published in mechanism and machine theory, vol. 37, no. 11 ...shai/pdfs/dual2.pdf · utilization of...

Published in Mechanism and Machine Theory, Vol. 37, No. 11, pp. 1307-1323, November, 2002.

1

UTILIZATION OF THE DUALISM BETWEEN DETERMINATE TRUSSES AND MECHANISMS

Dr. Offer Shai

Department of Mechanics Material and Systems Tel Aviv University

ABSTRACT

Current paper presents a continuation of a previously published one, in which a mutual duality connection between determinate trusses and mechanisms has been established and proved. The dualism argued in these papers states that for every determinate truss there exists a corresponding dual mechanism and vice versa. This results in coincidence of the statical analysis procedure of the former with the kinematical analysis procedure of the latter. The new relation has opened up new ways of research and practical application, to which the current paper is dedicated. Among the applications presented in the paper there are: Establishing connections between known methods in statics and kinematics; Deriving new methods in structural mechanics from machine theory: a method for truss decomposition to components, dual vector resolution method, methods for checking the stability of structures; Deriving methods in machine theory from structural analysis: dual Henneberg’s method, a method for checking the mobility of mechanisms and new systematic design techniques based on the dualism connection.

Keywords: duality relation, trusses, structural analysis, mechanisms, graph theory, Multidisciplinary Combinatorial Approach

INTRODUCTION

This paper shows applications of one of the aspects of the Multidisciplinary Combinatorial Approach (MCA) currently being carried out and studied by the author. The idea behind MCA is to build discrete mathematical representations, called Combinatorial Representations (CR), with which diverse engineering systems are then represented. Afterwards, combinatorial theorems embedded in the CR together with the properties derived from the connections between the CR are all applied to analyze the represented engineering systems. The combinatorial representations are based on graph theory, matroid theory and discrete linear programming. Up until today, MCA has been applied to the following engineering fields: structural analysis (Shai 2001c), machine theory (Shai 2001a), integrated engineering systems (Shai 2001b), representations in Artificial Intelligence (Shai and Preiss 1999) and more for which the reports are currently being in preparation.

The current paper is based on one of the most significant accomplishments of MCA, which is the establishment of the mutual duality relation between determinate trusses and mechanisms. This new relation was derived from the properties embedded in two CR

Flow Graph Representation (FGR) and Potential Graph Representation (PGR) that are used to represent determinate trusses and mechanisms respectively. These two CR were proved to be dual (Shai 2001a), and thus it was concluded and proved that determinate trusses and mechanisms are dual as well. It is noteworthy that employing MCA has also enabled to establish another duality - the duality between two known engineering analysis methods – Force and Displacement methods of structural mechanics (Shai 2001c).


2

The duality relation lies on the fact that for each determinate truss there exists a dual mechanism and vice versa. Each rod in the truss has a corresponding link in the dual mechanism. The result of analysis of both systems yields that the forces in the truss rods are equal to the relative velocities of the corresponding links in the dual mechanism.

The paper is organized in the following manner:

Section 1 provides the reader with a brief review on the results obtained in (Shai 2001a). It explains how to represent mechanisms with PGR and trusses with FGR and proves briefly the mutual duality relation between them.

Section 2 employs methods from machine theory in structural analysis. Doing so exposes the fact that some methods in machine theory are dual to the known methods in structural analysis. For example, the image velocity diagram for kinematical analysis of a mechanism is actually the same as the Maxwell-Cremona diagram for its dual determinate truss. Moreover, the approach enables to derive new methods in structural analysis. This is demonstrated by introducing a new method for truss analysis on the basis of its dual method from machine theory, called “vector resolution”. The dualism is also used to develop methods for checking the stability of structures on the basis of the mobility of their dual mechanisms.

Section 3 employs knowledge and known methods from structural mechanics to machine theory. This issue is demonstrated by deriving a new method for kinematical analysis of mechanisms consisting of high order Assur groups dual to Henneberg’s method for complex truss analysis. Another example is given, showing a way of checking the mobility of the mechanism in a given position, on the basis of the stability of its dual truss.

Section 4 introduces a new technique in design, which is also based on the dualism. It presents a case, where there is a need to design a truss possessing some special statical properties. Applying the technique introduced in this section, the design is derived from the existent mechanism with similar kinematical properties. Same technique may yield results also when it is applied in the opposite manner.

