50 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Mathematical Modelling of a Rotating Deformable Space Vehicle Lace Attitude Motions Saraka Jean Modeste Kouassia, Moussa Syllab* aUFR Mathématiques et Informatique Université Félix Houphouët-Boigny Laboratoire de Mécanique bUFR Mathématiques et Informatique Université Félix Houphouët-Boigny Laboratoire de Mécanique, Abidjan, 22 BP 582, Côte d’Ivoire aEmail: ksarakajm@yahoo.fr bEmail: ba_mouss@yahoo.fr Abstract A rotating elastic space vehicle composed of a central rigid body connected to two solar arrays, subjected to external torques, is studied using the spectral expansion continuum method of Rayleigh-Ritz. The lace attitudes motion of the rotating space vehicle can be described by a linear combination of the global modes different from the usual ordinary modes. By application of the spectral expansion continuum approach of Rayleigh-Ritz, the impedance function of the space vehicle in terms of the global modes of the whole system is derived. The numerical simulation of these global modes of great interest in this study, are performed by means of this impedance function relating the external torques to the rotating space vehicle lace attitudes. Keywords: vibration; global mode; spectral expansion; impedance function. 1. Introduction Several papers using the spectral decomposition continuum approach of Rayleigh-Ritz [1-12] are interested by the attitudes motions of elastic multibody system in dynamics. Many authors use different mathematical models to perform the modal analysis of the mechanical system attitudes motion. These attitudes motions are supposed to be coupled and expanded generally with the choice of the non-rotating base cantilever modes [2, 3, 5, 7, 8, 9, 11, 12] or the rotating base cantilever modes [1, 4, 6, 7, 10]. The modes used by the authors to describe the distributed flexibility of the mechanical system, leads to the spectral expansion of the reduced impedance matrix of the multibody system deriving from this analysis. The external torques exerted on the multibody system are related to its coupled attitudes by means of this matrix. ----------------------------------------------------------------------- * Corresponding author. http://asrjetsjournal.org/ mailto:%20aEmail: mailto:%20bEmail: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 51 The present investigation uses the spectral decomposition continuum approach of Rayleigh-Ritz. One describes the dynamics of a simplified elastic space vehicle composed of a central rigid body connected to two flexible solar arrays. The particularity in this work, is the decoupling between the attitudes motions of the spacecraft, which phenomena can be observed physically due to the effect of attitudes control. In this case, the interest is put on the lace attitude motions interacting with the flexible parts global modes [12]. From the spectral decomposition theory performed, one gets the impedance function of the whole system. Hence, by means of numerical algorithm using this impedance function, the global modes frequencies and the natural vibrations frequencies of the whole space vehicle are computed. In the modal analysis, one supposes that the elastic space vehicle undergoes small lace attitudes motions and elastic vibrations motions interacting with its rotating orbital motion, around an equilibrium corresponding to some gravity-gradient stabilized position where the flexible appendages are undeformed [7, 13]. 