THE DEFINITE QUESTIONS OF SIMULATION OF TRANSFORMABLE SPACE STRUCTURES DYNAMICS

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Introduction. The intensive development of space technology poses the task of creating of fundamentally new large space structures [1; 2]. A necessary stage of their development is a preliminary investigation of the characteristics of both free and controlled such structures' motion by means of mathematical modeling of their dy- namics. Mathematical modeling of dynamics of such struc- tures is a subject of wide research. The main methods of analysis of the mechanical behavior of large space struc- tures are described in [3]. Transformable large space structures have different configurations in the transport state and in the operating position in orbit. As a rule, de- ployment of the developed transformable structures, is unique process for each considered system, however, in some cases it is possible to find a common way of model- ing the dynamics of their deployment. A model that is simple but well enough considering the special features of structures consisting of tens, hundreds and even thousands elements interconnected by hinges is accepted as a design scheme [4-9]. System of differential equations of trans- formable structure elements motion is usually known as a model of dynamics of such structure, and a numerical solution of these equations is usually understood as the mathematical modeling of structure dynamics. Calculation model for the analysis of the deploy- ment of the circuit antenna frame. Among the large space structures, a special place is occupied by the multi- link transformable circuit antennas, the load-bearing frame of which provides the minimal dimensions of the structure in the folded transport position and required rigidity of the structure in the unfolded working one (fig. 1). The frame consists of one-type elements con- nected by elastic hinges. The elements can be made of both metals and composite materials (fig. 2). The hinges contain one-sided stops to provide of deployment of the reflector into the opened working position, making de- sired shape of the reflecting surface its maintenance dur- ing operating life. Deployment of considered structures occurs automatically due to potential energy accumulated in the elastic elements of the hinges when the reflector had been being folded into the transport state. Computer modeling of steps of the multi-link load-bearing frame deployment gives us an opportunity to consider various schemes of this process and evaluate correctness of the technical solutions incorporated in the structure's scheme at the project level. Modeling of deployment of the load-bearing frame with a diameter of 5 m using MSC.Adams and MSC. Patran/Nastran software is under consideration [10]. Fig. 1. The spacecraft with circular transformable antennas with diameters of 20 m Рис. 1. Космический корабль с трансформируемыми кольцевыми антеннами диаметром 20 м а b Fig. 2. The elements of transformable space structures: a - shaped rectangular panels; b - thin-walled quill rods of unidirectional carbon fiber reinforced polymer with longitudinal fibers orientation Рис. 2. Элементы трансформируемых космических конструкций: а - прямоугольные профилированные панели; б - тонкостенные трубчатые стержни из однонаправленного углепластика с продольным расположением волокон The folding load-bearing frame consists of two pack- ets of shaped rectangular panels 550 ´ 230 ´ 6 mm (13 panels per package) one side of which is pivotally connected with the spacecraft, and another side the same way is connected with short closing panel. As a design scheme for studying of dynamics of the antenna folding frame deployment using MSC.Adams, a system of rigid bodies interconnected by hinges is ac- cepted. At a certain relative position of the adjacent panangle of adjacent panels, when the placing on the stop is getting; j& i is the relative angular velocity of the adjacent panels. When moving, adjacent panels can rotate towards each other and touch each other. The model contains stops preventing a contact between the panels. They are pre- sented by massless elastic elements with nonlinear de- pendency of the moment from the opening angle els during the deployment, restrictions limiting mutual M (j ) = ìï 0, if ji > jcont , Подпись: i angular displacement of panels are applied to provide conti i í-c (j - j ), if j £ j , i holding of the antenna frame in the operating unfolded îï conti i conti i conti position. The panels of the foldable antenna frame are connected with each other by external and internal hinges where ccont is the stiffness coefficient of the i-th elastic modeled by means of “axial joint” elements (Revolute element; jcont is value of the i-th opening angle, when i Joint in terms of Adams software). Torsion springs in the hinges are modeled by means of elastic and dissipative elements with linear dependencies of the moments from the opening angles and the relative angular velocities of the panels (Torsion Spring in terms of Adams software), the stiffness coefficient, viscous resistance coefficient and preliminary angles of twist of the springs for which are given. The external and internal hinges