Kinematic Analysis of Geared Mechanisms Using the Concept of Kinematic Fractionation

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Kinematic analysis of geared mechanisms using the concept

of kinematic fractionation

Chia-Pin Liu, Dar-Zen Chen

*

_{, Yu-Tsung Chang}

Department of Mechanical Engineering, National Taiwan University, Taipei 10660, Taiwan Received 14 May 2002; received in revised form 3 May 2004; accepted 28 May 2004

Abstract

A systematic approach to the determination of kinematic relations between input(s) and output(s) in geared mechanisms is presented based on the concept of kinematic fractionation. It is shown that kinematic unit (KU) can be viewed as functional building block of geared mechanisms, and kinematic propagation path from input to output can be determined systematically according to the interface among KUs. The local gain between the local input and output of each KU can be systematically formulated. Along the propagating path connecting input and output, global kinematic relation can then be evaluated by collect-ing local gains of KUs. It is believed that this unit-by-unit evaluation procedure provides a better insight of the eﬀects of each KU on the interactions among input(s) and output(s). An epicyclic-type automatic trans-mission mechanism is used to illustrate the procedure.

1. Introduction

Geared mechanisms have been used widely as power transmission and force ampliﬁcation de-vices in machines and vehicles. The input power is transmitted to output through a path com-posed of meshing gear pairs and corresponding carriers. Through kinematic analysis, dependent relations among input(s) and output(s) of the mechanism are evaluated.

*_{Corresponding author. Fax: +886 2363 1755.}

E-mail address:dzchen@ccms.ntu.edu.tw(D.-Z. Chen).

www.elsevier.com/locate/mechmt

Mechanism and Machine Theory 39 (2004) 1207–1221

Mechanism

and

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Many research efforts had been devoted to develop efficient approaches to the kinematic anal-ysis of geared mechanisms. Some basic methods, such as the tabular method and formula method have been widely known and elaborated in the textbooks[1–3]. Although these methods provide basic skills to investigate the kinematic relations among input(s) and output(s), it can be labori-ous as these procedures are applied to complex gear trains. Based on the application of graph theory[4], the concept of fundamental circuit was applied to the kinematic analysis of gear trains [5,6]. However, the determination of the kinematic relations needs to solve a set of linear equa-tions simultaneously. The mathematical manipulation cannot provide much insight into the kin-ematic structure of the mechanism. Chatterjee and Tsai [7] established the concept of fundamental geared entity (FGE) for automatic transmission mechanisms and applied the con-cept to associated speed ratio analysis and power loss analysis [8]. However, the concept of FGE can only be applied to reverted type epicyclic gear trains and is specialized in determining kinematic relations among coaxial links. Chen and Shiue[9]showed that a geared robotic mech-anism can be regarded as a combination of input units and transmission units. Chen[10]verified the forward and backward gains of each unit and proposed a unit-by-unit evaluation procedure for the kinematic analysis of geared robotic mechanisms. Although this approach is straight-forward and provides clear kinematic insight in the torque transmission, it is restricted to geared robotic mechanisms.

Based on the concept of kinematic fractionation developed by Liu and Chen [11], a method to determine the kinematic propagation path from input to output links in geared mecha-nisms will be established in this paper. It will be shown that a geared mechanism can be re-garded as a combination of kinematic units (KUs). The connection among KUs reveals the kinematic propagating path in the mechanism, and the kinematic relationship between input and output links can be formulated eﬃciently by combining local gain of each KU along the path. The kinematic modules in turn serve as an eﬃcient tool to determine complicated kin-ematic relations among input(s) and output(s). It is believed that the concept of kinkin-ematic fractionation can provide lucid perspective to determine kinematic relations in a geared mechanism.

2. Concept of kinematic fractionation

In the graph representation of geared mechanisms, links are represented by vertices, gear pairs by heavy edges, turning pairs by thin edges, and each thin edge is labeled according to the asso-ciated axis location. Liu and Chen [11] deﬁned the KU as a basic kinematic structure in geared mechanisms. Each KU is composed of a carrier and all the gears on it. A graph-based procedure to identify the KUs in a geared mechanism[12] is brieﬂy described as follows with an illustration on the graph representation of epicyclic gear train (EGT) inFig. 1:

Step 1: Construct the displacement graph [4]. Fig. 1(b) shows the displacement graph of Fig. 1(a).

