Influence of the Machining Process on the Thrust Force and Mechanical Characteristics for the Direct Drive System

Abstract

This paper investigates the effects of the machining process on the thrust force and mechanical characteristics for the direct drive feed system driven by the flat permanent magnet synchronous linear motor (PMSLM) in machine tools considering the electromechanical couplings. Firstly, the cutting force in the machining process is researched. Then, the analytical model of the direct drive feed system is established and analyzed. The electromechanical couplings between the mechanical system and servo system in the direct drive feed system are studied. Furthermore, the influences of the cutting force on different couplings are analyzed, and the thrust force characteristic is analytically represented. Finally, the validity of the theoretical analysis is verified by the experiments, and the effects of the machining process on the dynamic precision of the feed system are discussed. The results show that the electromechanical couplings in the direct drive system will aggravate the effects of the machining process on the thrust force and mechanical characteristics of the feed system. A large number of new paired thrust harmonics will be produced. The influence of the machining process on the mechanical system will be extended from the discrete frequency point caused by the cutting force to the approximate frequency band caused by the thrust force, affecting the dynamic precision of the feed system and the cutting stability of the machine tool.

1. Introduction

The direct drive feed system incorporates the merits of simple structure, fast response, high velocity, and high acceleration, which has a bright future in railway transportation, robotic systems, micro aerial vehicle, electromagnetic launchers, and machine tools [1,2].

Due to the non-linearity of the drive circuit and the motor structure, there are lots of harmonics in the thrust force. The thrust fluctuation is one of the significant restrictions for the direct-driven feed system, which has been concerned and studied by scholars. Different kinds of structure optimization and compensation methods have been proposed [3,4,5,6,7,8,9,10], which have important significance for reducing the thrust fluctuation and improving the performance of the feed system. Although the direct drive feed system has outstanding performance advantages, its application is still far less than the traditional lead screw until now in the high-speed machine tool. There are obvious thrust harmonics and displacement fluctuations, especially in the machining process. In order to improve the machining accuracy, much research about the modeling of the cutting force and the optimization of the machining parameters has been conducted [11,12,13,14,15]. The influence of the machining process on the mechanical system is mainly concerned with the mechanical vibration caused by the cutting force itself under different cutting parameters and cutting stability. The dynamic precision of the feed system has an important influence on the final surface quality and the dimensional accuracy of parts [16]. The integrated modeling and optimization design method have been produced for the ball screw feed system in machine tools considering the machining process [17,18]. However, the coupling between the servo system and the mechanical system has not been paid enough attention in the current research. The dynamic characteristics of the mechanical system are ignored for the direct drive feed system. Although many scholars have paid attention to the electromechanical coupling problem in the direct drive feed system [19,20,21], the influence of the machining process has not been considered. In the machining process, there are many more disturbances caused by various non-ideal factors, such as cutting force, cooling pump, complex mechanical vibrations, and so on. The disturbances may aggravate the electromechanical couplings in the direct drive system, putting forward higher requirements for the dynamic precision and the control system of the feed system.

Therefore, in this paper, the effects of the machining process on the thrust force and mechanical characteristics are investigated for the direct drive feed system in machine tools considering the dynamic electromechanical couplings. Firstly, the cutting force in the milling process and the dynamic characteristic of the direct drive system are analyzed, respectively. Then the electromechanical couplings between the mechanical system and servo system in the direct drive feed system are studied. Furthermore, the influences of the cutting force on different couplings are analyzed, and the thrust force characteristic is analytically represented. Finally, the validity of the previous theoretical analysis is verified by the experiments, and the effects of the machining process on the dynamic precision of the feed system are discussed.

2. Analysis of the Cutting Force and the Dynamic Characteristics of the Direct Drive System

2.1. Analysis of the Cutting Force in the Milling Process

The milling is a discontinuous cutting process involving multiple teeth. The cutting force is ever-changing in the milling process. In order to improve the machining accuracy, it is necessary to predict the milling force for improving the tool life and maintaining the stability of machining quality. Furthermore, an accurate force model is also the basis for optimizing milling parameters and maintaining process stability. Considerable research has been conducted to build a reliable cutting force prediction model [22,23,24]. This paper mainly focuses on the influence of cutting force on the direct drive feed system. Therefore, the cutting force itself is not further studied. Based on the previous research, in the milling process, the cutting force can be expressed as:

$$\{\begin{array}{c}{F}_{cx}\left(t\right)={\displaystyle \sum _{j=1}}{F}_{cxj}\sin \left({\omega }_{cxj}t+{\varphi }_{cxj}\right)\\ F_{cy}\left(t\right)={\displaystyle \sum _{j=1}}{F}_{cyj}\sin \left({\omega }_{cyj}t+{\varphi }_{cyj}\right)\\ F_{cz}\left(t\right)={\displaystyle \sum _{j=1}}{F}_{czj}\sin \left({\omega }_{czj}t+{\varphi }_{czj}\right)\end{array}$$

(1)

It can be seen from Equation (1) there are lots of harmonics in the cutting force with different frequencies which are related to the machining parameters.

2.2. Analysis of Dynamic Characteristic of the Direct Drive System

The analytical model of the direct drive feed system is established as shown in Figure 1. The drive circuit is consisting of modulator circuit, rectifier, filter circuit and inverter. In order to produce the ideal thrust force, the three phase currents are expected to be ideal sine waveforms. However, there are different kinds of the nonlinear factors in the drive circuit (such as dead time effect, error of the sensor, back electromotive force), leading to lots of harmonics.

