INTRODUCTION Involute gear transmission has the advantages of good machining process, stable transmission, small vibration, constant output speed and no fluctuation. The center distance of the two wheels allows a certain installation error, which will not cause fluctuations in output speed. Aircraft, vehicles, ships and other industrial fields. The involute gear is a three-dimensional entity with complex shapes. The shape and precision of the involute gear will directly affect the subsequent finite element analysis, simulation results and directly affect the manufacturing precision when manufacturing gears by 3D modeling. The key to the three-dimensional modeling of the gear is to generate the involute tooth profile that meets the requirements. However, since the commercial three-dimensional software does not provide the direct modeling function of the involute gear, it is often difficult to directly realize the involute modeling. In response to this problem, Guo Yue proposed using the modeling function to generate the shape of the involute and the tooth. This method realizes the parametric modeling of the spur gear, but there are also some defects: the involute gear drawing process It is difficult to guarantee high precision by forming a curve by using a spline to form a curve to replace the involute. Parametric modeling is used by most designers for its complicated size calculation and re-modeling process; Some software does not directly draw the involute command. Hu Chibing et al. proposed using VisuaC to program a series of coordinates on the tooth profile curve, store it in a text file, and then let Solidworks use this file to draw a spline and implement the tooth. The drawing of the profile curve. Using a programming method can achieve many more accurate involutes that are difficult to implement manually, but requires the user to have some programming knowledge.
Aiming at the above problems, this paper proposes a modeling method based on the formation principle of the involute helicoid and its relationship with the involute to form the involute profile. Combined with the example, the modeling process of the spur gear is introduced in detail: firstly use the "curve / spiral", "surface / scan" command to draw the involute spiral surface, and then any one in the middle of the involute spiral surface The position is an end face plane, and the intersection line between the end face plane and the involute helicoid is an involute, and then the shape of the involute gear is completed by stretching or scanning features. This modeling process directly utilizes the computational functions inside the commercial 3D software, eliminating complex calculations, simple operation, and accurate gear drawing.
1 The principle of the formation of the involute of the gear 1.1 The principle of the formation of the involute When a straight line BK is purely rolled along the circumference of the radius rb, the trajectory (AK) of any point on the straight line is the involute of the circle.
Deriving the Cartesian coordinate equation of the involute: x=rbcosφ rbφsinφy=rbsinφ-rbφcosφ where: rk―——the diameter of the K point; φ―——the angle of the involute; rb—the radius of the base circle.
1.2 The principle of the formation of the involute helicoid is the spiral line as the wire. When the busbar is moving, the curved surface formed by the tangent to the spiral is called the tangent spiral surface, which is also called the involute spiral surface. According to the formation principle of the involute spiral surface, when the bus bar ML is cut and the cylindrical spiral AE is at the point M, the straight bus bar performs continuous motion, moving from the starting position A to the M1L1 position, and the curved wire cylindrical spiral AE is at the M1. The straight busbar is continuously moved so as to remain tangent to the curved wire spiral AE, and the curved surface formed is an involute helix.
In order to more vividly illustrate the formation of the involute helicoid, according to another important property of the involute helicoid: if the involute helicoid is truncated by any plane tangential to the base circle, the section line on this plane is Straight line, another way to form the involute helicoid. As shown, the paper strip is cut into a right-angled triangle AFE, so that the right-angled edge AE coincides with the straight line of the cylinder, the paper strip surrounds the cylindrical surface, the tip point F coincides with A, and then the paper strip is unfolded, so that the strip is The trajectory of the hypotenuse forms an involute helix.
1.3 Relationship between the involute and the involute helicoid A plane curve that the tip point F traverses in the space is the involute of the circle. The occurrence of the involute helicoid is a diagonal line in a plane tangential to the base cylinder, and the angle between the oblique line and the end surface is the base circle helix angle α of the helicoid.
Since the spiral is a spatial curve, the involute helix is ​​a tangent surface and is a developable surface.