1 SUMMARY OF THE DUALITY RELATION BETWEEN TRUSSES AND MECHANISMS.

Definitions and signs

Current paper employs classification of determinate trusses into three types (Hibbeler 1985):

Simple trusses – constructed by starting with a basic triangular element, and adding two connected rods at a time.

Compound truss - formed by connecting two or more simple trusses together.

Complex truss – is a truss that cannot be classified as being either simple or compound.

Following is the review of the principal definitions and theorems required for understanding and utilizing the duality relations between mechanisms and determinate trusses. All can be found in greater detail in (Shai 2001a).

Definition 1. Network graph – is a directed graph G = <V, E>, where V is the vertex set and E is the edge set. The vertex upon which the arrow is directed is called the ”head vertex” and the other is called the “tail vertex”. Each edge e is assigned a vector called ‘flow’ )e(F

r and each vertex v

is assigned a vector called ‘potential’ )v(πr . Subtraction of the potential of the tail vertex from the

potential of the head vertex for a specific edge is called the ‘potential difference’ - )e(∆r

of that edge.

Table 1 summarizes the two combinatorial representations (CR) called Flow Graph Representation (FGR) and Potential Graph Representation (PGR) and their relations to determinate trusses and mechanisms.


3

Combinatorial Representations (CR)

Flow Graph Representation - GF

(FGR)

Potential Graph Representation- G∆

(PGR)

Main property Flow law - The vector sum of the flows in every cutset of GF is equal to zero.

0FQ =⋅ )G()G( FF

rr

Potential law – The vector sum of the potential differences in every circuit of G∆ is equal to zero.

0B =⋅ ∆∆ )G()G( ∆rr

Represented engineering system

Determinate truss. Mechanism.

Edge Truss element: rod, reaction, external force in the truss.

The flow in the edge is interpreted as the force in the corresponding truss element.

Link in the mechanism. The potential difference is interpreted as the relative velocity between the end joints of the link.

Engineering interpretation of the graph elements

Vertex Pinned joint of the truss Joint in the mechanism. The potential is interpreted as the linear velocity of the joint.

Table 1. Flow and Potential graphs and their usage.

Duality between flow and potential graphs (Shai 2001a). Given a flow graph GF execute the following steps: build its dual graph *

FG , equate the potential differences in the edges of *FG to

the flows in the corresponding edges of GF. It then follows from the properties of dual graphs (Swamy and Thulasiraman, 1981) that these potential differences satisfy the potential law in *

FG . Thus, *

FG can be considered as a valid potential graph G∆. Finally, it can be postulated that for each flow graph GF there exists a dual potential graph G∆ and vice versa.

From the duality between the flow and potential graph representations one can deduce the duality relation between trusses and mechanisms, as is outlined in the diagram in Figure 1.

Flow Graph

Representation FGR

Potential Graph

Representation PGR

Mechanisms Determinate trusses

mutual dualism *)G()G( BQ

rr=

mutual dualism

Represents Represents

Figure 1. Diagram explaining the mutual dualism between trusses and mechanisms.

Thus for every determinate truss there exists a dual mechanism satisfying:

- there is a link in the mechanism for each rod, mobile support reaction or external force in the truss.


4

- the vector of the relative velocity of each mechanism link is equal to the vector of the force acting in the corresponding element in the truss.

Same statements can also be formulated in the vice versa form

2 EMPLOYING METHODS FROM MACHINE THEORY IN STRUCTURAL ANALYSIS.

Current section uses the duality connection between mechanisms and structures for obtaining structural analysis methods from machine theory. Next section, on the other hand, carries out an opposite task: it utilizes methods from structural analysis in machine theory.

2.1 Applying Assur groups decomposition of mechanisms to truss analysis.

One of the known methods for decomposition of mechanisms into kinematical groups, is called Assur Groups decomposition (Manolescu 1968). This issue is important and useful since it enables developing specific algorithms for each type of kinematical Assur group. This subsection employs the fact that when the dual mechanism is decomposable into certain Assur groups, the primal truss is decomposable to their dual groups upon which efficient algorithms are then applied.

One of such cases is when the dual mechanism can be decomposed exclusively into dyads. For this type of mechanisms there are several known analysis methods, all of which having the following form:

- start with the dyad that forms a contour (a circle) with the driving link and the ground. Its rather simple analysis gives the relative velocities of its links.

