The method of singular value decomposition is shown to have useful application to the problem of reducing the equations of motion for a class of constrained dynamical systems to their minimum dimension. This method is shown to be superior to classical Gaussian elimination for several reasons: {i) The resulting equations of motion are assured to be of full rank, (ii) The process is more amenable to automation, as may be appropriate in the development of a computer program for application to a generic class of systems. (Hi) The analyst is spared the responsibility for the selection of specific coordinates to be eliminated by substitution in each individual case, a selection that has no physical justification but presents abundant risk of mathematical contradiction. This approach is shown to be very efficient when the governing dynamical equations are derived via Kane's method.
Floating reference frames which move with the flexible body under dynamic analysis offer the advantages of a linear vibration analysis in the presence of large sytem rotations. When the deformations of an elastic continuum are expanded in terms of the free-free modes on an unconstrained system, the rigid-body modes are found to be fixed in a reference frame called the Tisserand frame, with respect to which the relative momentum is zero. This result also guarantees the independence of small variations of frame motions and coordinates for all modes with nonzero natural frequencies, a condition which can greatly simplify the formulation of equations of motion. A modified Tisserand constraint is introduced in order to define a floating reference frame with similar properties for an elastic body which contains spinning rotors.
Rigid-body approximations for turbulent motion in a liquid-filled, spinning and precessing, spherical cavity are presented. The first model assumes the turbulent liquid to spin and precess as a rigid solid sphere coupled to the cavity wall by a thin layer of massless viscous liquid. The second model replaces the layer of massless viscous liquid by a series of n concentric rigid spherical shells. The number and thickness of the shells can be varied so that the interior sphere varies from a negligible diameter to nearly the diameter of the cavity. Although these models do not provide solutions of the fluid equations of motion, they yield steady-state energy dissipation rates that compare favorably with existing experimental data associated with turbulent flow in such a cavity. The models also duplicate several other important features of rotating fluid flow theory. In particular, the motions of the concentric shells exhibit characteristics associated with a classic Ekman layer structure.
Starting from basic physical considerations, this paper develops a concept of the degree of controllability of a control system, and then develops numerical methods to generate approximate values of the degree of controllability for any linear time-invariant system. In many problems, such as the control of future, very large, flexible spacecraft and certain chemical process control problems, the question of how to choose the number and locations of the control system actuators is an important one. The results obtained here offer the control system designer a tool which allows him to rank the effectiveness of alternative actuator distributions, and hence to choose the actuator locations on a rational basis. The degree of controllability is shown to take a particularly simple form when the dynamic equations of a satellite are in second-order modal form. The degree of controllability concept has still other fundamental uses-it allows one to study the system structural relations between the various inputs and outputs of a linear system, which has applications to decoupling and model reduction.
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