Keywords: duality relation, combinatorial representations, graph theory, trusses, beams, pillar systems.

The approach introduced in the current paper enables to transfer knowledge from various domains of engineering to structural mechanics and vise versa. The paper is concentrated on transferring knowledge by means of a duality relation between statical and kinematical systems.

The work presented in the paper is one of the outcomes of a general research called Multidisciplinary Combinatorial Approach (MCA) [1] aimed to assist in obtaining a general perspective over engineering systems. The approach is focused on developing global mathematical models, called Combinatorial Representations and associating them with different engineering systems. Once the engineering system is associated with a specific combinatorial representation, analysis and other forms of engineering reasoning can be computerized and conducted solely upon the combinatorial representation.

The duality relations between statical and kinematical systems are established in the paper through systematic mathematical processes based on relations between combinatorial representations. The first duality presented in the paper is between determinate trusses and mechanisms [2]. It is shown that a combinatorial representation of a mechanism is a potential graph representation, whereas the representation of a truss is the flow graph representation. It is then proved that the potential and flow graph representations are mutually dual, thus making the trusses and mechanisms dual as well, as is depicted in the example of Figure 7.1(a). Another duality relation that has been recently established and will be introduced here for the first time is the duality between statical beams and gear systems (Figure 7.1(b)).

Establishing the duality between statics and kinematics provides a new channel for knowledge and information transfer, yielding immediate practical and theoretical implications, as is described below.

A new insight on the analysis process is obtained when new connections in engineering are revealed, for example, finding that Maxwell-Crimona diagram for forces in trusses and velocity polygon of mechanisms are actually dual methods.

In addition to the theoretical value provided by the duality relation, the paper outlines its practicality to engineering research in general and structural mechanics in particular. To emphasize this issue, a new method for decomposing a truss into special groups will be derived on the basis of Assur group decomposition [3] known in machine theory.

Further importance of the duality relation is demonstrated by developing rules for validity checking. It is shown that methods for checking the mobility of mechanisms are applicable to checking the stability of trusses and vice versa. One of the most promising practical applications of the approach is opening a new avenue of research for facilitating the process of design of engineering systems. Employing the dualism, civil engineer faces an opportunity to search for a solution to his problem also among known mechanisms [4]. Once a mechanism performing the desired task is found, the designer is just left to transform it to a corresponding static system.

1

O. Shai, "The Multidisciplinary Combinatorial Approach and its Applications in Engineering" AIEDAM - AI for Engineering Design, Analysis and Manufacturing, Vol. 15, No. 2, 2001. doi:10.1017/S0890060401152030

2

O. Shai, "The Duality Relation between Mechanisms and Trusses", Mechanism and Machine Theory, Vol. 36(3), pp. 343-369, 2001. doi:10.1016/S0094-114X(00)00050-1

3

N.I. Manolescu, "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. doi:10.1016/0022-2569(68)90353-4

4

O. Shai, "Utilization of the Dualism between Determinate Trusses and Mechanisms", accepted for publication, Mechanism and Machine Theory, February, 2002. doi:10.1016/S0094-114X(02)00038-1

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