Step 2: Separate the displacement graph into sub-graph(s) each with only one carrier label. Fig. 1(c) shows the separated displacement graph ofFig. 1(b).

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Step 3: Add a carrier vertex to each segment of the separated displacement graph and connect the gear–carrier pairs by thin edges. Each thin edge is then labeled with axis orientations. The example result is shown to the left ofFig. 1(d)in which vertices 1 and 5 are common to both sub-graphs.

Step 4: In each sub-graph obtained in Step 3, identify the vertices which are shared as common links, and connect these vertices with a thin edge. Each resultant sub-graph is referred to as a KU. Since vertices 1 and 5 in Fig. 1(d) are coaxial, a thin edge can be formed by coaxial re-arrangement without changing kinematic charac-teristics of the mechanism [13]. Fig. 1(d) shows the KUs of Fig. 1(a) on the right hand side.

By applying above procedure, EGTs with 1-dof 5-link enumerated by Freudenstein [4] and Tsai [14] and EGTs with 2-dof 6-link enumerated by Tsai and Lin [13] can be fractionated systematically. Fig. 2(a) shows 1-dof 5-link EGTs with only one KU, and Fig. 2(b) shows EGTs with multiple KUs. Fig. 3 shows 2-dof 6-link EGTs with 3 KUs. In Figs. 2 and 3, it can be seen that there are 10 distinct KUs can be identiﬁed and shown in Table 1. In Table 1, each KU is labeled with Kn-# where n is the number of links and # is the serial number.

Fig. 1. Graph representation of 5-link EGT. (a) Graph representation, (b) displacement graph, (c) separated displacement graph, (d) resultant KUs.

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3. Internal conversion

3.1. Admissible internal conversion modes

Liu and Chen [11] showed that each KU can be regarded as a 1-dof sub-mechanism in the geared mechanism since the kinematic relations among links can be determined by a single input. In each KU, motion is initiated by the local input, which is either a contained input or the com-mon linkage connecting to the preceding KU(s). The local input of a KU is then modulated, and transmitted to the local output, which is either the global output or the common linkage connect-ing to the succeedconnect-ing KU(s). This process of transformconnect-ing and transmittconnect-ing from local input to local output within a KU is referred to as the internal conversion.

Both the local input and output in the KU can be expressed as the relative angular displace-ment between a turning pair, which is corresponding to a thin edge in graph representation. Accord-ing to the types of adjacent vertices, thin edges in a KU can be classiﬁed into two diﬀerent types:

(1) gear–carrier (g–c) type: One end of the thin edge is a gear vertex, and the other end is a carrier vertex. A g–c type thin edge is denoted by a thin line as shown in Table 1.

(2) gear–gear (g–g) type: Both ends of the thin edge are gear vertices. A g–g type thin edge is distinguished from the g–c type thin edges by a double line representation.

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Note that a g–g type thin edge in a KU can be formed by coaxial re-arrangement as the thin edges connecting each of the two gear vertices and the carrier have the same axis label. As a g–g type thin edge is added to a KU, one of the coaxial g–c type thin edges should be deleted.

Fig. 3. Kinematic fractionation of 2-dof 6-link EGTs.

Table 1

Local gains for up-to-5 link KUs with g–c vs. g–c internal conversion mode KUs

Local gain Gðy; k; x; kÞ ¼hy;k hx;k¼ Ex;y

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The KUs, which can include g–g type thin edge(s), are collected as shown in Tables 2 and 3. Each KU in Table 2is labeled as Kn-#S which indicates that the KU is originated from Kn-# with single g–g type thin edge. Similarly, each KU inTable 3is labeled as Kn-# D which indicates that the KU is originated from Kn-# with double g–g type thin edges. Note that one of the coaxial g–c type thin edges of KUs inTable 2and two of the coaxial g–c type thin edges of KUs inTable 3 can be removed arbitrarily.

Among these thin edges, a KU can have at least one of the following three internal conversion modes:

Case 1. g–c vs. g–c type: The internal conversion is between two g–c type thin edges. Both the local input and output of the KU are located on g–c type thin edges. For KUs with up to ﬁve links, this internal conversion mode can take place between any two thin edges inTable 1.

