Vol. 6, No. 1-4, 2011

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ISSN: 1559-3959
A remarkable structure of Leonardo and a higher-order infinitesimal mechanism

Tibor Tarnai and András Lengyel

Vol. 6 (2011), No. 1-4, 591–604
Abstract

This paper is concerned with the static and kinematic behavior of two chain-like bar-and-joint assemblies which have the same topology. One is a structure which is both statically and kinematically indeterminate, and constitutes a higher-order infinitesimal mechanism. The other is a structure which is both statically and kinematically determinate, introduced by Leonardo da Vinci in the Codex Madrid. Proceeding along the internal joints from the bottom to the top of the assembly, the lateral components of the displacements of the internal joints of the infinitesimal mechanism show an exponential decay, and the forces in the internal bars of Leonardo’s structure show an exponential growth. It is pointed out that, in the elastic model of Leonardo’s structure, the propagation of displacements of internal joints and the propagation of forces in internal bars also show an exponential character in a modified form. This work also provides some hints for overcoming difficulties arising in higher-order infinitesimal mechanisms, and corrects minor mistakes made by Leonardo.

Dedicated to the memory of Marie-Louise Steele

Keywords
bar-and-joint structure, both statically and kinematically indeterminate structure, rigidity, infinitesimal mechanism, force propagation, Leonardo da Vinci
Milestones
Received: 29 March 2010
Revised: 29 June 2010
Accepted: 7 July 2010
Published: 28 June 2011
Authors
Tibor Tarnai
Department of Structural Mechanics
Budapest University of Technology and Economics
Müegyetem rkp. 3
Budapest
H-1521
Hungary
András Lengyel
Department of Structural Mechanics
Budapest University of Technology and Economics
Müegyetem rkp. 3
Budapest
H-1521
Hungary