Effect of the Stiffness of the Turntable Bearing Joint on the Dynamic Characteristics of the Five-Axis Machine Tool Rotary System

Abstract

The positioning accuracy and machining performance of the five-axis machine tool are significantly impacted by the dynamic characteristics of the rotary system of the two-axis rotary table machine. In this study, first, the stiffness of the joint bearing of the rotary table system is calculated, and the effect of bearing clearance and external load on the stiffness is analyzed. Second, considering the stiffness characteristics of the joint, the natural frequency and mode shapes of the turntable system are calculated. Finally, the influence of turntable angle and bearing stiffness on the dynamic characteristics of the turntable system is analyzed. The results show that the natural frequency of the rotary table system does not change obviously with the axial stiffness of the turntable bearing joint, but increases significantly with the increase in the radial stiffness. The first order natural frequency of the turntable decreases with the increase in the swing angle, and the change in the first order natural frequency is 77.43 Hz. The research results provide theoretical basis and guidance for machine tool design and use.

1. Introduction

The static deformation of table surface is influenced by the support stiffness of turntable. The static stiffness of the turntable bearing directly affects dynamic characteristics of the workbench.

Common turntable bearings are double volleyball bearings, three-row cylindrical roller bearings and other four forms. When studying the stiffness of single-row cylindrical roller bearing, scholars discovered that for cylindrical roller bearing, its stiffness is not only affected by the structure and installation process, but also closely related to the actual working conditions (such as speed, lubrication, temperature and external load) [1,2,3,4]. In the existing research of three-row cylindrical roller bearings, the main focus is on the calculation and stress analysis of roller load as well as the analysis of the life and strength of bearings [5,6,7,8,9]. For the effective analysis of cylindrical roller bearings, the actual stress state of rollers needs to be determined. Göncz et al. [5] used the symmetrical three-dimensional finite element model of three-row cylindrical roller bearing to numerically calculate the distribution of internal contact force and determine the stress field in the contact area between roller and raceway. Li and Jiang [6] calculated the internal stress distribution of three-row cylindrical roller bearings by using finite element analysis and a mixed finite element model combining solid and spring elements. Li et al. [7] carried out finite element analysis on the elastic–plastic characteristics of contact between cylindrical roller and raceway, and calculated the contact stress distribution. Wang et al. [8] borrowed a set of nonlinear springs to simulate each roller in this way so as to establish an equivalent finite element model with certain reliability, and obtained load–deformation characteristics. He et al. [9] used the substitution method to replace the solid rollers with the nonlinear springs, and then analyzed the actual load conditions to obtain the maximum contact load between rollers and raceways, and established a local contact CAD model between rollers and raceways. Based on engineering experience and the calculation formula of strain measurement, the stiffness model of three-row cylindrical roller bearings is analyzed in the literature [10,11]. He and Wang [10] analyzed the effect of different bolt types, bolt pre-tightening force, and roller diameter on the structural stiffness and strength of three-row cylindrical roller bearings. Wei et al. [11] calculated the static stiffness matrix of the three-row cylindrical roller table bearings by considering the load distribution of a radial cylindrical roller and its interaction with axial roller load. However, research on the static stiffness of three-row cylindrical roller turntable bearings and its effect on the dynamic characteristics of the turntable is lacking. Therefore, the static deformation and dynamic characteristics of the turntable are very important to the analysis of the machine tool turntable. In order to accurately analyze its characteristics, it is necessary to analyze the stiffness of the turntable bearing in the static state and its impact on the overall dynamic characteristics of the turntable.

In the process of research, modeling the joint surface of the machine tool and identifying the dynamic characteristic parameters are the main focus of the work. Li et al. [12] analyzed the dynamic characteristics of different machining positions of machine tools based on Kriging method. Chen et al. [13] studied the dynamic contact characteristics and dynamic stiffness of single-nut ball screw pairs, and established the dynamic stiffness model of all ball raceway contact pairs. Chen et al. [14] used the classical contact theory to establish the contact stiffness model of ball screw pair and guide pair. Yu et al. [15] used the contact characteristics of linear guide rail to establish the dynamic stiffness model of the system under different load conditions and operating time. Liu et al. [16] analyzed the spindle/tool holder contact stiffness model based on the Majumdar and Bhushan model (MB model) [17]. In recent years, many modifications of the MB model were proposed based on multi-scale fractal contact theory [18,19,20].