Case 2. g–c vs. g–g type: The internal conversion is between a g–c type thin edge and a g–g type thin edge. The local input and output of the KU are located on diﬀerent types of thin edges. For KUs with up to ﬁve links, this internal conversion mode can take place between the g–g type thin edge and any one of the g–c type thin edges inTable 2. Case 3. g–g vs. g–g type: The internal conversion is between two g–g type thin edges. Both the

local input and output of the KU are located on g–g type thin edges. For KUs with up to ﬁve links, this internal conversion mode can take place between the two g–g type thin edges in Table 3.

Table 2

Local gains for up-to-5 link KUs with g–c vs. g–g internal conversion mode KUs

Local gain Gðx; y; y; kÞ ¼hx;y

hy;k¼ ðEy;x 1Þ; Gðx; y; r; kÞ ¼ hx;y

hr;k¼ ðEy;x 1ÞEr;y

Table 3

Local gains for up-to-5 link KUs with g–g vs. g–g internal conversion mode KUs Local gain Gðx; q; y; pÞ ¼hx;q hy;p¼ Ep;xEp;q Ep;y1 Gðx; p; y; pÞ ¼ hx;p hy;p¼ Ep;x1 Ep;y1

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3.2. Local gain

The local gain of a KU is the gear ratio from local input to local output. According to the inter-nal conversion mode, associated local gain can be derived as follows:

3.2.1. g–c vs. g–c type conversion

Since there is a unique carrier in each KU inTable 1, the kinematic relation between the two ends of the heavy-edged path can be derived by combining associated fundamental circuit equa-tions as follows:

hy;k¼ ex;xþ1 ey1;yhx;k ¼ Ex;yhx;k ð1Þ

where hy, kis the relative angular displacement between gear vertex y and the carrier k, x + 1

rep-resents the vertex on the right hand side of vertex x, y 1 represents the vertex on the left hand side of vertex y, ex + 1, xis the gear ratio between vertices x + 1 and x, and Ex, y is the product of

gear ratios on the heavy-edged path from x to y. According to Eq.(1), we have:

Rule 1: The local gain of g–c vs. g–c type conversion can be expressed as:

Gðy; k; x; kÞ ¼ hy;k hx;k

¼ Ex;y ð2Þ

where x and y are gear vertices, and k is the unique carrier in the KU.

3.2.2. g–c vs. g–g type conversion 1. Conversion among coaxial vertices

For each KU inTable 2, the coaxial relation between two gear vertices, x and y, and the carrier k can be written as:

hx;k ¼ hx;k hy;k ð3Þ

From Eqs.(2) and (3), we have:

Rule 2: The local gain of g–c vs. g–g type conversion among coaxial vertices can be expressed as:

Gðx; k; y; kÞ ¼ hx;k hy;k

¼ ðEy;x 1Þ ð4Þ

where x and y are coaxial gear vertices, and k is the unique carrier. 2. Conversion including non-coaxial vertices

From Eq.(1), the kinematic relation between two gear vertices, r and y, and the carrier k can be derived as

hy;k¼ Er;yhr;k ð5Þ

Combining Eqs.(4) and (5)yields

Rule 3: The local gain of g–c vs. g–g type conversion among non-coaxial vertices can be expressed as:

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Gðx; y; r; kÞ ¼hx;y hr;k

¼ ðEy;x 1ÞEr;y ð6Þ

where x and y are coaxial gear vertices, r is another gear vertex which is connected to the carrier k with a thin edge with diﬀerent axis label.

3.2.3. g–g vs. g–g type conversion

For the left hand side KU in Table 3, the local gains can be derived from Eq.(4)as

hx;q ¼ ðEq;x 1Þhq;k ð7aÞ

hy;q ¼ ðEp;y 1Þhp;k ð7bÞ

From Eq.(1), hq,kand hp,k can be related by

hp;k ¼ Eq;phq;k ð8Þ

By substituting Eq.(8) into Eq.(7b) and then eliminating hq,kin Eqs. (7a) and (7b), we have:

Rule 4: The local gain of g–g vs. g–g type conversion between two g–g type thin edges (x, q) and (y, p), which have diﬀerent axis labels can be expressed as

