Boom Integrated to a CubeSat
VALERIY SHEPENKOV
Degree project in
Boom Integrated to a CubeSat
Valeriy Shepenkov
School 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
I am truly thankful to my parents and relatives for their compassion and support during the two years of my studies at KTH. I would like to thank my thesis advisor Dr. Gunnar Tibert who provided me with a valuable leadership with which I was able to complete the project. I would like to thank Gunnar Tibert for giving me the opportunity to work in the CubeSat project at KTH, and for being a great coordinator in Engineering Mechan- ics MSc program. It was my pleasure to work together with Julien Servais and Pau Mallol on the satellite project at Mechanics Department. I would like to thank my friends who made my life full of interesting events during the two years in Stockholm. I am thankful to Will Reid who has been a great project manager and a great colleague in the RAINproject which I participated in at KTH during 2010 – 2012. Thank you Will for teaching me LaTeX and MATLAB programming. Overall my graduate studies at KTH were carried out thanks to the generous Visby Scholarship provided by Swedish Institute and the financial aid is gratefully acknowledged.
CubeSat or Cubic Satellite is an effective method to study the space around the Earth thanks to its low cost, easy maintenance and short lead time.
However, a great challenge of small satellites lies in achieving technical and scientific requirements during the design stage. In the present work primary focus is given to dynamic characterization of the deployable tape- spring boom in order to verify and study the boom deployment dynamic effects on the satellite. The deployed boom dynamic characteristics were studied through simulations and experimental testing. The gravity off- loading system was used to simulate weightlessness environment in the experimental testing and simulations showed that the deployment of the system influence the results in a different way depending on the vibration mode shape.
En CubeSat eller kubisk satellit är effektivt för att studera rymden runt jorden på grund av dess låga kostnad, enkla underhåll och korta ledtid. En stor utmaningen i utformningen av små satelliter är att uppnå de tekniska och vetenskapliga kraven. Detta arbete har analyserat de dynamiska egen- skaperna hos en utfällbar band-fjäder bom i syfte att verifera och för att studera bommens utfällningsdynamiska effekter på satellitens bana och at- tityd. Den utfällda bommens dynamiska egenskaper har studerats genom simuleringar och experimentella tester. Ett tyngdkraftskompenserande sy- stem har använts för att simulera tyngdlöshet i de experimentella testerna och simuleringar visar att utformningen av detta system påverkar resulta- ten olika beroende på svängingsmodens form.
Contents v
List of Figures vii
1 Introduction 1
1.1 Background . . . 1
1.2 Objectives . . . 2
2 Theory of Structural Dynamics 5 2.1 Introduction . . . 5
2.2 Equation of motion and natural frequency . . . 7
2.3 Modal analysis . . . 7
2.4 Steady-state vibrations . . . 8
2.5 Damping . . . 9
2.6 Root–mean–square value . . . 12
2.7 Modal parameter extraction: complex exponential method . . . 12
3 Design of the Structure 14 3.1 Concept design of the CubeSat . . . 14
3.2 Structure . . . 16
3.3 Materials . . . 16
3.4 Gravity off-loading systems as a mean to obtain dynamic characteris- tics of structures . . . 17
4 Measurements 22 4.1 Introduction . . . 22
4.2 Instrumentation . . . 22
4.3 Measurement procedure . . . 24
4.4 Signal processing . . . 26
4.5 Calibration . . . 27
4.6 Modal parameter extraction . . . 27
4.7 Results . . . 30
4.7.1 Results of the measurements for the free free vibration test 1 . 30 4.7.2 Results of the measurements for the free free vibration test 2 . 31 4.7.3 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 38 5.1 Chapter overview . . . 38
5.2 Geometry of the structure . . . 39
5.3 Material properties for components of the model . . . 40
5.4 Analysis . . . 41
5.5 Eigenfrequencies from the model . . . 41
5.6 Vibration modes of the free–free vibrating boom . . . 42
5.7 Vibration modes of the boom in the gravity off-loading system . . . . 42
5.8 Sensitivity analysis . . . 43
6 Discussion 53 6.1 Mode shapes comparison and discussion . . . 53
6.2 Mass participation . . . 54
6.3 Summary . . . 55
7 Conclusions and Recommendations 56 7.1 Chapter overview . . . 56
7.2 Research conclusions . . . 56
7.3 Lessons learned . . . 57
7.4 Recommendations . . . 58
References 59
1.1 SwissCube-1[8]. . . 2
1.2 SIMPLE boom shown in the (a) stowed configuration, (b) partially deployed and (b) fully deployed states [14]. . . 3
1.3 Exploded hub view of the SIMPLE boom in initial stages of deploy- ment, [14]. . . 4
2.1 A signal in time domain. . . 6
2.2 A signal in frequency domain. . . 6
2.3 Mechanical model for a simple SDOF system and its free body dia- gram, [12]. . . 7
2.4 A damped free vibration. . . 10
2.5 Dynamic magnification factor (DMF) versus frequency ratio for vari- ous levels of damping [12]. . . 11
2.6 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]. . . 16
3.3 Hub [courtesy of Pau Mallol]. . . 17
3.4 Tip plate with electronics [courtesy of Pau Mallol]. . . 18
3.5 KTH boom prototype in its (a) stowed configuration and (b) deployed configuration [courtesy of Pau Mallol]. . . 19
