Grinding the side surfaces of the spiroid cylindrical gear worm turns with a tapered wheel

. Information about the worms’ production technology of spiroid cylindrical gears is given. A method of the profile analytical calculation of a disk abrasive tool is proposed, providing the possibility to profile grinding wheels for various types of ruled helical surfaces: Archimedean, convolute, involute, etc. The use of the calculation methodology makes it possible to achieve high accuracy in processing the active (working) side surfaces of a hardened grinding worm; it sharply reduces the wear intensity of the spiroid wheel associated with it, the ring gear of which is made of bronze.


Introduction
A sufficiently large number of various spiroid gears are known, differing in the shape and arrangement of links, the profile of engaging elements, the number and location of engagement zones, and other features [1][2].
The two most common varieties of spiroid gears are shown in Figure 1.
When choosing the type of gears, it is necessary to take into account the technological availability and labour intensity of its production, and the availability of technological equipment at the enterprise. As in worm gears, the spiroid wheel is cut with a tool that copies the worm in shape, position, and relative motion. For the production of spiroid gears, gear-cutting universal equipment, which is widely used at machine-building plants, is used, which distinguishes them favourably, for example, from globoid gears [1,[3][4][5].
Of the two types of spiroid gears shown in Figure 1, the gear with a conical worm (see Figure 1-a) is most used abroad due to its greater load capacity. The advantages of this gear are not very great, therefore, in Russia; a spiroid cylindrical gear is used (see Fig. 1-b).
The highest performance indicators for spiroid gears, as well as for worm gears, are provided for a combination of steel-bronze transmission links in the case of using hardened and profile-ground worms [6]. a) b)

Fig. 1. Types of spiroid gears
A typical technological process for manufacturing spiroid worms includes the following operations [1]: -cutting the rolled material on a band saw machine; -trimming the outermost ends, centering and turning the short base neck on one side of the intermediate workpiece on a universal lathe (screw-turning lathe) with program control (usually in two operations for machining from two different sides); -pre-cutting of turns on a screw-cutting lathe using a cutter or (in mass production) a vortex end head; -machining of auxiliary elements of the worm structure (threads, keyways, etc.); -metalworking -cutting off the sharp ends of the worm turns; -hardening thermal or chemical-thermal treatment (in the second case, it is advisable to protect against excessive embrittlement during hardening of necks with sharp edges, such as keyways) to the hardness of the surface at least 50 HRCe; -straightening of center bores; -grinding of necks and ends on a face grinding or circular grinding machine; -grinding of coils on a worm or thread grinding machine; -control of worm parameters. Due to some specificity, in the proposed article, more attention is paid to the option of grinding the profile of the worm turns with a tapered abrasive wheel.
Out of three known varieties (cylindrical, traditionally tapered and reverse-tapered) spiroid cylindrical gears are the most technologically advanced. As already noted, this circumstance was decisive when choosing the type of transmission for the cable assembly mechanism of an electric loader.
A significant advantage of spiroid cylindrical gears in terms of manufacturability is low sensitivity to manufacturing and assembly errors, as well as the possibility of manufacturing on the metal-cutting equipment that is widespread in the industry. In many cases it is enough to have screw-cutting and gear-cutting lathe available for the manufacture of spiroid gears.
The main difference between spiroid cylindrical worms and worms of cylindrical gears is the asymmetric profile. The noted feature is due to the position of the engagement zone relative to the center transmission line and is taken into account both in the design and in the manufacture of the gears.
The cutting of gear rims of spiroid wheels in single, small-scale and serial production is carried out by running in with spiroid hobs on conventional gear-cutting machines (for example, models 5K301, 5KZ10, 5DZ2, 5E32, 5K32, etc.). In cases where the kinematics of the machine provides an axial feed of the tool spindle, spiroid wheels can be cut with flying cutters or cutters with a chamfer cone. Figure 2 shows the arrangement of the spiroid cutter relative to the wheel crown at the end of gear cutting and the main movements in the cutting process, and cutting on the machine is shown in Figure 3. The designations adopted in Fig. 1: Bl is the distance from the center transmission line to the nearest end of the cutter, which is taken equal to the distance from the center transmission line to the nearest end of the spiroid worm; h is the height of the wheel tooth, h=2.25m; аw -deviation of the center distance in processing is selected according to GOST 3675-81.
