Vibration Modal Analysis of a DeployableBoom Integrated to a CubeSat VALERIY SHEPENKOV Degree project in Mechanics Second cycle Stockholm, Sweden 2013 Vibration Modal Analysis of a Deployable 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 Dedicated to my mother - Liubov Esaulova. It was my pleasure to work together with Julien Servais and Pau Mallol on the satellite project at Mechanics Department. 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. Gunnar Tibert who provided me with a valuable leadership with which I was able to complete the project. . and for being a great coordinator in Engineering Mechanics MSc program. I would like to thank Gunnar Tibert for giving me the opportunity to work in the CubeSat project at KTH. I would like to thank my friends who made my life full of interesting events during the two years in Stockholm. Thank you Will for teaching me LaTeX and MATLAB programming. I would like to thank my thesis advisor Dr. I am thankful to Will Reid who has been a great project manager and a great colleague in the RAIN project which I participated in at KTH during 2010 – 2012.Acknowledgements I am truly thankful to my parents and relatives for their compassion and support during the two years of my studies at KTH. a great challenge of small satellites lies in achieving technical and scientific requirements during the design stage.Abstract CubeSat or Cubic Satellite is an effective method to study the space around the Earth thanks to its low cost. In the present work primary focus is given to dynamic characterization of the deployable tapespring boom in order to verify and study the boom deployment dynamic effects on the satellite. However. easy maintenance and short lead time. The gravity offloading 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. . The deployed boom dynamic characteristics were studied through simulations and experimental testing. Ett tyngdkraftskompenserande system har använts för att simulera tyngdlöshet i de experimentella testerna och simuleringar visar att utformningen av detta system påverkar resultaten olika beroende på svängingsmodens form. Detta arbete har analyserat de dynamiska egenskaperna hos en utfällbar band-fjäder bom i syfte att verifera och för att studera bommens utfällningsdynamiska effekter på satellitens bana och attityd. En stor utmaningen i utformningen av små satelliter är att uppnå de tekniska och vetenskapliga kraven. enkla underhåll och korta ledtid. . Den utfällda bommens dynamiska egenskaper har studerats genom simuleringar och experimentella tester.Sammanfattning En CubeSat eller kubisk satellit är effektivt för att studera rymden runt jorden på grund av dess låga kostnad. . . . . . . . . . . . . . 3. . . . . . . . . . . . . .3 Materials . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . 2. . . . . . . . . . . . 2. . . . . . . . . . . . . . . . 4. . . . 2. . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . 4. . . . 2 Theory of Structural Dynamics 2. . . . . . .6 Modal parameter extraction 22 22 22 24 26 27 27 . . . . . . . .1 Concept design of the CubeSat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . .Contents Contents v List of Figures vii 1 Introduction 1. . .7 Modal parameter extraction: complex exponential method 3 4 . . . . . . . . . . . . . . . . . . . .5 Damping . . . . . . . . . . . . . . . . . 14 14 16 16 Measurements 4. . . . . . 4. . . . . . . . . .2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . 2. . . .6 Root–mean–square value . . . . . . . . . . . . . . . . .1 Background . . . . . . . . . . . . . 1. .2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Calibration . . 3. .2 Equation of motion and natural frequency . . . . . . . . . . .1 Introduction .3 Measurement procedure . . . . . . . . . . . . . . 2. . 1 1 2 5 5 7 7 8 9 12 12 Design of the Structure 3. . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . .4 Gravity off-loading systems as a mean to obtain dynamic characteristics of structures . .2 Structure . . . . . . . . . . . . . . . . . . .4 Steady-state vibrations . . . . . .4 Signal processing . 4 Recommendations . . . . 7. . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . .7. . . .2 Research conclusions . . 5.7 . . Finite Element Modal Analysis of the Boom Integrated to a CubeSat 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Chapter overview . . . . . . . . . Mode shapes extracted using the results of the vibration testing . . . . . . . . .1 Results of the measurements for the free free vibration test 1 4. . .4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . .1 Chapter overview . . . . . . 38 38 39 40 41 41 42 42 43 6 Discussion 6. . . . . . . 6. .8 5 Results . . . . . 5. . . . . . . . . . . . . . . . . . 53 53 54 55 7 Conclusions and Recommendations 7. . . . .8 Sensitivity analysis . . . . . . . . .5 Eigenfrequencies from the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Mass participation . . . . . . . . . . . . . . . . . . . . .1 Mode shapes comparison and discussion . .7. . 5. . . . . . . . . . . . . . . . . . . 4. . . .7 Vibration modes of the boom in the gravity off-loading system . . . . . .6 Vibration modes of the free–free vibrating boom . . . . . . .3 Summary . . . . . . . 6. . . 56 56 56 57 58 4.7. . . . .3 Material properties for components of the model . . . . . . . . . . . . . . . . 30 30 31 31 32 . . . . .2 Geometry of the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Results of the measurements for the free free vibration test 2 4. . . . 5. . . . 5. . . . . . . . .CONTENTS 4. .3 Lessons learned . . . . . . 59 vi . . . References . . . . . . . . .3 Results of the measurements for cantilever down testing . . 7. . . . . . 5. . . . . . Typical measurements set–up. . Positioning of the shaker in the in-boom-plane vibration test. . . . . . . . . . . . Exploded hub view of the SIMPLE boom in initial stages of deployment. . . . .4 2. . . . . . . . . . . . . . . . . . . . . . . . . Mechanical model for a simple SDOF system and its free body diagram. . . . . . . . . . . . . . KTH boom prototype in its (a) stowed configuration and (b) deployed configuration [courtesy of Pau Mallol]. . . . . . . . .3 2. . . . . . . . . .1 2. . . . . . . . . . . . .List of Figures 1.a satellite dummy with a deployed boom [courtesy of Pau Mallol]. . . . . . . . . .3 3. . . . . . . . . . . . . . . . . . . . . . Tip plate with electronics [courtesy of Pau Mallol]. . . . . . . . . . . . . . . . . . . . . . . . .2 2. . . . . . .5 3. . . . . . . . . . . . . . . . . Damping ratio versus frequency for Rayleigh damping. . . . . . . . . Marionette system (left picture) and the satellite structure hanging at three points (pictures to the right). . . . . . . .6 3. . . . . . . . . . . . . . . .1 4. . . . . . . . . . . Satellite dummy [courtesy of Pau Mallol]. . Kristoffersson & M. . . . . . . . . . . . . . . . . . . . . . . . . 2 A signal in time domain. . . .4 SwissCube-1 . . . . . . Positioning of the accelerometers. . . . . . . . .1 1. . . 6 6 Overview of the structure . Scheme of the experimental set-up. . .4 3. . . . . . .3 2. . . . . .2 3. . . . . . . . . . . . . . Hub [courtesy of Pau Mallol]. vii 3 4 7 10 11 12 15 16 17 18 19 20 21 23 23 25 26 . . . . . . . . . . . . . . . . . . . . . SIMPLE boom shown in the (a) stowed configuration. A signal in frequency domain. . . . . . . . . . . . . . . . . . . .2 1. . . . . .2 4. . . . . . .6 3. . . . .1 3. . . . . . . . . . . A damped free vibration. . . . . . . (b) partially deployed and (b) fully deployed states . . . . . . . . .7 4. . . . Larsson]. . . . . . . . .3 4. . . Scheme of the experimental set-up for the boom vertical orientation vibration test [courtesy of J. . . .5 2. . . . . . . . . . . Dynamic magnification factor (DMF) versus frequency ratio for various levels of damping . . . . . . . M. . .15 4. . 