具有黏滞阻力时的最速降线

摘要

针对实际应用中存在黏滞阻力的最速降线的问题,首先推导出适于此类问题的解除约束的广义变分原理,它适用于具有摩擦阻尼和多自由度系统优化的问题.得到描述有黏滞阻力情况下最速降线相关函数的微分方程,它在黏滞阻力为零时即退化为滚轮线.利用MATLAB数值计算给出了最速降线受黏滞阻力的影响:在黏滞阻力系数较小时最速降线趋于变凹,当阻力系数增大到一定值之后最速降线趋于平缓,当阻力系数很大时最速降线趋于直线.%The brachistochrone with consideration of viscous resistance may be an issue in engineering situ ations. A generalized variational principle with constraints removed is deduced first, and it is also applicable to other forms of resistance and optimization of systems with multiple degrees of freedom. We then obtain the ordinary differential equations describing the functions of the brachistochrone with viscous resistance, whose solution is reduced to roulette when the viscous resistance is neglected. The effects of the resistance on the brachistochrone are quantitatively analyzed on the basis of numerical computations by using MATLAB. When the viscous resistance is not very large, the brachistochrone becomes more concave. With the increase of the resistance, the brachistochrone will become flatter. A very large resistance will finally lead to a straight-line brachistochrone.