Noether’s theorems of variable mass systems on time scales

This paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

Tracking Control Design for Variable Mass and Configuration Robotic Systems

The paper addresses control of variable mass and configuration mechanical systems subjected to holonomic or nonholonomic constraints, which are imposed due to systems desired performance, tracking specified motions or other control needs. The control design is model-based and an analytical dynamics modeling framework underlying controller design is presented. The framework novelty is that constraints, including nonholonomic ones and these on variable mass, can be merged into variable mass system dynamics and final motion equations are free of the constraint reaction forces so they can be used directly to control design. Many mechanical systems change their mass or configuration when they move, e.g. inertia-based propelled underwater vehicles, mobile robots and manipulators transporting loads or space vehicles flying their space missions. The dynamics modeling framework presented in the paper can be applied to all variable mass system examples mentioned above. An underwater inertia-based propelled vehicle model dynamics and control performance illustrate the theoretical development presented in the paper. The paper contribution is two folded. It presents a unified approach to constrained variable mass or configuration systems modeling and introduces analytical dynamics methods to the nonlinear control domain.

Analyzing the Vibration System with Time-Varying Mass

Considering that impact motion is not only a function of time but also dependent on phase-angle, we suppose the non-linear response of the vibro-impact system with time-varying mass is a function dependent not only on different time scale but also on the phase parameter. An approximate analytical solution of second order for the vibration is obtained by using Multi-Scales Method. The feasibility is verified by the numerical solution using Runge-Kutta algorithms. It is shown that the motion of the variable mass system has periodic behavior with the period of the changing mass, and the mass variation can only influence the system amplitude but not its cycle. However, the bigger the mass factor varies, the more intensive the response enlarges, and vice versa. The method and findings may be useful to analyze similar vibration systems with impact dampers or design the vibration control strategy.