5. . . . . . . . . . . . . . . viii 26 28 29 30 31 34 34 35 35 36 36 37 40 40 41 43 44 45 46 47 48 49 50 51 . . . . . . . . . . . . . . . . . . . . . . Banach]. .8 First eigen mode for the gravity off-loaded boom prototype. . . . . . . . . The coherence for the accelerometers calculated from one measurement [courtesy of J. . . . .12 4. Coherence for the laser measurement points [courtesy of J.11 Second eigen mode for the gravity off-loaded boom prototype. . . . . . . . . . . . . . . . . .0 Hz − (a) Front view (b) Side view (c) Top view . . . . . . Shapes for modes from 1 to 9 Hz [courtesy of C. . . . 5. . .2 Hz − (a) Front view (b) Side view (c) Top view . . . . .3 Hz − Overall side view of the gravity off-loading system . . . . . Shapes for modes from 2 to 7 Hz [courtesy of C. . C. . . . . . . . . . . . .8 Hz − Overall side view of the gravity off-loading system .2 5. . . . . . . . . .1 Hz − (a) Front view (b) Side view (c) Top view . . Curve-fitting for the point 7 [courtesy of P. . . .5 4. . . . . . . .10 Second eigen mode for the gravity off-loaded boom prototype. . . .8 4. . 5. . . . . . . . . . . . .6 4. . . . Zapka]. . . . . . . . . Kristoffersson & M. . . . . . 21. . . . . . . . Banach]. . .7 Hz − (a) Front view (b) Side view (c) Top view Second eigenfrequency. . Frangoudis. . . . . . . . . . . . . . . . . . . .8 Hz − (a) Front view (b) Side view (c) Top view . 16. . . . Curve-fitting for the point 11 [courtesy of P. .12 Third eigen mode for the gravity off-loaded boom prototype. . . . . . . Kastby. . Laser measurement point mobilities when the dummy and boom are turned 90 degrees angle [courtesy of J. 5. . . Mode animations [courtesy of P. Larsson]. . Kristoffersson & M. . . . . . . . . . . . . . . . .10 4. . . .13 4. . . . . . . . . Satellite dummy with the boom. . . . C.9 First eigen mode for the gravity off-loaded boom prototype. . . . . . . . . . . . 5. . . . . . Calibration of the laser. . . . . . . . . . .1 5. 5. . Larsson]. . . . .6 Third eigenfrequency. .16 Measurement points on the lateral faces (left picture) and on the hub lower face (the picture to the right). . . M. . . . .14 4. . 11. 11. . . . .LIST OF FIGURES 4. . . . . .11 4. . Meshed geometry of the boom with gravity off-loading strings. . . . . . Larsson]. . 10. . . . . . . . . .5 Cross section view of the tape springs positioning. Kastby. . . . . . . . . . . . . Mode Indicator Function [courtesy of P. . . . . . . . . . . .3 5. 5. . 7. . . . . Larsson]. . . . . . . . . . . . . . Kristoffersson & M. . . . . .7 Fourth eigenfrequency. . . . . 7. . . . . . Zapka]. . . . . Frangoudis. . . Banach]. . . . 7. . Banach]. . . . . . .3 Hz − (a) Front view (b) Side view (c) Top view . . . .9 4. . . . . .7 4. . . 5. 16. . . . First eigenfrequency.5 Hz − (a) Front view (b) Side view (c) Top view . . . .4 5. . Kristoffersson & M. . . . Laser measurement point mobilities when the dummy and boom are turned 0 degree angle [courtesy of J. . . boom with levelled tape springs . .(a) Side view. .4 First eigenfrequency. 4. . . . . .2 6. . 16. . . . .2 Hz . . . . ρ0 denotes a region of a transition zone in a tape spring.8 degrees for the principal axis of inertia of the boom cross-section . The rotation 39. . . . . . . 1. .13 Third eigen mode for the gravity off-loaded boom prototype.LIST OF FIGURES 5. . . Transition zone. . . . . . .  . . . . . . . . Comparison of the shapes for the first mode in the FEA analysis. . . .Top view .76 Hz (b) . 7. . . . . . .3 6. . . . . . . . . (b) . . . . . . . . . .1 Hz − Overall side view of the gravity off-loading system . . . . . ix 52 53 54 54 55 .7 Hz (a) and the vibration testing. . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . .1 6. . deployable booms based on bi-stable composite tape springs are proposed for CubeSat deployable antennas .e. 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  as well as at various industrial research organizations. The project at KTH includes the design. Russia. The tape-springs are made of glass fibre reinforced epoxy with an embedded copper alloy conductor. For instance. was a cost effective solution to study the nightglow within the Earth’s atmosphere. Norwegian University of Science and Technology and the University of Tokyo. The first successful launches of CubeSats began in the mid-2003 from Plesetsk. 1. the satellites are predominantly being developed by Stanford University. For instance.1). University of Florida and the Inter–American University of Puerto Rico.Chapter 1 Introduction 1. Deployable booms are used in today’s CubeSat missions.1 Background CubeSats are standardized complex systems used for space research by universities and have special properties such as a volume of 10 cm3 and a mass not exceeding 1.33 kg . Swiss satellite. Boeing . i. Several types of CubeSats exist: 0. 2U. Bi-stability properties enable the antenna to be elastically stable in both deployed and stowed configurations .5U. These remarkable size and mass make these satellites appropriate for cost-effective space exploration in the low Earth orbits. California PolyTechnic University. 3U. KTH. Since then. Dynamic characterization of the deployable structures during their deployment and in their deployed state are of primary importance because of how those factors influ- 1 . 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. development and verification of the SIMPLE deployable boom prototype . 1U. 5U and 6U. the CubeSat SwissCube-1 (Fig. Introduction Figure 1. The analysis will be performed both on the experimental structure and on a finite element model of the structure.1: SwissCube-1 . The proposed boom design can be seen in Figure 1. The deployable structure using bi-stability properties of the tape-springs is presented in  and depicted in Fig. The vibrations of the boom structure are recorded using accelerometers and laser 2 .2. ence the behaviour of the satellite. The project focus is on the dynamic characterization of a SIMPLE boom prototype and on numerical analysing the means of the boom vibration testing. In many cases. .1. 1. 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. 1.3.2 Objectives The objective of this research is to analyse the dynamic characteristics of the boom deployed from the CubeSat. the deployment mechanisms involve large kinematic. however the angular and linear momentum will not change. unconstrained rotations when deployed. and Inter–American University of Puerto Rico was divided into three primary focus areas. and University of Florida. The development of the Deployable Boom for the CubeSat project at KTH in collaboration with AFRL. (b) partially deployed and (b) fully deployed states .2: SIMPLE boom shown in the (a) stowed configuration. The finite element model of the boom is created and post-processed in a software application for finite element analysis called Abaqus CAE. whilst the actual calculations are carried out in Abaqus Standard. The primary focus of the KTH research and development was to study: • dynamic characterization of the gravity off-loading system for the boom vibration testing. vibrometer and the obtained experimental data is processed and analysed with Matlab. Introduction (a) Stowed configuration (b) Partially Deployed Configuration (c) Deployed State Figure 1. • boom deployment dynamics. 3 . The prerequisites for the posterior design improvements of the boom are investigated in this thesis. • dynamic characterization of the deployed boom.1. USA. 4 .1.3: Exploded hub view of the SIMPLE boom in initial stages of deployment. Introduction Figure 1. . loads that alternate in time. The dynamic loads pose new problems compared to the static case. the magnitude of the load might not be as pivotal as the frequency it is alternating with. i.. 