9.2 Structures 9.2.1 Modeling In the early days of space exploration, Spacecraft were relatively small, compact, and mechanically simple. They were modeled as rigid bodies for purposes of motion simulation, stability determination, and active control design. Even then this approximation, introduced by neglecting flexibility, was sometimes unwarranted as demonstrated by the instability of the Explorer I Spacecraft in 1958. The abnormal behavior of this satellite was attributed to energy dissipation induced by vibration of the long wire turnstile antennas, which protruded from the cylindrical housing of the satellite. These and subsequent experiences led to a vigorous program of research in multibody systems with flexible components. The approximate analytic and numerical techniques developed in the course of this research proved to be quite successful in designing Spacecraft with modest size and flexibility. However, large flexible structures required for solar power generation in space present new challenges to accurately model their dynamics and develop control procedures. In general, these structures are characterized by interconnected flexible bodies having small structural damping and low, closely spaced frequency spectra. The tasks of controlling the rigid body rotations (librations) for pointing accuracy and stabilizing the flexible structure vibrations pose dynamical and control design problems, never encountered before. This is even more the case for the unprecedented size of space solar power program structures. A question arises: why not conduct ground based experiments before deploying a structure or its subassemblies in space? Unfortunately, ground-based experiments have their limitations as accurate representation of the gravitational, magnetic, solar radiation, free molecular and other fields has proven to be elusive. Thus, refined mathematical models and comprehensive control simulation techniques will be necessary to accurately and reliably predict complex dynamical interactions in large space structures. Moreover, as Spacecraft become more complex and architecturally metamorphical, development of precise dynamic models and derivation of the corresponding equations of motion for transient and evolutionary stages become overwhelming. Hence, considerable attention has been directed towards development of computer algorithms to automate the dynamic simulation process for complex systems. In an effort to make these programs applicable to a large class of systems, the number of structural members constituting a system is considered a variable, i.e. left arbitrary. The phrase “multi-body computer program” has been coined to denote applicability of the code to a system with an arbitrary topology. Multibody Dynamics In the field of Spacecraft dynamics, the first paper describing a general multibody dynamics formulation was published by Hooker and Margulies in 1965. This work was based on Newton- Euler equations and is applicable to a point connected set of rigid bodies in a topological tree, where the constraint torques are obtained via Lagrange's multipliers. At about the same time, Roberson and Wittenburg treated a system with n-rigid bodies independently and derived the dynamic equations in the matrix form. Ever since, a number of multibody formalisms have been reported in the literature. Ness and Farrenkopf, and Ho and Gluck extended the above models to the flexible n-body system. Ness and Farrenkopf chose the unified approach to deal with the nonlinear equations for the total motion of the system, while Ho and Gluck opted for the perturbation approach to deal with the flexibility dynamics. Kane and Levinson employed D'Alembert's principle and the concept of angular momentum for derivation of the equations of motion for a flexible tree type topological system. Vu-Quoc and Simo have proposed multibody formulations for both open chain and closed-loop structures. Modi and Lips have presented a general Lagrangian formulation for the librational dynamics of cluster type Spacecraft with an arbitrary number of deploying flexible appendages. Modi and Ibrahim extended the above model to include shift in the center of mass, changing central rigid body inertia and offset of the appendage attachment point. Modi and Ng further extended the multibody formulation to include an all flexible two-tier tree type configuration, incorporating thermal deformations and appendage deployment maneuvers.
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