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GB 3826.4, GB 3826.9 standards, inspected by the National Quality
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In the modeling process, after the "transformation"--projection by the ball facing the back-cone plane, the formed tooth shape is only an approximate tooth shape, although the error is within the "allowed" range, but after all, the theoretical tooth shape exists. With a certain gap, the accuracy has been discounted. Obviously, it is difficult to draw a correct conclusion by comparing and detecting the actual tooth shape based on such a tooth shape. This paper attempts to explore the precise modeling and modeling method of conical helical gears. Based on the principle of conical helical gear tooth profile surface, the mathematical model is derived and the mask is directly taken from the mathematical model to obtain a precise conical helical gear model to meet the requirements. The need for actual production.
The profile of the bevel gear is formed on a base cone of radius rb, and there is a plane S which is a pure rolling without sliding on the surface of the cone. The linear motion track that coincides with any busbar of the cone on the plane is the gear. Tooth profile surface. The trajectory of any point P on a straight line is the tooth profile curve at that point -- the spherical involute.
The spherical involute equation is based on the principle of the involute of the spherical surface. In the polar coordinate system with the cone apex as the pole, the equation of the involute of the spherical surface can be expressed as: rp=Rδ=1sinθbarccos(cosθpcosθb)-arccos(tanθbtanθp) In the formula (1), R is the taper of the cone, rp is the distance from the arbitrary point to the top of the cone. Obviously it is equal to the taper of the cone, δ is the research angle of the pole model at any point, and θb is the base cone. The half cone angle, θp is the half cone angle of the cone at any point. Taking the apex of the cone as the origin, taking the apex of the cone and the line perpendicular to the bottom of the cone as the Z axis, leaving the cone as the positive direction of the Z axis, and establishing the right-handed space rectangular coordinate system as shown, the spherical surface is gradually formed. The open line can be transformed into a Cartesian coordinate equation, that is: x = Rsin θpcos δy = Rsin θ psin δz = Rcos θp (2) where R is the spherical radius, that is, the length of the conical bus bar.
Cartesian coordinate system 1.3 Conical spiral surface forming principle The tooth profile surface of the conical helical gear is a spherical involute curved surface, and its tooth line is an equidistant conical spiral. It has its characteristics, and any point M on the involute of the spherical surface is located. The conical surface x2 y2=z2 rotates around the z-axis at an equal angular velocity ω, and simultaneously moves linearly in a positive direction parallel to the z-axis at a constant velocity v. The trajectory of the point M is a conical spiral. All the points on the involute of the spherical surface move according to the M point motion law, and the result forms the contour curved surface of the conical helical gear. Different conical faces have different conical helices, and the angle between the tangential direction at any point of each helix and the conical busbar passing through the apex of the cone at that point is the conical helix angle at that point. The helix angles on different cones are also different. The conical helix angle at the base cone is denoted by βb, and the helix angle at the indexing cone is denoted by β.
The conic spiral surface equation is calculated by the equation (2). The point of the spherical involute is calculated according to the position of the spherical surface (measured by the distance between the cones) and rotated around the Z axis by a certain angle to form a conical spiral involute. The point of the line.
Therefore, it is necessary to rotate the coordinates x, y, and z of the point in the formula (2), and set the rotation angle to βp, and the transformed coordinates are x', y', and z', and the transformation factor is A1=Rot(z, Βp), the homogeneous coordinate [x'y'z'1] of the transformed point can be expressed as: [x'y'z'1]T=[xyz1]A1=[xyz1]cosβpsinβp00-sinβpcosβp00010001(3) After simplification of equation (3), the Cartesian coordinate equation for the point on the conical spiral surface is: x'=Rsinθpcos(δ βp)y'=Rsinθpsin(δ βp)z'=Rcosθp(4) where βp is any point The helix angle at which it is a function of the large end spherical indexing cone diameter r and the conical taper distance Rp. That is: βp=f(r, Rp) (5) Equation (4) is the mathematical model of the conic spiral surface of the involute conical helical gear, which is actually the general formula for the calculation of the conical gear profile. Obviously, if βp=0 in the equation, the calculation result is the coordinate of the point of the straight bevel gear, and if βp≠0, it is the coordinate of the point of the conical helical gear.
Geometric Modeling of Conical Spiral Gears The teeth of a bevel gear taper from the big end to the small end, ie their dimensions on each spherical surface are proportional to their distance from the apex of the cone. Because the big end size is the largest, the measurement is convenient, and the relative error of the data is the smallest. Therefore, the parameters of the specified conical gear are subject to the big end.
The geometric modeling of the conical helical gear is also started from the big end, as follows.
The basic parameters of the input and calculation of the basic parameters of the conical helical gear are: the number of teeth z, the modulus m, the pressure angle α, the crest height coefficient ha, the head clearance coefficient c, the conical helix angle β, the taper distance R, tooth width b, and the like. From this, the other main parameters of the bevel gear tooth shape are calculated: the index circle diameter d, the tip circle diameter da, the root circle diameter df, the base circle diameter db, etc. These geometric dimensions are proportional to the mean modulus. In addition, there are gear cone angle θ, root cone angle θf, tip cone angle θa, base cone angle θb, etc. These parameters are independent of the modulus.
Determining the big end tooth profile For a standard bevel gear, the tooth thickness s and the inter-tooth width e on the indexing cone should be equal, ie s=e, then the pitch on the indexing circle p=2s=2e=dπz The point on the indexing circle is firstly positioned as a reference point. According to equation (4), the coordinate of each point is calculated from the indexing circle up to the top of the tooth, the indexing circle down to the root, and a large end tooth of the bevel gear is generated. Shape, that is, the reference tooth shape. Here, the helix angle βp=0. wherein the involute of the root segment is limited to the index circle to the base circle, whether the base circle is larger than the root circle or the root circle is larger than the base circle and its profile is connected under different conditions. There is a special research literature on the treatment, and will not be repeated here.
It is determined that the other large-end tooth profiles of the other tooth shapes are identical to the reference tooth profiles. Here, using the method of rotating the array, all other tooth shapes of the big end are reproduced, so that a complete bevel gear tooth shape is formed on the large end spherical surface.
It is determined that the tooth shape at the arbitrary spherical surface of the arbitrary spherical tooth shape is similar to that of the large end tooth, but the modulus is different and the urine is different. Let the modulus of the arbitrary tooth shape be mp, mp=RpRm where Rp is the spherical radius at any position. Obviously, if m and R are constant values, then mp and Rp are in a proportional relationship. Here, the tooth shape corresponding to the large end has a relative rotation angle increment βp, which can be derived from the conical spiral angle definition: βp=β2d2R(R-Rp) will calculate the modulus, spiral angle increment and other This derived parameter is substituted into the formula (4), and the tooth form of the arbitrary spherical surface can be made by following the generation method of the big end tooth shape.
The method of generating the conical helical gear tooth profile repeat 2.4, and expanding the arbitrary spherical surface from the big end to the small end, can generate an infinite number of (actually limited) tooth shapes on different spherical surfaces, and the tooth shapes are sequentially arranged. Then, the skeleton of the conical helical gear is formed, and on the skeleton, the skin processing of the conical spiral gear can be obtained by using the three-dimensional CAD for the skin treatment.
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1). GB/T 9081-2008 motor vehicles fuel dispensers
2).JJG443-2006 fuel dispenser national metrological verification regulations
3).GB 22380.1-2008 gas station explosion-proof sechnology Part 1: fuel
dispenser explosion-proof security technological requirements.
Practice of numerical control architecture of tapered curved gear