EXPERIMENTAL ANALYSIS OF THE INFLUENCE OF BALLSCREW STIFFNESS ON THE POSITIONING PRECISION OF KINEMATIC FEED CHAINS USED ON CNC MACHINE TOOLS

Recent trends in the machine tools domain have focused on improving the manufacturing performances by increasing the feed rates and acceleration values. In this context, the stiffness of the constructive elements which form the mechanical part of the kinematic feed chains need to be studied in order to determine the machine tool dynamic behavior. This study plays a very important role in the correct dimensioning of the feed chain structure and in establishing an optimum control law for the closed-loop system. The present paper presents an experimental study which demonstrates the influence of the whole kinematic feed chain stiffness on the positioning precision of the moving table. Experimental tests were carried out on a test bed which uses the indirect position measuring system, in which case all the external disturbances affecting the mechanical part are found in the values of the positioning precision parameters.


INTRODUCTION
Recent trends in machine tools domain have focused on improving the manufacturing performances by increasing the feed rates and acceleration values.In this context, the stiffness of the constructive elements which form the mechanical part of the kinematic feed chains need to be studied in order to determine the machine tool dynamic behaviour.This study plays a very important role in the correct dimensioning of the feed chain structure and in establishing an optimum control law for the closed-loop system [1].
When designing high-performance machine tools, it is essential to take into account the kinematic feed chain stiffness, in order to obtain a high positioning precision and also high efficiency when transforming the rotational movement given by the servomotor into the linear movement of the machine tool table.Beside the stiffness characteristic of the kinematic feed chain, a major importance is given to its capacity of vibrations damping, which are created and transmitted once the stiffness is increased through different means [2,3].
Thorough studies require the analysis of this phenomena, especially in the ballscrew supporting points and also in the ballnut, in order to determine the influence of the total stiffness of the ballscrew system on the positioning precision of the moving table [4].
The influence of the ballscrew system stiffness on the positioning precision of the kinematic feed chain is found in the deadzone value.Deadzone or dead stroke terms are used for indicating a condition in which the commanded position is not reached.This phenomenon is caused mainly by the elastic deformations of the mechanical parts and the clearance between assembled parts.The deadzone is a combination of these factors.
The deadzone variation along the stroke of a numerically controlled axis represents a very important problem especially when using the indirect position measuring system [5].
The stiffness of a kinematic feed chain which uses the ballscrew mechanism as the motion transmitting system is given in proportion of 80 percent by the stiffness of this mechanism.In this case, the total stiffness of the kinematic feed chain is given by the elastic deformations of all the component parts, the stiffness of an individual element being given by the relationship between the applied load and the resulted deformation.Total ballscrew system stiffness is given by the following relationship, according to the majority of ballscrew manufacturers [6,7,8]: where: K T is the total system stiffness; K H is the ballscrew stiffness; K U represents the ballnut stiffness and K C is ballscrew bearing support stiffness.This relationship is valid only for axial loads, in order to illustrate the variation and the influence of the ballscrew system on the positioning precision of the moving table, in relation to the axis stroke.The stiffness is different in function of the bearing types on which the ballscrew is mounted.In the case of the researched axis, the ballscrew is supported by two angular contact bearings on the motor side and a polymer sliding bearing on the opposite side.In this case, the ballscrew stiffness is given by the following relationship: where: d c is the bottom screw thread diameter and l s1 is the ballnut stroke.
The deadzone value in this case will be: Having the maximum value given by the relationship: where: l s is the deadzone value, and F A is the axial force applied to the moving table.
Variations of the deadzone values along the axis stroke will result in modified transient regimes at high speeds and also for the positioning stage, these modifications deriving from the stability criterion of the kinematic feed chain.Taking into account the variations of the deadzone along the axis stroke, the dimensioning of the kinematic feed chain must consider its maximum value.Also, due to the fact that during the transient regimes, the driving torque of the servomotor is 2 to 6 times bigger than the rated torque and that the deadzone value is proportional to the driving torque, high stiffness of the mechanical structure of kinematic feed chains is required.