The three phase windings of PMSLM are a star connection without the mid-line, which does not contain three times harmonics. Ignoring the transient response and setting the initial phase equal to zero, the servo current can be obtained as:

$$\{\begin{array}{l}{i}_a\left(t\right)={\displaystyle \sum _{m=1,5,7,\dots ,6i\pm 1}}{I}_{am}\sin \left[m\left(\frac{\pi v}{\tau }t+\theta \right)\right]\\ i_b\left(t\right)={\displaystyle \sum _{m=1,5,7,\dots ,6i\pm 1}}{I}_{bm}\sin \left[m\left(\frac{\pi v}{\tau }t+\theta -2\pi /3\right)\right]\\ i_b\left(t\right)={\displaystyle \sum _{m=1,5,7,\dots ,6i\pm 1}}{I}_{cm}\sin \left[m\left(\frac{\pi v}{\tau }t+\theta -4\pi /3\right)\right]\end{array}$$

(2)

In Figure 1, x₀o₀z₀ and xoz are the coordinate system of the mover and the stator, respectively. x represents the position of the mover in the coordinate system of the stator. Based on the Maxwell’s equations, the magnetic field distributions in air-gap are calculated and given by:

$$B_m\left(x,z\right)=\frac{4B_r}{\pi }{\displaystyle \sum }_{n=1,3,5}^{\infty }\{\frac{{\left(-1\right)}^{\frac{n+3}{2}}}{n}\cdot \frac{\sin \frac{n\pi w_p}{2\tau }sh\left(\frac{n\pi }{\tau }h\right)}{sh\left[\frac{n\pi }{\tau }\left(h+g\right)\right]}ch\left[\frac{n\pi }{\tau }\left(h+g-z\right)\right]\cos \left(\frac{n\pi }{\tau }z\right)\}$$

(3)

The SC transformation is introduced to analyze the detent force. Based on the analysis in [25,26], the magnetic field distributions in air-gap considering the slotting effect and end effect can be obtained as:

$$B_{ma}\left(x,z\right)=B_m\left(x,z\right)\cdot {\lambda }_s\left(x,z\right)\cdot {\lambda }_e\left(x,z\right)$$

(4)

where, λ_s(x,z) and λ_e(x,z) represent the relative permeance function considering the slotting effect and end effect, respectively.

The thrust force of PMSLM can be calculated as follows using the electromagnetic energy method.

$$\begin{array}{ll}\hfill F_m\left(t\right)=& \frac{1}{v}\left\{\frac{\partial }{\partial t}\left[\left(L_a+M_{ab}+M_{ac}\right)\cdot i_a\left(t\right)\right]-Nl\frac{\partial }{\partial t}\left[{\int }_{x-\tau /2}^{x+\tau /2}{B}_a\left(x\right)dx\right]\right\}\cdot i_a\left(t\right)+\\ & \frac{1}{v}\left\{\frac{\partial }{\partial t}\left[\left(L_b+M_{ab}+M_{bc}\right)\cdot i_b\left(t\right)\right]-Nl\frac{\partial }{\partial t}\left[{\int }_{x-\tau /2}^{x+\tau /2}{B}_b\left(x\right)dx\right]\right\}\cdot i_b\left(t\right)+\\ & \frac{1}{v}\left\{\frac{\partial }{\partial t}\left[\left(L_c+M_{bc}+M_{ac}\right)\cdot i_c\left(t\right)\right]-Nl\frac{\partial }{\partial t}\left[{\int }_{x-\tau /2}^{x+\tau /2}{B}_c\left(x\right)dx\right]\right\}\cdot i_c\left(t\right)\end{array}$$

(5)

Ignoring the harmonic components of the self-inductance and mutual inductance in the coil, substituting Equations (2) and (4) into Equation (5), the thrust force can be obtained as:

$$F_m\left(t\right)=F_0\left(t\right)+F_{mr}\left(t\right)=F_0\left(t\right)+{\displaystyle \sum }_i{F}_{mi}\left(t\right)$$

(6)

The mechanical structure of the direct drive feed system consists of worktable, linear guide, motor mover, permanent magnet, linear encoder and machine bed. The flexibility of the worktable and the machine bed is ignored. The mechanical vibrations of the worktable can be assumed as the linear displacements along three axis x, y, and z, and the angle displacements around three axes, θx, θy and θz, respectively. Considering the cutting force, the dynamic equation of the mechanical system can be obtained as follows.

$$M\ddot{\mathit{X}}+C\dot{\mathit{X}}+K\mathit{X}=H_c{F}_{\mathrm{c}}+H_m{F}_{mr}$$

(7)

where, $X={\left[\begin{array}{cccccc}x& y& z& {\theta }_x& {\theta }_y& {\theta }_z\end{array}\right]}^T$, M, C, K are mass matrix, damping matrix and stiffness matrix, respectively.

$$H_m={\left[\begin{array}{cccccc}{H}_{xm}& H_{ym}& H_{zm}& H_{{\theta }_x}& H_{{\theta }_y}& H_{{\theta }_z}\end{array}\right]}^T, H_c=\left[\begin{array}{ccc}{H}_{xc}& 0& 0\\ 0& H_{yc}& 0\\ 0& 0& H_{zc}\\ 0& H_{xyc}& H_{xzc}\\ H_{yxc}& 0& H_{yzc}\\ H_{zxc}& H_{zyc}& 0\end{array}\right], F_c=\left[\begin{array}{c}{F}_{cx}\\ F_{cy}\\ F_{cz}\end{array}\right].$$