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. Besides the assumptions about physical laws one have to decide whether to view the system as continuous or discrete. the response can be substantially larger than for a load with the same magnitude but different frequency. In this case.1 Introduction Vibrations occur in structures due to the dynamic loads.2 During analysis of a mechanical system there is always a matter of how scrupulous one shall be. When analysing a structure that is subjected to a static load.e. how much one can idealize the system and still simulate its behaviour. To convert a signal between the two domains a Fourier analysis algorithm such as Fast Fourier Transform can be used. For a load with an excitation frequency close to a so called natural frequency (or eigen frequency) of the structure. The situation is not as obvious when it comes to dynamic loads acting on the structure. “Resonance is an operating condition where an excitation frequency is near a natural frequency of a machine structure. Vibrations can be described as the oscillations around the equilibrium of a structure. a larger force will in general cause larger displacements and a stiffer construction is therefore needed. An example of a signal in the time domain and its response spectrum is shown in Figures 2.1 and 2. It is practically impossible to analyse a complex 5 . which is the position the structure comes back to when no external forces are acting on it. A natural frequency is the frequency at which a structure will vibrate if deflected and then let go” .Chapter 2 Theory of Structural Dynamics 2. The last alternative can be useful when analysing the dynamic behaviour of a structure and is called the response spectra. structure as continuous and therefore it is transferred into a number of discrete counter parts having number of DOFs (Degrees Of Freedom). . For references on this chapter and more detailed description. A DOF represents either a displacement or a rotation and together they represent the behaviour of the discrete system.2. 6 . Figure 2.2: A signal in frequency domain. see e.1: A signal in time domain. Theory of Structural Dynamics Figure 2.g. If there are n degrees of freedom in the model. Eq.3 Modal analysis A structure has an infinite number of eigenfrequencies as it has an infinite number of DOFs if it is not constrained. The mass moves with negligible friction in the horizontal direction and the degree of freedom is the displacement x from the equilibrium. Figure 2. These are the frequencies the structure will oscillate 7 . .3: Mechanical model for a simple SDOF system and its free body diagram. For a system having small displacements (neglecting any non-linear behaviour). A time-dependent load F(t) excites the mass.2) m 2 + c + kx = F(t) dt dt The displacement x(t) is given by solving the equation of motion.3) dt dt In such a case M is the mass matrix. It consists of a mass m connected to the wall with a damper and a spring (both having mass m = 0).3. 2.1) results in the equation of motion for a SDOF model: dx d2x (2. An example of the system is illustrated in Figure 2. the matrices will be of size n × n and the vectors n × 1. which represents the system by only one degree of freedom. This results in a multi-degree-of-freedom model (MDOF).3) d2x dx M 2 +C +Kx= F(t) (2. (2. (2. In case of a complex structure it is necessary to include more degrees of freedom to describe the dynamic behaviour.2. Analysing the free body diagram in Figure 2. x is the displacement vector and F(t) the load vector.2 Equation of motion and natural frequency The most straightforward method of modelling a dynamic system is to use a singledegree-of-freedom model (SDOF).1) dt dt Reorganizing the terms in Eq. (2.1) d2x dx F(t) − c − ku = m 2 (2.3 and using Newton’s second law of motion leads to Eq. the equation of motion for a SDOF system can be converted to the multi– dimensional case. C is the damping matrix and K is the stiffness matrix. Theory of Structural Dynamics 2. Theory of Structural Dynamics at when it is allowed to vibrate without the influence of any external loads. These are the angular eigen frequencies and mode shapes of the structure. . Studying a structure in free vibration (with no external forces acting on the structure) and assuming that no damping is introduced. ω2 . ωn (2. Each eigenfrequency has a matching mode shape.5) Differentiation of x(t) and insertion into Eq. (2. The mode vectors form an orthogonal basis and the solution can be rearranged as a sum using all eigenvalues and eigenvectors: n x(t) = ∑ qi (t)Φi . a harmonic solution is assumed: x(t) = A cos(ωt)Φ + B sin(ωt)Φ (2.8) .5) that satisfies the differential equation. Together. Each eigenvalue ωi has a matching eigenvector Φi . there are n eigenvalues.7) 1 2. (K − ω2 M)Φ = 0 ⇒ det(K − ω2 M) = 0 ⇒ ω1 . the equation of motion is reduced to: d2x (2. it will after a starting transient phase oscillate with the frequency of the input load. Applying a harmonic load to the undamped system gives the following equation of motion: M d2x +Kx = F(t).4).2.. it will oscillate at the corresponding frequency.6).4) gives the eigenvalue problem in Eq. an eigenvalue and eigenvector pair form a solution to (2.. qi (t) = Ai cos(ωit) + Bi sin(ωit) (2. This is called vibrations with a steadystate characteristic. (2.6) When a structure is discretized into n degrees of freedom. dt 2 8 F(t) = F0 sin(ωt) (2.4) M 2 +Kx = 0 dt To solve Eq. (2.. If the structure is deflected into a mode shape and released from rest.4 Steady-state vibrations When a structure is subjected to a harmonic load. A and B are given by the initial conditions x(0) = x0 and the same condition for initial velocity. This results in n uncoupled equations of the form: Mi d 2 qi + Ki qi = Fi (t). ωD = ωn 1 − ζ2 (2. or internal properties of materials.12. 2. ωn = . But if the level of damping is low. This is of course not the case in reality.10) dt 2 Insertion of a harmonic ansatz solution qi (t) = q0i sin(ωt) gives the following result: q0i = P0i 1 Ki . Pi = ΦTi P(t) = P0i sin(ωt) (2.13) 9 .9) can be derived: n ∑ 1 d ΦTr MΦi 2q i (t) + dt 2 n ∑ ΦTr KΦiqi(t)=ΦTr F(t) (2.5 Damping Damping is included in mathematical models to represent the energy dissipation in structural dynamics and is always present in real structures. If F(t) = 0 in Equation 2. 2 −ζωn t x(t) = e x(0) cos(ωDt) + ( ddt 2x )t=0 + ζωn x(0) ωD ! q sin(ωDt) .ζ = (2. ω = i Mi ωi 2 − ω2 Mi (2. It can for example be friction in joints. (2. the equation of motion can be rewritten as: r dx k c d2x + 2ζωn + ωn 2 x = 0. Theory of Structural Dynamics n by using modal coordinates.9) 1 The eigenvectors are orthogonal in the scalar products ΦTr KΦi and ΦTr MΦi hence only terms with i = r are non-zero. x(t) = ∑ qi (t)Φi and multiplying with ΦTr from the 1 left Eq. there will still be a peak in the displacement amplitude spectrum at the eigenfrequencies. where any kind of damping in the structure will prevent such a phenomenon. Solution to the equation is the response for a damped free vibrating structure. Ki = ΦTi KΦi . Mi = ΦTi MΦi .12) 2 dt dt m 2mωn ζ is the so called damping ratio and ωn the eigenfrequency for the undamped case.2.11) The result shows that an excitation frequency ω equal to an eigenfrequency will give an infinite amplitude. Figure 2. There are two types of damping matrices.4 for an example of damped free vibration. like in the undamped case. For a MDOF model. On the contrary. the damping properties of materials are not as well agreed upon and also the energy dissipation in joints needs to be taken into account. Rayleigh method is a method of constructing the damping matrix. It is not as simple as assembling the stiffness matrix. See Figure 2.2). For relatively small damping ratios (ζ ≺ 0. As mentioned in the steadystate vibrations section. and it is therefore becomes possible to perform classical modal analysis of the structure.2. which delivers a classical damping matrix. classical and non-classical. The difference between them is that classical damping matrices are diagonal and non − classical damping matrices are not. the response amplitude of a harmonic load will not go to infinity if there is some kind of damping present in the structure. For example. a damping matrix needs to be assembled. Figure 2. A diagonal matrix makes it possible to separate the equation system into n uncoupled equations. Theory of Structural Dynamics p ωD is the eigenfrequency for damped vibrations and is related to ωn by a factor 1 − ζ2 . 