However, there is still a lack of research on the static stiffness of three-row cylindrical roller turntable bearings and the dynamic characteristics of the turntable under different machining attitudes. In order to accurately analyze the static deformation and dynamic characteristics of the turntable, it is important to consider the static stiffness of the turntable bearing and the dynamic characteristics of the turntable under the influence of different machining swing angles. In this study, based on the modified Palmgren cylindrical roller elasticity approximation formula, the stiffness of bearing joints is calculated, the effects of bearing clearance and external load on the stiffness are discussed, and the finite element model of the rotary table system is established. Based on this, the theoretical modal analysis of the turntable is carried out, and then the swing angle and bearing stiffness of the turntable are studied, and its influence on the dynamic characteristics of the turntable system is obtained.

2. Basic Structure of Turntable Rotary System

In this study, the AC shaft two-axis rotary table of the five-axis machine tool is adopted as the research object. Its specific structure is shown in Figure 1. The main mechanical joint parts of the worktable include turntable bearing joint part, worm gear and worm joint part, fixed joint part and swing arm bearing joint part.

The YRT turntable bearing of the two-axis rotary table is composed of a shaft ring, a seat ring, two rows of axial cylindrical rollers, a row of radial cylindrical rollers, and a cage, as shown in Figure 2. To avoid convergence difficulties and enhance calculation efficiency in subsequent finite element calculation, the bearing model is simplified. First, it is assumed that when calculating the bearing stiffness theoretically, the race, washer and collar are considered as rigid bodies and will not deform under the action of force. The deformation area includes axial roller, radial roller and their contact area. The stay ring is fixed. Under the action of external force, the collar will have relative displacement. The rotating speed of the table does not exceed $12\mathrm{r}/\min$, and the centrifugal force is ignored. As shown in Figure 2, the axial direction of the YRT turntable bearing is composed of two rows of cylindrical rollers, which contact with two planes. The number of upper and lower rows of rollers is generally equal. Figure 3 shows the force and torque of the turntable bearing.

2.1. Axial Stiffness Modeling of Turntable Bearing

The axial force on the turntable bearing is illustrated in Figure 3b. According to the modified Palmgren cylindrical roller elasticity approximation formula [21], the load of a single axial roller can be expressed as

$${\delta }_a=4.83\times {10}^{-5}\frac{Q_a{}^{0.9}}{L_a^{0.74}{D}_{aw}^{0.1}}$$

(1)

where $Q_a$ is the load of a single axial roller, $L_a$ is the length of the roller, $D_{aw}$ is the effective diameter of the axial roller, and ${\delta }_a$ is the axial elasticity-approaching quantity.

In the state of axial preload negative clearance G_a of YRT turntable bearing, the deformation of two rows of axial cylindrical rollers on each contact surface is equal; therefore, the elasticity-approaching quantity ${\delta }_a=-G_a/4$ for the axial cylindrical rollers on each contact surface can be obtained from Equation (1) as follows:

$$Q_a={\left(\frac{L_a^{0.74}{D}_{aw}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}{\left(\frac{-G_a}{4}\right)}^{1.11}$$

(2)

When the total elasticity-approaching quantity change in the axial direction of the bearing is $1\mathsf{\mu }\mathrm{m}$, the stiffness of the bearing is $K_a\left(\mathrm{N}/\mathsf{\mu }\mathrm{m}\right)$. At this time, the elasticity approach increase between the cylindrical roller and a contact surface is $\Delta {\delta }_a=1 \mathsf{\mu }\mathrm{m}/4=0.00025\mathrm{mm}$, and the increased load $\Delta Q_a$ of a single roller is

$$\Delta Q_a={\left(\frac{L_a^{0.74}{D}_{aw}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}{\left(\frac{-G_a}{4}+0.00025\right)}^{1.11}-{\left(\frac{L_a^{0.74}{D}_{aw}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}{\left(\frac{-G_a}{4}\right)}^{1.11}$$

(3)

The forces on each roller are equal in the state of negative axial clearance of the cylindrical roller bearing; in this state, the axial stiffness $K_a\left(\mathrm{N}/\mathsf{\mu }\mathrm{m}\right)$ of the bearing can be calculated as follows:

$$K_a=Z_a·\Delta Q_a=Z_a·{\left(\frac{L_a^{0.74}{D}_{aw}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}\left[{\left(\frac{-G_a}{4}+0.00025\right)}^{1.11}-{\left(\frac{-G_a}{4}\right)}^{1.11}\right]$$

(4)

where $Z_a$ is the number of single-row axial cylindrical rollers.

