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Dynamic characteristic of ball screw feed system in a milling machine is studied numerically in this work. In order to avoid the difficulty in determining the stiffness of rolling joints theoretically, a dynamic modeling method for analyzing the feed system is discussed, and a stiffness calculation method of the rolling joints is proposed based on the Hertz contact theory. Taking a 3-axis computer numerical control (CNC) milling machine set ermined as a research object, the stiffness of its fixed joint between the column and the body together with the stiffness parameters of the rolling joints is evaluated according to the Takashi Yoshimura method. Then, a finite element (FE) model is established for the machine tool. The correctness of the FE model and the stiffness calculation method of the rolling joints are validated by theoretical and experimental modal analysis results of the machine tool’s workbench. Under the two modeling methods of joints incorporating the stiffness parameters and rigid connection, a theoretical modal analysis is conducted for the CNC milling machine. The natural frequencies and modal shapes reveal that the joints’ dynamic characteristic has an important influence on the dynamic performance of a whole machine tool, especially for the case with natural frequency and higher modes.

Ball screw feed drive system has become a key part of CNC machine tools owning to its advantages of high positioning accuracy and transmission efficiency, long operating life, and less internal heat [

There are two types of joints in the feed system: fixed joints as bolted joints and rolling joints including ball screw assembly, rolling guide paired, and bearing jointed. So far, the identification methods of joints’ dynamic characteristic parameters can be summarized into three kinds: theoretical calculation method, experimental test method, and theoretical calculation combined with experimental test method. As for the fixed joints, [

As for the identification of rolling joints’ dynamic characteristic parameters, [

In allusion to the above problem, with a 3-axis CNC milling machine taken as research object, this paper discusses the dynamic modeling method of the feed system and proposes a stiffness calculation method of the rolling joints based on the Hertz contact theory. A finite element model of the CNC milling machine is established, and the correctness of the FE model and the stiffness calculation method of the rolling joints are verified by the comparison of theoretical and experimental modal analysis of machine workbench. On this basis, under the two modelling methods of the joints taking stiffness into consideration and connecting joints rigidly, theoretical modal analysis of the CNC milling machine is implemented, respectively, and the results are compared and analyzed.

Typical structure of a CNC machine tool feed system [

Structure diagram of the feed system.

The fixed joints in the feed system mainly include bolted joints between workbench and slide block, between workbench and nut base, between body and bearing base, and between body and motor base, while the rolling joints mainly include the joints of ball screw assembly, rolling guide pair, and bearing. The most common method for modeling the joints is to substitute several spring-damping units for the connection between joints’ substructures.

In order to establish the dynamic model of feed system more conveniently, the fixed joints can be simplified as rigid connection on account of their high stiffness. Although the AC servo motor, screw, bearings and other related parts are not modeled in the analysis, their axial stiffness is carefully incorporated. Based on the above simplified conditions, dynamic modeling method of the feed system [

Dynamic modeling of the feed system.

Hertz contact theory is the classical theory to calculate contact deformation and contact stress of elastic body. As shown in Figure

Hertz contact model.

As for the rolling joints of ball screw assembly, rolling guide pairs, and bearings in the feed system, the common feature is that they all contain the contact between ball and groove; therefore the stiffness of rolling joints in feed system can be calculated based on the Hertz contact theory.

Rolling guide pair is comprised of slide block, balls, guides, and so on, assuming that the contacts between ball and slide block groove, between ball and guide groove satisfy the four conditions of the Hertz contact theory; in addition, load is evenly distributed among each backing ball in the same row; then the normal and tangential stiffness of the rolling guide pair can be calculated based on the Hertz contact theory.

Typical structure of the rolling guide pair [

Force analysis of rolling guide joint.

Assuming that the pretightened force

Dynamic model of the axial feed unit is shown in Figure

Dynamic model of ball screw feed unit.

The screw is supported fixed at one end and simply supported at the other, which is shown in Figure

Schematic diagram of screw support mode.

