Analytic Geometry

The French mathematician Rene Descartes (1596-1650) published, in 1637, his Discours de la methode (Discussions on the Method)-of finding scientific truth by good reasoning, that is. In a hundred-page appendix to this book Descartes combined algebra and geometry. He pointed out that if one drew two perpendicular straight lines, marked the intersection 0, and laid off units on each line, positive numbers to the right and up, negative numbers to the left and down, then every point in the plane could be pre...

Groove modeling and digital simulation for intersecting structures of circular tubes based on coplanarity of vectors

In order to establish the groove model for intersecting structures of circular tubes,mathematical model of the intersecting line is established by the method of analytic geometry,and parametric equations are thus determined.The dihedral angle,groove angle and actual cutting angle for any position of the intersecting line are derived as well.In order to identify groove vectors for two pipes,a new analytical method,i.e.coplanarity of vectors,is further proposed to complete the groove model.The established model is virtually verified by programming and simulation calculation in the MATLAB environment.The results show that groove vectors of intersecting structures simulated by MATLAB are consistent with the theoretical groove model,indicating that the theoretical groove model established in this paper is accurate,and further proves that the proposed coplanarity of vectors for solving groove vectors is correct and feasible.Finally,a graphical user interface(GUI)is developed by MATLAB software to independently realize functions such as model drawing,variable calculation and data output.The research outcome provides a theoretical foundation for the actual welding of circular intersecting structures,and lays an essential basis for weld bead layout and path planning.

Analytical geometry method of planetary gear trains

This paper presents an analytical geometry method for kinematics and efficiency of planetary gear trains (PGTs). The novel method which is capable of evolution and contrast analysis of mechanism kinematics, can be applied to any typical one-and two-degree-of-freedom plane PGTs containing any number of simple, compound or complex-compound planetary gear sets. The efficiency analysis of this method features a systematized and programmed process and its independence of the speed ratio. The primary contribution of this work lies in the integration of quantitative calculation, qualitative evolution and comparative analysis of kinematics of PGTs into one diagram, and in the integration of kinematics and efficiency analysis into a single method system. First, the analytical geometry method is defined, its basic properties are given, and the systematization procedure to perform kinematic analysis is demonstrated. As an application, analytical geometry diagrams of common PGTs are exhibited in the form of a list, whose kinematic characteristics and general evolution tendency are discussed. Then, with the mapping of PGTs onto the angular speed plane, the efficiency formula of analytical geometry, which has an extremely concise form, and a simple method for power flow estimation are put forward. Moreover, a general procedure is provided to analyze the efficiency and power flow. Finally, four numerical examples including a complicated eleven-link differential PGTs are given to illustrate the simpleness and intuitiveness of the analytical geometry method.

NEW SCALAR-TYPE EQUILIBRIUM EQUATIONS OF THE GENERAL SPACE FORCE SYSTEM

Using the principle of analytical geometry, several properties of the space straight lille are proved. Based on these properties, the equilibrium of general space force system is considered and its four new scalar-type equilibrium equations are derived which are equivalent to the vector-type necessary and sufficient conditions far equilibrium.