Vibration Modal Analysis of a Deployable

Boom Integrated to a CubeSat

VALERIY SHEPENKOV

Degree project in

Mechanics

Second cycle

Stockholm, Sweden 2013

Vibration Modal Analysis of a DeployableBoom Integrated to a CubeSat

Valeriy ShepenkovSchool of Engineering Sciences, Department of Mechanics

KTH Royal Institute of Technology

A thesis submitted for the degree of

Master of Science in Engineering Mechanics

February 2013, Stockholm, Sweden

Dedicated to my mother - Liubov Esaulova.

Acknowledgements

I am truly thankful to my parents and relatives for their compassion andsupport during the two years of my studies at KTH. I would like to thankmy thesis advisor Dr. Gunnar Tibert who provided me with a valuableleadership with which I was able to complete the project. I would like tothank Gunnar Tibert for giving me the opportunity to work in the CubeSatproject at KTH, and for being a great coordinator in Engineering Mechan-ics MSc program. It was my pleasure to work together with Julien Servaisand Pau Mallol on the satellite project at Mechanics Department. I wouldlike to thank my friends who made my life full of interesting events duringthe two years in Stockholm. I am thankful to Will Reid who has been agreat project manager and a great colleague in the RAIN project which Iparticipated in at KTH during 2010 – 2012. Thank you Will for teachingme LaTeX and MATLAB programming. Overall my graduate studies atKTH were carried out thanks to the generous Visby Scholarship providedby Swedish Institute and the financial aid is gratefully acknowledged.

Abstract

CubeSat or Cubic Satellite is an effective method to study the space aroundthe Earth thanks to its low cost, easy maintenance and short lead time.However, a great challenge of small satellites lies in achieving technicaland scientific requirements during the design stage. In the present workprimary focus is given to dynamic characterization of the deployable tape-spring boom in order to verify and study the boom deployment dynamiceffects on the satellite. The deployed boom dynamic characteristics werestudied through simulations and experimental testing. The gravity off-loading system was used to simulate weightlessness environment in theexperimental testing and simulations showed that the deployment of thesystem influence the results in a different way depending on the vibrationmode shape.

Sammanfattning

En CubeSat eller kubisk satellit är effektivt för att studera rymden runtjorden på grund av dess låga kostnad, enkla underhåll och korta ledtid. Enstor utmaningen i utformningen av små satelliter är att uppnå de tekniskaoch vetenskapliga kraven. Detta arbete har analyserat de dynamiska egen-skaperna hos en utfällbar band-fjäder bom i syfte att verifera och för attstudera bommens utfällningsdynamiska effekter på satellitens bana och at-tityd. Den utfällda bommens dynamiska egenskaper har studerats genomsimuleringar och experimentella tester. Ett tyngdkraftskompenserande sy-stem har använts för att simulera tyngdlöshet i de experimentella testernaoch simuleringar visar att utformningen av detta system påverkar resulta-ten olika beroende på svängingsmodens form.

Contents

Contents v

List of Figures vii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory of Structural Dynamics 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Equation of motion and natural frequency . . . . . . . . . . . . . . . 72.3 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Steady-state vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Root–mean–square value . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Modal parameter extraction: complex exponential method . . . . . . 12

3 Design of the Structure 143.1 Concept design of the CubeSat . . . . . . . . . . . . . . . . . . . . . 143.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Gravity off-loading systems as a mean to obtain dynamic characteris-

tics of structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Measurements 224.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Measurement procedure . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Modal parameter extraction . . . . . . . . . . . . . . . . . . . . . . . 27

v

CONTENTS

4.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.7.1 Results of the measurements for the free free vibration test 1 . 304.7.2 Results of the measurements for the free free vibration test 2 . 314.7.3 Results of the measurements for cantilever down testing . . . 31

4.8 Mode shapes extracted using the results of the vibration testing . . . . 32

5 Finite Element Modal Analysis of the Boom Integrated to a CubeSat 385.1 Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Geometry of the structure . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Material properties for components of the model . . . . . . . . . . . . 405.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.5 Eigenfrequencies from the model . . . . . . . . . . . . . . . . . . . . 415.6 Vibration modes of the free–free vibrating boom . . . . . . . . . . . . 425.7 Vibration modes of the boom in the gravity off-loading system . . . . 425.8 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6 Discussion 536.1 Mode shapes comparison and discussion . . . . . . . . . . . . . . . . 536.2 Mass participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Conclusions and Recommendations 567.1 Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.2 Research conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 567.3 Lessons learned . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

References 59

vi

List of Figures

1.1 SwissCube-1 [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 SIMPLE boom shown in the (a) stowed configuration, (b) partially

deployed and (b) fully deployed states [14]. . . . . . . . . . . . . . . 31.3 Exploded hub view of the SIMPLE boom in initial stages of deploy-

ment, [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 A signal in time domain. . . . . . . . . . . . . . . . . . . . . . . . . 62.2 A signal in frequency domain. . . . . . . . . . . . . . . . . . . . . . 62.3 Mechanical model for a simple SDOF system and its free body dia-

gram, [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 A damped free vibration. . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Dynamic magnification factor (DMF) versus frequency ratio for vari-

ous levels of damping [12]. . . . . . . . . . . . . . . . . . . . . . . . 112.6 Damping ratio versus frequency for Rayleigh damping, [10]. . . . . . 12

3.1 Overview of the structure - a satellite dummy with a deployed boom[courtesy of Pau Mallol]. . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Satellite dummy [courtesy of Pau Mallol]. . . . . . . . . . . . . . . . 163.3 Hub [courtesy of Pau Mallol]. . . . . . . . . . . . . . . . . . . . . . 173.4 Tip plate with electronics [courtesy of Pau Mallol]. . . . . . . . . . . 183.5 KTH boom prototype in its (a) stowed configuration and (b) deployed

configuration [courtesy of Pau Mallol]. . . . . . . . . . . . . . . . . . 193.6 Scheme of the experimental set-up. . . . . . . . . . . . . . . . . . . . 203.7 Marionette system (left picture) and the satellite structure hanging at

three points (pictures to the right). . . . . . . . . . . . . . . . . . . . 21

4.1 Scheme of the experimental set-up for the boom vertical orientationvibration test [courtesy of J. Kristoffersson & M. Larsson]. . . . . . . 23

4.2 Typical measurements set–up, [4]. . . . . . . . . . . . . . . . . . . . 234.3 Positioning of the shaker in the in-boom-plane vibration test. . . . . . 254.4 Positioning of the accelerometers. . . . . . . . . . . . . . . . . . . . 26

vii

LIST OF FIGURES

4.5 Measurement points on the lateral faces (left picture) and on the hublower face (the picture to the right). . . . . . . . . . . . . . . . . . . 26

4.6 Calibration of the laser. . . . . . . . . . . . . . . . . . . . . . . . . . 284.7 Mode Indicator Function [courtesy of P. Banach]. . . . . . . . . . . . 294.8 Curve-fitting for the point 7 [courtesy of P. Banach]. . . . . . . . . . . 304.9 Curve-fitting for the point 11 [courtesy of P. Banach]. . . . . . . . . . 314.10 Mode animations [courtesy of P. Banach]. . . . . . . . . . . . . . . . 344.11 Laser measurement point mobilities when the dummy and boom are

turned 0 degree angle [courtesy of J. Kristoffersson & M. Larsson]. . . 344.12 Laser measurement point mobilities when the dummy and boom are

turned 90 degrees angle [courtesy of J. Kristoffersson & M. Larsson]. 354.13 Coherence for the laser measurement points [courtesy of J. Kristoffers-

son & M. Larsson]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.14 The coherence for the accelerometers calculated from one measure-

ment [courtesy of J. Kristoffersson & M. Larsson]. . . . . . . . . . . 364.15 Shapes for modes from 2 to 7 Hz [courtesy of C. Frangoudis, C. Kastby,

M. Zapka]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.16 Shapes for modes from 1 to 9 Hz [courtesy of C. Frangoudis, C. Kastby,

M. Zapka]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 Cross section view of the tape springs positioning. . . . . . . . . . . . 405.2 Satellite dummy with the boom. . . . . . . . . . . . . . . . . . . . . 405.3 Meshed geometry of the boom with gravity off-loading strings. . . . . 415.4 First eigenfrequency, 7.7 Hz − (a) Front view (b) Side view (c) Top view 435.5 Second eigenfrequency, 10.0 Hz − (a) Front view (b) Side view (c)