The above method vividly shows that the end face of the involute helicoid is an involute, and the geometrical method is used to prove that the end face of the involute is truncated: the involute can be regarded as the straight bus ML. The helix plane formed by the spiral motion has a radial ON of a point N: ON=OK KM MN(1) where: MN=u―- a section of the straight busbar, that is, the straight busbar is the same as the base cylinder Cut the point M to the line segment of any point N on the involute helix surface; OK=rb--base cylinder radius; KM=pφ--the axial displacement when the angle φ is turned in the spiral motion.
Projecting the vector ON onto the coordinate axis yields: ON=(rbcosφ ucosαsinφ)i (rbsinφ-ucosαsinφ)j (pφ-usinα)k(2) where: p=rbtanα; α―——the angle of the straight bus MN It is equal to the angle of the helix of the base cylinder.
The form written as a Cartesian coordinate: x = rbcos φ ucos αsin φ y = rbsin φ - ucos α sin φ z = p φ - usin α (3) The involute plane is cut by the plane z = constant is the involute. For convenience, we take z=0, that is, pφ-usinα=0(4) and because: p=rbtanα: u=pφsinα=rbcosαφ(5) Substituting (5) into equation (3): x=rbcosφ rbφsinφy= Rbsinφ-rbφcosφ(6) Equation (6) is an involute expression. From this formula, it can be seen that the end truncated shape of the involute helicoid is an involute, and the helix rise with the involute helicoid The angle α does not matter. When the plane of z≠0 is used to cut the involute helicoid, the magnitude of the helix angle α only affects the starting position of the involute, regardless of the shape of the involute.
It is known from the formation principle of the involute that the end section of the involute helicoid is an involute. Therefore, the involute of the gear can be drawn by the end cut of the involute helix, thereby completing the three-dimensional modeling of the involute gear.
2 Modeling of Involute Spur Gears 2.1 Basic Parameters The key to three-dimensional modeling of gears is to generate involutes and gears that meet the requirements. The geometry of a standard involute cylindrical gear depends on five important parameters of the gear: the number of teeth z, the modulus m, the pressure angle α, the tip height coefficient ha and the head clearance coefficient c. The standard tooth height coefficient and the head clearance coefficient of the spur gear are taken as 1 and 0.25 according to the normal gear. The geometric parameters of each part of the gear are calculated by using these basic parameters: the height of the tooth top: ha=ham; the height of the root: hf= (ha c)m; index circle diameter: d=mz; base circle diameter: db=dcosα; tip circle diameter: da=(z 2ha)m; root circle diameter: df=(z-2ha-2c) m.
Below we introduce the three-dimensional modeling of involute gears in Solidworks according to a specific example: a gear parameter is known: z=20, m=4, α=20, ha=1, c=0.25; according to the above relationship The corresponding index circle, tip circle, base circle and root circle diameter can be calculated, which are 80, 88, 75, 70 respectively.
2.2 Specific modeling process 2.2.1 Formation of involute (1) Open Solidworks, create a new file, select the front reference plane, and enter the sketch drawing mode. Draw a circle centered on the origin of the coordinate, which is 75 in diameter, which is the diameter of the base circle. Exit the sketch mode.
(2) Use the "Curve/Involute/Spiral Line" command to generate a spiral. By defining the pitch and the number of turns to complete the spiral drawing, since the pitch refers to the axial distance of the corresponding points of the adjacent teeth on the thread, there is no influence on the end face truncation of the involute helicoid, so the value can be customized as needed. . The involute is the end face of the involute helix. The number of turns, the starting angle and the direction of the helix have no effect on the shape of the involute, so there is no special requirement.
(3) Select the datum tool in the reference geometry to create a datum plane parallel to the right datum plane and pass an end point of the helix 1. Select the datum face 1 to enter the sketch mode, and the view direction to select the right view, so that the datum face and The desktop is parallel. Draw a line segment and then use the Sketch Constraint to establish the constraint that the end of the line coincides with the end of the spiral; the line segment is tangent to the spiral. Exit the sketch command, select the Surface/Scan Surface command, and select the outline and path as shown to create the involute helix.