2. Dynamical equations of the space vehicle 2.1. Formulation and parameterization of the space vehicle motion The mechanical system (𝔅𝔅) studied is composed of a central rigid body (𝔅𝔅0) linked symmetrically to two solar arrays (𝔅𝔅1) and (𝔅𝔅2) by means of two rigid pieces (𝛾𝛾1) and (𝛾𝛾2) in two points 𝑂𝑂1and 𝑂𝑂2 respectively. Each array (𝔅𝔅𝑖𝑖) (𝑖𝑖 = 1 or 2) is composed of a supporting piece (𝑆𝑆𝑆𝑆𝑖𝑖), a membrane (𝐵𝐵𝐵𝐵𝑖𝑖) and a tip piece (𝑇𝑇𝑆𝑆𝑖𝑖). The solar arrays have the same physical characteristics. One notes ℜ(𝑡𝑡) = �𝑂𝑂0; �⃗�𝑋0,𝑌𝑌�⃗0,𝑍𝑍0� the central reference frame which is rigidly joined to (𝔅𝔅0) of mass center 𝑂𝑂0. ℜ𝐺𝐺 = �𝐺𝐺; �⃗�𝑎, 𝑏𝑏�⃗ , 𝑐𝑐� is the orthonormal orbiting reference of origin 𝐺𝐺 which is the mass center of the whole system deformed. The solar arrays are modelled by rectangular membranes of constant surface density 𝜎𝜎. The width and length of each solar array in its underformed position are denoted by 𝑒𝑒 and 𝑙𝑙 respectively. The plane �𝑂𝑂0; �⃗�𝑋0,𝑍𝑍0� designates the whole system symmetric plane in its underformed position (see figure 1). The inertial frame reference is denoted by ℜ𝑇𝑇 = �𝑇𝑇; �⃗�𝑋,𝑌𝑌�⃗ ,𝑍𝑍�. Relative to ℜ𝑇𝑇, the frame ℜ𝐺𝐺 is spinning around the axis �𝑇𝑇;𝑍𝑍� with a constant angular speed 𝑛𝑛0. During its orbital motion, the frame ℜ𝐺𝐺 drives the whole space vehicle (𝔅𝔅) which mass center 𝐺𝐺 is supposed to describe a circular orbit of center 𝑇𝑇 in the plane �𝑇𝑇; �⃗�𝑋,𝑌𝑌�⃗ � (see figure 2). The equilibrium position of the space vehicle (𝔅𝔅) with respect to the orbital reference frame is perturbed by its small lace attitude motions of angle 𝜃𝜃(𝑡𝑡) interacting with the solar arrays elastic deformations, represented by the vector 𝑑𝑑𝑖𝑖 (see figure 4): 𝑑𝑑𝑖𝑖(𝑀𝑀, 𝑡𝑡) = (𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡), 0, 0)𝑇𝑇; 𝑠𝑠 ∈ [0 ; 𝑙𝑙]; on (𝑆𝑆𝑆𝑆𝑖𝑖); 𝑑𝑑𝑖𝑖(𝑀𝑀, 𝑡𝑡) = (𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡), 0, 0)𝑇𝑇 (𝑥𝑥, 𝑠𝑠) ∈ 𝑅𝑅; on (𝐵𝐵𝐵𝐵𝑖𝑖). On (𝑇𝑇𝑆𝑆𝑖𝑖) the displacements field 𝑑𝑑𝑖𝑖 are obtained by the boundary condition applied on (𝐵𝐵𝐵𝐵𝑖𝑖) and (𝑆𝑆𝑆𝑆𝑖𝑖). The vector 𝑑𝑑𝑖𝑖 is measured in the frame components ℜ(𝑡𝑡). 𝑅𝑅 is the integration domain representing the rectangular membrane (𝐵𝐵𝐵𝐵𝑖𝑖) defined as follows: 𝑅𝑅 = �(𝑥𝑥, 𝑠𝑠) / −𝑒𝑒 2 ≤ 𝑥𝑥 ≤ +𝑒𝑒 2 , 0 ≤ 𝑠𝑠 ≤ 𝑙𝑙�. For the remainder, for any function F depending on the spatial variable 𝑠𝑠 and the time 𝑡𝑡, one notes: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 52 �̇�𝐹 = 𝜕𝜕𝐹𝐹 𝜕𝜕𝑡𝑡 , �̈�𝐹 = 𝜕𝜕2𝐹𝐹 𝜕𝜕𝑡𝑡2 ,𝐹𝐹′ = 𝜕𝜕𝐹𝐹 𝜕𝜕𝑠𝑠 ,𝐹𝐹′′ = 𝜕𝜕2𝐹𝐹 𝜕𝜕𝑠𝑠2 . 2.2. Equations of the space vehicle motion By the use of the variational principle of Hamilton, one gets the equation of the global motion and the local equations of the flexible appendages motion (see appendix 2): • Equation of the global motion 𝐶𝐶�̈�𝜃 + 3𝑛𝑛02(𝐵𝐵 − 𝐴𝐴)𝜃𝜃 + �̈�𝑁𝑢𝑢∗ + 𝑛𝑛0�̇�𝑁𝛼𝛼∗ + 3𝑛𝑛02𝑁𝑁𝑢𝑢∗ = Υ. (1) Where 𝐴𝐴, 𝐵𝐵 and 𝐶𝐶 are moments of inertia of the space vehicle and Υ is the external forces applied to the space vehicle: � 𝑁𝑁𝛼𝛼∗ = 2∑ �∫ 𝜎𝜎𝑥𝑥𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡)𝑑𝑑𝑠𝑠𝑙𝑙 0 �2 𝑖𝑖=1 𝑁𝑁𝑢𝑢∗ = ∑ (−1)𝑖𝑖 �∫ 𝜌𝜌(𝑏𝑏 + 𝑠𝑠)𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡)𝑑𝑑𝑠𝑠 + 𝜎𝜎∬ (𝑏𝑏 + 𝑠𝑠)𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡)𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 + 𝑚𝑚�(𝑏𝑏 + 𝑙𝑙)𝑢𝑢𝑖𝑖(𝑙𝑙, 𝑡𝑡) 𝑙𝑙 0 �2 𝑖𝑖=1 . (2) • Local motions equations of each solar array (𝐵𝐵𝑖𝑖 , 𝑖𝑖 = 1,2) 𝜌𝜌��̈�𝑢𝑖𝑖 + (−1)𝑖𝑖(𝑏𝑏 + 𝑠𝑠)��̈�𝜃 + 3𝑛𝑛02𝜃𝜃�� + [𝑆𝑆�(𝑠𝑠)𝑢𝑢𝑖𝑖′]′ + 𝐸𝐸𝐸𝐸𝑢𝑢𝑖𝑖 (4) = 0 into (𝑆𝑆𝑆𝑆𝑖𝑖), (3) 𝜎𝜎��̈�𝛼𝑖𝑖 − 2𝑛𝑛0𝑥𝑥�̇�𝜃 + (−1)𝑖𝑖(𝑏𝑏 + 𝑠𝑠)��̈�𝜃 + 3𝑛𝑛02𝜃𝜃�� − ��𝐺𝐺� + 𝑇𝑇�(𝑠𝑠)�𝛼𝛼𝑖𝑖′� ′ = 0 into (𝐵𝐵𝐵𝐵𝑖𝑖), (4) 𝑚𝑚���̈�𝑢𝑖𝑖(𝑙𝑙, 𝑡𝑡) + (−1)𝑖𝑖(𝑏𝑏 + 𝑙𝑙)��̈�𝜃 + 3𝑛𝑛02𝜃𝜃�� − 𝐸𝐸𝐸𝐸𝑢𝑢𝑖𝑖 (3)(𝑙𝑙, 𝑡𝑡) − 𝑆𝑆𝑢𝑢𝑖𝑖′(𝑙𝑙, 𝑡𝑡) + �𝐺𝐺�𝑒𝑒 + 𝑆𝑆 + 3𝑛𝑛02𝑚𝑚�(𝑏𝑏 + 𝑙𝑙)�𝛼𝛼𝑖𝑖′(𝑙𝑙, 𝑡𝑡) = 0 on (𝑇𝑇𝑆𝑆𝑖𝑖). (5) Here: 𝑆𝑆�(𝑠𝑠) is the dynamic stiffness exerted at any material point 𝑀𝑀(𝑠𝑠) of (𝑆𝑆𝑆𝑆𝑖𝑖): 𝑆𝑆�(𝑠𝑠) = 𝑆𝑆 − 3𝜌𝜌 𝑛𝑛0 2 2 [(𝑏𝑏 + 𝑙𝑙)2 − (𝑏𝑏 + 𝑠𝑠)2]. (6) P is the constant compression applied to the supporting piece (𝑆𝑆𝑆𝑆𝑖𝑖). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 53 𝑇𝑇�(𝑠𝑠) is the dynamic stiffness exerted to any material point 𝑀𝑀 of (𝐵𝐵𝐵𝐵𝑖𝑖): 𝑇𝑇�(𝑠𝑠) = 𝑃𝑃 𝑒𝑒 + 3 𝑛𝑛0 2 2 �𝜎𝜎(𝑏𝑏 + 𝑙𝑙)2 − 𝜎𝜎(𝑏𝑏 + 𝑠𝑠)2 + 2 𝑚𝑚� 𝑒𝑒 (𝑏𝑏 + 𝑙𝑙)�. (7) 𝐺𝐺� = 𝐸𝐸ℎ 4(1 + 𝜈𝜈). E is the Young modulus, 𝜈𝜈 is the Poisson ratio of the membranes, h is the thickness of the blanket (𝐵𝐵𝐵𝐵𝑖𝑖). • Boundary conditions of the problem The geometrical conditions on the boundary between the rigid interface (𝛾𝛾𝑖𝑖) and the solar array (𝐵𝐵𝑖𝑖), relative to the displacement fields 𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡) and 𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡) of (𝑆𝑆𝑆𝑆𝑖𝑖) and (𝐵𝐵𝐵𝐵𝑖𝑖) respectively, are