Top view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.6 Third eigenfrequency, 16.2 Hz − (a) Front view (b) Side view (c) Top

view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.7 Fourth eigenfrequency, 21.5 Hz− (a) Front view (b) Side view (c) Top

view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.8 First eigen mode for the gravity off-loaded boom prototype, 7.8 Hz −

(a) Front view (b) Side view (c) Top view . . . . . . . . . . . . . . . 475.9 First eigen mode for the gravity off-loaded boom prototype, 7.8 Hz −

Overall side view of the gravity off-loading system . . . . . . . . . . 485.10 Second eigen mode for the gravity off-loaded boom prototype, 11.3 Hz

− (a) Front view (b) Side view (c) Top view . . . . . . . . . . . . . . 495.11 Second eigen mode for the gravity off-loaded boom prototype, 11.3 Hz

− Overall side view of the gravity off-loading system . . . . . . . . . 505.12 Third eigen mode for the gravity off-loaded boom prototype, 16.1 Hz

− (a) Front view (b) Side view (c) Top view . . . . . . . . . . . . . . 51

viii

LIST OF FIGURES

5.13 Third eigen mode for the gravity off-loaded boom prototype, 16.1 Hz− Overall side view of the gravity off-loading system . . . . . . . . . 52

6.1 First eigenfrequency, boom with levelled tape springs , 4.2 Hz - (a)Side view, (b) - Top view . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 Transition zone. ρ0 denotes a region of a transition zone in a tapespring, [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.3 The rotation 39.8 degrees for the principal axis of inertia of the boomcross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.4 Comparison of the shapes for the first mode in the FEA analysis, 7.7Hz (a) and the vibration testing, 1.76 Hz (b) . . . . . . . . . . . . . . 55

ix

Chapter 1

Introduction

1.1 BackgroundCubeSats are standardized complex systems used for space research by universitiesand have special properties such as a volume of 10 cm3 and a mass not exceeding 1.33kg [6]. The main goal of CubeSat research and development projects is to stimulateinterest in science and increase competence in space technology among students andeducational institutions [8] as well as at various industrial research organizations, i.e.Boeing [7]. Several types of CubeSats exist: 0.5U, 1U, 2U, 3U, 5U and 6U. Theseremarkable size and mass make these satellites appropriate for cost-effective spaceexploration in the low Earth orbits. For instance, the CubeSat SwissCube-1 (Fig. 1.1),Swiss satellite, was a cost effective solution to study the nightglow within the Earth’satmosphere.The first successful launches of CubeSats began in the mid-2003 from Plesetsk, Russia.Since then, the satellites are predominantly being developed by Stanford University,California PolyTechnic University, Norwegian University of Science and Technologyand the University of Tokyo. KTH, the Royal Institute of Technology is making anattempt to build a system for the CubeSat mission in collaboration with the US AirForce Research Laboratory, University of Florida and the Inter–American Universityof Puerto Rico. The project at KTH includes the design, development and verificationof the SIMPLE deployable boom prototype [14].Deployable booms are used in today’s CubeSat missions. For instance, deployablebooms based on bi-stable composite tape springs are proposed for CubeSat deploy-able antennas [13]. The tape-springs are made of glass fibre reinforced epoxy withan embedded copper alloy conductor. Bi-stability properties enable the antenna to beelastically stable in both deployed and stowed configurations [13].

Dynamic characterization of the deployable structures during their deployment andin their deployed state are of primary importance because of how those factors influ-

1

1. Introduction

Figure 1.1: SwissCube-1 [8].

ence the behaviour of the satellite. In many cases, the deployment mechanisms involvelarge kinematic, unconstrained rotations when deployed, however the angular and lin-ear momentum will not change. [13].

The deployable structure using bi-stability properties of the tape-springs is pre-sented in [14] and depicted in Fig. 1.2.

1.2 ObjectivesThe objective of this research is to analyse the dynamic characteristics of the boomdeployed from the CubeSat. The project focus is on the dynamic characterizationof a SIMPLE boom prototype and on numerical analysing the means of the boomvibration testing. The proposed boom design can be seen in Figure 1.3. The analysiswill be performed both on the experimental structure and on a finite element model ofthe structure. The aim is to retrieve the eigenfrequencies and mode shapes from theexperimental data and to create an FE model that with satisfying resemblance simulatesthe behaviour of the structure.

The vibrations of the boom structure are recorded using accelerometers and laser

2

1. Introduction

(a) Stowed configuration (b) Partially Deployed Configura-tion

(c) Deployed State

Figure 1.2: SIMPLE boom shown in the (a) stowed configuration, (b) partially de-ployed and (b) fully deployed states [14].

vibrometer and the obtained experimental data is processed and analysed with Matlab.The finite element model of the boom is created and post-processed in a software ap-plication for finite element analysis called Abaqus CAE, whilst the actual calculationsare carried out in Abaqus Standard.

The development of the Deployable Boom for the CubeSat project at KTH in col-laboration with AFRL, and University of Florida, USA, and Inter–American Universityof Puerto Rico was divided into three primary focus areas. The primary focus of theKTH research and development was to study:

• dynamic characterization of the gravity off-loading system for the boom vibra-tion testing;

• dynamic characterization of the deployed boom;

• boom deployment dynamics.

The prerequisites for the posterior design improvements of the boom are investi-gated in this thesis.

3

1. Introduction

Figure 1.3: Exploded hub view of the SIMPLE boom in initial stages of deployment,[14].

4

Chapter 2

Theory of Structural Dynamics

2.1 IntroductionVibrations occur in structures due to the dynamic loads, i.e. loads that alternate in time.The dynamic loads pose new problems compared to the static case. When analysinga structure that is subjected to a static load, a larger force will in general cause largerdisplacements and a stiffer construction is therefore needed. The situation is not asobvious when it comes to dynamic loads acting on the structure. In this case, themagnitude of the load might not be as pivotal as the frequency it is alternating with.For a load with an excitation frequency close to a so called natural frequency (or eigenfrequency) of the structure, the response can be substantially larger than for a load withthe same magnitude but different frequency.

“Resonance is an operating condition where an excitation frequency is near a nat-ural frequency of a machine structure. A natural frequency is the frequency at which astructure will vibrate if deflected and then let go” - [5].

Vibrations can be described as the oscillations around the equilibrium of a structure,which is the position the structure comes back to when no external forces are actingon it. The displacements around the equilibrium are measured in order to analysethe vibrations and can be considered both as functions in the time domain and in thefrequency domain. The last alternative can be useful when analysing the dynamicbehaviour of a structure and is called the response spectra. To convert a signal betweenthe two domains a Fourier analysis algorithm such as Fast Fourier Transform can beused. An example of a signal in the time domain and its response spectrum is shownin Figures 2.1 and 2.2

During analysis of a mechanical system there is always a matter of how scrupulousone shall be, how much one can idealize the system and still simulate its behaviour.Besides the assumptions about physical laws one have to decide whether to view thesystem as continuous or discrete. It is practically impossible to analyse a complex

5

2. Theory of Structural Dynamics

Figure 2.1: A signal in time domain.

Figure 2.2: A signal in frequency domain.

structure as continuous and therefore it is transferred into a number of discrete counterparts having number of DOFs (Degrees Of Freedom). A DOF represents either a dis-placement or a rotation and together they represent the behaviour of the discrete sys-tem.

For references on this chapter and more detailed description, see e.g. [12].

6

2. Theory of Structural Dynamics

2.2 Equation of motion and natural frequencyThe most straightforward method of modelling a dynamic system is to use a single-degree-of-freedom model (SDOF), which represents the system by only one degree offreedom. An example of the system is illustrated in Figure 2.3. It consists of a mass mconnected to the wall with a damper and a spring (both having mass m = 0). The massmoves with negligible friction in the horizontal direction and the degree of freedom isthe displacement x from the equilibrium. A time-dependent load F(t) excites the mass.

Figure 2.3: Mechanical model for a simple SDOF system and its free body diagram,[12].

Analysing the free body diagram in Figure 2.3 and using Newton’s second law ofmotion leads to Eq. (2.1)

F(t)− cdxdt− ku = m

d2xdt2 (2.1)

Reorganizing the terms in Eq. (2.1) results in the equation of motion for a SDOFmodel:

md2xdt2 + c

dxdt

+ kx = F(t) (2.2)

The displacement x(t) is given by solving the equation of motion.In case of a complex structure it is necessary to include more degrees of free-

dom to describe the dynamic behaviour. This results in a multi-degree-of-freedommodel (MDOF). For a system having small displacements (neglecting any non-linearbehaviour), the equation of motion for a SDOF system can be converted to the multi–dimensional case, Eq. (2.3)

Md2xdt2 +C

dxdt

+Kx= F(t) (2.3)

In such a case M is the mass matrix, C is the damping matrix and K is the stiffnessmatrix; x is the displacement vector and F(t) the load vector. If there are n degrees offreedom in the model, the matrices will be of size n×n and the vectors n×1.

