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1、1.0 INTRODUCTION This section presents a technical coverage of gear fundamentals. It is intended as a broad coverage written in a manner that is easy to follow and to understand by anyone interested in knowing how gear systems function. Since gearing involves specialty components it is expected that
2、 not all designers and engineers possess or have been exposed to all aspects of this subject However, for proper use of gear components and design of gear systems it is essential to have a minimum understanding of gear basics and a reference source for details. For those to whom this is their first
3、encounter with gear components, it is suggested this section be read in the order presented so as to obtain a logical development of the subject. Subsequently, and for those already familiar with gears, this material can be used selectively in random access as a design reference. 2.0 BASIC GEOMETRY
4、OF SPUR GEARS The fundamentals of gearing are illustrated through the spur-gear tooth, both because it is the simplest, and hence most comprehensible, and because it is the form most widely used, particularly in instruments and control systems. 2.1 Basic Spur Gear Geometry The basic geometry and nom
5、enclature of a spur-gear mesh is shown in Figure 1.1. The essential features of a gear mesh are: 1. center distance 2. the pitch circle diameters (or pitch diameters) 3. size of teeth (or pitch) 4. number of teeth 5. pressure angle of the contacting involutes Details of these items along with their
6、interdependence and definitions are covered in subsequent paragraphs. 2.2 The Law of Gearing A primary requirement of gears is the constancy of angular velocities or proportionality of position transmission, Precision instruments require positioning fidelity. High speed and/or high power gear trains
7、 also require transmission at constant angular velocities in order to avoid severe dynamic problems. Constant velocity (i.e. constant ratio) motion transmission is defined as “conjugate action” of the gear tooth profiles. A geometric relationship can be derived (1,7)* for the form of the tooth profi
8、les to provide cojugate action, which is summarized as the Law of Gearing as follows: “A common normal to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point.” Any two curves or profiles
9、 engaging each other and satisfying the law of gearing are conjugate Curves. _ *Numbers in parenthesis refer to references at end of text. T25 T26 2.3 The Involute Curve There are almost an infinite number of curves that can be developed to satisfy the law of gearing, and many different curve forms
10、have been tried in the past. Modem gearing (except for clock gears) based on involute teeth. This is due to three major advantages of the involute curve: 1. Conjugate action is independent of changes in center distance. 2. The form of the basic rack tooth is straight-sided, and therefore is relative
11、ly simple and can be accurately made; as a generating tool ft imparts high accuracy to the cut gear tooth. 3. One cutter can generate all gear tooth numbers of the same pitch. The involute curve is most easily understood as the trace of a point at the end of a taut string that unwinds from a cylinde
12、r. It is imagined that a point on a string, which is pulled taut in a fixed direction, projects its trace onto a plane that rotates with the base circle. See Figure 1.2. The base cylinder, or base circle as referred to in gear literature, fully defines the form of the involute and in a gear it is an
13、 inherent parameter, though invisible. The development and action of mating teeth can be visualized by imagining the taut string as being unwound from one base circle and wound on to the other, as shown in Figure 1.3a Thus, a single point on the string simultaneously traces an involute on each base
14、circles rotating plane. This pair of involutes is conjugate, since at all points of contact the common normal is the common tangent which passes through a fixed point on the line-of-centers. It a second winding/unwinding taut string is wound around the base circles in the opposite direction, Figure
15、1 .3b, oppositely curved involutes are generted which can accommodate motion reversal. When the involute pairs are properly spaced the result is the involute gear tooth, Figure 1.3c. 2.4 Pitch Circles Referring to Figure 1.4 the tangent to the two base circles is the line of contact, or line-of-acti
16、on in gear vernacular. Where this line crosses the line-of-centers establishes the pitch point, P. This in turn sets the size of the pitch circles, or as commonly called, the pitch diameters. The ratio of the pitch diameters gives the velocity ratio: Velocity ratio of gear 2 to gear 1 = Z = D1 (1) D