defined by: � 𝑢𝑢𝑖𝑖(0, 𝑡𝑡) = 0 𝛼𝛼𝑖𝑖(𝑥𝑥, 0, 𝑡𝑡) = 0 𝑢𝑢𝑖𝑖′(0, 𝑡𝑡) = 0 . (8) The continuity of the displacement fields on the boundary between (𝐵𝐵𝐵𝐵𝑖𝑖) and (𝑆𝑆𝑆𝑆𝑖𝑖), is expressed by: 𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑙𝑙, 𝑡𝑡) = 𝑢𝑢𝑖𝑖(𝑙𝑙, 𝑡𝑡). (9) Consequently, one has: 𝜕𝜕𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑙𝑙, 𝑡𝑡) 𝜕𝜕𝑥𝑥 = 𝜕𝜕𝑢𝑢𝑖𝑖(𝑙𝑙, 𝑡𝑡) 𝜕𝜕𝑥𝑥 = 0. From (9), one considers an admissible function 𝛼𝛼𝑖𝑖 independent of the variable 𝑥𝑥, of the form: 𝛼𝛼𝑖𝑖 = 𝛼𝛼𝑖𝑖(𝑠𝑠, 𝑡𝑡). The natural condition in absence of external torques exerted on (𝑆𝑆𝑆𝑆𝑖𝑖) tip, is determined by: 𝑢𝑢𝑖𝑖′′(𝑙𝑙, 𝑡𝑡) = 0. (10) 3. Resolution of the global modes equations 3.1. Analytical forms of the shape functions The global modes are the solutions obtained by setting the following conditions in the equations (1), (3), (4) and (5): American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 54 Υ = 0, 𝑛𝑛0 = 0. (11) In this case, the space vehicle attitudes motion are decoupled. Here, one recalls that the interest is put on the lace attitude motion of the space vehicle interacting with the solar arrays bending in plane. The global modes solutions proposed are of the forms: � 𝜃𝜃(𝑡𝑡) = 𝜃𝜃𝑛𝑛 𝑐𝑐𝑐𝑐𝑠𝑠(𝜑𝜑𝑛𝑛𝑡𝑡) 𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡) = 𝑈𝑈𝑛𝑛𝑖𝑖(𝑠𝑠) 𝑐𝑐𝑐𝑐𝑠𝑠(𝜑𝜑𝑛𝑛𝑡𝑡) 𝛼𝛼𝑖𝑖(𝑠𝑠, 𝑡𝑡) = 𝛼𝛼𝑛𝑛𝑖𝑖(𝑠𝑠) 𝑐𝑐𝑐𝑐𝑠𝑠(𝜑𝜑𝑛𝑛𝑡𝑡) . (12) with: � 𝑈𝑈𝑛𝑛𝑖𝑖(𝑠𝑠) = (−1)𝑖𝑖𝜃𝜃𝑛𝑛𝑈𝑈𝑛𝑛∗(𝑠𝑠) 𝛼𝛼𝑛𝑛𝑖𝑖(𝑠𝑠) = (−1)𝑖𝑖𝜃𝜃𝑛𝑛𝛼𝛼𝑛𝑛∗(𝑠𝑠) ; 𝑛𝑛 = 1, … … ,∞. (13) One gets, from the resolution of the equations (3) and (4), the following solutions: ⎩ ⎪ ⎨ ⎪ ⎧𝑈𝑈𝑛𝑛 ∗(𝑠𝑠) = 𝑎𝑎𝑛𝑛 𝑐𝑐𝑐𝑐𝑠𝑠ℎ(𝜓𝜓𝑛𝑛𝑠𝑠) + 𝑏𝑏𝑛𝑛 𝑠𝑠𝑖𝑖𝑛𝑛ℎ(𝜓𝜓𝑛𝑛𝑠𝑠) + (𝑏𝑏 − 𝑎𝑎𝑛𝑛) 𝑐𝑐𝑐𝑐𝑠𝑠(𝜙𝜙𝑛𝑛𝑠𝑠) + 1 − 𝑏𝑏𝑛𝑛𝜓𝜓𝑛𝑛 𝜙𝜙𝑛𝑛 𝑠𝑠𝑖𝑖𝑛𝑛(𝜙𝜙𝑛𝑛𝑠𝑠) − 𝑏𝑏 − 𝑠𝑠 𝛼𝛼𝑛𝑛∗(𝑠𝑠) = 𝑐𝑐𝑛𝑛 𝑠𝑠𝑖𝑖𝑛𝑛(𝜉𝜉𝑛𝑛𝑠𝑠) + 𝑏𝑏 𝑐𝑐𝑐𝑐𝑠𝑠(𝜉𝜉𝑛𝑛𝑠𝑠) − 𝑏𝑏 − 𝑠𝑠 , where: 𝜓𝜓𝑛𝑛 = �− 𝑆𝑆 2𝐸𝐸𝐸𝐸 + �� 𝑆𝑆 2𝐸𝐸𝐸𝐸 � 2 + 𝜌𝜌𝜏𝜏𝑛𝑛2 𝐸𝐸𝐸𝐸 ; 𝜙𝜙𝑛𝑛 = � 𝑆𝑆 2𝐸𝐸𝐸𝐸 + �� 𝑆𝑆 2𝐸𝐸𝐸𝐸 � 2 + 𝜌𝜌𝜏𝜏𝑛𝑛2 𝐸𝐸𝐸𝐸 ; and 𝜉𝜉𝑛𝑛 = 𝜑𝜑𝑛𝑛� 𝜎𝜎𝑒𝑒 �𝑒𝑒𝐺𝐺� + 𝑆𝑆� . 𝜑𝜑𝑛𝑛 is the global eigenfrequency associated with the global eigenmodes 𝜃𝜃𝑛𝑛, 𝑈𝑈𝑛𝑛𝑖𝑖(𝑠𝑠) and 𝛼𝛼𝑛𝑛𝑖𝑖(𝑠𝑠). 3.2. Equation of the eigenfrequencies The solutions of formula (12) are introduced into the equations of the global motion (1), and taking into account the conditions (11), one obtains from laborious calculations the following equation of the eigenfrequencies: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 55 ∫ 𝜌𝜌(𝑏𝑏 + 𝑠𝑠)𝑈𝑈𝑛𝑛∗(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + 𝜎𝜎𝑒𝑒 ∫ (𝑏𝑏 + 𝑠𝑠)𝛼𝛼𝑛𝑛∗(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + 𝑚𝑚�(𝑏𝑏 + 𝑙𝑙)𝑈𝑈𝑛𝑛∗(𝑙𝑙) = −𝐶𝐶 2 . (14) The global eigenfrequencies 𝜑𝜑𝑛𝑛 corresponding to bending in plane, are computed by means of the equation (14). 3.3. Calculation of the lace attitude 𝜽𝜽𝒏𝒏 One considers a set of global eigenmodes denoted by (𝑈𝑈𝑛𝑛𝑖𝑖 ,𝑈𝑈𝑝𝑝𝑖𝑖 ), and (𝛼𝛼𝑛𝑛𝑖𝑖,𝛼𝛼𝑝𝑝𝑖𝑖 ), satisfying the following normalized orthogonality property (see appendix 3): 𝜃𝜃𝑛𝑛𝜃𝜃𝑝𝑝 �2 �𝜌𝜌 ∫ 𝑈𝑈𝑛𝑛∗(𝑠𝑠)𝑈𝑈𝑝𝑝∗(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + 𝜎𝜎𝑒𝑒 ∫ 𝛼𝛼𝑛𝑛∗(𝑠𝑠)𝛼𝛼𝑝𝑝∗(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + 𝑚𝑚�𝑈𝑈𝑛𝑛∗(𝑙𝑙)𝑈𝑈𝑝𝑝∗(𝑙𝑙)� − 𝐶𝐶� = 𝛿𝛿𝑛𝑛𝑝𝑝. (15) 𝛿𝛿𝑛𝑛𝑝𝑝 = �1 𝑖𝑖𝑖𝑖 𝑛𝑛 = 𝑝𝑝 0 𝑖𝑖𝑖𝑖 𝑛𝑛 ≠ 𝑝𝑝 ; 𝑛𝑛 = 1, … … ,∞; 𝑝𝑝 = 1, … … ,∞, 𝛿𝛿𝑛𝑛𝑝𝑝 designates Kronecker symbol. Using the formula (15), one gets the following expression of the lace attitude 𝜃𝜃𝑛𝑛: 𝜃𝜃𝑛𝑛 = �2 �𝜌𝜌 ∫ 𝑈𝑈𝑛𝑛∗2(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + 𝜎𝜎𝑒𝑒 ∫ 𝛼𝛼𝑛𝑛∗2(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + 𝑚𝑚�𝑈𝑈𝑛𝑛∗2(𝑙𝑙)� − 𝐶𝐶� −12 . (16) 4. General motion of the space vehicle in term of the global modes superposition The general motion of the space vehicle is governed by the global equation (1) and the local equation (3) and (4) under the following conditions: Υ ≠ 0 ; 𝑛𝑛0 ≠ 0 ; 𝜃𝜃(𝑡𝑡) ≠ 0 ; 𝛼𝛼𝑖𝑖(𝑠𝑠, 𝑡𝑡) ≠ 0 ; 𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡) ≠ 0. According to the Rayleigh-Ritz superposition method [1-12], the deformations of the flexible bodies and the lace attitudes of the space vehicle, solutions of these equations are represented by a linear combination of the global eigenmodes known associated with functions called generalized coordinates. It follows: ⎩ ⎪ ⎨ ⎪ ⎧𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡) = ∑ (−1)𝑖𝑖𝜃𝜃𝑛𝑛𝑈𝑈𝑛𝑛∗(𝑠𝑠)𝑔𝑔𝑛𝑛(𝑡𝑡)∞ 𝑛𝑛=1 𝛼𝛼𝑖𝑖(𝑠𝑠, 𝑡𝑡) = ∑ (−1)𝑖𝑖𝜃𝜃𝑛𝑛𝛼𝛼𝑛𝑛∗(𝑠𝑠)𝑔𝑔𝑛𝑛(𝑡𝑡)∞ 𝑛𝑛=1 𝜃𝜃(𝑡𝑡) = 𝜆𝜆(𝑡𝑡) + ∑ 𝜃𝜃𝑛𝑛𝑔𝑔𝑛𝑛(𝑡𝑡)∞ 𝑛𝑛=1 , (17) 𝑔𝑔𝑛𝑛(𝑡𝑡) is the generalized coordinates to be determined by the modal analysis. Here, 𝜆𝜆(𝑡𝑡) represents the rigid American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 56 response of the whole spacecraft if it were rigid. 