Chapter 4
Structural Analysis of Mechanisms
4.1 Introduction
Structural analysis is the study of the nature of connection among the members
of a mechanism and its mobility. It is concerned primarily with the fundamental
relationships among the degrees of freedom, the number of links, the number of
joints, and the type of joints used in a mechanism. It should be noted that structural
analysis only deals with the general functional characteristics of a mechanism and not
with the physical dimensions of the links. A thorough understanding of the structural
characteristics is very helpful for enumeration of mechanisms.
In this text, graph theory will be used as an aid in the study of the kinematic structure
of mechanisms. Except for a few special cases, we limit ourselves to those mecha-
nisms whose corresponding graphs are planar. Although there are a few mechanisms
whose corresponding graphs are not planar, these mechanisms usually contain a large
number of links. In addition, we also limit ourselves to graphs that contain no artic-
ulation points or bridges. A graph with an articulation point or a bridge represents a
mechanism that is made up of two mechanisms connected in series with a common
link but no common joint, or with a common joint but no common link. These types of
mechanisms can be treated as two separate mechanisms and, therefore, are excluded
from the study.
A thorough understanding of the structural topology can be helpful in several
ways. First of all, mechanisms can be classified into families of similar structural
characteristics. Various families of mechanisms can be quickly evaluated during the
conceptual design phase. Secondly, a systematic methodology can be developed for
enumeration of mechanisms according to certain prescribed structural characteristics.
4.2 Correspondence Between Mechanisms and Graphs
Since the topological structure of a kinematic chain can be represented by a graph,
many useful characteristics of graphs can be translated into the corresponding char-
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acteristics of a kinematic chain. Table 4.1 describes the correspondence between
the elements of a kinematic chain and that of a graph. Table 4.2 summarizes some

Total number of loops (
L + 1
)
˜
L
Total number of loops (
L + 1
)
˜
L
Number of loops with
i
edges
L
i
Number of loops with
i
joints
L
i
Table 4.2
Structural Characteristics of Mechanisms and Graphs.
Graphs Mechanisms
L = e − v + 1 L = j − n + 1
e − v + 2 ≥ d
i
≥ 2 j − n + 2 ≥ d
i
≥ 2


2
≥ 3v − 2en
2
≥ 3n − 2j

i
L
i
=
˜
L = L + 1

i
L
i
=
˜
L = L + 1

i
iL
i
= 2e

i
iL
i
= 2j
Isomorphic graphs Isomorphic mechanisms
4.3 Degrees of Freedom

Intuitively, the degrees of freedom of a mechanism is equal to the degrees of
freedom of all the moving links diminished by the degrees of constraint imposed by
the joints. If all the links are free from constraint, the degrees of freedom of an n-link
mechanism with one link fixed to the ground would be equal to λ(n − 1). Since the
total number of constraints imposed by the joints are given by

i
c
i
, the net degrees
of freedom of a mechanism is
F = λ(n − 1) −
j

i=1
c
i
. (4.1)
The constraints imposed by a joint and the degrees of freedom permitted by the joint
are related by
c
i
= λ − f
i
. (4.2)
Substituting Equation (4.2) into Equation (4.1) yields
F = λ(n − j − 1) +
j

i=1

E − E
Sliding along an axis parallel to the line of intersection of the planes of
the two E pairs. If the two planes are parallel, three passive dof exist.
Passive degrees of freedom cannot be used to transmit motion or torque about
an axis. When such joint pairs exist, one degree of freedom should be subtracted
from the degrees of freedom equation. We exclude the E − E combination as being
impractical, because a link (or links) with an E − E pair can slide freely along an
axis parallel to the line of intersection of the two E planes. Let f
p
be the number of
passive degrees of freedom in a mechanism, then Equation (4.3) can be modified as
F = λ(n − j − 1) +
j