The initial clearance of the axial roller is determined by the preload of the fastening screw. Under the action of an external load $F_a$, the axial clearance $G_a$ [11] of the axial roller under the given tightening torque of the screw is

$$G_a=-1.54\times {10}^{-4}{\left(\frac{Z_{b}T}{Z_a{K}_b{d}_b}\right)}^{0.9}{L}_a^{-0.8}=-1.54\times {10}^{-4}{\left(\frac{F_a}{Z_a}\right)}^{0.9}{L}_a^{-0.8}.$$

(5)

In Equation (5), Z_b is the number of screws, T is the tightening torque, d_b is the screw diameter, K_b is the screw torque coefficient, and K_b is generally 0.2.

2.2. Radial Stiffness Modeling of Turntable Bearing

The radial force of the bearing is shown in Figure 3c. The elasticity-approaching quantity ${\delta }_r$ of the roller and the raceway of the cylindrical roller bearing with a line contact can be obtained from Equation (1):

$${\delta }_r=4.83\times {10}^{-5}\frac{Q_r}{L_r^{0.74}{D}_{rw}^{0.1}}{\left(1\pm k\right)}^{0.1}=4.83\times {10}^{-5}\frac{4.08F_r}{L_r^{0.74}{D}_{rw}^{0.1}{Z}_r}{\left(1\pm k\right)}^{0.1},$$

(6)

where Q_r is the load of the single radial roller, L_r is the length of the roller, D_rw is the effective diameter of the roller, Z_r is the number of radial cylindrical rollers, F_r is the radial load, k is the ratio of the effective diameter of the roller to the diameter of the raceway D_rw/D_r, and D_r is the diameter of the raceway.

In the same way, axial stiffness of the bearing can be obtained as follows:

$$K_r=\frac{Z_r}{4.08}{\left(\frac{L_r^{0.74}{D}_{rw}^{0.1}}{4.83\times {10}^{-5}\left({\left(1+k\right)}^{0.1}+{\left(1-k\right)}^{0.1}\right)}\right)}^{1.11}\left({\left(\frac{-G_r}{2}+0.001\right)}^{1.11}-{\left(\frac{-G_r}{2}\right)}^{1.11}\right),$$

(7)

where G_r is the initial clearance of the radial roller.

2.3. Angular Stiffness Modeling of Turntable Bearing

When the YRT turntable bearing is subjected to overturning moment $M$, the inner ring will have a certain inclination angle relative to the outer ring owing to the uneven force of the circumferential roller. Both rows of thrust cylindrical rollers bear the overturning torque load. The torque load is shown in Figure 3d and its load distribution is shown in Figure 4.

As shown in Figure 4, D_a is the axial roller raceway diameter. When the inclination angle is $\theta =\Delta \delta /\left(D_a/2\right)=1\mathrm{mrad}=0.001\mathrm{rad}$, i.e., $\Delta \delta =0.0005D_a$. Equation (2) indicates that the load of the two raceways on the roller at the angular position φ is expressed as follows:

$$Q_{\phi }={\left(\frac{L_a{}^{0.74}{D}_{aw}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}{\left(\frac{-G_a}{4}+0.0005D_a\cos \phi \right)}^{1.11},$$

(8)

$${Q_{\phi }}^{\prime }={\left(\frac{L_a{}^{0.74}{D}_{aw}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}{\left(\frac{-G_a}{4}-0.0005D_a\cos \phi \right)}^{1.11}.$$

(9)

Then, at angular position φ, the roller load difference between the upper and lower raceways and the torque generated by the roller load difference are, respectively, as follows:

$$\Delta Q_{\phi }=Q_{\phi }-{Q_{\phi }}^{\prime },$$

(10)

$$M_{\phi }=\Delta Q_{\phi }\frac{D_a}{2}\cos \phi .$$

(11)

