Helical cylindrical gear pair for uniform power transmission
The invention in question is a helical cylindrical gear pair shown in the front view in Fig. 1. It consists of a driving gear (1) with multiple teeth and a driven gear (2) with multiple teeth where power is transmitted uniformly from the driving to the driven gear. The gears in such a gear pair are characterised by their module, usually a different number of teeth and the same helix angles but opposite direction.
Areas of application and the core of the problem
Power transmission from power machines to different devices occurs in all the areas of mechanical engineering and is essentially the driving force of modern civilization. The tendencies in the development of power machines are to achieve greater rotational speeds (gas turbines, internal-combustion engines, electric motors), greater power transmission and improved efficiency. However, the velocities of devices need to be adapted to working conditions, so speed converters, including gearings - a more specific focus of this invention - are used. Gearings are used in practically all fields of engineering for transforming rotational speeds from the smallest to the biggest powers, from low to high transmission ratios and from low to very high rotational speeds. Within this wide field of engineering special attention will be paid to helical cylindrical gears, which are
used in heavy industrial drives, large devices, oil pumps and driving big wind power plants.
The results of extensive scientific research and technological achievements in the field of gearings have been published in literature written by authors from the most technologically developed countries and are also described in textbooks by authors from renowned European, Japanese and American universities. Under the auspices of the international ISO organization, professional associations in the industrially developed countries have developed extensive ISO standards for gear verification and control. However, disagreements on the verification of scuffing resistance (integral or flash temperature method) as a result of transforming some friction into heat have not been resolved yet. This invention refers to uniform frictional loading of teeth flanks, which is the result of friction or sliding velocities. With this invention we propose teeth flanks shaped in a way that enables a uniform distribution of load, uniform sliding, less friction and smaller contact load.
Non involute gears with conformal contact1 generally relate to helical gear teeth with cycloidical gear tooth profiles to provide conformal contact between adjacent teeth. Each tooth flank incorporate a relief at the region of pitch circle areas to separate a tooth addendum from tooth dedendum of the same tooth flank, so that the power transmission from driving gear to driven gear could be transmitted from convexly shaped addendum to concavely shaped dedendum. The relief area precludes contact in the areas where convex-convex contact would occur.
Gear Tooth Profile2 patented by John Colbourne refers to a gear and method for producing the gear. The gear has a gear tooth profile conjugate to a gear basic- cutter tooth-profile having an addendum with a convex portion having an
1 US Patent No.: 6,837,123. R.M.Hawkins, R.M. West Point (NY), 2005.
2 US Patent No.: 6,964,210. J. R. Colbourne, St. Albert (CA), 2005
addendum point proximal to a pitch line and a dedendυm with a concave portion having a dedendum point proximal to the pitch line. The convex portion is complementary with a corresponding portion of a mating-gear basic-cutter tooth- profile dedendum. The concave portion is complementary with a corresponding portion of the mating-gear basic-cutter tooth-profile addendum. A transition zone between the addendum point and the dedendum point has a predetermined width. The gear basic-cutter tooth-profile has a predetermined half pitch relief at the pitch line and continuity of profile and continuity of slope at the addendum point.
Novikov spur gears3 with double line of action, Basic rack, is a Russian standard defining gears with a line of action in tooth addendum and dedendum, limiting to hardness of 320 HB, modules of less then 16 mm and velocities below 20 m/s.
Many varieties of Novikov-Wildhaber gear drives have been developed due to their good features. The recent version was proposed by Litvin et al.4 The advantages of the developed gear drive are reduction of noise and vibration caused by errors of alignment, the possibility of grinding and application of hardened materials, and reduction of stresses. These advantages are possible due to application of geometry, based on application of parabolic rack-cutters, double-crowning of pinion, and parabolic type of transmission errors. Helical gears of new geometry can be applied in high-speed transmissions.
Description of the new solution
The helical cylindrical gear pair, shown in front view in Fig. 1 , includes a driving (1) and a driven (2) gear. This invention features gear teeth profiles (3) which are in a radial plane composed of addendum (4) and dedendum (5) circular arcs where the addendum arc forms a part of the addendum circle (6), and the dedendum arc a part of the dedendum circle (7). The driving gear features a driving kinematic
3 GOST 15023-76.
4 Litvin, F.L. et al. New version of Novikov-Wildhaber helical gears: computerized design, simulation of meshing and stress analysis. Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 49-50, 2002,pp. 5707-5740.
cylinder (9), while the driven gear features a driven kinematic cylinder (10). They coincide in the pitch point C which is traversed by the rack's datum line (11). The shape of the tooth flank in the region of the kinematic cylinder (12) is determined by the gear manufacturing tool in accordance with the basic rack profile (13) shown in Fig 2.
