Hybrid multibody system method for the dynamic analysis of an ultra‐precision fly‐cutting machine tool

The dynamics of an ultra‐precision machine tool determines the precision of the machined surface. This study aims to propose an effective method to model and analyze the dynamics of an ultra‐precision fly‐cutting machine tool. First, the dynamic model of the machine tool considering the deformations of the cutter head and the lathe head is developed. Then, the mechanical elements are classified into M subsystems and F subsystems according to their properties and connections. The M‐subsystem equations are formulated using the transfer matrix method for multibody systems (MSTMM), and the F‐subsystem equations are analyzed using the finite element method and the Craig–Bampton reduction method. Furthermore, all the subsystems are assembled by combining the restriction equations at connection points among the subsystems to obtain the overall transfer equation of the machine tool system. Finally, the vibration characteristics of the machine tool are evaluated numerically and are validated experimentally. The proposed modeling and analysis method preserves the advantages of the MSTMM, such as high computational efficiency, low computational load, systematic reduction of the overall transfer equation, and generalization of its computational capability to general flexible‐body elements. In addition, this study provides theoretical insights and guidance for the design of ultra‐precision machine tools.

dynamics of the ultra-precision fly-cutting machine tool to improve the machining precision.
In the current literature, the finite element method (FEM) is the most widely used method for the dynamic analysis of machine tools. Liang et al. 3,4 investigated the relations between the machined surface and the dynamics of an ultra-precision machine tool by combining ABAQUS with MATLAB. Based on FEM, Liang et al. 5 also proposed a state-space model of an ultra-precision machine tool and calculated the machined waviness along the cutting direction and feed direction excited by the cutting force. Zhang 6 and Yao et al. 7 combined ANSYS with the dynamic simulation software ADAMS to simulate the dynamic and static characteristics of a machine tool system to identify the weak parts. Chen et al. 8 proposed a simplified method based on FEM to study the interaction between the machining process and the quality of the machined surface to obtain the interaction relationships. An et al. 9 integrated the virtual material method with FEM to investigate the dynamic performance of an ultra-precision fly-cutting machine tool. However, in all the FEM-based dynamic analysis of machine tool systems, the orders of the global dynamic equations are so high that deriving and solving the equations are computationally expensive. 10 Research efforts have been devoted to the dynamic modeling of the machine tools, mostly focusing on the single part of the machine tools due to the lack of efficient dynamic modeling methods. An et al. 11 utilized the rigid-body dynamic method to predict the motion of the spindle part in an ultra-precision fly-cutting machine tool. An et al. 12 observed a low-frequency vibration signal (about 1/6 of the spindle rotation frequency), which is considered to be the source of the medium-frequency ripple on the surface. Chen et al. 13 developed a complete degree-of-freedom (DOF) dynamic model of the bolt coupling system to analyze the effect of misalignment coupling faults. Yang et al. 14 developed a surface topography model of the whole machine tool and analyzed the air spindle using ANSYS Workbench. Badiger et al. [15][16][17][18] studied the characterization and tribological behavior of coatings and found that the surface roughness of the workpiece and the wear of the tool could be significantly improved by using coated tools.
To enhance the computational efficiency, the classical transfer matrix method (TMM) was used for the dynamic analysis of the spindle component in the machine tool. 19 While the theory of TMM is suitable for a single element, challenges are encountered when using TMM for multibody systems (MSTMM), referred to as the "Rui Method." 10,20,21 Compared to conventional dynamic methods, MSTMM has several advantages, such as replacing the global dynamic equations with loworder transfer equations, high computational speed, and simplified computational implementation. 22 In our previous studies, [23][24][25] the dynamic models of an ultra-precision fly-cutting machine tool consisting of rigid body elements, beam elements, and hinge elements were developed by MSTMM. It was also found that the deformation of the cutter head and the lathe bed may influence the quality of the machined surface. 25 However, the deformations of these two components have not been investigated in previous studies. Recently, at the theoretical level, a hybrid dynamic method was developed, extending MSTMM with FEM and the Craig-Bampton (CB) reduction method for dynamic modeling and analysis of multibody systems with flexible-body elements. 26 In this paper, the proposed hybrid dynamic method is used to develop a novel model of the ultra-precision fly-cutting machine tool, where the cutter head and the lathe bed are treated as general flexible elements, and the overall transfer equation is formulated.
Compared with previous MSTMM studies, this study improves MSTMM computational performance in the general case of flexible elements while maintaining the above-mentioned advantages.
In Section 2, the novel model of an ultra-precision fly-cutting machine tool system is introduced. In Section 3, the mechanical elements in the model are categorized into M subsystems and F subsystems. In Section 4, M subsystems and F subsystems are analyzed using MSTMM and FEM-CB, respectively. In Section 5, all subsystems are recombined, and the overall transfer equation of the whole system is derived. In Section 6, the vibration characteristics of the machine tool system are simulated and validated using the modal test. Conclusions are presented in Section 7.