Ball screw assembly is comprised of screw, balls, screw nut, and so on, assuming that the contacts between ball and screw groove, between ball and screw nut groove satisfy the four conditions of the Hertz contact theory; in addition (1) centrifugal force and gyroscopic moment induced by balls rotation are so small that can be ignored; (2) axial load is evenly distributed between all backing balls; thus ball screw assembly axial stiffness can be calculated based on the Hertz contact theory.

Ball screw assembly of feed system is pretightened by applying a preload, and external axial load can be neglected; force analysis of the ball screw assembly under the force of preload

Force analysis of ball screw joint.

Synthetic curvature at contact point between each ball and screw nut groove and synthetic curvature at contact point between each ball and screw groove are [

Bearing is comprised of inner ring, balls, outer ring, and so on. Assuming that the contacts between ball and inner ring, between ball and outer ring satisfy the four conditions of the Hertz contact theory; in addition (1) centrifugal force and gyroscopic moment induced by ball rotation are so small that can be ignored; (2) axial load is evenly distributed between all backing balls; thus bearing axial stiffness can be calculated based on the Hertz contact theory.

Taking the angular contact bearings used at fixed end as an example, the pretightened force

Force analysis of bearing joint.

A CNC milling machine with three axes,

Structure diagram of CNC milling machine.

In order to establish FE model of the milling machine more easily, bolted joint between column and body is emphasized while the fixed joints between guide and body, between slide block and workbench, and between servo motor and spindle box are connected rigidly with MPC184 element in ANSYS.

Takashi Yoshimura studied bolted joints in machine tools. It was concluded that all these joints possess the same dynamic characteristic data per unit area if the average contact pressures at fixed joints are equal. As for the bolted joint between column and body, the contact pressure can be considered as being uniformly distributed when neglecting the influence of bolts distribution and structures physical deformation. Therefore, stiffness of the bolted joints can be obtained through calculating contact pressure and the Takashi Yoshimura method. The column is connected to the body with 16 parallel MATRIX27 stiffness elements; stiffness of the bolted joints and parameters of the stiffness elements are shown in Table

Stiffness of bolted joints and parameters of stiffness elements.

Direction | Stiffness (N/m) | Number of stiffness elements | Parameters of stiffness elements |
---|---|---|---|

Normal | 5.07 × 10^{10} |
16 | 3.12 × 10^{9} |

Tangential | 2.02 × 10^{10} |
16 | 1.26 × 10^{9} |

Stiffness of the rolling joints in

Stiffness of rolling joints in

Normal stiffness of rolling guide |
Tangential stiffness of rolling guide |
Axial stiffness of screw |
Stiffness of ball screw |
Axial stiffness of bearings |
Axial equivalent stiffness |
---|---|---|---|---|---|

156 | 112 | 443 | 290 | 503 | 130 |

Stiffness of rolling joints in

Normal stiffness of rolling guide |
Tangential stiffness of rolling guide |
Axial stiffness of screw |
Stiffness of ball screw |
Axial stiffness of bearings |
Axial equivalent stiffness |
---|---|---|---|---|---|

182 | 131 | 591 | 290 | 503 | 141 |

Stiffness of rolling joints in

Normal stiffness of rolling guide |
Tangential stiffness of rolling guide |
Axial stiffness of screw |
Stiffness of ball screw |
Axial stiffness of bearing |
Axial equivalent stiffness |
---|---|---|---|---|---|

167 | 120 | 752 | 165 | 373 | 225 |

Geometric model of the CNC milling machine is built and then is transmitted to HyperMesh to establish its finite element model, during which some points should be focused on as follows:

In order to verify the validity of the finite element model established above, theoretical and experimental modal analysis of the workbench are conducted. The finite element model of the workbench taking joints’ stiffness parameters into consideration is transmitted to ANSYS software, theoretical modal calculation is accomplished based on the Block Lanczos algorithm, and the first four orders natural frequencies and modal shapes are shown in Table

Comparison of theoretical natural frequencies and experimental natural frequencies.

Modal orders | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Measured results (Hz) | 279.0 | 360.3 | 396.4 | 429.7 |

Throretical results (Hz) | 281.9 | 363.1 | 388.6 | 417.5 |

Relative error (%) | −1.03 | −0.77 | 2.07 | 2.92 |

Theoretical vibration modes of workbench.

