A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations,for which it is quite difficult to find analytical solutions.The initial motions method proposed by Whittaker is...A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations,for which it is quite difficult to find analytical solutions.The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems,including the nonholonomic systems discussed in the first part of this paper,where in addition to differential equations of motion,nonholonomic constraints apply.The final equations of motion for these systems are obtained in the form of corresponding power series.Also,an alternative,direct method to determine the initial values of higher-order derivatives(q)0,(q)0,…is proposed,being different from that of Whittaker.The second part of this work analyzes the stability of equilibrium of less complex,nonholonomic mechanical systems represented by gradient systems.We discuss the stability of equilibrium of such systems based on the properties of the gradient system.The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems.Finally,these theoretical considerations are illustrated through four examples.展开更多
This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of constraints.The first step is to establish the autonomous and non-autonomous differential equations ...This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of constraints.The first step is to establish the autonomous and non-autonomous differential equations of motion of the system,based on Pfaff-Birkhoff principle.Secondly,the existence of constraint multipliers are exhaustively discussed.Thirdly,the definition of one kind motion of the system,called free motion,is given,which is described and analyzed by the absence of constraints that are determined by constraint multipliers.Lemma 2 illustrates that one system can realize its free motion by selecting proper control parameters.In particular,theorem 2 provides that one system can naturally realize free motion when we consider the integral of the unconstrained Birkhoffian system as the constraints of constrained Birkhoffian system.Finally,the results obtained are illustrated by several examples.展开更多
基金the National Natural Science Foundation of China(Grants 11572145,11472124,and 11572034).
文摘A large proportion of constrained mechanical systems result in nonlinear ordinary differential equations,for which it is quite difficult to find analytical solutions.The initial motions method proposed by Whittaker is effective to deal with such problems for various constrained mechanical systems,including the nonholonomic systems discussed in the first part of this paper,where in addition to differential equations of motion,nonholonomic constraints apply.The final equations of motion for these systems are obtained in the form of corresponding power series.Also,an alternative,direct method to determine the initial values of higher-order derivatives(q)0,(q)0,…is proposed,being different from that of Whittaker.The second part of this work analyzes the stability of equilibrium of less complex,nonholonomic mechanical systems represented by gradient systems.We discuss the stability of equilibrium of such systems based on the properties of the gradient system.The advantage of this novel method is its avoidance of the difficulty of directly establishing Lyapunov functions aimed at such unsteady nonlinear systems.Finally,these theoretical considerations are illustrated through four examples.
基金supported by the National Natural Science Foundation of China(Grants 11272050,11572034,11872030 and 11972177).
文摘This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of constraints.The first step is to establish the autonomous and non-autonomous differential equations of motion of the system,based on Pfaff-Birkhoff principle.Secondly,the existence of constraint multipliers are exhaustively discussed.Thirdly,the definition of one kind motion of the system,called free motion,is given,which is described and analyzed by the absence of constraints that are determined by constraint multipliers.Lemma 2 illustrates that one system can realize its free motion by selecting proper control parameters.In particular,theorem 2 provides that one system can naturally realize free motion when we consider the integral of the unconstrained Birkhoffian system as the constraints of constrained Birkhoffian system.Finally,the results obtained are illustrated by several examples.
