Dynamics of Mechanical Systems 2009 Part 13 docx

Thelocation of the pitch circle on a gear tooth is determined by the location of the point ofcontact between meshing teeth on the line connecting the gear centers, as illustrated inFigur

Unlike the root circle, the base circle, and the addendum circle, the pitch circle is notfixed relative to the tooth Instead, the pitch circle is determined by the center distancebetween the mating gears Hence, the addendum, dedendum, and tooth thickness mayvary by a small amount depending upon the specific location of the pitch circle Thelocation of the pitch circle on a gear tooth is determined by the location of the point ofcontact between meshing teeth on the line connecting the gear centers, as illustrated inFigure 17.5.2.

An advantageous property of involute spur gear tooth geometry is that the gears willoperate together with conjugate action at varying center distances with only the provisionthat 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 inFigure 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

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 meters Then, the module and diametral pitch are related by the expression:

milli-(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 dimensional representation of a gear tooth illustrating the width, top land, face, flank, andbottom land

three-Finally, involute spur gear tooth designers have the following tooth proportions

depend-ing upon the diametral pitch P [17.3]:

Dedendum = 1.157/P (17.5.6) Clearance = 0.157/P (17.5.7) Fillet radius = 0.157/P (17.5.8)

17.6 Kinematics of Meshing Involute Spur Gear Teeth

In this section, we consider the fundamentals of the kinematics of meshing involute spurgear 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 elementarydescription of spur gear kinematics We will follow his outline here, and the reader maywant to consult the reference itself for additional details

In our discussion, we will consider the interaction of a pair of mating teeth from thetime they initially come into contact, then as they pass through the mesh (the region ofcontact), and, finally, as they separate We will assume that the gears always have at leastone pair of teeth in contact and that each pair of mating teeth is the same as each otherpair 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 respectiveangles 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 meshingspur gear pair Contact between teeth will occur only within the region of overlap of thecircles The extent of the overlap region is dependent upon the gear radii The location ofthe 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 andrecess 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

As noted in the foregoing sections, the rotation of meshing gears may be modeled byrolling 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 r1, r2 and d1, d2 are the radii and diameters, respectively, of the pitch circles of thegears From Eq (17.5.2) we may express the angular speed ratio as:

(17.6.3)

where, as before, N1 and N2 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 leastone pair of teeth be in contact at all times (otherwise there will be intermittent contactwith 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 Fromthe 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.

Path of Approach

Direction of Rotation

P

O Gear 1

cord being unwrapped about the base circle (see Figure 17.4.4), we see that the normalpitch may also be expressed as the distance between points of adjacent tooth surfacesmeasured along the pressure line as in Figure 17.6.5 Observing that the path of contactalso lies along the pressure line (see Figure 17.6.1) we see that the length of the path ofcontact 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:

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 analyticalrepresentation of the contact ratio To see this, consider an enlarged and more detailedrepresentation 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 A1 and A2 (pressure line/addendum circle intersections) is the length of the path of contact Consider the segment A2P; let Q

be at the intersection of the line through A2 perpendicular to the line through the gear centers as in Figure 17.6.7 Then A2QO2 is a right triangle whose sides are related by:

(17.6.7)

where r2a is the addendum circle radius of gear 2.

Let 2 be the length of segment A2P (the path of approach of Figure 17.6.2) Then, fromFigure 17.6.7, we have the relations:

(17.6.8)(17.6.9)(17.6.10)

where, as before, θ is the pressure angle, and r2p 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 2, we obtain:

(17.6.12)Similarly, for gear 1 we have:

(17.6.13)

where 1 is the length of the contact path segment from P to A1 (the path of recess in

Figure 17.6.2), and r1p and r1a 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 1 and 2 aregiven by Eqs (17.6.12) and (17.6.13) Spotts and Shoup [17.10] state that for smooth gearoperation 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 motionwith a 2-to-1 speed ratio between axes separated by 9 inches If 20° pressure angle gearswith 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:

1P

2P 2A

2 2 1 2

= −r psinθ+[r a−r pcos θ]/

2 1

(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 ted between contacting involute spur gear teeth To this end, consider the contact of twoteeth as they pass through the pitch point as depicted in Figure 17.7.1 It happens that atthe pitch point there is no sliding between the teeth Hence, the forces transmitted between

transmit-the teeth are equivalent to a single force N, normal to transmit-the contacting surfaces and thus

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 theconcept 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 nitude of the driving moment For most gears, the pressure angle is designed to be either14.5 or 20°, with a recent trend toward 20° The pressure angle, however, is also dependentupon the gear positioning That is, because the position of the pitch circles depends uponthe gear center separation (see Section 17.5 and Figure 17.5.2), the pressure angle will not,

mag-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 thepitch point the tooth surfaces slide relative to each other This sliding (or “rubbing”) canlead 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 ofthe 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 P1 be the point of initial mesh (contact) of the gear teeth and let P2 be the point of final contact If P1f is that point of the follower gear coinciding with P1, then the velocity of P1f

may be expressed as:

where ωf is the angular velocity of the follower gear and Of P1f is the position vector locating P1f relative to the follower gear center O f Then, in terms of unit vectors shown in Figure17.8.1, Of P1f 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, ωf may be expressed as:

(17.8.3)

where n3 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)

Consider now the rubbing itself: First, for the follower gear, as the meshing begins the

contact point is at the tip P1f 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 Vs is in the n⊥⊥direction This means that, because Vs is the sliding velocity of thefollower gear relative to the driver gear, the upper portion of the follower gear tooth hasthe 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 P2f During this movement, however, ξ is negative;

thus, we see from Eq (17.8.7) that the sliding velocity Vs is now in the –n⊥⊥direction This

in turn means that the rubbing on the lower portion of the follower gear tooth is alsodirected toward the pitch point

Finally, for the driver gear tooth the rubbing is in the opposite directions When thecontact is on the lower portion of the tooth (below the pitch point), the rubbing is directedaway 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 toothwear (or degradation), this rubbing pattern has the tendency to pull the driver toothsurface away from the pitch point On the follower gear tooth the rubbing tends to pushthe 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 (perfectlydeformable) wheel, or gear blank, rolling on its pitch circle over the rack, as in Figure17.9.2 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.

straight-sided rack cutter generates an involute spur gear tooth is somewhat beyondour scope; however, a relatively simple proof using elementary procedures of differentialgeometry may be found in Reference 17.12.

17.10 Gear Drives and Gear Trains

As we have noted several times, gears are used for the transmission of forces and motion

Thus, a pair of meshing gears is called a transmission Generally speaking, however, a transmission usually employs a series of gears, and is sometimes called a gear train Figure

17.10.1 depicts a gear train of parallel shaft gears If the pitch diameter of the first gear is

d1 and the pitch diameter of the nth gear is d n, then by repeated use of Eq (17.2.1) we findthe angular speed ratio to be:

The gears of a gear train producing an angular speed ratio, as in Eq (17.10.1), thusproduce either an angular speed increase or an angular speed reduction between the shafts

of the first and the last gears In an ideal system, with no friction losses, there will be acorresponding reduction or increase in the moments applied to the first and last shafts

That is, if the moment applied to the shaft of the first gear is M1 and if the moment produced at the last shaft is M n, then we have the ratios:

(17.10.2)

An efficient method of speed and moment reduction (or increase) may be obtained by

using a planetary gear system — so called because one or more of the gears does not have

a fixed axis of rotation but instead has an axis that rotates about the other gear axis That

is, although the axes remain parallel, the axis of one or more of the gears is itself allowed

to rotate To illustrate this, consider the system of Figure 17.10.2 consisting of two gears

A and B, whose axes are connected by a link C If gear A is fixed, then as gear B engages gear A in mesh, the axis of B will move in a circle about the axis of A The connecting link

C will rotate accordingly Because B moves around A, B is often called a planet gear and then A is called the sun gear.

Using our principles of elementary kinematics, we readily discover that the angular

speeds of B and C are related by the expressions:

(17.10.3)or

(17.10.4)

where r A and r B are the pitch circle radii of A and B.

To see this, let O A and O B be the centers of gears A and B Then because O A is fixed and

because C rotates about O A , the speed of O B is:

(17.10.5)

where r C is the effective length of the connecting link C However, because the pitch circle

of B rolls on the pitch circle of A, we also have (see Eq (4.11.5)):

Hence, we have:

(17.10.7)or

(17.10.8)

As a second illustration, consider the system of Figure 17.10.4, where the fixed sun gear

A is external to the planet gear B The external sun gear is often called a ring gear Again,

using the principles of elementary kinematics as above, we readily see that, if the ring

gear A is fixed, the angular speeds of the planet gear B and the connecting link C are

related by the expression:

(17.10.9)

Planetary gear systems generally have both a sun gear and a ring gear as represented

in Figure 17.10.5 In this case, the system has four members, A, B, C, and D, with C being

a connecting link between the centers of the sun gear D and the planet gear B.

Generally, in applications, either the sun gear or the ring gear is fixed If the sun gear

D is fixed, we can, by again using the principles of elementary kinematics, find relations between the angular velocities of the ring gear A, the planet gear B, and the connecting link C For example, the angular velocities of A and C are related by the expression:

A simple planetary gear system with an

external (or ring) gear.

17.11 Helical, Bevel, Spiral Bevel, and Worm Gears

Thus far we have focused our attention and analyses on parallel shaft gears and specifically

on spur gears Although these are the most common gears, there are many other kinds ofgears and other kinds of gear tooth forms In the following sections, we will brieflyconsider some of the more common of these other types of gears Detailed analyses ofthese gears, however, is beyond our scope Indeed, the geometry of these gears makestheir analyses quite technical and complex In fact, comprehensive analyses of many ofthese gears have not yet been developed, and research on them is continuing Nevertheless,the fundamental principles (conjugate action, pitch points, rack forms, etc.) are the same

or very similar to those for involute spur gears

To explore the geometry of helical gears a bit further, consider the basic rack withinclined teeth as shown in Figure 17.12.2 If the rack is deformed into a cylindrical shape,the gear form of Figure 17.12.1 is obtained When formed in this way, the teeth have aninvolute form in planes parallel to the plane of the gear

If we visualize the rack being deformed and wrapped into a helical gear, the teeth formhelix segments along the gear cylinder To see this, consider a rectangular sheet as in

Figure 17.12.3 having a diagonal line AB as shown If the sheet is wrapped, or spindled, into a cylinder, as in Figure 17.12.4, the line AB becomes a circular helix.

The inclination angle of the rack gear teeth of Figure 17.12.2a is called the helix angle.

In practice, the helix angle can range from just a few degrees up to 45° [17.6] The greaterthe helix angle, the greater the gear tooth length and thus the greater the time of contact

Unfortunately, however, the inclined tooth surface creates an axial thrust for gears inmesh, as depicted in Figure 17.12.5 Specifically, the normal force N will have an axialcomponent Nsinθ This axial force can then in turn reduce the efficiency of meshing helicalgears by introducing friction forces to be overcome by the driving gear.

Also, in the absence of thrust-bearing constraint, the axial force component can causethe meshing helical gears to tend to separate axially To eliminate this separation tendency,helical gears are often used in pairs with opposite helix angles as in Figure 17.12.6 Such

gears are generally called herringbone gears.

17.13 Bevel Gears

Unlike spur and helical gears, bevel gears transmit forces and motion between nonparallelshafts With bevel gears, the shaft axes intersect, usually at 90° Figure 17.13.1 depicts atypical bevel gear pair As is seen in the figure, bevel gears have a conical shape Theirgeometrical characteristics are thus somewhat more complex than those of spur or helicalgears The kinematic principles, however, are essentially the same

Bevel gears have tapered teeth The geometric properties of these teeth (for example,pressure angle, pitch, backlash, etc.) are generally measured at the mean cone position —that is, halfway between the large and small ends (“heel and toe”) of the cone frustrum.

In their profile, bevel gear teeth are like involute spur gear teeth The teeth themselves

may be either straight or curved Bevel gears with curved teeth are commonly called spiral bevel gears The curvature of a spiral bevel gear tooth may vary somewhat, but typically

at the mid-tooth position the tangent line to the tooth will make an angle θ with a conical

element, as depicted in Figure 12.13.2 This angle, which is typically 35°, is called the spiral angle.

Straight bevel gears are analogous to involute spur gears, whereas spiral bevel gearsare analogous to helical gears The advantages of spiral bevel gears over straight bevelgears are analogous to the advantages of helical gears over spur gears — that is, strongerand smoother acting teeth with longer tooth contact The principal disadvantage of spiralbevel and helical gear teeth is their need for precision manufacture Also, if their precisegeometry is distorted under load, the kinematics of the gears can be adversely affected.The geometry of bevel gears makes their analysis and design more difficult than forparallel shaft gears Also, bevel gears are generally less efficient than parallel shaft gears

As a consequence, engineers and designers use parallel shaft gears wherever possible (as,for example, with front-wheel-drive cars with engines mounted parallel to the drive axles).Details of the geometry and kinematics of bevel gears are beyond our scope, but theinterested reader may want to contact the references for additional information

17.14 Hypoid and Worm Gears

Hypoid and worm gears are used to transmit forces and motion between nonparallel and

nonintersecting shafts Generally, the shafts are perpendicular In this sense, hypoid andworm gears are similar to bevel gears

Figure 17.14.1 depicts a hypoid gear set As with other gear pairs, the larger member is

called the gear and the smaller is called the pinion Hypoid gears have the same form and

shape as spiral bevel gears; however, the nonintersecting shafts produce hyperbolic, asopposed to conical, gear shapes

FIGURE 17.13.1

A bevel gear pair.

Figure 17.13.2

Spiral angle.

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 gearrack Figure 17.14.3 provides a profile view of a worm and common terminology [17.10].Similarly, Figure 17.14.4 provides a profile of a worm gear set in mesh

Hypoid and worm gears are generally used when a large speed-reduction ratio is neededwith smooth action, or with little or no backlash Hypoid and worm gears are employedwhen it is impractical or impossible to use intersecting shafts Indeed, a principal advan-tage of hypoid and worm gears — in addition to their high speed reduction and smoothoperation — is that bearings for both shafts may be used on both sides of the gear elements,thus providing structural rigidity and stability On the other hand, a disadvantage ofhypoid and worm gears (as with bevel gears) is that they are not nearly as efficient asparallel shaft gears The inclined and curved surfaces, while strong, also induce slidingbetween the mating surfaces, leading to friction losses Readers interested in additionaldetails should consult the references

17.15 Closure

In this chapter, we have briefly considered the fundamentals of gearing, with a focus uponspur gear geometry and kinematics The dynamic principles we have developed in earlierchapters are directly applicable with gearing systems In spite of the relative simplicity ofspur gear geometry, the complex geometry of other gear forms (that is, helical, bevel,spiral bevel, hypoid, and worm gears) makes elementary analyses impractical, eventhough the basic principles are essentially the same as those as spur gears Research ongears and gearing systems is continuing and expanding in response to increasing demandsfor greater precision and longer-lived transmission systems Although details of thisresearch are beyond our scope, interested readers may want to consult the references foradditional information about this work We conclude our chapter in the following sectionwith a glossary of gearing terms

17.16 Glossary of Gearing Terms

The following is a partial listing of terms commonly used in gearing technology togetherwith a brief definition and the chapter section where the term is first discussed:

Addendum — height of a spur gear tooth above the pitch circle (see Figure 17.5.1)[17.5]

Addendum circle — external or perimeter circle of a gear (see Figure 17.5.1) [17.5]

Angle of action — angle turned through by a gear as a typical tooth passes throughthe path of contact (see also angle of contact) [17.6]

Angle of approach — angle turned through by a gear from the position of initialcontact 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 throughthe path of contact (see also angle of action) [17.6]

Angle of recess — angle turned through by a gear from contact of a pair of teeth atthe pitch point up to the end of contact (see Figure 17.6.1) [17.6]

Axial pitch — distance between corresponding points of adjacent screw surfaces of

a worm, measured axially (see Figure 17.14.3) [17.3]

Backlash — looseness or rearward separation of meshing gear teeth (see Figure17.5.3) [17.5]

Base circle — circle of rolling wheel pulley (see Figure 17.4.2) or generating involutecurve circle (see Figures 17.4.4 and 17.4.5) [17.4]

Bevel gear — gear in the shape of a frustrum of a cone and used with intersectingshaft axes [17.12]

Bottom land — inside or root surface of a spur gear tooth (see Figure 17.5.5) [17.5]

Circular pitch — distance, measured along the pitch circle, between correspondingpoints of adjacent teeth (see Figure 17.5.3 and Eq (17.5.1)) [17.5]

Clearance — difference between root circle and base circle radii; the elevation of thesupport base of a tooth (see Figure 17.5.1) [17.5]

Conjugate action — constant angular speed ratio between meshing gears [17.3] Conjugate gears — pair of gears that have conjugate action (constant angular speed

ratio) when they are in mesh [17.3]

Contact ratio — average number of teeth in contact for a pair of meshing gears [17.6] Dedendum — depth of a spur gear tooth below the pitch circle (see Figure 17.5.1)

[17.5]

Diametral pitch — number of gear teeth divided by the diameter of the pitch circle

(see Eq (17.5.2)) [17.5]

Driver — gear imparting or providing the motion or force [17.2]

Face — surface of a spur gear tooth above the pitch circle (see Figure 17.5.4) [17.5] Fillet — gear tooth profile below the pitch circle (see Figure 17.5.1) [17.5]

Flank — surface of a spur gear tooth below the pitch circle (see Figure 17.5.5) [17.5] Follower — gear receiving the motion or force [17.2]

Gear — larger of two gears in mesh [17.2]

Gear train — transmission usually employing several gears in mesh in a series

[17.10]

Heel — large end of a bevel gear or of a bevel gear tooth [17.12]

Helical gear — parallel shaft gear with curved teeth in the form of a helix (see Figure

17.12.1) [17.12]

Helix angle — inclination of a helix gear tooth (see Figure 17.12.2a) [17.12]

Herringbone gears — pair of meshing helical gears with opposite helix angles (see

Figure 17.12.6) [17.12]

Hob — reciprocating cutter in the form of a rack gear tooth [17.9]

Involute function — function Invφ defined as (tanφ) – φ (see Eq (17.4.17)) [17.4]

Involute of a circle — curve formed by the end of an unwrapping cable around a

circle (see Figure 17.4.4) [17.4]

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 action, pressure line, and path of contact) [17.6]

Mesh — interaction and engaging of gear teeth

Module — pitch circle diameter divided by the number of teeth of a gear

Normal circular pitch — distance between corresponding points of adjacent screw

surfaces of a worm, measured perpendicular or normal to the screw (see Figure17.13.4) [17.3]

Normal pitch — distance between corresponding points on adjacent teeth measured

along the base circle (see Figure 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]

Planet gear — gear of a planetary gear system whose center moves in a circle about

the center of the sun gear [17.10]

Planetary gear system — gear train or transmission where one or more of the gears

rotate on moving axes [17.10]

Pressure angle — inclination of line of action (see Figure 17.4.3) [17.4]

Pressure line — line normal to contacting gear tooth surfaces at this point of contact (see also line of action, line of contact, and path of contact) [17.4]

Rack gear — gear with infinite radius [17.9]

Ring gear — external sun gear of a planetary gear system (see Figure 17.10.4) [17.10] Root circle — boundary of the root, or open space, between spur gear teeth (see

Figure 17.5.1) [17.5]

Spiral angle — angle between the tangent to a spiral bevel gear tooth and a conical

element (see Figure 17.12.2) [17.12]

Spiral bevel gear — bevel gear with curved teeth [17.12]

Sun gear — central gear of a planetary gear system with a fixed center [17.10] Toe — small end of a bevel gear or of a bevel gear tooth [17.12]

Tooth thickness — distance between opposite points at the pitch circle for an involute

spur gear tooth (see Figure 17.5.1) [17.5]

Top land — outside surface of a spur gear tooth (see Figure 17.5.4) [17.5]

Transmission — pair of gears in mesh [17.1]

Whole depth — total height of a spur gear tooth (see Figure 17.6.1) [17.3]

Width — axial thickness of a spur gear tooth (see Figure 17.5.4) [17.5]

Working depth — height of the involute portion of a spur gear tooth (see Figure

17.1 Buckingham, E., Analytical Mechanics of Gears, Dover, New York, 1963.

17.2 Townsend, D P., Ed., Dudley’s Gear Handbook, 2nd ed., McGraw-Hill, New York, 1991 17.3 Lent, D., Analysis and Design of Mechanisms, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1970,

chap 6.

17.4 Oberg, E., Jones, F D., and Horton, H L., Machinery’s Handbook, 23rd ed., Industrial Press,

New York, pp 1765–2076.

17.5 Dudley, D W., Handbook of Practical Gear Design, McGraw-Hill, New York, 1984.

17.6 Drago, R J., Fundamentals of Gear Design, Butterworth, Stoneham, MA, 1988.

17.7 Michalec, G W., Precision Gearing Theory and Practice, Wiley, New York, 1966.

17.8 Jones, F D., and Tyffel, H H., Gear Design Simplified, 3rd ed., Industrial Press, New York, 1961 17.9 Litvin, F L., Gear Geometry and Applied Theory, Prentice Hall, Englewood Cliffs, NJ, 1994 17.10 Spotts, M F., and Shoup, T E., Design of Machine Elements, 7th ed., Prentice Hall, Englewood

Cliffs, NJ, 1998, p 518.

17.11 Coy, J J., Zaretsky, E V., and Townsend, D P., Gearing, NASA Reference Publication 1152,

AVSCOM Technical Report 84-C-15, 1985.

17.12 Chang, S H., Huston, R L., and Coy, J J., A Computer-Aided Design Procedure for Generating Gear Teeth, ASME Paper 84-DET-184, 1984.

Problems

Section 17.5 Spur Gear Nomenclature

P17.5.1: The gear and pinion of meshing spur gears have 35 and 25 teeth, respectively Letthe pressure angle be 20° and let the teeth have involute profiles Let the diametral pitch

be 5 Determine the following:

a Pitch circle radii for the gear and pinion

b Base circle radii for the gear and pinion

P17.5.2: Repeat Problem P17.5.1 if the gear and pinion have 40 and 30 teeth, respectively

P17.5.3: Repeat Problems P17.5.1 and 17.5.2 if the module m, given by Eq (17.5.3), is 5.

Express the answers in millimeters

P17.5.4: Suppose two meshing spur gears are to have an angular speed ratio of 3 to 2.Suppose further that the distance between the centers of the gears is to be 5 inches For

a diametral pitch of 6, determine the number of teeth in each gear (Hint: The angular

speed ratio is the inverse of the ratio of the pitch circle radii [see Eq (17.6.2)].)

P17.5.5: See Problem P17.5.4 Let the pressure angle of the gears be 20° Determine thecircular pitch and the base circle radii of the gears

P17.5.6: Repeat Problem P17.5.5 if the pressure angle is 14.5°

P17.5.7: See Problem P17.5.4 Determine the module of the gears and pitch radii in bothinches and centimeters

Section 17.6 Kinematics of Meshing Involute Spur Gear Teeth

P17.6.1: Consider three parallel shaft spur gears in mesh as represented in Figure P17.6.1.Develop expressions analogous to Eqs (17.6.2) and (17.6.3) for the angular speed ratio ofgear 1 to gear 3

P17.6.2: Generalize the results of Problem P17.6.1 for n gears in mesh.

P17.6.3: Two meshing spur gears have an angular speed ratio of 2.0 to 1 Suppose thepinion (smaller gear) has 32 teeth and that the distance separating the gear centers is 12inches Determine: (a) the number of teeth in the gear (larger gear); (b) the diametral pitch;(c) the circular pitch; and (d) the pitch circle radii of the gears.

P17.6.4: Repeat Problem P17.6.3 if the center-to-center distance is 30 cm Instead of thediametral pitch, find the module

P17.6.5: Verify Eqs (17.6.12) and (17.6.13)

P17.6.6: Meshing 20° pressure angle spur gears with 25 and 45 teeth, respectively, havediametral pitch 6 Determine the contact ratio

P17.6.7: Repeat Problem P17.6.6 if the gears have 30 and 50 teeth, respectively

P17.6.8: Repeat Problem P17.6.6 if the diametral pitch is (a) 7 and (b) 8

Section 17.8 Sliding and Rubbing between Contacting Involute Spur Gear Teeth P17.7.1: See Problem P17.6.6 where 20° pressure angle gears with 25 and 45 teeth anddiametral pitch 6 are in mesh Determine the maximum distance from the pitch point to

a point of contact between the teeth

P17.7.2: Repeat Problem P17.7.1 for the data of Problem P17.6.6

P17.7.3: See Problem P17.7.1 If the pinion is the driving gear with an angular speed of

350 rpm, determine the magnitude of the maximum sliding velocity between the gears

P17.7.4: Repeat Problem P17.7.3 for the data and results of Problems P17.5.5 and P17.7.2

Section 17.10 Gear Drives and Gear Trains

P17.10.1: In the simple planetary gear system of Figure P17.10.1 the connecting arm C has

an angular speed of 150 rpm, rotating counterclockwise If the stationary gear A has 85 teeth and if gear B has 20 teeth, determine the angular speed of gear B.

P17.10.2: Repeat Problem P17.10.1 if gear A, instead of being stationary, is rotating

(a) counterclockwise at 45 rpm, and (b) clockwise at 45 rpm

P17.10.3: Repeat Problems P17.10.1 and P17.10.2 if the connecting arm C is rotating

clock-wise at 100 rpm

P17.10.4: Consider the planetary gear systems of Figure P17.10.4 Let the angular speed

of gear B be the angular speed of the connecting arm C if the fixed ring gear A has 100 teeth and gear B has 35 teeth.

P17.10.5: Repeat Problem P17.10.4 if gear A, instead of being stationary, is rotating

(a) clockwise at 25 rpm, and (b) counterclockwise at 25 rpm

P17.10.6 Repeat Problems P17.10.4 and P17.10.5 if the angular speed of gear B is rotating

counterclockwise at 75 rpm

FIGURE P17.10.4

A planetary gear system with a fixed

external (ring) gear.

C

A

B (Fixed)

Technically, a multibody system is simply a collection of bodies The bodies themselvesmay be either connected to each other or free to translate (or separate) relative to eachother The bodies may be either rigid or flexible They may or may not form closed loops.Multibody systems consisting entirely of connected rigid bodies and without closedloops are called open-chain or open-tree systems Figure 18.1.1 shows such a system Alter-natively, a multibody system may have large separation between the bodies, closed loops,and flexible members, as depicted in Figure 18.1.2.

As we noted earlier, multibody systems may be used to model many physical systems

of interest and of practical importance In the next two chapters, we will consider twosuch systems that have recently received considerable attention from analysts — specifi-cally, robots and biosystems

To keep our analysis simple, and at least moderate in length, we will focus our attentionupon open-chain systems with connected rigid bodies Extension to other systems havingflexible bodies, closed loops, and relative separation between the bodies may be considered

by using procedures documented in References 18.1 to 18.3

18.2 Connection Configuration: Lower Body Arrays

A characteristic of multibody systems, particularly large systems, is that the multitude ofbodies creates unwieldy geometric complexity An effective way to work with this com-plexity is to use a lower body array, which is an array of body numbers (or labels) that

606 Dynamics of Mechanical Systems

define the connection configuration of the multibody system The lower body arrays may

be used to develop a simplified approach to the kinematics of the system

To define and develop the lower body array, consider the multibody system of Figure18.2.1 This is an open-chain (or open-tree) multibody system moving in an inertial refer-ence frame R Let the bodies of the system be numbered and labeled as follows: arbitrarilyselect a body, perhaps one of the larger bodies, as a reference body and call it Body 1, or

B1, and label it 1 as in Figure 18.2.2 Next, number and label the other bodies of the system

in ascending progression away from B1 through the branches of the tree system cally, consider the representation of the multibody system of Figures 18.2.1 and 18.2.2 as

Specifi-a projection of the imSpecifi-ages of the bodies onto Specifi-a plSpecifi-ane Next, select Specifi-a body Specifi-adjSpecifi-acent to B1,call it B2, and label it 2 Then, continue to number the bodies in a serial manner throughthe branch of bodies containing B2 until the extremity of the branch (B5) is reached, as inFigure 18.2.3 Observe that two extremity bodies branch off of B4 Let the other body becalled B6 and label it 6 Next, return to B2, which is also a branching body, and numberthe bodies in the other branch of B2 in a similar manner, as in Figure 18.2.4 Then labelthe remaining bodies in the second branch in ascending progression, moving clockwise

in the projected image of the system Finally, return to B1 and number and label the bodies

in the remaining branch, leading to a complete numbering and labeling of the system, as

3 4 5

6

10 11

1