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. The amplitude of the spike in the frequency response spectrum will depend on the damping ratio.4: A damped free vibration. which is organized by considering the stiffness properties of individual elements.5 shows the deformation response factor for frequencies around an eigenfrequency. The Rayleigh damping matrix is a linear combination of the 10 . the damping matrix may be constructed from the modal damping ratios of the structure. ωD ≈ ωn . (2.12) . If ζ is assumed to be constant in modes i and j. a0 = ζ 2ωi ω j 2 .14) Using the formula for the damping ratio stated in Eq.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. (2.16). a0 and a1 is given by expressions (2. a1 = ζ ωi + ω j ωi + ω j (2.14) . Theory of Structural Dynamics Figure 2.15) a0 and a1 can be obtained from two known damping ratios ζi and ζ j .6. it can be shown that the damping ratio of the nth mode is given by: ζn = a1 ω n a0 + 2ωn 2 (2. as shown in Eq.2.5: Dynamic magnification factor (DMF) versus frequency ratio for various levels of damping . mass matrix and the stiffness matrix. 11 . The mass matrix damps the lower frequencies and the stiffness matrix damps the higher frequencies. C =a0 M+a1 K (2. 2. Theory of Structural Dynamics Figure 2.6: Damping ratio versus frequency for Rayleigh damping, . 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. s Z 1 t0 +∆t 2 x (t)dt (2.17) xRMS = ∆t t0 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 Prony method , or the complex exponential method, was chosen. Once the measurements are done and the results are treated (velocities are converted into acceleration by multiplying by iω), 14 accelerance functions 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: q ∑ bi z−i R= i=0 − Hr (z) p 1 + ∑ ai (2.18) z−i i=1 Multiplying R with the denominator polynomial gives the modified polynomial R0 : 12 2. Theory of Structural Dynamics ! q R0 = ∑ biz−i − Hr (z) i=0 p 1 + ∑ ai z−i ! (2.19) i=1 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 . 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: 35 ∑ Im(Hi (ω)) MIF(ω) = i=1 35 (2.20) 2 ∑ Hi (ω) i=1 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. 13 Chapter 3 Design of the Structure 3.1 Concept design of the CubeSat The standard 0.1 × 0.1 × 0.1 m3 basic CubeSat is often called a “1U” CubeSat meaning 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 vehicle 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 however, 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. 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 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 therefore it requires special means of investigation. Most of the choices regarding the experimental set-up and the procedure to acquire the data are directly based on the specifications 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. 14 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. The total length of the boom is one meter. Regarding their physical properties.3 kg.1. accelerometers can be used on the structure itself since it is stiff and heavy enough. one needs to point out that the dummy used for the experiment is lighter than the real satellite since some equipments are missing. Design of the Structure Figure 3. 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.3. 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.1: Overview of the structure . When the satellite has reached the mission orbit. However. The alternative that has been chosen is to use a Laser Doppler Vibrometer (LDV). Consequently. Nevertheless. As shown in the picture. 15 . The next step is to find a way to simulate the absence of gravity in which the satellite will operate. The boom can be seen in Fig 3. Its mass was measured to 2. there are 2 × 2 composite tape springs. their axial stiffness is less critical than their rotational stiffness and hence the fundamental mode might be a torsion mode. the boom is deployed. more details will be given in a later section. 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.a satellite dummy with a deployed boom [courtesy of Pau Mallol]. To avoid risks of failure.4 kg). The composite tape springs have the particularity of being curved in the transversal plane. 3. The structure consists of a satellite arm manufactured of altogether four deployable carbon fibre reinforced plastic tape springs.3) in the other end. The other two tape springs were connected to the deployment drum in one end and to the end piece. 16 . The tip is shown in Fig.2 Structure The satellite dummy with its dimensions and adjacent satellite systems mock-ups is shown in Fig.5. 3. Figure 3. called tip.3 Materials Data for all structural parts and materials are shown in Table 3.2: Satellite dummy [courtesy of Pau Mallol]. The satellite was balanced by additional led masses to resemble the mass distribution in the actually designed satellite. The prototype of the boom in its stout and deployed configuration is shown in Fig.4. 3.1.2. see Fig. Two tape springs were connected to a dummy satellite in one end and to the deployment drum (hub. The satellite dummy during testing was comprised of a square profile 3 mm thick aluminium hollowed plates.3. Design of the Structure 3. 3. that is meant to contain the measurement electronics in the other end. 3. 04x0.04 1 0.1x0.04x0. t=0.3. In such sus- 17 .3: Hub [courtesy of Pau Mallol].1.00025 4 Gravity off-loading systems as a mean to obtain dynamic characteristics of structures For cost effective hardware verification of large space structures high fidelity and sensitivity is required of the laboratory simulation of weightlessness.04x0. t=0.04x0.1: Parts and materials used in the structure for vibration testing Part Satellite Dummy Attachment Plate Hub Tip plate Tape Springs 3.003 1 0.04x0.4 Material Aluminium Alloy 6061 PVC PVC PVC PrePreg HexPly Dimensions.25x0. Design of the Structure Figure 3.01 1 0. 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.04x0. m Quantity 0.01 1 r=0.0067. Table 3. is low. fault tolerance. friction. Design of the Structure Figure 3. and the simulation of inertial loading conditions in weightlessness. Kinematics can naturally involve up to moderate specimen displacements and deformations in both the vertical and horizontal directions. The basic idea of this method is that each part of the studied structure should be balanced 18 . pension device which is described in  by Greschik mass overhead. The concept can also be generalized to accommodate some adaptive model geometries. This unique combination of high performance. It is obvious that an artificial weightlessness cannot be easily recreated in a laboratory. the absence of gravity that occurs in orbit can not be neglected. The difficulty lies in how to simulate this absence of gravity without biasing the results. Damping. is a simple and efficient way to fulfil the previous objective ().4: Tip plate with electronics [courtesy of Pau Mallol]. for example. and specimen response is not affected by deleterious material stiffness. 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. with few limitations on its overall design and with high tolerance against both specimen and support system imperfections. Thus. The architecture to achieve these qualities is simple. steady state acceleration. The above mentioned Marionette system. contributed to by light fly beams and suspension cords only. often negligible. is also possible. In the case of a satellite. mechanical simplicity. and design flexibility are an attractive alternative to classic gravity compensation schemes. when designed properly. Deleterious stiffness effects can be eliminated with a precision limited only by the accuracy of geometric measurement. 