3.6 Scheme of the experimental set-up. . . 20
3.7 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 orientation vibration test [courtesy of J. Kristoffersson & M. Larsson]. . . 23
4.2 Typical measurements set–up, [4]. . . 23
4.3 Positioning of the shaker in the in-boom-plane vibration test. . . 25
4.4 Positioning of the accelerometers. . . 26
4.5 Measurement points on the lateral faces (left picture) and on the hub
lower face (the picture to the right). . . 26
4.6 Calibration of the laser. . . 28
4.7 Mode Indicator Function [courtesy of P. Banach]. . . 29
4.8 Curve-fitting for the point 7 [courtesy of P. Banach]. . . 30
4.9 Curve-fitting for the point 11 [courtesy of P. Banach]. . . 31
4.10 Mode animations [courtesy of P. Banach]. . . 34
4.11 Laser measurement point mobilities when the dummy and boom are turned 0 degree angle [courtesy of J. Kristoffersson & M. Larsson]. . . 34
4.12 Laser measurement point mobilities when the dummy and boom are turned 90 degrees angle [courtesy of J. Kristoffersson & M. Larsson]. 35 4.13 Coherence for the laser measurement points [courtesy of J. Kristoffers- son & M. Larsson]. . . 35
4.14 The coherence for the accelerometers calculated from one measure- ment [courtesy of J. Kristoffersson & M. Larsson]. . . 36
4.15 Shapes for modes from 2 to 7 Hz [courtesy of C. Frangoudis, C. Kastby, M. Zapka]. . . 36
4.16 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. . . 40
5.2 Satellite dummy with the boom. . . 40
5.3 Meshed geometry of the boom with gravity off-loading strings. . . 41
5.4 First eigenfrequency, 7.7 Hz − (a) Front view (b) Side view (c) Top view 43 5.5 Second eigenfrequency, 10.0 Hz − (a) Front view (b) Side view (c) Top view . . . 44
5.6 Third eigenfrequency, 16.2 Hz − (a) Front view (b) Side view (c) Top view . . . 45
5.7 Fourth eigenfrequency, 21.5 Hz − (a) Front view (b) Side view (c) Top view . . . 46
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
5.9 First eigen mode for the gravity off-loaded boom prototype, 7.8 Hz − Overall side view of the gravity off-loading system . . . 48
5.10 Second eigen mode for the gravity off-loaded boom prototype, 11.3 Hz − (a) Front view (b) Side view (c) Top view . . . 49
5.11 Second eigen mode for the gravity off-loaded boom prototype, 11.3 Hz − Overall side view of the gravity off-loading system . . . 50
5.12 Third eigen mode for the gravity off-loaded boom prototype, 16.1 Hz − (a) Front view (b) Side view (c) Top view . . . 51
5.13 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 tape
spring, [14] . . . 54 6.3 The rotation 39.8 degrees for the principal axis of inertia of the boom
cross-section . . . 54 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) . . . 55
Introduction
1.1 Background
CubeSats are standardized complex systems used for space research by universities and have special properties such as a volume of 10 cm3and a mass not exceeding 1.33 kg [6]. The main goal of CubeSat research and development projects is to stimulate interest in science and increase competence in space technology among students and educational 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. These remarkable size and mass make these satellites appropriate for cost-effective space exploration in the low Earth orbits. For instance, the CubeSat SwissCube-1 (Fig. 1.1), Swiss satellite, was a cost effective solution to study thenightglowwithin the Earth’s atmosphere.
The first successful launches of CubeSats began in the mid-2003 fromPlesetsk, Russia.
Since then, the satellites are predominantly being developed by Stanford University, California PolyTechnic University, Norwegian University of Science and Technology and the University of Tokyo. KTH, the Royal Institute of Technology is making an attempt to build a system for the CubeSat mission in collaboration with the US Air Force Research Laboratory, University of Florida and the Inter–American University of Puerto Rico. The project at KTH includes the design, development and verification of the SIMPLE deployable boom prototype [14].
Deployable booms are used in today’s CubeSat missions. For instance, deployable booms 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 with an embedded copper alloy conductor. Bi-stability properties enable the antenna to be elastically stable in both deployed and stowed configurations [13].
Dynamic characterization of the deployable structures during their deployment and in their deployed state are of primary importance because of how those factors influ-
Figure 1.1:SwissCube-1[8].
ence the behaviour of the satellite. In many cases, the deployment mechanisms involve large 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 Objectives
The objective of this research is to analyse the dynamic characteristics of the boom deployed from the CubeSat. The project focus is on the dynamic characterization of a SIMPLE boom prototype and on numerical analysing the means of the boom vibration testing. The proposed boom design can be seen in Figure 1.3. The analysis will be performed both on the experimental structure and on a finite element model of the structure. The aim is to retrieve the eigenfrequencies and mode shapes from the experimental data and to create an FE model that with satisfying resemblance simulates the behaviour of the structure.