4.1. Determination of generalized coordinates Injecting the expansions (17) of the elastic deformations and the lace attitudes into the local equations (3), (4) and (5) and taking into account the boundary conditions (8), (9) and (10) the normalized orthogonality property (15) and the natural frequencies equation (14), one gets the following differential equation of the generalized coordinates: �̈�𝑔𝑘𝑘(𝑡𝑡) + 𝜑𝜑𝑘𝑘2𝑔𝑔𝑘𝑘(𝑡𝑡) + 3𝑛𝑛02 ∑ 𝑚𝑚𝑛𝑛𝑘𝑘𝑔𝑔𝑛𝑛(𝑡𝑡)∞ 𝑛𝑛=1 = 𝐶𝐶𝜃𝜃𝑘𝑘��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�, 𝑘𝑘 = 1,2, …. (18) here: 𝑚𝑚𝑛𝑛𝑘𝑘 = 𝜃𝜃𝑛𝑛𝜃𝜃𝑘𝑘 �∫ 𝑒𝑒𝑇𝑇�1∗(𝑠𝑠)𝛼𝛼𝑛𝑛∗ ′(𝑠𝑠)𝛼𝛼𝑘𝑘∗ ′(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 + ∫ 𝑆𝑆�1∗(𝑠𝑠)𝑈𝑈𝑛𝑛∗ ′(𝑠𝑠)𝑈𝑈𝑘𝑘∗ ′(𝑠𝑠)𝑑𝑑𝑠𝑠𝑙𝑙 0 − 𝐶𝐶�, (19) with 𝑇𝑇�1∗(𝑠𝑠) = 𝜎𝜎(𝑏𝑏 + 𝑙𝑙)2 − 𝜎𝜎(𝑏𝑏 + 𝑠𝑠)2 + 2 𝑚𝑚� 𝑒𝑒 (𝑏𝑏 + 𝑙𝑙); 𝑆𝑆�1∗(𝑠𝑠) = 𝜌𝜌[(𝑏𝑏 + 𝑙𝑙)2 − (𝑏𝑏 + 𝑠𝑠)2]. One supposes that the generalized coordinates and the rigid response are harmonic of the form: 𝑔𝑔𝑛𝑛(𝑡𝑡) = 𝑔𝑔�𝑛𝑛𝑒𝑒𝑗𝑗𝑗𝑗𝑗𝑗 , 𝜆𝜆(𝑡𝑡) = �̂�𝜆𝑒𝑒𝑗𝑗𝑗𝑗𝑗𝑗 , (20) here: 𝛺𝛺 is a real parameter to be determined and 𝑗𝑗2 = −1. One defines the following column vectors: 𝐺𝐺�𝑇𝑇𝑁𝑁 = (𝑔𝑔�1,⋯⋯ ,𝑔𝑔�𝑁𝑁), 𝜃𝜃 𝑁𝑁 𝑇𝑇 = (𝜃𝜃1,⋯⋯ ,𝜃𝜃𝑁𝑁), and the following matrix: 𝑀𝑀 𝑁𝑁 = ⎝ ⎜ ⎜ ⎛ 𝜑𝜑12 + 3𝑛𝑛02𝑚𝑚11 3𝑛𝑛02𝑚𝑚21 ⋮ 3𝑛𝑛02𝑚𝑚𝑘𝑘1 ⋮ 3𝑛𝑛02𝑚𝑚𝑁𝑁1 3𝑛𝑛02𝑚𝑚12 𝜑𝜑22 + 3𝑛𝑛02𝑚𝑚22 ⋯ ⋯ ⋯ ⋯ ⋱ … … 3𝑛𝑛02𝑚𝑚1𝑘𝑘 3𝑛𝑛02𝑚𝑚2𝑘𝑘 ⋮ 𝜑𝜑𝑘𝑘2 + 3𝑛𝑛02𝑚𝑚𝑘𝑘𝑘𝑘 ⋮ 3𝑛𝑛02𝑚𝑚𝑁𝑁𝑘𝑘 ⋯ … ⋱ … 3𝑛𝑛02𝑚𝑚1𝑁𝑁 3𝑛𝑛02𝑚𝑚2𝑁𝑁 ⋮ 3𝑛𝑛02𝑚𝑚𝑘𝑘𝑁𝑁 ⋮ 𝜑𝜑𝑁𝑁2 + 3𝑛𝑛02𝑚𝑚𝑁𝑁𝑁𝑁⎠ ⎟ ⎟ ⎞ , �𝑀𝑀 𝑁𝑁 � 𝑛𝑛𝑘𝑘 = 𝜑𝜑𝑛𝑛2𝛿𝛿𝑛𝑛𝑘𝑘 + 3𝑛𝑛02𝑚𝑚𝑛𝑛𝑘𝑘. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 57 Equations (18) are written in the following matrix forms: �𝑀𝑀 𝑁𝑁 − 𝛺𝛺2𝐸𝐸 𝑁𝑁 � 𝐺𝐺�𝑁𝑁 = 𝐶𝐶(3𝑛𝑛02 − 𝛺𝛺2)𝜃𝜃 𝑁𝑁 �̂�𝜆. (21) 𝐸𝐸 𝑁𝑁 is the unit matrix 𝑁𝑁 × 𝑁𝑁. The symmetrical matrix 𝑀𝑀 𝑁𝑁 with respective eigenvalues denoted ℎ𝑘𝑘 (𝑘𝑘 = 1,⋯ ,𝑁𝑁) can be diagonalized as: 𝑀𝑀 𝑁𝑁 = 𝐵𝐵 𝑁𝑁 𝐻𝐻 𝑁𝑁 𝐵𝐵𝑇𝑇 𝑁𝑁 , (22) here: �𝐵𝐵 𝑁𝑁 � 𝑛𝑛𝑘𝑘 = 𝑏𝑏𝑛𝑛𝑘𝑘 , 𝐻𝐻 𝑁𝑁 = � ℎ1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ ℎ𝑁𝑁 �. The matrix 𝐵𝐵 𝑁𝑁 is orthogonal matrix. Substituting the factorizations (22) into the equation (21) and assuming that: ℎ𝑘𝑘 ≠ 𝛺𝛺2. One gets finally the generalized coordinates: 𝑔𝑔�𝑛𝑛 = 𝑔𝑔�𝑛𝑛∗ �̂�𝜆, (23) here 𝑔𝑔�𝑛𝑛∗ = 𝐶𝐶(3𝑛𝑛02 − 𝛺𝛺2)�� 𝑏𝑏𝑛𝑛𝑘𝑘 ℎ𝑘𝑘 − 𝛺𝛺2 �𝑏𝑏𝑘𝑘𝑘𝑘𝜃𝜃𝑘𝑘 𝑁𝑁 𝑘𝑘=1 � 𝑁𝑁 𝑘𝑘=1 , 𝑛𝑛 = 1,2, … .. 4.2. Determination of the impedance function of the system By using the expressions (2), the solutions (17), the equations of eigenfrequencies (14), and the Fourier transformation, the equations (1) of the global motion becomes: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 58 {3𝑛𝑛02[𝐵𝐵 − 𝐴𝐴 + (𝐵𝐵 − 𝐴𝐴 − 𝐶𝐶)∑ 𝜃𝜃𝑛𝑛𝑔𝑔�𝑛𝑛∗ 𝑁𝑁 𝑛𝑛=1 ] − 𝛺𝛺2𝐶𝐶}�̂�𝜆 = Υ�. (24) One defines the following terms: 𝑍𝑍0(𝛺𝛺) = 3𝑛𝑛02(𝐵𝐵 − 𝐴𝐴) − 𝛺𝛺2𝐶𝐶, 𝑍𝑍𝑛𝑛(𝛺𝛺) = 3𝑛𝑛02(𝐵𝐵 − 𝐴𝐴 − 𝐶𝐶)𝜃𝜃𝑛𝑛𝑔𝑔�𝑛𝑛∗ , (25) 𝑍𝑍(𝛺𝛺) = 𝑍𝑍0(𝛺𝛺) + ∑ 𝑍𝑍𝑛𝑛𝑁𝑁 𝑛𝑛=1 (𝛺𝛺). (26) One can rewrite the equation (24) in the form: 𝑍𝑍(𝛺𝛺)�̂�𝜆 = Υ�. (27) Here: 𝑍𝑍(𝛺𝛺) is the impedance function connecting the external forces Υ� and the response �̂�𝜆 of the space vehicle. Formula (26) is the spectral expansion of the impedance function 𝑍𝑍(𝛺𝛺). 4.3. Equation of global frequencies In the absence of external forces, the global motion of the space vehicle is governed by the following equation: 𝑍𝑍(𝛺𝛺)�̂�𝜆 = 0. (28) One assumes non trivial solution �̂�𝜆 of equation (28). Hence the function 𝑍𝑍(𝛺𝛺) must necessarily verifies: 𝑍𝑍(𝛺𝛺) = 0. (29) Equation (29) is the equation of the global eigenfrequencies 𝛺𝛺 when the space vehicle undergoes natural lace attitude motion interacting with elastic deformations. 