On the nonlinear dynamics of elastically interacting asymmetric rigid bodies

Abstract

A symmetric mathematical model is developed to describe the spatial motion of a system of space vehicles whose structure is represented by regular geometrical figures (Platonic bodies). The model is symmetrized by using the Euler-Lagrange equations of motion, the Rodrigues-Hamilton parameters, and quaternion matrix mathematics. The results obtained enable us to model a wide range of dynamic, control, stabilization, and orientation problems for complex systems and to solve various problems of dynamic design for such systems, including estimation of dynamic loading on the basic structure during maneuvers in space Introduction. Dynamic models in the form of discrete mechanical systems of asymmetric rigid bodies connected by elastic bars arise in dynamic problems for space vehicles (SVs) joined by standard adapter modules into open ( Fig. 1à) or closed chains. Closed SV chains may be either plane (triangle, square, or any other regular polygon (Fig. 1b)) or spatial (Platonic bodies: tetrahedron, octahedron, icosahedron, etc. (Fig. 1c)). The nodes of such geometrical figures are space vehicles and their edges are adapter modules. The space vehicles are modeled by asymmetric rigid bodies that experience follower forces generated by the control, stabilization, and orientation system. The adapter modules are modeled by inertialess rectilinear elastic bars of certain length and cross section embedded in the bodies. Such dynamic models are of importance since they allow us to analyze the dynamics of an SV system with allowance for the elastic interaction between them during maneuvers and to estimate the dynamic loading on both the space vehicle [1] and the adapter module [2].Design models in the form of connected rigid bodies are widely used to solve applied dynamic problems for complex engineering structures [3,4,[11][12][13][14]. Such dynamic problems can mainly be solved by conducting a computational experiment [5]. To this end, it is necessary to set up adequate mathematical models of physical processes. These models should be easily implemented on state-of-the-art computers. The key element in this model design concept is the symmetry of a model, which was understood by H. Weyl as order, perfection [6].Following the concept of symmetrization, we will develop a mathematical model to describe the spatial motion of a system of asymmetric rigid bodies connected by elastic bars into open or closed ordered chains in the form of regular polygons or polyhedrons. The model will be symmetrized by using the Euler-Lagrange equations of motion, the Rodrigues-Hamilton parameters, and quaternion matrix mathematics [7,8].1. Problem Formulation. According to the principle of removing constraints (and replacing them by their reactions), the whole variety of structures of SV systems under consideration can be reduced to a standard unified substructure consisting of an isolated spacecraft and an adapter module. This substructure is taken to be a superelement and is modeled by an inertialess elastic bar embedded in an asymmetric rigid bod...

Evaluation of the centrifugal, coriolis, and gyroscopic forces on a railroad vehicle moving at high speed

“…The problem posed is of current importance [15][16][17][18][19] and is directly related to high-speed traffic safety [7].…”