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1、Table 13-1 Equations of Speed Ratio for a Planetary Type No.Description Sun Gear A Za Planet Gear B Zb Internal Gear C Zc Carrier D 1 Rotate sun gear A once while holding carrier + 1 - Za Zb - Za Zc 0 2 System is fixed as a whole while rotating +(Za/Zc) + Za Zc + Za Zc + Za Zc + Za Zc 3Sum of 1 and
2、2 1 + Za Zc + Za - Za Zc Zb 0 (fixed) + Za Zc Table 13-2 Equations of speed Ratio for a Solar Type No.Description Sun Gear A Za Planet Gear B Zb Internal Gear C Zc Carrier D 1 Rotate sun gear A once while holding carrier + 1 - Za Zb - Za Zc 0 2 System is fixed as a whole while rotating +(Za/Zc) - 1-
3、 1- 1- 1 3Sum of 1 and 2 0 (fixed) - Za - 1 Zb - Za - 1 Zc - 1 SECTION 14 BACKLASH Up to this point the discussion has implied that there is no backlash. If the gears are of standard tooth proportion design and operate on standard center distance they would function ideally with neither backlash nor
4、 jamming. Backlash is provided for a variety of reasons and cannot be designated without consideration of machining conditions. The general purpose of backlash is to prevent gears from jamming by making contact on both sides of their teeth simultaneously. A small amount of backlash is also desirable
5、 to provide for lubricant space and differential expansion between the gear components and the housing. Any error in machining which tends to increase the possibility of jamming makes it necessary to increase the amount of backlash by at least as much as the possible cumulative errors. Consequently,
6、 the smaller the amount of backlash, the more accurate must be the machining of the gears. Runout of both gears, errors in profile, pitch, tooth thickness, helix angle and center distance all are factors to consider in the specification of the amount of backlash. On the other hand, excessive backlas
7、h is objectionable, particularly if the drive is frequently reversing or if there is an overrunning load. The amount of backlash must not be excessive for the requirements of the job, but it should be sufficient so that machining costs are not higher than necessary. In order to obtain the amount of
8、backlash desired, it is necessary to decrease tooth thickness. See Figure 14-1. This decrease must almost always be greater than the desired backlash because of the errors in manufacturing and assembling. Since the amount of the decrease in tooth thickness depends upon the accuracy of machining, the
9、 allowance for a specified backlash will vary according to the manufacturing conditions. It is customary to make half of the allowance for backlash on the tooth thickness of each gear of a pair, although there are exceptions. For example, on pinions having very low numbers of teeth, it is desirable
10、to provide all of the allowance on the mating gear so as not to weaken the pinion teeth. In spur and helical gearing, backlash allowance is usually obtained by sinking the hob deeper into the blank than the theoretically standard depth. Further, it is true that any increase or decrease in center dis
11、tance of two gears in any mesh will cause an increase or decrease in backlash. Thus, this is an alternate way of designing backlash into the system. In the following, we give the fundamental equations for the determination of backlash in a single gear mesh. For the determination of backlash in gear
12、trains, it is necessary to sum the backlash of each mated gear pair. However, to obtain the total backlash for a series of meshes, it is necessary to take into account the gear ratio of each mesh relative to a chosen reference shaft in the gear train. For details, see Reference 10 at the end of the
13、technical section. 14.1 Definition Of Backlash Backlash is defined in Figure 14-2(a) as the excess thickness of tooth space over the thickness of the mating tooth. There are two basic ways in which backlash arises: tooth thickness is below the zero backlash value; and the operating center distance i
14、s greater than the zero backlash value. 387 If the tooth thickness of either or both mating gears is less than the zero backlash value, the amount of backlash introduced in the mesh is simply this numerical difference: j = Sstd - Sact = S (14-1) where: j = linear backlash measured along the pitch ci
15、rcle (Figure 14-2(b) Sstd = no backlash tooth thickness on the operating pitch circle, which is the standard tooth thickness for ideal gears Sact = actual tooth thickness Backlash, Along Line-of-Action = jn = jcos When the center distance is increased by a relatively small amount, a, a backlash spac
16、e develops between mating teeth, as in Figure 14-3. The relationship between center distance increase and linear backlash jn along the line-of-action is: jn = 2 a sin (14-2) This measure along the line-of-action is useful when inserting a feeler gage between teeth to measure backlash. The equivalent
17、 linear backlash measured along the pitch circle is given by: j = 2 a tan (14-3a) where: a = change in center distance = pressure angle Hence, an approximate relationship between center distance change and change in backlash is: a = 1.933j for 14.5 pressure angle gears (14-3b) a = 1.374j for 20 pres
18、sure angle gears (14-3c) Although these are approximate relationships, they are adequate for most uses. Their derivation, limitations, and correction factors are detailed in Reference 10. Note that backlash due to center distance opening is dependent upon the tangent function of the pressure angle.