As shown in Figure 4, the moments generated by the left and right rollers are equal. By integrating the roller moment at each angular position, the moment balance equation is obtained as

$$M=2{\displaystyle {\int }_{-\pi /2}^{\pi /2}\Delta Q_{\phi }}\frac{D_a}{2}\cos \phi d\phi .$$

(12)

By substituting Equations (8)–(10) into Equation (12), the moment load M when the inclination angle of the bearing is 1 m/rad can be calculated; that is, the angular stiffness K_M in the negative clearance state (unit, Nm/rad) is obtained as follows:

$$K_M=2{\displaystyle {\int }_{-\pi /2}^{\pi /2}{\left(\frac{L_a{}^{0.74}{D}_{aw}{}^{0.1}}{4.83\times {10}^{-5}}\right)}^{1.11}}\left[{\left(\frac{-G_a}{4}+0.0005D_a\cos \phi \right)}^{1.11}-{\left(\frac{-G_a}{4}-0.0005D_a\cos \phi \right)}^{1.11}\right]\frac{D_a}{2}\cos \phi d\phi .$$

(13)

From Equation (5), the axial clearance under the influence of external load $M_{x,y}$ is obtained as follows:

$$G_a=-1.54\times {10}^{-4}{\left(2{\displaystyle {\int }_{-\pi /2}^{\pi /2}2M/\left(D_a\cos \phi \right)}\mathrm{d}\phi /Z_a\right)}^{0.9}{L}_a{}^{-0.8}.$$

(14)

3. Analysis of Influencing Factors of Bearing Stiffness of Turntable

3.1. Effect of Clearance on Bearing Stiffness of Turntable

The initial parameters of the turntable bearing are listed in

Table 1. Initial parameters of the turntable bearing.

Parameters	Value
Number of axial radial rollers Za, Zr	70, 115
Axial radial roller diameter Daw, Drw/mm	5, 5
Axial radial roller length La, Lr/mm	8, 8
Diameter of axial radial roller raceway Da, Dr/mm	199, 183.5
Number of screws Zb	36
Screw diameter db/mm	7

. According to Equations (5) and (14), the change in the actual clearance size $\left|G_a\right|$ of the axial roller under different screw tightening torque T, external load axial force $F_a$, and external load torque $M_{x,y}$ can be obtained as shown in Figure 5. It can be seen that $\left|G_a\right|$ increases significantly with T, $F_a$, and $M_{x,y}$. Figure 6 shows the effect of the axial clearance of the turntable bearing on the stiffness of the turntable. When the axial clearance $\left|G_a\right|$ increases with screw tightening torque T, both the axial stiffness $K_a$ and angular stiffness $K_M$ increase accordingly; specifically, $K_M$ increases more significantly, while $K_a$ increases gradually when $\left|G_a\right|$ increases to 20 µm. The radial clearance $\left|G_{\mathrm{r}}\right|$ considerably affects the radial stiffness $K_{\mathrm{r}}$. However, more radial rollers are present than axial rollers; hence, the effect of $\left|G_{\mathrm{r}}\right|$ on $K_{\mathrm{r}}$ is not as obvious as that of $\left|G_a\right|$ on $K_a$.

3.2. Effect of External Load on Bearing Stiffness of Turntable

Under different screw tightening torques T, the variation law of axial stiffness $K_a$ of turntable bearing with external load and the variation law of angular stiffness $K_M$ with external load are shown in Figure 7e,f. It can be seen from Figure 7b,d,f that $K_M$ is insensitive to the change of $F_a$, M_x,y and T, and the change is less than 2%. As shown in Figure 7a,c,e, $K_a$ increases with the increase in $F_a$, M_x,y and T, and $K_a$ increases more obviously with the increase in M_x,y and T than with the increase in $F_a$.