EXPERIMENTAL DESIGN AND SETUP
Experimental researches have been conducted on a test stand, having the structure presented in Figure 1.The kinematic feed chain is mainly composed of the electromechanical actuator 1, having the rod 5 fixed against the moving element 8.The table has the dimensions of 250x400 mm and the axial guiding is achieved using two linear rolling cylindrical guide rails 4, with the dimensions of Ø 20x1000 mm, provided with several supporting elements 3, with the purpose of assuring a high stiffness of the table-guideway assembly.
The structure of the electromechanical actuator is given in Figure 2. The main components are represented by the precision ballscrew 4 (precision class 7, in conformity with ISO 3408), having the diameter of 20 mm and 5 mm pitch.The ballscrew is supported by two sets of angular contact ball bearings 3 on the motor side.Using this type of bearings allows the development of high axial forces, reaching a maximum of 9300 N, in both moving ways and also leads to obtaining a minimum backlash when reversing the moving direction.On the side opposite to the motor, the ballscrew is supported by a polymer sliding bearing 7, which has the advantages of high service life and vibration free functioning.The ballscrew mechanism is actuated by an AC servomotor 1, MH10560089192I65A74 series, having a rated torque of 1.4 Nm and which is electrically controlled by a servo drive.The servomotor and the ballscrew are coupled using an elastic coupling 2. The double preloaded ballnut 5 is fixed on the movable rod 6, supported by a radial sliding bearing 8, which allows taking over radial forces as high as 100 N.

DESCRIPTION OF THE EXPERIMENTAL METHOD
In order to study the influence of the ballscrew mechanism stiffness on the positioning precision of the moving table, an elastic element was introduced in the mechanical structure of the kinematic feed chain, under the form of a disk spring, having the dimensions given in Figure 3.For determining the influence of the elastic deformation of the disk spring on the ballscrew mechanism stiffness, under a given axial load, the loaddeformation curve was experimentally determined, using a Lloyd EZ50 universal testing machine, being presented in Figure 4. Analysis of the experimental data was achieved using the Spider-8 data acquisition system, and the processing and read-out of the data were made by using the Nexigen software.
Determination of the positioning precision of numerically controlled axes from CNC machine tools is standardized by a series of international standards, from which the most commonly used in the present time are VDI/DGQ 3441 and ISO 230:2.These standards establish the methodology regarding the testing, testing conditions and evaluation procedure for processing the measuring results.Testing procedure is based on repeated measurements of the effective position of the tested feed axis, discreted in several points (target positions), placed at equal distances along the table stroke [9].Measuring the positioning precision of the researched feed axis was carried out using a calibrated measuring system (Renishaw ML 10 laser interferometer).The arrangement of the measuring system on the experimental setup is given in Figure 5.In order to evaluate the positioning precision of the moving table, several parameters were used: positioning inaccuracy P, mean repeatability P s and mean reversal error U.It is necessary for the precision parameters to be determined using statistical methods, because of the large number of measuring points and measuring runs for each point.This is required in order to evaluate the evolution of the position deviations with high accuracy.

EXPERIMENTAL RESULTS AND DISCUSSION
Measuring and determining the component parameters of the positioning precision was made taking into account three values of the axially applied force, given in Table 1.For the three experimental sets, corresponding positioning precision diagrams were plotted, represented in Figure 6.The statistically determined parameters of the moving table positioning precision are given in Table 2.By analysing the preceding diagrams, a significant increase of the positioning inaccuracy P can be observed, from 29.8 µm for a 350 N axial force, to 36.9 µm for an increased axial force of 1088 N, which corresponds to a deformation of the disk spring of 0.369 mm.The mean positioning repeatability is less influenced by the deformation of the disk spring, recording an increase of approximately 2 µm when varying the axial force from 350 to 1088 N. A significant influence can be observed in the case of the mean reversal value, which increases from 4 µm to 11 µm, once the axial force is increased.

CONCLUSIONS
In this paper, a method of experimental analysis for determining the values of the parameters used to evaluate the positioning precision of CNC machine tools was presented.The main parameters are given by the positioning inaccuracy, repeatability and reversal error.Experimental researches were made, regarding the influence of the ballscrew mechanism stiffness on the moving table positioning precision, by introducing an elastic element in the structure of the kinematic feed chain.From the experimental analysis resulted that the phenomenon of deadzone caused by the disk spring has a significant influence on the positioning precision in the way of decreasing it, as the axial force increased.Also, the presence of the disk spring results in a high value of the dead zone, materialized in an increased reversal error.In order to optimize the controlled axis, taking into account the variations of the deadzone along the axis stroke, the dimensioning of the kinematic feed chain must consider the maximum deadzone value for obtaining a high stiffness of the kinematic feed chain.

Fig. 1 .
Fig. 1.General view of the experimental test stand.

Fig. 3 .
Fig. 3. Dimensions of the disk spring used during the experimental tests.

Fig. 6 .
Positioning precision diagrams, in conformity with VDI/DGQ 3441: a. experimental results for F A = 350 N; b. experimental results for F A = 710 N; c. experimental results for F A = 1088 N.

Table 1 .
Servomotor parameters for different axial force values.

Table 2 .
Positioning precision parameters resulted from experiments.