelerometers $30.00 DigiKey NiCd Battery $20.00 Local Hobby Store

G-Switch $10.00 DigiKey Experimental Apparatus $150.00 BigBlueSaw.com

Photogates (4) $80.00 DigiKey SD Card Supplied by UNC In House

Data Logger $60.00 SparkFun Power System $20.00 DigiKey

Atmega32 $10.00 DigiKey Mechanical Restraints $20.00 In House

Latch $15.00 DigiKey Travel costs to Virginia (4 people) $2500 per person

TOTAL $10,415.00