基于Udwadia-Kalaba理论的Hamel嵌入法研究
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摘要
Hamel嵌入法直接将约束嵌入到非约束运动的动能中去,从而避免使用Lagrange(拉格朗日)乘子.但这个简单、直观的方法却并不总是正确.Hamel认为这种方法可能导致错误的结果,然而他并没有给出Hamel嵌入法正确性的适用条件.在利用Udwadia-Kalaba理论的基础上,提出了Hamel嵌入法成立的充要条件;指出了Rosenberg在Hamel嵌入法正确性研究中的不足,通过给出的具体算例可以看出,在完整约束下Hamel嵌入法可能不正确,而在非完整约束下也可能得出正确的结果;理论和实例分析表明,Hamel嵌入法是否成立除了与约束有关以外还与系统模型相关.
Hamel embedded the constraint directly into the kinetic energy of unconstrained motion to avoid the use of Lagrange multiplier,which made a simple,straightforward,but incompletely correct method. Hamel stated that this method may lead to wrong results,but did not point out the applicable conditions for its correctness. Based on the Udwadia-Kalaba theory,the necessary and sufficient condition for Hamel's embedding method was found. Besides,examples show that Rosenberg's work on the validity of Hamel's embedding method is insufficient. Hamel's embedding method may be correct under nonholonomic constraint and may be incorrect under holonomic constraint. According to the theoretical and exemplary analysis,the correctness of Hamel's embedding method is not only determined by the constraints,but also determined by the mechanical system model.
Hamel embedded the constraint directly into the kinetic energy of unconstrained motion to avoid the use of Lagrange multiplier,which made a simple,straightforward,but incompletely correct method. Hamel stated that this method may lead to wrong results,but did not point out the applicable conditions for its correctness. Based on the Udwadia-Kalaba theory,the necessary and sufficient condition for Hamel's embedding method was found. Besides,examples show that Rosenberg's work on the validity of Hamel's embedding method is insufficient. Hamel's embedding method may be correct under nonholonomic constraint and may be incorrect under holonomic constraint. According to the theoretical and exemplary analysis,the correctness of Hamel's embedding method is not only determined by the constraints,but also determined by the mechanical system model.
引文
[1]Hamel G.Theoretische Mechanik[M].Berlin:Springer,1949.
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[4]CHEN Ye-hwa.Equations of motion of mechanical systems under servo constraints:the Maggi approach[J].Mechatronics,2008,18(4):208-217.
[5]乔永芬,赵淑红.Poincaré-Chetaev变量下变质量非完整动力学系统的运动方程[J].物理学报,2001,50(5):805-810.(QIAO Yong-fen,ZHAO Shu-hong.Equations of motion of variable mass nonholonomic dynamical system in Poincaré-Chetaev variables[J].Acta Physica Sinica,2001,50(5):805-810.(in Chinese))
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[7]Papastavridis J G.Analytical Mechanics:A Comprehensive Treatise on the Dynamics of Constrained Systems;for Engineers,Physicists,and Mathematicians[M].New York:Oxford University Press,2002.
[8]Udwadia F E,Wanichanon T.Hamel’s paradox and the foundations of analytical dynamics[J].Applied Mathematics and Computation,2010,217(3):1253-1265.
[9]Rosenberg R M.Analytical Dynamics of Discrete Systems[M].New York:Plenum,1977.
[10]CHEN Ye-hwa.Hamel paradox and Rosenberg conjecture in analytical dynamics[J].Journal of Applied Mechanics,2013,80(4):041001-1-041001-8.
[11]CHEN Ye-hwa.Mechanical systems under servo constraints:the Lagrange’s approach[J].Mechatronics,2005,15(3):317-337.
[12]宋端.一阶Lagrange系统平衡稳定性对参数的依赖关系[J].应用数学和力学,2014,35(6):692-696.(SONGDuan.Dependence of equilibrium stability of first order Lagrange systems on parameters[J].Applied Mathematics and Mechanics,2014,35(6):692-696.(in Chinese))
[13]Kalaba R E,Udwadia F E.Lagrangian mechanics,Gauss’s principle,quadratic programming,and generalized inverses:new equations for nonholonomically constrained discrete mechanical systems[J].Quarterly of Applied Mathematics,1994,52(2):229-241.
[14]Udwadia F E,Kalaba R E.An alternate proof for the equation of motion for constrained mechanical systems[J].Applied Mathematics and Computation,1995,70(2):339-342.
[15]Udwadia F E,Kalaba R E,Eun H C.Equations of motion for constrained mechanical systems and the extended d’Alembert’s principle[J].Quarterly of Applied Mathematics,1997,56(2):321-331.
[2]黄友钦,林俊宏,傅继阳,等.多重约束下空间桁架结构抗风优化[J].应用数学和力学,2013,34(8):824-835.(HUANGYou-qin,LIN Jun-hong,FU Ji-yang,et al.Wind resistant optimization of spatial truss structures under multi-constraints[J].Applied Mathematics and Mechanics,2013,34(8):824-835.(in Chinese))
[3]Blajer W.Projective formulation of Maggi’s method for nonholonomic systems analysis[J].Journal of Guidance Control Dynamics,2015,15(2):522-525.
[4]CHEN Ye-hwa.Equations of motion of mechanical systems under servo constraints:the Maggi approach[J].Mechatronics,2008,18(4):208-217.
[5]乔永芬,赵淑红.Poincaré-Chetaev变量下变质量非完整动力学系统的运动方程[J].物理学报,2001,50(5):805-810.(QIAO Yong-fen,ZHAO Shu-hong.Equations of motion of variable mass nonholonomic dynamical system in Poincaré-Chetaev variables[J].Acta Physica Sinica,2001,50(5):805-810.(in Chinese))
[6]Brogliato B.Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[J].Multibody System Dynamics,2013,32(2):175-216.
[7]Papastavridis J G.Analytical Mechanics:A Comprehensive Treatise on the Dynamics of Constrained Systems;for Engineers,Physicists,and Mathematicians[M].New York:Oxford University Press,2002.
[8]Udwadia F E,Wanichanon T.Hamel’s paradox and the foundations of analytical dynamics[J].Applied Mathematics and Computation,2010,217(3):1253-1265.
[9]Rosenberg R M.Analytical Dynamics of Discrete Systems[M].New York:Plenum,1977.
[10]CHEN Ye-hwa.Hamel paradox and Rosenberg conjecture in analytical dynamics[J].Journal of Applied Mechanics,2013,80(4):041001-1-041001-8.
[11]CHEN Ye-hwa.Mechanical systems under servo constraints:the Lagrange’s approach[J].Mechatronics,2005,15(3):317-337.
[12]宋端.一阶Lagrange系统平衡稳定性对参数的依赖关系[J].应用数学和力学,2014,35(6):692-696.(SONGDuan.Dependence of equilibrium stability of first order Lagrange systems on parameters[J].Applied Mathematics and Mechanics,2014,35(6):692-696.(in Chinese))
[13]Kalaba R E,Udwadia F E.Lagrangian mechanics,Gauss’s principle,quadratic programming,and generalized inverses:new equations for nonholonomically constrained discrete mechanical systems[J].Quarterly of Applied Mathematics,1994,52(2):229-241.
[14]Udwadia F E,Kalaba R E.An alternate proof for the equation of motion for constrained mechanical systems[J].Applied Mathematics and Computation,1995,70(2):339-342.
[15]Udwadia F E,Kalaba R E,Eun H C.Equations of motion for constrained mechanical systems and the extended d’Alembert’s principle[J].Quarterly of Applied Mathematics,1997,56(2):321-331.