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. and slip are eliminated.3. and if not. since there is a single external support.5: KTH boom prototype in its (a) stowed configuration and (b) deployed configuration [courtesy of Pau Mallol]. at least easy to 19 . This should be carried out using a set of fly beams integrated into a single hierarchy.3. the structure is not subject to any constraint except that its center of gravity is maintained stationary in space. Design of the Structure Figure 3. often negligible. by the other parts. This condition is equivalent to weightlessness with a fidelity defined by the geometry of support hierarchy. Thus. Moreover. the mass overhead of the system is low. suspended from one external location (in this case the ceiling of the laboratory). since the fly beams and suspension cords are very light. 3.5 mm. an external diameter of 15 mm. a length of 0. Figure 3. The upper beam is an aluminium tube with an internal diameter of 12. Design of the Structure measure precisely. 3. and slip are eliminated. A feature of this system is that damping. friction. The cords are made of fishing lines. 20 .7.55 m and a mass of 42 g. Pictures of the final set–up are shown in Fig. a length of 0.6: Scheme of the experimental set-up.85 m and a mass of 105 g. and specimen response is not affected by deleterious material stiffness from the suspending system. The lower beam is also in aluminium with a rectangular section of 10 × 3 mm2 . 7: Marionette system (left picture) and the satellite structure hanging at three points (pictures to the right). 21 .3. Design of the Structure Figure 3. three accelerometers were used to measure the response of the satellite dummy structure. For that purpose. Kristoffersson. The choice of a shaker as the excitation source was obvious: since the investigated frequency range is not wide (from 1 to about 22 . 4. and therefore only the most important data processing were made. M. Kastby. The analyses were limited to 18 first natural frequencies of the structure. C. a shaker provided the excitation and keeping a single excitation point and measure the response of several measurement points. M. Larsson. L.2 Instrumentation Experimental modal analysis aims at finding the modal parameters of the structure by measuring the receptance matrix . The measurements were carried out at KTH in a course of Experimental Dynamics by students J. Banach. transverse to the boom and vertical orientation. the structure has to be discretised in several measurement points or excitation points. C. Zapka and myself. but not for all measurements and accelerometers. Gediminas. The measurement was done at one measurement point at a time with manual movement of the laser in between measurements.Chapter 4 Measurements 4. Eigenfrequencies were evaluated for all axis. mode shapes and damping of the structure. Processing and analysing measurement data is time-consuming though.1 Introduction The objective of the measurements is to obtain information about eigenfrequencies. P. Frangoudis. Mode shapes were only evaluated for the complete boom structure. The measurements were carried out for the the boom in three orientations: in plane to the boom. J. As mentioned previously. Brondex. In this case. a laser vibrometer was used to perform the experimental modal analysis. In each section that are to follow the procedure and results of measurements in each of orientations mentioned above is presented. In addition to the optical measurement points. efficient and fast solution to excite the structure. Measurements 100 Hz).2: Typical measurements set–up. One drawback is that leakage errors are likely to occur but they can be compensated by windowing (see the “signal processing” section). it is a simple.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  Figure 4. 4. Larsson]. Below a list of the instrumentation used is presented . Kristoffersson & M. 23 .4. Fig.1: Scheme of the experimental set-up for the boom vertical orientation vibration test [courtesy of J. Figure 4. . it is necessary to measure at least one row or one column of the response matrix. Serial 904206 7. Model-No V401. Model-No OFV 3001. Regarding the positioning of the shaker. 1997 2. Data collection system Agilent 8491A and Agilent E1432A 4.3). Bruel Kjaer Power Preamplifier. To avoid problems of symmetry. Since there is a single point of excitation and several points of measurement. One needs to point out that accelerometers give data in terms of acceleration whereas the laser measure velocities. Date of Manufact. (a4) Art Nr 4507 B 005 Serial Nr 10163. Model-No OFV 303. Polytech-Laser Vibrometer. Hfg data Adr. Tripod for Laser Vibrometer 5. 22/1/97 and Forcetransducer. Piezo Electric IEPE Accelerometer. (a2) Art Nr 4507 B 005 Serial Nr 10071. Dytran 5192. Type 1405. it is a column of the matrix that is measured. 4. Hfg data June 1996 3. (a1) Art Nr 4507 B 005 Serial Nr 10069. D-76337. Ling Dynamic Systems-Shaker.4. Serial nr 469 0411. Serial Nr 1970441. a push-rod is mounted between the shaker and the force transducer to reduce undesired moment excitation. PCP-854 9. Type 2706. it was decided to convert velocities into accelerations in order to get the accelerance functions rather than mobility functions. Polytech-Vibrometer Controller. Bruel Kjaer. Serial Nr 9606016. The transducer is cemented 24 . the force transducer was mounted on a bottom corner of the face of the dummy structure corresponding to the boom-room (see Fig. 6. As can be seen in the picture. For simplicity. 4. (a5) Art Nr 4507 B 005 Serial Nr 10162. MWL UNO 8-channel Preamplifier. Serial 660120 8.3 Measurement procedure To be able to extract the modal parameters of the structure. Measurements 1. 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. Bruel Kjaer Noise Generator. (a3) Art Nr 4507 B 005 Serial Nr 10072. the measurement devices have to be calibrated . 4.4. The shaker is fed with a white noise signal in the frequency range between 0 to 100 Hz. Fig. Eventually. Then each sticker is successively targeted with the laser to get the time history of the velocity of the measured point. stickers with reflective properties were placed on the boom as follows : 1.5 shows the measurement points (except the two points of the lower face of the tip). Two accelerometers were mounted on the top face of the dummy and the third was mounted on the back face (see Fig. Finally. 3 stickers on the lateral face of the dummy Measurements are made by aiming the laser at each of these stickers. 2 stickers on the lateral face of the hub 4. 11 measurements are required (as many as there are stickers). The force transducer and the accelerometers give the time history of the force and the accelerations respectively for three points of the structure.3: Positioning of the shaker in the in-boom-plane vibration test. 2 stickers on the lateral face of the tip 2. to the excitation point to ensure a stiff coupling. Measurements Figure 4. 2 stickers on the lower face of the tip 3. 2 stickers on the lower face of the hub 5. 25 . Thus.4). 4. an operation of windowing must be performed to avoid leakage errors.4.5 fs .4: Positioning of the accelerometers. Finally some averaging is performed to 26 . Therefore. 4. each measured signal is amplified to get a high signal to noise ratio. The FFT-analyser contains an A/D converter which ensures the conversion of the analog signal into a digital signal.5: Measurement points on the lateral faces (left picture) and on the hub lower face (the picture to the right). a Hanning window is used to force the signals to 0 at the beginning and at the end of the time record. Figure 4. This must be done with great care to avoid overloading the inputs of the data acquisition system. Measurements Figure 4. it also contains a low pass filter denoted anti-aliasing filter which removes all frequency components above 0.4 Signal processing First of all. Finally the FFT analyser estimates the Fast Fourier Transform of the signal. Before that. Since the signal is not periodic in the time window. the sampling frequency was set to fs = 500 Hz. A straight cylinder with a mass of 2425 g was used as a well defined reference object.6 shows the calibration procedure for the laser.3) . the other for the acceleration) need to be calibrated. Then. 