MT Chap7 Design of Gear Mechanisms

2020-02-27 173浏览

  • 1.
  • 2.7.1 Classification of Gear Mechanisms 1. Parallel Shaft Gears (1) Spur gears The teeth are straight and parallel to the gear axis. Thi s gear is spur gear. If the gears have external teeth on the outer surfac e of the cylinders, they rotate in the opposite direction, as shown in Fig 7-1a. An internal gear can mesh with a external pinion only and they rot ate in the same direction, as shown in Fig 7-1b. The pinion and rack co mbination converts rotary motion into translational motion or vice versa, as shown in Fig 7-1c. Fig.7-1 Spur gears (直齿圆柱齿轮)
  • 3.(2) Helical gears Like spur gears, helical gears can also be used to connect parallel shafts, but their teeth are not parallel to their shafts axes, each being helical in shape. Two meshing gears have the same helix angle, but have teeth of opposite hands, as shown in Fig 7-2. Helical gear can also be classified as external contact, internal contact, helical pinion and rack. Fig.7-2 Helical gears (斜齿圆柱齿轮)
  • 4.(3) Herringbone gears Herringbone gears are equivalent to a pair of helical gears with opposite helix angles mounted side by side, as shown in Fig 7-3. The axial thrust forces of the two rows of teeth cancel each other. So they can be used at high speeds with less noise and vibrations. Fig.7-3 Herringbone gears (人字齿轮)
  • 5.2. Spatial Gears (1) Intersecting shaft gears If the teeth are formed on the cone surface, this gea r is called a bevel gear. If the teeth are straight and coincident with their cone el ements, this gear is called straight teeth bevel gear shown in Fig 7-4a. When the teeth of bevel gear are inclined at an angle to the face of bevel, this gear is kno wn as helical bevel gear or spiral bevel gear, as shown in Fig 7-4b. The curved t eeth bevel gear has teeth that are curved, but with a zero degree spiral angle, as shown in Fig 7-4c. Fig.7-4 Bevel gears (锥齿轮)
  • 6.(2) Skew shaft gears To transmit motion between the skew shafts, the spiral gears or crossed helical gears, worm gears and Hypoid gears are often used in machines. The spiral gears or crossed helical gears are limited to high load, and their shafts can be set at any angle, as shown in Fig 7-5. (3) Worm gears Fig 7-6 shows worm gears in which a few teeth of the smaller gear are wrapped around its circumference a number of times and form screw threads, and the larger wheel has concave shape in its diameter direction so that to envelope the portion of the smaller gear. The smaller gear is called worm; the larger gear is called worm gear. Fig.7-5 Crossed helical gears (交错轴斜齿轮) Fig.7-6 Worm gears (蜗轮蜗杆)
  • 7.7.2 Fundamental Law of Gearing[ 1. Law of Gearing During meshing of gears, the profile of a tooth on gear 1 is meshing at point K with another profile on that gear 2 to produce a rotary motion. Fig7-7 shows two contacting profiles with common normal n—n, and this normal intersects the line of the two gear centers O1O2 at point P. Fig.7-7 Fundamental law of gearing (齿廓啮 合的基本定律)
  • 8.If the pitch point varies for all phase of the gearing, the ratio is not a constant. This type of gear is called noncircular gears. Fig 7-8 shows a pair of noncircular gears. Fig.7-8 Noncircular gears (非圆齿轮)
  • 9.2. Conjugate Profiles When all the common normal lines for every instantaneous point of contact c an pass through the pitch point, this is the fundamental law of gearing; any mes hing profiles which satisfy the law of gearing can be called the conjugate profile s. In order to maintain the fundamental law of gearing to be true, the gear tooth profiles on meshing gears must be conjugate of one another. There are infinite n umbers of possible conjugate pairs that can be used, but only a few curves have been practically applied as gear teeth. The cycloid profile is still used in watche s and clocks as a tooth form, but most other gears use the involute curve for thei r shape.
  • 10.7.3 Involute Properties and Involute Tooth Profiles 1. Involute (1) Development of the involute Fig 7-9 show s an involute generated by a line rolling on the circumference of a circle with center at O; Wh en the line rolls, the path generated by the poin t K on the line is the involute curve. Fig.7-9 Development of the involute (渐开线的形成)
  • 11.(2) Involute properties 1) The length of the generating line is equal to the a rc length which the generating line rolls without slip ping on the base circle. 2) The generating line is always tangent to the bas e circle, and it is always a normal of the involute at a point K. 3) The length NK is the curvature radius of the inv olute at point K. Point N is the center of the curvatur e of the involute at point K. Fig.7-10 Shape of involutes 4) The shape of the involute depends upon the rad and radius of base circles (渐开线的形状与基圆半径) ius of the base circle. The smaller the radius of the b ase circle is, the steeper the involute is. If the radius of the base circle is infinite, the involute curve beco mes a straight line. This is shown in Fig 7-10.
