Keywords: equations : differentielles ; equations : lineaires ; elements : finis Reference Record created on 2005-11-18, modified on 2016-08-08

Numerical solution partial differential equations in science and engineering

Generic models of flexible space structures are investigated from the infinite discrete-spectrum viewpoint of distributed parameter systems. The models are simple enough to treat analytically, yet complicated enough to demonstrate the practical usefulness of the transcendental-transfer-function modeling for the purposes of preliminary control synthesis. Transfer functions of the various generic models are derived analytically, and their pole-zero patterns are investigated. The alternating pole-zero pattern of a transfer function from an actuator to the colocated sensor is well-known. It is, however, newly found that in certain colocated cases, each mode has an associated zero of higher frequency than the pole; in fact, the rigid-body mode has an associated zero very close to the origin. This direct transmission property must be taken into consideration when designing large space structures such as the dual-keel Space Station, which has a pole-zero pattern very similar to that of the generic models. The practical significance of such pole-zero patterns on colocated control is discussed.

Pole-zero modeling of flexible space structures

Coupled vibration analysis for equivalent dynamic model of the space antenna truss

A homogenized modeling technique is developed in this paper to account for viscous damping properties in beam-like lattice structures of repeated patterns. Necessary assumptions regarding the local free deformations, shear deformation beam theory, and compatibility conditions are made to obtain an equivalent continuum model of a three-dimensional lattice. A dissipated energy equivalence approach is then used to relate the energy dissipation for a general case of damping matrix in a lattice element to an equivalent proportionally damped model. As a result, a continuum model with Kelvin–Voigt damping is obtained. Damped bending natural frequencies, frequency response functions, and damping ratios are found using this method and compared to the results of a finite-element analysis for several structures for the purpose of validation.

Damping in Periodic Structures: A Continuum Modeling Approach

Optimal vibration control of curved beams using distributed parameter models

In this paper we consider the construction of hybrid models for flexible structures, including both the natural spatially distributed dynamics of low-mass density structures and the lumped models associated with rigid-body dynamics. For the purposes of active control of structural vibrations we argue that by using transform methods, the standard approach to transfer function (or mechanical impedance) modeling can be extended to provide a complete formulation of a hybrid model with localized control. We proceed to give a systematic procedure for obtaining the required transfer functions and Green's functions for the hybrid model dynamics consisting of one-dimensional elastic elements (such as beams, rods, and cables) together with rigid-body (lumped) models. We focus on a well-known family of models for elastic elements together with various reductions and indicate the construction of well-posed state-space models for these systems. Using the resulting state-space, we next demonstrate the construction of a complete frequency domain characterization of the elemental system response using transfer functions and Green's functions. Hybrid constructions are discussed for various interconnections of distributed and lumped elements, and a simple illustrative example is given. We highlight the extent to which the required computations can be performed automatically on a digital computer using a computer algebra system.

H. G. Kwatny

Papers

Continuum modeling of flexible structures with application to vibration control