The vibrations of the boom structure are recorded using accelerometers and laser
(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.
Thefinite 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 calculations are 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 University of Puerto Rico was divided into three primary focus areas. The primary focus of the KTH 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.
Figure 1.3: Exploded hub view of the SIMPLE boom in initial stages of deployment, [14].
Theory of Structural Dynamics
2.1 Introduction
Vibrations 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 analysing a structure that is subjected to a static load, a larger force will in general cause larger displacements and a stiffer construction is therefore needed. The situation is not as obvious when it comes to dynamic loads acting on the structure. In this case, the magnitude 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 eigen frequency) of the structure, the response can be substantially larger than for a load with the 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 a structure 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 acting on it. The displacements around the equilibrium are measured in order to analyse the vibrations and can be considered both as functions in the time domain and in the frequency domain. The last alternative can be useful when analysing the dynamic behaviour of a structure and is called the response spectra. To convert a signal between the two domains a Fourier analysis algorithm such as Fast Fourier Transform can be used. An example of a signal in the time domain and its response spectrum is shown in Figures2.1and2.2
During analysis of a mechanical system there is always a matter of how scrupulous one 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 the system as continuous or discrete. It is practically impossible to analyse a complex
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 counter parts 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].
2.2 Equation of motion and natural frequency
The 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 of freedom. An example of the system is illustrated in Figure2.3. It consists of a mass m connected to the wall with a damper and a spring (both having mass m = 0). The mass moves with negligible friction in the horizontal direction and the degree of freedom is the 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 Figure2.3and using Newton’s second law of motion leads to Eq. (2.1)
F(t) − cdx
dt − ku = md2x
dt2 (2.1)
Reorganizing the terms in Eq. (2.1) results in the equation of motion for a SDOF model:
md2x dt2 + cdx
dt + 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-freedom model (MDOF). For a system having small displacements (neglecting any non-linear behaviour), the equation of motion for a SDOF system can be converted to the multi–
dimensional case, Eq. (2.3)
Md2x dt2+Cdx
dt+Kx= F(t) (2.3)
In such a case M is the mass matrix, C is the damping matrix and K is the stiffness matrix; x is the displacement vector and F(t) the load vector. If there are n degrees of freedom in the model, the matrices will be of size n × n and the vectors n × 1.
2.3 Modal analysis
A structure has an infinite number of eigenfrequencies as it has an infinite number of DOFs if it is not constrained. These are the frequencies the structure will oscillate
at when it is allowed to vibrate without the influence of any external loads. Each eigenfrequency has a matching mode shape. If the structure is deflected into a mode shape 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 reduced to:
Md2x
dt2+Kx = 0 (2.4)
To solve Eq. (2.4), a harmonic solution is assumed:
x(t) = A cos(ωt)Φ + B sin(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. Each eigenvalue ωihas a matching eigenvector Φi. These are the angular eigen frequencies and mode shapes of the structure. Together, an eigenvalue and eigenvector pair form a solution to (2.5) that satisfies the differential equation. A and B are given by the initial conditions x(0) = x0and the same condition for initial velocity. The mode vectors form an orthogonal basis and the solution can be rearranged as a sum using all eigenvalues and eigenvectors:
x(t) =
n
∑
1
qi(t)Φi; qi(t) = Aicos(ωit) + Bisin(ωit) (2.7)
2.4 Steady-state vibrations
When a structure is subjected to a harmonic load, it will after a starting transient phase oscillate 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 of motion:
Md2x
dt2+Kx = F(t); F(t) = F0sin(ωt) (2.8)
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
ΦTrMΦid2qi(t) dt2 +
n
∑
1
ΦTrKΦiqi(t)=ΦTrF(t) (2.9) The eigenvectors are orthogonal in the scalar products ΦTrKΦiand ΦTrMΦihence only terms with i = r are non-zero. This results in n uncoupled equations of the form:
Mid2qi
dt2 + Kiqi= Fi(t); Ki= ΦTiKΦi; Mi= ΦTi MΦi; Pi= ΦTi P(t) = P0isin(ωt) (2.10) Insertion of a harmonicansatzsolution qi(t) = q0isin(ωt) gives the following result:
q0i= P0i Mi
1
ωi2− ω2; ωi= Ki
Mi (2.11)
The result shows that an excitation frequency ω equal to an eigenfrequency will give an infinite amplitude. This is of course not the case in reality, where any kind of damping 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 the eigenfrequencies.
2.5 Damping
Damping is included in mathematical models to represent the energy dissipation in structural dynamics and is always present in real structures. It can for example be friction in joints, or internal properties of materials.
If F(t) = 0 in Equation2.12, the equation of motion can be rewritten as:
d2x
dt2 + 2ζωn
dx
dt + ωn2x= 0; ωn= rk
m, ζ = c
2mωn (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) +(d2x
dt2)t=0+ ζωnx(0) ωD
sin(ωDt)
!
, ωD= ωn
q 1 − ζ2
(2.13)
ωDis the eigenfrequency for damped vibrations and is related to ωnby a factorp 1 − ζ2. For relatively small damping ratios (ζ ≺ 0.2), ωD≈ ωn.
