In order to simulate the cutting performance of a spindle mounted in the machine tool, the finite element (FE) model of spindles is required to be coupled with machine tool. However, the unknown joint dynamics (e.g., bolts) between the spindle and machine tool column limit the accuracy of the model. In this paper, an FE model updating method is proposed based on the identification of joint dynamics in both translational and rotational degrees-of-freedom (DOF). The receptance coupling (RC) technique is enhanced to estimate frequency response functions (FRFs) corresponding to rotational DOFs. The joint stiffness is identified through the iteration process by minimizing the difference between the simulated FRF and the measured FRF of the assembly. The proposed method is verified with a machine-tool spindle system. The good agreement between simulation and experiment shows the effectiveness of the method.

Apart from excessive tool wear/breakage, chatter is still the biggest obstacle in increasing material removal rates in high-speed machining. In order to predict chatter vibrations, the accurate prediction or measurement of the machine-tool dynamics is critical. Among all the components of a machine tool, the spindle-tool system is usually the most flexible part that limits the overall dynamic stiffness. Therefore, a full understanding of the dynamic response of the spindle-tool system is required in order to avoid or suppress the chatter problem.

In the past, many researchers have investigated the dynamic modeling of high-speed spindles with consideration of the nonlinear behavior of bearings. Recently, Abele et al. reviewed the historical development, recent challenges, and future trends of the machine-tool spindle unit in detail [

While finite element analysis (FEA) is a well-established and accepted technique for the dynamic analysis of structural systems at the design stage, typically the joint dynamics in the machine are not taken into account [

Differently, the FRF-based model updating methods make use of the measured FRFs directly which circumvent the need to identify the modal parameters from the experiments [

Receptance Coupling (RC) method divides complex systems into several simple substructures, the FRFs of which can be respectively obtained by analytical or experimental methods. The system response can be gained by composing the subsystems, according to the coupling relationship between the subsystems, that is, equilibrium condition and compatibility condition on the common border. In 2000, Schmitz introduced the RC technique into kinetic analysis of the machine-tool field to predict the FRFs at the tool tip in the spindle-tool system. Compared with the conventional methods which measure the tool tip dynamic response after each tool change, the RC technique can largely improve the efficiency [

In this paper, the FRF-based iteration algorithm is introduced and improved. The RC technique is enhanced to predict the FRFs of the rotational DOFs, and then a general FE model updating method is proposed. Next, the coupled model of a machine-tool spindle system is updated using the proposed method. The experimental results show that the updated model can represent the dynamic behavior of the spindle mounted in the machine-tool column fairly accurately when the dynamics of the joints are largely unknown.

The rest of the paper is organized as follows. In Section

The principle of FRF-based iterative algorithm can be described in the form of

Equation (

If the updated parameters are

Figure

RC model.

A force

Similarly, if the external force

For Timoshenko beam elements, the motion of nodes at a specific surface is composed of translational and rotational DOFs. The input force vector

By substituting (

As (

The FRFs of each rotational DOF can be estimated by using (

The proposed method is based on the assumption that the analytical FE model of a mechanical structure (

The flowchart of the proposed method.

The FE model updating process starts from analytical modeling of the mechanical structure. The mass, stiffness, and damping matrices are used to represent the properties of each element and then assembled by nodes to obtain the systematical matrices

After data is prepared, the analytical FRFs are compared with experimental data. The residual between the analytical and measured FRFs is calculated. As the initial FE model is usually not accurate, the 2-norm of the residual (

The coupled model of the machine-tool spindle system was presented by Cao and Altintas [

Coupled modeling of the machine-tool spindle system.

The spindle assembly

Bolts connection

In the coupled model of the machine-tool spindle system, the mass parameter of the joint surface is ignored and the damping parameter is determined by the experimental modal analysis method. Thus only the stiffness parameter of the joint surface needs to be identified. The stiffness parameter of the joint surface between the spindle and the spindle-head is expressed as equivalent springs. In the following parts, the joint stiffness between the spindle and the spindle-head will be estimated systematically by using the proposed FE model updating method. Without considering the joint dynamics between the spindle and machine tool, the global stiffness matrix of the machine-tool spindle system is

If the node

The joint damping can be added in the same manner as the joint stiffness. However, due to the complexity of the nonlinear damping, the damping matrix is ignored when modeling the system. The modal damping ratios are obtained by experimental modal analysis.

In the coupled model, the two ends of the equivalent spring separately connect with the 40th node (substructure of the spindle) and the 51st node (substructure of the spindle-head); that is,

Utilize the RC technique to estimate the FRF of the rotational DOF

The schematic diagram of the spindle modal test.

By using least squares algorithm to solve the overdetermined equation (

Iterative procedure of the updated parameters.

Translational stiffness of the joint surface

Coupled stiffness of the joint surface

Rotational stiffness of the joint surface

Then the updated coupled model of the machine-tool spindle system is applied to simulate the FRFs at the spindle nose and the tool tip and the simulated results are compared with experimental results so as to verify the validity of the updated model. Figures

FRFs measurement.

At the spindle nose

At the tool tip

The FRF at the spindle nose in radial direction is simulated and compared with the measured value, as shown in Figure

Comparison of the FRF at the spindle nose between simulation and measurement.

When the tool holder and the tool are installed on the spindle, the mode of the tool dominates the FRF. In the modeling, the Timoshenko beam elements were still used to build the FE model of the tool, assuming that the connection between the tool and tool holder was rigid. In this way, as shown in Figure

Comparison of the FRF at the tool tip between simulation and measurement.

In this work, a general FE model updating method has been proposed to update mechanical structure with unknown joints. The FRF-based iterative algorithm is improved to lower the requirement for the experimental FRFs and the RC technique is used to predict the FRFs of the rotational DOFs. The translational stiffness, rotational stiffness, and coupled stiffness of joints are identified after iterations, and the coupled model of the machine-tool spindle system is updated. The results show that the simulated FRFs at both the spindle nose and the tool tip match very well with measurements. With the updated model, it is possible to simulate the cutting performance of a high-speed spindle mounted on the machine tool.

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

The authors wish to express their heartfelt gratitude to Professor Yusuf Altintas from Manufacturing Automation Laboratory (MAL), The University of British Columbia. All the experiments of this paper were carried out in MAL. This work is jointly supported by National Natural Science Foundation of China (no. 51421004), the National Science and Technology Major Project (2014ZX04001-191-01), and the Fundamental Research Funds for the Central University (CXTD2014001).