Gðx; q; y; pÞ ¼hx;q hy;p

¼Ep;x Ep;q Ep;y 1

ð9aÞ

For the right hand side KU inTable 3, the local gain can be derived along a similar procedure from Eqs.(7) to (9), and the following rule can be concluded:

Rule 5: The local gain of g–g vs. g–g type conversion between two coaxial g–g type thin edges (x, p) and (y, p) can be expressed as

Gðx; p; y; pÞ ¼hx;p hy;p

¼Ep;x 1 Ep;y 1

ð9bÞ

Tables 1–3show local gains of KUs with diﬀerent internal conversion modes. WithTables 1–3, associated local gains of a KU can be formulated accordingly as the locations of local input and output are speciﬁed.

4. Global propagation

4.1. Common linkage

A common linkage is referred to as the interface among KUs and is composed of links and con-necting thin edges shared by each other. FromFigs. 2 and 3, two kinds of common linkages can be identiﬁed:

(1) 2-link-chain type: This kind of common linkage exists between two KUs, and the relative angular displacement between links on the common linkage is used as the communicating medium between KUs. As shown inFig. 1(d), KU1and KU2share a 2-link chain with vertices

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(2) Coaxial-triangle type: This kind of common linkage exists among three KUs in which each pair of KUs shares a common vertex. As shown on the right hand side ofFig. 3, those thin edges forming the coaxial triangle are specially marked with short lines.

4.2. Kinematic propagation

For geared mechanisms with only one KU, kinematic propagation from input to output is com-pleted in the same KU as shown inFig. 4(a). Hence, the kinematic relation between input and output links can be described by Rules 1–5, which means that the global propagation is exactly equivalent to internal conversion.

For two KUs sharing a 2-link chain type common linkage, the output of a preceding KU is received directly by the succeeding KU as the input, the kinematic propagation path is shown as Fig. 4(b). Considering the mechanism in Fig. 1(d), h3,5 and h4,5 can be assigned as output

and input, respectively. It can be observed that the lower KU labeled as KU1in which local input

h4,5is transmitted to local output, h1,5, on the common linkage through a g–c vs. g–c type

con-version. Then, h1,5 is received by the upper KU, which is labeled as KU2, as local input from

the common linkage and is subsequently converted into output, h3,5 through a g–c vs. g–g type

conversion. With the propagation through the common linkage, the motion is transmitted from KU1to KU2.

As shown inFig. 3, it is known that KUs around a coaxial-triangle type common linkage forms a 2-dof EGT. The coaxial relations result in a 2-input, 1-output interface among KUs, the kine-matic propagation path is shown inFig. 4(c). For instance, the graph inFig. 3(a), which is com-posed of three K3-1 type KUs, can represent a 2-dof EGT by using h1,4and h6,4as input and h3,5

as output. In the lower left KU, which is labeled as KU1, local input h1,4is transmitted to local

output h2,4 on the common linkage. On the other hand, local input of the upper KU, which is

Fig. 4. Global propagation. (a) Single KU type, (b) 2-link chain type, (c) coaxial triangle type.

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labeled as KU2, h6,4, is transmitted to local output h5,4on the common linkage. According to the

coaxial condition, h2,4and h5,4are combined to form the resultant motion h2,5according to the

following equation:

h2;5¼ h2;4þ ðh5;4Þ ð10Þ

The resultant motion is then received by the lower right KU, which is labeled as KUs, as local input and converted into output h3,5. Hence, the propagation through a coaxial-triangle type

com-mon linkage needs two independent motions to initiate a resultant motion in the remaining KU around a coaxial triangle.

For geared mechanisms with multiple KUs, kinematic relations among input(s) and out-put(s) can be symbolically determined by virtue of a trace-back procedure from the KU with the global output. The procedure can be demonstrated as follows with the graph represen-tation of EGT shown in Fig. 1(d) and the graph representation of EGT shown in Fig. 3(a) as examples.

Step 1: Express the global output in terms of local input of the associated KU. The result can be generally expressed as:

hout ¼out½Kx #_in hinÞL ð11Þ

where houtis the output, hin)Lis the local input andout[Kx#]inrepresents the local gain associated

with the conversion from hin)L to houtin Kx#.