  • 12.2. Meshing of Involute Profiles (1) The instantaneous angular velocity ratio is consta nt Fig 7 -11 shows two base circles with centers at O 1, O2 and radii rb1, rb2. (2) Separability of the center distance Any change in center distance will have no effect upon the involute profiles; the gear ratio can not be varied. (3) Stationary action force The line of action in case of involute teeth is along the common normal at the c ontact point, and the common normal is the common tangent of the base circles. Fig.7-11 Gearing of involute profiles (渐开线齿廓的啮合)
  • 13.7.4 Nomenclatures of Standard Spur Gear and Ge ar Sizes 1. Gear Teeth Nomenclatures Fig.7-12 Spur gear nomenclatures (渐开线直齿圆柱齿轮名词术语)
  • 14.(1) Addendum circle It is a circle passing through the topes of the teeth. The radius and diameter are denoted as ra and da. (2) Dedendum circle It is a circle passing through the roots of the teeth. The radius and diameter are denoted as rf and df. (3) Reference circle It is a datum circle in gear design and measurement. T he radius and diameter are denoted as r and d. Note that the subscript will be omitted when expressing the dimensions of a gear, such as r, d, p, etc. (4) Base circle It is a circle generating the involute curves. The radius and d iameter are denoted as rb and db. (5)Tooth thickness, space width and circular pitch The tooth thickness is the thickness of the tooth measured along the circumference, such as sK. The spac e width is the space between the adjacent teeth measured along the circumfere nce, such as eK
  • 15.(6) Addendum, dedendum and full depth The addendum is the radial heigh t from the reference circle to the addendum circle, denoted as ha. The deden dum is the radial height from the reference circle to the root circle, denoted as hf. The full depth is the radial distance between the addendum circle and t he root circle, denoted as hf. (7) Normal pitch It is the circular pitch measured along their normal, denote d as pb. It is equal to the corresponding pitch on the base circle. (8) Face width It is the length of the tooth parallel to the gear axis, denote d as B.
  • 16.2. Parameters of Involute Gear (1) Number of teeth It is the total number of the teeth the gear possesses, and it i s always an integer, denoted as z. (2) Module The length of circumference of the reference circle is equal to the su m of the number of the circular pitches on the reference circle, so wehave:πd=pz. (3) Pressure angle The definition of pressure angle has been demonstrated and h ere we emphasize that there are different angles on the different circumferences of the gear. (4) Coefficient of addendum A standard value of the addendum is ha=h*am. h*a is called the coefficient of addendum, usually, h*a=1, for normal teeth; h*a=0.8, for shorter teeth. (5) Coefficient of clearance Fig.7-13 Gear size with same number of teeth and different module of teeth (同齿数、不同模数齿轮尺寸)
  • 17.3. Geometrical Sizes of Standard Spur Gears If a gear has the standard module and pressure angle on the reference circle, standard addendum and dedendum, also the tooth thickness is equal to the space width on the reference circle, it is called the standard gear. Tab 7-2 shows the formula of calculating the sizes of standard gears.
  • 18.4. Tooth Thickness Along an Arbitrary Circle In gear design and manufacturing, the tooth thickness on the arbitrary circle, such as tooth thickness on the gear top, sometimes may be necessary. Fig 7-14 shows a gear tooth that has thickness sK at radical location rK. Fig.7-14 Tooth thickness along an arbitrary circle (任意圆齿厚)
  • 19.5. Terminology for Internal Gears An internal gear has its teeth cut on the inside of the rim rather than on the outside, and it has concave tooth profiles, while the tooth profiles of the external gear are convex. Fig 7-15 shows a typical internal gear, and the followings are different from the external gear. Fig.7-15 Terminology for internal gears (渐开线内齿圆柱齿轮术语)
  • 20.6. Terminology for a Rack A rack is a portion of a gear having an infinite base diameter, thus its refe rence circle, addendum circle, dedendum circle are all straight lines. The involut e profile of the rack becomes a straight line and is perpendicular to the line of a ction, see the Fig 7-16. Fig.7-16 Terminology for a rack (齿条术语)
  • 21.The characteristics of a rack are as follows. 1) The profiles are all skew lines, and they are parallel on the same side of the teeth. The pressure angles at different point o n the profile are all the same and they are equal to the nominal pressure angle of 20°. 2) The pitch remains unchangeable on the reference line, add endum line, and so on. Its value is p=πm; the base pitch is pb= πmcosα. 3) The addendum and dedendum are the same with the extern al gears.