4.4. Convergence of the global modes gain One defines the global modes gain as the following series: �𝜃𝜃𝑛𝑛2 ∞ 𝑛𝑛=1 . The modal analysis of the space vehicle equations motion performed using the superposition method of American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 59 Rayleigh-Ritz, establishes the convergence of the global modes gain above (see appendix 4): ∑ 𝜃𝜃𝑛𝑛2∞ 𝑛𝑛=1 = 2ℒ 𝐶𝐶(𝐶𝐶−2ℒ) , (30) here: ℒ = 𝑚𝑚�(𝑏𝑏 + 𝑙𝑙)2 + (𝜎𝜎𝑒𝑒 + 𝜌𝜌) �𝑏𝑏2𝑙𝑙 + 𝑏𝑏𝑙𝑙2 + 𝑙𝑙3 3 �. This convergence property gives an approximation of the modal gain with a finite number N of the global modes selected in order to perform the dynamical model of the space vehicle. 5. Digital simulations The simulations are performed with the following data: 𝐴𝐴 = 9.5 × 106 𝑘𝑘𝑔𝑔.𝑚𝑚2; 𝐵𝐵 = 106 𝑘𝑘𝑔𝑔.𝑚𝑚2; 𝐶𝐶 = 4 × 106 𝑘𝑘𝑔𝑔.𝑚𝑚2; 𝐸𝐸 = 103 𝑁𝑁.𝑚𝑚2; 𝐸𝐸 = 103 𝑁𝑁.𝑚𝑚−2; ℎ = 0,01 𝑚𝑚; 𝑒𝑒 = 3,2 𝑚𝑚; 𝑙𝑙 = 28,8 𝑚𝑚; 𝑏𝑏 = 1,8 𝑚𝑚; 𝑚𝑚� = 6 𝑘𝑘𝑔𝑔; 𝑆𝑆 = 140 𝑁𝑁; 𝜎𝜎 = 1,4 𝑘𝑘𝑔𝑔.𝑚𝑚−2; 𝜌𝜌 = 204 𝑁𝑁.𝑚𝑚−3; 𝜈𝜈 = 0,3 𝐸𝐸 is the moment of inertia 𝑁𝑁: Newton; 𝑚𝑚: meter; 𝑘𝑘𝑔𝑔: kilogram; 5.1. Global modes gain The convergence of the global modes gain is given by the equation (30). One defines the numerical sequence 𝑖𝑖𝑁𝑁: 𝑖𝑖𝑁𝑁 = �𝜃𝜃𝑛𝑛2 𝑁𝑁 𝑛𝑛=1 . As the global modes gain is convergent, the number N selected achieves a good approximation of formula (30). 𝑁𝑁 = 35 is suitable to perform the numerical simulations in the following. The Figure 5 represents the simulation of the numerical sequence 𝑖𝑖𝑁𝑁. 5.2. Computation of the global eigenfrequencies In the absence of external forces and orbital motion, the space vehicle lace attitude motion 𝜃𝜃(𝑡𝑡) is interacting with array bending in plane. From equations (14), one computes the global modes eigenfrequencies 𝜑𝜑𝑛𝑛 corresponding to arrays bending in plane and the lace attitude of the space vehicle. In accordance with the modal gain approximation (30), 35 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 60 eigenfrequencies are selected in each case of global modes (see table 1). Table 1: Global eigenfrequencies bending in plane in the bandwidth [0 – 4 Hz] 𝜑𝜑𝑛𝑛/2𝜋𝜋(𝐻𝐻𝐻𝐻) 0.100 0.179 0.201 0.397 0.495 0.579 0.597 0.695 0.298 0.793 0.893 0.992 1.091 1.184 1.199 1.289 1.389 1.487 1.686 1.785 1.884 1.587 1.983 2.017 2.084 2.182 2.281 2.380 2.479 2.578 2.678 2.776 2.876 2.974 3.068 5.3. Calculation of the space vehicle global frequencies Table 2: Global frequencies in band width [0 – 2 Hz] 𝑛𝑛0 (𝑟𝑟𝑎𝑎𝑑𝑑/𝑠𝑠) 10−8 10−7 10-6 10-5 10-4 10-3 Ω/2π (𝐻𝐻𝐻𝐻) 5.436×10-9 5.43×10-8 5.436×10-7 5.436×10-6 5.456×10-5 2.777×10-4 1.412×10-8 1.412×10-7 1.412×10-6 1.412×10-5 1.419×10-4 4.667×10-4 2.311×10-8 1.875×10-7 1.875×10-6 1.875×10-5 1.879×10-4 8.938×10-4 2.602×10-8 2.311×10-7 2.311×10-6 2.311×10-5 2.331×10-4 3.051×10-8 2.602×10-7 2.602×10-6 2.605×10-5 2.808×10-4 3.473×10-8 3.051×10-7 3.051×10-6 3.052×10-5 3.120×10-4 3.950×10-8 3.473×10-7 3.473×10-6 3.476×10-5 3.553×10-4 4.359×10-8 3.950×10-7 3.950×10-6 3.962×10-5 4.098×10-4 5.071×10-8 4.359×10-7 4.359×10-6 4.369×10-5 4.465×10-4 5.276×10-8 5.071×10-7 5.071×10-6 5.101×10-5 5.155×10-4 5.996×10-8 5.276×10-7 5.276×10-6 5.313×10-5 5.454×10-4 6.722×10-8 5.996×10-7 5.996×10-6 6.040×10-5 6.110×10-4 7.428×10-8 6.722×10-7 6.722×10-6 6.779×10-5 8.136×10-8 7.134×10-7 7.134×10-6 7.143×10-5 8.592×10-8 7.428×10-7 7.428×10-6 7.458×10-5 9.445×10-8 8.136×10-7 8.136×10-6 8.174×10-5 9.910×10-8 8.592×10-7 8.592×10-6 8.621×10-5 1.080×10-8 9.445×10-7 9.445×10-6 9.475×10-5 1.142×10-7 9.910×10-7 9.910×10-6 9.938×10-5 1.331×10-7 1.080×10-6 1.080×10-6 1.081×10-4 5.503×10-7 1.142×10-6 1.142×10-6 1.145×10-4 1.114×10-5 1.331×10-6 1.331×10-5 1.334×10-4 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 61 One considers that the space vehicle undergoes orbital motion interacting with the lace attitude motion and the arrays deformations. In this case, one uses the spectral expansion (26) of the truncated impedance function 𝑍𝑍(𝛺𝛺) (𝑁𝑁 = 35) in accordance with the modal gain approximation (30). The natural global frequencies 𝛺𝛺 of the whole space vehicle in absence of external torques, are computed from equation (29). Different values of the orbital parameter 𝑛𝑛0 are considered to simulate the global frequencies of the whole space vehicle (see table 2). In the case of small angular speed 𝑛𝑛0 of orders 10−8 , 10−7, 10−6 and 10−5 in the bandwidth considered, one obtains for each order a spectrum of 22 eigenfrequencies 𝛺𝛺 computed. On the other hand, the spectrum of eigenfrequencies obtained in work [12] where the attitudes motion are coupled, is very reduced. Since then, one notes a great interaction between the lace attitude motion and the vibrations of the space vehicle. Consequently, the model of attitude control suitable of the space vehicle should take into account the spectrum of lace attitude eigenfrequencies to avoid the resonance phenomena. 