have different preliminary angles of twist, which have been chosen the way to provide a circle shape of the folding load-bearing frame in the operating unfolded position. The panels are modeled by rigid rectangular parallele- pipeds with dimensions of 550 ´ 230 ´ 6 mm. The density of the equivalent material of the panels in Adams is cho- sen so that the mass of the simulated non-profiled panels is equal to the mass of the real profiled panels. The mo- ments of inertia of the panels in Adams are taken equal to the moments of inertia of the real profiled panels. Free ends of the panels placing in the base of the left and right packets are considered to be hinged. The power characteristic of each of the elastic and dis- sipative elements modeling the torsion spring in the hinges [11-13] is determined by the following ratio i Mi (ji , j& i ) = ci (jtw - ji ) - mi j& i , where ci is the stiffness coefficient of the i -th spring; mi is the coefficient of viscous resistance of the i -th spring; i ji is the current opening angle of adjacent panels; jtw is the preliminary angle of twist of the i -th spring. At a certain relative position of the adjacent panels when they are opening, and the opening angle ji reaches the adjacent elements are in contact. As a result of calculations, dependencies of coordi- nates, speeds and accelerations of centres of mass of the load-bearing frame panels from time, as well as depend- encies their angular speeds and accelerations from time were received. Analysis of stressed state of the frame elements. The stressed state of the elements of the load-bearing frame during its deployment is determined by the impact loads arising when the adjacent frame panels are placing on the stops. The impact loads are obtained from the analysis of the frame deployment dynamics using the MSC. Adams software. Simulation shows that the panels of the load-bearing frame are placed on the stops at the different time points. At each time point, various groups of panels in different places of the frame reach their stops. Several time points are chosen when relative velocities of the adjacent panels from the certain group are ultimate. To investigate the stressed state of the load-bearing frame panels during deployment, in MSC. Patran/Nastran software the frame shape at the selected moments is adopted as a design scheme. At each considered moment it is supposed that panels are placed on their stops and the structure behaves as an elastic rod with specified charac- teristics. This approach gives the safety margin, since the mobility of frame panels relative to each other leads to reduction of real stresses due to the loss of kinetic energy in the joints. Knowing value of the angular velocity of the hinged first panel of one of the packets and knowing values of the angular velocities of the other panels of the packet, the a certain value jstop corresponding to operating position velocity field at the certain points of each panel of the i of panels, restrictions limiting the mutual angular dispacket is determined by the formula v n = v p + w ´ r , placement of the panels are applied to panels by the cer- tain making of their hinges. Physically the restrictions are made as various stops, which are modeled by massless elastic and damping elements with nonlinear dependency of the moment from the opening angle and can be written where v p is the speed of the poles, for which each joint on the antenna frame from the attachment base point to the short closing panel is consistently accepted; w is the angular velocity vector of the corresponding panel; as r = rn - rp , rp is the radius vector of the pole, rn is the M (j , j& ) = ìï 0, if ji < jstop i , radius vector of the point, where the velocity is deterstopi i i í-c (j - j ) -m j& , if j ³ j mined. The radius vectors are specified in the inertial îï stopi i stopi stopi i i stopi frames of reference whose origins are located at the where cstop is the stiffness coefficient of the i-th elastic places of hinge fastening for each frame packet on a rigid base. Velocities of the opposite packets panels are taken i i element; mstop is the coefficient of viscous resistance of equal in magnitude and opposite in direction. Thus, the i the i-th damping element; jstop is value of the opening opening of the frame occurs symmetrically. The angular velocities of the panels as well as the ve- locities of their centres of mass (hereinafter used to test calculation of velocities of the selected model points) obtained for each time step of the deployment calculation are exported from Adams to a certain file as a table. To find velocities of the model selected points from the deanalysis time integration step should be selected taking the conditions as at least ten steps per response period time for the maximal frequency of interest. Then, if we take into account first ten frequencies of natural oscilla- tions of the antenna frame in the plane of deployment, in our case the time integration step will be equal to pendency given above for each selected moment of time a C++ program was written. To perform stress analysis of the frame panels, finite Dt = T10 = 1 = 1 10 10 × f10 10 × 9 Hz » 0, 01sec. element model of the frame was built