Dynamics of the orbital deployment of an elastic ring-shaped antenna

2011

Int Appl Mech

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The natural frequencies and modes of a flexible elastic ring fixed at one point are determined numerically by solving an eigenvalue problem for the differential operator of the equation of its motion. The ring models a large circular antenna that slowly expands under zero gravity. Bending and torsional vibrations in a plane perpendicular to the antenna plane are studied Introduction. The problem of determining the natural frequencies and modes of either in-plane or out-of-plane vibrations 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 [1,4,7].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. The majority of publications address a free ring for which periodicity conditions play the role of boundary conditions. However, the accuracy of these data is low because they were mainly obtained with simplified methods [2,3,9]. An extensive bibliography on the subject can be found in [2,3,11]. We failed to find any discussion of the case of an elastic ring with one fixed point in the literature.The present paper continues the studies of the modes and frequencies of flexural in-plane vibrations of a circular ring fixed at one point [13,14]. In the present paper, we will discuss the natural modes and frequencies of such a ring under zero gravity that were obtained by solving a boundary-value problem for the equations of motion of an elastic ring with direct numerical methods easily implementable on modern computers. A similar problem was addressed in [12].1. Problem Formulation. Let us determine the natural modes and frequencies of free flexural in-plane vibrations of a ring fixed at one point. Without loss of generality, we assume that the ring is fixed at the points j = 0 and j p = 2 , where j is the

mentioning

confidence: 80%

Natural frequencies and modes of out-of-plane vibrations a fixed ring

2011

Int Appl Mech

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“…(2) and (3) and supplementing them with suitable kinematic equations, we can formulate a Cauchy problem for studying the behavior of the spacecraft during the deployment of the pantograph. Omitting details, we choose the Rodrigues-Hamilton parameters to define the angular position of the body-fixed frame in the orbital frame (see [25] for details and for the transformation between quaternions and Krylov angles).…”

Section: Mathematical Modelmentioning

confidence: 99%

Dynamics of an Unstabilized Spacecraft During the Deployment of an Elastic Pantograph Structure

2014

Int Appl Mech

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An unstabilized spacecraft with a pantograph structure deployed in orbit to carry solar batteries is the subject of study. The objective of the study is to construct a mathematical model of this system taking into account the elastic properties of elements of the pantograph in longitudinal and transverse directions. This model is based on the Lagrangian formalism as applied to a mechanical system with rheonomic constraints. The expressions for the coefficients of the equations of motions are obtained using Mathematica 5 © . A Fortran application software package is used for numerical simulation of dynamic processes. This package can be adapted, if necessary, to study other structures. The behavior of the spacecraft during the deployment of the pantograph is numerically analyzed using different values of spacecraft parameters and the parameters of the deployment process in the gravity field.Introduction. The dynamics of reconfigurable systems is one of the promising and important research areas in mechanics [9,[11][12][13]23]. Modern spacecraft incorporate various components that are deployed in orbit (solar batteries (SB), gravity-gradient booms, antennas, etc.). Such mechanical systems are delivered to orbit compactly packaged and then are deployed there. There is literature on the deployment of elastic elements from a fixed base and from a spinning spacecraft [9,[11][12][13]23], including the deployment of a gravity-gradient boom [15,19]. In these studies, the maximum bending moments, deflections of booms, and optimal deployment time were assessed using various simplifying assumptions. Cherchas [14] studied the dynamics of spin-stabilized satellites during the extension of long flexible booms. He has determined the maximum nutation and precession angles after the deployment of the boom and its maximum bending moments and deflections, derived the equations of motion using Lagrange's equations, and discretized the elastic degrees of freedom by a modal analysis. In [20,21,24], the dynamics of deployable elastic elements is also described in terms a modal analysis with time-dependent natural modes.Banerjee and Kane [7] present a new method for simulating the motion of a beam that is being extruded from, or retracted into a rotating rigid body. In essence, the method consists of modeling the beam as a series of elastically connected rigid links and then working with equations of motion linearized in the modal coordinates for the links outside the rigid body. In [6,8], the analysis was extended to large deflections and used an order-n formulation for a variable number of bodies. The modeling has revealed that the tip deflections are very sensitive to the extrusion/retraction rate, the retraction process is less stable than the extrusion process, and the behavior of the beam is strongly dependent on the angular rate of the body.The deployment of such elements is a substantial perturbation to the motion of the spacecraft around its center of mass. It is impossible to model such perturbations within the framework of the...

“…The parameters of Rodrigues-Hamilton were chosen as kinematic parameters, which determine the attitude of the main body in the orbital reference frame in order to anticipate possible singularities in the kinematical equations of the spacecraft while conducting the simulation. Reference [17] gives details of mutual transformations between quaternion and conventional attitude angles.…”

Section: Mathematical Model Of Systemmentioning

confidence: 99%

“…Only a few of the studies, which deal with the dynamics of the SC of variable configuration, take into account the dynamics of the deployment actuating mechanism. Zakrzhevskii and Khoroshilov [17] studied the dynamics of the orbital deployment of an elastic ring-shaped antenna taking into account dynamics of actuating drive. Creamer [7] studied the transverse motion of the truss boom deployed along an axis, in order to compare the theory and experiment.…”

Section: Introductionmentioning

confidence: 99%

Modelling of Spacecraft Dynamics at Deployment of Large Elastic Structure

Modelling and Simulation in Engineering

2012

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In this paper, a new approach to the modelling of the deployment dynamics of a flexible multi-body system with the time dependent configurations is demonstrated in the frame of the study the dynamics of a spacecraft with the gyro-gravitational system of stabilization. Primarily the gravitational stabilizer that is made as a pantograph structure is in a compact form. The deployment of a flexible pantograph structure is performed after placing the spacecraft into orbit and completion of the preliminary damping by a special jet-propelled system, and after uncaging the gyros. After its deployment, the pantograph turns into an elongated structure that serves as a gravitational stabilizer and carrier of solar batteries. The objective of the study is the creation of the generalized mathematical model and the conducting of the computational modelling of the spacecraft dynamics. The equations of motion are derived with the use of the Eulerian-LaGrangian formalism and symbolic computing. Numerical simulations of the typical operational mode of the system are conducted taking into account various control profiles for the deployment. Numerical results indicate that the system used for attitude stabilization ensures the shape of the deployed design and prescribed accuracy of the orientation.