2.3 Modal analysisA structure has an infinite number of eigenfrequencies as it has an infinite number ofDOFs if it is not constrained. These are the frequencies the structure will oscillate

7

2. Theory of Structural Dynamics

at when it is allowed to vibrate without the influence of any external loads. Eacheigenfrequency has a matching mode shape. If the structure is deflected into a modeshape and released from rest, it will oscillate at the corresponding frequency.

Studying a structure in free vibration (with no external forces acting on the struc-ture) and assuming that no damping is introduced, the equation of motion is reducedto:

Md2xdt2 +Kx = 0 (2.4)

To solve Eq. (2.4), a harmonic solution is assumed:

x(t) = Acos(ωt)Φ+Bsin(ωt)Φ (2.5)

Differentiation of x(t) and insertion into Eq. (2.4) gives the eigenvalue problem in Eq.(2.6).

(K−ω2M)Φ = 0⇒ det(K−ω2M) = 0⇒ ω1,ω2, ...,ωn (2.6)

When a structure is discretized into n degrees of freedom, there are n eigenvalues. Eacheigenvalue ωi has a matching eigenvector Φi. These are the angular eigen frequenciesand mode shapes of the structure. Together, an eigenvalue and eigenvector pair form asolution to (2.5) that satisfies the differential equation. A and B are given by the initialconditions x(0) = x0 and the same condition for initial velocity. The mode vectors forman orthogonal basis and the solution can be rearranged as a sum using all eigenvaluesand eigenvectors:

x(t) =n

∑1

qi(t)Φi; qi(t) = Ai cos(ωit)+Bi sin(ωit) (2.7)

2.4 Steady-state vibrationsWhen a structure is subjected to a harmonic load, it will after a starting transient phaseoscillate with the frequency of the input load. This is called vibrations with a steady-state characteristic.

Applying a harmonic load to the undamped system gives the following equation ofmotion:

Md2xdt2 +Kx = F(t); F(t) = F0 sin(ωt) (2.8)

8

2. Theory of Structural Dynamics

by using modal coordinates, x(t) =n∑1

qi(t)Φi and multiplying with ΦTr from the

left Eq. (2.9) can be derived:

n

∑1

ΦTr MΦi

d2qi(t)dt2 +

n

∑1

ΦTr KΦiqi(t)=Φ

Tr F(t) (2.9)

The eigenvectors are orthogonal in the scalar products ΦTr KΦi and ΦT

r MΦi henceonly terms with i = r are non-zero. This results in n uncoupled equations of the form:

Mid2qi

dt2 +Kiqi = Fi(t);Ki = ΦTi KΦi;Mi = Φ

Ti MΦi;Pi = Φ

Ti P(t) = P0i sin(ωt) (2.10)

Insertion of a harmonic ansatz solution qi(t) = q0i sin(ωt) gives the following result:

q0i =P0i

Mi

1ωi2−ω2 ;ωi =

Ki

Mi(2.11)

The result shows that an excitation frequency ω equal to an eigenfrequency willgive an infinite amplitude. This is of course not the case in reality, where any kind ofdamping in the structure will prevent such a phenomenon. But if the level of damp-ing is low, there will still be a peak in the displacement amplitude spectrum at theeigenfrequencies.

2.5 DampingDamping is included in mathematical models to represent the energy dissipation instructural dynamics and is always present in real structures. It can for example befriction in joints, or internal properties of materials.If F(t) = 0 in Equation 2.12, the equation of motion can be rewritten as:

d2xdt2 +2ζωn

dxdt

+ωn2x = 0;ωn =

√km,ζ =

c2mωn

(2.12)

ζ is the so called damping ratio and ωn the eigenfrequency for the undamped case.Solution to the equation is the response for a damped free vibrating structure.

x(t) = e−ζωnt

(x(0)cos(ωDt)+

(d2xdt2 )t=0 +ζωnx(0)

ωDsin(ωDt)

),ωD = ωn

√1−ζ2

(2.13)

9

2. Theory of Structural Dynamics

ωD is the eigenfrequency for damped vibrations and is related to ωn by a factor√

1−ζ2.For relatively small damping ratios (ζ≺ 0.2), ωD ≈ ωn.See Figure 2.4 for an example of damped free vibration. As mentioned in the steady-state vibrations section, the response amplitude of a harmonic load will not go to in-finity if there is some kind of damping present in the structure. The amplitude of thespike in the frequency response spectrum will depend on the damping ratio. Figure 2.5shows the deformation response factor for frequencies around an eigenfrequency. Thedeformation response factor is a quota between the amplitude of the dynamic responsefor a harmonic load and the amplitude of the static response for a static load of thesame magnitude.

Figure 2.4: A damped free vibration.

For a MDOF model, a damping matrix needs to be assembled. It is not as sim-ple as assembling the stiffness matrix, which is organized by considering the stiffnessproperties of individual elements. For example, the damping properties of materialsare not as well agreed upon and also the energy dissipation in joints needs to be takeninto account. On the contrary, the damping matrix may be constructed from the modaldamping ratios of the structure. There are two types of damping matrices, classicaland non-classical. The difference between them is that classical damping matrices arediagonal and non− classical damping matrices are not. A diagonal matrix makes itpossible to separate the equation system into n uncoupled equations, like in the un-damped case, and it is therefore becomes possible to perform classical modal analysisof the structure.

Rayleigh method is a method of constructing the damping matrix, which delivers aclassical damping matrix. The Rayleigh damping matrix is a linear combination of the

10

2. Theory of Structural Dynamics

Figure 2.5: Dynamic magnification factor (DMF) versus frequency ratio for variouslevels of damping [12].

mass matrix and the stiffness matrix, as shown in Eq. (2.14) .

C =a0M+a1K (2.14)

Using the formula for the damping ratio stated in Eq. (2.12) , it can be shown thatthe damping ratio of the nth mode is given by:

ζn =a0

2ωn+

a1ωn

2(2.15)

a0 and a1 can be obtained from two known damping ratios ζi and ζ j. If ζ is assumedto be constant in modes i and j, a0 and a1 is given by expressions (2.16).

a0 = ζ2ωiω j

ωi +ω j,a1 = ζ

2ωi +ω j

(2.16)

The damping as a function of frequency in the case of constant damping ratios inmodes i and j is shown in Figure 2.6. The mass matrix damps the lower frequenciesand the stiffness matrix damps the higher frequencies.

11

2. Theory of Structural Dynamics

Figure 2.6: Damping ratio versus frequency for Rayleigh damping, [10].

2.6 Root–mean–square valueThe RMS value (root–mean–square value), defined in Eq. (2.17), is used to create anaverage magnitude of a signal over time. Since the sign of the displacements variesin time during vibrations, it is practicable to use the RMS value instead of the meanvalue.

xRMS =

√1∆t

∫ t0+∆t

t0x2(t)dt (2.17)

2.7 Modal parameter extraction: complex exponentialmethod

There are several ways to extract the modal parameters using the data collected. Inthis case, a numerical method called the Prony method [3], or the complex exponen-tial method, was chosen. Once the measurements are done and the results are treated(velocities are converted into acceleration by multiplying by iω), 14 accelerance func-tions are obtained. Each of them corresponds to one of the measurement points. Theidea of this method is to minimize the function R, which is the difference between thefrequency response function modelled as a ratio between polynomials in Z−1 and theZ transform of the measured frequency response function Hr(z), as given below:

R =

q∑

i=0biz−i

1+p∑

i=1aiz−i

−Hr(z) (2.18)

Multiplying R with the denominator polynomial gives the modified polynomial R′:

12

2. Theory of Structural Dynamics

R′ =

(q

∑i=0

biz−i−Hr(z)

)(1+

p

∑i=1

aiz−i

)(2.19)

This modified polynomial R′ enables to get a linear problem that can be solved bya computer. The program calculates the coefficient of the denominator (ai such as1 ≤ i ≤ p ) of a single transfer function using Shanks transformation [16]. Thesecoefficients are theoretically the same for all transfer functions and give access to thepoles of the system. Note that if the transfer function used for the calculations of polesis situated on a node line, some modes could be missed. The program calculates thepoles for an increasing p (number of poles). Since the physical poles must be stableindependently of p, it is then possible to eliminate the non-physical poles of the postprocessing analysis. Note that the mathematical modes are needed for the curve fittingin order to correct the influence of out-of-band poles. The mode indicator functionMIF is also a good indicator to detect modes:

MIF(ω) =

35∑

i=1Im(Hi(ω))

35∑

i=1Hi(ω)

2(2.20)

There are several ways to calculate a mode indicator function but in this case, alow value of MIF corresponds to a mode. The next step is to calculate the coefficients(bi 0 ≺ i ≺ q) of the frequency response function. In practice the program solves alinear system of 14 equations. At this stage of the modal extraction H(z) is entirelydetermined. It is then possible to plot H(z) and Hr(z) in order to check if the curvesfit closely. The denominator enables to calculate the mode shape vectors. In order tocheck the accuracy of the results the orthogonality of the mode shape vectors can alsobe checked. The results of the modal parameter extraction are given in the Chapter onMeasurements.