In Fig. 1(d), the output is located in KU2and its kinematic relation corresponding to Eq.(11)

can be written as:

h3;5¼35½K4 1S15 h1;5 ð12Þ

InFig. 3(a), the output is located in KU3and the relation corresponding to Eq.(11)can be written

as:

h3;5¼35½K3 125 h2;5 ð13Þ

where 35[K3 1]25 is the local gain associated with the conversion from local input, h2,5, to the

local output, h3,5in KU3 in the EGT inFig. 3(a).

Step 2: Transform the local input in Eq. (11) into local output of preceding KU(s).

According to Fig. 4(b) and (c), the transformation can be determined by the following cases:

(a) For a 2-link-chain type common linkage, there is only one preceding KU, and the local out-put of the preceding KU is identical to the local inout-put of its succeeding KU. Hence, there is no modiﬁcation required for Eq.(11).

For the EGT inFig. 1(d), the common linkage is a 2-link chain, and thus local input of KU2,

h1, 5, is equal to the local output of KU1. Hence, Eq. (12) needs no modiﬁcation.

(b) For a coaxial-triangle type common linkage, there are two preceding KUs from which two distinct local output merge into their succeeding KU. According to Eq. (10), Eq. (11) can be modiﬁed as:

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hout¼out½Kx #_in ðhoutÞp1þ houtÞp2Þ ð14Þ

where hout)p1 and hout)p2are the two local output of preceding KUs.

For the EGT inFig. 3(a), since KU3shares a coaxial-triangle type common linkage with

pre-ceding KUs, Eq. (13)can be rewritten according to Eqs. (10) and (14) as:

h3;5¼35½K3 1₂₅ ½h2;4þ ðh5;4Þ ð15Þ

Step 3: Apply Eq. (11) to convert the local output(s) derived in Step 2 into associated local input(s) and repeat Steps 2 and 3 until all the local inputs are from KUs with input links. The ﬁnal relation among global output and input can be generally expressed:

hout¼ X m Y_out ½Kx #_in hinÞm ð16Þ

where (Õout[Kx #]in) represents the product of involved local gains from the input to output

and hin)mis the input contained in KUm.

By applying Eq. (16)to the EGT inFig. 1(d), Eq. (12) can be expanded as:

h3;5¼35½K4 1S15

15_{½K3 1}

45h4;5 ð17Þ

where15[K3 1]45 is the local gain associated with the conversion from local input, h4,5, to the

local output, h1,5in KU1 inFig. 1(d).

By applying Eq. (16)to the EGT inFig. 3(a), Eq. (15) can be expanded as: h3;5¼35½K3 125 f 24_{½K3 1} 14 h1;454½K3 164 h6;4Þg ¼35_{½K3 1} 25 24_{½K3 1} 14 h1;4 35_{½K3 1} 25 54_{½K3 1} 64 h6;4 ð18Þ

where24[K3 1]14 is the local gain associated with the conversion from local input, h1,4, to the

local output, h2,4in KU1 inFig. 3(a), and54[K3 1]64is the local gain associated with the

con-version from local input, h6,4, to the local output, h4,5in KU2 inFig. 3(a).

Eqs.(17) and (18)provide the global kinematic relation between the input and output as a pol-ynomial in terms of local gains. The form of Eq.(17) implies that only one sequential kinematic propagating path exists in the EGT inFig. 1(d)while multiple terms in Eq.(18)represents that the EGT inFig. 3(a)contains two distinct propagating paths which merge at the coaxial-triangle type common linkage.

The local gains in Eqs.(17) and (18)can be substituted with the forms expressed inTables 1–3. For the EGT in Fig. 1(d), 35[K4 1S]15can be determined by Table 2as

35 ½K4 1S₁₅¼h3;5 h1;5 ¼ h1;5 h5;3 1 ¼ ½Gð1; 5; 5; 3Þ1¼ 1 ðE5;1 1Þ ð19Þ 15_[K3

1]45 can be determined fromTable 1as 15_{½K3 1}

45¼

h1;5

h4;5

¼ Gð1; 5; 4; 5Þ ¼ E4;1 ð20Þ

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By substituting Eqs. (19) and (20)into Eq. (17), we have h3;5¼ E4;1 1 E5;1 h4;5¼ e4;1 1 e5;2e2;1 h4;5 ð21Þ