See Figure2.4for 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 the spike in the frequency response spectrum will depend on the damping ratio. Figure2.5 shows the deformation response factor for frequencies around an eigenfrequency. The deformation response factor is a quota between the amplitude of the dynamic response for a harmonic load and the amplitude of the static response for a static load of the same 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 stiffness properties of individual elements. For example, the damping properties of materials are not as well agreed upon and also the energy dissipation in joints needs to be taken into account. On the contrary, the damping matrix may be constructed from the modal damping ratios of the structure. There are two types of damping matrices, classical and non-classical. The difference between them is that classical damping matrices are diagonal and non − classical damping matrices are not. A diagonal matrix makes it possible 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 analysis of the structure.
Rayleigh method is a method of constructing the damping matrix, which delivers a classical damping matrix. The Rayleigh damping matrix is a linear combination of the
Figure 2.5: Dynamic magnification factor (DMF) versus frequency ratio for various levels 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 that the damping ratio of the nth mode is given by:
ζn= a0
2ωn+a1ωn
2 (2.15)
a0 and a1can be obtained from two known damping ratios ζi and ζj. If ζ is assumed to be constant in modes i and j, a0and a1is given by expressions (2.16).
a0= ζ 2ωiωj ωi+ ωj
, a1= ζ 2 ωi+ ωj
(2.16) The damping as a function of frequency in the case of constant damping ratios in modes i and j is shown in Figure 2.6. The mass matrix damps the lower frequencies and the stiffness matrix damps the higher frequencies.
Figure 2.6: Damping ratio versus frequency for Rayleigh damping, [10].
2.6 Root–mean–square value
The RMS value (root–mean–square value), defined in Eq. (2.17), is used to create an average magnitude of a signal over time. Since the sign of the displacements varies in time during vibrations, it is practicable to use the RMS value instead of the mean value.
xRMS= s
1
∆t
Z t0+∆t t0
x2(t)dt (2.17)
2.7 Modal parameter extraction: complex exponential method
There are several ways to extract the modal parameters using the data collected. In this case, a numerical method called the Pronymethod[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. The idea of this method is to minimize the function R, which is the difference between the frequency response function modelled as a ratio between polynomials in Z−1 and the Z transform of the measured frequency response function Hr(z), as given below:
R=
q
∑
i=0
biz−i 1 +
p
∑
i=1
aiz−i
− Hr(z) (2.18)
Multiplying R with the denominator polynomial gives the modified polynomial R0:
R0=
q i=0
∑
biz−i− Hr(z)
! 1 +
p i=1
∑
aiz−i
!
(2.19) This modified polynomial R0 enables to get a linear problem that can be solved by a computer. The program calculates the coefficient of the denominator (ai such as 1 ≤ i ≤ p ) of a single transfer function using Shanks transformation [16]. These coefficients are theoretically the same for all transfer functions and give access to the poles of the system. Note that if the transfer function used for the calculations of poles is situated on a node line, some modes could be missed. The program calculates the poles for an increasing p (number of poles). Since the physical poles must be stable independently of p, it is then possible to eliminate the non-physical poles of the post processing analysis. Note that the mathematical modes are needed for the curve fitting in order to correct the influence of out-of-band poles. The mode indicator function MIF is also a good indicator to detect modes:
MIF(ω) =
35
∑
i=1
Im(Hi(ω))
35
∑
i=1
Hi(ω)2
(2.20)
There are several ways to calculate a mode indicator function but in this case, a low 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 a linear system of 14 equations. At this stage of the modal extraction H(z) is entirely determined. It is then possible to plot H(z) and Hr(z) in order to check if the curves fit closely. The denominator enables to calculate the mode shape vectors. In order to check the accuracy of the results the orthogonality of the mode shape vectors can also be checked. The results of the modal parameter extraction are given in the Chapter on Measurements.
Design of the Structure
3.1 Concept design of the CubeSat
The standard 0.1 × 0.1 × 0.1 m3basic CubeSat is often called a “1U” CubeSat mean- ing one unit. CubeSats are scalable in 1U increments and larger. CubeSats such as a 2U 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 both built 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 be launched and deployed using a common deployment system. CubeSats are typically launched 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 received from 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 two 1U CubeSats are the same size as one 2U CubeSat, the P-POD can deploy 1U, 2U, or 3U CubeSats in any combination up to a maximum volume of 3U.
Figure3.1illustrates the overall view of the structure under analysis in the thesis.
The structure was built at KTH Mechanics Department. This structure resembles the actual version of the CubeSat with deployable boom which will be launched into low earth 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 mass into orbit, it is a lightweight structure without any ferrous materials to avoid magnetic interferences. Basically, the satellite is a rectangular parallelepiped of dimensions 300
× 100 × 100 mm3. It contains, electronics, batteries, a sensor called WINCS, and the boom folded on itself and located in the boom room as can be seen in Fig. 3.1.