In order to facilitate subsequent calculations, the coupling between each modal and higher-order modes of mechanical vibrations is ignored. The stiffness and damping in the vertical and tangential directions are large, leading to tiny amplitudes of the vibrations [27]. The main vibration responses in different directions can be obtained as:

$$X=\left[\begin{array}{c}x\\ {\theta }_x\\ {\theta }_y\\ {\theta }_z\end{array}\right]=\left[\begin{array}{c}{x}_m+x_c\\ {\theta }_{mx}+{\theta }_{cx}\\ {\theta }_{my}+{\theta }_{cy}\\ {\theta }_{mz}+{\theta }_{cz}\end{array}\right]=\left[\begin{array}{c}{\displaystyle \sum _i{x}_{mi}\sin \left({\omega }_{xmi}t+{\phi }_{xmi}\right)+{\displaystyle \sum _i{x}_{ci}\sin \left({\omega }_{xci}t+{\phi }_{xci}\right)}}\\ {\displaystyle \sum _i{\theta }_{mxi}\sin \left({\omega }_{\theta mxi}t+{\phi }_{\theta mxi}\right)+{\displaystyle \sum _i{\theta }_{cxi}\sin \left({\omega }_{\theta cxi}t+{\phi }_{\theta cxi}\right)}}\\ {\displaystyle \sum _i{\theta }_{myi}\sin \left({\omega }_{\theta myi}t+{\phi }_{\theta myi}\right)+{\displaystyle \sum _i{\theta }_{cyi}\sin \left({\omega }_{\theta cyi}t+{\phi }_{\theta cyi}\right)}}\\ {\displaystyle \sum _i{\theta }_{mzi}\sin \left({\omega }_{\theta mzi}t+{\phi }_{\theta mzi}\right)}+{\displaystyle \sum _i{\theta }_{czi}\sin \left({\omega }_{\theta czi}t+{\phi }_{\theta czi}\right)}\end{array}\right]$$

(8)

It can be obtained from Equation (8) that there are complex multi-frequency mechanical vibrations in the direct drive feed system considering the excitation of the thrust force and cutting force.

3. Analysis of the Dynamic Electromechanical Couplings in the Direct Drive System Considering the Cutting Force

In the direct drive feed system, the mover of PMSLM is directly connected with the mechanical system. The couplings between the servo system and mechanical system become more complex and prominent. Multiple coupling loops, which are overlapped and influenced by each other, determine the dynamic precision of the feed system. The coupling condition and influence factor of each coupling are different. However, in electromechanical couplings, the intermediate coupling parameters all are motor thrust harmonics [21]. In the machining process, the cutting force harmonics with multiple frequencies act on the mechanical system of the direct drive feed system, which will affect the electromechanical couplings as shown in Figure 2. Therefore, in this section, the effects of the cutting force on the thrust force are mainly analyzed considering different kinds of couplings.

3.1. Effects of the Cutting Force on the Thrust Force Considering the Air-Gap Fluctuation

In the direct drive feed system, the mechanical vibration in the normal direction of the worktable will produce the mover eccentricity, leading to air-gap fluctuation. Especially in the machining process, the torsional vibration of the mechanical system will be aggravated by the cutting forces in three directions.

Considering the mechanical pitch and roll vibrations, the relative permeance functions between actual and ideal air-gap are introduced to represent the effects of the air-gap fluctuations on the magnetic field distributions in the air-gap.

In the roll vibration, the high order terms are ignored in three coefficients, and the whole relative permeability function caused by the air-gap fluctuation can be calculated as shown in Equation (9).

$${\lambda }_r\left(t\right)=L_{mb}+\frac{L_{mb}}{6}{\left(\frac{L_{mb}}{2g}\right)}^2{A}_r^2-\frac{L_{mb}}{6}{\left(\frac{L_{mb}}{2g}\right)}^2{A}_r^2\cos \left(2{\omega }_{r0}t\right)$$

(9)

In the pitch vibration, the relative permeance function between actual and ideal air-gap is equal to

$${\lambda }_p\left(x_0,t\right)=a_0+{\displaystyle \sum }_{i=1}^{\infty }\left[a_i\cos \left(\frac{2i\pi }{L}{x}_0\right)+b_i\sin \left(\frac{2i\pi }{L}{x}_0\right)\right]$$

(10)

where, $a_0=1+\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2-\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2\cos \left(2{\omega }_{p0}t\right)$, $a_i=3{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^3\sin \left({\omega }_{p0}t\right)-{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^3\sin \left(3{\omega }_{p0}t\right)$, $b_i={\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2+{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2\cos \left(2{\omega }_{p0}t\right)$.