Cutting mode parameters V (cutting speed, m/min) and SB (vertical feed, mm/rev) can be assigned with sufficient accuracy in accordance with the recommendations contained in [7,8].
The control of spiroid wheels is similar to the control of worm and, in part, bevels wheels. With careful identity control of the spiroid cutters and worms' main parameters, the correct relative position of the cutter and the cut spiroid wheel at the time of gear cutting completion, the wheel cannot be subjected to special control. To control the size and nature of the location of the contact spots on the teeth when they are engaged, you can use bevel gear tester (for example, mod. 5A725, etc.). A number of parameters, as practice has shown, can be controlled using the B1 OM bench center, the BV-584 gear measuring device, the LIZ angular pedometer, etc.
Since the technology for manufacturing spiroid gears is similar in many respects to the technology for manufacturing worm gears, the design of the tool is carried out in a similar way. However, it is necessary to take into account the profile geometric features of spiroid worms, and therefore the cutter must be a copy of the worm, differing from it in the presence of cutting elements and allowance for regrinding the tool.
As for the technological capabilities of gear-cutting machines when used for cutting spiroid wheels, they can be significantly expanded [9,10]. This refers to the fact that the crown of the spiroid wheel is in the area located either under the tool (cutter, flying cutter) or above it.
We have implemented two methods for producing spiroid wheels: with a spiroid cutter and a flying cutter. The best quality of the side surface is achieved when cutting with a milling cutter. When cutting with a flying cutter, a large cut of the teeth is obtained. In the case of using bronze for wheel rims, the cut disappears after 10-15 hours of gear work.
When using spiroid cylindrical gears, a different profile of the worm turns is used: Archimedean, convolute, involute, convex-concave, etc. The geometry of the profile of the worm turns is chosen taking into account the purpose of the transmission and the technological capabilities of the manufacturer. The greatest effect from the use of spiroid gears can be achieved in the case of using heat-treated worms with grinding and polished side surfaces of the turns.
Grinding multi-thread worms has a number of features. Therefore, the issue requires more detailed consideration.
Among the methods for calculating the profile of a disk tool, the most widely used are analytical methods for calculating the profile of a disk tool.
The most common analytical methods for determining the tool profile [6,7,[11][12][13] are characterized by the fact that at first the coordinates of the points of the contact line of the tool surface with the surface of the workpiece are found, and then the tool profile is determined directly.
To find the coordinates of the contact line points, we use the theorem [6] -the contact line of the product surface with the conjugate instrumental surface, if the latter is a surface of rotation, is a set of such product surface points, the normals to which intersect the tool axis.
Let us turn to the scheme of a disk tool installation in relation to the workpiece presented in Fig. 4. The tool is connected with the Su(Xu, Yu, Zu) coordinate system, located in such a way that the Zu axis of rotation of the tool turns out to be rotated relative to the Zl axis of the worm by an angle u and is separated from it at a distance Аu. The connection between the systems Su and S1 is expressed by the equations: The Zu axis equation in the Su system is: Considering (2) together with the normal equation  (3) to the machined surface of the product, we obtain the equation of that normal (3), which passes through the axis of the tool and, therefore, through the contact point of the tool surface with the machined surface of the product: Equation (6), considered together with the equations of the processed helical surface, will determine the line of contact of the tool with the specified surface in the coordinate system S1(X1; Y1;Z1) associated with the worm. To determine the position of this line in the system Su(Xu; Yu;Zu) one should use formulas (1) for the transition of S1 to Su. Since the instrumental surface in the case under consideration is a surface of revolution, its axial profile can be determined by the Zu and Ru, coordinates, where u n n R x y .  22 (7) Thus, the axial profile of a disk tool designed for processing a helical surface is determined by equations (1), (6) and (7).
Let us turn to the equation derivation of the ruled helical surface of the spiroid worm turns and the projection of the normal vector to this surface.
To derive the equation of the ruled helical surface of the turns of the spiroid worm, we use the following coordinate systems (Fig. 5): the fixed system S(X; Y; Z), the movable system S1(X1; Y1; Z1), associated with the worm, and the auxiliary system coordinates Sв(Xв; Yв; Zв), associated with the tool.
