Amatching Methodology For Static And Dynamic Stiffness Of Transmission Chain Employed In Rotary Table Of CNC Machine Tools

As one of the key component of CNC machine tools, the performance of rotarytable plays a significant role in a whole machine tool’s performance and consequentlyin part’s quality. To improve the static and the dynamic rotary feeding accuracy of arotary table, a matching methodology for both static and dynamic stiffness in rotaryfeeding direction of a transmission chain is proposed in this work. The followingcontributions are included:Aunified static torsional stiffness model for the transmission chain used in rotarytable is established. After extracting the expressions for stiffness of eachcomponent in the chain, general formulations relating adjacent angulardisplacements are derived; according to the definition of static stiffness, underthe equilibrium conditions, the unified models for typical transmission chainsemployed in rotary table are presented. Comparisons with the results from finiteelement simulations indicate the validity of the models. The proposed stiffnessmodels could be directly used to estimate static stiffness of transmission chaininstead of FEM simulation and expriment.Based on the mapping between the static stiffness and design parameters,analytical expressions for sensitivity analysis are derived by conductingcontinuous differentiating operations. The validity of the analytical model isverified by comparing results from the difference method. The presentedanalytical models for sensitivity analysis could be utilized to find out thecomponent with low stiffness, by which modifications could be made, andtedious computation through stiffness model could be avoided.In the view of design, a matching rule for static stiffness of transmission chain isproposed. The influences of meshing stiffness and torsional stiffness on thestiffness of a transmission chain are revealed through conducting sensitivityanalysis. Then the rule that transmission ratio is allocated firstly and nexttorsional stiffness of shafts has to be checked is proposed. Moreover, a matchingcase is studied and the resultant stiffness is satisfied, which indicates thefeasibility of the matching rule.A matching method for dynamic stiffness is also proposed. In terms ofexpressions for multi-degree-of-freedom system with modal coordinates, the most sensitive component to natural frequency could be found out by analyticalsensitivity method. Based on simulation results, static stiffness matching shouldbe executed firstly, by which an area where shafts’ parameters satisfy staticstiffness requirement could be specified; then a matching chart between thedesign parameters and the first-order natural frequency could be established fromlow-speed stage to the higher one, by which parameters of each shaft could bedetermined. Case studies verify the proposed matching strategy.Optimizations are conducted to minimize volume of the transmission. Taking thevolume of a transmission as the objective function, both static and dynamicstiffness as constraints, a light-weight optimization problem is formulated andsolved.A measurement method for torsional stiffness of rotary table is proposed. Theresults from both experiment and simulation agree well, which indicates thefeasibility of the proposed method.The above achievements provide theoretical supports for the digital design ofstiffness in rotary feeding direction and also the matching design of stiffness ofmachine tools.