Introduction to 0
Gear transmission is one of the most important mechanical transmissions. More than 300 years BC, the ancient Greek philosopher Aristotle expounded the problem of transmitting rotary motion with bronze or cast iron gears in Mechanical Problems. From the end of 17 to the beginning of 18, people began to study the strength of gears. After industrial revolution in europe, gear technology developed rapidly, and gear transmission was widely used in mechanical transmission and the whole mechanical field. Gear design has become one of the important design contents in mechanical design. At present, the commonly used gear strength design formulas in the world mainly include: the calculation method of the International Organization for Standardization (ISO); American Gear Manufacturers Association (AGMA) standard calculation method; German industrial standard (DIN) calculation method; Japan Gear Industry Association (JGMA) calculation method; British BS calculation method, etc. In the teaching of mechanical design, especially gear design, the author found that the knowledge points in many places are relatively simple and difficult to understand. Therefore, this paper deeply analyzes several problems in gear design, such as the failure mode of gear, the history and present situation of gear strength design, and discusses the historical origin of gear strength design in China and some puzzles in gear design. Through in-depth analysis, it is helpful for everyone to better understand the significance and context of gear design formula.
Discussion on failure mode of 1 gear
Gears will fail in various forms during transmission, and even lose transmission capacity. The failure mode of gear transmission is related to gear material, heat treatment mode, lubrication condition, load size, load change law and speed. People's understanding of gear failure is a developing process. /kloc-in the middle of 0/8th century, people began to study the failure of gears. A preliminary understanding of gear friction and wear, pitting formation and tooth surface bonding is obtained. 1928, Buckingham published a paper on gear wear, which divided the tooth surface failure into six failure forms: pitting, abrasive wear, gluing, peeling, scratching and biting. In 1939, Rideout classified gear damage into eight forms, including normal wear, pitting, peeling, gluing, scratching, cutting, rolling and hammering. In 1953, Borsoff and Sorem classified gear damage into six categories. In 1967, according to a large number of experiments, Niemann drew the bearing capacity limit diagram for four failure forms of involute gears, and pointed out that when the gear speed is low, the main factor affecting the bearing capacity of soft tooth surface gears is pitting corrosion, while the bearing capacity of hard tooth surface gears is broken teeth; For high-speed and heavy-load transmission gears, the influencing factor is often gluing. Since 1950s, some countries have classified gear damage forms in a standard form, and specified terms, performance characteristics and reasons. For example, 195 1 year, the United States divides gear damage into two categories, one is tooth surface damage, including wear, plastic deformation, gluing, surface fatigue and so on. And the other is broken teeth. The first kind of tooth surface damage is the surface damage caused by gear as a high pair; The fracture of the latter gear tooth is that the gear tooth as a stress member is damaged due to insufficient volume strength. 1968 Austrian national standard stipulates the term of gear damage.
1983, China promulgated the national standard "Terms, characteristics and causes of gear tooth damage" (GB /T348 1-83), which divided the gear damage forms into five categories, namely, wear, tooth surface fatigue (including pitting and peeling), plastic deformation, gear tooth fracture and other damages, and ***26 kinds of failures. 1997, China promulgated the revised national standard GB/T348 1- 1983. At present, most mechanical design textbooks and manuals in China simplify gear failure modes, which are generally divided into five categories, namely, gear tooth fracture, tooth surface fatigue pitting, tooth surface bonding, tooth surface wear and plastic deformation.
Discussion on Strength Design of Gear 2
2. 1 Calculation of bending strength of gear teeth
1785, Watt, England, put forward a method for calculating the bending strength of the tooth root, which regarded the tooth as a flat cantilever beam with rectangular cross section, and then a variety of formulas for calculating the bending strength appeared. 1893, Louis published a formula for calculating the bending strength of gear teeth, and found the dangerous section of gears by inscribed parabola method, which is called "parabola method" [12], as shown in figure 1. Lewis deduced the formula of tooth root bending stress from the load acting on the tooth top, but for gear transmission with coincidence ratio greater than 1 less than 2, theoretically, only when a single pair of teeth are engaged, the load is borne by one tooth. For the gears with enough accuracy with coincidence ratio greater than 2 and less than 3, the maximum bending stress point is low because there are more than two pairs of gears meshing at the same time.