5.4. Graphical simulation of the array membrane deformations Using the eigenfrequencies bending in plane 𝜑𝜑𝑛𝑛 , one simulates the different deformations of the array membrane. This simulations are represented in the figure 6, 7 and 8. 6. Conclusion A previous work [12] has investigated the space vehicle using the spectral decompositions method. The global modes of the investigated space vehicle considering its attitudes motions coupled, are performed [12]. A reduced spectrum of the eigenfrequencies is obtained in this case. This fact is characterized by a small interaction between the coupled attitudes motions and the elastic deformations. Therefore, it is necessary to simulate a decoupling of the space vehicle attitudes motions in order to get a great interaction with the elastic deformations. Here, the decoupling of the space vehicle attitudes motion brings out the interaction between the lace attitude motions and the elastic deformations of the space vehicle. Consequently, one gets in this present study a great spectrum of the eigenfrequencies contrary to work [12]. This wide spectrum of the eigenfrequencies obtained here is necessary to perform a suitable model attitude control of the space vehicle in the practice. References [1] H. B. Hablani. "Modal Analysis of Gyroscopic Flexible Spacecraft, a Continuum Approach." J. Guid 5 (5), pp. 448 – 457, 1982. [2] P. C. Hughes. "Dynamics of Flexible Space Vehicles with Active Attitudes Control." Celest. Mech. 9, pp. 21-39, 1974. [3] L. Meirovitch. "A Stationary Principle for the Eigenvalue Problem for Rotating Structures." AIAAJ. 14 (10), pp. 1387 – 1394, 1976. [4] M. Pascal. "Modal Analysis of a Rotating Flexible Space by a Continuum Approach." Z. Flug – wiss. Weltraumforsch. 18, pp. 395 – 401, 1994. [5] M. Pascal, M. Sylla. "Dynamic Model of a Large Space Structure by a Continuous Approach." American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 62 Recherche Aerospatiale N° 2, pp. 67 – 77, 1993. [6] A. Sandu, D. Negrut, E. J. Haug, F.A. Potra, C. Sandu. "A Rosenbrock-Nystrom State Space Implicit Approach for the Dynamic Analysis of Mechanical System." Theoretical Formulation, J. Multibody Dyn. 217(4), 263 – 271, 2003. [7] M. Sylla. "Spectral Decomposition Analysis Using the Global Modes of a Rotating Elastic Space Vehicle." Far East Journal of applied Mathematics 37(1), pp. 69 – 89, 2009. [8] M. Sylla and B. Asseke. "Dynamics of a Rotating Flexible and Symmetric Spacecraft Using Impedance Matrix in Terms of the Flexible Appendages Cantilever Modes." Multibody System Dynamics. 19, pp. 345 – 364, 2008. [9] M. Sylla and R. A. P. Barou. "A Mathematical Model of a Helicopter Standing and Rotating Rotor Blade." Far East Journal of applied Mathematics. 31(1), pp. 1 – 25, 2008. [10] M. Sylla, M. Dosso and R. A. P. Barou. "Dynamical Analysis of a Flexible Mechanism by a Continuum Approach Using the Rotating Base Cantilever Modes." Far Fast Journal of Applied Mathematics. 33(2), pp. 187 – 204, 2008. [11] M. Sylla and L. Gomat. "Modal Analysis of a Rotation Flexible Manipulator Arm." Afrika Matematika. Serie 3(19), 7 – 28, 2008. [12] M. Sylla and S. J. M. KOUASSI. "Method of Modal Synthesis in the Dynamics of a Deformable Spatial Structure." Far East Journal of Dynamical Systems Volume 30(3), 103-133, 2018. [13] M. Pascal. "La Stabilité d'Attitude d'un Satellite Muni de Panneaux Solaires." Acta Astronaut. 