Power or force F, which is transmitted from the driving gear (1) to the driven gear (2), passes through two concave-convex contact areas, that is through contact points Pa and Pd. Contact Pd is formed by the convex profile of the addendum of the driven gear and the concave profile of the dedendum of the driving gear. Contact P3 is formed by the convex profile of the addendum of the driving gear and the concave profile of the dedendum of the driven gear. Contact surfaces Pa and Pd lie on the sliding circle (8), have the same normal and are diametrically opposite to the kinematic point C. The distance between the contact zones is defined by:
PaPd = mπcosa where m - module; α - angle, which limits the addendum and dendendum arc (Fig. 2); m π = p = e + s, p - circular pitch, further divided into s - tooth thickness and e - the tooth space width.
The gears that are the subject of this invention can be manufactured on any common gear cutting machines with a cutting tool which corresponds to the basic rack profile (13) shown in Fig. 2. The rack space width corresponds to the gear tooth thickness s and the rack tooth thickness (1 - k) k m π corresponds to the gear tooth space width e. Arc ED, which is part of the addendum circle (6), forms part of the dedendum tooth flank of the rack (4) and arc GF1 which is part of the dedendum circle (7), forms part of the addendum tooth flank of the rack (5). The circular arc (14) with the centre in point Oi is in point 1 tangentially connected to dedendum circular arc EI of the rack and in point 2 with arc 2G5 of the addendum of the rack. Arc (17) with the radius p connects the right and the left tooth flank of the rack. Tangential contact of all the three arcs is the smooth edge of the cutting tool. If we want a deeper interspace DF between the rack addendum arc (4) and
rack dedendum arc (6) a connecting arc (15) with a diameter of the addendum circle (6) is made through points D and F. The bottom of the rack tooth space is limited by a straight line in depth h, which equals or is bigger than the gear module. The difference between the gear tooth thickness s and tooth space width e is established with coefficient k < 0,15.
With this invention gear teeth are formed by successive cutting of the workpiece with a tool whose basic profile corresponds to the basic rack profile (13) in Fig. 2, so that after each cut the rack datum line (11) rolls over the refence circle (9) of the manufactured gear for the thickness of one cut, followed by the next cut. The rolling process is shown in Fig. 3 where the discrete positions of the rack profile (16) are marked with a dotted line. When the pitch point C on the rack datum line (11) reaches the pitch point »C« on the reference circle (9), the cutters of the rack circular arcs form the shape of the gear tooth addendum (4) and dedendum (5). This is the position of the rack (tool) in which the manufacturing of the gear tooth addendum arc, which is identical to the dedendum circular arc of the tool, is completed. At the same time the dedendum tooth circular arc of the gear, which is the same as the addendum circular arc of the tool, is also shaped. It is for this reason that all gears manufactured with the same tool (the same rack) have identical addendum (4) and dedendum (5) arcs. A part of the cutting edge which is shaped by the connecting arc (14) or (15) on the rack, shapes the tooth flank profile (12) (see Fig. 1) between the addendum (4) and dedendum (5) arc. The tooth flank profile (12) reduces the tooth thickness in the area of the rolling circle and prevents the contact of the teeth flanks in the radial plane on the path from contact Pd to contact Paas shown in Fig. 4.
Fig. 1 shows that force F is transmitted from the driving to the driven gear through the contacts Pd and Pa. Since the gears are helical, the shape of the tooth flanks is that of a helix, so each of both contact points are located on two helices shown in Fig. 5. Therefore, the helix (18) which runs on the dedendum tooth flank of the driving gear corresponds to contact point Pd and the helix (20) which runs on the addendum tooth flank of the driven gear. Similarly, the helix (19), which runs on
the addendum tooth flank of the driving gear and the helix (21), which runs on the dedendum tooth flank of the driven gear. With rotation the helices of the driving gear (18) and (19) push the helices of the driven gear (20) and (21) in the direction of the rotation. Due to the rolling of the helices of the driving gear on the helices of the driven gear the contact points Pd and P3, which are actual contact points of the corresponding helices, move from the front radial plane to the rear radial plane with velocity vr. As the tooth flank of the driving gear in contact points Pa and Pa rolls on the flank of the driven tooth in axial direction with velocity vr, the tooth flank of the driving gear slides on the sliding circle on the tooth flank of the driven gear with sliding velocity vg.