  • 22.7.5 Meshing Drive of Standard Spur Gears 1. Conditions of Correctly Meshing for Involute Gears Gears transmit motion by means of successively engaging teeth, but not both gears are to be meshed together correctly. Fig 7-17 shows a pair of meshing gears in which all the contact points between the two gears with the involute profiles must lie on the line of action, so that the pitch point remain fixed. Fig.7-17 Meshing of teeth (齿轮啮合)
  • 23.2. Conditions of Continuous Transmission of Gears (1) Meshing process of a pair of gears In Fig- 7-18, two gears 1 and 2 with rotating centers at O1 and O2 respectively are in contact at point B2 and K. (2) Conditions of continuous transmission of g ears Fig 7-18 shows that the later pair of teet h is just coming into contact at the initial point B2 and the previous pair of teeth is in contact a t the point K, and the contact will not yet have r eached final point B1. Thus, for a short time th ere will be two pairs of teeth in contact, the con Fig.7-18 Meshing process 啮合过程) of teeth (轮齿的 tinuous transmission is satisfied.
  • 24.(3)Value of contact ratio 1) Contact ratio for external spur gears. Fig 7-19 shows a pair o f meshing gears having involute teeth. The length B2B1 can be c alculated from the following relationship. Rearranging the above equations, weobtain:Fig.7-19 Contact ratio for external gears (外啮合齿轮重合度计算)
  • 25.2) Contact ratio for internal spur gears. The contact ratio for internal gear is illustrated in Fig 7-20. In the same method, we can obtain the following formula. Fig.7-20 Contact ratio for internal gear (内啮合齿轮重合度计算)
  • 26.3) Contact ratio for a pinion and a rack. The contact ratio for a pinion and a rack is illustrated in Fig 7-21. Where PB1 is as the same as befor e and PB2 depends upon the relative position of the pinion and the rac k. If the pinion meshes with a rack without changing the distance of th e center, that is to say, the reference line of the rack is tangent to the re ference circle of the pinion, the length of PB2 is asfollows:Fig.7-21 Contact ratio for a pinion and a rack (齿轮齿条啮合的重合度计算)
  • 27.The contact ratio means the average number of pairs of teeth which are in contact usually is not an integer. If the ratio is 1 2, as shown in Fig 7-22, it does not mean that there are 1 2 pairs of teeth in contact. It means that there are alternately one pair and two pairs of teeth in contact, or one pair of teeth is always in contact, and two pairs of gears are in contact 20 percent of times. From Fig 7-22, we know that there are two pairs of teeth in contact on the segments of B2K′ and KB1, and on the segment KK′ only one pair of teeth is in contact. Fig.7-22 Nature of teeth action (重合度的意义)
  • 28.3. Relative Slide Between Contact Teeth From Fig 7-22 we know that the working profile of gear 1 is from point B2 to its tooth top, and the working profile of gear 2 is from point B1 to its tooth top. When the pair of teeth contacts at an arbitrary point, such as point K′, the slide will occur between the teeth surfaces along their tangent direction. This is because that the velocity of point K′ is not the same on the two gears.
  • 29.4. Center Distance of Gears When the shafts of a pair of gears are mounted correctly, the following condi tions must be satisfied. 1) The radical clearance between the addendum circle of a gear and the dedendu m circle of the meshing gear must be the standard value, that is c=c*m; see the Fig7-23. 2) The backlash must be zero theoretically; the tooth thickness on the pitch circle is equal to the space width on the pitch circle of the meshing gear. Fig.7-23 Normal center distance (无侧隙啮合的径向间隙)
  • 30.5. Pinion and Rack Example 7-1 Two involute gears in mesh have a module of 2.5mm and a pr essure angle of 20°. The numbers of teeth are z1=22, z2=33. The coefficient of addendum is h*a=1, and c*=0.25. Find thefollowings:1) Sizes of the two gears; 2) Contact ratio; 3) If the center distance is increased 1mm, find the contact ratio. Fig.7-24 Meshing of a pinion and a rack (齿轮与齿 条啮合)
  • 31.7.6 Forming and Undercutting of Gear Teeth 1. Gear Teeth Forming (1) Forming Cutting Probably the oldest method of cutting gear teeth is milli ng. A form milling cutter corresponding to the shape of the tooth space is used to cut one tooth space at a time, after the gear is indexed through one circular pitch to the next position. There are two kinds of millingcutter:one is the dis c cutter shown in Fig 7-25a; the other is the finger cutter shown in Fig 7-25b. Fig.7-25 Forming cutting (仿形加工)
  • 32.(2) Generating Cutting In generating, a tool having a shape different from the tooth profile is moved relative to the gear blank to obtain the proper tooth sha pe. The most common methods of generating gear teeth are shaping method a nd hobbing method. 1) Shaping teeth. Shaping is a highly favored me thod of generating teeth of gear. The cutting tool used in the shaping method is either a rack cutte r or a pinion cutter. Fig 7-26 shows shaping teet h with a pinion cutter; Fig 7-27 shows shaping t eeth with a rack cutter. Fig.7-27 Sha ping teeth with a rack cutter (齿条插刀) Fig.7-26 Shaping teeth with a pinion cutter (齿轮插刀)
