Modeling and Simulation on Errors of Feed Unit by Considering Change of Force Bearing Point

Abstract: The linear feed unit is a type of precision linear motion component that is widely used in computer numerical control (CNC) machine tools. The contact stiffness and error influences the performance of the feed unit directly. Thus, investigating contact stiffness and error is important in optimizing the design and improving the performance of the linear feed unit. In this study, the contact mechanics and the deformation of the roller between the roller and rail are analyzed. Calculation model of the contact stiffness and error based on the Hertz theory and multi-body kinematics are established and the change of the contact angle is also considered. Errors and the contact stiffness curve of five directions and the changes of the slope of the stiffness curve after the load increases to a certain size are obtained. The Motion precision errors of The roller linear feed unit are analyzed. The effectiveness of the proposed models on contact stiffness and error is verified through simulation on a specialized test system of the linear feed unit.


Introduction
rigidity and is applicable to large CNC machine tools and machining centers. In recent years, the growing trend of high speed, high precision, and high durability in CNC machine tools has propelled high performance and reliability to be the most important index for linear feed unit. Thus, in-depth researches on the dynamic characteristics of linear feed unit are greatly needed. This research can lay a good foundation for enhancing the performance of linear feed unit.
Existing researches on the dynamic characteristics of linear feed unit were focused on contact stiffness, vibration during operation, and friction characteristics of roller linear guides. Wu [1,2] and Liu [3] studied the contact stiffness between the roller and raceway and its related influence factors were analyzed and calculated based on the Hertz theory. In addition, vibration synthesis and prediction by the method of employing the finite element were performed. After Shimizu's study [4,5] on the characterization of frictional hysteresis in roller-bearing guide ways, In article Jui Pin Hung [6], the linear rolling guide under different loading finite element model was established based on Hertz contact theory, the influence of external load to guide stiffness was studied and the finite element model was modified based on test results. In Wei Jun T [7], the relationship of the force between roller and raceway was analyzed and the stiffness model of horizontal, vertical and deflection angle around the spindle rotational was built based on Palmgren formula. The hertz contact model of roller among nails of linear roller guide is established firstly. Then the coordinate transformation matrix between the two coordinate systems is obtained with multi-body kinematic theory. Due to the analysis the stress of the roller linear motion unit, five mechanical equilibrium equations are obtained, then the amount of deformation of its five directions is determined. Thereafter, the analysis of roller linear feed units is simulated based on contact stiffness model proposed and the motion precision errors of the roller linear feed unit feed unit are analyzed. Finally, the proposed models are simulated by computer.

Contact stiffness of linear feed unit
The roller linear feed unit (hereinafter referred to as the table) contains four sliders; each slider also contains a number of rollers. The feed unit produces three rotation errors and two straightness errors with the role of the force of three directions, which are represented respectively as: δy δz εx εy εz. δy δz are the deformation along Y and Z axis respectively, and εx εy εz are the deflection angle around the X,Y, Z axis respectively.

Figure.1 Coordinate system of Linear feed unit
The transformation matrix 0 T 1 from the coordinate system of O 0 to the coordinate system of O 1 is: In the rolling linear guide, the roller, the slider and the track have contact surface. When the slider is subjected to external load, the contact stress and the elastic deformation are generated in the contact area at the same time. Under the condition of applying the pre pressure, the cross section of the slider 1 is analyzed. In Figure 2, the position of each column roller in slider and force are showed below. ( Where, n is the number of every column of guide. By the coordinate conversion, the coordinates obtained in the coordinate system can be expressed as equation (4): Also the curvature center coordinate of the j-th roller in the i-th column in the coordinate After receiving the loading, the distance between the curvature center of the slider and the

IC4M & ICDES 2017
12 curvature center of the guide is expressed below equation (6): The elastic deformation of each roller is equation (7): The model of contact force and roller elastic deformation base on hertz theory are showed below: 2 3 =KQ G 8 Where, K is parameters related to the curvature. The contact angle of the roller after deformation is:   4. When the load moves far from the geometry center of the linear feed unit, three rotational errors and two straightness accuracy errors all become bigger in different degrees.