| MODEL OF AN ULTRA-PRECISION FLY-CUTTING MACHINE TOOL SYSTEM
In this section, a novel model of an ultra-precision fly-cutting machine tool system considering the deformations of the cutter head and lathe bed is developed.

| Dynamic modeling
In the dynamic model, all the components are regarded as individual mechanics elements, including body and hinge elements.
The body elements include rigid components, beam-like components, and general flexible components. The hinge elements are used to link the body elements, such as elastic hinges and fixed hinges.
The dynamic model and the elements of the machine tool system are depicted in Figure 2, including 19 body elements and 27 hinge elements. All the elements are numbered uniformly, with details shown in Tables 1 and 2, respectively.
The transfer matrix of hinge element j is expressed as where the linear and rotational stiffnesses are  At virtual cuts at points P 11,1 , P 11,3 , P 13,2 , P 13,6 , P 15,28 , P 20,17 , P 41,27 , P 44,45 (2) , P 44,45(3) , P 44,45 (4) , and P 44,45 (5) , the ultra-precision fly-cutting machine tool system becomes a tree system shown in Figure 4. At the "cutting points," the following relations can be formulated: where Z i j , is defined as the state vector at the point (i, j). According to MSTMM, 20 The machine tool system is further analyzed using the hybrid dynamic method in the following sections. As shown in Figure 5, the machine tool system is divided into 14 subsystems, including 12 M subsystems and 2 F subsystems, and the members of the subsystems are shown in Table 3.
M subsystems 1, 3, and 4 are connected to F subsystem 9 at points C 1 , C 2 , and C 3 , namely, the connection points; M subsystems 2, 5, 6, 1, 7, 8, 11, 12, 13, and 14 are connected to F subsystem 10 at connection points C 4 , C 5 , C 6 , C 7 , C 8 , C 9 , C 10 , C 11 , C 12 , and C 13 . The state vectors of the machine tool system are defined as The state vectors of the ends and the topology structure of M subsystems 1-8, 11-14 are presented in Table 4. The overall state vectors of F subsystems 9 and 10 are defined as the modal coordinates of the flexible elements after CB reduction, that is,

| Dynamic analysis of M subsystems
According to Table 4 where the overall state vectors are

| Overall transfer equation of tree subsystem 1
According to the input and output ends of subsystem 1 in Table 4 and the constraints in Equation (3), the overall state vectors of subsystem 1 are defined as The main transfer equations of M subsystem 1 are derived as T   T   T   T   T   T   T   T   C  I  I  I  I  I  I   I  I  I  I   I  I   I  I   40 39 38  I 26  I 21 Similarly, the geometric equations of element 25 (two input ends), element 35 (two input ends), and element 37 (five input ends) are written as where Arranging the main transfer equation in Equation (11) and the geometric equations in Equations (12) and (13) in the sequence of the overall state vectors in Equation (10), the overall transfer equation of subsystem 1 is given as  The geometric equations of element 9 with four input ends are where The overall transfer equation, the overall state vector, and the overall transfer matrix of subsystem 2 are