- continue the process each time taking the dyad forming a contour exclusively with those links, whose relative velocities are known.

This algorithm can be applied to the corresponding dual trusses. Since each contour in the mechanism corresponds to a pinned joint in the truss, at each iteration of the truss analysis (force equilibrium at a pinned joint) there are only two rods whose internal forces are unknown. Therefore, such trusses can be analyzed sequentially by solving the force equilibrium at one pinned joint at a time. According to the terminology adopted in this paper, such trusses are called ‘simple trusses’.

The mechanism (truss) presented in Figure 2a(b) can be solved as follows: the equations of force equilibrium (compatibility of relative velocities) at joint I‘ (contour 1,3,2) provide the forces (relative velocities) in rods (links) 2’ (2) and 3’ (3) , then equations for joint II’ (contour 2,6,7) give the forces (relative velocities) at 6’(6) and 7’(7) and finally from equilibrium at joint III’ (contour 3,4,5,6) the forces (relative velocities) in 4’(4) and 5’(5) are found.


5

'P1

r V1r

=

2’

6’

(b)

3’

4’5’

7’

1’I'

II'

III'

(a)

2

4

1

37

5

6III

II

I

V1r

Figure 2. Example of decomposing a mechanism and its dual truss into dyads. a) Mechanism. (b) The dual truss.

An example of Assur group decomposition of a more complicated mechanism and the corresponding groups in the dual truss appear in Figure 3.

1V r

1

2

3

54

911

6

7

10

12

16

1413

17

15

8

(a)

V

VIVII

VIII

III

II

I

OIV

IV'

111 VP 'rr

=

1 ’

7 ’

5 ’

4 ’

8 ’

6 ’

16 ’

14 ’

15 ’

13 ’

9 ’

11 ’

12 ’

10 ’

2 ’ 3 ’

( b)

V' VII'

VI'

III'

I'

II' 17 ’

VIII'

Figure 3. Example of a dual truss decomposition into Assur groups.

(a) Mechanism and its decomposition. (b) The dual truss and the corresponding decomposition

2.2 The mutual dualism between Maxwell-Cremona and image velocity diagrams.

Image velocity diagram is a known graphical method for velocity analysis of mechanisms (Norton 1992). Once the image velocity diagram of a mechanism is constructed, the relative velocities of every mechanism link can be measured directly from it.

Knowing that the relative velocities in the mechanism links are equal to the internal forces in the rods of the dual truss, it can be concluded that the image velocity can be used to perform the static analysis of a truss. Furthermore, one can build the diagram directly from the truss, without even considering its dual mechanism as is explained in the current subsection. This process becomes clear when summarizing all the properties of the image velocity diagram while simultaneously rewriting them in the terminology of structural analysis , as is done in Table 2.


6

Image velocity properties

In mechanism terminology In terminology of the dual truss

Each point corresponds to a joint in a mechanism.

Each point corresponds to a non-bisected area in the truss closed by truss elements.

The driving link is represented by a line in its velocity direction and length proportional to its velocity value. The line connects the points corresponding to the end joints of the link.

The external force is represented by a line in its direction with length proportional to its magnitude. The line connects the points corresponding to the areas separated by the external force.

The relative velocity of a link is represented by a line parallel to the link relative velocity, which connects the points corresponding to the end joints of the link.

The axial force in a rod is represented by a line parallel to the rod, connecting the points corresponding to the areas separated by it.

Table 2. Properties of the image velocity diagram in the terminology of both mechanisms and the dual trusses.

From Table 2, it follows that the image velocity method completely coincides with the known Maxwell-Cremona diagram algorithm for static analysis of determinate structures (Timoshenko and Young 1969). Consequently, Maxwell-Cremona and Image velocity methods are mutually dual methods.

Figure 4 presents a four bar chain, its dual truss and the image velocity diagram. One can verify that this diagram also presents the static analysis diagram of the dual truss, namely its Maxwell-Cremona diagram.

A

O

2

1

0/AVr

I

O

II

2’

1’

A,I

O B,II 2

1 'PV 0/A

rr=

(a) (b)

(c)

B

'Pr

Figure 4. The correspondence between image velocity and Maxwell-Cremona diagrams. (a) The four bar chain. (b) Its dual truss. (c) The image velocity/ Maxwell-Cremona

diagram.

