Design and Analysis of Kinematic Couplings for Modular Machine ...

Figure 2.5: Quasi-kinematic contactor. Figure 2.6: Quasi-kinematic target with θ CT =90 deg.Because of the arc-shaped line contact of quasi-kinematic couplings, the exact force-equilibrium solutionis non-deterministic. To give a good approximation of the exact solution, first displacements areimposed to determine the normal contact stiffnesses, then the forces are calculated. In the case of elasticplasticcontact, nonlinear behavior due to plastic flow dictates the use of finite element analysis (FEA) toestimate a power-law force-deflection relationship for usage in the analytical stiffness relations. For FEAsimulation of contact problems, Culpepper showed that the mesh size of contacting elements should be nolarger than 5% the width of the contact region.Presentation of the exact force-deflection relationship for the spherical target and conical contactor arebeyond the scope of this thesis; however, based on an initial design geometry, straightforward calculationsshow the boundary between elastic and plastic deformation of the contacts. The force per unit length(f nYIELD ) at which plastic flow begins is:f nYIELD2.8πR e σ=y---------------------- , (2.10)E ewhere R e and E e are the equivalent radius and modulus of the contact, calculated in the traditional Hertzianfashion. Now the contact displacement that induces plastic flow is known from:29