4. Analysis of Dynamic Characteristics of the Rotary System of the Turntable

In order to build the geometric model of the rotary system in the 3D software, the rotary system was imported into the finite element analysis software and the finite element model was built. In order to achieve this, the following points need to be considered. To realize a reasonable shape of the unit generated by the finite element mesh division and improve the accuracy and efficiency of calculation and analysis, the structural model of the rotating system needs to be simplified before building the finite element model. Therefore, the following points are ensured: the main body of the rotary table is in the middle of the two-axis rotary table, including the table, the rotary table bearings, the worm gear and the main body; the holes, slots and chamfers of the rotary table are ignored. Figure 8a shows the structural model of the turntable.

During the finite element analysis, according to the solid modeling of the vertical machining center and its structural mechanical characteristics, the SOLID45 element is used in this analysis to set the boundary conditions of each joint surface and component. The fixed joint surface of the rotary table system is mainly bolted connection, and the default form is set. As shown in

Table 2. Turntable joint stiffness.

Type of Junction	Axial Stiffness (N/m)	Radial Stiffness (N/m)	Angular Stiffness (N/m)
Turntable supports Bearing joint	$1.32\times {10}^8\mathrm{N}/\mathrm{m}$	$6.23\times {10}^7\mathrm{N}/\mathrm{m}$	-
	$1.32\times {10}^8\mathrm{N}/\mathrm{m}$	$6.23\times {10}^7\mathrm{N}/\mathrm{m}$	-
Turntable bearing joint	$4.95\times {10}^9\mathrm{N}/\mathrm{m}$	$3.95\times {10}^9\mathrm{N}/\mathrm{m}$	$4.17\times {10}^7\mathrm{N}/\mathrm{m}$
Worm gear and worm Pair joint	$3.38\times {10}^8\mathrm{N}/\mathrm{m}$	$5.11\times {10}^8\mathrm{N}/\mathrm{m}$	-

, the stiffness of the swing support bearing is set as axial stiffness 1.32 × 10⁸ N/m and radial stiffness 6.20 × 10⁷ N/m. The stiffness of the joint of the turntable bearing is set as axial stiffness 4.27 × 10⁹ N/m and radial stiffness 3.33 × 10⁹ N/m and angular stiffness 1.87 × 10⁷ N/m. Then, the table rotary system is meshed, and 124,354 grid units and 216,604 nodes are obtained. The grid model of the rotary table system is shown in Figure 8b.

4.1. Dynamic Characteristics Analysis of Rotary System at Different Swing Angles of the Turntable

The rotary table shown in Figure 8 was used as the object of study. The dynamic characteristics (swing angle of the rotary table) of the rotary system of the five-axis machine tool were analyzed for the negative clearance of the turntable bearing. The modes of the rotary table at θ = 0° are shown in Figure 9, and the results are shown in

Table 3. First six-order natural frequencies of the turntable.

Mode Order	1th	2th	3th	4th	5th	6th
Theoretical value/Hz	338.84	350.93	420.32	519.45	576.65	803.49
Theoretical vibration mode	X-direction bending vibration of worktable	Y-direction bending vibration of worktable	Z-direction bending vibration of worktable	X-direction torsional vibration of table	Z-direction torsional vibration of table	Y-direction torsional vibration of table

.

To analyze the effect of the dynamic characteristics of the rotary system under different swing angles of the turntable, the turntable shown in Figure 8 is considered as the research object. The dynamic characteristics of the turntable are analyzed at $\theta =0^{\circ }$, $\theta =\pm {30}^{\circ }$, $\theta =\pm 60^{\circ }$, and $\theta =\pm {90}^{\circ }$ angles with or without considering the bearing stiffness of the turntable. The first six-order natural frequencies of the turntable under different angles of the turntable are calculated in the same way as shown in Figure 10 and

Table 4. First six-order natural frequencies of the turntable at different swing angles.

Mode Order Theoretical Value/Hz	1th	2th	3th	4th	5th	6th
0° (the swing angles of the turntable)/Hz	338.84	350.93	420.32	519.45	576.65	803.49
30°	320.24	352.23	462.45	517.91	569.32	805.48
−30°	313.13	348.52	463.74	522.77	554.97	823.51
60°	291.93	350.75	501.87	545.23	562.45	833.19
−60°	280.96	349.02	498.07	530.73	568.04	870.91
90°	271.83	329.28	495.88	533.42	677	855.99
−90°	261.41	328.54	488.72	540.1	679.81	907.91

. The structure and the swing angle of the turntable have great influence on its dynamic characteristics. The first-order natural frequencies of the turntable decrease with the turntable swing angle, and the natural frequencies of the positive angle of the turntable are larger than those of the corresponding negative angle. The maximum natural frequency is 338.84 Hz when the turntable swing angle is 0°. The minimum natural frequency is 261.41 Hz when the turntable swing angle is −90°. With varying angle, the first-order natural frequency of the turntable changes by 77.43 Hz.