A more compound demonstration of the correspondence between image velocity and Maxwell-Cremona diagrams is given in Figure 5.


7

IV' V2

V6

V7

V3

'PV1

rr=

V4

V5

II'

O

III'

I' III'

O3

1

(a)

F2’

I

F6'

F7’

F3’

Pr

F4’

F5’

II

0

III

IV

(c) (d)

I'

2

3

45

O1

'Pr

(b)

I

II

IV

III

5’

4’

3’

2’7’

6’

IV'II'

7

6

O

1Vr

Figure 5. Complex example for the relation between image velocity and Maxwell-Cremona

diagrams. (a), (c) The mechanism and its image velocity diagram. (b),(d) The dual truss and its

Maxwell-Cremona diagram.

2.3 The dual vector resolution method in statics.

In this section, an additional relation between variables of trusses and those of their dual mechanisms is derived. As it was explained above, each joint in a mechanism corresponds to a non-bisected area (face) closed by the truss elements. Furthermore, each vertex in the potential graph is associated with the proper potential, which is equal to the velocity of the corresponding joint. In the dual graph (FGR), the potential associates with the so-called “face flow” – flow in the corresponding graph face, or the “face force” in the corresponding face of the represented truss. Face flow can be thought of as a multidimensional generalization of the “mesh current” in electrical circuits (Balabanian and Bickart 1969). Consequently, the force of each truss element equals the difference between the forces in the faces separated by it.

Figure 6(a) shows an example of a set of face forces in a truss, that are equal to the linear velocities of the corresponding joints in the dual mechanism (Figure 6(b)).


8

O’

PV 'rr

=VI

V'

II'III'

IV'

VI'

I'

O’O’

O’Pr

I IIIII

IV

V

VI O

Figure 6. The dualism between joint potentials in a mechanism and the face forces in the dual truss.

(a) The primal truss and its face forces. (b) The corresponding dual mechanism.

2.3.1 Utilization of the vector resolution method for truss analysis.

The vector resolution method for mechanism analysis is based on the properties of the joint velocities in the mechanism, therefore the joint velocities are its primary variables. The principle underlying the method is as follows: Let AB be a mechanism link and A and B be its end joints, then the projection of the linear velocity of the joint A on AB (or its continuation), should be equal to the projection of the linear velocity of joint B on that line, i.e.:

ABAB |B|A VVrr

= ( 1)

This property is outlined in Figure 7, where the highlighted segments are supposed to be equal.

O/BVr

O/AVr

A

BVB|AB

VA|AB

Figure 7. The principle underlying the vector resolution method.

Employing the explanations given above and equation 1 to the dual trusses yields: for each truss element that separates two arbitrary faces A and B, the projections of the face forces of faces A and B on the line perpendicular to the rod AB are equal, i.e.:

ABAB |B|A FF⊥⊥

=rr

( 2)

The dual vector resolution analysis algorithm based on the above principle, is then as follows:

a) One of the faces is chosen to be the – “reference face” and its face force is set to zero.

b) For each face which has a rod common with the reference face, the direction of its face force is known: it is the direction of the rod.


9

c) Let KFr

and DFr

be the face forces in two adjacent faces such that the magnitude and

direction of KFr

and the direction of DFr

are known. Then, employing equation 2 gives the

magnitude of force DFr

.

d) Let uFr

be a face force, whose magnitude and direction are unknown, but which is adjacent

to two faces whose face force vectors are known, then uFr

can be calculated by employing twice equation 2.

An example for applying this algorithm to analyze a truss is given in Table 3.


10

Simple truss and division to faces:

1 2

3

4

5

6

7 Pr

oFr

IVFr

IFr

IIFr

IIIFr

Step 1 – Rod 2 separates faces O and II

1 2

3

4

5

6

7 Pr

oFr

IVFr

IFr

IIFr

IIIFr

IIFr

1) 0=0Fr

by definition. 2) PFI

rr= since P

r separates I and O.

3) 2||FII

r since 2 separates II and O.

4) 3||)FF( III

rr− since 3 separates I and

II 33 ⊥⊥ = |F|F III

rr.

IIFr

is found by raising a perpendicular from the projection 3⊥|FI

r to the line of rod 2.