After that, 30 tangent method, Niemann method, Buckingham method and so on appeared. 1980, ISO put forward the Basic Principle of Bearing Capacity of Involute Cylindrical Gear (ISO 6336- 1980), and published the calculation methods of gear tooth bending strength and tooth contact strength.
In the past, the calculation method of gear strength in China was chaotic and there was no unified standard, which brought many problems to production, scientific research and teaching. Therefore, in 198 1, the national standard research group of "calculation method of involute cylindrical gear" was established in China, and a comprehensive research work was carried out based on ISO 6336- 1980. 1983 promulgated the national standard for calculating the bearing capacity of involute cylindrical gears (GB /T3480— 1983).
At present, the design formula of gear bending strength in China basically adopts the 30 tangent method, that is, the oblique line that forms a 30 angle with the symmetrical center line of the gear tooth and is tangent to the fillet of the tooth root, and the connecting line of the two tangent points is the dangerous section position of the tooth root. Moreover, the highest point in the meshing area of a single pair of teeth is the most unfavorable load point, and the bending stress generated at this time is the largest, as shown in Figure 2. In addition, in the calculation formula of bending fatigue strength, the tooth profile coefficient is only related to the number of teeth in many mechanical designs, and has nothing to do with the modulus, which is not easy to understand. The following is a detailed analysis of related issues. As shown in fig. 2, the tooth root bending stress σ f = MW = fnhfcos αfbs2f/6 = 6kfthfcos α fbs2f cos α = kftbm6 (hfm) cos α f (SFM) 2cosα (1), where α f is the pressure angle of the tooth tip circle. YF =6( hFm) cos αF( SFm)2cos α In the formula (1), YF is called the tooth profile coefficient, which was first quoted by Louis in his formula for calculating the bending strength of gear teeth. It can be seen that YF is a coefficient related to the geometric parameters of tooth profile. Because, according to the principle of gear formation, the change of tooth number will cause the change of parameters such as hF, SF and aF on gear teeth. Since hF, SF and aF are all proportional to the gear modulus, the modulus in the tooth profile coefficient can be omitted. Therefore, the tooth profile coefficient is not affected by the modulus, but only related to the number of teeth. The more teeth, the smaller YF, and vice versa. This is why it is often seen in mechanical design textbooks that the tooth profile coefficient of standard gears is only related to the number of teeth and has nothing to do with the modulus.
2.2 Influence of gear compressive stress on bending stress
According to 30 tangent method and gear stress analysis. The normal force Fn moves to the center line of the gear teeth and is decomposed into two vertical components, namely, the circumferential force Ft and the radial force Fr. According to mechanical theory, the bending stress produced by Ft is σF, while the compressive stress produced by Fr is σ y.. Therefore, the stress on the dangerous section of the tooth root is a composite stress composed of bending and compression, which leads to the unequal stress on both sides of the tooth root. However, in the relevant mechanical design data, the compressive stress caused by radial force is not calculated in the calculation formula of gear bending strength, and in most related teaching materials, it is considered that the compressive stress is relatively small relative to the maximum bending stress of tooth root and can be ignored. However, few people have calculated how big the compressive stress is and why it can be ignored. The following is a discussion about compressive stress and bending stress. As shown in fig. 2, the bending stress σF generated by Ft is shown in formula (1). The compressive stress σy generated by Fr is σy = Fnsin αFbSF( 2). σyσF= SF6hFtan αF = h', then SF = 2h' tan30, so σyσF= tan 30tan αF3h'hF Suppose the standard gear modulus is m and the number of teeth is Z. Then the pressure angle of the tooth top circle is cos αF = rbra= zz+2cos α, because h'hF.
2.3 Calculation of Contact Fatigue Strength of Tooth Surface
Fig. 4 The contact fatigue strength calculation of tooth surface of Hertz contact stress model is the strength calculation for calculating the fatigue pitting failure of gear tooth surface. In 188 1, Hertz put forward the load distribution formula on the contact surface when two cylinders are in contact as the theoretical basis for calculating the tooth surface strength, as shown in Figure 4. According to Hertz contact stress theory, the maximum contact stress in the contact zone under load is σ h = fn π b1ρ1ρ 21-μ 21e1+1-μ 22e2 (. B is the contact length; μ 1 and μ2 are Poisson's ratios of two cylindrical materials respectively; E 1 and E2 are the elastic modulus of two cylindrical materials. ρ 1 and ρ2 are the contact radii of two cylinders, where "+"is used for external contact and "-"is used for internal contact. 1898, Lasser applied the "pressure" principle to study the contact fatigue strength of tooth surface according to the normal force. In 1908, Witteki of Austria applied Hertz's contact stress theory of two cylinders to calculate the tooth surface stress of gear teeth, and drew the maximum contact stress diagram along the meshing line. 1932, British BS put forward a calculation method based on experimental data, which takes the basic surface stress as the tooth surface strength. American AGMA adopted the calculation method of the maximum contact stress at the heaviest load point of tooth surface strength in 1940.