The surface scan forms an involute helix (4) Select "Base 1" to enter the sketch mode, draw a line segment that intersects the involute helix and is perpendicular to the axis of the helix, and exits the sketch mode. Select the Surface/Extreme Surface command to stretch to the appropriate width: that is, completely intersect the involute helix, then click OK to complete the surface stretch.
(5) Select the “Curve/Split Line†command, select the “Crossover Point†for the segmentation type, and then select the surface 1 and the involute helicoid in the selection items below, and determine that the dividing line of the two surfaces is the involute. As shown.
The formation of the involute part of the gear tooth profile 2.2.2 Generating the gear (1) In 2.1, the corresponding diameter has been obtained by the calculation formula of the index circle, the addendum circle, and the root circle diameter. Select the extruded surface where the involute is located, enter the sketch mode, and convert the involute to a sketch entity using the Convert Entity Reference tool. Then draw three concentric circles with the center of the base circle as the center, that is, the index circle, the tip circle, and the root circle, the dimensions are 80, 88, 70 respectively.
Select the Draw Point command, draw a point on the involute, and then create the following constraint by adding geometry: the index circle is fixed; the created point and the index circle coincide; the index circle is used as the construction line. Tools/Sketch Tools/Circumferential Array, with an equally spaced array of points, the number is 40.
Then make a vertical line (construction line) between the point on the involute and the line connecting the adjacent points. Use the "Mirroring Entity Command" to center the vertical line and mirror the involute. Then use "Cut the solid/cut to the nearest end" to subtract the part of the outer tip circle of the involute profile, and use the "cutting solid/strong cut" to subtract the involute part of the outer circumference of the addendum circle to obtain the tooth profile. Involute part, (2) Drawing of the rest of the tooth profile: The number of teeth of the involute gear determines the relationship between the radius of the root circle and the radius of the base circle, which in turn affects the shape of the tooth profile. For example, when the involute gear pressure angle α is 20°, the tip height coefficient ha is 1, and the top coefficient c is 0.25, when the number of teeth z≥42, the root circle radius is larger than the base circle radius, and the tooth profile curve It is completely involute; when the number of teeth of the involute gear is z<42, the radius of the root circle is smaller than the radius of the base circle, and an approximate curve of an involute between the root circle and the base circle. In the example of the present invention, the number of teeth of the gear is 20, so the radius of the root circle is smaller than the radius of the base circle, so the shape of the tooth profile is not completely involute.
The end point of the involute at the base circle and the root circle are connected by a spline curve as a transition arc of the root portion of the involute gear. Although the root transition arc does not contribute to the gear meshing process, it has an important influence on the strength of the gear, especially on the bending strength of the gear. This paper focuses on the drawing of the involute part of the gear profile, and has proposed a method to establish an accurate root transition arc, which will not be discussed in detail in this paper.
The excess of the base circle and the transition arc is trimmed separately to obtain one side of the gear transition arc profile. Then take the construction line of the center of the base circle as the center line, use the "mirror entity" command to draw the tooth profile on the other side of the tooth root, and chamfer the top of the tooth to obtain a complete end face profile of the tooth.
(3) The end profile of the "skeleton circumferential array" array of teeth, the number of arrays is 20, and then the excess portion of the root circle is cut off, as shown.
(4) After exiting the sketch, the entire gear profile is stretched as a section, the stretching depth is the gear thickness, and the stretching is completed. Similarly, a three-dimensional model of the involute helical gear can be obtained by scanning along the spiral.
3 Conclusion The shape of the involute gear is more complicated, which has always been the difficulty of 3D CAD design. This paper introduces the path of the involute gear with the involute spur gear as an example. For the helical gear, the basic principle is the same. After completing the tooth profile of the involute helical gear, a spiral is generated, and the profile of the end face is used as a contour by the scanning method. gear.
The method of forming the involute gear based on the forming principle of the involute spiral surface is a relatively simple and easy-to-understand method. The gear obtained by the method can ensure the accuracy of the shape, and the modeling speed is fast, and the manual point-taking shape is avoided. Complex process.
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