In order to simplify the calculation process, the slotting effect and end effect are ignored. Using the Equations (4), (9) and (10), the magnetic field distributions considering the air-gap fluctuation can be obtained as:

$$B\left(x+x_0,z\right)={\lambda }_r\left(x_0,t\right){\lambda }_p\left(x_0,t\right)B_{ma}\left(x,z\right)$$

(11)

Substituting Equations (2) and (11) into Equation (5), the thrust force can be given by

$$F_{cag}={F^{\prime }}_0\left(t\right)+{F^{\prime }}_r\left(t\right)+F_{cr}\left(t\right)+F_{cp}\left(t\right)+F_{crp}\left(t\right)$$

(12)

where, ${F_0}^{\prime }\left(t\right)$ represents the new nominal thrust force, and

$${F_0}^{\prime }\left(t\right)=\left\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[1+\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2\right]\cdot \left[\frac{\tau }{n\pi }\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]+\\ {\displaystyle {\displaystyle \sum }_{i=1,2,3\dots }^{\infty }}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2\cdot \left[\frac{4i{L}_a{\tau }^2\pi {\left(-1\right)}^i{B}_n}{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\end{array}\right\}\cdot F_0\left(t\right)$$

${F^{\prime }}_r\left(t\right)$ represents the original thrust harmonics, and

$${F_r}^{\prime }\left(t\right)={\displaystyle \sum }_{n,m,j}^{\infty }\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[1+\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2\right]\cdot \left[\frac{\tau }{n\pi }\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]+\\ {\displaystyle \sum _{i=1,2,3,\dots }^{\infty }}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2\left[\frac{4i{L}_a{\tau }^2\pi {\left(-1\right)}^i}{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}{B}_n\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\end{array}\}{F}_{rn}\cos \left(6j{\omega }_{0}t\right)$$

F_cr(t) represents the new thrust harmonics caused by the roll vibrations, and

$$F_{cr}\left(t\right)={\displaystyle \sum }_{n,m,j}^{\infty }\left\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[1+\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2\right]\cdot \left[\frac{\tau }{n\pi }\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]+\\ {\displaystyle \sum _{i=1,2,3\dots }^{\infty }}\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot {\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2\cdot \\ \left[\frac{4i{L}_a{\tau }^2\pi }{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}{\left(-1\right)}^i{B}_n\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\end{array}\end{array}\right\}·\left[\begin{array}{c}{F}_{rn}\cos \left(6j{\omega }_{0}t+2{\omega }_{r0}t\right)+\\ F_{rn}\cos \left(6j{\omega }_{0}t-2{\omega }_{r0}t\right)\end{array}\right]$$

F_cp(t) represents the new thrust harmonics caused by the pitch vibrations, and

$$F_{cp}\left(t\right)=F_{cp1}\left(t\right)+F_{cp2}\left(t\right)+F_{cp3}\left(t\right)$$

$$F_{cp1}\left(t\right)={\displaystyle \sum }_{n,m,j}^{\infty }{F}_{cp}{\displaystyle \sum }_{i=1,2,3\dots }^{\infty }\left\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[3{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^3\right]\cdot \\ \left[\frac{2n{L}_a^2\tau \pi {\left(-1\right)}^i{B}_n}{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\end{array}\right\}\cdot \left[\begin{array}{c}\sin \left(6j{\omega }_{0}t+{\omega }_{p0}t\right)+\\ \sin \left(6j{\omega }_{0}t-{\omega }_{p0}t\right)\end{array}\right]$$

$$F_{cp2}\left(t\right)={\displaystyle \sum }_{n,m,j}^{\infty }\left\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[-\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2\right]\cdot \left[\frac{B_n\tau }{n\pi }\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]+\\ {\displaystyle \sum _{i=1,2,3\dots }^{\infty }}\left\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot {\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2\cdot \\ \left[\frac{4i{L}_a{\tau }^2\pi }{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}{\left(-1\right)}^i{B}_n\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\end{array}\right\}\end{array}\right\}·\left[\begin{array}{c}{F}_{cp}\cos \left(6j{\omega }_{0}t+2{\omega }_{p0}t\right)+\\ F_{cp}\cos \left(6j{\omega }_{0}t-2{\omega }_{p0}t\right)\end{array}\right]$$

$$F_{cp3}\left(t\right)={\displaystyle \sum }_{n,m,j}^{\infty }{F}_{cp}{\displaystyle \sum }_{i=1,2,3\dots }^{\infty }\left\{\begin{array}{c}\left[L_b+\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[-{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^3\right]\cdot \\ \left[\frac{2n{L}_a^2\tau \pi {\left(-1\right)}^i{B}_n}{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\end{array}\right\}\left[\begin{array}{c}\sin \left(6j{\omega }_{0}t+3{\omega }_{p0}t\right)+\\ \sin \left(6j{\omega }_{0}t-3{\omega }_{p0}t\right)\end{array}\right]$$

F_crp(t) represents the new thrust harmonics caused by the couplings between the roll and pitch, and

$$F_{crp}\left(t\right)=F_{crp1}\left(t\right)+F_{crp2}\left(t\right)+F_{crp3}\left(t\right)$$

$$F_{crp1}\left(t\right)={\displaystyle \sum _{n,m,j}^{\infty }{F}_{crp}{\displaystyle \sum _{i=1,2,3\dots }^{\infty }\left\{\left[-\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot 3{\left(\frac{L_a{A}_p}{2i\pi g}\right)}^3\cdot \left[\frac{2n{L}_a^2\tau \pi {\left(-1\right)}^i}{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}{B}_n\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\right\}}\cdot \left[\begin{array}{l}\sin \left(2{\omega }_{r0}t+{\omega }_{p0}t+6j{\omega }_{0}t\right)+\\ \sin \left(2{\omega }_{r0}t+{\omega }_{p0}t-6j{\omega }_{0}t\right)+\\ \sin \left(2{\omega }_{r0}t-{\omega }_{p0}t+6j{\omega }_{0}t+\pi \right)+\\ \sin \left(2{\omega }_{r0}t-{\omega }_{p0}t-6j{\omega }_{0}t+\pi \right)\end{array}\right]}$$

