最小二乘法在高斯拘束函数求解中的应用

APPLICATION OF LEAST SUQARE METHOD IN SOLVING GAUSS CONSTRAINT FUNCTION

高斯最小拘束原理是一种典型的微分变分原理,以加速度为变量,通过寻求拘束函数极值的变分方法直接得出系统的运动规律.目前,国内常用的求解高斯拘束函数的方法为拉格朗日乘子法,通过引入拉格朗日乘子将高斯拘束函数的条件极值问题转化为带有拉格朗日乘子的无条件极值问题,这种求解方法会增加未知变量的个数.为减少变量个数,进一步提高运算效率,文章首先对高斯拘束函数进行简单变形,引入加速度形式的约束方程将高斯拘束函数化为最小二乘形式,直接运用最小二乘法导出使高斯拘束函数取极小值时系统真实加速度的表达式.最后通过对曲柄滑块机构正动力学问题的分析和计算,验证了该方法的有效性.

Gauss′s minimum constraint principle is a typical differential variation principle, where the acceleration is the variable, and the motion law of the system can be obtained directly by the variation method of seeking the extreme of the constraint function. At present, Lagrange multiplier method is widely used for solving the Gauss constraint function. Through the introduction of Lagrange multipliers, the conditional extreme value problem of Gauss constraint function is transformed to the unconditional extreme problem. However, this method increase the number of unknown variables. In order to reduce the number of variables, it need further study to improve the operation efficiency. In this paper, the deformation on Gauss constraint function is firstly simplified. The constraint equations of the acceleration form are also introduced into the Gauss constraint equations to get the least square form of equations. The least square method is then used to export the expression of the true acceleration of the system, which can make the Gauss binding function to take the minimum value. In the end, the validity of the method is verified by analyzing and calculating the normal dynamics of the crank slider mechanism.