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Yi Zhangwith Susan FingerStephannie BehrensTable of Contents 7 Gears Gears are machine elements that transmit motion by means ofsuccessively engaging teeth. The gear teeth act like small levers. 7.1 Gear Classification Gears may be classified according to the relative position of theaxes of revolution. The axes may beparallel,intersecting,neither parallel nor intersecting.Here is a brief list of the common forms. We will discuss eachin more detail later.Gears for connecting parallel shaftsGears for connecting intersecting shaftsNeither parallel nor intersecting shaftsGears for connecting parallel shaftsSpur gears The left pair of gears makes external contact, and the right pairof gears makes internal contactParallel helical gears Herringbone gears (or double-helical gears) Rack and pinion (The rack is like a gear whose axis is at infinity.) Gears for connecting intersecting shaftsStraight bevel gears Spiral bevel gears Neither parallel nor intersecting shaftsCrossed-helical gears Hypoid gears Worm and wormgear 7.2 Gear-Tooth Action7.2.1 Fundamental Law of Gear-Tooth ActionFigure 7-2 shows two mating gear teeth,in whichTooth profile 1 drives toothprofile 2 by acting at the instantaneous contact point K.N1N2 is the common normal of the two profiles.N1 is the foot of the perpendicular fromO1 to N1N2N2 is the foot of the perpendicular fromO2 to N1N2.Figure 7-2 Two gearing tooth profilesAlthough the two profiles have different velocitiesV1 and V2 at pointK, their velocities along N1N2 areequal in both magnitude and direction. Otherwise the two toothprofiles would separate from each other. Therefore, we have(7-1)or(7-2)We notice that the intersection of the tangencyN1N2 and the line of centerO1O2 is point P, and(7-3)Thus, the relationship between the angular velocities of the drivinggear to the driven gear, or velocity ratio, of a pair of matingteeth is(7-4)Point P is very important to the velocity ratio, and it iscalled the pitch point. Pitch point divides the line betweenthe line of centers and its position decides the velocity ratio of thetwo teeth. The above expression is the fundamental law ofgear-tooth action. 7.2.2 Constant Velocity RatioFor a constant velocity ratio, the position of P should remainunchanged. In this case, the motion transmission between two gears isequivalent to the motion transmission between two imagined sliplesscylinders with radius R1 and R2 ordiameter D1 and D2. We can get twocircles whose centers are at O1 andO2, and through pitch point P. These twocircle are termed pitch circles. The velocity ratio is equal tothe inverse ratio of the diameters of pitch circles. This is thefundamental law of gear-tooth action. The fundamental law of gear-tooth action may now also be statedas follow (for gears with fixed center distance) (Ham 58):The common normal to the tooth profiles at the point of contact mustalways pass through a fixed point (the pitch point) on the line ofcenters (to get a constant velocity ration).7.2.3 Conjugate Profiles To obtain the expected velocity ratio of two tooth profiles,the normal line of their profiles must pass through the correspondingpitch point, which is decided by thevelocity ratio. The two profiles which satisfy this requirementare called conjugate profiles. Sometimes, we simply termed thetooth profiles which satisfy the fundamental law of gear-toothaction the conjugate profiles. Although many tooth shapes are possible for which a mating tooth couldbe designed to satisfy the fundamental law, only two are in generaluse: the cycloidal and involute profiles. The involutehas important advantages -- it is easy to manufacture and the centerdistance between a pair of involute gears can be varied withoutchanging the velocity ratio. Thus closetolerances between shaft locations are not required when using theinvolute profile. The most commonly used conjugate tooth curveis the involute curve (Erdman & Sandor 84).7.3 Involute Curve The following examples are involute spur gears. We use the wordinvolute because the contour of gear teeth curves inward.Gears have many terminologies, parameters and principles. One of theimportant concepts is the velocity ratio, which is the ratio ofthe rotary velocity of the driver gear to that of the driven gears. The SimDesign file for these gears is simdesign/gear15.30.sim. The number of teeth in these gears are 15 and 30, respectively. Ifthe 15-tooth gear is the driving gear and the 30-teeth gear is thedriven gear, their velocity ratio is 2. Other examples of gearsare in simdesign/gear10.30.sim and simdesign/gear20.30.sim7.3.1 Generation of the Involute CurveFigure 7-3 Involute curveThe curve most commonly used for gear-tooth profiles is the involuteof a circle. This involute curve is the path traced by a pointon a line as the line rolls without slipping on the circumference of acircle. It may also be defined as a path traced by the end of a stringwhich is originally wrapped on a circle when the string is unwrappedfrom the circle. The circle from which the involute is derived iscalled the base circle. In Figure 7-3, let