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 avoid risks of failure, it is not a motor that is used to deploy the boom but composite tape springs storing strain energy that is released during the deployment. The boom can be seen in Fig 3.1. As shown in the picture, there are 2 × 2 composite tape springs. The two first ones, going from the satellite to the middle part called the hub and the two second ones going from the hub to the SMILE sensor plate. The total length of the boom is one meter. The composite tape springs have the particularity of being curved in the transversal plane. Regarding their physical properties, their axial stiffness is less critical than their rotational stiffness and hence the fundamental mode might be a torsion mode. They are also very light in comparison with the remaining of the satellite (the mass of the whole boom is estimated at 150 g whereas the mass of the satellite with all its equipment is estimated at 3.4 kg). Nevertheless, one needs to point out that the dummy used for the experiment is lighter than the real satellite since some equipments are missing. Its mass was measured to 2.3 kg. Consequently, it is clear that accelerometers cannot be used to take measurements on the boom which are described in the next chapter since the accelerometers have no capibility to capture low frequencies. 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 be used on the structure itself since it is stiff and heavy enough. The next step is to find a way to simulate the absence of gravity in which the satellite will operate.
3.2 Structure
The satellite dummy with its dimensions and adjacent satellite systems mock-ups is shown 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 additional led 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 deployable carbon fibre reinforced plastic tape springs. Two tape springs were connected to a dummy satellite in one end and to the deployment drum (hub, see Fig. 3.3) in the other end. The other two tape springs were connected to the deployment drum in one end and to the end piece, called tip, that is meant to contain the measurement electronics in the 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 Materials
Data for all structural parts and materials are shown in Table3.1.
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 Quantity
Satellite Dummy Aluminium Alloy 6061 0.25x0.1x0.1, t=0.003 1
Attachment Plate PVC 0.04x0.04x0.01 1
Hub PVC 0.04x0.04x0.04 1
Tip plate PVC 0.04x0.04x0.01 1
Tape 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 the specimen is balanced by the other parts is called a Marionette paradigm. In such sus-
Figure 3.4: Tip plate with electronics [courtesy of Pau Mallol].
pension device which is described in [9] by Greschik mass overhead, contributed to by light fly beams and suspension cords only, is low, often negligible. Damping, friction, and slip are eliminated, and specimen response is not affected by deleterious material stiffness. Deleterious stiffness effects can be eliminated with a precision limited only by the accuracy of geometric measurement. The architecture to achieve these qualities is simple, with few limitations on its overall design and with high tolerance against both specimen and support system imperfections. Kinematics can naturally involve up to moderate specimen displacements and deformations in both the vertical and hori- zontal directions. The concept can also be generalized to accommodate some adaptive model geometries, and the simulation of inertial loading conditions in weightlessness, for example, steady state acceleration, is also possible. This unique combination of high performance, fault tolerance, mechanical simplicity, and design flexibility are an attractive alternative to classic gravity compensation schemes.
To get relevant results it is necessary to model the environment in which the system is going to operate as close to reality as possible. In the case of a satellite, the absence of 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 artificial weightlessness cannot be easily recreated in a laboratory. Thus, the only alternative is to hang the satellite to the ceiling but this must be done in such a way that the hanging system 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]). The basic idea of this method is that each part of the studied structure should be balanced
Figure 3.5: KTH boom prototype in its (a) stowed configuration and (b) deployed configuration [courtesy of Pau Mallol].
by the other parts. This should be carried out using a set of fly beams integrated into a single hierarchy, suspended from one external location (in this case the ceiling of the laboratory). Thus, since there is a single external support, the structure is not subject to any constraint except that its center of gravity is maintained stationary in space. This condition is equivalent to weightlessness with a fidelity defined by the geometry of support 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
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 the suspending 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 an internal diameter of 12.5 mm, an external diameter of 15 mm, a length of 0.85 m and a 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 shown in Fig. 3.7.
Figure 3.7: Marionette system (left picture) and the satellite structure hanging at three points (pictures to the right).
Measurements
4.1 Introduction
The objective of the measurements is to obtain information about eigenfrequencies, mode shapes and damping of the structure. Processing and analysing measurement data is time-consuming though, and therefore only the most important data processing were made. Eigenfrequencies were evaluated for all axis, but not for all measurements and 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. Mode shapes 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 follow the procedure and results of measurements in each of orientations mentioned above is presented. The measurements were carried out at KTH in a course of Experimental Dynamics by students J. Kristoffersson, M. Larsson, P. Banach, J. Brondex, L. Gedim- inas, C. Frangoudis, C. Kastby, M. Zapka and myself.
4.2 Instrumentation
Experimental modal analysis aims at finding the modal parameters of the structure by measuring 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 provided the excitation and keeping a single excitation point and measure the response of sev- eral measurement points. As mentioned previously, a laser vibrometer was used to perform 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. In addition to the optical measurement points, three accelerometers were used to measure the response of the satellite dummy structure. The choice of a shaker as the excitation source was obvious: since the investigated frequency range is not wide (from 1 to about
100 Hz), it is a simple, efficient and fast solution to excite the structure. One drawback is 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 experimental set-up that was implemented except that the structure was the satellite dummy and a laser 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].