13

Chapter 3

Design of the Structure

3.1 Concept design of the CubeSatThe standard 0.1 × 0.1 × 0.1 m3 basic CubeSat is often called a “1U” CubeSat mean-ing one unit. CubeSats are scalable in 1U increments and larger. CubeSats such as a2U CubeSat (0.2 x 0.1 x 0.1 m3) and a 3U CubeSat (0.3 x 0.1 x 0.1 m3) have been bothbuilt and launched. A 1U CubeSat typically weighs about 1kg.

Since CubeSats have all the same 0.1 × 0.1 m2 cross-section, they can all belaunched and deployed using a common deployment system. CubeSats are typicallylaunched and deployed from a mechanism called a Poly-PicoSatellite Orbital Deployer(P-POD), also developed and built by CalPoly. P-PODs are mounted to a launch ve-hicle and carry CubeSats into orbit and eject them once the proper signal is receivedfrom the launch vehicle. The P-POD Mk III has capacity for three 1U CubeSats how-ever, since three 1U CubeSats are exactly the same size as one 3U CubeSat, and two1U CubeSats are the same size as one 2U CubeSat, the P-POD can deploy 1U, 2U, or3U CubeSats in any combination up to a maximum volume of 3U.

Figure 3.1 illustrates the overall view of the structure under analysis in the thesis.The structure was built at KTH Mechanics Department. This structure resembles theactual version of the CubeSat with deployable boom which will be launched into lowearth orbit.

Because of its mission, the SWIM satellite is not a conventional structure and there-fore it requires special means of investigation. Most of the choices regarding the ex-perimental set-up and the procedure to acquire the data are directly based on the speci-fications of the investigated structure. First of all, since it is so expensive to send massinto orbit, it is a lightweight structure without any ferrous materials to avoid magneticinterferences. Basically, the satellite is a rectangular parallelepiped of dimensions 300× 100 × 100 mm3 . It contains, electronics, batteries, a sensor called WINCS, and theboom folded on itself and located in the boom room as can be seen in Fig. 3.1.

14

3. Design of the Structure

Figure 3.1: Overview of the structure - a satellite dummy with a deployed boom [cour-tesy of Pau Mallol].

When the satellite has reached the mission orbit, the boom is deployed. To avoidrisks of failure, it is not a motor that is used to deploy the boom but composite tapesprings storing strain energy that is released during the deployment. The boom can beseen in Fig 3.1. As shown in the picture, there are 2 × 2 composite tape springs. Thetwo first ones, going from the satellite to the middle part called the hub and the twosecond ones going from the hub to the SMILE sensor plate. The total length of theboom is one meter. The composite tape springs have the particularity of being curvedin the transversal plane. Regarding their physical properties, their axial stiffness isless critical than their rotational stiffness and hence the fundamental mode might bea torsion mode. They are also very light in comparison with the remaining of thesatellite (the mass of the whole boom is estimated at 150 g whereas the mass of thesatellite with all its equipment is estimated at 3.4 kg). Nevertheless, one needs topoint out that the dummy used for the experiment is lighter than the real satellite sincesome equipments are missing. Its mass was measured to 2.3 kg. Consequently, it isclear that accelerometers cannot be used to take measurements on the boom which aredescribed in the next chapter since the accelerometers have no capibility to capture lowfrequencies. The alternative that has been chosen is to use a Laser Doppler Vibrometer(LDV), more details will be given in a later section. However, accelerometers can beused on the structure itself since it is stiff and heavy enough. The next step is to find away to simulate the absence of gravity in which the satellite will operate.

15

3. Design of the Structure

3.2 StructureThe satellite dummy with its dimensions and adjacent satellite systems mock-ups isshown in Fig. 3.2. The satellite dummy during testing was comprised of a square pro-file 3 mm thick aluminium hollowed plates. The satellite was balanced by additionalled masses to resemble the mass distribution in the actually designed satellite.

Figure 3.2: Satellite dummy [courtesy of Pau Mallol].

The structure consists of a satellite arm manufactured of altogether four deployablecarbon fibre reinforced plastic tape springs. Two tape springs were connected to adummy satellite in one end and to the deployment drum (hub, see Fig. 3.3) in the otherend. The other two tape springs were connected to the deployment drum in one endand to the end piece, called tip, that is meant to contain the measurement electronics inthe other end. The tip is shown in Fig. 3.4.

The prototype of the boom in its stout and deployed configuration is shown in Fig.3.5.

3.3 MaterialsData for all structural parts and materials are shown in Table 3.1.

16

3. Design of the Structure

Figure 3.3: Hub [courtesy of Pau Mallol].

Table 3.1: Parts and materials used in the structure for vibration testing

Part Material Dimensions, m QuantitySatellite Dummy Aluminium Alloy 6061 0.25x0.1x0.1, t=0.003 1Attachment Plate PVC 0.04x0.04x0.01 1Hub PVC 0.04x0.04x0.04 1Tip plate PVC 0.04x0.04x0.01 1Tape Springs PrePreg HexPly r=0.0067, t=0.00025 4

3.4 Gravity off-loading systems as a mean to obtain dy-namic characteristics of structures

For cost effective hardware verification of large space structures high fidelity and sen-sitivity is required of the laboratory simulation of weightlessness.

A known solution for the hardware verification where the weight of each part of thespecimen is balanced by the other parts is called a Marionette paradigm. In such sus-

17

3. Design of the Structure

Figure 3.4: Tip plate with electronics [courtesy of Pau Mallol].

pension device which is described in [9] by Greschik mass overhead, contributed to bylight fly beams and suspension cords only, is low, often negligible. Damping, friction,and slip are eliminated, and specimen response is not affected by deleterious materialstiffness. Deleterious stiffness effects can be eliminated with a precision limited onlyby the accuracy of geometric measurement. The architecture to achieve these qualitiesis simple, with few limitations on its overall design and with high tolerance againstboth specimen and support system imperfections. Kinematics can naturally involve upto moderate specimen displacements and deformations in both the vertical and hori-zontal directions. The concept can also be generalized to accommodate some adaptivemodel geometries, and the simulation of inertial loading conditions in weightlessness,for example, steady state acceleration, is also possible. This unique combination ofhigh performance, fault tolerance, mechanical simplicity, and design flexibility are anattractive alternative to classic gravity compensation schemes.

To get relevant results it is necessary to model the environment in which the systemis going to operate as close to reality as possible. In the case of a satellite, the absenceof gravity that occurs in orbit can not be neglected. The difficulty lies in how to sim-ulate this absence of gravity without biasing the results. It is obvious that an artificialweightlessness cannot be easily recreated in a laboratory. Thus, the only alternative isto hang the satellite to the ceiling but this must be done in such a way that the hangingsystem does not bias the results. The above mentioned Marionette system, when de-signed properly, is a simple and efficient way to fulfil the previous objective ([9]). Thebasic idea of this method is that each part of the studied structure should be balanced

18

3. Design of the Structure

Figure 3.5: KTH boom prototype in its (a) stowed configuration and (b) deployedconfiguration [courtesy of Pau Mallol].

by the other parts. This should be carried out using a set of fly beams integrated into asingle hierarchy, suspended from one external location (in this case the ceiling of thelaboratory). Thus, since there is a single external support, the structure is not subject toany constraint except that its center of gravity is maintained stationary in space. Thiscondition is equivalent to weightlessness with a fidelity defined by the geometry ofsupport hierarchy. Moreover, since the fly beams and suspension cords are very light,the mass overhead of the system is low, often negligible, and if not, at least easy to

19

3. Design of the Structure

measure precisely. A feature of this system is that damping, friction, and slip are elim-inated, and specimen response is not affected by deleterious material stiffness from thesuspending system.

