Annals of DAAAM for 2012 & Proceedings of the 23rd International DAAAM Symposium, Volume 23, No.1, ISSN 2304-1382 ISBN 978-3-901509-91-9, CDROM version, Ed. B. Katalinic, Published by DAAAM International, Vienna, Austria, EU, 2012 Make Harmony between Technology and Nature, and Your Mind will Fly Free as a Bird Annals & Proceedings of DAAAM International 2012 OPTIMISED KINEMATIC MOUNT CONFIGURATION FOR HIGH-PRECISION APPLICATIONS ZELENIKA, S[asa]; MARKOVIC, K[ristina] & RUBESA, J[osipa] Abstract: The generic kinematic mount design configuration, designated as the Maxwell-type kinematic mount, is constituted by three V-grooves on one end and three balls on the other so as to achieve an exact constraint of all six degrees of freedom. The analysis of this coupling configuration comprises force and moment balance equations, as well as expressions for stress- strain and error motion calculations. For a determined external load, the geometry of the mount will thus imply the loads at each groove-ball interface and the respective contact point reactions. The calculation comprises the necessity to deal with the non-linear Hertzian theory of point contacts. This work recalls the limits of applicability of the available analytical approaches for the calculation of ball-V groove couplings employed in ultra-high precision positioning. The analytical results are validated experimentally. In the whole range of elastic deformations the correspondence of the theoretical values with the experimental ones is within the intervals of uncertainty of the latter. The calculation procedure is then used to assess the optimal characteristics of a kinematic mount employed to support a large vacuum chamber of a particle accelerator facility. Stability conditions for different design configurations are established. Keywords: kinematic mount, stability, model, optimisation 1. INTRODUCTION Kinematic mounts are used in high-precision applications since they are self-locating and free from backlash, allow sub-micrometric re-positioning in static and dynamic applications, can accommodate differential thermal expansions and their behaviour can be represented in a closed form solution [1]. The main drawback of kinematic mounts is constituted by the high contact stresses that can be analysed only by employing the nonlinear Hertz theory [2]. Fig. 1. Maxwell kinematic mount The most common kinematic mount design configuration is the Maxwell-type mount constituted by three V-grooves on the support and three balls on the supported piece, so as to achieve an exact constraint of all six spatial degrees of freedom (Fig. 1). The aim of this work is to analyse the influence of mechanical parameters on the behaviour of the considered class of kinematic mounts and especially on their positioning precision and stability. An example of a mount used to support a large structure at a particle accelerator facility is then considered. Stability conditions for different design configurations are established. 2. MODEL OF BEHAVIOUR AND ITS VALIDATION The analysis of a Maxwell-type kinematic mount comprises force and moment balance equations, expressions for the calculation of stresses and deflections at the contact points and error motion calculation. Knowing the external loads (including the preload acting on the coupling) and the geometry, the loads at each groove-ball interface and the respective contact point reactions can be computed from the overall force and moment balances [1]. Hertz theory describes the nonlinear behaviour of point contacts between elastic isotropic solids loaded perpendicular to the surface, where the contact area is small compared to the radii of curvature and the dimensions of the involved bodies. The respective analytical model entails a lengthy iterative evaluation of transcendental equations involving elliptic integrals [3, 4, 5]. The approximated methods suggested in literature, where the need to calculate the elliptic integrals is obviated by introducing polynomial [1], tabular [6, 7, 8] or graphical [8, 9] approximations of the characteristic parameters, are appropriate for most high-precision applications. In fact, as depicted in Fig. 2, the introduced errors are always smaller than ± 2%. Given the small entity of the stresses and strains involved in most high-precision applications, these errors can hence be considered negligible in all but those cases where true nanometric accuracies are sought. Only in the case when the curvature of the grooves approaches that of the balls, the errors tend to become appreciable. In this case, however, Hertz theory itself starts to break down [3]. - 0319 -