  • 33.2) Hobbing teeth. Fig 7-28 illustrates the generating process with a hob. A hob is a cylindrical cutter with one or more helical threads quite like a scre w thread tap, and has straight sides like a rack. The hob and the blank are rota te continuously at the proper angular velocity ratio, and the hob is then fed sl owly across the face of the blank from one end of teeth to the other. All teeth have been cut. Fig.7-28 Hobbing teeth (滚齿加工)
  • 34.2. Undercutting When gear teeth are produced by a generating process, the top of cutting tool removes the portion of the involute profile near the root teeth. This is called u ndercutting. The undercutting weakens the tooth by removing material at its ro ot shown in Fig 7-29. Severe undercutting will promote early tooth failure. It may also reduce the length of contact and result in rougher and noisier gear ac tion. In machine design, the undercutting must be avoided or eliminated by th e designer. Fig.7-29 Undercutting (根切现象)
  • 35.(1) Causation of undercutting The difference between the rack c utter and the rack is that the addendum of the rack cutter is great er than that of the rack, a distance c*m, as shown in Fig 7-30. Fig.7-30 Rack cutter profile (齿条插刀的齿廓)
  • 36.Fig 7-31 illustrates a generating process of a standard gear with the rack cutter. The initial point of cutting is at point B1 which is the intersection between the line of action and the right edge of the rack cutter. When the rack cutter is moved from position 1 to position 2, the involute profile of the gear will completely be cut. Because the addendum line of the rack exceeds the extreme point N, the generating process can not be stopped, such as at position 3, obviously, the involute profile of the root tooth on the left edge of the cutter will be cut away. Fig.7-31 Undercutting process (根切的形成)
  • 37.(2) Minimum number of teeth to avoid undercutting The methods to eliminate undercutting are asfollows:1) Reduce the height of the tooth of the cutter. These gears to be cut are called short tooth gears ; they have a small contact ratio. 2) Increase the pressure angle. This results in a smaller base circle, so that more of the tooth prof ile becomes involute. 3) Reduce the number of teeth. When the numb er of teeth of a gear to be cut is reduced, the radi us of base circle is reduced too, and the extreme point N is likely to fall on the right of the point B 2 where the addendum line of the rack cutter inte rsects the line of action. See the Fig 7-32. Fig.7-32 Minimum number of teeth to avoid undercutting (避免根切的最少齿数)
  • 38.7.7 Nonstandard Spur Gears 1. Concept of Nonstandard Gears In generating a standard gear with a rack cutter, if the addendum line of the rack cutter excesses the extreme point N where the addendum line of the cutter intersects the line of action, the undercutting will occur. To solve it, the rack cutter can be moved a distance xm outwards until the addendum line of the cutter falls down the extreme point N, as shown in Fig 7-33. In this case, the reference line is no longer tangent to the reference circle of the gear, the line which is tangent to the reference circle of the gear is the pitch line, and the tooth thickness and tooth space on the pitch line of the rack is not equal. The tooth thickness and tooth space on the reference circle of the gear to be cut is not equal too. This gear is called the nonstandard gear or modified gear.
  • 39.2. Minimum Coefficient of Offset When the addendum line passes through the extreme point N, there is just no undercutting to occur, as shown in Fig 7-33. There for, wehave:Fig.7-33 Smallest coefficient of offset (最 小变位系数)
  • 40.3. Comparison of Standard Gear and Nonstandard Gear (1) No varied sizes The base circle, reference circle, circular pitch and base pitch are not varied. The involute curve is not changed; it is used on the diff erent portion of the involute. (2)Varied sizes The tooth thickness and width space are varied. Fig 7-34 shows a generating process of a standard gear and a positive modified gear. Fig.7-34 Tooth thickness of a positive modified gear (正变位齿轮齿 厚)
  • 41.The addendum and dedendum are varied. From the Fig 7-35, we can observe that the addendum of a p ositive modified gear is larger than that of the standard gear, and its dedendum is smaller than that of the standard gear. In the sam e method, we can find the difference between the standard gear and negative modified gear in their addendum and dedendu m. Fig.7-35 Tooth profiles of the standard gear and modified gears (变位齿轮与标准齿轮的齿廓)
  • 42.4. Brief Introduction of Nonstandard Gears Supposing that the coefficients of offset of the two gears are x1 and x2, th e types of gear transmission can be classified as follows. 1) The coefficients of offset x1 and x2 are all zero, thatis:'>is: