Dynamics of Mechanical Systems 2009 Part 13 docx

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
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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
592 Dynamics of Mechanical Systems
straight-sided rack cutter generates an involute spur gear tooth is somewhat beyond
our scope; however, a relatively simple proof using elementary procedures of differential
geometry 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
d
1
and the pitch diameter of the nth gear is d
n
, then by repeated use of Eq. (17.2.1) we find
the angular speed ratio to be:
(17.10.1)
Theoretically, there is no limit to this speed ratio; however, the larger the number of
gears, the greater is the friction loss. Also, from a practical viewpoint, the number of gears
is often limited by the space available in a given mechanical system.
FIGURE 17.9.1

Basic involute rack.
FIGURE 17.9.2
A plastic gear blank rolling on its pitch circle over a
rack gear.
FIGURE 17.9.3
Involute teeth formed on a plastic rolling wheel.
FIGURE 17.9.4
A reciprocating hob cutter cutting a tooth on a
gear blank.
θ
Speed Ratio =
1
ωω
nn
dd=
1
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Mechanical Components: Gears 593
The gears of a gear train producing an angular speed ratio, as in Eq. (17.10.1), thus
produce 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 a
corresponding 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 M
1
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)):
(17.10.6)
FIGURE 17.10.1
A gear train with n gears.
FIGURE 17.10.2
A simple planetary gear system.

1
2
3
n
C
A
B
Fixed
MM dd
nnn11 1
==ωω
ωω
BC AB
rr=+
[]
1
ωω
BC AB
rr=+

()
1
Vr rr
O
CC A B C
B
==+
()
ωω
Vr
O
BB
B
=ω
0593_C17_fm Page 593 Tuesday, May 7, 2002 7:12 AM
594 Dynamics of Mechanical Systems
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:
(17.10.10)
Similarly, if the ring gear A is fixed, the angular velocities of the sun gear D and the
connecting link C are related as:
(17.10.11)
FIGURE 17.10.4
A simple planetary gear system with an
external (or ring) gear.
FIGURE 17.10.5
A planetary gear system with both a sun gear
and a ring gear.
C
A
B
D
C
A
B
rrr
BB A B C
ωω=+
()
ωω
BC ABB AB
rrr rr=+
()
=+
()

1
ωω
BC AB
rr=−
()
1
ωω
ADAC
rr=+
()
[]
1
ωω
DADC
rr=+
()
[]
1
0593_C17_fm Page 594 Tuesday, May 7, 2002 7:12 AM
Mechanical Components: Gears 595
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 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 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.

17.12 Helical Gears
Helical gears, like spur gears, are gears for parallel shafts. They may be viewed as a
modification of spur gears where the teeth are curved in the axial direction. The objective
of the curved tooth is to provide a smoother and quieter mesh than is obtained with spur
gears. Specifically, the teeth form helical curves along the cylinder of the pitch circles.
Figure 17.12.1 shows a sketch of a helical gear.
To explore the geometry of helical gears a bit further, consider the basic rack with
inclined 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 an
involute 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 form
helix 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 greater
the helix angle, the greater the gear tooth length and thus the greater the time of contact.
FIGURE 17.12.1
A helical gear.
FIGURE 17.12.2
Basic rack of a helical gear.
a. Top View
b. Front View
0593_C17_fm Page 595 Tuesday, May 7, 2002 7:12 AM
596 Dynamics of Mechanical Systems
Unfortunately, however, the inclined tooth surface creates an axial thrust for gears in
mesh, as depicted in Figure 17.12.5. Specifically, the normal force N will have an axial
component Nsinθ. This axial force can then in turn reduce the efficiency of meshing helical
gears by introducing friction forces to be overcome by the driving gear.
Also, in the absence of thrust-bearing constraint, the axial force component can cause

the 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 nonparallel
shafts. With bevel gears, the shaft axes intersect, usually at 90°. Figure 17.13.1 depicts a
typical bevel gear pair. As is seen in the figure, bevel gears have a conical shape. Their
geometrical characteristics are thus somewhat more complex than those of spur or helical
gears. The kinematic principles, however, are essentially the same.
FIGURE 17.12.3
A rectangular sheet with diagonal line AB.
FIGURE 17.12.4
A circular helix.
FIGURE 17.12.5
Normal force N on helix gear tooth surface.
FIGURE 17.12.6
A pair of meshing helical gears — herringbone
gears.
A
B
θ
0593_C17_fm Page 596 Tuesday, May 7, 2002 7:12 AM
Mechanical Components: Gears 597
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 gears
are analogous to helical gears. The advantages of spiral bevel gears over straight bevel
gears are analogous to the advantages of helical gears over spur gears — that is, stronger
and smoother acting teeth with longer tooth contact. The principal disadvantage of spiral
bevel and helical gear teeth is their need for precision manufacture. Also, if their precise
geometry 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 for
parallel 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 the
interested 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 and
worm 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, as
opposed to conical, gear shapes.
FIGURE 17.13.1
A bevel gear pair.
Figure 17.13.2
Spiral angle.
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598 Dynamics of Mechanical Systems
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
rack. 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 needed
with smooth action, or with little or no backlash. Hypoid and worm gears are employed
when 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 smooth
operation — 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 of
hypoid and worm gears (as with bevel gears) is that they are not nearly as efficient as
parallel shaft gears. The inclined and curved surfaces, while strong, also induce sliding
between the mating surfaces, leading to friction losses. Readers interested in additional
details should consult the references.
FIGURE 17.14.1
A hypoid gear set.
FIGURE 17.14.2
A worm gear set.
FIGURE 17.14.3
A profile sketch of a worm.
FIGURE 17.14.4
Profile of a worm gear set in mesh.
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Mechanical Components: Gears

