The consequences for the static and dynamic system behaviour of misalignment in an overconstrained direction are analysed. Therefore, a relatively simple parallel leaf spring guidance, which is overdetermined only once, serves as a case to gain insight. A multibody program using flexible beam theory is used to determine the change in vibration mode frequencies and stiffnesses due to misalignment. A previously developed beam element for modelling the leaf springs is shown to be able to describe these phenomena with a limited number of elements. Buckling loads and associated buckling modes are also determined analytically. An instrument has been fabricated to measure the change of vibration mode frequencies due to a rotational misalignment in the overconstrained direction. Experiments using wire spark eroded leaf springs are in good agreement with the calculations. Differences between the experimental and calculated results are attributed to the imperfections in the hardware model, in particular residual stresses, and the assumptions used for the beam element in the numerical model. A small misalignment, in this case 0.8–5.8mrad, causes strong change of static and dynamic system behaviour. The decreased stiffnesses perpendicular to the compliant direction are disturbing, because these are designed to be large to enhance precision manipulation. The negative effects of overconstrained design are largest if relatively thin and wide leaf springs are used. If the misalignment is kept below 50% of the angle of which a bifurcation occurs, the overconstrained design of the parallel leaf spring mechanism does not significantly influence the system natural frequencies and stiffness.
Fig. 1 Parallel leaf spring guidance with adjustable misalignment
Fig. 2 Measured and calculated radial frequencies for the set of leaf springs with t = 0.37mm
Fig. 3 Calculated compliances for the set of leaf springs with t = 0.20 mm. Note the scale factors 1,000 for fy and 100,000 for fz.
Fig. 4 Photograph of the test system
For more information:
J.P. Meijaard, D.M. Brouwer and J.B. Jonker, Analytical and experimental investigation of a parallel leaf spring guidance, Multibody System Dynamics: Volume 23, Issue 1, 2010, pp. 77-97, doi:10.1007/s11044-009-9172-4