An Investigaton of Minimum-Weight Dual-Material Symmetrically Loaded Wheels and Torsion Arms

A cylindrically symmetric layout of two opposite families of logarithmic spirals is shown to define the layout of minimum-weight, symmetrically loaded wheel structures, where different materials are used for the tension and compression members, respectively; referred to here as dual-material structures. Analytical solutions are obtained for both structure weight and deflection. The symmetric solutions are shown to form the basis for torsion arm structures, which when designed to accept the same total load, have identical weight and are subjected to identical deflections. The theoretical predictions of structure weight, deflection, and support reactions are shown to be in close agreement to the values obtained with truss designs, whose nodes are spaced along the theoretical spiral layout lines. The original Michell solution based on 45 deg equiangular spirals is shown to be in very close agreement with layout solutions designed to be kinematically compatible with the strain field required for an optimal dual-material design.