The use of principal points and principal vectors in the formulation of the equations of motion of a general 4R planar four-bar linkage is shown with two kinds of methods, one that opens kinematic loops and one that does not. The opened kinematic loop approach analyses the moving links as a system with a tree connectivity, introducing reaction forces for closing the loops. Compared with the conventional Newton–Euler method, this approach results in fewer equations and constraint forces, whereas the mass matrix entries remain meaningful, but there is a stronger coupling between the equations. Two equivalent mass formulations for the closed kinematic loop approach are presented, which need not open the loop and introduce loop constraint forces in the equations of motion. With the method of complex joint masses, the mass of the links closing the loops is represented by real and virtual equivalent masses, defining the principal points. The principle of virtual work with the inclusion of inertia terms is used to derive the equations of motion. As an example the dynamic balance conditions of the four-bar linkage are derived. With the method of the equivalent mass matrix it is shown how a constant mass matrix can be used to describe the dynamics of binary links with an arbitrary mass distribution. A four-bar linkage could be modelled with only three truss elements instead of the conventional result with three or more beam elements, which gives a significant reduction of the computational complexity.

中文翻译：

---

平面四杆机构运动方程的主矢量和等效质量描述

通过两种方法显示了在一般的4R平面四连杆机构的运动方程的公式化中使用主点和主矢量的方法，一种方法打开运动循环，而另一种方法则不。开放式运动回路方法将活动链接分析为具有树连通性的系统，从而引入反作用力来闭合回路。与传统的牛顿-欧拉方法相比，该方法产生的方程式和约束力更少，而质量矩阵项仍然有意义，但是方程式之间存在更强的耦合性。针对闭环运动学方法，提出了两种等效的质量公式，它们无需打开环并将运动约束力引入运动方程中。用复杂的关节质量的方法 闭合环路的链节的质量由定义主要点的实际和虚拟等效质量表示。包含惯性项的虚拟功原理被用于导出运动方程。作为示例，得出了四连杆机构的动平衡条件。通过等效质量矩阵的方法，显示了如何使用恒定质量矩阵来描述具有任意质量分布的二进制链接的动力学。可以仅用三个桁架单元来模拟四连杆机构，而不用三个或更多梁单元来模拟常规结果，从而大大降低了计算复杂度。包含惯性项的虚拟功原理被用于导出运动方程。作为示例，得出了四连杆机构的动平衡条件。通过等效质量矩阵的方法，显示了如何使用恒定质量矩阵描述具有任意质量分布的二进制链接的动力学。可以仅用三个桁架单元来模拟四连杆机构，而不用三个或更多梁单元来模拟常规结果，从而大大降低了计算复杂度。包含惯性项的虚拟功原理被用于导出运动方程。作为示例，得出了四连杆机构的动平衡条件。通过等效质量矩阵的方法，显示了如何使用恒定质量矩阵来描述具有任意质量分布的二进制链接的动力学。可以仅用三个桁架单元来模拟四连杆机构，而不用三个或更多梁单元来模拟常规结果，从而大大降低了计算复杂度。