4.2. Dynamic Characteristics Analysis of Rotary Table System under Different Bearing Stiffness of the Turntable

According to the analysis in Section 3.2, the stiffness of the turntable bearing is affected by the external load, and $K_M$ is insensitive to the change in the external load. Therefore, the influence of the angular stiffness of the bearing stiffness is ignored when studying the dynamic characteristics analysis of the turntable under different external loads. Take the turntable shown in Figure 8 as the research object to explore the dynamic characteristics of the swing system of the table under different bearing stiffness of the table. Set the swing angle as 0°, set the axial stiffness and radial stiffness of the turntable bearing of different sizes, and also simulate the dynamic characteristic parameters of the rotary table system in the way described in the previous section. The natural frequency of the turntable varies with the bearing stiffness of the turntable as shown in Figure 11. Similarly, when different sizes of external load axial force $F_a$ and radial force $F_{\mathrm{r}}$ of turntable bearing are set, the natural frequency of the turntable changes with the external load of the turntable bearing, as shown in Figure 12.

As shown in Figure 11, in the process of changing the stiffness of the turntable bearing, the change in radial stiffness of the joint of the turntable bearing has a great influence on the natural frequency of the turntable. When the axial stiffness of the turntable bearing changes from 4.95 × 10⁵ N/mm to 4.95 × 10⁹ N/mm, and the radial stiffness changes from 3.95 × 10⁵ N/mm to 3.95 × 10⁹ N/mm, the first, second, fourth and fifth natural frequencies of the turntable will not change significantly with the axial stiffness of the turntable bearing joint, and the change in the first natural frequency is 30.3 Hz. The third and sixth order natural frequencies of the turntable increase with the increase in the axial and radial stiffness of the turntable bearing joint, and the increase in the radial stiffness is more obvious.

As shown in Figure 12, in the process of external axial force 0 to 120 kN and radial force 0 to 120 kN of the turntable bearing, the change in external load of turntable bearing has little influence on the natural frequency of the turntable, and the natural frequency of the turntable changes by less than 1 Hz. By comparing Figure 11 and Figure 12, it can be observed that under the same swing angle, the natural frequency of turntable is more dependent on the change in bearing stiffness than the change in limited external load on the turntable bearing.

5. Analysis of Dynamic Characteristics of Two Axis Rotary Table Five-Axis Machine Tool

In this study, the stiffness parameters of the equivalent joint surface of the two-axis rotary table five-axis machine tool are set on the basis of spring damping equivalence. Fixed binding has little effect on the dynamic characteristics and is set as the binding constraint.

First, the three-dimensional model of five-axis machine tool Figure 13a are imported into the finite element simulation software, and the boundary conditions of each joint surface are set. The fixed joint surface of the five-axis machine tool with the two-axis rotary table is mainly connected by bolts. The stiffness of the joint part of the turntable bearing is set as described in the previous Section 4. The setting of the stiffness of the main motion joint surface is shown in

Table 5. Machine tool joint stiffness.

Type of Junction	Interface Structure	Axial Stiffness (N/µm)	Normal Stiffness (N/µm)
Linear guide pair	Cradle and bed	-	348
	Slide rails and beams	-	326
	Spindle and slide rails	-	304
	Slide rails and columns	-	305
Ball screw pair	Cradle and bed	267	-
	Slide rails and spindle	262	-
	Slide rails and beams	253	-
	Slide rails and columns	257	-

. Next, as shown in Figure 13b, the whole machine is meshed into 524,232 mesh units and 831,997 nodes.