Similarly, the global kinematic relation of the EGT in Fig. 3(a)can be derived by converting Eq. (18) according toTable 1as:

h3;5¼ E2;3 E1;2 h1;4 E2;3 E6;5 h6;4¼ e2;3 ½e1;2 h1;4 e6;5 h6;4 ð22Þ

5. An application to automatic transmission mechanisms

Fig. 5(a) shows the functional representation of a typical transmission mechanism, which is used as an example by Hsieh and Tsai[8]. FromFig. 5(a), it can be observed that the mechanism has three sets of sun-planetary-ring gear systems which corresponds to the three FGEs as shown inFig. 5(b), in which the unlabeled vertex represents the housing. According to the connection between FGEs, a unique FGE diagram can be constructed for the mechanism, and then the over-all gear ratio is determined by identifying the operation modes of associated FGEs [8].

In contrast to the three FGEs inFig. 5(b), the mechanism has only two KUs as shown in Fig. 5(c)according to the concept of kinematic fractionation. The gear ratio analysis can be performed as follows with given location of ground, input and output links ([G, I, O]):

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1stgear with [G, I, O]¼ [5, 2, 8].

Since both the input and output links are in KU2, the relation between the input and output is

simply a g–c vs. g–c type internal conversion with the local gain as the overall gear ratio. From Table 1, the overall gear ratio at this operation mode is determined as:

h8;5

h2;5

¼ E2;8¼ e2;8 ð23Þ

2ndgear with [G, I, O] = [1, 2, 8].

The transmission from input, h21, to the output, h81, is a g–g vs. g–g type internal conversion in

KU2. FromTable 3, the overall gear ratio at this operation mode is determined as:

h8;1 h2;1 ¼E1;8 1 E1;2 1 ¼e1;8 e7;8 1 e1;6 e6;2 1 ð24Þ 3rdgear with [G, I, O] = [1, 4, 8].

In KU2, the output h81 can be expressed in terms of the local input, h25, as follows:

h8;1¼81½KU225 h2;5 ð25Þ

where81[KU2]25 is the local gain associated with a g–c vs. g–g type conversion in KU2.

Note that although KU2has six links, the connecting condition between local input and output

is identical to those KUs inTable 2. Hence,81[KU2]25can also be determined by ﬁtting the

expres-sion inTable 2.

It can be observed that input h41involves both the two KUs inFig. 5(c). According to the

fol-lowing coaxial condition, h4,1can be decomposed into two dependent terms which lie in diﬀerent

KUs:

h4;1¼ h4;2þ h2;1 ð26Þ

Eq. (26) can be related to the local output of KU1 as

h4;1¼ f42½K4 1S25þ

21_{½K4 1S}

25g h2;5 ð27Þ

where both terms of local gains in Eq.(27) are associated with the g–c vs. g–g type conversion in KU-4 1S.

Re-arranging Eq. (27)yields

h2;5¼ f42½K4 1S₂₅þ21½K4 1S₂₅g1 h4;1 ð28Þ

By substituting Eq.(28)into Eq.(25), the overall gear ratio at the third gear are determined as h8;1 h4;1 ¼81_½KU 225 f 42_{½K4 1S} 25þ 21_{½K4 1S} 25g 1 ð29Þ

where the local gains can be further expanded in terms of gear ratios according toTable 2. It can be seen that the concept of FGEs are obtained from structural aspects rather than from kinematic characteristics, over decomposition may be occurred. As shown inFig. 5(b), the second and third FGEs should be considered as a single KU since they share a carrier. Hence, it is

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believed that KUs represent more direct and eﬃcient modules than FGEs in conducting kinematic analysis of geared mechanisms.

6. Conclusion

The concept of kinematic fractionation is introduced to identify the kinematic modules in geared mechanisms. The concept of kinematic fractionation exposes the kinematic propagation in the mechanism and facilitates the determination of global kinematic relation between input and output links. Admissible internal conversion modes and associated local gains are determined for KUs with up to ﬁve links. According to the internal conversion mode in each KU, input and output can be correlated by sequential substitution along the global kinematic propagating path(s). It is believed that the proposed approach provides much kinematic insight into the inter-actions in geared mechanisms.

Acknowledgment

The ﬁnancial support of this work by the National Science Council of the Republic of China under the Grant NSC 90-2212-E-002-166 is gratefully acknowledged.

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