$$F_{crp2}\left(t\right)={\displaystyle \sum _{n,m,j}^{\infty }\left\{\begin{array}{l}\left[-\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot \left[-\frac{1}{6}{\left(\frac{L_a{A}_p}{2g}\right)}^2\right]\cdot \left[B_n\frac{\tau }{n\pi }\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]+\\ {\displaystyle \sum _{i=1,2,3\dots }^{\infty }\left\{\left[-\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot {\left(\frac{L_a{A}_p}{2i\pi g}\right)}^2\cdot \left[\frac{4i{L}_a{\tau }^2\pi {\left(-1\right)}^i}{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}{\left(-1\right)}^i{B}_n\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\right\}}\end{array}\right\}\cdot \left[\begin{array}{l}{F}_{crp}\cos \left(2{\omega }_{r0}t+2{\omega }_{p0}t+6j{\omega }_{0}t\right)+\\ F_{crp}\cos \left(2{\omega }_{r0}t+2{\omega }_{p0}t-6j{\omega }_{0}t\right)+\\ F_{crp}\cos \left(2{\omega }_{r0}t-2{\omega }_{p0}t+6j{\omega }_{0}t\right)+\\ F_{crp}\cos \left(2{\omega }_{r0}t-2{\omega }_{p0}t-6j{\omega }_{0}t\right)\end{array}\right]}$$

$$F_{crp3}\left(t\right)={\displaystyle \sum _{n,m,j}^{\infty }{F}_{crp}{\displaystyle \sum _{i=1,2,3\dots }^{\infty }\left\{\left[\frac{L_b}{6}{\left(\frac{L_b}{2g}\right)}^2{A}_r^2\right]\cdot {\left(\frac{L_a{A}_p}{2i\pi g}\right)}^3\cdot \left[\frac{2n{L}_a^2\tau \pi }{{\left(2i\pi \tau \right)}^2-{\left(n{L}_a\pi \right)}^2}{\left(-1\right)}^i{B}_n\sin \left(\frac{n{L}_a\pi }{2\tau }\right)\right]\right\}}\cdot \left[\begin{array}{l}\sin \left(2{\omega }_{r0}t+3{\omega }_{p0}t+6j{\omega }_{0}t\right)+\\ \sin \left(2{\omega }_{r0}t+3{\omega }_{p0}t-6j{\omega }_{0}t\right)+\\ \sin \left(2{\omega }_{r0}t-3{\omega }_{p0}t+6j{\omega }_{0}t+\pi \right)+\\ \sin \left(2{\omega }_{r0}t-3{\omega }_{p0}t-6j{\omega }_{0}t+\pi \right)\end{array}\right]}$$

$$F_{rn}=\frac{12Nl{B}_r}{\pi }\frac{1}{n}\sin \frac{n\pi w_p}{2\tau }\frac{\mathrm{sh}\left(\frac{n\pi }{\tau }h\right)}{\mathrm{sh}\left[\frac{n\pi }{\tau }\left(h+g\right)\right]}\cdot \mathrm{ch}\left(\frac{n\pi h_s}{2\tau }\right)I_m\sin \left(\frac{n\pi }{2}\right), F_{crp}=0.75Nl{I}_m\sin \left(\frac{n\pi }{2}\right)$$

$$B_n=\frac{4B_r}{\pi }\frac{1}{n}\sin \frac{n\pi w_p}{2\tau }\frac{\mathrm{sh}\left(\frac{n\pi }{\tau }h\right)}{\mathrm{sh}\left[\frac{n\pi }{\tau }\left(h+g\right)\right]}\cdot \mathrm{ch}\left(\frac{n\pi h_s}{2\tau }\right), F_{cp}=1.5Nl{I}_m\sin \left(\frac{n\pi }{2}\right)$$

It can be obtained from Equation (12) that the air-gap fluctuation will produce lots of new unbalanced magnetic thrust harmonics. Meanwhile, there is complex coupling between the effects of the roll and pitch vibrations, leading to more new coupled thrust harmonics. The frequencies of these new coupled harmonics are related to the frequencies of the original thrust force harmonics and these of the mechanical torsional vibrations. The amplitudes of these new unbalanced magnetic thrust harmonics depend on the structure parameters of PMSLM and amplitudes of the mechanical vibrations. Only the ripple thrust harmonics are considered in previous calculation. The slot effect and end effect will produce more new coupling thrust harmonics. Their influences on motor thrust can be obtained by using a calculation method similar to the ripple thrust, which ismethods not discussed here.

3.2. Effects of the Cutting Force on the Thrust Force Considering the Feedback Control

In the direct drive feed system, the linear encoder, which is mounted on the worktable, is generally adopted to realize the full-closed loop control. In the machining process, the mechanical vibrations in the feed direction caused by the cutting force and other disturbances will be collected by the reading head of the encoder to produce the feedback harmonics. The feedback harmonics will produce the regulating thrust force to suppress the disturbances through the servo control system. However, because of the non-linearity of the drive circuit and PMSLM, the feedback control will intensify the coupling between the different harmonics, affecting the thrust force characteristics.