As shown in Figure 13a, the coordinate origin of the machining space of the machine tool in the reference system of the workbench is established, considering the coordinate origin (center point P1) as the center. A cube of side length of 300 mm is set as the machining space; each space has nine observation points. When the pendulum is at $\theta =0^{\circ }$, $\theta =\pm {30}^{\circ }$, and $\theta =\pm 60^{\circ }$, the observation position points are 9 × 5 = 45.

The double-pendulum five-axis machine tool’s spindle end’s 45 spatial positions are subjected to a modal analysis. Taking the spindle end at the coordinate origin X = 0 mm, Y = 0 mm, Z = 0 mm, θ = 0° as an example, the simulation results are shown in Figure 14, and the description of the modal parameters and modes is shown in

Table 6. First six-order natural frequencies of a two-axis rotary table five-axis machine tool.

Mode Order	1th	2th	3th	4th	5th	6th
Theoretical value/Hz	35.55	45.24	88.97	96.90	99.47	112.73
Theoretical vibration mode	Y-direction bending vibration of column	X-direction bending vibration of spindle box	Y-direction bending vibration of worktable	Y-direction bending vibration of spindle box	Z-direction torsional vibration of column	X-direction bending vibration of worktable

.

Following the same method, as shown in

Table 7. Machine tool modal parameters at different positions of the spindle end.

	Spindle End Position Parameters (mm)			f1 (Hz)	f2 (Hz)	f3 (Hz)	f4 (Hz)	f5 (Hz)	f6 (Hz)
0°	0	0	0	35.55	45.24	88.97	96.90	99.47	112.73
0°	−150	−150	150	34.05	43.35	86.82	94.19	98.89	106.78
0°	−150	150	150	34.04	43.33	83.54	94.16	99.26	105.38
0°	150	150	150	34.04	43.33	86.72	94.34	99.30	110.78
0°	150	−150	150	34.05	43.35	89.79	94.47	99.05	112.24
0°	−150	−150	−150	37.06	46.95	86.88	98.32	99.13	106.95
0°	−150	150	−150	37.05	46.93	83.58	98.32	99.56	105.45
0°	150	150	−150	37.05	46.93	86.82	98.47	99.60	110.83
0°	150	−150	−150	37.06	46.95	89.98	98.49	99.32	112.39
30°	0	0	0	35.54	45.25	88.61	96.89	99.45	112.88
30°	−150	−150	150	34.04	43.37	86.38	94.19	98.86	106.81
30°	−150	150	150	34.03	43.35	83.27	94.17	99.25	105.46
30°	150	150	150	34.03	43.35	86.36	94.32	99.29	110.86
30°	150	−150	150	34.04	43.37	89.37	94.42	99.04	112.34
30°	−150	−150	−150	37.04	46.95	86.43	98.29	99.09	106.99
30°	−150	150	−150	37.03	46.93	83.30	98.30	99.54	105.55
30°	150	150	−150	37.03	46.93	86.45	98.43	99.58	110.91
30°	150	−150	−150	37.04	46.95	89.52	98.44	99.29	112.48
60°	0	0	0	35.54	45.25	86.83	96.80	99.43	112.24
60°	−150	−150	150	34.04	43.37	84.67	94.13	98.78	106.10
60°	−150	150	150	34.03	43.35	81.71	94.12	99.21	104.74
60°	150	150	150	34.03	43.35	84.67	94.23	99.28	110.19
60°	150	−150	150	34.04	43.37	87.54	94.25	99.01	111.69
60°	−150	−150	−150	37.04	46.95	84.70	98.25	98.99	106.30
60°	−150	150	−150	37.03	46.93	81.73	98.26	99.49	104.83
60°	150	150	−150	37.03	46.93	84.73	98.36	99.57	110.25
60°	150	−150	−150	37.04	46.95	87.61	98.35	99.25	111.82
−30°	0	0	0	35.54	45.25	88.01	96.85	99.47	112.27
−30°	−150	−150	150	34.04	43.37	85.78	94.16	98.87	106.20
−30°	−150	150	150	34.03	43.35	82.73	94.15	99.26	104.85
−30°	150	150	150	34.03	43.35	85.74	94.29	99.31	110.36
−30°	150	−150	150	34.04	43.37	88.73	94.35	99.07	111.73
−30°	−150	−150	−150	37.04	46.95	85.81	98.28	99.09	106.38
−30°	−150	150	−150	37.03	46.93	82.76	98.29	99.55	104.93
−30°	150	150	−150	37.03	46.93	85.81	98.41	99.61	110.41
−30°	150	−150	−150	37.04	46.95	88.84	98.41	99.32	111.86
−60°	0	0	0	35.54	45.25	85.80	96.76	99.47	111.27
−60°	−150	−150	150	34.04	43.37	83.62	94.10	98.80	105.10
−60°	−150	150	150	34.03	43.35	80.78	94.10	99.23	103.75
−60°	150	150	150	34.03	43.35	83.62	94.19	99.31	109.37
−60°	150	−150	150	34.04	43.37	86.44	94.19	99.06	110.71
−60°	−150	−150	−150	37.04	46.95	83.64	98.24	99.00	105.29
−60°	−150	150	−150	37.03	46.93	80.79	98.25	99.50	103.85
−60°	150	150	−150	37.03	46.93	83.66	98.33	99.60	109.42
−60°	150	−150	−150	37.04	46.95	86.48	98.32	99.31	110.84

