Abstract
Tensegrity structures are classified as prestressed pin-jointed structures, and they have distinct properties compared to other pin-jointed structures: (1) they are free-standing, without any support; and (2) they have both tensile and compressive members. Prior to further studies on tensegrity structures in the following chapters, this chapter presents the formulations of (self-)equilibrium for general prestressed pin-jointed structures. The equilibrium equations are formulated in two ways: (1) using the equilibrium matrix associated with prestresses or axial forces, and (2) using the force density matrix associated with nodal coordinates. Conditions for static as well as kinematic determinacy of a prestressed pin-jointed structure are then given in terms of rank of the equilibrium matrix. Furthermore, the non-degeneracy condition for a prestressed free-standing pin-jointed structure is presented in terms of rank deficiency of the force density matrix.
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Notes
- 1.
‘Prestressed’ means that prestresses are introduced to the structure a priori. Prestresses are the internal forces in the members when no external load is applied.
- 2.
Definition of member directions is not unique. Using opposite definition necessarily leads to the same equilibrium equation.
- 3.
The existence of unique solution is subjected to independence of the equilibrium equations.
- 4.
Rank and null-space of a matrix \(\mathbf{D}\) can be found by using, for example, the commands \(rank(\mathbf{D})\) and \(null(\mathbf{D})\) in Octave or Matlab, respectively.
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Zhang, J.Y., Ohsaki, M. (2015). Equilibrium. In: Tensegrity Structures. Mathematics for Industry, vol 6. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54813-3_2
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DOI: https://doi.org/10.1007/978-4-431-54813-3_2
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