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The object to be studied is a spacecraft with a deployable pantograph structure as a solar-battery carrier. The objective of research is to design a mathematical model of this structure taking the elasticity of pantograph elements into account. The Lagrangian formalism is followed. To model the dynamic processes in the system, a software package has been developed, which can be adapted, if necessary, to study deployable structures of other types. The behavior of the structure during deployment, collapse, and redeployment under the action of various perturbations is modeled numerically. Plots illustrating the variation of characteristic variables are presented Introduction. Dynamics of reconfigurable systems is one of the promising and important areas of research in mechanics [1,2,5,[8][9][10][11]. Modern spacecraft incorporate elements that transform when in orbit (solar batteries, gravity-gradient booms, antennas, etc.). Such mechanical systems are delivered into orbit compactly packaged and then are deployed into the operational configuration.Issues of current importance in research of reconfigurable systems are minimization of deployment time and energy and the effect of reconfigurable structures on the motion and attitude of the spacecraft. Also, of vital importance is mass reduction for such systems. Their elements should be considered as elastic bodies.1. Mechanical Model. The object of study is a pantograph structure consisting of two connected parallelogram-type linkages (Fig. 1). The mechanical model is a reconfigurable structure with a pantograph-like deployment mechanism mounted on a rigid body.
The natural modes and frequencies of an elastic ring fixed at a point are determined by numerically solving a boundary-value problem for a differential operator of the sixth order. The ring models a large circular antenna that slowly expands under zero gravity. The in-plane flexural vibrations of the ring are analyzed. Numerical results are presented Keywords: natural frequency, natural mode, flexible ring, vibrations, differential operator Introduction. The problem of determining the natural frequencies and modes of a ring arose in studying the dynamics of a spacecraft that carries an antenna of variable geometry (a compact roll of an elastic ribbon slowly deployed, in a programmed manner, into a ring). Assuming that the deployment duration is much longer than the period of vibration of the ring in the first mode, we can neglect the perturbations of natural modes of the ring caused by the variation in its radius with time.Structures as a circular antenna can be modeled by thin curved bars. If the elongation is small, the question whether such a model is applicable can be answered only after solving the appropriate problem of elasticity. If it is, then it remains to be seen whether the model involves factors that disturb the vibrations. The linear analysis of the free and forced vibrations of curved bars is based on the assumption that the displacements and angles of rotation are small. In this case, the Bernoulli hypothesis applies well and vibrations are described by a system of differential equations for the vector of displacements and angular rates of rotation [1,3,7]. The solution can only be found if the coefficients of this system are constant. There are three such cases [3]: a straight bar , ring, and helix, the first two being special cases of the third. For the coefficients of the equations of motion to be constant, it is particularly necessary that the bar have constant cross-section. In the case of a closed ring, the vector system breaks up into two scalar subsystems describing in-plane and out-of-plane vibrations of the ring. There are Hoppe's approximate solutions for the natural frequencies of in-plane flexural vibrations of a closed ring [7] and Michell's solutions for the natural frequencies of out-of-plane vibrations of a closed ring [7]. There are also two more axisymmetric approximate solutions obtained by Hoppe [7] for the torsional in-plane vibrations of a ring and by Basset [7] for the torsional out-of-plane vibrations of a ring.The analytic solution for an open ring appears so complex that most authors preferred approximate analytic and numerical methods to find its natural frequencies and modes [5].Though the natural modes and frequencies of a ring were already studied hundred years ago, modern computers allow us to approach this problem from a qualitatively different level. There are published numerical data on the several first natural frequencies and modes of in-plane and out-of-plane vibrations of closed rings with certain boundary conditions. However, the accuracy of these data is l...
A generalized mathematical model of an unguided spacecraft that carries an elastic body of variable geometry (a deployable ring-shaped antenna) is constructed. Its dynamics is simulated numerically
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