, the modal analysis is performed for the 45 different machining positions of the spindle end.

The first-order natural frequencies of machine tools vary in the working space. As shown in Table 7, when the swing angle is $\theta =0^{\circ }$, it is observed that the frequency of x-axis and y-axis changes slightly, while that of z-axis changes greatly. As the value of z-axis increases, the first-order natural frequency gradually decreases, with the variation at 3.01 Hz. When the table rotates at $\theta {=-60}^{\circ }$, $\theta =-30^{\circ }$, $\theta =0^{\circ }$, $\theta {=30}^{\circ }$, and $\theta {=60}^{\circ }$, it is observed that the swing table angle of the machine tool has little influence on the first-order natural frequency of the machine tool, and the variation is approximately 3.00 Hz.

Therefore, the dynamic characteristics of the two-axis rotary table five-axis machine tool are mainly determined by the changes in its Z-axis. The higher position of the z-axis guide near the workbench, the higher first-order natural frequency of the machine tool. The swing angle of the turntable has no obvious influence on the dynamic characteristics of the whole machine tool.

6. Conclusions

(1)

The stiffness of the joint bearing of the rotary table system is calculated, and the effect of bearing clearance and external load on the stiffness is analyzed. The increase in the screw tightening torque T and external load F_a and M_x,y increases $\left|G_a\right|$, which in turn increases K_a and K_M accordingly. The increase in K_M is significant, while K_a increases gradually when $\left|G_a\right|$ increases to 20 μm. As the radial clearance K_r increases, the radial stiffness $\left|G_r\right|$ first increases significantly and then tends to be stable.

(2)

When the swing angle of the turntable changes from −90° to 90°, the first-order natural frequency of the turntable decreases with the increase in the swing angle. The natural frequency of the positive angle is larger than that of the corresponding negative angle. When the swing angle is 0°, the natural frequency reaches the maximum (338.84 Hz), and when the swing angle is −90°, the natural frequency reaches the minimum (261.41 Hz). With the change in swing angle, the change in the first order natural frequency of the turntable is 77.43 Hz.

(3)

In the process of changing the turntable bearing stiffness, the change in the radial stiffness of the turntable bearing joint has a great influence on the natural frequency of the turntable. When the axial stiffness of the turntable bearing changes from 4.95 × 10⁵ N/mm to 4.95 × 10⁹ N/mm, and the radial stiffness changes from 3.95 × 10⁵ N/mm to 3.95 × 10⁹ N/mm, the first order natural frequency of the turntable does not change significantly with the axial stiffness of the turntable bearing joint, but increase with the increase in the radial stiffness of the turntable bearing joint, and the change in the first order natural frequency is 30.3 Hz. However, in the process of external axial force 0 to 120 kN and radial force 0 to 120 kN of the turntable bearing, the change in external load of turntable bearing has little influence on the natural frequency of the turntable, and the natural frequency of the turntable changes less than by 1 Hz.

(4)

The dynamic characteristic of the two-axis turntable five-axis machine tools mainly decided by the change in the Z-axis. The higher the Z-axis guide stays near the workbench, the higher is the first-order natural frequency of the machine tool.