Figure 3.6: Scheme of the experimental set-up.

The cords are made of fishing lines. The upper beam is an aluminium tube with aninternal diameter of 12.5 mm, an external diameter of 15 mm, a length of 0.85 m anda mass of 105 g. The lower beam is also in aluminium with a rectangular section of 10× 3 mm2, a length of 0.55 m and a mass of 42 g. Pictures of the final set–up are shownin Fig. 3.7.

20

3. Design of the Structure

Figure 3.7: Marionette system (left picture) and the satellite structure hanging at threepoints (pictures to the right).

21

Chapter 4

Measurements

4.1 IntroductionThe objective of the measurements is to obtain information about eigenfrequencies,mode shapes and damping of the structure. Processing and analysing measurementdata is time-consuming though, and therefore only the most important data processingwere made. Eigenfrequencies were evaluated for all axis, but not for all measurementsand accelerometers. The measurements were carried out for the the boom in three ori-entations: in plane to the boom, transverse to the boom and vertical orientation. Modeshapes were only evaluated for the complete boom structure. The analyses were lim-ited to 18 first natural frequencies of the structure. In each section that are to followthe procedure and results of measurements in each of orientations mentioned above ispresented. The measurements were carried out at KTH in a course of ExperimentalDynamics by students J. Kristoffersson, M. Larsson, P. Banach, J. Brondex, L. Gedim-inas, C. Frangoudis, C. Kastby, M. Zapka and myself.

4.2 InstrumentationExperimental modal analysis aims at finding the modal parameters of the structure bymeasuring the receptance matrix [2]. For that purpose, the structure has to be discre-tised in several measurement points or excitation points. In this case, a shaker providedthe excitation and keeping a single excitation point and measure the response of sev-eral measurement points. As mentioned previously, a laser vibrometer was used toperform the experimental modal analysis. The measurement was done at one measure-ment point at a time with manual movement of the laser in between measurements. Inaddition to the optical measurement points, three accelerometers were used to measurethe response of the satellite dummy structure. The choice of a shaker as the excitationsource was obvious: since the investigated frequency range is not wide (from 1 to about

22

4. Measurements

100 Hz), it is a simple, efficient and fast solution to excite the structure. One drawbackis that leakage errors are likely to occur but they can be compensated by windowing(see the “signal processing” section). Fig. 4.1 shows a scheme of the experimentalset-up that was implemented except that the structure was the satellite dummy and alaser vibrometer was used in addition to the three accelerometers [11]

Figure 4.1: Scheme of the experimental set-up for the boom vertical orientation vibra-tion test [courtesy of J. Kristoffersson & M. Larsson].

Figure 4.2: Typical measurements set–up, [4].

Below a list of the instrumentation used is presented [1].

23

4. Measurements

1. Polytech-Laser Vibrometer, D-76337, Model-No OFV 303, Serial Nr 1970441,Hfg data Adr. 1997

2. Polytech-Vibrometer Controller, Model-No OFV 3001, Serial Nr 9606016, Hfgdata June 1996

3. Data collection system Agilent 8491A and Agilent E1432A

4. Tripod for Laser Vibrometer

5. Ling Dynamic Systems-Shaker, Model-No V401, Serial nr 469 0411, Date ofManufact. 22/1/97 and Forcetransducer, Dytran 5192.

6. Bruel Kjaer Noise Generator, Type 1405, Serial 904206

7. Bruel Kjaer Power Preamplifier, Type 2706, Serial 660120

8. MWL UNO 8-channel Preamplifier, PCP-854

9. Piezo Electric IEPE Accelerometer, Bruel Kjaer,

(a1) Art Nr 4507 B 005 Serial Nr 10069,

(a2) Art Nr 4507 B 005 Serial Nr 10071,

(a3) Art Nr 4507 B 005 Serial Nr 10072,

(a4) Art Nr 4507 B 005 Serial Nr 10163,

(a5) Art Nr 4507 B 005 Serial Nr 10162,

4.3 Measurement procedureTo be able to extract the modal parameters of the structure, it is necessary to measureat least one row or one column of the response matrix. Since there is a single point ofexcitation and several points of measurement, it is a column of the matrix that is mea-sured. One needs to point out that accelerometers give data in terms of accelerationwhereas the laser measure velocities. For simplicity, it was decided to convert veloc-ities into accelerations in order to get the accelerance functions rather than mobilityfunctions. Regarding the positioning of the shaker, it had to be done with great careto get a good excitation and to ensure a stiff connection between the force transducerand the excitation point. To avoid problems of symmetry, the force transducer wasmounted on a bottom corner of the face of the dummy structure corresponding to theboom-room (see Fig. 4.3).

As can be seen in the picture, a push-rod is mounted between the shaker and theforce transducer to reduce undesired moment excitation. The transducer is cemented

24

4. Measurements

Figure 4.3: Positioning of the shaker in the in-boom-plane vibration test.

to the excitation point to ensure a stiff coupling. Two accelerometers were mounted onthe top face of the dummy and the third was mounted on the back face (see Fig. 4.4).

Finally, stickers with reflective properties were placed on the boom as follows :

1. 2 stickers on the lateral face of the tip

2. 2 stickers on the lower face of the tip

3. 2 stickers on the lateral face of the hub

4. 2 stickers on the lower face of the hub

5. 3 stickers on the lateral face of the dummy

Measurements are made by aiming the laser at each of these stickers. Fig. 4.5shows the measurement points (except the two points of the lower face of the tip).The shaker is fed with a white noise signal in the frequency range between 0 to 100Hz. The force transducer and the accelerometers give the time history of the forceand the accelerations respectively for three points of the structure. Then each stickeris successively targeted with the laser to get the time history of the velocity of themeasured point. Thus, 11 measurements are required (as many as there are stickers).Eventually, the measurement devices have to be calibrated [15].

25

4. Measurements

Figure 4.4: Positioning of the accelerometers.

Figure 4.5: Measurement points on the lateral faces (left picture) and on the hub lowerface (the picture to the right).

4.4 Signal processingFirst of all, each measured signal is amplified to get a high signal to noise ratio. Thismust be done with great care to avoid overloading the inputs of the data acquisition sys-tem. The FFT-analyser contains an A/D converter which ensures the conversion of theanalog signal into a digital signal. Before that, it also contains a low pass filter denotedanti-aliasing filter which removes all frequency components above 0.5 fs. Finally theFFT analyser estimates the Fast Fourier Transform of the signal. Since the signal is notperiodic in the time window, an operation of windowing must be performed to avoidleakage errors. Therefore, a Hanning window is used to force the signals to 0 at thebeginning and at the end of the time record. Finally some averaging is performed to

26

4. Measurements

reduce the influence of the noise. To enable the investigation of the structure on thefrequency range between 0 and 100 Hz, some choice regarding the parameters of thesignal processing were made. First, the sampling frequency was set to fs = 500 Hz.Then, to get results accurate enough, the number of samples was set to Ns = 10. Hence,using the relation below

Ts =Ns−1

fs,∆ f =

1Ts

(4.1)

we obtain a frequency resolution of ∆ f = 0.03 Hz:

4.5 CalibrationAccelerances are acceleration quantities normalised with the exciting force. Thus, onlythe ratio between the pair of channels (one for the force, the other for the acceleration)need to be calibrated. A straight cylinder with a mass of 2425 g was used as a welldefined reference object. If this reference object is submitted to an excitation force F,Newton′s second law gives the relation:

µ(ω) =F(ω)

a(ω)(4.2)

where µ(ω) is the apparent mass. For low frequencies (below say 30% of the lowestelastic eigenfrequency), the apparent mass should be equal to the known static mass.Therefore, the exciting force is applied in the measured DOF only, it is possible tocalibrate the ratio between force and acceleration by imposing the apparent mass equalto the static mass. Fig. 4.6 shows the calibration procedure for the laser. As it can beseen on the picture, it consists in exciting one side of the cylinder and measuring thevelocity (that will be converted in acceleration) of the other side with the laser.

Note that the same operation applies to the three accelerometers mounted on thefree side of the cylinder, though these can be made at the same time as it is assumedthat the cylinder is moving along the direction of its axis only.