In order to reduce the calculation process, the motion error caused by one of the cutting force harmonics is considered here, and the feedback error is assumed to be

$$x_{ce}\left(t\right)=x_{ce0}\cos \left(2\pi f_{c0}t+{\alpha }_{ce0}\right)$$

(13)

The three-phase current of the drive circuit can be calculated as follows considering the feedback harmonics.

$$\{\begin{array}{l}{i}_a\left(t\right)={\displaystyle \sum _m}{I}_{am}\sin \left[m\left(\frac{\pi v}{\tau }t+\theta \right)\right]+i_{qce0}\cos \left(2\pi f_{c0}t+{\gamma }_{ce0}\right)\sin \left(\omega t\right) \\ i_b\left(t\right)={\displaystyle \sum _m}{I}_{bm}\sin \left[m\left(\frac{\pi v}{\tau }t+\theta -2\pi /3\right)\right]+i_{qce0}\cos \left(2\pi f_{c0}t+{\gamma }_{ce0}\right)\sin \left(\omega t-2\pi /3\right)\\ i_b\left(t\right)={\displaystyle \sum _m}{I}_{cm}\sin \left[m\left(\frac{\pi v}{\tau }t+\theta -4\pi /3\right)\right]+i_{qce0}\cos \left(2\pi f_{c0}t+{\gamma }_{ce0}\right)\sin \left(\omega t-4\pi /3\right)\end{array}$$

(14)

Substituting Equations (4) and (14) into Equation (5), the thrust force can be obtained as:

$$F_{mf}\left(t\right)=F_0\left(t\right)+F_{mr}\left(t\right)+F_{ce}\left(t\right)$$

(15)

where, F₀(t) represents the nominal thrust force, F_mr(t) represents the original thrust harmonics, and F_ce(t) represents the new thrust harmonics caused by the feedback harmonics, $F_{ce}\left(t\right)=F_{ce0}\left(t\right)+F_{cer}\left(t\right)+F_{ces}\left(t\right)+F_{cers}\left(t\right)$.

F_ce0(t) represents the nominal regulating thrust force caused by the feedback harmonics, and $F_{ce0}\left(t\right)=a_0{F}_{c0}\cos \left(2\pi f_{c0}t+{\eta }_{ce0}\right)$.

F_cer(t) represents the new thrust harmonics caused by the couplings between the feedback harmonics and ripple thrust, and

$$F_{cer}\left(t\right)={\displaystyle \sum }_{\begin{array}{c}n+1=6j or n-1=6j\\ n=1,3,5\dots ,j=1,2,3,\dots \end{array}}^{\infty }1.5Nl{a}_0{B}_n{I}_{qr0}\sin \left(\frac{n\pi }{2}\right)\left\{\begin{array}{c}\cos \left[\left(\frac{6j\pi v}{\tau }+2\pi f_{c0}\right)t+{\eta }_{ce0}\right]+\\ \cos \left[\left(\frac{6j\pi v}{\tau }-2\pi f_{c0}\right)t-{\eta }_{ce0}\right]\end{array}\right\}$$

F_ces(t) represents the new thrust harmonics caused by the couplings between the feedback harmonics and slot force, and

$$F_{ces}\left(t\right)={\displaystyle \sum }_{i=1}^{\infty }0.75Nl{a}_i{B}_1{I}_{qr0}\cos \left(\frac{i\tau \pi }{{\tau }_s}\right)\left[\begin{array}{c}\cos \left(2\pi f_{c0}t+\frac{2i\pi v}{{\tau }_s}t+{\eta }_{ce0}\right)+\\ \cos \left(2\pi f_{c0}t-\frac{2i\pi v}{{\tau }_s}t+{\eta }_{ce0}\right)\end{array}\right]$$

F_cers(t) represents the new thrust harmonics caused by the couplings, and

$$F_{cers}\left(t\right)={\displaystyle \sum _{\begin{array}{l}n+1=6j \\ or n-1=6j\\ n=1,3,5\dots ,\\ j=1,2,3,\dots \end{array}}^{\infty }{\displaystyle \sum _{i=1}^{\infty }0.75Nl{a}_i{B}_n{I}_{qr0}\sin \left(\frac{n\pi }{2}\right)\cos \left(\frac{i\tau \pi }{{\tau }_s}\right)}\left\{\begin{array}{l}\cos \left[\left(\frac{6j\pi v}{\tau }+\frac{2i\pi v}{{\tau }_s}+2\pi f_{c0}\right)t+{\eta }_{ce0}\right]+\\ \cos \left[\left(\frac{6j\pi v}{\tau }+\frac{2i\pi v}{{\tau }_s}-2\pi f_{c0}\right)t-{\eta }_{ce0}\right]+\\ \cos \left[\left(\frac{6j\pi v}{\tau }-\frac{2i\pi v}{{\tau }_s}+2\pi f_{c0}\right)t+{\eta }_{ce0}\right]+\\ \cos \left[\left(\frac{6j\pi v}{\tau }-\frac{2i\pi v}{{\tau }_s}-2\pi f_{c0}\right)t-{\eta }_{r0}\right]\end{array}\right\}}$$

$$a_0=\frac{1}{K_c}\left[1-1.6\frac{w_s}{{\tau }_s}{\beta }_c\right], a_i=-\frac{4}{i\pi }{\beta }_c\left[0.5+\frac{{\left(\frac{w_s}{{\tau }_s}i\right)}^2}{0.78215-2{\left(\frac{w_s}{{\tau }_s}i\right)}^2}\right]\sin \left(1.6\pi \frac{w_s}{{\tau }_s}i\right)$$

It can be obtained from Equation (15) that the couplings between the feedback harmonics and the non-linearity of the drive circuit and PMSLM will produce a great number of the coupled thrust harmonics.