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, Hfg data 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 of Manufact. 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 procedure
To be able to extract the modal parameters of the structure, it is necessary to measure at least one row or one column of the response matrix. Since there is a single point of excitation 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 acceleration whereas the laser measure velocities. For simplicity, it was decided to convert veloc- ities into accelerations in order to get the accelerance functions rather than mobility functions. Regarding the positioning of the shaker, it had to be done with great care to get a good excitation and to ensure a stiff connection between the force transducer and the excitation point. To avoid problems of symmetry, the force transducer was mounted on a bottom corner of the face of the dummy structure corresponding to the boom-room (see Fig. 4.3).
As can be seen in the picture, a push-rod is mounted between the shaker and the force transducer to reduce undesired moment excitation. The transducer is cemented
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 on the 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. 4.5 shows 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 100 Hz. The force transducer and the accelerometers give the time history of the force and the accelerations respectively for three points of the structure. Then each sticker is successively targeted with the laser to get the time history of the velocity of the measured point. Thus, 11 measurements are required (as many as there are stickers).
Eventually, the measurement devices have to be calibrated [15].
Figure 4.4: Positioning of the accelerometers.
Figure 4.5: Measurement points on the lateral faces (left picture) and on the hub lower face (the picture to the right).
4.4 Signal processing
First of all, each measured signal is amplified to get a high signal to noise ratio. This must 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 the analog signal into a digital signal. Before that, it also contains a low pass filter denoted anti-aliasing filterwhich removes all frequency components above 0.5 fs. Finally the FFT analyser estimates the Fast Fourier Transform of the signal. Since the signal is not periodic in the time window, an operation of windowing must be performed to avoid leakage errors. Therefore, a Hanning windowis used to force the signals to 0 at the beginning and at the end of the time record. Finally some averaging is performed to
reduce the influence of the noise. To enable the investigation of the structure on the frequency range between 0 and 100 Hz, some choice regarding the parameters of the signal 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 = 1
Ts (4.1)
we obtain a frequency resolution of ∆ f = 0.03 Hz:
4.5 Calibration
Accelerances are acceleration quantities normalised with the exciting force. Thus, only the 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 well defined reference object. If this reference object is submitted to an excitation force F, Newton0s 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 lowest elastic 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 to calibrate the ratio between force and acceleration by imposing the apparent mass equal to the static mass. Fig. 4.6shows the calibration procedure for the laser. As it can be seen on the picture, it consists in exciting one side of the cylinder and measuring the velocity (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 the free side of the cylinder, though these can be made at the same time as it is assumed that the cylinder is moving along the direction of its axis only.
4.6 Modal parameter extraction
As mentioned previously, at the end of the experiment each measurement device must be calibrated in order to get the calibration factors. The laser gives an apparent mass of 0.0061 kg but it is known that the actual mass of the cylinder is 2.425 kg the calibration factor can be calculated as:
γ = µ
m (4.3)
Figure 4.6: Calibration of the laser.
Hence, the calibration factor of the laser is about 0.0025. In practise, this means that the FRF measured via the laser must be divided by 0.0025. The same operations was carried 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 accelerometers has to be divided by the corresponding calibration factor.
The first thing that must be said is that the analysed structure was complex and hence the obtained results are not highly accurate. Indeed, mainly because of the lightness of the structure, the meshing of measurement points was not very dense and therefore some complex modes are not perfectly described. On the other hand, the frequency range under investigation was narrow and certain modes were separated by less than 1 Hz so there are problems of overlapping modes. Nevertheless, the complex exponential 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 in Fig. 4.7.
Figure 4.7: Mode Indicator Function [courtesy of P. Banach].
As it can be seen on the MIF, some eigenfrequencies are obvious whereas some others 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 rather good 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 and4.9. Note also that above 3 Hz the results are getting worse as the frequency increases.
As a consequence of this lack of precision, the extracted modal parameters are not highly reliable. The results are detailed in Table 4.1. The table clearly illustrates the aforementioned problem with modes that are very close to one another. Thus, some of 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 with modes 11, 12, 13 and 14. This impression tends to be confirmed by the mode shape animations which shows that the deformations are very similar for the modes quoted above. However, this could also be due to the fact that the meshing is not dense enough to give a highly detailed representation of the deformations. The general conclusion that should be drawn is that those results should be exploited with a great care and
Figure 4.8: Curve-fitting for the point 7 [courtesy of P. Banach].
critical judgement is required. Fig. 4.10shows the animation of the interesting modes.
For the reason explained above the modes 8, 10, 12, 13 and 14 were skipped. From this figure it can be seen that all modes seem to have a physical meaning. On the other hand, the lack of measurement points to get accurate deformations is proven by the fact that the last torsion modes are very similar.
4.7 Results
In this section the results from the measurements are presented. In the discussion chapter explains these results further together with a presentation of the limitations of the measurement set-up.
4.7.1 Results of the measurements for the free free vibration test 1
The obtained by experimental method eigenfrequencies later will be compared with the modes obtained by numerical FEM calculations for the boom structure. Table4.1 and Fig4.10show the results for the first free–free vibration test.
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 2
Table 4.2 below lists the modes identified together with the damping ratios and Fig 4.16depicts them.