5, pp. 817 – 844, 1978. • Appendix 1 Model description Figure 1: Satellite and solar arrays. American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 63 Figure 2: Satellite orbiting the earth. Figure 3: Attitude 𝜃𝜃 motion. The angular velocity vector Λ��⃗ of the reference frame ℜ with respect to the reference frame ℜ𝑇𝑇 in �𝐺𝐺; �⃗�𝑋0,𝑌𝑌�⃗0,𝑍𝑍0� components is (figure 3): Λ��⃗ = � 0 0 𝑛𝑛0 + �̇�𝜃 � �𝑋𝑋�⃗ 0,𝑌𝑌�⃗0,�⃗�𝑍0� . The elastic displacement fields modelling of the solar arrays investigated in works [4, 5, 8], is used here. Consequently, for each solar array (𝔅𝔅𝑖𝑖) 𝑖𝑖 = 1, 2 the elastic displacement field 𝑑𝑑𝑖𝑖 is: on (𝑆𝑆𝑆𝑆𝑖𝑖), 𝑑𝑑𝑖𝑖(𝑀𝑀, 𝑡𝑡) = (𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡), 0,0)𝑇𝑇 𝑠𝑠 ∈ [0 ; 𝑙𝑙], on (𝐵𝐵𝐵𝐵𝑖𝑖), 𝑑𝑑𝑖𝑖(𝑀𝑀, 𝑡𝑡) = (𝛼𝛼𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡), 0,0)𝑇𝑇 (𝑥𝑥, 𝑠𝑠) ∈ 𝑅𝑅. Since the solar arrays are inextensible, the longitudinal displacement 𝑣𝑣𝑖𝑖(𝑠𝑠, 𝑡𝑡) and 𝛽𝛽𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡) is of second order and neglected in the linear theory [2, 3, 4]: 𝑣𝑣𝑖𝑖(𝑠𝑠, 𝑡𝑡) = − 1 2 � �� 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝜕𝜕 � 2 � 𝑑𝑑𝜕𝜕 𝑙𝑙 0 ; 𝛽𝛽𝑖𝑖(𝑥𝑥, 𝑠𝑠, 𝑡𝑡) = − 1 2 ��� 𝜕𝜕𝛼𝛼𝑖𝑖 𝜕𝜕𝜕𝜕 � 2 � 𝑑𝑑𝜕𝜕 𝑙𝑙 0 . American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 64 Figure 4: Bending in plane 𝛼𝛼𝑖𝑖, 𝑢𝑢𝑖𝑖 of the solar arrays and lace attitude motion of the space vehicle. • Appendix 2 The following variational principle of Hamilton is used to establish the motion equations [4, 5, 7, 8, 13]. 𝛿𝛿 � (𝐾𝐾 − 𝑉𝑉)𝑑𝑑𝑡𝑡 𝑗𝑗2 𝑗𝑗1 = 0, where 𝐾𝐾 and 𝑉𝑉 are, respectively, the kinetic and the total potential energy (including gravitational potential energy and the work of external torques) of the space vehicle. 𝑉𝑉 = 𝑛𝑛02 2 [3(𝐴𝐴 − 𝐵𝐵)𝜃𝜃2 + 6𝑁𝑁𝑢𝑢∗𝜃𝜃] + 𝛱𝛱�𝐷𝐷 + Υ𝜃𝜃, 𝛱𝛱�𝐷𝐷 = �� 𝐸𝐸𝐸𝐸 2 � 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝑠𝑠 � 2 − 𝑆𝑆� 2 � 𝜕𝜕𝑢𝑢𝑖𝑖 𝜕𝜕𝑠𝑠 � 2 𝑑𝑑𝑠𝑠 𝑙𝑙 0 2 𝑖𝑖=1 + �� Γ 2 �� 𝜕𝜕𝛼𝛼𝑖𝑖 𝜕𝜕𝑥𝑥 � 2 + 1 2 �𝐺𝐺� + 𝑇𝑇�(𝑠𝑠)� � 𝜕𝜕𝛼𝛼𝑖𝑖 𝜕𝜕𝑠𝑠 � 2 � 𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 2 𝑖𝑖=1 , 𝐾𝐾 = 1 2 �𝑉𝑉�⃗ (𝐺𝐺/ℜ𝑇𝑇)� 2 + 𝜔𝜔��⃗ . 𝐽𝐽𝐺𝐺(𝜔𝜔��⃗ ) + � 𝐺𝐺𝑆𝑆�����⃗ 2𝑑𝑑𝑚𝑚 𝑃𝑃∈(𝔅𝔅) + 2𝜔𝜔��⃗ . � 𝐺𝐺𝑆𝑆�����⃗ ∧ 𝐺𝐺𝑆𝑆�����̇⃗ 𝑑𝑑𝑚𝑚 𝑃𝑃∈(𝔅𝔅) , �𝑉𝑉�⃗ (𝐺𝐺/ℜ𝑇𝑇)� 2 = (𝑛𝑛0𝑟𝑟0)2. 𝑛𝑛0 is constant orbital parameter. Here 𝐽𝐽𝐺𝐺 designates the inertia tensor in 𝐺𝐺 𝐽𝐽𝐺𝐺 = � 𝐴𝐴 + 𝑁𝑁1 𝑁𝑁4 0 𝑁𝑁4 𝐵𝐵 + 𝑁𝑁2 0 0 0 𝐶𝐶 + 𝑀𝑀3 �, 𝑡𝑡𝑟𝑟(𝐽𝐽𝐺𝐺) = 𝐴𝐴 + 𝐵𝐵 + 𝐶𝐶 + 𝑁𝑁1 + 𝑁𝑁2 + 𝑁𝑁3, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 65 𝑁𝑁1 = ��−� 𝑆𝑆�1(𝑠𝑠) 3𝑛𝑛02 � 𝑑𝑑𝑢𝑢𝑖𝑖 𝑑𝑑𝑠𝑠 � 2 𝑑𝑑𝑠𝑠 𝑙𝑙 0 −� 𝑇𝑇�1(𝑠𝑠) 3𝑛𝑛02 � 𝜕𝜕𝛼𝛼𝑖𝑖 𝜕𝜕𝑠𝑠 � 2 𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 � 2 𝑖𝑖=1 , 𝑁𝑁2 = ��� 𝜌𝜌𝑢𝑢𝑖𝑖2𝑑𝑑𝑠𝑠 𝑙𝑙 0 + �𝜎𝜎[𝛼𝛼𝑖𝑖2 + 2𝑥𝑥𝛼𝛼𝑖𝑖]𝑑𝑑𝑠𝑠𝑑𝑑𝑥𝑥 𝑅𝑅 + 𝑚𝑚�𝑢𝑢𝑖𝑖2(𝑙𝑙)� 2 𝑖𝑖=1 − 𝑚𝑚∑𝑋𝑋𝐺𝐺2, 𝑁𝑁3 = ��� 𝜌𝜌𝑢𝑢𝑖𝑖2𝑑𝑑𝑠𝑠 𝑙𝑙 0 + �𝜎𝜎[𝛼𝛼𝑖𝑖2 + 2𝑥𝑥𝛼𝛼𝑖𝑖]𝑑𝑑𝑠𝑠𝑑𝑑𝑥𝑥 𝑅𝑅 − � 𝑆𝑆�1(𝑠𝑠) 3𝑛𝑛02 � 𝑑𝑑𝑢𝑢𝑖𝑖 𝑑𝑑𝑠𝑠 � 2 𝑑𝑑𝑠𝑠 𝑙𝑙 0 −� 𝑇𝑇�1(𝑠𝑠) 3𝑛𝑛02 � 𝜕𝜕𝛼𝛼𝑖𝑖 𝜕𝜕𝑠𝑠 � 2 𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 + 𝑚𝑚�𝑢𝑢𝑖𝑖2(𝑙𝑙)� 2 𝑖𝑖=1 − 𝑚𝑚∑𝑋𝑋𝐺𝐺2, 𝑁𝑁4 = ∑ (−1)𝑖𝑖 �∬ 𝑥𝑥 𝑃𝑃1(𝑠𝑠) 3𝑛𝑛0 2 � 𝜕𝜕𝛼𝛼𝑖𝑖 𝜕𝜕𝑠𝑠 � 2 𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 + ∫ 𝜌𝜌(𝑏𝑏 + 𝑠𝑠)𝑢𝑢𝑖𝑖𝑑𝑑𝑠𝑠 𝑙𝑙 0 + ∬ 𝜎𝜎(𝑏𝑏 + 𝑠𝑠)𝛼𝛼𝑖𝑖𝑑𝑑𝑠𝑠𝑑𝑑𝑥𝑥 𝑅𝑅 + 𝑚𝑚�(𝑏𝑏 + 𝑠𝑠)𝑢𝑢𝑖𝑖(𝑙𝑙)�2 𝑖𝑖=1 . Here: 𝑆𝑆�1(𝑠𝑠) = 3𝜌𝜌 𝑛𝑛02 2 [(𝑏𝑏 + 𝑙𝑙)2 − (𝑏𝑏 + 𝑠𝑠)2], 𝑆𝑆1(𝑠𝑠) = 𝑛𝑛02 2 �𝜎𝜎(𝑙𝑙 − 𝑠𝑠) + 𝑚𝑚� 𝑒𝑒 �, 𝑇𝑇�1(𝑠𝑠) = 3 𝑛𝑛02 2 �𝜎𝜎(𝑏𝑏 + 𝑙𝑙)2 − 𝜎𝜎(𝑏𝑏 + 𝑠𝑠)2 + 2 𝑚𝑚� 𝑒𝑒 (𝑏𝑏 + 𝑙𝑙)�, ⎩ ⎪⎪ ⎨ ⎪⎪ ⎧𝑋𝑋𝐺𝐺 = 1 𝑚𝑚∑ ��� 𝜌𝜌𝑢𝑢𝑖𝑖𝑑𝑑𝑠𝑠 + 𝜎𝜎�𝛼𝛼𝑖𝑖𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 + 𝑚𝑚�𝑢𝑢𝑖𝑖(𝑙𝑙, 𝑡𝑡) 𝑙𝑙 0 � 2 𝑖𝑖=1 𝑌𝑌𝐺𝐺 = 1 𝑚𝑚∑ ��� 𝜌𝜌𝑣𝑣𝑖𝑖𝑑𝑑𝑠𝑠 + 𝜎𝜎�𝛽𝛽𝑖𝑖𝑑𝑑𝑥𝑥𝑑𝑑𝑠𝑠 𝑅𝑅 + 𝑚𝑚�𝑣𝑣𝑖𝑖(𝑙𝑙, 𝑡𝑡) 𝑙𝑙 0 � 2 𝑖𝑖=1 𝑍𝑍𝐺𝐺 = 0 , 𝑚𝑚∑ is the mass of the space vehicle. Hence, the application of Hamilton's variational principle above, gives the linearized equations (1) to (10). • Appendix 3 One introduces the expression of the deformations 𝑢𝑢𝑖𝑖(𝑠𝑠, 𝑡𝑡) (17) in the equation (3) of local motion (𝐵𝐵𝐵𝐵𝑖𝑖). Taking into account the conditions (11), the expressions (13) and (6) one gets: 𝜌𝜌∑ (−1)𝑖𝑖𝜃𝜃𝑛𝑛[𝑈𝑈𝑛𝑛∗(𝑠𝑠) + (𝑏𝑏 + 𝑠𝑠)][�̈�𝑔𝑛𝑛(𝑡𝑡) + 𝜑𝜑𝑛𝑛2𝑔𝑔𝑛𝑛(𝑡𝑡)]∞ 𝑛𝑛=1 + 3𝑛𝑛02𝜌𝜌(−1)𝑖𝑖 ∑ �(𝑏𝑏 + 𝑠𝑠) − 1 2 �[(𝑏𝑏 + 𝑙𝑙)2 −∞ 𝑛𝑛=1 (𝑏𝑏 + 𝑠𝑠)2]𝑈𝑈𝑛𝑛∗′(𝑠𝑠)�′� 𝜃𝜃𝑛𝑛𝑔𝑔𝑛𝑛(𝑡𝑡) = −𝜌𝜌(−1)𝑖𝑖(𝑏𝑏 + 𝑠𝑠)��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�. (a) One multiplies the