599

17.15 Closure

In this chapter, we have briefly considered the fundamentals of gearing, with a focus upon

spur gear geometry and kinematics. The dynamic principles we have developed in earlier
chapters are directly applicable with gearing systems. In spite of the relative simplicity of
spur gear geometry, the complex geometry of other gear forms (that is, helical, bevel,
spiral bevel, hypoid, and worm gears) makes elementary analyses impractical, even
though the basic principles are essentially the same as those as spur gears. Research on
gears and gearing systems is continuing and expanding in response to increasing demands
for greater precision and longer-lived transmission systems. Although details of this
research are beyond our scope, interested readers may want to consult the references for
additional information about this work. We conclude our chapter in the following section
with a glossary of gearing terms.

17.16 Glossary of Gearing Terms

The following is a partial listing of terms commonly used in gearing technology together
with 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 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 through by a gear from contact of a pair of teeth at
the 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 Figure
17.5.3) [17.5]

Base circle

— circle of rolling wheel pulley (see Figure 17.4.2) or generating involute
curve 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 intersecting
shaft 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 corresponding
points 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 the
support base of a tooth (see Figure 17.5.1) [17.5]

0593_C17_fm Page 599 Tuesday, May 14, 2002 10:46 AM
600 Dynamics of Mechanical Systems
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 Figure
17.13.4) [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]
0593_C17_fm Page 600 Tuesday, May 7, 2002 7:12 AM
Mechanical Components: Gears 601
Pitch point — point of contact between rolling pitch circles [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
17.5.1) [17.5]
Worm — smaller member of a worm gear set, in the form of a screw (see Figure
17.13.2) [17.3]
Worm wheel — larger member of a worm gear set, similar to a helical gear (see
Figure 17.3.2) [17.3]
References
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.
0593_C17_fm Page 601 Tuesday, May 7, 2002 7:12 AM
602 Dynamics of Mechanical Systems
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. Let
the 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
c. Center-to-center distance
d. Circular pitch
e. Module
f. Addendum
g. Dedendum
h. Clearance
i. Fillet radius
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 the
circular 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 both
inches 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 of
gear 1 to gear 3.
P17.6.2: Generalize the results of Problem P17.6.1 for n gears in mesh.
0593_C17_fm Page 602 Tuesday, May 7, 2002 7:12 AM
Mechanical Components: Gears 603
P17.6.3: Two meshing spur gears have an angular speed ratio of 2.0 to 1. Suppose the
pinion (smaller gear) has 32 teeth and that the distance separating the gear centers is 12
inches. 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 the
diametral 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, have
diametral 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 and
diametral 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.
FIGURE P17.6.1
A train of three spur gears.
FIGURE P17.10.1
A simple planetary gear system.
1
2
3
r
r
r
1
3
2
C
A
B
Fixed
O
A
0593_C17_fm Page 603 Tuesday, May 7, 2002 7:12 AM
604 Dynamics of Mechanical Systems
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)
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605

18

Introduction to Multibody Dynamics

18.1 Introduction

In the foregoing chapters — indeed, in the major portions of this book — we have focused
upon relatively simple mechanical systems. Our objective has been to illustrate funda-
mental 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 chapter, we develop
procedures for extending our analysis techniques 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.

Technically, a multibody system is simply a collection of bodies. The bodies themselves
may be either connected to each other or free to translate (or separate) relative to each
other. 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 closed
loops 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 two
such 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 attention
upon open-chain systems with connected rigid bodies. Extension to other systems having
flexible 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 of
bodies 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

0593_C18_fm Page 605 Tuesday, May 7, 2002 8:50 AM

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 Figure
18.2.1. 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: arbitrarily
select a body, perhaps one of the larger bodies, as a reference body and call it Body 1, or

B

1

, 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

B

1

through the branches of the tree system. Specifi-
cally, consider the representation of the multibody system of Figures 18.2.1 and 18.2.2 as
a projection of the images of the bodies onto a plane. Next, select a body adjacent to

B

1

,
call it

B

2

, and label it 2. Then, continue to number the bodies in a serial manner through
the branch of bodies containing

B

2

until the extremity of the branch (

B

5

) is reached, as in
Figure 18.2.3. Observe that two extremity bodies branch off of

B

4

. Let the other body be
called

B

6

and label it 6. Next, return to

B

2

, which is also a branching body, and number
the bodies in the other branch of

B

2

in a similar manner, as in Figure 18.2.4. Then label
the remaining bodies in the second branch in ascending progression, moving clockwise
in the projected image of the system. Finally, return to

B

1

and number and label the bodies
in the remaining branch, leading to a complete numbering and labeling of the system, as
in Figure 18.2.5.
Once the bodies of the system are numbered and labeled, the connection configuration
may be described by a lower body array defined as follows: First, observe that except for

B

1

each body of the system has a

unique

adjacent lower numbered body. (A body may
have more than one adjacent higher numbered bodies [such as

B

2

,

B

4

,

B

8

,

B

9

, and

B

14

of

FIGURE 18.1.1

An open-chain, open-tree multibody system.

FIGURE 18.1.2

A multibody system with a closed loop, separation
between the bodies, and long flexible members.

FIGURE 18.2.1

A multibody system.

FIGURE 18.2.2

Reference body

B

1

for the system of Figure 18.2.1.
R
R
1
B
15
16
14
13
2
7
8
9
12
3
4
5

6
10
11
1

0593_C18_fm Page 606 Tuesday, May 7, 2002 8:50 AM