Study On Principle Of Bertrand Conjugating Surfaces And Loxodrome Normal Circular-arc Spiral Bevel Gear Transmission

In this dissertation, for the need of engineering, Berchand surface is put forward and studied which is refined and enhanced from the common characteristics of gyration surfaces, developed surfaces and normal circular-arc helical surfaces. According to which, many kinds of important surfaces can be structured. As one of the applications in transmission area, the principle of Bertrand conjugating surface is studied. On this base, it is studied that the general principle of normal circular-arc gear transmission, and with the instruction of which, the principle and technology of the loxodrome normal circular-arc spiral bevel gear transmission is studied.Starting with Bertrand curve in differential geometry, Bertrand curve with broad sense is put forward, and Bertrand surface is refined from the practice experience. Their geometry nature and inner relationship are studied. Bertrand curves with broad sense can be regarded as the equidistants of special line, which form the family of Bertrand curve with broad sense, and be convenient to the choice. According to the restriction of the generatrix line, Bertrand surface can be constructed with the Bertrand curve with broad sense group. Surfaces used widely in engineering, such as gyration surface, developed surface, the normal circular-arc surface and so on, are the typical Bertrand surface. There are some geometry characters about it, for example, it is simple in structure, and the normal lines along generatrix line are in one plane, so there are the extensive applying prospects hi mechanical transmission and surface processing. Relevant contents have established the foundation for the following study.As one of the most important contents, the basic principles of Bertrand conjugating surface has been systematically studied. Based on the geometry characteristics of Bertrand surfaces, the complicated surface conjugation issue can be discussed with their directrix line. According to the different generatrix line which can be divided into common plane curve, circular-arc curve and straight line, Bertrand conjugation surfaces are parted into three typical types, and the basic equation and differential formula are established, then the conjugation conditions are found. Aiming at the inclusive problem, the structure condition is given. So far, the main frame of the conjugation of Betrand are established, On the basis of which, it is given that the primary condition the directrix line must be satisfied, the relative curvature of the conjugating surfaces, the relations between non-interventional condition and curvature axle of the directrix line, and so on.The transmission of normal circular-arc gear is a typical Bertrand conjugation. In order to promote the transformation from theory to technology, the general principle of this kind of transmission is studied. For convenient to be analyzed, datum surface is defined, and the research route from datum surface to directrix line is established. Their characteristics for various transmission forms are given. Then the non-interventional condition is that the relative geodesic curvature is zero. Furthermore, for the transmission of the parallel or crossing axles, if the datum surface is the pitch one, the non-interventional condition is naturally satisfied. Considering the transmission of the normal circular-arc gear about parallel axles is ripe, the others are studied emphatically. Basing on the premise that the capability of the transmission and the technique are synthetically considered, three new kinds of gear pairs is given, that is, loxodromic normal circular-arc spiral bevel gears, geodetic normal circular-arc spiral bevel gear, and hyperbolic normal circular-arc gear. The directrixes of theinformer two transmissions are the loxodrome and the geodesic on the pitch cones, that of the latter is an equal-pitch line along the direction of straight generatrixes in the hyperboloid surface, also is the vertical trajectory of the straight generatrixes. those fine geometrical nature make them be the first selection.