The current position of the Sв system is determined, firstly, by the parameter p of the helical motion and, secondly, by the angle  through which the Sв system is rotated relative to the S1 system and which is conveniently measured in the end plane between two radii, one of which is parallel to the axis Х1, and the other axis Хв..   The transition from one coordinate system to another using the matrix method of coordinate transformation [6,17] is expressed by the following equations: ss вв r M r ,  11 (8) ss ss s sвв r M r M M r .
  1 1 1 1 (9) In these equations в ⃗ ⃗ , 1 ⃗⃗ ⃗ , -column matrices that determine the position of a point in the systems Sв, S1, S , respectively, and the transition matrices from one system to another have the form: Sв, S1, S , is the transition matrix from the system Sв to the system S1: cos -sin 0 sin sin cos 0 cos Mss1 transition matrix from system S1 to system S: The product of matrices Ms1sв and Mss1 is the transition matrix Mssв from system Sв to system S: As a result, we get: cos( + ) -sin( + ) 0 sin( + ) sin( + ) cos( + ) 0 cos( + ) Relationship equations between Xв coordinates; Yв; Zв and X1; Y1; Z1 based on (8) and (10) are written as follows: Entering the designation 1 and using (9) and (12), the equations of connection between the Xв coordinates; Yв; Zв and X1; Y1; Z1 will be obtained in the form: cos sin sin sin cos cos Turning to the equation derivation of the worm turns linear helical surface, it should be noted that, as is known [7,11,17], surfaces that are formed as a result of helical motion in the space of a straight line are called ruled helical surfaces. The most obvious and common technological schemes of forming surfaces of this type are characterized by the fact that the lateral cutting edge of the lathe cutter is considered as a formative straight line, and the relative helical motion is represented as a uniform rotation of the workpiece, on which the helical cutting is performed, and uniform forward movement of the cutter together with the longitudinal slide of the lathe parallel to the axis of the workpiece.
There are three types of ruled helical surfaces -convolute (general case), Archimedean and involute (special cases). Let's turn to Fig. 5. The generatrix of the line ОвМ , which has the designation U, and ОвМ=U , is located in the ХвZв plane of the auxiliary coordinate system Sв(Хв;Yв; Zв).
The point Ов is the point of the line U closest to the axis Z1 of the worm and, thus, is located on the surface of a cylinder of radius rц , which is called the guide cylinder, since it characterizes to a certain extent the direction of the line U, which always remains tangent to it. The Zв axis always coincides with any generatrix of the guide cylinder, and the Хв axis is always perpendicular to the Zl axis of the worm. During the helical motion of the system Sв containing the straight line U, the point Ов describes a helical line L on the guiding cylinder, characterized by the parameter p, which, in accordance with the diagram shown in Fig. 5 has a positive (negative) sign if the helix is right (left). The beginning of the line L Coincides with the initial position of the point Ов, at which the Yв axis is collinear in the same direction with the Y axis, and the Xв axis is parallel to the X axis (and has the same direction) of the coordinate system Sв(Хв;Yв; Zв), rigidly connected to the worm.
Let us designate the angle of elevation of the helix through L through ц . Its value can be found from the relation ц ц p tg . r   (15) If we turn to the diagram, this angle can be imagined, for example, as the angle between the Xв axis and the tangent to the line L at the point Ов . The angle  between the same Xв axis and the straight line U is the angle of the helical surface profile in section with its plane tangent to the guide cylinder.
The position of the current point M of the straight line U in the Sв system can be fixed by the coordinates: Considering together the equations (13) of the transition from the Sв system to the Sl system with expressions (16), we obtain the equations of a ruled helical surface of a general form -a convoluted helical surface in the S1, system associated with the worm: cos cos sin cos sin cos The equations of the convolute helical surface in the fixed coordinate system S(X; Y; Z) will be obtained using expressions (14) and (16) Assuming that the angle  of the profile is located in Fig. 5 in the first quarter and taking into account the accepted direction of increase in its value, we can say that the systems of equations (17) and (18)  Particular cases of a convolute surface are, as already noted, Archimedean and involute helical surfaces. The first is obtained when the generating line U passes through the axis Z1 of the worm. In this case, the radius rц of the guide cylinder is equal to zero. Assuming in (18) rц =0, we obtain the equations of the Archimedean helical surface in the fixed coordinate system S(X; Y; Z). Another special case -an involute helical surface is obtained when the generating line U touches the helix L on the guide cylinder, which in this case is called the main one. In this case,  and turn out to be equal and formula (15) takes the form: