582 Dynamics of Mechanical Systems Unlike the root circle, the base circle, and the addendum circle, the pitch circle is not fixed relative to the tooth. Instead, the pitch circle is determined by the center distance between the mating gears. Hence, the addendum, dedendum, and tooth thickness may vary by a small amount depending upon the specific location of the pitch circle. The location of the pitch circle on a gear tooth is determined by the location of the point of contact between meshing teeth on the line connecting the gear centers, as illustrated in Figure 17.5.2. An advantageous property of involute spur gear tooth geometry is that the gears will operate together with conjugate action at varying center distances with only the provision that contact is always maintained between at least one pair of teeth. Next, consider a closer look at the interaction of the teeth of meshing gears as in Figure 17.5.3. If the gears are not tightly pressed together, there will be a separation between noncontacting teeth. This separation or looseness is called the backlash. The distance measured along the pitch circle between corresponding points on adjacent teeth is called the circular pitch. Circular pitch is commonly used as a measure of the size of a gear. From Figure 17.5.3 we see that, even with backlash, unless two gears have the FIGURE 17.5.2 Meshing involute spur gears determining the pitch circles. FIGURE 17.5.3 Profile of meshing gear teeth. 0593_C17_fm Page 582 Tuesday, May 7, 2002 7:12 AM Mechanical Components: Gears 583 same circular pitch they will not mesh or operate together. If a gear has N teeth and a pitch circle with diameter d, then the circular pitch p is: (17.5.1) Another commonly used measure of gear size is diametral pitch P, defined as the number of teeth divided by the pitch circle diameter. That is, (17.5.2) Usually the diametral pitch is computed with the pitch circle diameter d measured in inches. Still another measure of gear size is the module, m, which is the reciprocal of the diametral pitch. With the module, however, the pitch circle diameter is usually measured in milli- meters. Then, the module and diametral pitch are related by the expression: (17.5.3) From Eqs. (17.5.1) and (17.5.2) we also have the relations: (17.5.4) For terminology regarding spur gear depth, consider Figure 17.5.4, which shows a three- dimensional representation of a gear tooth illustrating the width, top land, face, flank, and bottom land. Finally, involute spur gear tooth designers have the following tooth proportions depend- ing upon the diametral pitch P [17.3]: Addendum = 1/P (17.5.5) Dedendum = 1.157/P (17.5.6) Clearance = 0.157/P (17.5.7) Fillet radius = 0.157/P (17.5.8) Figure 17.5.4 Perspective view of a spur gear tooth. pdN=π PNd= mP= 25 4. pP p m==ππand 0593_C17_fm Page 583 Tuesday, May 7, 2002 7:12 AM 584 Dynamics of Mechanical Systems 17.6 Kinematics of Meshing Involute Spur Gear Teeth In this section, we consider the fundamentals of the kinematics of meshing involute spur gear teeth. We will focus upon ideal gears — that is, gears with exact geometry. In practice, of course, gear geometry is not exact, but the closer the geometry is to the theoretical form, the more descriptive our analysis will be. Lent [3] provides an excellent elementary description of spur gear kinematics. We will follow his outline here, and the reader may want to consult the reference itself for additional details. In our discussion, we will consider the interaction of a pair of mating teeth from the time they initially come into contact, then as they pass through the mesh (the region of contact), and, finally, as they separate. We will assume that the gears always have at least one pair of teeth in contact and that each pair of mating teeth is the same as each other pair of mating teeth. With involute gear teeth, the locus of points of contact between the teeth is a straight line (see Figures 17.3.4, 17.4.2, and 17.5.2). This line is variously called the line of contact, path of contact, pressure line, or line of action. The angle turned through by a gear as a typical tooth travels along the path of contact is called the angle of contact. (Observe that for gears with different diameters, thus having different numbers of teeth, that their respective angles of contact will be different — even though their paths of contact are the same.) Figure 17.6.1 illustrates the path and angles of contact for a typical pair of meshing spur gears. The angle of contact is sometimes called the angle of action. The path of contact AB is formed from the point of initial contact A to the point of ending contact B. The path of contact passes through the pitch point P. The angle turned through during the contact, measured along the pitch circle, is the angle of contact. The angle of approach is the angle turned through by the gear up to contact from the pitch point P. The angle of recess is the angle turned through by the gear during tooth contact at the pitch point to the end of contact at B. Next, consider Figure 17.6.2, which shows an outline of addendum circles of a meshing spur gear pair. Contact between teeth will occur only within the region of overlap of the circles. The extent of the overlap region is dependent upon the gear radii. The location of the pitch point within the overlap region is dependent upon the size or length of the addendum. Observe further that the location of the pitch point P determines the lengths of the paths of approach and recess. Observe moreover that the paths of approach and recess are generally not equal to each other. FIGURE 17.6.1 Path and angle of contact for a pair of meshing involute teeth. Angle of Contact Path of Contact A Direction of Rotation Pitch Circle B P Angle of Recess Angle of Approach 0593_C17_fm Page 584 Tuesday, May 7, 2002 7:12 AM Mechanical Components: Gears 585 As noted in the foregoing sections, the rotation of meshing gears may be modeled by rolling wheels whose profiles are determined by the pitch circles as in Figure 17.6.3. Then, for there to be rolling without slipping, we have from Eq. (17.2.1): (17.6.1) Then, (17.6.2) where r 1 , r 2 and d 1 , d 2 are the radii and diameters, respectively, of the pitch circles of the gears. From Eq. (17.5.2) we may express the angular speed ratio as: (17.6.3) where, as before, N 1 and N 2 are the numbers of teeth in the gears. Observe that the angular speed ratio is inversely proportional to the tooth ratio. Observe that for meshing gears to maintain conjugate action it is necessary that at least one pair of teeth be in contact at all times (otherwise there will be intermittent contact with kinematic discontinuities). The average number of teeth in contact at any time is called the contact ratio. The contact ratio may also be expressed in geometric terms. To this end, it is helpful to introduce the concept of normal pitch, defined as the distance between corresponding points on adjacent teeth measured along the base circle, as shown in Figure 17.6.4. From the property of involute curves being generated by the locus of the end point of a belt or FIGURE 17.6.2 Overlap of addendum circles of meshing gear teeth. FIGURE 17.6.3 Rolling pitch circles. FIGURE 17.6.4 Normal pitch. Line of Centers Path of Recess Path of Approach Direction of Rotation P O Gear 1 Gear 2 O P r r 1 1 2 2 rr 11 22 ωω= ωω 12 21 21 ==rr dd ωω 12 2 1 = NN 0593_C17_fm Page 585 Tuesday, May 7, 2002 7:12 AM 586 Dynamics of Mechanical Systems cord being unwrapped about the base circle (see Figure 17.4.4), we see that the normal pitch may also be expressed as the distance between points of adjacent tooth surfaces measured along the pressure line as in Figure 17.6.5. Observing that the path of contact also lies along the pressure line (see Figure 17.6.1) we see that the length of the path of contact will exceed the normal pitch if more than one pair of teeth are in contact. Indeed, the length of the path of contact is proportional to the average number of teeth in contact. Hence, we have the verbal equation: (17.6.4) Observe further that the definitions of normal pitch and circular pitch are similar. That is, the normal pitch is defined relative to the base circle, whereas the circular pitch is defined relative to the pitch circle. Hence, from Eq. (17.5.1), we may express the normal pitch p n as: (17.6.5) where d b is the diameter of the base circle. Then, from Eqs. (17.5.1) and (17.4.2), the ratio of the normal pitch to the circular pitch is: (17.6.6) where θ is the pressure angle. In view of Figures 17.6.2 and 17.6.5, we can use Eq. (17.6.4) to obtain an analytical representation of the contact ratio. To see this, consider an enlarged and more detailed representation of the addendum circle and the path of contact as in Figure 17.6.6, where P is the pitch point and where the distance between A 1 and A 2 (pressure line/addendum circle intersections) is the length of the path of contact. Consider the segment A 2 P; let Q be at the intersection of the line through A 2 perpendicular to the line through the gear centers as in Figure 17.6.7. Then A 2 QO 2 is a right triangle whose sides are related by: (17.6.7) where r 2a is the addendum circle radius of gear 2. Let ᐉ 2 be the length of segment A 2 P (the path of approach of Figure 17.6.2). Then, from Figure 17.6.7, we have the relations: (17.6.8) (17.6.9) (17.6.10) FIGURE 17.6.5 Normal pitch measured along the pressure line. Contact Ratio = Path of Contact Normal Pitch pdN n b =π pp dd n b ==cosθ O Q QA O A r a2 2 2 2 22 2 2 2 () + () = () = QA 22 = l cosθ PQ = l 2 sinθ OQ r PQ r pp22 22 =+ =+l sinθ 0593_C17_fm Page 586 Tuesday, May 7, 2002 7:12 AM Mechanical Components: Gears 587 where, as before, θ is the pressure angle, and r 2p is the pitch circle radius of gear 2. Then, by substituting from Eqs. (17.6.8) and (17.6.10) into (17.6.7), we have: (17.6.11) Solving for ᐉ 2 , we obtain: (17.6.12) Similarly, for gear 1 we have: (17.6.13) where ᐉ 1 is the length of the contact path segment from P to A 1 (the path of recess in Figure 17.6.2), and r 1p and r 1a are the pitch and addendum circle radii for gear 1. From Eqs. (17.6.4) and (17.6.6), the contact ratio is then: (17.6.14) where p is the circular pitch, and the lengths of the contact path segments ᐉ 1 and ᐉ 2 are given by Eqs. (17.6.12) and (17.6.13). Spotts and Shoup [17.10] state that for smooth gear operation the contact ratio should not be less than 1.4. To illustrate the use of Eq. (17.6.14), suppose it is desired to transmit angular motion with a 2-to-1 speed ratio between axes separated by 9 inches. If 20° pressure angle gears with diametral pitch 7 are chosen, what will be the contact ratio? To answer this question, let 1 and 2 refer to the pinion and gear, respectively. Then, from Eq. (17.6.2), we have: (17.6.15) FIGURE 17.6.6 Addendum circles and path of contact for meshing gears. FIGURE 17.6.7 Geometry surrounding the contact path. O r r r O A A Addendum Circles Path of Contact 1P 2P 2A 2 1 2 1 P Q P A O ᐉ r r Contact Path Segment 2 2a 2 2p 2 θ rr pa22 2 2 2 2 + () + () =llsin cosθθ θ l 22 2 2 2 22 12 =− + − [] rrr pap sin cos / θθ l 11 1 2 1 22 12 =− + − [] rrr pap sin cos / θθ Contact Ratio D = = + CR p ll 12 cosθ rr r r pp p p21 1 1 29and then += 0593_C17_fm Page 587 Tuesday, May 7, 2002 7:12 AM 588 Dynamics of Mechanical Systems or (17.6.16) From Eqs. (17.5.2) and (17.5.5), we then also have: (17.6.17) and (17.6.18) Hence, from Eqs. (17.6.12) and (17.6.13) the contact path segment lengths are found to be: (17.6.19) Finally, from Eq. (17.5.4), the circular pitch p is: (17.6.20) The contact ratio is then: (17.6.21) 17.7 Kinetics of Meshing Involute Spur Gear Teeth As noted earlier, gears have two purposes: (1) transmission of motion, and (2) transmission of forces. In this section, we consider the second purpose by studying the forces transmit- ted between contacting involute spur gear teeth. To this end, consider the contact of two teeth as they pass through the pitch point as depicted in Figure 17.7.1. It happens that at the pitch point there is no sliding between the teeth. Hence, the forces transmitted between the teeth are equivalent to a single force N, normal to the contacting surfaces and thus FIGURE 17.7.1 Contacting gear teeth at the pitch point. rr pp12 36== . .in and in Nr Nr pp11 22 7 2 42 7 2 84= () == () =and rr rr ap ap11 2 2 1 7 3 143 1 2 6 143=+ = =+ =. .in. and ll 12 0 363 0 387==. . . in and in. p ==π 7 0 449. in. CR p=+ () =+ ()()() =ll 12 0 363 0 387 0 449 0 940 1 777cos . . . . .θ 0593_C17_fm Page 588 Tuesday, May 7, 2002 7:12 AM Mechanical Components: Gears 589 along the pressure line, which is also the path of contact. (This is the reason the path of contact, or line of action, is also called the pressure line.) As we noted earlier, the angle θ between the pressure line and a line perpendicular to the line of centers is called the pressure angle (see Section 17.4). The pressure angle is thus the angle between the normal to the contacting gear surfaces (at their point of contact) and the tangent to the pitch circles (at their point of contact). We also encountered the concept of the pressure angle in our previous chapter on cams (see Section 16.3). The pressure angle determines the magnitude of component N T of the normal force N tangent to the pitch circle, generating the moment about the gear center. That is, for the follower gear, the moment M f about the gear center O f is seen from Figure 17.7.2 to be: (17.7.1) where N is the magnitude of N. Equation (17.7.1) shows the importance of the pressure angle in determining the mag- nitude of the driving moment. For most gears, the pressure angle is designed to be either 14.5 or 20°, with a recent trend toward 20°. The pressure angle, however, is also dependent upon the gear positioning. That is, because the position of the pitch circles depends upon the gear center separation (see Section 17.5 and Figure 17.5.2), the pressure angle will not, in general, be exactly 14.5 or 20°, as designed. 17.8 Sliding and Rubbing between Contacting Involute Spur Gear Teeth When involute spur gear teeth are in mesh (in contact), if the contact point is not at the pitch point the tooth surfaces slide relative to each other. This sliding (or “rubbing”) can lead to wear and degradation of the tooth surfaces. As we will see, this rubbing is greatest at the tooth tip and tooth root, decreasing monotonically to the pitch point. The effect of the sliding is different for the driver and the follower gear. To see all this, consider Figure 17.8.1 showing driver and follower gear teeth in mesh. Let P 1 be the point of initial mesh (contact) of the gear teeth and let P 2 be the point of final contact. If P 1f is that point of the follower gear coinciding with P 1 , then the velocity of P 1f may be expressed as: (17.8.1) FIGURE 17.7.2 Force acting onto the follower gear. N θ Follower r O f f MrNrN ff T f ==cosθ VOP P fff f1 1 =×ωω 0593_C17_fm Page 589 Tuesday, May 7, 2002 7:12 AM 590 Dynamics of Mechanical Systems where ω f is the angular velocity of the follower gear and O f P 1f is the position vector locating P 1f relative to the follower gear center O f . Then, in terms of unit vectors shown in Figure 17.8.1, O f P 1f may be expressed as: (17.8.2) where r f is the pitch circle radius of the follower gear and ξ is the distance from the pitch point P to P 1f . The unit vector n is parallel to the line of contact (or pressure line) and is thus normal to the contacting gear tooth surfaces. Also, ωω ωω f may be expressed as: (17.8.3) where n 3 is normal to the plane of the gears. By substituting from Eqs. (17.8.2) and (17.8.3) into (17.8.1) we obtain: (17.8.4) where n ⊥⊥ ⊥⊥ is perpendicular to n and parallel to the plane of the gear as in Figure 17.8.1. Similarly, if P 1d is that point of the driver gear coinciding with P 1f , the velocity of P 1d is: (17.8.5) The difference in these velocities is the sliding (or rubbing) velocity. From Eq. (17.2.1) we see that: (17.8.6) Hence, the rubbing (or sliding) velocity V s is: (17.8.7) Equation (17.8.7) shows that the rubbing is greatest at the points of initial and final contact (maximum ξ) and that the rubbing vanishes at the pitch point (ξ = 0). FIGURE 17.8.1 Driver and follower gears in mesh. P P P Driver Follower n n n n n y 3 x ⊥ ⊥ 1 2 OP n n ff f y r 1 =+ξ ωω ff =ω n 3 Vnnn nn P ff y ff x f f r r 1 3 =×+ () =− + ⊥ ωξ ωωξ Vnn P dd x d d r 1 =− − ⊥ ωωξ rr ff dd ωω= VV V n s P P f d f d =−=+ () ⊥ 1 1 ωωξ 0593_C17_fm Page 590 Tuesday, May 7, 2002 7:12 AM Mechanical Components: Gears 591 Consider now the rubbing itself: First, for the follower gear, as the meshing begins the contact point is at the tip P 1f . The contact point then moves down the follower tooth to the pitch point P. During this movement we see from Eq. (17.8.7) that with ξ > 0 the sliding velocity V s is in the n ⊥⊥ ⊥⊥ direction. This means that, because V s is the sliding velocity of the follower gear relative to the driver gear, the upper portion of the follower gear tooth has the rubbing directed toward the pitch point. Next, after reaching the pitch point, the contact point continues to move down the follower gear tooth to the root point P 2f . During this movement, however, ξ is negative; thus, we see from Eq. (17.8.7) that the sliding velocity V s is now in the –n ⊥⊥ ⊥⊥ direction. This in turn means that the rubbing on the lower portion of the follower gear tooth is also directed toward the pitch point. Finally, for the driver gear tooth the rubbing is in the opposite directions. When the contact is on the lower portion of the tooth (below the pitch point), the rubbing is directed away from the pitch point and toward the root. When the contact is on the upper portion of the tooth (above the pitch point), the rubbing is also directed away from the pitch point, but now toward the tooth tip. Figure 17.8.2 shows this rubbing pattern on the driver and follower teeth. For tooth wear (or degradation), this rubbing pattern has the tendency to pull the driver tooth surface away from the pitch point. On the follower gear tooth the rubbing tends to push the tooth surface toward the pitch point. If the gear teeth are worn to the point of fracture, the fracture will initiate as small cracks directed as shown in Figure 17.8.3. 17.9 Involute Rack Many of the fundamentals of involute tooth geometry can be understood and viewed as being generated by the basic rack gear. A basic rack is a gear of infinite radius as in Figure 17.9.1. For an involute spur gear the basic rack has straight-sided teeth. The inclination of the tooth then defines the pressure angle as shown. A rack can be used to define the gear tooth geometry by visualizing a plastic (perfectly deformable) wheel, or gear blank, rolling on its pitch circle over the rack, as in Figure 17.9.2. With the wheel being perfectly plastic the rack teeth will create impressions, or footprints on the wheel, thus forming involute gear teeth, as in Figure 17.9.3. The involute rack may also be viewed as a reciprocating cutter forming the gear teeth on the gear blank as in Figure 17.9.4. Indeed, a reciprocating rack cutter, as a hob, is a common procedure for involute spur gear manufacture [17.2]. The proof that the FIGURE 17.8.2 Sliding or rubbing direction for meshing drives and follower gear teeth. FIGURE 17.8.3 Fracture pattern for meshing driver and follower gear teeth. P Driver Follower P 0593_C17_fm Page 591 Tuesday, May 7, 2002 7:12 AM [...]... relatively simple mechanical systems Our objective has been to illustrate fundamental principles Often our systems have been as simple as a single particle or body Mechanical systems of practical importance, however, are generally far more complex than such simple systems Nevertheless, the procedures of analysis for the more complex systems are essentially the same as those of the simple systems In this... 17.6.4) [17.6] Path of contact — locus of points of contact of a pair of meshing spur gear teeth (see also line of action, pressure line, and line of contact) [17.6] Pinion — smaller of two gears in mesh [17.2] Pitch circles — perimeters of rolling