line MN roll in thecounterclockwise direction on the circumference of a circle withoutslipping. When the line has reached the position M'N', itsoriginal point of tangent A has reached the position K,having traced the involute curve AK during the motion. As themotion continues, the point A will trace the involute curveAKC.7.3.2 Properties of Involute Curves The distance BK is equal to the arc AB, becauselink MN rolls without slipping on the circle.For any instant, the instantaneous center of the motion ofthe line is its point of tangent with the circle.Note: We have not defined the term instantaneous centerpreviously. The instantaneous center or instant centeris defined in two ways (Bradford& Guillet 43): When two bodies have planar relative motion, the instantcenter is a point on one body about which the other rotates at theinstant considered.When two bodies have planar relative motion, the instant center isthe point at which the bodies are relatively at rest at the instantconsidered.The normal at any point of an involute is tangent to the basecircle. Because of the property (2) of the involute curve, the motion ofthe point that is tracing the involute is perpendicular to the line at anyinstant, and hence the curve traced will also be perpendicular to theline at any instant.There is no involute curve within the base circle.7.4 Terminology for Spur Gears Figure 7-4 shows some of the terms for gears.Figure 7-4 Spur GearIn the following section, we define many of the terms used in theanalysis of spur gears. Some of the terminology has been definedpreviously but we include them here for completeness. (See (Ham 58) for more details.) Pitch surface : The surface of the imaginary rollingcylinder (cone, etc.) that the toothed gear may be considered toreplace.Pitch circle: A right section of the pitch surface.Addendum circle: A circle bounding the ends of the teeth,in a right section of the gear.Root (or dedendum) circle: The circle bounding the spacesbetween the teeth, in a right section of the gear.Addendum: The radial distance between the pitch circle andthe addendum circle.Dedendum: The radial distance between the pitch circle andthe root circle.Clearance: The difference between the dedendum of one gearand the addendum of the mating gear.Face of a tooth: That part of the tooth surface lyingoutside the pitch surface.Flank of a tooth: The part of the tooth surface lyinginside the pitch surface.Circular thickness (also called the tooththickness) : The thickness of the tooth measured on the pitchcircle. It is the length of an arc and not the length of a straightline.Tooth space: The distance between adjacent teeth measuredon the pitch circle.Backlash: The difference between the circle thickness ofone gear and the tooth space of the mating gear.Circular pitch p: The width of a tooth and a space,measured on the pitch circle.Diametral pitch P: The number of teeth of a gear perinch of its pitch diameter.A toothed gear must have an integral number of teeth. The circularpitch, therefore, equals the pitch circumference divided by thenumber of teeth. The diametral pitch is, by definition, thenumber of teeth divided by the pitch diameter. That is,(7-5)and(7-6)Hence(7-7)wherep = circular pitchP = diametral pitchN = number of teethD = pitch diameterThat is, the product of the diametral pitch and the circular pitchequals . Module m: Pitch diameter divided by number of teeth. Thepitch diameter is usually specified in inches or millimeters; in theformer case the module is the inverse of diametral pitch.Fillet : The small radius that connects the profile of atooth to the root circle.Pinion: The smaller of any pair of mating gears. Thelarger of the pair is called simply the gear.Velocity ratio: The ratio of the number of revolutions ofthe driving (or input) gear to the number of revolutions of the driven(or output) gear, in a unit of time.Pitch point: The point of tangency of the pitch circles ofa pair of mating gears.Common tangent: The line tangent to the pitch circle atthe pitch point.Line of action: A line normal to a pair of mating toothprofiles at their point of contact.Path of contact: The path traced by the contact point ofa pair of tooth profiles.Pressure angle : The angle between the common normal at the point of tooth contact andthe common tangent to the pitch circles. It is also the angle betweenthe line of action and the common tangent.Base circle :An imaginary circle used in involute gearingto generate the involutes that form the tooth profiles.Table 7-1 lists the standard tooth systemfor spur gears. (Shigley &Uicker 80) Table 7-1 Standard tooth systems for spur gearsTable 7-2 lists the commonly used diametral pitches. Coarse pitch22.252.53468101216Fine pitch20243240486496120150200Table 7-2 Commonly used diametral pitches Instead of using the theoretical pitch circle as an index of tooth size, the base circle, which is a more fundamental circle,can be used. The result is called the base pitchpb, and it is related to the circular pitch pby the equation(7-8)7.5 Condition for Correct Meshing Figure 7-5 shows two meshing gears contacting atpoint K1 and