In addition, the mechanical vibrations may change the optical path between the reading head mounted on the worktable and the scale mounted on the machine bed, leading to encoder errors. Based on the results in Ref. [28], the different kinds of encoder errors caused by the mechanical vibrations in the machining process can be represented as:

$$\{\begin{array}{l}\Delta {\delta }_{ye}=A_{\Delta {\delta }_y}\sin \left({\omega }_{\Delta {\delta }_y}t+{\varphi }_{\Delta {\delta }_y}\right)\\ \Delta {\delta }_{re}=\Delta {\delta }_{r0}+A_{\Delta {\delta }_r}\cos \left({\omega }_{\Delta {\delta }_r}t+{\varphi }_{\Delta {\delta }_r}\right) \\ \Delta {\delta }_{pe}=\Delta {\delta }_{p0}+A_{\Delta {\delta }_p}\cos \left({\omega }_{\Delta {\delta }_p}t+{\varphi }_{\Delta {\delta }_p}\right)\end{array}$$

(16)

The three-phase servo currents considering encoder errors can be given by

$$\{\begin{array}{l}{i}_a\left(t\right)=\left[\begin{array}{c}{i}_{q\Delta {\delta }_{r0}}+i_{q\Delta {\delta }_{p0}}+{i^{\prime }}_{\Delta {\delta }_y}\sin \left({\omega }_{\Delta {\delta }_y}t+{\phi }_{\Delta {\delta }_y}\right)+\\ {i^{\prime }}_{\Delta {\delta }_r}\cos \left({\omega }_{\Delta {\delta }_r}t+{\phi }_{\Delta {\delta }_r}\right)+{i^{\prime }}_{\Delta {\delta }_p}\cos \left({\omega }_{\Delta {\delta }_p}t+{\phi }_{\Delta {\delta }_p}\right)\end{array}\right]\sin \left(\omega t\right)\\ i_b\left(t\right)=\left[\begin{array}{c}{i}_{q\Delta {\delta }_{r0}}+i_{q\Delta {\delta }_{p0}}+{i^{\prime }}_{\Delta {\delta }_y}\sin \left({\omega }_{\Delta {\delta }_y}t+{\phi }_{\Delta {\delta }_y}\right)+\\ {i^{\prime }}_{\Delta {\delta }_r}\cos \left({\omega }_{\Delta {\delta }_r}t+{\phi }_{\Delta {\delta }_r}\right)+{i^{\prime }}_{\Delta {\delta }_p}\cos \left({\omega }_{\Delta {\delta }_p}t+{\phi }_{\Delta {\delta }_p}\right)\end{array}\right]\sin \left(\omega t-2\pi /3\right)\\ i_b\left(t\right)=\left[\begin{array}{c}{i}_{q\Delta {\delta }_{r0}}+i_{q\Delta {\delta }_{p0}}+{i^{\prime }}_{\Delta {\delta }_y}\sin \left({\omega }_{\Delta {\delta }_y}t+{\phi }_{\Delta {\delta }_y}\right)+\\ {i^{\prime }}_{\Delta {\delta }_r}\cos \left({\omega }_{\Delta {\delta }_r}t+{\phi }_{\Delta {\delta }_r}\right)+{i^{\prime }}_{\Delta {\delta }_p}\cos \left({\omega }_{\Delta {\delta }_p}t+{\phi }_{\Delta {\delta }_p}\right)\end{array}\right]\sin \left(\omega t-4\pi /3\right)\end{array}$$

(17)

The thrust force considering the encoder error can be obtained as follows.

$$F_{cee}=F_{0\Delta {\delta }_{r0}}\left(t\right)+F_{0\Delta {\delta }_{p0}}\left(t\right)+F_{\Delta \delta }\left(t\right)+F_{\Delta \delta r}\left(t\right)+F_{\Delta \delta sr}\left(t\right)$$

(18)

where, $F_{0\Delta {\delta }_{r0}}\left(t\right)$ and $F_{0\Delta {\delta }_{p0}}\left(t\right)$ represent the nominal thrust force caused by the encoder errors due to pitch and roll.

$F_{\Delta \delta }\left(t\right)$ represents the regulating thrust force caused by the encoder errors, and

$$F_{\Delta \delta }\left(t\right)=F_{\Delta {\delta }_y}\sin \left({\omega }_{\Delta {\delta }_y}t+{\varsigma }_{\Delta {\delta }_y}\right)+F_{\Delta {\delta }_r}\cos \left({\omega }_{\Delta {\delta }_r}t+{\varsigma }_{\Delta {\delta }_r}\right)+F_{\Delta {\delta }_p}\cos \left({\omega }_{\Delta {\delta }_p}t+{\varsigma }_{\Delta {\delta }_p}\right)$$

$F_{\Delta \delta r}\left(t\right)$ represents the thrust harmonics caused by the couplings between the encoder errors and ripple thrust force, and

$$F_{\mathsf{\Delta }\delta r}\left(t\right)=1.5Nl{a}_0\left(y_s\right)B_n\sin \left(\frac{n\pi }{2}\right)$$

