Nonlinear Mathematical Model of the Systems Dynamics with Geometric Constraints and its Simplification Due to the Justified Linearization of Constraints

Success in examining the dynamics of any controlled object is determined to a decisive degree by the complexity of the study, which primarily depends on the quality of modeling. Analytical mechanics offers a wide range of coordinate types to describe the dynamics of mechanical components of mechatronic systems. Modeling of the dynamics of systems with geometrical constraints requires a rigor which is unusual for technical practice. This is determined by the necessity of taking into account not only linear but also at least quadratic terms in the expansion of nonlinear constraint equations when identifying the first approximation of the equations of perturbed motion in stability and stabilization problems. The method of rigorous modeling of the dynamics of systems with geometric constraints is presented in detail in the article. Possible simplifications of the study are substantiated on the basis of these results. The effectiveness of the proposed approach is demonstrated by considering two variants of the stand Ball & Beam equilibrium.