Dynamics of geometrically nonlinear sliding beams




Behdinan, Kamran

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The elasto-dynamics of flexible frame structures is of interest in many areas of engineering. In certain structural systems the deflections can be large enough to warrant a nonlinear analysis. For example, offshore structures, long suspension bridges and other relatively slender structures used in space applications require a geometrically nonlinear analysis. In addition, if the structure has deployable elements, as in some space structures, the required analysis becomes even more complex. Typical examples are spacecraft antennae, radio telescopes, solar panels and space-based manipulators with deployable elements. The main objective of the present work is to formulate the problem of sliding beams undergoing large rotations and small strains. Further we aim to develop efficient finite element technique for analysis of such complex systems. Finally we wish to examine the nature of the motion of sliding beams and point out its salient features. We start with two well known approaches in the nonlinear finite element static analysis of highly flexible structures, namely, the updated Lagrangian and the consistent co-rotational methods and extend these techniques to dynamic analysis of geometrically nonlinear beam structures. We analyse several examples by the same methods and compare the performance of each for efficiency and accuracy. Next, using McIver's extension of Hamilton's principle, we formulate the problem of geometrically flexible sliding beams by two different approaches. In the first the beam slides through a fixed rigid channel with a prescribed sliding motion. In this formulation which we refer to as the sliding beam formulation, the material points on the beam slide relative to a fixed channel. In the second formulation the material points on the fixed beam are observed by a moving observer on a sliding channel and the beam is axially at rest. The governing equations of motion for the two formulations describe the same physical problem and by mapping both to a fixed domain, using proper transformations, we show that the two sets of governing equations become identical. It is not, possible to find analytical solutions to our problem and we choose the Galerkin numerical method to obtain the transient response of the problem for the special case axially rigid beam. Next we follow a more elegant approach wherein we use the developed incremental nonlinear finite element approaches (the updated Lagrangian and the consistent co-rotational method) in conjunction with a variable time domain beam finite elements (where the number of elements is fixed and as mass enters the domain of interest, but the sizes of elements change in a prescribed manner in the undeformed configuration). To verify the formulation and its computational implementation we analyse many examples and compare our findings with those reported in the literature when possible. We also use these illustrative examples to identify the importance of various terms such as axial flexibility and foreshortening effects. Finally we look into the problem of parametric resonance for the beam with periodically varying length and we show that the regions of stability obtained in the literature, using a linear analysis, do not hold when a more realistic nonlinear analysis is undertaken.



Beam dynamics, Particle beams, Nonlinear functional analysis