i=1
f
i
− f
p
. (4.4)
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In general, if the Grübler criterion yields F>0, the mechanism has F degrees
of freedom. If the criterion yields F = 0, the mechanism becomes a structure with
zero degrees of freedom. On the other hand, if the criterion yields F<0, the mech-
anism becomes an overconstrained structure. It should be noted, however, that there
are mechanisms that do not obey the degrees of freedom equation. These overcon-
strained mechanisms require special link length proportions to achieve mobility. The
Bennett [5] mechanism is a well-known overconstrained spatial 4R linkage. It con-
tains four links connected in a loop by four revolute joints. The opposite links have
equal link lengths and twist angles, and are related to that of the adjacent link by a

Five-bar linkage.
Example 4.4 Spur-Gear Drive
For the spur-gear set shown in Figure 1.10, we have n= 3 and j
1
= 2,j
2
= 1.
Equation (4.3) gives F = 3(3 − 3 − 1) + 4 = 1. Therefore, the spur-gear drive is a
one-dof mechanism.
Example 4.5 Spatial RCSP Mechanism
For the spatial RCSP mechanism shown in Figure 3.17, we have n= 4,j
1
=
2,j
2
= 1, and j
3
= 1. Equation (4.3) yields F = 6(4−4−1)+2×1+1×2+1×3 = 1.
Hence, the RCSP linkage is a one-dof mechanism.
Example 4.6 Swash-Plate Mechanism
For the swash-plate mechanism shown in Figure 1.12, we have n= 4,j
1
=
2,j
2
= 0,j
3
= 2,j= j
1
+ j

˜
L = j − n + 2 . (4.6)
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Equation (4.5) is known as Euler’s equation. Combining Equation (4.5) with Equa-
tion (4.3) yields
j

i=1
f
i
= F + λL . (4.7)
Equation (4.7) is known as the loop mobility criterion. The loop mobility criterion is
useful for determining the number of joint degrees of freedom needed for a kinematic
chain to possess a given number of degrees of freedom.
Example 4.7 Four-Bar Linkage
For the planar four-bar linkage shown in Figure 1.8, we have n= 4,j= 4.
Equation (4.5) yields L = 1. For F = 1, Equation (4.7) yields

f
i
= 1+3×1 = 4.
Hence, the total number of joint degrees of freedom should be equal to four to achieve
a one-dof mechanism.
Example 4.8 Humpage Gear Reducer
The Humpage gear reduction unit shown in Figure 3.14 is a five-bar spherical
mechanism, in which links 1, 2, and 5 are three coaxial bevel gears, link 3 is a
compound planet gear, and link 4 is the carrier. In this mechanism, link 1 is fixed
to the ground, link 5 is the input link, and link 2 serves as the output link. The
compound planet gear 3 meshes with gears 1, 2, and 5. Overall, the mechanism has
four revolute joints and three gear pairs. With λ = 3, n = 5,j

where
˜
L = L + 1. Combining Equations (4.8) and (4.9) yields
˜
L ≥ d
i
≥ 2 . (4.10)
In other words, the minimum number of joints on each link of a closed-loop chain
is 2 and the maximum number is limited by the total number of loops.
Example 4.9 Stephenson Six-Bar Linkage
Figure4.5showsthekinematicstructureandgraphrepresentationoftheStephenson
six-bar linkage. The number of joints on the links are: d
1
= d
3
= d
4
= d
6
= 2, and
d
2
= d
5
= 3. Since there are six links and seven joints, the number of independent
loops is given by L = j − n + 1 = 7 − 6 + 1 = 2. Hence, the number of joints on
any link is bounded by 3 ≥ d
i
≥ 2.
FIGURE 4.5

FIGURE 4.6
Binary, ternary, and quaternary links.
Let n
i
denote the number of links with i joints, that is, n
2
denotes the number of
binary links, n
3
the number of ternary links, n
4
the number of quaternary links, and
so on. Clearly,
n
2
+ n
3
+ n
4
+···+n
r
= n, (4.12)
where r =
˜
L denotes the largest number of joints on a link.
Since each of the n
i
links contains i joints and each joint connects exactly two
links, the following equation holds.
2n