wheels used to model gears in mesh [17.3] 0593_C17_fm Page 601 Tuesday, May 7, 2002 7:12 AM Mechanical Components: Gears 601 Pitch point — point of contact between... passes through the path of contact (see also angle of contact) [17.6] Angle of approach — angle turned through by a gear from the position of initial contact of a pair of teeth up to contact at the pitch point (see Figure 17.6.1) [17.6] Angle of contact — angle turned through by a gear as a typical tooth travels through the path of contact (see also angle of action) [17.6] Angle of recess — angle turned... Law of conjugate action — requirement that the normal line of contacting gear tooth surfaces passes through the pitch point [17.3] Line of action — line normal to contacting gear tooth surfaces at their point of contact (see also line of contact, pressure line, and path of contact) [17.4] Line of contact — line passing through the locus of contact points of meshing spur gear teeth (see also line of. .. Observe the transitive nature of Eq (18.3.17) By repeated use of this expression, we can obtain a transformation matrix between the unit vectors of a typical body, say Bk, and the unit vectors of the inertial frame R For example, consider again the multibody system of 0593_C18_fm Page 612 Tuesday, May 7, 2002 8:50 AM 612 Dynamics of Mechanical Systems 11 10 9 12 8 15 6 7 3 14 13 16 1 4 2 5 R FIGURE 18.3.3... to multibody systems As the name implies, multibody systems contain many bodies Multibody systems may be used to model virtually all physical systems Although the fundamental principles used with simple systems can also be used with multibody systems, it is necessary to develop procedures for organizing the complex geometry of the systems These organizational procedures are the focus of this chapter... and B14 of 0593_C18_fm Page 607 Tuesday, May 7, 2002 8:50 AM Introduction to Multibody Dynamics 607 10 9 8 6 7 2 1 3 4 5 R FIGURE 18.2.3 A numbering and labeling of the bodies in a branch of the multibody system FIGURE 18.2.4 A numbering and labeling of the bodies of two branches of the multibody system 11 10 9 12 8 15 6 7 14 3 13 16 4 5 2 1 R FIGURE 18.2.5 A complete numbering and labeling of the multibody... gears, there are many other kinds of gears and other kinds of gear tooth forms In the following sections, we will briefly consider some of the more common of these other types of gears Detailed analyses of these gears, however, is beyond our scope Indeed, the geometry of these gears makes their analyses quite technical and complex In fact, comprehensive analyses of many of these gears have not yet been... from components of the angular velocity matrix In this context, ωk is the dual vector [18.4] of WK 18.5 Euler Parameters With large multibody systems, the solution of the governing dynamical equations will generally require the use of numerical procedures and computational techniques The objective of such procedures is to obtain a time history of the dependent variables of the system, most of which are... 17.14.3 A profile sketch of a worm Dynamics of Mechanical Systems FIGURE 17.14.2 A worm gear set FIGURE 17.14.4 Profile of a worm gear set in mesh Figure 17.14.2 depicts a worm gear set With worm gears, the smaller member is called the worm and the larger the worm wheel A worm is similar to a screw, and the worm wheel is similar to a helical gear The geometry of a worm is analogous to that of a helical gear . 17.6.4) [17.6] Path of contact — locus of points of contact of a pair of meshing spur gear teeth (see also line of action, pressure line, and line of contact) [17.6] Pinion — smaller of two gears in. typical pair of meshing spur gears. The angle of contact is sometimes called the angle of action. The path of contact AB is formed from the point of initial contact A to the point of ending contact. paths of approach and recess are generally not equal to each other. FIGURE 17.6.1 Path and angle of contact for a pair of meshing involute teeth. Angle of Contact Path of Contact A Direction of