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A kinematic coupling design incorporates an array of concepts and equations. One of the key fundamental points is exact constraint that goes back to 1880's with James Clerk Maxwell and anecdotal evidence suggests potentially 30 years before with Profl Robert Willis. There are several books that discuss exact constraint design. One option is to read Exact constraint: Machine Design Using Kinematic Principles by Douglass L. Blanding. Now given a kinematic design, the process consists of five steps: 1. Hertzian Contact 2. Force & Moment Equilibrium 3. Solve Systems of Equations 4. Calculate Errors of Motions 5. Homogeneous Transformation Matrices Note that all of these concepts and equations are covered in depth in Precision Machine Design by Alexander Slocum
When repeatibility is crucial, kinematic interfaces can be essential!
Although kinematic constraints / kinematic couplings existed since at least the 1880s, a closed form analysis was first published in 1988 calculating performance for high load applications (Slocum 1988). A second paper in 1992 (Slocum 1992) outlines the six equations for static equilibrium for the kinematic coupling configuration. There system of six equations is the sum of the forces in three directions and the sum of the moments in three directions resulting in six equations. Equation
Solve System of Equations The force moment equilibrium system of equations is organized in the form A · F = B where A is a matrix 6x6 for the geometry configurations (see PMD pg 405 or 1992 Slocum Paper), F are the unknown six forces at the contacts, and B are the loading parameters (both preload and load). Using linear algebra one can solve for F, the six orthogonal forces to the contact points. Equation
Calculate Contact Areas & deflection For each orthogonal force at the contact, calculate the contact area ellipse using the hertzian contact elliptical coefficients. The λ coefficient is used to calculate elastic deflection. Equation
Calculate Contact Pressures & Max Shear Using the calculated contact area for each contact, the contact pressure q can be calculated as well as the stress (shear and normal). Read PMD pg 228-232 for details on material limitations considerations. Equation
Depth of Maximum Shear The depth of maximum shear z is a function of the equivalent contact radius (a) and the poisson value η of the material. Equation
Build HTM A homogeneous transformation matrix (HTM) is used to account for the displacements at the contact points. PMD pg 407-408. Given a point of interest, one can calculate the expected error based on all of the parameters of the kinematic coupling.
Solution Via Excel The excel analysis worksheet created by Prof. Slocum is available on this website for free. Currently the excel is configured for a planar equilateral triangle. Download Excel
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