using At the same time, it is recommended to take the time MSC.Patran/Nastran software. Analysis of the stressed integration step finding within Dt = 10-4...10-3 sec . interand deformed state of antenna frame was carried out by means of Direct Transient Response analysis (SOL 109, study of transient process) both without taking into ac- count damping and with consideration of damping. Each panel is divided into ten “beam” finite elements. In total, the model consists of 270 such elements. All fi- nite elements including elements at the panel boundaries are rigidly connected to each other. The panels placed at the base of the left and right packet are considered to be clamped by their free ends. As a dynamic model of the folding frame the finite element model is accepted M u&& + D u& + K u = P , where M, D, K are mass, damping and stiffness matrices respectively, P is the external loads vector, u is the vec- tor of nodal displacements, u& is the vector of nodal ve- locities. In our case P = 0 . Calculated values of velocities in the selected points of the frame model are taken as initial conditions for cal- culation of the transient process using MSC. Nastran soft- ware, that is, when t = 0: u (0) = u0 , u& (0) = u& 0, where u0 , u& 0 are vectors of the initial nodal displacements and velocities. In accordance with the recommendations presented in the MSC. Nastran user manuals, transient response val, so to obtain more accurate stress values, the time in- tegration step during the calculations was taken equal to Dt = 10-3 sec. Structural damping in Direct Transient Response analysis is taken into account by usage of the equivalent viscous damping. For the sinusoidal displacement re- sponse with constant amplitude, the structural damping force is constant and the viscous damping force is propor- tional to the frequency. For such a response, the two damping forces are equal at some frequency. Having done calculations without damping it was clear that the fre- quency of the structure oscillations under the impact had closed value to the second natural oscillations frequency in the plane of the opening of the frame. So the second natural oscillations frequency was chosen as a frequency of equivalence of structural damping and viscous one. As a results of investigation of stressed and deformed states of antenna circuit foldable frame elements both without taking into account damping and with considera- tion of damping, stresses arising in the foldable frame elements at the certain time points during deployment are found. Fig. 3 shows the stressed state of the antenna frame at the time point of t = 5.925 sec., when small stresses occur, looking all considered cases. The values of the stresses occurring in the panels are shown the same. a b Fig. 3. Stressed state of the antenna frame at time point of t = 5.925 sec.: a - without consideration of damping; b - with consideration of damping Рис. 3. Напряжения в момент времени t = 5,925 с.: a - без демпфирования; б - с демпфированием Conclusion. It is possible to reduce or even exclude impact loads during the process of uncontrolled deploy- ment by reducing the initial potential energy of the springs, i.e. by changing the elastic characteristics of the spring elements included in the structure. At the same time, the angular velocities at the time of placing of adja- cent structural elements on the stops will accordingly de- crease. This will lead to decreasing of the velocities de- termining the magnitude of the impact impulse acting on the considered elastic transformable structure. However, this way is not always possible, since besides the smooth and reliable deployment of the transformable structures, it is necessary to guarantee their following operation in or- bit. Therefore, in order to provide a controlled transforma- tion of the system from the transport folded position to the operating opened one, it is necessary either to develop the structure with an additional cable synchronization system, or to replace the springs by operating drives. This will significantly complicate the structure and increase its mass. It is possible to create large transformable space structures in orbit, using for their deployment actuators made of materials with a shape memory effect [14; 15]. There are several types of actuators based on materials with shape memory effect. One of them can be called a one-dimensional direct-acting actuator, the another one - an actuator with aftereffect, which uses a spring or differ- ential drive making an active retarding force and damping for creating force of actuator itself.

About the authors

Zhang Zikun

Bauman Moscow State Technical University

5, 2-nd Baumanskaya St., building 1, Moscow, 105005, Russian Federation

V. N. Zimin

Bauman Moscow State Technical University

Email: zimin@bmstu.ru
5, 2-nd Baumanskaya St., building 1, Moscow, 105005, Russian Federation

A. V. Krylov

Bauman Moscow State Technical University

5, 2-nd Baumanskaya St., building 1, Moscow, 105005, Russian Federation

S. A. Churilin

Bauman Moscow State Technical University

5, 2-nd Baumanskaya St., building 1, Moscow, 105005, Russian Federation

References

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