Step 2 – Rod 6 separates faces II and IV

1 2

3

4

5

6

7 Pr

oFr

IVFr

IFr

IIFr

IIIFr

1) IIFr

is known from step 1. 2) 7||FIV

r since 7 separates IV and O.

3) 66 ⊥⊥ = |F|F IVII

rr since 6 separates IV

and II. 4) IVF

r is obtained by raising up a

perpendicular from the projection 6⊥|FII

r to the line of rod 7.

IIFr

IVFr

Step 3 – Face III is adjacent to faces IV and I

1 2

3

4 5

6

7 Pr

oFr

IVFrIF

r

IIFr

IIIFr

1) IFr

and IVFr

are known from previous steps.2) 55 ⊥⊥ = |F|F IIIIV

rr since 5 separates faces IV

and III. 3) 44 ⊥⊥ = |F|F IIII

rr since 4 separates faces I and

III. 4) IIIF

r is found in the intersection of the

perpendiculars to the projections 5⊥|FIII

r

and 4⊥|FI

r.

IVFr

IIIFr

Table 3. Analyzing simple determinate truss using the dual vector resolution method.


11

Having found all the face forces of the truss, the forces in the truss rods can be found by simply subtracting the face forces of the two faces separated by a rod.

2.4 Applying Machine theory methods to checking stability of trusses.

There are many cases, in which it is not sufficient to use the mobility criterion (Timoshenko and Young 1965) in order to verify the stability of determinate trusses. In some cases, despite the fact, that the mobility criterion returns zero degrees of freedom, the truss may still be unstable. This may happen due to both topological and geometrical factors.

In this section it is suggested to employ the duality relation between trusses and mechanisms, thus transforming the problem of checking the truss stability to problem of checking the mobility of its dual mechanism. This issue is based on the main outcome of the duality relation saying that the result of the static analysis of a truss is precisely the result obtained from the kinematical analysis of its dual mechanism. From the fact that the static (kinematic) analysis of the truss (mechanism) is a necessary and sufficient criterion for the validness of trusses and mechanisms, the following rule is derived:

The dualism validity rule: determinate truss is valid if and only if its dual mechanism is valid, in other words: determinate truss is stable if and only if its dual mechanism is in a mobile position.

Hence, instead of checking the stability of the truss, one can build its dual mechanism and to check its mobility. In many cases, checking the instant mobility of mechanisms can be carried out more efficiently, since it enables one to employ known theorems from machine theory.

Consider as an example the two trusses presented in Figures 8a and 8b.


12

P'Vrr

=

Pr

1 ’

2 ’

3 ’

8 ’

11 ’

10 ’ 9 ’

7 ’ 6 ’

5 ’ 4 ’

R’

P’

R’

1 '

2 '

3 '

11 ’

8 ' 10 ' 9 '

7 ' 6 '

5 ' 4 'PV 'P

rr=

( a) ( b)

1

2

3

8

11

10 9

7 6

5 4

1

2

3

8

11

10 9

7 6

5 4

Pr

( c) ( d)

P'

Figure 8. Example of trusses, whose stability is checked using their dual mechanisms.

(a,b) The primal trusses (c,d) and their dual mechanisms.

Only an exhaustive analysis procedure can reveal whether these two trusses are stable or not. On the other hand, the dual mechanism of Figure 8c is obviously locked, since links 1’ and 3’ are on the same line. Thus, from the dualism validity rule it follows that the primal truss is mobile, i.e. not stable. On the contrary, in the mechanism in Figure 8d, the links 1’ and 3’ are not located on the same line, therefore it can move, which means that their primal truss (Figure 8b) is stable.

3 USING METHODS FROM STRUCTURAL ANALYSIS IN MACHINE THEORY.

In the latter section, methods from machine theory were applied to structural analysis. Since the duality relation between trusses and mechanisms is mutual, methods from structural analysis can be as well applied to machine theory, as is done in the current section.


13

3.1 Henneberg’s method applied to machine theory.

Henneberg’s method (Timoshenko and Young 1965) is a known method in structural analysis that enables to facilitate the static analysis process of compound and complex trusses. The idea of this method is to transform a complex truss into two simple trusses, and from the analysis of the these two trusses to obtain the solution of the original one.

Current section introduces the method dual to this method, this time applied to analyze the known mechanism called: “Stephenson mechanism type II”, which is widely used for different industrial purposes (Erdman and Sandor 1997).

