Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints

Based on the dynamical theory of multi-body systems with nonholonomic constraints and an algorithm for complementarity problems, a numerical method for the multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints is presented. In particular, a wheeled multi-body system is considered. Here,the state transition of stick-slip between wheel and ground is transformed into a nonlinear complementarity problem(NCP). An iterative algorithm for solving the NCP is then presented using an event-driven method. Dynamical equations of the multi-body system with holonomic and nonholonomic constraints are given using Routh equations and a constraint stabilization method. Finally, an example is used to test the proposed numerical method. The results show some dynamical behaviors of the wheeled multi-body system and its constraint stabilization effects.

Lagrangian Formulation of Fractional Nonholonomic Constrained Damping Systems

Fractional Euler Lagrange equations for fractional nonholonomic constrained damping systems have been presented. The equations of motion are obtained using fractional Euler Lagrange equations in a similar manner to the usual technique. The results of fractional method reduce to those obtained from classical method when μ →0 and α,β →1 are equal unity only. This work is discussed using illustrative example.

Mei symmetry and new conserved quantities of Tzénoff equations for higher-order nonholonomic system

In this paper,the Mei symmetry of Tzénoff equations for the higher-order nonholonomic system and the new conserved quantities derived from that are researched,and the function expression of new conserved quantities and criterion equation which deduces these conserved quantities are presented.This result establishes the theory basis for further researches on conservation laws of Tzénoff equations of the higher-order nonholonomic constraint system.

Instability of equilibrium of nonholonomic systems with dissipation and circulatory forces

The paper discusses the equilibrium instability problem of the scleronomic nonholonomic systems acted upon by dissipative,conservative,and circulatory forces.The method is based on the existence of solutions to the differential equations of the motion which asymptotically tends to the equilibrium state of the system as t tends to negative infinity.It is assumed that the kinetic energy,the Rayleigh dissipation function,and the positional forces in the neighborhood of the equilibrium position are infinitely differentiable functions.The results obtained here are partially generalized the results obtained by Kozlov et al.(Kozlov,V.V.The asymptotic motions of systems with dissi-pation.Journal of Applied Mathematics and Mechanics,58(5),787-792(1994).Merkin,D.R.Introduction to the Theory of the Stability of Motion(in Russian),Nauka,Moscow(1987).Thomson,W.and Tait,P.Treatise on Natural Philosophy,Part I,Cambridge University Press,Cambridge(1879)).The results are illustrated by an example.

Bicycle dynamics and its circular solution on a revolution surface

In this paper,we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution.We employ symbolic manipulations to derive the contact constraint equations from an ordered process,and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations(DAEs),leaving nine coupled differential equations,six contact equations,two holonomic constraint equations and four nonholonomic constraint equations.We then present a complete description of hands-free circular motions,in which the time-dependent variables are eliminated through a rotation transformation.We find that the circular motions,similar to those of the bicycle moving on a horizontal surface,nominally fall into four solution families,characterized by four curves varying with the angular speed of the front wheel.Then,we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface.Furthermore,we directly linearize the nonlinear DAEs,from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates.The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position.We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families.Finally,based on the numerical simulation of the original nonlinear DAEs system,we show that the stable circular motion is not asymptotically stable.

Control design of an unmanned hovercraft for agricultural applications

The efficient and precise application of agricultural materials such as fertilizer or herbicide can be greatly facilitated by autonomous operation.This is especially important under difficult conditions at remote sites.The purpose of this work is to develop an accurate nonlinear controller using a direct Lyapunov approach to ensure stability of an unmanned hovercraft prototype used for the execution of these agricultural tasks.Such a craft constitutes an underactuated system which has fewer actuators than degrees of freedom.The proposed closed loop system is simulated to demonstrate that a control law can stabilize both the actuated and unactuated degrees of freedom of the hovercraft.It is shown that the position and orientation of the hovercraft achieve high dynamic and steady performance.