In 1949, Buckingham proposed a method for calculating the contact stress of tooth surface on pitch circle, which was later adopted by many calculation methods. 1954, Niemann adopts the rolling pressure at the maximum load point. Up to now, the theoretical basis for calculating the contact fatigue strength of tooth surface in China is Hertz formula, that is, Hertz stress is used as the judgment index of pitting corrosion. Generally, let 1ρ σ = 1ρ 1ρ 2, and ρ σ is called comprehensive curvature. For standard gears,1ρ σ = 2d1sinα i1i. And let ze =1π1-μ 21e1+1-μ 22e in formula (3) be the elastic influence coefficient. Therefore, the basic formula for obtaining contact fatigue strength of involute spur gear is σ h = zezh2kt1BD 21I1σ h (4), where ZH = 2 sin αcos α, which is called area coefficient, is equal to 20 for pressure angle α. There are many versions of gear contact fatigue strength formulas in mechanical design manuals or mechanical design textbooks, among which the most common one is to simplify the checking formula of gear tooth surface contact strength of a pair of steel standard gears, taking E 1 = E2 =2. 06 × 105MPa，μ 1 =μ2 =0。 3. Then the commonly used checking formulas in mechanical design are obtained. σh = 67 1kt 1bd 2 1I 1I ≤[σ]h(5)
2.4 Calculation of Tooth Surface Bonding Strength
Another common fault of gears is tooth surface gluing. There is a unified view on gear gluing: two meshing metal tooth surfaces contact directly under certain pressure, and at the same time, with the movement of tooth surfaces, the metal is torn off from the tooth surfaces, resulting in adhesion and wear. Glue is divided into cold glue and hot glue For high-speed and heavy-load gear transmission, thermal bonding is easy to occur when the instantaneous temperature of the tooth surface is high and the relative sliding speed is high. For the heavy-duty gear transmission with low speed and heavy load, the oil film on the tooth surface is destroyed because of the excessive pressure between the tooth surfaces. Although the tooth surface temperature is not high, it is easy to glue, which is called cold glue.
At present, the research on the calculation of gear tooth surface bonding strength is mainly based on two theories. One is based on Pv value (product of pressure and velocity) or PTV value (t is the distance from the meshing point to the node) as an index to calculate adhesion. The other is Brock algorithm, which takes tooth surface temperature as the criterion to judge bonding. 1975, Winter proposed the integral temperature method. Now these two methods are mainly used in ISO standard. In 2003, China promulgated the national standard "Calculation Method of Bonding Bearing Capacity of Cylindrical Gear, Bevel Gear and Hypoid Gear" (GB-Z 64 13. 1-2003 and GB-Z 64 13. 2-2003). This standard is equivalent to ISO/TR 13989-2000 "Calculation Method of Bond Bearing Capacity of Cylindrical Gear, Bevel Gear and Hypoid Gear". Some people have tried to calculate the thickness of oil film between tooth surfaces according to the theory of elastic hydrodynamic lubrication as a basis for judging adhesion.
In most mechanical design textbooks in China, gear strength design generally only provides two calculation methods of tooth surface contact fatigue strength and tooth root bending fatigue strength, but does not provide the strength calculation formula of tooth surface bonding.
3 Conclusion
In this paper, the problems of gear strength design in mechanical design teaching are analyzed and discussed respectively, and the historical evolution and present situation of gear strength design in China are explained in detail, as well as the puzzling problems and solutions in the calculation process of gear strength design. It is pointed out that in the calculation of gear bending fatigue strength, the influence of compressive stress on bending stress is limited and can generally be ignored, and its influence should be considered only when accurate calculation is needed. The research in this paper can help gear designers and students better understand the related contents in gear design and lay a good foundation for future mechanical design work.
;