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. The laser gives an apparent mass of 0.∆ f = fs Ts (4. some choice regarding the parameters of the signal processing were made. using the relation below Ts = 1 Ns − 1 . Hence.6 Modal parameter extraction As mentioned previously. 4.425 kg the calibration factor can be calculated as: γ= µ m 27 (4. it is possible to calibrate the ratio between force and acceleration by imposing the apparent mass equal to the static mass. Fig. to get results accurate enough. Thus. 4. Newton0 s second law gives the relation: F(ω) a(ω) µ(ω) = (4. 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. the apparent mass should be equal to the known static mass. Measurements reduce the influence of the noise. the exciting force is applied in the measured DOF only.0061 kg but it is known that the actual mass of the cylinder is 2. First.03 Hz: 4. the number of samples was set to Ns = 10. at the end of the experiment each measurement device must be calibrated in order to get the calibration factors. As it can be seen on the picture. only the ratio between the pair of channels (one for the force. If this reference object is submitted to an excitation force F. For low frequencies (below say 30% of the lowest elastic eigenfrequency).1) we obtain a frequency resolution of ∆ f = 0. To enable the investigation of the structure on the frequency range between 0 and 100 Hz.4.5 Calibration Accelerances are acceleration quantities normalised with the exciting force. Therefore. Note that the same operation applies to the three accelerometers mounted on the free side of the cylinder.2) where µ(ω) is the apparent mass. 0025.6: Calibration of the laser. Measurements Figure 4. the complex exponential method explained above was used to get the modal parameters. the frequency range under investigation was narrow and certain modes were separated by less than 1 Hz so there are problems of overlapping modes. In practise. the calibration factor of the laser is about 0. On the other hand. mainly because of the lightness of the structure. 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. this means that the FRF measured via the laser must be divided by 0.4. 28 . the meshing of measurement points was not very dense and therefore some complex modes are not perfectly described. The obtained mode indicator function on the frequency range from 1. 4. Nevertheless.0025.5 to 4 Hz is plotted in Fig. Indeed. Hence.7. The first thing that must be said is that the analysed structure was complex and hence the obtained results are not highly accurate. 12. 13 and 14. Note also that above 3 Hz the results are getting worse as the frequency increases. 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. This impression tends to be confirmed by the mode shape animations which shows that the deformations are very similar for the modes quoted above.4. For example. Banach].9. some of those modes might actually be only combination of other modes. 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. As it can be seen on the MIF.8 and 4. Measurements Figure 4. The general conclusion that should be drawn is that those results should be exploited with a great care and 29 . Figs 4. respectively.1. It is the same with modes 9 and 10 and with modes 11. Thus.7: Mode Indicator Function [courtesy of P. Using this function. As a consequence of this lack of precision. some eigenfrequencies are obvious whereas some others are not very sharp. the extracted modal parameters are not highly reliable. The table clearly illustrates the aforementioned problem with modes that are very close to one another. the curve-fitting procedure was performed with more or less accuracy depending on the measurement point. The results are detailed in Table 4. For example. modes 7 and 8 are not clearly distinct. Fig. 13 and 14 were skipped. 4. 4. 12. 10. 4. 30 .10 shows the animation of the interesting modes.10 show the results for the first free–free vibration test. In the discussion chapter explains these results further together with a presentation of the limitations of the measurement set-up. On the other hand. Banach]. For the reason explained above the modes 8. From this figure it can be seen that all modes seem to have a physical meaning.1 and Fig 4. critical judgement is required. Table 4. the lack of measurement points to get accurate deformations is proven by the fact that the last torsion modes are very similar. Measurements Figure 4.8: Curve-fitting for the point 7 [courtesy of P.4.1 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.7 Results In this section the results from the measurements are presented.7. 7.16 depicts them.3 Results of the measurements for cantilever down testing Table 4. 4.9: Curve-fitting for the point 11 [courtesy of P.15 show the results of the cantilever down testing.4.3 and Fig 4.2 below lists the modes identified together with the damping ratios and Fig 4. Measurements Figure 4.2 Results of the measurements for the free free vibration test 2 Table 4. Banach].7. 31 . 4. 52 weak torsion The coherence for the accelerometers and the laser can be seen in Figs.22 1.78 torsion Mode 17 18.15 and 4.74 0.77 2.04 0.78 1.00 0.04 bending and torsion Mode 9 9.4.8 Mode shapes extracted using the results of the vibration testing The mode shapes corresponding to the eigen frequencies can be seen in Figs 4.14 32 .35 bending and torsion Mode 10 10.62 bending and torsion Mode 3 3. 4.58 torsion Mode 12 14.16.86 bending and torsion Mode 6 7. 7 and 8 respectively. 4.41 1.87 0.73 0. see Fig.93 0. The coherence has been determined for one measurement per measurement point and then plotted together.13 1.41 1.2 mainly torsion Mode 7 8.46 2.98 0.1: Results of the measurements for the free free vibration test 1 Mode number Frequency [Hz] ζ[%] comment Mode 1 1.1 bending and torsion Mode 5 6.23 2.18 bending and torsion Mode 8 8. Measurements Table 4.33 0.98 torsion Mode 16 17.18 weak torsion Mode 13 15.41 bending Mode 4 4.14 1.19 weak torsion Mode 14 15.10.23 bending and torsion Mode 11 14.55 weak torsion Mode 15 16. 4.08 bending and torsion Mode 2 2. 9 torsion 7 8.9 0.3 Rigid 3.6 0.5 Rigid 4.0 0.5 Elastic 8.6 0.4.4 Elastic 7.5 Elastic Table 4.7 bending and torsion 3 4.72 bending 2 4.4 0.8 Elastic 5.3 Elastic 11.4 1.1 Elastic 7.8 bending 4 5.9 0.5 2.9 Rigid 2. Measurements Table 4.1 2.3: Results of the measurements for cantilever down testing Mode Frequency [Hz] Damping Type of mode 1 2.6 0.1 2.3 longitudinal 33 .0 2.2: Results of the measurements for the free free vibration test 2 Mode 1 2 3 4 5 6 7 8 9 Frequency [Hz] Damping Type of mode 1.5 1.0 2.6 2.5 2.2 torsion 5 6.4 bending 6 7. 10: Mode animations [courtesy of P. Figure 4. Larsson]. Measurements Figure 4.11: Laser measurement point mobilities when the dummy and boom are turned 0 degree angle [courtesy of J. Banach].4. 34 . Kristoffersson & M. Larsson]. Figure 4. Measurements Figure 4. Larsson]. Kristoffersson & M. Kristoffersson & M.4.12: Laser measurement point mobilities when the dummy and boom are turned 90 degrees angle [courtesy of J. 35 .13: Coherence for the laser measurement points [courtesy of J. Measurements Figure 4.14: The coherence for the accelerometers calculated from one measurement [courtesy of J. Zapka]. M. Kristoffersson & M. C. Figure 4.4. Frangoudis. Larsson]. 