4.7.3 Results of the measurements for cantilever down testing
Table4.3and Fig4.15show the results of the cantilever down testing.
Table 4.1: Results of the measurements for the free free vibration test 1 Mode number Frequency [Hz] ζ[%] comment
Mode 1 1.77 2.08 bending and torsion
Mode 2 2.87 0.62 bending and torsion
Mode 3 3.41 1.41 bending
Mode 4 4.98 0.1 bending and torsion
Mode 5 6.14 1.86 bending and torsion
Mode 6 7.04 0.2 mainly torsion
Mode 7 8.33 0.18 bending and torsion
Mode 8 8.93 0.04 bending and torsion
Mode 9 9.73 0.35 bending and torsion
Mode 10 10.00 0.23 bending and torsion
Mode 11 14.46 2.58 torsion
Mode 12 14.78 1.18 weak torsion
Mode 13 15.22 1.19 weak torsion
Mode 14 15.74 0.55 weak torsion
Mode 15 16.41 1.98 torsion
Mode 16 17.23 2.78 torsion
Mode 17 18.13 1.52 weak torsion
The coherence for the accelerometers and the laser can be seen in Figs. 7 and 8 respectively.
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 Figs4.10,4.15 and4.16. The coherence has been determined for one measurement per measurement point and then plotted together, see Fig. 4.14
Table 4.2: Results of the measurements for the free free vibration test 2 Mode Frequency [Hz] Damping Type of mode
1 1.6 2.9 Rigid
2 2.1 2.3 Rigid
3 3.4 1.5 Rigid
4 4.6 0.8 Elastic
5 5.6 0.4 Elastic
6 7.5 2.1 Elastic
7 7.9 0.5 Elastic
8 8.9 0.3 Elastic
9 11.6 0.5 Elastic
Table 4.3: Results of the measurements for cantilever down testing Mode Frequency [Hz] Damping Type of mode
1 2.4 0.72 bending
2 4.0 2.7 bending and torsion
3 4.5 2.8 bending
4 5.0 0.2 torsion
5 6.0 2.4 bending
6 7.1 2.9 torsion
7 8.5 1.3 longitudinal
Figure 4.10: Mode animations [courtesy of P. Banach].
Figure 4.11: Laser measurement point mobilities when the dummy and boom are turned 0 degree angle [courtesy of J. Kristoffersson & M. Larsson].
Figure 4.12: Laser measurement point mobilities when the dummy and boom are turned 90 degrees angle [courtesy of J. Kristoffersson & M. Larsson].
Figure 4.13: Coherence for the laser measurement points [courtesy of J. Kristoffersson
& M. Larsson].
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].
Figure 4.16: Shapes for modes from 1 to 9 Hz [courtesy of C. Frangoudis, C. Kastby, M. Zapka].
Finite Element Modal Analysis of the Boom Integrated to a CubeSat
5.1 Chapter overview
In this chapter the finite element model is presented. The finite element model of the satellite 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 of the gravity off-loading system is justified. The basic idea behind introduction of the gravity off-loading system is to simulate the zero gravity condition while also trying to eliminate any influence of the gravity off loading system on the dynamic analysis of the boom integrated to the CubeSat. The eigen frequencies of the gravity off loading system used should be at least 10 times lower than the expected eigen frequency of the boom [9].
In order to identify eigenfrequencies and reference mode shapes of any structure the modal analysis is used. The modal analysis is linear and may account for damping effects, but ignores a plastic deformation, creep or contact stiffness. In case of dynamic loading and analysis of vibrations and transient responses in structures it is crucial to identify 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 operations are 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:
• 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 the model,
8. specify the parameters for modal frequencies identification.
5.2 Geometry of the structure
In order to carry out a numerical eigenfrequency analysis of the structure in this thesis the software Abaqus/CAE was used. In order to quickly prepare the geometry CAD software NX 6.0 was used and later this geometry was imported to the Abaqus/CAE graphics 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 gravity off-loading system
A simplified model of the CubeSat dummy with the inbuilt boom is shown in Figure 5.2. The geometry was created using the software NX Unigraphix and then imported to the Abaqus software. The cross section view of the tape spring connections to the hub 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 the Tie constraints need to be specified for the surfaces of parts which touch each other.
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 model
The material properties are specified for each component. The material data can be summarized 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 1780
kg/m3
3. Nylon Strings - Modulus of elasticity 5 GPa, Poisson’s ratio 0.3, density 800 kg/m3
4. Aluminium Bars - Modulus of elasticity 69 GPa, Poisson’s ratio 0.3, density 2850 kg/m3
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, density 9400 kg/m3. This density is given a number 9400 kg/m3 in order to allow for the masses that are not included in the satellite dummy, the volume of the satellite dummy was computed and known mass of 2.5 kg was used to compute equal density)
6. Springs - Modulus of elasticity 210 GPa, Poisson’s ratio 0.3, density 7800 kg/m3
5.4 Analysis
The linear eigenfrequency analysis was performed on the structures using Abaqus CAE. Lanczos algorithm was used. The eigenfrequencies for the first 10 modes were extracted. The model was not constrained during the analysis and the first 6 rigid body motion modes which give 0 Hz frequencies are omitted in the following text.