Figure 9 presents the original truss and its dual mechanism.

Pr

10

97 6

54

3

2

(a)

PV '1

rr=

10’

9’

7’

5’

4’

3’

2’

1’

(b)

6'

Figure 9. The truss and its dual mechanism to be analyzed with the Henneberg’s method. (a) Original complex truss. (b) The dual mechanism (Stephenson type II mechanism).

At the first stage of the solution, one of the rods in the truss designated by t and referred here as the 'transformed rod’, is removed, another rod designated by v and called ‘the virtual rod’ is inserted instead at such a location, so that the truss becomes simple. Since the transformed truss is simple, it follows that its dual mechanism is decomposable into dyads. A possible transformation is shown in Figure 10. The rod 9 (transformed rod) was removed and rod v was added instead, creating a simple transformed truss whose corresponding dual mechanism is accordingly decomposable into dyads. Now the analysis can be performed either on the mechanism or on the truss – both are simple. The resulting forces (velocities) are designated )'V('F ii , whereas

'V'F ii = due to the duality property.


14

P'Prr

=

6’10

v7

6

5 4

3

2

10’

v’

7’

5’

4’

3’

2’

1’

(a)

(b)

P'V 1

rr=

Figure 10. The first transformation of Henneberg’s method and its dual mechanism.

a) Simple truss. b) The dual mechanism.

The second step is to remove the original external force from the truss and to insert back the transformed rod 9. Then the calculation is performed again, while this time a unit internal force is applied in the transformed rod. The transformation and its dual mechanism are presented in Figure 11. Now, the dual mechanism has a new driving link, labeled 9”, its relative velocity also equals unity. In this configuration the mechanism can also be decomposed into dyads. The resulting forces/velocities are designated ''V/''F ii , whereas ''V''F ii = again due to the duality property.


15

10

v

7

6

5 4

3

2

(a)

1F"9 =

(b)

6’

10’

v’

7’

5’

4’

3’

2’

9’1"

'9V =9

Figure 11. The second transformation of Henneberg’s method and its dual mechanism. a) The transformed truss with a unity force in its transformed rod 9. b) the dual mechanism.

According to the Henneberg’s method the forces in the primal truss are calculated by the following equation:

0''F0'F''F'F''F'FF 1t

v

viii ==−=αα+=

Accordingly, the relative velocities in the dual mechanism are calculated by the same equation, as follows:

0''V0'V''V'V''V'VV 1t

v

viii ==−=αα+=

In conclusion, we have obtained a method, by which a mechanism known to be non-decomposable into dyads, is analyzed as a linear combination of solutions of two simple mechanisms (decomposable to dyads). This idea was derived on the basis of the example of Stephenson type II mechanism and can be applied to any of its positions and geometries. In the same way the method can be developed for a very wide range of compound and complex mechanisms.

3.2 Checking the mobility of mechanisms by structural analysis methods

Subsection 2.3 has employed the principle, that a truss is valid if and only if its dual mechanism is also valid. The principle was used for checking the stability of trusses by checking the mobility of their dual mechanisms instead. Current subsection demonstrates the converse possibility: it checks the mobility of a mechanism, through the stability of its dual truss. Consider for example the mechanism presented in Figure 12:


16

'PV11

rr= 11

8 10

9

7

6

1232

14 15

1

4 13

5

11V'Prr

=

8’

11’

7’

14’

15’

12’

10’9’

6’

4’5’ 1’

3’2’

13’

(a) (b)

Figure 12. Example of revealing a mechanism mobility problem through the topology of its dual truss.

a) A mechanism in a locking position due to geometrical reasons (b) The dual truss whose non-stability is due to topological reasons.

It is very difficult to decide whether the mechanism in Figure 12a is mobile or locked, while for the dual truss (Figure 12b) it is obvious to deduce that it is not stable by just checking its topology. This truss obviously possesses redundancy in its internal part (rods 1',2',3',4’,5',6'), so its external part lacks rods and hence the whole truss is unstable. Thus, employing the dual validity rule yields that the original mechanism is not valid, i.e. locked. This is one of the interesting cases where a geometric problem in the mechanism is transformed into a topological problem in the dual truss.

