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Abstract Positioning systems for machine tools are generally driven by ball screws due to their high stiffness and low sensitivity to external perturbations. However, as modern machine tools increase their velocity and acceleration of positioning, the resonant modes of these systems could be excited degrading the trajectory tracking accuracy. Therefore, a dynamic model including the vibration modes is required for machine design as well as for controller selection and tuning. This work presents a high-frequency dynamic model of a ball screw drive. The analytical formulation follows a comprehensive approach, where the screw is modeled as a continuous subsystem, using Ritz series approximation to obtain an model. Based on this model, the axial and angular components of each mode function are studied for different
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transmission ratios to determine the degree of coupling between them. After that, the frequency variation of each mode was studied for different carriage positions and different moving masses. Finally, an analysis of these results applied to controller design and parameter estimation is also presented.21954
Keywords Feed drive • Ball screw • Vibration modes • Machine tools
1 Introduction
High-performance feed drives are required in modern machine tools to achieve high-quality parts and cycle time reduction [1]. Ball screw feed drives are widely
used for positioning due to their relative low cost and low sensitivity to external forces and inertia variations, compared to linear motors. However, the high-frequency dynamics of a ball screw must be considered for high accelerations, where the system modes can be excited degrading the position accuracy. This is the case of high speed machining, where the feed between the cutting tool and the workpiece increases proportionally to the increased spindle speed [2]. This represents a problem, particularly in machining parts that require short and repetitive movements resulting in demanding acceleration profiles. Therefore, traditional models must be augmented with higher-order dynamics to assist the controller design.
Since low-order models are preferable for controller design and tuning, a model with 2 degrees of freedom can be used, which is able to capture the rigid body mode and the first vibration mode with some degree of accuracy [3–5]. This kind of model is useful when the position loop is closed with the direct carriage position. However, higher resonant frequencies become important when closing the loop with rotary sensors at the motor position [2, 6].
On the other hand, modern tendencies of virtual machine design require more accurate models to predict the system dynamics, in order to anticipate its interaction with the machining process [7]. In this sense, an accurate model for a positioning system must include high-frequency vibration modes in order to capture the interaction of design parameters and operating conditions with respect to dynamic response.
A comprehensive model was presented by Varanasi [3], where the screw was considered as a distributed parameter system in which the axial and torsional deformation fields were simplified to vary linearly with the axial coordinate of the screw. However, the model includes only two generalized coordinates, making it
suitable for specific control purposes but not allowing higher-frequency analysis.
Smith [2] modeled the ball screw with finite element method (FEM) beam formulation to draw conclusions about the behavior of the system in each mode
and to predict the vibration frequencies. Alternatively, Erkorkmaz and Kamalzadeh [6] measured screw torsional deformations at particular locations to experimentally verify the mode shapes prediction from a FEM. The authors of both works agree that the first mode is mainly axial whereas the second and third are mainly torsional. However, these conclusions are drawn for their particular systems under a fixed operating condition.
Chen et al. [8] used a mechanical model of a ball screw to study the contouring error due to the compliance effect. The model has 5 degrees of freedom, and it predicts the rigid rotation, axial, and torsional vibration of the ball screw and axial and rotational vibration of the whole table. Although the model is suitable for the particular analysis done by the authors, the formulation uses only lumped parameters, and it does not predict the screw deformation in each mode. In addition, the table rotation becomes important only when the machining area is at a high vertical distance with respect to the table base.