Growing super stable tensegrity frameworks

This paper discusses methods for growing tensegrity frameworks akin to what are now known as Henneberg constructions, which apply to bar-joint frameworks. In particular, the paper presents tensegrity framework versions of the three key Henneberg constructions of vertex addition, edge splitting and framework merging (whereby separate frameworks are combined into a larger framework). This is done for super stable tensegrity frameworks in an ambient two or three-dimensional space. We start with the operation of adding a new vertex to an original super stable tensegrity framework, named vertex addition. We prove that the new tensegrity framework can be super stable as well if the new vertex is attached to the original framework by an appropriate number of members, which include struts or cables, with suitably assigned stresses. Edge splitting can be secured in R2 (R3) by adding a vertex joined to three (four) existing vertices, two of which are connected by a member, and then removing that member. This procedure, with appropriate selection of struts or cables, preserves super-stability. In d dimensional ambient space, merging two super stable frameworks sharing at least d + 1 vertices that are in general positions, we show that the resulting tensegrity framework is still super stable. Based on these results, we further investigate the strategies of merging two super stable tensegrity frameworks in IRd; (d 2 f2; 3g)that share fewer than d + 1 vertices, and show how they may be merged through the insertion of struts or cables as appropriate between the two structures, with a super stable structure resulting from the merge

Theoretical Approaches in Non-Linear Dynamical Systems

From Preface: The 15th International Conference „Dynamical Systems - Theory and Applications” (DSTA 2019, 2-5 December, 2019, Lodz, Poland) gathered a numerous group of outstanding scientists and engineers who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without great effort of the staff of the Department of Automation, Biomechanics and Mechatronics of the Lodz University of Technology. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our event was attended by over 180 researchers from 35 countries all over the world, who decided to share the results of their research and experience in different fields related to dynamical systems. This year, the DSTA Conference Proceedings were split into two volumes entitled „Theoretical Approaches in Non-Linear Dynamical Systems” and „Applicable Solutions in Non-Linear Dynamical Systems”. In addition, DSTA 2019 resulted in three volumes of Springer Proceedings in Mathematics and Statistics entitled „Control and Stability of Dynamical Systems”, „Mathematical and Numerical Approaches in Dynamical Systems” and „Dynamical Systems in Mechatronics and Life Sciences”. Also, many outstanding papers will be recommended to special issues of renowned scientific journals.Cover design: Kaźmierczak, MarekTechnical editor: Kaźmierczak, Mare