36 .15: Shapes for modes from 2 to 7 Hz [courtesy of C. Kastby. C.16: Shapes for modes from 1 to 9 Hz [courtesy of C. Zapka]. Frangoudis. Measurements Figure 4. Kastby. M. 37 .4. Chapter 5 Finite Element Modal Analysis of the Boom Integrated to a CubeSat 5. creep or contact stiffness. convert units of measurements.modal.1 Chapter overview In this chapter the finite element model is presented. import it to Abaqus/CAE 3. Aim of analysis calculation of eigen frequencies and reference mode shapes. 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. step by step: 1. In case of dynamic loading and analysis of vibrations and transient responses in structures it is crucial to identify the characteristics of eigen frequencies. if necessary 4. assign material properties: 38 . Here is an example of modal analysis: Type of analysis . 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 . creat a model of the structure in any CAD/CAE software 2. 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 finite element model of the satellite dummy and integrated boom is prepared in order to identify the eigenfrequencies of the structure using numerical method and in order to conclude if the use of the gravity off-loading system is justified. but ignores a plastic deformation. The following operations are necessary. 1. 5. specify the parameters for modal frequencies identification. The meshed geometry is shown in Fig.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.5.5. 6. The cross section view of the tape spring connections to the hub and attachment plates is shown in Fig. there are four types of structures that are under investigation in this thesis: • boom with satellite dummy. Thus. The geometry was created using the software NX Unigraphix and then imported to the Abaqus software. 39 . 7. In order to quickly prepare the geometry CAD software NX 6.3. ties between the assembly parts and constrain the model. create necessary connections. 5. choose an element type.2. 8. Finite Element Modal Analysis of the Boom Integrated to a CubeSat • Young modulus • Density • Poisson’s ratio 5. 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. mesh the model.0 was used and later this geometry was imported to the Abaqus/CAE graphics module. • boom with satellite dummy with levelled positioning of tape-springs • boom with satellite dummy without extension springs in gravity off-loading system • 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. Nylon Strings .3 Material properties for components of the model The material properties are specified for each component. Poisson’s ratio 0. Plastic . Poisson’s ratio 0. density 2850 kg/m3 40 . 1.Modulus of elasticity 69 GPa.5.Modulus of elasticity 63 GPa. density 1200 kg/m3 2. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. Poisson’s ratio 0.3.3.2: Satellite dummy with the boom. Aluminium Bars .Modulus of elasticity 5 GPa. Poisson’s ratio 0. density 800 kg/m3 4. density 1780 kg/m3 3. Tape Springs . The material data can be summarized as follows.3. Figure 5. 5.1: Cross section view of the tape springs positioning.3.Modulus of elasticity 32 GPa. 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. the volume of the satellite dummy was computed and known mass of 2. Poisson’s ratio 0.4 Analysis The linear eigenfrequency analysis was performed on the structures using Abaqus CAE. density 7800 kg/m3 5.3: Meshed geometry of the boom with gravity off-loading strings.5 kg was used to compute equal density) 6. This density is given a number 9400 kg/m3 in order to allow for the masses that are not included in the satellite dummy. 5. Springs .3.1 were obtained.5 Eigenfrequencies from the model After running the eigenfrequency analysis for four types of structures mentioned above the data represented in Table 5. Satellite Dummy . Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. The eigenfrequencies for the first 10 modes were extracted. 5. density 9400 kg/m3 .Modulus of elasticity 210 GPa. Poisson’s ratio 0.Modulus of elasticity 69 GPa.3. where: 41 .5. Lanczos algorithm was used. 79 (+1.78 16.42 2 10.2%) 16.31 (−15. the first 6 free body motion modes are omitted together with not important gravity off-loading system modes • I . Hz I II III IV 1 7.09%) 13. Finite Element Modal Analysis of the Boom Integrated to a CubeSat • ∗ .05 11.* Eigenfrequency. 42 .11 (−15.17%) 11.24 4 21.Modal analysis of free-free boom with satellite dummy • II .Modal analysis for gravity off-loaded boom with satellite dummy with extension springs in gravity off-loading system • IV .13 show the first three important mode shapes of the gravity off-loaded boom.modes are given only for the boom with satellite dummy. Figs 5.1: Eigenfrequencies for the boom model configurations calculated in Abaqus CAE Mode.18%) 7.26 21.2%) 7.59 5.10 (−4.11%) 21.4 to 5.2%) 9.7 5.5.Modal analysis of free-free boom with satellite dummy with levelled position of tape-springs.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 Figs 5. Table 5.73 7.18 (−1.Modal analysis for gravity off-loaded boom with satellite dummy without extension springs in gravity off-loading system • III .05 (−4.09 (−1.78 (+1.7 Vibration modes of the boom in the gravity off-loading system This section presents the mode shapes for the gravity off-loaded boom.08 3 16.2%) 4.8 to 5. 2 shows the variation of the first eigenfrequency with respect this changes. Table 5.2 denotes the number of the eigen frequency of the boom 43 .5. 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 correctly 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.4: First eigenfrequency. The list below outlines the configurations of the structure for the sensitivity analysis. where i in Table 5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. 10.5: Second eigenfrequency.0 Hz − (a) Front view (b) Side view (c) Top view I II III IV V VI the free free vibrating central position of the fishing lines shifted 50 mm to the boom fishing line of the satellite dummy shifted 50 mm away from the boom fishing line of the satellite dummy shifted 150 mm away to the satellite dummy fishing line of the aluminium bar shifted 150 mm away from the satellite dummy fishing line of the aluminium bar 44 .5. 18 (+9.5%) 45 .6: Third eigenfrequency.57 (-1.2: Sensitivity analysis i I II III IV V VI frequency.73 7.77 (+0. 16. Hz 7.2%) 7.6%) 8. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5.8%) 7.79 (+1.5.2 Hz − (a) Front view (b) Side view (c) Top view Table 5.7%) 7.76 (+0. 7: Fourth eigenfrequency. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5.5.5 Hz − (a) Front view (b) Side view (c) Top view 46 . 21. 8 Hz − (a) Front view (b) Side view (c) Top view 47 . 7. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5.8: First eigen mode for the gravity off-loaded boom prototype.5. 5.9: First eigen mode for the gravity off-loaded boom prototype. 7. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5.8 Hz − Overall side view of the gravity off-loading system 48 . 11.5.3 Hz − (a) Front view (b) Side view (c) Top view 49 .10: Second eigen mode for the gravity off-loaded boom prototype. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. 11: Second eigen mode for the gravity off-loaded boom prototype.5.3 Hz − Overall side view of the gravity off-loading system 50 . 11. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5.1 Hz − (a) Front view (b) Side view (c) Top view 51 .5.12: Third eigen mode for the gravity off-loaded boom prototype. 16. 1 Hz − Overall side view of the gravity off-loading system 52 . Finite Element Modal Analysis of the Boom Integrated to a CubeSat Figure 5. 16.13: Third eigen mode for the gravity off-loaded boom prototype.5. 4.Chapter 6 Discussion 6.Top view 53 .1: First eigenfrequency. In the figure below the first mode shape is shown for such boom with levelled tape-springs: Figure 6. 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.1 Mode shapes comparison and discussion Due to the least resistance principle. (b) .(a) Side view. 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. boom with levelled tape springs .2 Hz . 