4.6 Modal parameter extractionAs mentioned previously, at the end of the experiment each measurement device mustbe calibrated in order to get the calibration factors. The laser gives an apparent mass of0.0061 kg but it is known that the actual mass of the cylinder is 2.425 kg the calibrationfactor can be calculated as:

γ =µm

(4.3)

27

4. Measurements

Figure 4.6: Calibration of the laser.

Hence, the calibration factor of the laser is about 0.0025. In practise, this means thatthe FRF measured via the laser must be divided by 0.0025. The same operations wascarried out for the three accelerometers and it gives calibration factors of respectively:

• 0:0058 for the first accelerometer

• 0:0054 for the second accelerometer

• 0:0061 for the third accelerometer

thus each one of the three FRF estimated via the data measured by the accelerometershas to be divided by the corresponding calibration factor.

The first thing that must be said is that the analysed structure was complex andhence the obtained results are not highly accurate. Indeed, mainly because of thelightness of the structure, the meshing of measurement points was not very dense andtherefore some complex modes are not perfectly described. On the other hand, thefrequency range under investigation was narrow and certain modes were separated byless than 1 Hz so there are problems of overlapping modes. Nevertheless, the complexexponential method explained above was used to get the modal parameters. The ob-tained mode indicator function on the frequency range from 1.5 to 4 Hz is plotted inFig. 4.7.

28

4. Measurements

Figure 4.7: Mode Indicator Function [courtesy of P. Banach].

As it can be seen on the MIF, some eigenfrequencies are obvious whereas someothers are not very sharp. Using this function, the curve-fitting procedure was per-formed with more or less accuracy depending on the measurement point. For example,the fitting for the measurement point 7 (point on the lower face of the hub) is rathergood whereas it is much less precise for the point 11 (point on the lower face of the tip)as is showed in, respectively, Figs 4.8 and 4.9. Note also that above 3 Hz the resultsare getting worse as the frequency increases.

As a consequence of this lack of precision, the extracted modal parameters are nothighly reliable. The results are detailed in Table 4.1. The table clearly illustrates theaforementioned problem with modes that are very close to one another. Thus, someof those modes might actually be only combination of other modes. For example,modes 7 and 8 are not clearly distinct. It is the same with modes 9 and 10 and withmodes 11, 12, 13 and 14. This impression tends to be confirmed by the mode shapeanimations which shows that the deformations are very similar for the modes quotedabove. However, this could also be due to the fact that the meshing is not dense enoughto give a highly detailed representation of the deformations. The general conclusionthat should be drawn is that those results should be exploited with a great care and

29

4. Measurements

Figure 4.8: Curve-fitting for the point 7 [courtesy of P. Banach].

critical judgement is required. Fig. 4.10 shows the animation of the interesting modes.For the reason explained above the modes 8, 10, 12, 13 and 14 were skipped. Fromthis figure it can be seen that all modes seem to have a physical meaning. On the otherhand, the lack of measurement points to get accurate deformations is proven by thefact that the last torsion modes are very similar.

4.7 ResultsIn this section the results from the measurements are presented. In the discussionchapter explains these results further together with a presentation of the limitations ofthe measurement set-up.

4.7.1 Results of the measurements for the free free vibration test 1The obtained by experimental method eigenfrequencies later will be compared withthe modes obtained by numerical FEM calculations for the boom structure. Table 4.1and Fig 4.10 show the results for the first free–free vibration test.

30

4. Measurements

Figure 4.9: Curve-fitting for the point 11 [courtesy of P. Banach].

4.7.2 Results of the measurements for the free free vibration test 2Table 4.2 below lists the modes identified together with the damping ratios and Fig4.16 depicts them.

4.7.3 Results of the measurements for cantilever down testingTable 4.3 and Fig 4.15 show the results of the cantilever down testing.

31

4. Measurements

Table 4.1: Results of the measurements for the free free vibration test 1

Mode number Frequency [Hz] ζ[%] commentMode 1 1.77 2.08 bending and torsionMode 2 2.87 0.62 bending and torsionMode 3 3.41 1.41 bendingMode 4 4.98 0.1 bending and torsionMode 5 6.14 1.86 bending and torsionMode 6 7.04 0.2 mainly torsionMode 7 8.33 0.18 bending and torsionMode 8 8.93 0.04 bending and torsionMode 9 9.73 0.35 bending and torsionMode 10 10.00 0.23 bending and torsionMode 11 14.46 2.58 torsionMode 12 14.78 1.18 weak torsionMode 13 15.22 1.19 weak torsionMode 14 15.74 0.55 weak torsionMode 15 16.41 1.98 torsionMode 16 17.23 2.78 torsionMode 17 18.13 1.52 weak torsion

The coherence for the accelerometers and the laser can be seen in Figs. 7 and 8respectively.

4.8 Mode shapes extracted using the results of the vi-bration testing

The mode shapes corresponding to the eigen frequencies can be seen in Figs 4.10, 4.15and 4.16. The coherence has been determined for one measurement per measurementpoint and then plotted together, see Fig. 4.14

32

4. Measurements

Table 4.2: Results of the measurements for the free free vibration test 2

Mode Frequency [Hz] Damping Type of mode1 1.6 2.9 Rigid2 2.1 2.3 Rigid3 3.4 1.5 Rigid4 4.6 0.8 Elastic5 5.6 0.4 Elastic6 7.5 2.1 Elastic7 7.9 0.5 Elastic8 8.9 0.3 Elastic9 11.6 0.5 Elastic

Table 4.3: Results of the measurements for cantilever down testing

Mode Frequency [Hz] Damping Type of mode1 2.4 0.72 bending2 4.0 2.7 bending and torsion3 4.5 2.8 bending4 5.0 0.2 torsion5 6.0 2.4 bending6 7.1 2.9 torsion7 8.5 1.3 longitudinal

33

4. Measurements

Figure 4.10: Mode animations [courtesy of P. Banach].

Figure 4.11: Laser measurement point mobilities when the dummy and boom areturned 0 degree angle [courtesy of J. Kristoffersson & M. Larsson].

34

4. Measurements

Figure 4.12: Laser measurement point mobilities when the dummy and boom areturned 90 degrees angle [courtesy of J. Kristoffersson & M. Larsson].

Figure 4.13: Coherence for the laser measurement points [courtesy of J. Kristoffersson& M. Larsson].

35

4. Measurements

Figure 4.14: The coherence for the accelerometers calculated from one measurement[courtesy of J. Kristoffersson & M. Larsson].

Figure 4.15: Shapes for modes from 2 to 7 Hz [courtesy of C. Frangoudis, C. Kastby,M. Zapka].

36

4. Measurements

Figure 4.16: Shapes for modes from 1 to 9 Hz [courtesy of C. Frangoudis, C. Kastby,M. Zapka].

37

Chapter 5

Finite Element Modal Analysis of theBoom Integrated to a CubeSat

5.1 Chapter overviewIn this chapter the finite element model is presented. The finite element model of thesatellite dummy and integrated boom is prepared in order to identify the eigenfrequen-cies of the structure using numerical method and in order to conclude if the use ofthe gravity off-loading system is justified. The basic idea behind introduction of thegravity off-loading system is to simulate the zero gravity condition while also tryingto eliminate any influence of the gravity off loading system on the dynamic analysis ofthe boom integrated to the CubeSat. The eigen frequencies of the gravity off loadingsystem used should be at least 10 times lower than the expected eigen frequency of theboom [9].

In order to identify eigenfrequencies and reference mode shapes of any structurethe modal analysis is used. The modal analysis is linear and may account for dampingeffects, but ignores a plastic deformation, creep or contact stiffness. In case of dynamicloading and analysis of vibrations and transient responses in structures it is crucial toidentify the characteristics of eigen frequencies.

Here is an example of modal analysis: Type of analysis - modal. Aim of analysis -calculation of eigen frequencies and reference mode shapes. The following operationsare necessary, step by step:

1. creat a model of the structure in any CAD/CAE software

2. import it to Abaqus/CAE

3. convert units of measurements, if necessary

4. assign material properties:

38

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

• Young modulus

• Density

• Poisson’s ratio

5. choose an element type,

6. mesh the model,

7. create necessary connections, ties between the assembly parts and constrain themodel,

8. specify the parameters for modal frequencies identification.

5.2 Geometry of the structureIn order to carry out a numerical eigenfrequency analysis of the structure in this thesisthe software Abaqus/CAE was used. In order to quickly prepare the geometry CADsoftware NX 6.0 was used and later this geometry was imported to the Abaqus/CAEgraphics module.

