Research Program

Research Program of Prof. Offer Shai

Generally in engineering, including mechanical engineering, engineers invest significant efforts in designing new products, developing new methods, or revealing new concepts that potentially could be equivalent to existing devices, methods or theorems, already developed and available in other engineering domains. Unfortunately, they are unaware of that, as they lack tools necessary to reveal the equivalence. We often refer to it as “re-inventing the wheel.” Such equivalences between devices, methods, and concepts, although robust from a mathematical perspective, are usually difficult to detect or perceive even for experienced engineers or engineering researchers.

The vision behind my research is to provide a well-defined mathematical platform capable of supporting the analysis and design in several branches of engineering.

Discrete mathematics, in particular, graph and rigidity theories, was found fit to serve as a central tool underlying this platform. In the course of the theoretical development of this research, several important practical results were systematically devised in a number of engineering fields.

To achieve this long term vision, I have first developed combinatorial representations based on discrete mathematics with which different engineering domains and systems can be represented. Through these representations, I have succeeded to achieve the following diverse results:

(a) Developing new relations between existing engineering domains. A new relation between mechanisms and trusses was established in 2001 ([5] and [9]), which shown that for every mechanism there exists a corresponding determinate truss where the relative linear velocities of the former are the counterparts of the forces in the rods of the latter. Other dualities have also been revealed between gear trains and beams and between spatial linkages and multi platform robots [17].

(b) New perspectives on different engineering domains were created where the same graph representation has been applied simultaneously to two different engineering domains. For example, due to the existence of the same representation for both electricity and topics in structural mechanics, it was revealed that Betti's law and other known methods in structural mechanics are equivalent to Tellegen's theorem from electricity [8]. In addition, from the view point of matroid theory, the two known methods in structural analysis – force and displacement methods – were proved dual [7] .

(c) Entire new perspective on engineering problems was achieved when additional mathematical knowledge was augmented for solving engineering problems. For example, the topological synthesis of all the different topologies of the 2D linkages and determinate trusses was exposed (conference [42]) by relying on theorems from rigidity theory. The latter paper has received the A.T. Yang Memorial ASME Award in 2010.

(d) New concepts in engineering were exposed. In the classical well established domain of statics, a new entity – Face Force – was revealed and found to be the missing counterpart element of the absolute linear velocity in kinematics [9]. The exposure of new concepts has yielded the derivation of other new ones. For example, while searching for the locus of points upon which the face force acts, another new concept in statics – an equimomental line – has been revealed. The latter concept was proved to be the counterpart element of the instant center in kinematics [18] for which Prof. Pennock and I have received the A.T. Yang Memorial ASME Award in 2005.

(e) New methods were developed while working with these combinatorial representations. For example, due to the duality between statics and kinematics, methods from statics were transformed into kinematics yielding new methods, and vice versa [9]. A new method for finding the dead point positions of linkages relying on the new concept face force [19], new method for the synthesis of tensegrity systems relying on the equimomental lines (conference [43]) and additional synthesis methods appearing in [21]. With the collaboration of Prof. Reich, a new methodology in design, termed infused design, was developed, enabling the designer to use knowledge and known devices in other domains for designing a new device in his/her domain [13,14]. The latter methodology was also extended for creativity[22].

(f) Harnessing these theoretical concepts back to practical engineering. For that purpose, I selected adjustable deployable tensegrity structures and their applications as a driving product for demonstrating the value of the aforementioned theoretical ideas. A new type of deployable tensegrity structure was invented (now being patented) in my laboratory - a structure that can be both loose and rigid when needed (conference [45]). The mathematical foundation underlying this device consists of new theorems that were developed and proved with the collaboration of two mathematicians: Prof. Whiteley and Prof. Servatius and appear this year [24, 25].

Nowadays I am working on the following topics:

(a) Developing a new method for geometric synthesis in tensegrity structures, known as form finding, for which currently there is no efficient algorithm. Preliminary results indicate that the method will be founded on the new concept of “face force”.

(b) Extending the work of the tensegrity robot (conference [45]) to obtain a model for caterpillar locomotion in collaboration with a zoologist, Prof. Ayali, and M.Sc. student, Omer Orki (preparation [4]).

(c) Establishing the “Interdisciplinary Engineering Knowledge Genome” (submitted [1]), a framework enabling to transform knowledge between different domains in a systematic way. The work is done with the collaboration of Prof. Reich.

(d) Developing a computational method for decomposing any 2d and 3d linkages into Assur Graphs using the known pebble game (submitted[1], preparation [1]). This work is done with Prof. Whiteley and his Ph.d. student, Sljoka and relying on previous results of Dr. Shai (conference [40], [48]).

(e) Extending and completing the topological synthesis of all the different topologies of linkages and determinate trusses (pinned isostatic frameworks) in 3D. This work is a result of collaborative work with Prof. Whiteley (preparation [2]).

My long term research program is to proceed as follows:

(a) Practical applications – making the mathematical transformations systematic and available for practicing engineers in industry. To achieve this, I intend to develop software that will enable engineers to use the above combinatorial representations along with their mathematical relations without having the need to carry out the laborious work of transforming between the disciplines manually. Additionally, new types of devices are expected to be developed and derived through the combinatorial transformations, part of them will be patented (two are already in process).

(b) Theoretical work – the idea is to work in two main parallel ways: to extend the work to higher dimensions, such as: face force, equimomental lines and the map of all the topologies of structures, mechanisms and tensegrity structures in three and higher dimensions, while extending the scope of this work to other domains, such as Nanotechnology and Bioengineering.

In order to make these methods available to colleagues from different engineering domains and to engineers in industry, I intend to write at least one book and teach this material in undergraduate classes. For now, the material developed in this research work has been taught in graduate courses and special courses have been given to excellent students in high schools [7].