,n
3
,...,n
r
. All solutions, however,
must be nonnegative integers. The number of solutions can be treated as the number
of partitions of the integer 2j into parts 2, 3, ...,r with repetition permitted. This is a
well-known problem in combinatorial analysis. The solutions can be found by using
a nested-do loops computer algorithm to vary the values of n
i
. See Appendix A for a
description of the method. In the following, we study a heuristic algorithm developed
by Crossley [8].
1. Given the number of links and the number of joints, find the upper and lower
bounds on the number of joints on a link by Equation (4.10), and the minimal
number of binary links by Equation (4.15).
2. Find a particular solution to Equations (4.12) and (4.13). This can be done by
equating all but two variables, say n
2
and n
3
, to zero and solving the resulting
equations for these two variables. This produces one solution called a link
assortment.
3. For the link assortment obtained in the preceding step, apply Crossley’s op-
erator, (1, −2, 1) or its negative, wherever possible, to any three consecutive
numbers of n
i
s. Crossley’s operator effectively adds one link with i − 1 joints,
subtracts two links with i joints, and then adds another link with i + 1 joints.

Hence, with n = 8 and j = 10, Equations (4.12) and (4.13) reduce to
n
2
+ n
3
+ n
4
= 8 , (4.16)
2n
2
+ 3n
3
+ 4n
4
= 20 . (4.17)
A particular solution to Equations (4.16) and (4.17) is found to be n
2
= 4,n
3
= 4,
and n
4
= 0. Applying Crossley’s operator, we obtain the following three families of
link assortments:
Family
n
2
n
3
n

assume the values of 0, 2, and 4, one at a time, then
n
4
= 2, 1, and 0, respectively. Once n
3
and n
4
are known, we solve Equation (4.16)
for n
2
. This leads to the same results.
4.7 Partition of Binary Link Chains
Binary links in a mechanism may be connected in series to form a binary link chain.
The first and last links of a binary link chain are necessarily connected to nonbinary
links. We define the length of a binary link chain by the number of binary links in
that chain. Furthermore, we consider the special case for which two nonbinary links
are connected directly to each other as a binary link chain of zero length. Binary link
chains of length 0, 1, 2, and 3 are called the E, Z, D, and V chains, respectively, as
depicted in Figure 4.7.
Let b
k
denote the number of binary link chains of length k, and q denote the
maximal length of a binary link chain in a kinematic chain. Applying Equations (2.39)
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FIGURE 4.7
Various binary link chains.
and (2.40), we obtain
b
1
+ 2b

i
= F + λ,
where

k+1
i=1
f
i
denotes the total joint degrees of freedom associated with a binary
link chain of length k. It follows that the remaining links and joints of the mechanism
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would be overconstrained. Therefore, to avoid such degenerate cases, we impose the
condition
k+1

i=1
f
i
≤ F + λ − 1 . (4.21)
The maximum number of joints occurs when all the joints in a binary link chain are
one-dof joints. It follows from Equation (4.21) that the length of a binary link chain
is limited by
q ≤ F + λ − 2 . (4.22)
For each family of link assortment, we can solve Equations (4.19) and (4.20)
for various combinations of binary link chains. Since Equation (4.20) contains one
more variable than that of Equation (4.19), we can solve Equation (4.19) for b
k
for
k = 1, 2,...,q, and then Equation (4.20) for b
0

= 6 . (4.24)
Solving Equation (4.23) for nonnegative integers of b
1
and b
2
, and then Equa-
tion (4.24) for b
0
yields
Branch
b
0
b
1
b
2
1402
2321
3240
Hence, there are three branches of binary link chains. The first branch consists of
no binary link chains of length 1 and two binary link chains of length 2; the second
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