K2. Figure 7-5 Two meshing gearsTo get a correct meshing, the distance ofK1K2 on gear 1 should be the same as thedistance of K1K2 on gear 2. AsK1K2 on both gears are equal to the base pitch of their gears, respectively. Hence(7-9)Since(7-10)and(7-11)Thus(7-12)To satisfy the above equation, the pair of meshing gears must satisfy thefollowing condition:(7-13)7.6 Ordinary Gear Trains Gear trains consist of two or more gears for the purpose oftransmitting motion from one axis to another. Ordinary geartrains have axes, relative to the frame, for all gears comprisingthe train. Figure 7-6a shows a simpleordinary train in which there is only one gear for each axis. InFigure 7-6b a compound ordinary train isseen to be one in which two or more gears may rotate about a singleaxis. Figure 7-6 Ordinary gear trains 7.6.1 Velocity RatioWe know that the velocity ratio of a pair of gears is theinverse proportion of the diameters of their pitchcircle, and the diameter of the pitch circle equals to the numberof teeth divided by the diametral pitch. Also,we know that it is necessary for the to mating gears to have the samediametral pitch so that to satisfy the condition of correctmeshing. Thus, we infer that the velocity ratio of a pair ofgears is the inverse ratio of their number of teeth. For the ordinary gear trains in Figure 7-6a, we have(7-14)These equations can be combined to give the velocity ratio of thefirst gear in the train to the last gear:(7-15)Note:The tooth number in the numerator are those of the driven gears,and the tooth numbers in the denominator belong to the drivergears.Gear 2 and 3 both drive and are, in turn, driven. Thus, they arecalled idler gears. Since their tooth numbers cancel, idlergears do not affect the magnitude of the input-output ratio, but theydo change the directions of rotation. Note the directional arrows inthe figure. Idler gears can also constitute a saving of space andmoney (If gear 1 and 4 meshes directly across a long center distance,their pitch circle will be much larger.)There are two ways to determine the direction of the rotarydirection. The first way is to label arrows for each gear as in Figure 7-6. The second way is to multiplemth power of "-1" to the general velocity ratio. Wherem is the number of pairs of externalcontact gears (internal contact gear pairsdo not change the rotary direction). However, the second method cannotbe applied to the spatial gear trains. Thus, it is not difficult to get the velocity ratio of the gear trainin Figure 7-6b:(7-16)7.7 Planetary gear trains Planetary gear trains, also referred to as epicyclic geartrains, are those in which one or more gears orbit about thecentral axis of the train. Thus, they differ from an ordinary train byhaving a moving axis or axes. Figure 7-8 shows abasic arrangement that is functional by itself or when used as a partof a more complex system. Gear 1 is called a sun gear , gear2 is a planet, link H is an arm, or planetcarrier. Figure 7-8 Planetary gear trainsFigure 7-7 Planetary gears modeled using SimDesign The SimDesign file is simdesign/gear.planet.sim. Since thesun gear (the largest gear) is fixed, the DOF of the above mechanismis one. When you pull the arm or the planet, the mechanism has adefinite motion. If the sun gear isn't frozen, the relative motion isdifficult to control.7.7.1 Velocity Ratio To determine the velocity ratio of the planetary gear trains is slightly more complex ananalysis than that required for ordinary geartrains. We will follow the procedure: Invert the planetary gear train mechanism by imagining theapplication a rotary motion with an angular velocity of H to themechanism. Let's analyse the motion before and after the inversionwith Table 7-3:Table 7-3 Inversion of planetary gear trains. Note: H is the rotaryvelocity of gear i in the imagined mechanism. Notice that in the imagined mechanism, the armH is stationary and functions as a frame. No axis of gear moves anymore. Hence, the imagined mechanism is an ordinarygear train.Apply the equation of velocity ratio of the ordinarygear trains to the imagined mechanism. We get(7-17)or (7-18)7.7.2 ExampleTake the planetary gearing train in Figure 7-8as an example. Suppose N1 = 36, N2 = 18, 1 = 0, 2 = 30. What isthe value of N?With the application of the velocity ratio equation for the planetarygearing trains, we have the following equation: (7-19)From the equation and the given conditions, we can get the answer:N = 10. Table of Contents Complete Table of Contents1 Physical Principles2 Mechanisms and Simple Machines3 More on Machines and Mechanisms4 Basic Kinematics of Constrained Rigid Bodies5 Planar Linkages6 Cams7 Gears 7.1 Gear Classification 7.2 Gear-Tooth Action 7.2.1 Fundamental Law of Gear-Tooth Action 7.2.2 Constant Velocity Ratio 7.2.3 Conjugate Profiles 7.3 Involute Curve 7.3.1 Generation of the Involute Curve 7.3.2 Properties of Involute Curves 7.4 Terminology for Spur Gears 7.5 Condition for Correct Meshing 7.6 Ordinary Gear Trains 7.6.1 Velocity Ratio 7.7 Planetary gear trains 7.7.1 Velocity Ratio 7.7.2 Example 8 Other MechanismsIndexReferences sfinger@ri.cmu.edu 2b1af7f3a8