previous equation by (−1)𝑖𝑖𝜃𝜃𝑘𝑘𝑈𝑈𝑘𝑘∗(𝑠𝑠) then one integrates along (𝑆𝑆𝑆𝑆𝑖𝑖). After calculation one gets: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 66 �𝜃𝜃𝑛𝑛𝜃𝜃𝑘𝑘 ���𝜌𝜌𝑈𝑈𝑛𝑛∗(𝑠𝑠)𝑈𝑈𝑘𝑘∗(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 + �𝜌𝜌(𝑏𝑏 + 𝑠𝑠)𝑈𝑈𝑘𝑘∗(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 � [�̈�𝑔𝑛𝑛(𝑡𝑡) + 𝜑𝜑𝑛𝑛2𝑔𝑔𝑛𝑛(𝑡𝑡)] + 3𝑛𝑛02 �𝜌𝜌(𝑏𝑏 + 𝑠𝑠)𝑈𝑈𝑘𝑘∗(𝑠𝑠)𝑑𝑑𝑠𝑠𝑔𝑔𝑛𝑛(𝑡𝑡) 𝑙𝑙 0 � ∞ 𝑛𝑛=1 + 3 2 𝑛𝑛02�𝜃𝜃𝑛𝑛𝜃𝜃𝑘𝑘 �𝑆𝑆�1∗(𝑠𝑠)𝑈𝑈𝑛𝑛∗′(𝑠𝑠)𝑈𝑈𝑘𝑘∗′(𝑠𝑠)𝑑𝑑𝑠𝑠𝑔𝑔𝑛𝑛(𝑡𝑡) 𝑙𝑙 0 ∞ 𝑛𝑛=1 = −𝜃𝜃𝑘𝑘 �𝜌𝜌(𝑏𝑏 + 𝑠𝑠)𝑈𝑈𝑘𝑘∗(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 ��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�. One gets after some similar calculations the equations describing the motion of (𝐵𝐵𝐵𝐵𝑖𝑖) and the motion of (𝑇𝑇𝑆𝑆𝑖𝑖). One makes the summation of these three equations. Using the equations of eigenfrequencies (13) and making a summation on i, the equation above becomes: �𝜃𝜃𝑛𝑛𝜃𝜃𝑘𝑘 �2��𝜌𝜌𝑈𝑈𝑛𝑛∗(𝑠𝑠)𝑈𝑈𝑘𝑘∗(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 + �𝜎𝜎𝑒𝑒𝛼𝛼𝑛𝑛∗(𝑠𝑠)𝛼𝛼𝑘𝑘∗(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 + 𝑚𝑚�𝑈𝑈𝑛𝑛∗(𝑙𝑙)𝑈𝑈𝑘𝑘∗(𝑙𝑙)� − 𝐶𝐶� [�̈�𝑔𝑛𝑛(𝑡𝑡) + 𝜑𝜑𝑛𝑛2𝑔𝑔𝑛𝑛(𝑡𝑡)] ∞ 𝑛𝑛=1 + 3𝑛𝑛02�𝜃𝜃𝑛𝑛𝜃𝜃𝑘𝑘 ��𝑆𝑆�1∗(𝑠𝑠)𝑈𝑈𝑛𝑛∗′(𝑠𝑠)𝑈𝑈𝑘𝑘∗′(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 + �𝑒𝑒𝑇𝑇�1∗(𝑠𝑠)𝛼𝛼𝑛𝑛∗′(𝑠𝑠)𝛼𝛼𝑘𝑘∗′(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 − 𝐶𝐶�𝑔𝑔𝑛𝑛(𝑡𝑡) ∞ 𝑛𝑛=1 = 𝐶𝐶𝜃𝜃𝑘𝑘��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�. One uses normalized orthogonality property (15) and the expression of 𝑚𝑚𝑛𝑛𝑘𝑘 (19): �𝛿𝛿𝑛𝑛𝑘𝑘[�̈�𝑔𝑛𝑛(𝑡𝑡) + 𝜑𝜑𝑛𝑛2𝑔𝑔𝑛𝑛(𝑡𝑡)] ∞ 𝑛𝑛=1 + 3𝑛𝑛02�𝑚𝑚𝑛𝑛𝑘𝑘𝑔𝑔𝑛𝑛(𝑡𝑡) ∞ 𝑛𝑛=1 = 𝐶𝐶𝜃𝜃𝑘𝑘��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�. • Appendix 4 One multiplies the previous equation by (−1)𝑖𝑖(𝑏𝑏 + 𝑠𝑠) the equation (a) of the appendix 3, then one integrates along (𝑆𝑆𝑆𝑆𝑖𝑖). After calculation one gets: �𝜃𝜃𝑛𝑛 �𝜌𝜌�(𝑏𝑏 + 𝑠𝑠)𝑈𝑈𝑛𝑛∗(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 + 𝜌𝜌�(𝑏𝑏 + 𝑠𝑠)2𝑑𝑑𝑠𝑠 𝑙𝑙 0 � [�̈�𝑔𝑛𝑛(𝑡𝑡) + 𝜑𝜑𝑛𝑛2𝑔𝑔𝑛𝑛(𝑡𝑡)] ∞ 𝑛𝑛=1 + 3𝑛𝑛02𝜌𝜌�𝜃𝜃𝑛𝑛𝑔𝑔𝑛𝑛(𝑡𝑡) ∞ 𝑛𝑛=1 ��(𝑏𝑏 + 𝑠𝑠)2𝑑𝑑𝑠𝑠 𝑙𝑙 0 + 1 2 �[(𝑏𝑏 + 𝑙𝑙)2 − (𝑏𝑏 + 𝑠𝑠)2]𝑈𝑈𝑛𝑛∗′(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 � = −𝜌𝜌�(𝑏𝑏 + 𝑠𝑠)2𝑑𝑑𝑠𝑠 𝑙𝑙 0 ��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�. One gets after some similar calculations the equations describing the motion of (𝐵𝐵𝐵𝐵𝑖𝑖) and the motion of (𝑇𝑇𝑆𝑆𝑖𝑖). One makes the summation of these three equations. Using the equations of eigenfrequencies (13) and making a summation, the equation above becomes: American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 67 [2ℒ − 𝐶𝐶]�𝜃𝜃𝑛𝑛[�̈�𝑔𝑛𝑛(𝑡𝑡) + 𝜑𝜑𝑛𝑛2𝑔𝑔𝑛𝑛(𝑡𝑡)] ∞ 𝑛𝑛=1 + 3𝑛𝑛02�𝜃𝜃𝑛𝑛𝐶𝐶𝑛𝑛∗𝑔𝑔𝑛𝑛(𝑡𝑡) ∞ 𝑛𝑛=1 = −2ℒ��̈�𝜆(𝑡𝑡) + 3𝑛𝑛02𝜆𝜆(𝑡𝑡)�. with: 𝐶𝐶𝑛𝑛∗ = �𝑆𝑆�1∗(𝑠𝑠)𝑈𝑈𝑛𝑛∗′(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑙𝑙 0 + �𝑒𝑒𝑇𝑇�1∗(𝑠𝑠)𝛼𝛼𝑛𝑛∗′(𝑠𝑠)𝑑𝑑𝑠𝑠 + 𝑏𝑏𝑒𝑒𝑇𝑇�1∗(0)𝛼𝛼𝑛𝑛∗′(0) 𝑙𝑙 0 + 2ℒ. One uses the equation of generalized coordinates (18) and after some manipulations: �̈�𝜆(𝑡𝑡) �𝐶𝐶(2ℒ − 𝐶𝐶)�𝜃𝜃𝑛𝑛2 ∞ 𝑛𝑛=1 + 2ℒ� + 3𝑛𝑛02 �𝜆𝜆(𝑡𝑡) �𝐶𝐶(2ℒ − 𝐶𝐶)�𝜃𝜃𝑛𝑛2 ∞ 𝑛𝑛=1 + 2ℒ� + �𝜃𝜃𝑛𝑛 �𝐶𝐶𝑛𝑛∗𝑔𝑔𝑛𝑛(𝑡𝑡) − 𝐶𝐶(2ℳ − 𝐶𝐶)�𝑚𝑚𝑛𝑛𝑝𝑝𝑔𝑔𝑝𝑝(𝑡𝑡) ∞ 𝑝𝑝=1 � ∞ 𝑛𝑛=1 � = 0. The left side of the preceding equation which is a polynomial in 𝑛𝑛0 is equal to zero whatever 𝑛𝑛0. One deduces: �̈�𝜆(𝑡𝑡) �𝐶𝐶(2ℒ − 𝐶𝐶)�𝜃𝜃𝑛𝑛2 ∞ 𝑛𝑛=1 + 2ℒ� = 0 This equation is available whatever the parameter time 𝑡𝑡. Finally: �𝜃𝜃𝑛𝑛2 ∞ 𝑛𝑛=1 = 2ℒ 𝐶𝐶(𝐶𝐶 − 2ℒ). • Appendix 5 Figure 5: Representation of the function 𝑖𝑖𝑁𝑁 as a function of 𝑁𝑁. 0 5 10 15 20 25 30 35 0 1 2 3 4 x 10 -5 va lu e of s um m on s number of Frequency overall American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2019) Volume 55, No 1, pp 50-68 68 Figure 6: Deformation of the membrane solar arrays (𝜑𝜑𝑛𝑛/2𝜋𝜋 = 0.1 Hz). Figure 7: Deformation of the membrane solar arrays (𝜑𝜑𝑛𝑛/2𝜋𝜋 = 0.597 Hz). Figure 8: Deformation of the membrane solar arrays (𝜑𝜑𝑛𝑛/2𝜋𝜋 = 1.090 Hz). 0 5 10 15 20 25 30 -2 -1 0 1 2 Length of mast (meter) L en gt h of th e in te rfa ce (m et er ) 0 5 10 15 20 25 30 -2 -1 0 1 2 Length of mast (meter) L en gt h of th e in te rfa ce (m et er ) 0 5 10 15 20 25 30 -2 -1 0 1 2 Length of mast (meter) L en gt h of th e in te rfa ce (m et er ) References Model description