5.5 Eigenfrequencies from the model
After running the eigenfrequency analysis for four types of structures mentioned above the data represented in Table5.1were obtained, where:
• ∗ - modes are given only for the boom with satellite dummy, the first 6 free body motion modes are omitted together with not important gravity off-loading system modes
• I - Modal analysis of free-free boom with satellite dummy
• II - Modal analysis for gravity off-loaded boom with satellite dummy without extension 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 Abaqus CAE
Mode,* Eigenfrequency, Hz
- I II III IV
1 7.73 7.79 (+1.2%) 7.78 (+1.2%) 4.42
2 10.05 11.31 (−15.17%) 11.11 (−15.18%) 7.08 3 16.78 16.10 (−4.2%) 16.05 (−4.2%) 9.24 4 21.26 21.09 (−1.11%) 21.18 (−1.09%) 13.59
5.6 Vibration modes of the free–free vibrating boom
The first four modes and corresponding shapes for the free–free vibrating boom are shown in Figs5.4to5.7
5.7 Vibration modes of the boom in the gravity off-loading system
This section presents the mode shapes for the gravity off-loaded boom. Figs 5.8 to 5.13show the first three important mode shapes of the gravity off-loaded boom.
Figure 5.4: First eigenfrequency, 7.7 Hz − (a) Front view (b) Side view (c) Top view
5.8 Sensitivity analysis
In 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 line ending on the aluminium bar and fishing line attached to the satellite dummy was also moved away from the satellite dummy centre of gravity. The list below outlines the configurations of the structure for the sensitivity analysis.
Table5.2shows the variation of the first eigenfrequency with respect this changes.
where i in Table5.2denotes the number of the eigen frequency of the boom
Figure 5.5: Second eigenfrequency, 10.0 Hz − (a) Front view (b) Side view (c) Top view
I the free free vibrating
II central position of the fishing lines
III shifted 50 mm to the boom fishing line of the satellite dummy
IV shifted 50 mm away from the boom fishing line of the satellite dummy
V shifted 150 mm away to the satellite dummy fishing line of the aluminium bar VI shifted 150 mm away from the satellite dummy fishing line of the aluminium bar
Figure 5.6: Third eigenfrequency, 16.2 Hz − (a) Front view (b) Side view (c) Top view
Table 5.2: Sensitivity analysis i frequency, Hz
I 7.73
II 7.79 (+1.2%) III 7.77 (+0.8%) IV 7.76 (+0.7%) V 7.57 (-1.6%) VI 8.18 (+9.5%)
Figure 5.7: Fourth eigenfrequency, 21.5 Hz − (a) Front view (b) Side view (c) Top view
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
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
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
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
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
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
Discussion
6.1 Mode shapes comparison and discussion
Due to the least resistance principle, the tape springs in boom as can be seen in the first mode are bending diagonally with respect to the main orthogonal axis for the cross section. The bending and slight torsion in the first mode is caused by the principal axis of the second moment of inertia for the cross-section turn ca 40 degrees compare to the case when the tape springs in the boom are levelled. In the figure below the first mode shape is shown for such boom with levelled tape-springs:
Figure 6.1: First eigenfrequency, boom with levelled tape springs , 4.2 Hz - (a) Side view, (b) - Top view
Comparison of the mode shapes between experimental and finite element analysis results shows discrepancy in two of the measurements with respect to the finite element model. As shown in the results the natural frequencies received a higher estimate as the boom structure did not contain the transition zone. The transition zone is shown in Fig. 6.2. The transition zone is weaker compare to semi-circular cross section of the 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 and learn more about the shape of vibration of the boom.
Figure 6.2: Transition zone. ρ0denotes 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 boom cross-section
Fig6.4shows a comparison of the first eigen frequency, obtained through vibration testing and first eigen frequency for the free free vibrating boom from the FEA model.
6.2 Mass participation
Taking a closer look at the results for the eigenfrequencies of the boom presented in Table 5.1 we notice that the eigenfrequencies in the second mode for the gravity off-
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. The aluminum bar above the boom’s lightest part (tip plate and the hub) is participating in the boom vibration modes. The stiffness matrix for the boom in this mode having additional mass components and this gives a higher eigenfrequency estimate. It has been discussed in paper [14] by Greschik that the gravity off-loading system should be as light as possible besides having a 10 times lower eigenfrequency. Although the descipancy in our case is significant we still can draw sensible conclusions about how the gravity off-loading system works and describe quantitatively the influence the gravity off-loading system has. The gravity off-loading system has effect on the space structure but with reasonable degree of accuracy it allows us to study the dynamic charactertics of the boom.
6.3 Summary
The mathematical FEM model developed in Abaqus is one of the possible models which 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 the obtained data we can conclude that the use of gravity off loading system is justified as the eigen shapes are comparable in the free free vibration test case with free free vibration analysis model. Despite the problems with obtaining accurate results from the experimental modal analysis it is estimated that the first eigenfrequency is lower than 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 deployment tests and in finite element analysis.