Thus, there are four types of structures that are under investigation in this thesis:

• boom with satellite dummy,

• boom with satellite dummy with levelled positioning of tape-springs

• boom with satellite dummy without extension springs in gravity off-loading sys-tem

• gravity off loaded boom with satellite dummy with extension springs in gravityoff-loading system

A simplified model of the CubeSat dummy with the inbuilt boom is shown in Figure5.2. The geometry was created using the software NX Unigraphix and then importedto the Abaqus software. The cross section view of the tape spring connections to thehub and attachment plates is shown in Fig.5.1. The meshed geometry is shown in Fig.5.3. In order to assemble the structure in Abaqus and in order to run the analysis theTie constraints need to be specified for the surfaces of parts which touch each other.

39

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.1: Cross section view of the tape springs positioning.

Figure 5.2: Satellite dummy with the boom.

5.3 Material properties for components of the modelThe material properties are specified for each component. The material data can besummarized as follows.

1. Plastic - Modulus of elasticity 32 GPa, Poisson’s ratio 0.3, density 1200 kg/m3

2. Tape Springs - Modulus of elasticity 63 GPa, Poisson’s ratio 0.3, density 1780kg/m3

3. Nylon Strings - Modulus of elasticity 5 GPa, Poisson’s ratio 0.3, density 800kg/m3

4. Aluminium Bars - Modulus of elasticity 69 GPa, Poisson’s ratio 0.3, density2850 kg/m3

40

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.3: Meshed geometry of the boom with gravity off-loading strings.

5. Satellite Dummy - Modulus of elasticity 69 GPa, Poisson’s ratio 0.3, density9400 kg/m3. This density is given a number 9400 kg/m3 in order to allowfor the masses that are not included in the satellite dummy, the volume of thesatellite dummy was computed and known mass of 2.5 kg was used to computeequal density)

6. Springs - Modulus of elasticity 210 GPa, Poisson’s ratio 0.3, density 7800 kg/m3

5.4 AnalysisThe linear eigenfrequency analysis was performed on the structures using AbaqusCAE. Lanczos algorithm was used. The eigenfrequencies for the first 10 modes wereextracted. The model was not constrained during the analysis and the first 6 rigid bodymotion modes which give 0 Hz frequencies are omitted in the following text.

5.5 Eigenfrequencies from the modelAfter running the eigenfrequency analysis for four types of structures mentioned abovethe data represented in Table 5.1 were obtained, where:

41

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

• ∗ - modes are given only for the boom with satellite dummy, the first 6 freebody motion modes are omitted together with not important gravity off-loadingsystem modes

• I - Modal analysis of free-free boom with satellite dummy

• II - Modal analysis for gravity off-loaded boom with satellite dummy withoutextension springs in gravity off-loading system

• III - Modal analysis for gravity off-loaded boom with satellite dummy with ex-tension springs in gravity off-loading system

• IV - Modal analysis of free-free boom with satellite dummy with levelled posi-tion of tape-springs.

Table 5.1: Eigenfrequencies for the boom model configurations calculated in AbaqusCAE

Mode,* Eigenfrequency, Hz- I II III IV1 7.73 7.79 (+1.2%) 7.78 (+1.2%) 4.422 10.05 11.31 (−15.17%) 11.11 (−15.18%) 7.083 16.78 16.10 (−4.2%) 16.05 (−4.2%) 9.244 21.26 21.09 (−1.11%) 21.18 (−1.09%) 13.59

5.6 Vibration modes of the free–free vibrating boomThe first four modes and corresponding shapes for the free–free vibrating boom areshown in Figs 5.4 to 5.7

5.7 Vibration modes of the boom in the gravity off-loadingsystem

This section presents the mode shapes for the gravity off-loaded boom. Figs 5.8 to5.13 show the first three important mode shapes of the gravity off-loaded boom.

42

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.4: First eigenfrequency, 7.7 Hz − (a) Front view (b) Side view (c) Top view

5.8 Sensitivity analysisIn order to ensure that the finite element model for vibration analysis is set up cor-rectly the sensitivity analysis was performed with respect to the position of fishing lineending on the aluminium bar and fishing line attached to the satellite dummy was alsomoved away from the satellite dummy centre of gravity. The list below outlines theconfigurations of the structure for the sensitivity analysis.

Table 5.2 shows the variation of the first eigenfrequency with respect this changes.where i in Table 5.2 denotes the number of the eigen frequency of the boom

43

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.5: Second eigenfrequency, 10.0 Hz − (a) Front view (b) Side view (c) Topview

I the free free vibratingII central position of the fishing linesIII shifted 50 mm to the boom fishing line of the satellite dummyIV shifted 50 mm away from the boom fishing line of the satellite dummyV shifted 150 mm away to the satellite dummy fishing line of the aluminium barVI shifted 150 mm away from the satellite dummy fishing line of the aluminium bar

44

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.6: Third eigenfrequency, 16.2 Hz− (a) Front view (b) Side view (c) Top view

Table 5.2: Sensitivity analysis

i frequency, HzI 7.73II 7.79 (+1.2%)III 7.77 (+0.8%)IV 7.76 (+0.7%)V 7.57 (-1.6%)VI 8.18 (+9.5%)

45

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.7: Fourth eigenfrequency, 21.5 Hz − (a) Front view (b) Side view (c) Topview

46

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.8: First eigen mode for the gravity off-loaded boom prototype, 7.8 Hz − (a)Front view (b) Side view (c) Top view

47

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.9: First eigen mode for the gravity off-loaded boom prototype, 7.8 Hz −Overall side view of the gravity off-loading system

48

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.10: Second eigen mode for the gravity off-loaded boom prototype, 11.3 Hz− (a) Front view (b) Side view (c) Top view

49

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.11: Second eigen mode for the gravity off-loaded boom prototype, 11.3 Hz− Overall side view of the gravity off-loading system

50

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.12: Third eigen mode for the gravity off-loaded boom prototype, 16.1 Hz −(a) Front view (b) Side view (c) Top view

51

5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat

Figure 5.13: Third eigen mode for the gravity off-loaded boom prototype, 16.1 Hz −Overall side view of the gravity off-loading system

52

Chapter 6

Discussion

6.1 Mode shapes comparison and discussionDue to the least resistance principle, the tape springs in boom as can be seen in the firstmode are bending diagonally with respect to the main orthogonal axis for the crosssection. The bending and slight torsion in the first mode is caused by the principal axisof the second moment of inertia for the cross-section turn ca 40 degrees compare tothe case when the tape springs in the boom are levelled. In the figure below the firstmode shape is shown for such boom with levelled tape-springs:

Figure 6.1: First eigenfrequency, boom with levelled tape springs , 4.2 Hz - (a) Sideview, (b) - Top view

53

6. Discussion

Comparison of the mode shapes between experimental and finite element analysisresults shows discrepancy in two of the measurements with respect to the finite elementmodel. As shown in the results the natural frequencies received a higher estimate asthe boom structure did not contain the transition zone. The transition zone is shownin Fig. 6.2. The transition zone is weaker compare to semi-circular cross section ofthe untouched tape-spring and thus the eigenfrequencies of the real boom are lower.Although the tape springs of the boom in the model are stiffer we can observe andlearn more about the shape of vibration of the boom.

Figure 6.2: Transition zone. ρ0 denotes a region of a transition zone in a tape spring,[14]

Figure 6.3: The rotation 39.8 degrees for the principal axis of inertia of the boomcross-section

Fig 6.4 shows a comparison of the first eigen frequency, obtained through vibrationtesting and first eigen frequency for the free free vibrating boom from the FEA model.

6.2 Mass participationTaking a closer look at the results for the eigenfrequencies of the boom presented inTable 5.1 we notice that the eigenfrequencies in the second mode for the gravity off-

54

6. Discussion

Figure 6.4: Comparison of the shapes for the first mode in the FEA analysis, 7.7 Hz(a) and the vibration testing, 1.76 Hz (b)

loaded boom are having discrepancy of 15%. This has a simple explanation. Thealuminum bar above the boom’s lightest part (tip plate and the hub) is participatingin the boom vibration modes. The stiffness matrix for the boom in this mode havingadditional mass components and this gives a higher eigenfrequency estimate. It hasbeen discussed in paper [14] by Greschik that the gravity off-loading system shouldbe as light as possible besides having a 10 times lower eigenfrequency. Althoughthe descipancy in our case is significant we still can draw sensible conclusions abouthow the gravity off-loading system works and describe quantitatively the influence thegravity off-loading system has. The gravity off-loading system has effect on the spacestructure but with reasonable degree of accuracy it allows us to study the dynamiccharactertics of the boom.