19、Thus, 20 gears have 41% more backlash than 14.5 gears, and this constitutes one of the few advantages of the lower pressure angle. Equations (14-3) are a useful relationship, particularly for converting to angular backlash. Also, for fine pitch gears the use of feeler gages for measurement is imprac
20、tical, whereas an indicator at the pitch line gives a direct measure. The two linear backlashes are related by: j = jn (14-4) cos The angular backlash at the gear shaft is usually the critical factor in the gear application. As seen from Figure 14-2(a), this is related to the gears pitch radius as f
21、ollows: j = 3440 j (arc minutes) (14-5) R1 Obviously, angular backlash is inversely proportional to gear radius. Also, since the two meshing gears are usually of different pitch diameters, the linear backlash of the measure converts to different angular values for each gear. Thus, an angular backlas
22、h must be specified with reference to a particular shaft or gear center. Details of backlash calculations and formulas for various gear types are given in the following sections. 14.2 Backlash Relationships Expanding upon the previous definition, there are several kinds of backlash: circular backlas
23、h Jt, normal backlash jn, center backlash jr, and angular backlash J(), see Figure 14-4. Table 14-1 reveals relationships among circular backlash jt, normal backlash jn and center backlash Jr. In this definition, Jr is equivalent to change in center distance, a in Section 14.1. 388 Circular backlash
24、 jt has a relation with angular backlash j as follows: j = jt 360 (degrees) (14-6) d 14.2.1 Backlash of a Spur Gear Mesh From Figure 14-4 we can derive backlash of sour mesh as: (14-7) 14.2.2 Backlash of Helical Gear Mesh The helical gear has two kinds of backlash when referring to the tooth space.
25、There is a cross section in the normal direction of the tooth surface n, and a cross section in the radial direction perpendicular to the axis, t. Fig. 14-5 Backlash of Helical Gear Mesh jnn= backlash in the direction normal to the tooth surface jnt= backlash in the circular direction in the cross s
26、ection normal to the tooth jtn= backlash in the direction normal to the tooth surface in the cross section perpendicular to the axis jtt =backlash in the circular direction perpendicular to the axis These backlashes have relations as follows: In the plane normal to the tooth: jnn= jnt cosn (14-8) On
27、 the pitch surface: jnt=jttcos (14-9) In the plane perpendicular to the axis: (14-10) 14.2.3 Backlash of Straight Bevel Gear Mesh Figure 14-6 expresses backlash for a straight bevel gear mesh. In the cross section perpendicular to the tooth of a straight bevel gear, circular backlash at pitch line j
28、t, normal backlash jn and radial backlash jr have the following relationships: (14-11) The radial backlash in the plane of axes can be broken down into the components in the direction of bevel pinion center axis, jrt and in the direction of bevel gear center axis, jr2. (14-12) 14.2.4 Backlash of a S
29、piral Bevel Gear Mesh Figure 14-7 delineates backlash for a spiral bevel gear mesh. In the tooth space cross section normal to the tooth: jnn = jnt cos n (14-13) On the pitch surface: jnt = jtt cos m (14-14) 389 In the plane perpendicular to the generatrix of the pitch cone: (14-15) The radial backl
30、ash in the plane of axes can be broken down into the components in the direction of bevel pinion center axis, and in the direction of bevel gear center axis, jr2 (14-16) 14.2.5 Backlash of Worm Gear Mesh Figure 14-8 expresses backlash for a worm gear mesh. On the pitch surface of a worm: (14-17) In
31、the cross section of a worm perpendicular to its axis: (14-18) In the plane pependicular to the axis of the worm gear: (14-19) 14.3 Tooth Thickness And Backlash There are two ways to produce backlash. One is to enlarge the center distance. The other is to reduce the tooth thickness. The latter is mu
32、ch more popular than the former. We are going to discuss more about the way of reducing the tooth thickness. In SECTION 10, we have discussed the standard tooth thickness s. In the meshing of a pair of gears, if the tooth thickness of pinion and gear were reduced by S1 and S2 they would produce a ba
33、cklash of s1 + s2 in the direction of the pitch circle. Let the magnitude of s1 s2 be 0.1. We know that = 20 then: jt = S1+ S2= 0.1 +0.1 = 0.2 We can convert it into the backlash on normal direction: jn = jtcos = 0.2cos20 = 0.1879 Let the backlash on the center distance direction be jn then: jr = jt
34、 = 0.2 = 0.2747 2tan 2tan20 These express the relationship among several kinds of backlashes. In application, one should consult the JIS standard. There are two JIS standards for backlash - one is JIS B 1703-76 for spur gears and helical gears, and the other is JIS B 1705-73 for bevel gears. All the
