曲线约束摩擦力速度问题的求解
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摘要
大学物理课程中对含钢索类约束的质点动力学问题均未考虑滑动摩擦的影响,主要原因在于由此导出的方程具有非线性特征,普遍认为不易求解。本文首先证明了当约束体为二次型曲线时,原切向方程可化为以分离变量求解的形式;然后针对一般的情形,给出了线性化的办法,导出了一阶线性非齐次常微分方程,并给出解的积分表达式。最后以一具体问题为例进行计算,对比了两种办法的特点,讨论了摩擦力的影响。
The particle dynamics of the steel cable constraints in college physics courses do not consider the influence of sliding friction.The main reason is that the resulting equations present nonlinear characteristics,which generally is not easy to solve.The paper proves that when the constraint body is two times curve,the original tangent equation can be changed into the form of the solution by separation variables.Then,for the general situation,the linearization method is given,the first order linear non-homogeneous ordinary differential equation is derived,and the integral expression of the solution is given.Finally,a specific problem is taken as an example to calculate.The characteristics of the two methods are compared and the influence of friction is discussed.
The particle dynamics of the steel cable constraints in college physics courses do not consider the influence of sliding friction.The main reason is that the resulting equations present nonlinear characteristics,which generally is not easy to solve.The paper proves that when the constraint body is two times curve,the original tangent equation can be changed into the form of the solution by separation variables.Then,for the general situation,the linearization method is given,the first order linear non-homogeneous ordinary differential equation is derived,and the integral expression of the solution is given.Finally,a specific problem is taken as an example to calculate.The characteristics of the two methods are compared and the influence of friction is discussed.
引文
[1]周衍柏.理论力学教程[M].3版.北京:高等教育出版社,2013:30-31.
[2]程守珠,江之永.普通物理学[M].6版.北京:高等教育出版社,2005:47.
[3]马长占.对《自然坐标系中的符号规则》一文的异义[J].大学物理,2000(3):9-11.MA C Z.Objection to the article “Symbolic rules in the natural coordinate system”[J].University Physics,2000(3):9-11.(in Chinese)
[4]同济大学数学教研室.高等数学(下)[M].4版.北京:高等教育出版社,1996,12:342-343.
[2]程守珠,江之永.普通物理学[M].6版.北京:高等教育出版社,2005:47.
[3]马长占.对《自然坐标系中的符号规则》一文的异义[J].大学物理,2000(3):9-11.MA C Z.Objection to the article “Symbolic rules in the natural coordinate system”[J].University Physics,2000(3):9-11.(in Chinese)
[4]同济大学数学教研室.高等数学(下)[M].4版.北京:高等教育出版社,1996,12:342-343.