4 DESIGN OF ENGINEERING SYSTEMS USING THE DUALITY RELATION.

The duality connection between mechanisms and trusses can be applied for synthesis of new engineering systems. The main idea behind this approach lies in the fact that if a mechanism possesses some special engineering properties, then its dual truss possesses the exact same properties. In the following example the idea is employed to solve a static design problem.

Suppose one needs to design a static system, such that when a small force is applied to one of its joints, a much greater force is produced in one of its rods. Such a static system can be obtained immediately by using the duality between trusses and mechanisms. This is done by first finding a known mechanism having similar velocity characteristics, namely, a mechanism that for a small relative velocity in its driving link produces in its other link a much greater relative velocity. One of many known mechanisms satisfying this requirement is presented on Figure 13a. The velocity of link 1 of this mechanism is considerably larger than that of the link 5. The truss dual to this mechanism is presented in Figure 13b. According to the duality property, the truss possesses the same force characteristics as the velocity characteristics of the mechanism, i.e. a small external force P causes a much greater force in rod 1.


17

'1V

O1

O

'V5

Pr

The correspondingPGR

The correspondingtruss

Primal problem: design a static system such that the forceacting in rod 1 is highly amplified relative to the external force P.

Pr

The dual problem: find a kinematicalsystem such that the ratio of relativevelocities of its two links is large.This is a known problem in themechanism community, and one of theknown mechanisms is:

(a)(b)

(c)

(f)

(e)(d)

PGR and FGRare dual

Mechanisms anddeterminate trussesare dual

A1'

O'

31'

2

B 4'

5'C

C

O

B

A3

A2

II'I'

1'

4'5'

31'

32'2'

O'

I II

1

5

4

31

2 32O

I II

1

5

4

31

232

31

1 2

32

5

4

1

32'

O'

FCFS

PCPSFCFS

Figure 13.Driving a static system with special properties from a known mechanism.


18

5 CONCLUSIONS

The paper has introduced some of the theoretical and practical contributions of the dualism relation between mechanisms and determinate trusses. It was shown that this dualism enables to reveal new connections between methods in kinematics and statics; to derive new methods in a systematic way; to reveal special hidden properties through the dual system, where these properties are transparent; establishing a new idea for a systematic method for design of trusses and mechanisms.

The results that appear in this paper are due only to a connection between the two combinatorial representations corresponding to mechanisms and determinate trusses. In the general research encompassing this connection, there are more combinatorial representations whose interconnections are being investigated. Therefore, it is expected that the approach will be expanded and new connections between engineering fields will be systematically revealed, thus giving raise to additional practical and theoretical contributions in engineering analysis and design.

References Balabanian, N. and Bickart, T. A., Electrical Network Theory, John Wiley & Sons, NY, 1969.

Erdman A.G. and Sandor G.N., Mechanism Design – Analysis and Synthesis, Vol. 1, Prentice-Hall International, Third edition, London, 1997.

Hibbeler R.C., Structural Analysis, Macmillan Publishing Company, NY, 1985. Manolescu N.I., “For a United Point of View in the Study of the Structural Analysis of

Kinematic Chains and Mechanisms”, J. Mechanisms, 3, pp. 149-169, 1968.

R. L. Norton, Design of Machinery, McGRAW-HILL, 1992.

Shai O., “The duality relation between mechanisms and trusses”, Mechanism and Machine Theory, Vol. 36, No. 3, pp. 343-369, 2001(a).

Shai O., “The Multidisciplinary Combinatorial Approach and its Applications in Engineering”, AI for Engineering Design, Analysis and Manufacturing, Vol 15, No. 2, pp.109-144, April, 2001(b).

Shai O., "Combinatorial Representations in Structural Analysis", Computing in Civil Engineering, Vol. 15, No. 3, pp. 193-207, July, 2001.

Shai O. and Preiss K., “Graph Theory Representations of Engineering Systems and their Embedded Knowledge”, Artificial Intelligence in Engineering, Vol. 13, No. 3, pp. 273-284, 1999.

Swamy M.N. and Thulasiraman K., Graphs: Networks and Algorithms. John Wiley & Sons, NY, 1981.

Timoshenko, S.P. and Young, D.H.,. Theory of Structures. second edition, McGraw-Hill, 1965.

published in mechanism and machine theory, vol. 37, no. 11 ...shai/pdfs/dual2.pdf · utilization of...

Documents