17、2 T27 2.5 Pitch Essential to prescribing gear geometry is the size, or spacing of the teeth along the pitch circle. This is termed pitch and there are two basic forms. 2.5.1 Circular pitch A naturally conceived linear measure along the pitch circle of the tooth spacing. Referring to Figure 1.5 it is
18、 the linear distance (measured along the pitch circle ar between corresponding points of adjacent teeth. it is equal to the pitch-circle circumference divided by the number of teeth: pc = circular pitch = pitch circle circumference = D (2) number of teeth N 2.5.2 Diametral pitch A more popularly use
19、d pitch measure, although geometrically much less evident, is one that is a measure of the number of teeth per inch of pitch diameter. This is simply: expressed as: Pd = diametral pitch = N (3) D Diametral pitch is so commonly used with fine pitch gears that it is usually contracted simply to “pitch
20、“ and that it is diametral is implied. 2.5.3 Relation of pitches: From the geometry that defines the two pitches it can be shown that they are related by the product expression: Pd x Pe = (4) This relationship is simple to remember and permits an easy transformation from one to the other. T28 3.0 GE
21、AR TOOTH FORMS AND STANDARDS involute gear tooth forms and standard tooth proportions are specified in terms of a basic rack which has straight-sided teeth for involute systems. The American National Standards Institute (ANSI) and the American Gear Manufacturers Association (AGMA) have jointly estab
22、lished standards for the USA. Although a large number of tooth proportions and pressure angle standards have been formulated, only a few are currently active and widely used. Symbols for the basic rack are given in Figure 1.6 and pertinent standards for tooth proportions in Table 1.1. Note that data
23、 in Table 1.1 is based upon diametral pitch equal to one. To convert to another pitch divide by diametral pitch. 3.1 Preferred Pitches Although there are no standards for pitch choice a preference has developed among gear designers and producers. This is given in Table 1.2. Adherence to these pitche
24、s is very common in the fine- pitch range but less so among the coarse pitches. 3.2 Design Tables For the preferred pitches it is helpful in gear design to have basic data available as a function of the number of teeth on each gear, Table 1.3 lists tooth proportions common to a given diametral pitch
25、, as well as the diameter of a measuring wire. Table 1.6 lists pitch diameters and the over-wires measurement as a function of tooth number (which ranges from 18 to 218) and various diametral pitches, including most of the preferred fine pitches. Both tables are for 20 pressure-angle gears. 3.3 AGMA
26、 Standards In the United States most gear standards have been developed and sponsored by the AGMA. They range from general and basic standards, such as those already mentioned for tooth form, to specialized standards. The list is very long and only a selected few, most pertinent to fine pitch gearin
27、g, are listed in Table 1.4. These and additional standards can be procured from the AGMA by contacting the headquarters office at 1500 King Street; Suite 201; Alexandria, VA 22314 (Phone: 703-684-0211). a = Addendum b = Dedendum c = Clearance hk = Working Depth ht = Whole Depth Pc = Circular Pitch r
28、f = Fillet Radius t = circular Tooth Thickness = Pressure Angle Figure 1.6 Extract from AGMA 201.02 (ANSI B6.1 1968) T29 TABLE 1.1 TOOTH PROPORTIONS OF BASIC RACK FOR STANDARD INVOLUTE GEAR SYSTEMS Tooth Parameter Symbol in Rack Fig. 1.6 14-1/2 Full Depth involute System 20 Full Depth involute Syste
29、m 20 Coarse-Pitch involute Spur Gears 20 Fine-Pitch involute System 1. System Sponsors 2. Pressure Angle 3. Addendum 4. Dedendum 5. Whole Depth 6. Working Depth 7. Clearance. 8. Basic Circular Tooth Thickness on Pitch Line 9. Fillet Radius In Basic Rack 10. Diametral Pitch Range 11. Governing Standa
30、rd: ANSI AGMA a b ht hk C t rf - - - ANSI also, the radial distance between the pitch circle and the addendum circle (Figure 1.1); addendum can be defined as either nominal or operating. AXIAL PITCH (pa) is the circular pitch in the axial plane and in the pitch surface between corresponding sides of
31、 adjacent teeth, in helical gears and worms. The term axial pitch is preferred to the term linear pitch. (Figure 1.7) AXIAL PLANE of a pair of gears is the plane that contains the two axes. In a single gear, an axial plane may be any plane containing the axis and a given point. BASE DIAMETER (Db = g
32、ear, and db = pinion) is the diameter of the base cylinder from which involute tooth surfaces, either straight or helical, are derived. (Figure 1.1); base radius (Rb = gear, rb = pinion) is one half of the base diameter. BASE PITCH (pb) in an involute gear is the pitch on the base circle or along th
33、e line-of-action. Correspcndng sides of involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a common normal in a plane of rotation. (Figure 1.8) BASIC RACK is a rack that is adopted as the basis for a system of interchangeable gears
