18 The Dynamics of the Class 1 Shell Tensegrity Structure Robert E. Skelton University of California, San Diego Introduction Tensegrity Definitions A Typical Element • Rules of Closure for the Shell Class Jean-Paul Pinaud University of California, San Diego D. L. Mingori University of California, Los Angeles Dynamics of a Two-Rod Element Choice of Independent Variables and Coordinate Transformations Tendon Forces Conclusion Appendix Proof of Theorem Appendix Algebraic Inversion of the Q Matrix Appendix General Case for (n, m) = (i, 1) Appendix Example Case (n,m) = (3,1) Appendix Nodal Forces Abstract A tensegrity structure is a special truss structure. | 18 The Dynamics of the Class 1 Shell Tensegrity Structure Robert E. Skelton University of California San Diego Jean-Paul Pinaud University of California San Diego D. L. Mingori University of California Los Angeles Introduction Tensegrity Definitions A Typical Element Rules of Closure for the Shell Class Dynamics of a Two-Rod Element Choice of Independent Variables and Coordinate Transformations Tendon Forces Conclusion Appendix Appendix Appendix Appendix Appendix Proof of Theorem Algebraic Inversion of the Q Matrix General Case for n m i 1 Example Case n m 3 1 Nodal Forces Abstract A tensegrity structure is a special truss structure in a stable equilibrium with selected members designated for only tension loading and the members in tension forming a continuous network of cables separated by a set of compressive members. This chapter develops an explicit analytical model of the nonlinear dynamics of a large class of tensegrity structures constructed of rigid rods connected by a continuous network of elastic cables. The kinematics are described by positions and velocities of the ends of the rigid rods hence the use of angular velocities of each rod is avoided. The model yields an analytical expression for accelerations of all rods making the model efficient for simulation because the update and inversion of a nonlinear mass matrix are not required. The model is intended for shape control and design of deployable structures. Indeed the explicit analytical expressions are provided herein for the study of stable equilibria and controllability but control issues are not treated. Introduction The history of structural design can be divided into four eras classified by design objectives. In the prehistoric era which produced such structures as Stonehenge the objective was simply to oppose gravity to take static loads. The classical era considered the dynamic response and placed design constraints on the .