3: The rotation 39.6. 6.  Figure 6.2: Transition zone.2. ρ0 denotes a region of a transition zone in a tape spring.2 Mass participation Taking a closer look at the results for the eigenfrequencies of the boom presented in Table 5. 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.1 we notice that the eigenfrequencies in the second mode for the gravity off- 54 . Figure 6. The transition zone is shown in Fig.8 degrees for the principal axis of inertia of the boom cross-section Fig 6. As shown in the results the natural frequencies received a higher estimate as the boom structure did not contain the transition zone. obtained through vibration testing and first eigen frequency for the free free vibrating boom from the FEA model.4 shows a comparison of the first eigen frequency. 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. 6. Discussion 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. The aluminum bar above the boom’s lightest part (tip plate and the hub) is participating in the boom vibration modes.8 Hz) is very representative as this type of bending is observed in the deployment tests and in finite element analysis. Discussion Figure 6. The stiffness matrix for the boom in this mode having additional mass components and this gives a higher eigenfrequency estimate. 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. 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.76 Hz (b) loaded boom are having discrepancy of 15%.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.7 Hz (a) and the vibration testing. This has a simple explanation. 7. 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. 1. It has been discussed in paper  by Greschik that the gravity off-loading system should be as light as possible besides having a 10 times lower eigenfrequency.4: Comparison of the shapes for the first mode in the FEA analysis. 55 . 6.6. This models are mainly used to compare the mode-shapes of the free satellite dummy with deployed boom and the suspended in the gravity off loading system structure. Mode 1 (1. 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. 2 Research conclusions In this paper an experimental analysis of the boom of a satellite from a dynamical point of view was presented. The excitation of the structure was carried out by a shaker fed with a white noise signal and mounted on the dummy structure. a comparison of the mode shapes corresponding to those eigenfrequencies will be carried out.Chapter 7 Conclusions and Recommendations 7. The main objective was to find the modal parameters of the boom in a frequency range comprised between 0 to 100 Hz with a particular focus on the range from 0 to 20 Hz because the satellite should not be submitted to high frequency perturbations once in orbit. 10 – 17 modes were found in the frequency range between 0 to 19 Hz. Thus. The resemblance of the eigenmodes from the measurements and the simulations will be analysed. it was not possible to perform a very dense meshing of measurement points and the deformations of the composite tape-springs were not measured directly. the objective is to summarise the results of the measurements and the simulations of the boom structure integrated to the CubeSat and to compare the results. Because of the complexity of the structure under study some special measurement techniques were implemented to acquire the amplitudes. To establish that the modes from the measurements and from the model are actually from a similar eigenmode. 7.1 Chapter overview In this chapter. Some of them were obvious whereas 56 . Unfortunately. The measurement and simulation results of the simpler structures were used to simplify the mathematical modelling of the complete structure and will not be evaluated further than what was done in the previous chapters. 25 Hz. When modelling a structure of such complexity as the boom. the results given in this paper should be exploited with caution since they are not highly precise and accuracy is unverified beyond reasonable physical behaviour . However. Before performing any experiments. there are many possible sources of error. This meant that there was 57 . This is probably due to the lack of measurement points and to the fact that modes are very close to each other. When information was extracted about peak frequencies from the frequency sweep measurements. the tape-springs were joint to the hub. For the boom. This was probably successful since the connections in the real structure were made very firm. there would probably have been a need to experiment more with creating different partitioning of the surfaces or make special joint interfaces. the material properties of the boom and the boundary conditions between the tape springs and the hub and attachment plate is considered to be relatively representative. the attachment plate and the tip firmly and as symmetrical as possible. The material properties of the tape springs were not accurately modelled and only relative comparison was performed for boom with and without gravity off-loading system. since a lot of effort was made to obtain a good model of a single beam (with satisfying result). If that would not have been the case. Conclusions and Recommendations others seemed to be overlapping modes or combination of other modes rather than independent modes. In the simulations. It is hard to analyse whether or not these eigenmodes were successfully reproduced in the FEM model. the resolution of the RMS plots was poor for the lower frequencies (more than 1 Hz for the lowest frequencies). 7. The aim was to simulate the lower eigenmodes and the first two eigenmodes showed great similarity in mode shape between measurement and simulation. Finally. the resolution was set to 0. Asymmetrical of the real structure due to irregularities in the materials and the assemblage is another source of error. A major source of error when modelling an assembled structure may be how the connections between the different parts are modelled. There were also a couple of eigenmodes from simulations at higher frequencies that were close to the measured eigenmodes in terms of frequencies and with obvious similarity in mode shapes. a comparison with FEM results was performed and a correspondence between certain modes was found. However.7. tape-springs based structure good results were obtained by simply letting all DOFs on connecting surfaces be tied to each other. However. errors may exist since the material properties can vary.3 Lessons learned The objective of creating a model that has the same dynamic behaviour as the structure in terms of eigenfrequencies and mode shapes for frequencies up to 100 Hz turned out to be a bit optimistic since the displacement of the structure was too complicated to be measured with accelerometers for some of the higher frequencies. Conclusions and Recommendations no point in analysing the difference in frequencies down to decimals . One could also design a lighter gravity off-loading system using lightweight structures. it is recommended to use a more dense in resolution high precision capture system to capture the amplitudes of the boom as the experiments that were performed did not show intermediary boom structure amplitudes.7.4 Recommendations For those who are to follow. A more complex tree in the Marionette gravity off-loading system would probably give a better weightlessness environment simulation in the experimental testing. 7. More measurement points would produce a more precise result with regard to the boom mode shapes and frequencies. especially when it comes to the bars. The amplitudes were impossible to observe using the capture system used. 58 . We have seen the mass participation factor 15% influence in the second mode and this could be improved.  W. Kastby. http://marketing. JOURNAL OF SPACECRAFT AND ROCKETS. 5  Retrieved from www.org/images/developers/cds_rev12. 2012. 1990. 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