The 1st vibration mode

The 2nd vibration mode

The 3rd vibration mode

The 4th vibration mode

The hammering method with single-point exciting and multipoints vibration picking is used to conduct experiment modal test. Working principle of modal testing system is shown in Figure

Working principle of modal testing system.

A workbench structure model with measuring points arrangement is built in Modal View as shown in Figures

Measuring points arrangement of workbench.

A group of measured frequency response functions.

Basic process of modal test includes workbench, acquisition and processing of input and output signal, calculation of frequency response functions, and identification of modal parameters.

When the workbench is located at the neutral position, a group of measured FRF is shown in Figure

Experimental vibration modes of workbench.

The 1st vibration mode

The 2nd vibration mode

The 3rd vibration mode

The 4th vibration mode

It can be seen from Figures

The correctness of the finite element model is verified by the comparison of theoretical natural frequencies and modal shapes and their corresponding experimental results, which also indicates that the stiffness calculation method proposed in this paper is right and feasible.

The finite element model of the CNC milling machine is transmitted to ANSYS for modal calculation; the first four orders natural frequencies and modal shapes of the whole machine tool are shown in Table

Comparison of natural frequencies under the two modeling methods of joints.

Modal orders | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Flexible connection (Hz) | 44.4 | 49.4 | 85.5 | 98.2 |

Rigid connection (Hz) | 66.8 | 69.8 | 160.0 | 178.8 |

Absolute error (Hz) | 22.4 | 20.1 | 74.5 | 80.6 |

Vibration modes of the whole machine on side ring stiffness.

The 1st vibration mode

The 2nd vibration mode

The 3rd vibration mode

The 4th vibration mode

It can be seen from Figure

Furthermore, regardless of joints’ stiffness parameters and connecting joints rigidly in finite element model with MPC184 element, theoretical modal calculation is conducted on the model with rigid connection; the first four orders natural frequencies and modal shapes are shown in Table

Vibration modes of the whole machine with rigid connection.

The 1st vibration mode

The 2nd vibration mode

The 3rd vibration mode

The 4th vibration mode

It can be seen from Figure

The first four orders vibration modes of the whole machine tool have great difference under the two modeling methods of joints. Vibration modes considering joints stiffness mainly present as local vibration of the spindle box, the saddle, and the workbench, while vibration modes connecting joints rigidly focus on the overall vibration of the column and the spindle box, the saddle, and the workbench.

Comparison of the first orders natural frequencies is shown in Table

The comparison of natural frequencies and modal shapes indicates that joint’s dynamic characteristic is one of the key factors that influence the dynamic performance of machine tool, and the influence is much more significant when it comes to high order natural frequencies. Therefore, calculating stiffness parameters of joints in feed system accurately at designing stage of the machine tool is of decisive significance in obtaining right theoretical analysis results.

In allusion to the problem that stiffness of rolling joints is difficult to determine in theoretical modeling and analyzing of a CNC machine tool, a dynamic modeling method of the feed system is discussed, and a stiffness calculation method of the rolling joints is proposed based on the Hertz contact theory; a CNC milling machine finite element model is established, theoretical and experimental modal analysis of the workbench are conducted, theoretical vibration modes are completely in conformity with experimental vibration modes, and the error between theoretical natural frequency and experimental natural frequency is within 2.92%, which vertify the correctness of the finite element model and the stiffness calculation method of the rolling joints; under the two modelling methods of joints taking stiffness into consideration and connecting joints rigidly, theoretical modal analysis of the CNC milling machine is implemented, the comparison of natural frequencies and modal shapes indicates that joints’ dynamic characteristic parameter is one of the key factors that influence the dynamics performance of a machine tool, and the influence is much more significant when it comes to natural frequencies and modal shapes of high orders; the proposed stiffness calculation method of the rolling joints has the advantages of accurate, reliable, and good practicability, which laid a foundation for dynamic modeling of the feed system more accurately.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The project is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (2013 no. 8), and 2014 Shanghai “Overseas Outstanding Professor.” The authors would like to thank the editor and the reviewers for their constructive comments and suggestions which improved the quality of this paper.