Fig. 5 shows that in contact points Pd and Pa helical cylindrical gears for uniform power transmission and in steady conditions transmit motion and forces uniformly from the front to the back side of the gear. This is cyclically repeated from tooth to tooth with each mesh of the tooth pair. With involute gears the sliding direction changes in the kinematic point, while its velocity increases with the distance from the kinematic point as shown in Fig. 6a. With involute gears frictional work varies and with it also the value of contact temperature (0fla), which increases with distance from the kinematic point C. Under some working conditions there is a danger of flash temperature (0fla) exceeding the acceptable limit of scuffing resistance which leads to severe gear-tooth surface damage.
With helical cylindrical gears for uniform power transmission the tangential force is divided into two contact zones, so the pressure in individual contact points is lower and identical along the entire width (Fig. 6b). The distances between contact points and the kinematic point are shorter than the ones in involute gears, therefore, sliding velocity is lower as is friction and the possibility of gear damage.
A short description of figures
FIGURE 1 depicts a helical cylindrical gear pair for uniform power transmission. Teeth profiles of both gears are shown. Positions of both contact zones Pd and Pa
over which load is transmitted, are marked. The position of contact zones in relation to the kinematic point C and the sliding circle (8) over both contact points with the centre in point C is also presented.
FIGURE 2 shows the structure of the rack profile (13) for which the cutting tool for helical cylindrical gears for uniform power transmission must be suited. The flank profile of this rack (3) is comprised of a addendum circular arc (5), dedendum circular arc (4) and connecting arc 14 or 15. Rack tooth thickness (1 - k) m π implies the UPT gear tooth space, while the rack space thickness km π implies the gear tooth thickness.
FIGURE 3 shows how the rack cutter shapes the gear teeth by successive cutting of the workpiece and rolling of its datum line (11) on the kinematic circle (9). It can be seen how the dedendum of the rack shapes the addendum (4) of the helical cylindrical gear for uniform power transmission and how the addendum of the rack shapes the dedendum (5) of this gear. In this way, in gear cutting the addendum and dedendum profiles of the gear acquire the shape of the addendum and dedendum profile of the rack. Therefore, all gears manufactured with the same tool (the same rack) share the same addendum and dedendum profiles.
FIGURE 4 shows how the tooth flank of the driving and the tooth flank of the driven gear of the helical cylindrical gear pair for uniform power transmission move along the radial plane from contact zone Pd to contact zone Pa without touching. Since the teeth flanks of the driving and the driven gear are not in contact with each other on this path, there is no load applied to flanks, which consequently do not cause friction.
FIGURE 5 depicts kinematic circumstances in the transmission of motion and load between the driving and the driven gear. A view of the gear pair in the axial direction A and in the top view B is shown in this context. Considering that gear teeth for uniform power transmission are helical, both contact zones Pd in P3 have their own pair of helices which run along a tooth of each of both gears along the entire teeth
length from the front to the rear radial plane. Every helix has a specific contact width, which during power transmission travels with the contact zones Pd or P3 on its own path, which is parallel to the kinematic axis of the gears through point C. The helices lie each on its own base cylinder, so they have different helix angles βp with torsionial radius of curvature τ. The driving gear has helices 18 and 19, while the driven gear has helices 20 and 21. Both contact points travel with the same velocity in the direction perpendicular to the radial plane.
FIGURE 6 shows the course of contact loads and sliding velocities and consequently, also the course of the flash temperature (θfla) along the path of contact with involute gears and the conditions that arise along the helices with helical cylindrical gears for uniform power transmission. With involute gears the path of contact goes through the pitch point C where forces are transmitted only by rolling without sliding. At the beginning and at the end of the path the sliding velocities can be high. Accordingly, friction and the heating of sliding surfaces change (θfla). On the other hand, in helical cylindrical gears for uniform power transmission the loads between teeth flanks are transmitted uniformly, more with rolling of contact surfaces and less with sliding. Furthermore, the teeth flanks load is divided in two contact zones, therefore, the heating of surfaces (9fla) is reduced, steady and it does not include very hot points.