| Overall transfer equation of tree subsystem 5
The main transfer equation of subsystem 5 is The geometric equation of element 11 with two input ends is The overall transfer equation, the overall state vector, and the overall transfer matrix of subsystem 5 are The overall transfer equation, the overall state vector, and the overall transfer matrix of subsystem 6 are The geometric equations of element 45 with five input ends are The overall transfer equation, the overall state vector, and the overall transfer matrix of subsystem 8 are        are not diagonal matrices and need to be orthogonalized. The initial step of orthogonalization involves solving the vector ψ from the following equations:    Table 6.
The features of the connection points in the machine tool system are summarized and listed in Table 7. Points C 1 and C 4 are regarded as the F-M condition, and the other connection points belong to the M-F condition. Using Table 7 and Figure 5, the connection relations are formulated in the following two conditions.
(a) F-M condition:  Subsystem Point Subsystem F subsystem M-F F subsystem M-F T A B L E 6 Connection relations in three connection conditions where [ ] Similarly, substituting the force relations of points C 4 -C 13 into the overall transfer equation of subsystem 10 in Equation (38) gives where 43 , According to the boundary conditions in Equations (44) and (45)  6 | RESULTS AND VALIDATION

| Experimental modal test
The modal test of the ultra-precision fly-cutting machine tool system was designed and conducted to validate the dynamic model and the results. In the test, 364 measurement points were scattered around the whole machine tool, shown in Figure 7, among which 48 measurement points were in the cutter head and the lathe bed, respectively.
The machine tool was excited with a modal hammer (PCB086C03) to generate excitation with frequencies from 3 to 1500 Hz. The single-input-single-output (SISO) testing technique was used to measure the frequency response functions (FRFs).
Due to the reciprocity of FRF, 28 the "moving hammer" test and the "moving accelerometer" test were conducted according to the physical situation. More specifically, an accelerometer (PCB355B04)

| Results
According to the overall transfer matrix and the overall transfer equation in Section 5.2, the vibration characteristics, including the natural frequencies and the mode shapes of the machine tool system, are simulated by coding in C++. The first 20 modes simulated results by the proposed method are compared with the test ones, as shown in Table 8. The maximum relative error is 6.75%, confirming the accuracy of the dynamic model and the proposed method.
As shown in Table 8, all the components in the machine tool vibrate coherently in the first five mode shapes, indicating that the stiffnesses of the components and the coupling surface were selected to be high enough. In our previous work, 25 we found that the waviness on machined surfaces is affected by the mode shapes related to the vibrations of the tool tip. Herein, the relative vibration between the tool tip and the workpiece along the z-direction in the 15th, 16th, 17th, and 20th modes will significantly influence the quality of ultra-precision machining. Since the modal shapes of the 15th and 20th modes are translational vibrations, which were calculated by previous work, 25 these modes are not presented and discussed in detail in this study.
By using our MSTMMSim Software, 27 two typical modal shapes (16th and 17th modes) are visualized and are shown in Figures 9B and 10B. It can be seen in Figure 9 that the deformations of the lathe bed appear in the 16th mode in both experimental and simulated results. The cutter head bends on both sides in the 17th mode in Figure 10. These two modes will generate the relative vibrations between the tool tip and the workpiece, resulting in large waviness on the machined surfaces. To achieve higher machining precision, the structure and material of the lathe bed and the cutter head should be optimized with higher bending stiffness.

| CONCLUSIONS
This study presents a comprehensive modeling process of an ultraprecision fly-cutting machine tool using a hybrid dynamic method, which provides theoretical insights and guidance for ultra-precision The proposed modeling process of the ultra-precision fly-cutting machine tool is efficient, highly accurate, and easy to program. Once the system's overall transfer matrix is derived, as long as the structure of the multiterminal input elements remains unchanged, there is no need to re-derive it even if the structural parameters of the system change. The transfer matrix of the corresponding component can simply be added or deleted in the overall transfer matrix of the original system. The proposed method lays the foundation for our ongoing research efforts focused on the dynamic control and the optimization of an ultra-precision machine tool.