6.3 SummaryThe mathematical FEM model developed in Abaqus is one of the possible modelswhich predicts the eigen frequencies of the structure and the mode shapes. This mod-els are mainly used to compare the mode-shapes of the free satellite dummy with de-ployed boom and the suspended in the gravity off loading system structure. From theobtained data we can conclude that the use of gravity off loading system is justifiedas the eigen shapes are comparable in the free free vibration test case with free freevibration analysis model. Despite the problems with obtaining accurate results fromthe experimental modal analysis it is estimated that the first eigenfrequency is lowerthan 2 Hz and beyond 1 Hz and that the first eigenmode is a bending mode. Mode 1(1.8 Hz) is very representative as this type of bending is observed in the deploymenttests and in finite element analysis.

55

Chapter 7

Conclusions and Recommendations

7.1 Chapter overviewIn this chapter, the objective is to summarise the results of the measurements and thesimulations of the boom structure integrated to the CubeSat and to compare the results.The measurement and simulation results of the simpler structures were used to simplifythe mathematical modelling of the complete structure and will not be evaluated furtherthan what was done in the previous chapters. The resemblance of the eigenmodes fromthe measurements and the simulations will be analysed. To establish that the modesfrom the measurements and from the model are actually from a similar eigenmode, acomparison of the mode shapes corresponding to those eigenfrequencies will be carriedout.

7.2 Research conclusionsIn this paper an experimental analysis of the boom of a satellite from a dynamicalpoint of view was presented. The main objective was to find the modal parametersof the boom in a frequency range comprised between 0 to 100 Hz with a particularfocus on the range from 0 to 20 Hz because the satellite should not be submitted tohigh frequency perturbations once in orbit. Because of the complexity of the struc-ture under study some special measurement techniques were implemented to acquirethe amplitudes. Unfortunately, it was not possible to perform a very dense meshingof measurement points and the deformations of the composite tape-springs were notmeasured directly. The excitation of the structure was carried out by a shaker fed witha white noise signal and mounted on the dummy structure. Thus, 10 – 17 modes werefound in the frequency range between 0 to 19 Hz. Some of them were obvious whereas

56

7. Conclusions and Recommendations

others seemed to be overlapping modes or combination of other modes rather than in-dependent modes. This is probably due to the lack of measurement points and to thefact that modes are very close to each other. However, a comparison with FEM resultswas performed and a correspondence between certain modes was found. Finally, theresults given in this paper should be exploited with caution since they are not highlyprecise and accuracy is unverified beyond reasonable physical behaviour [11].

7.3 Lessons learnedThe objective of creating a model that has the same dynamic behaviour as the structurein terms of eigenfrequencies and mode shapes for frequencies up to 100 Hz turned outto be a bit optimistic since the displacement of the structure was too complicated to bemeasured with accelerometers for some of the higher frequencies. It is hard to analysewhether or not these eigenmodes were successfully reproduced in the FEM model.The aim was to simulate the lower eigenmodes and the first two eigenmodes showedgreat similarity in mode shape between measurement and simulation. There were alsoa couple of eigenmodes from simulations at higher frequencies that were close to themeasured eigenmodes in terms of frequencies and with obvious similarity in modeshapes. When modelling a structure of such complexity as the boom, there are manypossible sources of error. However, since a lot of effort was made to obtain a goodmodel of a single beam (with satisfying result), the material properties of the boomand the boundary conditions between the tape springs and the hub and attachmentplate is considered to be relatively representative. However, errors may exist sincethe material properties can vary. The material properties of the tape springs were notaccurately modelled and only relative comparison was performed for boom with andwithout gravity off-loading system.

A major source of error when modelling an assembled structure may be how theconnections between the different parts are modelled. For the boom, tape-springs basedstructure good results were obtained by simply letting all DOFs on connecting surfacesbe tied to each other. This was probably successful since the connections in the realstructure were made very firm. If that would not have been the case, there wouldprobably have been a need to experiment more with creating different partitioning ofthe surfaces or make special joint interfaces. Asymmetrical of the real structure due toirregularities in the materials and the assemblage is another source of error. Before per-forming any experiments, the tape-springs were joint to the hub, the attachment plateand the tip firmly and as symmetrical as possible. When information was extractedabout peak frequencies from the frequency sweep measurements, the resolution of theRMS plots was poor for the lower frequencies (more than 1 Hz for the lowest frequen-cies). In the simulations, the resolution was set to 0.25 Hz. This meant that there was

57

7. Conclusions and Recommendations

no point in analysing the difference in frequencies down to decimals [15].

7.4 RecommendationsFor those who are to follow, it is recommended to use a more dense in resolution highprecision capture system to capture the amplitudes of the boom as the experiments thatwere performed did not show intermediary boom structure amplitudes. The ampli-tudes were impossible to observe using the capture system used. More measurementpoints would produce a more precise result with regard to the boom mode shapes andfrequencies. One could also design a lighter gravity off-loading system using light-weight structures, especially when it comes to the bars. We have seen the mass partic-ipation factor 15% influence in the second mode and this could be improved. A morecomplex tree in the Marionette gravity off-loading system would probably give a betterweightlessness environment simulation in the experimental testing.

58

References

[1] M. Zapka C. Frangoudis, C. Kastby. Mode shape analysis of satellite boom.Experimental Structure Dynamics Project Report, Unpublished, KTH Royal In-stitute of Technology, Department of Vehicle Engineering, 2012. 23

[2] U. Carlsson. Experimental structure dynamics. Lecture Notes, KTH Royal Insti-tute of Technology, Department of Vehicle Engineering, 2009. 22

[3] J.F. Hauer et al. Initial results in prony analysis of power system response sig-nals. IEEE Transactions on Power Systems, 5, 1:80–89, 1990. Seminare Maurey-Schwartz (1975-1976). 12

[4] Z.-F. Fu et al. Modal Analysis, p.141. BUTTERWORTH HEINEMANN, Oxford,2001. vii, 23

[5] Retreived from www.mobiusinstitute.com. Be aware of resonance.Website, 2012. http://marketing.mobiusinstitute.com/acton/rif/2278/s-1c31-1207/-/l-sf-contact-003f:1589/g-02d4/showPreparedMessage. 5

[6] Retrieved from www.cubesat.org. CubeSat Design Specification. Rev.12, 2009.http://www.cubesat.org/images/developers/cds_rev12.pdf. 1

[7] Retreived from Space Daily. Boeing Completes CubeSat MissionTo Advance Nano-Satellite Technology. Website, August 17, 2007.http://www.spacedaily.com/reports/Boeing_Completes_CubeSat_Mission_To_Advance_Nano_Satellite_Technology_999.html. 1

[8] Retreived from Wikipedia. SwissCube-1. Website, 2011. http://en.wikipedia.org/wiki/SwissCube-1. vii, 1, 2

[9] W. K. Belvin G. Greschik. High-fidelity gravity offloading system for free – freevibration testing. JOURNAL OF SPACECRAFT AND ROCKETS, 44 N1, 2007.18, 38

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[10] A. K. Chopra. Dynamics of structures. Prentice Hall, 1995. vii, 12

[11] M. Larsson J. Kristoffersson. Satellite boom dynamics: Determination of eigen-frequencies and modes of a satellite boom. Experimental Structure DynamicsProject Report, Unpublished, KTH Royal Institute of Technology, Department ofVehicle Engineering, 2012. 23, 57

[12] C. A. Ross J. W. Tedesco, W. G. McDougal. Structural Dynamics. Theory andApplications. Addison-Wesley, 1998. vii, 6, 7, 11

[13] C. G. Christodoulou, J. B. J. Costantine, Y. Tawk and S. Lane. Cubesat de-ployable antenna using bistable composite tape-springs. IEEE ANTENNAS ANDWIRELESS PROPAGATION LETTERS, 11:285–288, 2012. 1, 2

[14] S. K. Jeon; T. W. Murphey. Design and analysis of a meter-class cubesat boomwith a motor-less deployment by bi-stable tape springs. American Institute ofAeronautics and Astronautics, pages 1–11, 2010. vii, ix, 1, 2, 3, 4, 54, 55

[15] L. Gediminas P. Banach, J. Brondex. Modal survey of a satellite boom - free-freevibration test. Experimental Structure Dynamics Project Report,Unpublished,KTH Royal Institute of Technology, Department of Vehicle Engineering, 2012.25, 58

[16] D. Shanks. Non-linear transformation of divergent and slowly convergent se-quences. Journal of Mathematics and Physics, 34:1–42, 1955. 13

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