34、. BACKLASH (B) is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth on the pitch circles. As actually indicated by measuring devices, backlash may be _ *Portions of this section are repented with permission from the Barber-Colman Co., Rockford, Ml. T31 determ
35、ined variously in the transverse, normal, or axial planes, and either in the direction of the pit circles or on the line-of-action. Such measurements should be corrected to corresponding values a transverse pitch circles for general comparisons. (Figure 1.9) CENTER DISTANCE (C), Distance between axe
36、s of rotation of mating spur or helical gears. CHORDAL ADDENDUM (ac) is the height from the top of the tooth to the chord subtending the circular-thickness arc. (Figure 1.10) CHORDAL THICKNESS (tc) is the length of the chord subtending a circular-thickness arc. (Figure 1.10) CIRCULAR PITCH (pc) is t
37、he distance along the pitch circle or pitch line between corresponding profiles of adjacent teeth. (Figure 1.1) CIRCULAR THICKNESS (t) is the length of arc between the two sides of a gear tooth on the p4 circle, unless otherwise specified. (Figure 1.10) CLEARANCE-OPERATING (c) is the amount by which
38、 the dedendum in a given gear exceeds addendum of its mating gear. (Figure 1.1) CONTACT RATIO (Spur) is the ratio of the length-of-action to the base pitch. CONTACT RATIO (Helical) is the contact ratio in the plane of rotation plus a contact portion a tributted to the axial advance. DEDENDUM (b) is
39、the depth of a tooth space below the pitch line; also, the radial distance beta, the pitch circle and the root circle. (Figure 1.1); dedendum can be defined as either nominal or operating. DIAMETRAL PITCH (Pd) is the ratio of the number of teeth to the number of inches in the pitch diameter. There i
40、s a fixed relation between diametral pitch (Pd) and circular pitch (pc): pc = / Pd FACE WIDTH (F) is the length of the teeth in an axial plane. FILLET RADIUS (r,) is the radius of the fillet curve at the base of the gear tooth. In generated this radius is an approximate radius of curvature. (Figure
41、1.13) FULL DEPTH TEETH are those in which the working depth equals 2000“ diametral pitch GENERATING RACK is a rack outline used to indicate tooth details and dimensions for the design of a hob to produce gears of a basic rack system. HELIX ANGLE () is the angle between any helix and an element of it
42、s cylinder. In helical gears a worms, it is at the pitch diameter unless otherwise specified. (Figure 1.7) INVOLUTE TEETH of spur gears, helical gears, and worms are those in which the active portion of the profile in the transverse plane is the involute of a circle. T32 LEAD (L) is the axial advanc
43、e of a helix for one complete turn, as in the threads of cylindrical worms and teeth of helical gears. (Figure 1.11) LENGTH-OF-ACTION (ZA) is the distance on an involute line of action through which the point of contact moves during the action of the tooth profiles. (Figure 1.8) LEWIS FORM FACTOR (Y
44、, diametral pitch; yc, circular pitch). Factor in determination of beam strength of gears. LINE-OF-ACTION is the path of contact in involute gears. It is the straight line passing through the pitch point and tangent to the base circles. (Figure 1.12) LONG- AND SHORT-ADDENDUM TEETH are those in which
45、 the addenda of two engaging gears are unequal. MEASUREMENT OVER PINS (M). Distance over two pins placed in diametrically opposed tooth spaces (even number of teeth) or nearest to it (odd number of teeth). NORMAL CIRCULAR PITCH, Pcn, is the circular pitch in the normal plane, and also the length of
46、the arc along the normal helix between helical teeth or threads. (Figure 1.7) NORMAL CIRCULAR THICKNESS (tn) is the circular thickness in the normal plane. In helical gears. it is an arc of the normal helix, measured at the pitch radius. NORMAL DIAMETRAL PITCH (Pdn) is the diametral pitch as calcula
47、ted in the normal plane. NORMAL PLANE is the plane normal to the tooth. For a helical gear this plane is inclined by the helix angle, , to the plane of rotation. OUTSIDE DIAMETER (Do gear, and do = pinion) is the diameter of the addendum (outside) circle (Figure 1.1); the outside radius (Ro gear, ro
48、 pinion) is one half the outside diameter. PITCH CIRCLE is the curve of intersection of a pitch surface of revolution and a plane of rotation. According to theory, it is the imaginary circle that rolls without slip with a pitch circle of a mating gear. (Figure 1.1) PITCH CYLINDER is the imaginary cy
49、linder in a gear that rolls without slipping on a pitch cylinder or pitch plane of another gear. PITCH DIAMETER (D = gear, d = pinion) is the diameter of the pitch circle. In parallel shaft gears, the pitch diameters can be determined directly from the center distance and the number of teeth by proportionality. Operating pitch diameter is the pitch diameter at which the gears opera