35、se standards regulate the standard backlashes in the direction of the pitch circle jt or jtt. These standards can be applied directly, but the backlash beyond the standards may also be used for special purposes. When writing tooth thicknesses on a drawing, it is necessary to specify, in addition, th
36、e tolerances on the thicknesses as well as the backlash. For example: Circular tooth thickness Backlash 0.100 . 0.200 14.4 Gear Train And Backlash The discussions so far involved a single pair of gears. Now, we are going to discuss two stage gear trains and their backlash. In a two stage gear train,
37、 as Figure 14-9 shows, j1 and j4 represent the backlashes of first stage gear train and second stage gear train respectively. If number one gear were fixed, then the accumulated backlash on number four gear jtT4 would be as follows: jtT4 = j1 d3 + j4 (14-20) d2 This accumulated backlash can be conve
38、rted into rotation in degrees: j = jtT4 360 (degrees) (14-21) d4 The reverse case is to fix number four gear and to examine the accumulated backlash on number one gear jtT1 jtT1 = j4 d2 + j1 (14-22) d3 This accumulated backlash can be converted into rotation in degrees: j = jtT1 360 (degrees) (14-23
39、) d1 390 14.5 Methods Of Controlling Backlash In order to meet special needs, precision gears are used more Frequently than ever before. Reducing backlash becomes an important issue. There are two methods of reducing or eliminating backlash - one a static, and the other a dynamic method. The static
40、method concerns means of assembling gears and then making proper adjustments to achieve the desired low backlash. The dynamic method introduces an external force which continually eliminates all backlash regardless of rotational position. 14.5.1 Static Method This involves adjustment of either the g
41、ears effective tooth thickness or the mesh center distance. These two independent adjustments can be used to produce four possible combinations as shown in Table 14-2. Case l By design, center distance and tooth thickness are such that they yield the proper amount of desired minimum backlash. Center
42、 distance and tooth thickness size are fixed at correct values and require precision manufacturing. Case ll With gears mounted on fixed centers, adjustment is made to the effective tooth thickness by axial movement or other means. Three main methods are: 1. Two identical gears are mounted so that on
43、e can be rotated relative to the other and fixed. See Figure 14-10a. In this way, the effective tooth thickness can be adjusted to yield the desired low backlash. 2. A gear with a helix angle such as a helical gear is made in two half thicknesses. One is shifted axially such that each makes contact
44、with the mating gear on the opposite sides of the tooth. See Figure 14-10b. 3. The backlash of cone shaped gears, such as bevel and tapered tooth spur gears, can be adjusted with axial positioning. A duplex lead worm can be adjusted similarly. See Figure 14-10c. Case lll Center distance adjustment o
45、f backlash can be accomplished in two ways: 1. Linear Movement - Figure 14-11a show adjustment along the line-of-centers in a straight or parallel axes manner. After setting to the desire value of backlash the centers are locked in place. 2. Rotary Movement- Figure 14-11b show an alternate way c ach
46、ieving center distance adjustment b rotation of one of the gear centers b means of a swing arm on an eccentric bushing. Again, once the desired backlash setting is found, the positioning arm is locked. Case IV Adjustment of both center distance and tooth thickness is theoretically valid, but is not
47、the usual practice. This would call for needless fabrication expense. 14.5.2 Dynamic Methods Dynamic methods relate to the static techniques. However, they involve a forced adjustment of either the effective tooth thickness or the center distance. 1. Backlash Removal by Forced Tooth Contact This is
48、derived from static Case 11 Referring to Figure 14-10a. a forcing spring rotates the two gear halves apart. This results in an effective tooth thickness that continually fills the entire tooth space in all mesh positions. 2. Backlash Removal by Forced Center Distance Closing This is derived from sta
49、tic Case lll. A spring force is applied to close the center distance; in one case as a linear force along the line-of-centers, and in the other case as a torque applied to the swing arm. In all of these dynamic methods, the applied external force should be known and properly specified. The theoretical relationship of the forces invo