$$\sum _{\begin{array}{c}n=1,3,5,\dots \\ n+1=6j \mathrm{or} n-1=6j\\ j=1,2,3,\dots \end{array}}^{\infty }\left\{\begin{array}{l}{i^{\prime }}_{\Delta {\delta }_y}\cos \left[\left(\frac{6j\pi v}{\tau }+{\omega }_{\Delta {\delta }_y}\right)t+{\varsigma }_{\Delta {\delta }_y}\right]+{i^{\prime }}_{\Delta {\delta }_y}\cos \left[\left(\frac{6j\pi v}{\tau }-{\omega }_{\Delta {\delta }_y}\right)t+{\varsigma }_{\Delta {\delta }_y}\right]+\\ {i^{\prime }}_{\Delta {\delta }_r}\cos \left[\left(\frac{6j\pi v}{\tau }+{\omega }_{\Delta {\delta }_r}\right)t+{\varsigma }_{\Delta {\delta }_r}\right]+{i^{\prime }}_{\Delta {\delta }_r}\cos \left[\left(\frac{6j\pi v}{\tau }-{\omega }_{\Delta {\delta }_r}\right)t+{\varsigma }_{\Delta {\delta }_r}\right]+\\ {i^{\prime }}_{\Delta {\delta }_p}\cos \left[\left(\frac{6j\pi v}{\tau }+{\omega }_{\Delta {\delta }_p}\right)t+{\varsigma }_{\Delta {\delta }_p}\right]+{i^{\prime }}_{\Delta {\delta }_p}\cos \left[\left(\frac{6j\pi v}{\tau }-{\omega }_{\Delta {\delta }_p}\right)t+{\varsigma }_{\Delta {\delta }_p}\right]\end{array}\right\}$$

$F_{\Delta \delta sr}\left(t\right)$ represents the thrust harmonics caused by the couplings between the encoder errors and slotting force, and

$$F_{\Delta \delta sr}\left(t\right)=0.75Nl{a}_i\left(y_s\right)B_1\cos \left(\frac{i\tau \pi }{{\tau }_s}\right)$$

$$\sum _{i=1,2,3,\dots }^{\infty }\left\{\begin{array}{l}{i^{\prime }}_{\Delta {\delta }_y}\cos \left[\left(\frac{2i\pi v}{{\tau }_s}+{\omega }_{\Delta {\delta }_y}\right)t+{\gamma }_{\Delta {\delta }_y}\right]+{i^{\prime }}_{\Delta {\delta }_y}\cos \left[\left(\frac{2i\pi v}{{\tau }_s}-{\omega }_{\Delta {\delta }_y}\right)t+{\gamma }_{\Delta {\delta }_y}\right]+\\ {i^{\prime }}_{\Delta {\delta }_r}\cos \left[\left(\frac{2i\pi v}{{\tau }_s}+{\omega }_{\Delta {\delta }_r}\right)t+{\gamma }_{\Delta {\delta }_r}\right]+{i^{\prime }}_{\Delta {\delta }_r}\cos \left[\left(\frac{2i\pi v}{{\tau }_s}-{\omega }_{\Delta {\delta }_r}\right)t+{\gamma }_{\Delta {\delta }_r}\right]+\\ {i^{\prime }}_{\Delta {\delta }_p}\cos \left[\left(\frac{2i\pi v}{{\tau }_s}+{\omega }_{\Delta {\delta }_p}\right)t+{\gamma }_{\Delta {\delta }_p}\right]+{i^{\prime }}_{\Delta {\delta }_p}\cos \left[\left(\frac{2i\pi v}{{\tau }_s}-{\omega }_{\Delta {\delta }_p}\right)t+{\gamma }_{\Delta {\delta }_p}\right]\end{array}\right\}$$

3.3. Analysis of the Thrust Force Characteristics under the Action of the Cutting Force

The thrust force of PMLM considering the cutting force can be obtained as:

$$F_{lm}=F_0\left(t\right)+F_r\left(t\right)+F_{ce}\left(t\right)+F_{cag}\left(t\right)+F_{cee}\left(t\right)+F_{\mathit{cother}}\left(t\right)$$

(19)

where, F_cother(t) represents the new thrust force caused by other factors.

It can be obtained from Equation (19) that the machining process will intensify the coupling between the mechanical system and the servo system in the direct drive feed system. The cutting force with multi-frequency will be coupled with the non-linearity of the drive circuit and motor structure through the mechanical vibrations, leading to plenty of new coupled thrust harmonics. The frequencies of these new coupled thrust harmonics are related to the frequencies of the original thrust force harmonics and these of the mechanical vibrations caused by the cutting force.

4. Experiments

4.1. Experimental Setup

The experiment is carried out in an independently designed vertical machining center as shown in Figure 3. The X axis and Y axis are driven by the PMLM (Heidenhain, LMP28-200-3WDE-232, Traunreut, Germany). The X axis is chosen as the tested object. The main parameters of PMLM are shown in

Table 1. The main parameters of direct drive feed system.

Name	Value
Position travel (mm)	600
Continuous force (N)	2920
Peak force (N)	11,500
Force constant (N/A)	165
Height of the air-gap (mm)	1.0
Length of mover (mm)	255
Pitch of permanent magnet (mm)	22
Tooth pitch (mm)	13
Acceleration (m/s/s)	15
Max velocity (m/min)	90

. The cutter parameters and milling parameters are shown in

Table 2. The main parameters in the machining process.

	Name	Value
Workpiece	Material	Al7075
Cutter parameters	Tooth number	3
	Diameter (mm)	16
Milling parameters	Type	Side milling
	Feed rate (mm/min)	1500
	Spindle speed (r/min)	3000
	ap (mm)	2
	ac (mm)	2

. The cutting force was measured by a dynamometer (model: Kistler 9265B, Winterthur, Switzerland) mounted between the aluminum alloy workpiece (model: 7075-T651) and machine table. The sampling frequency of cutting forces in the milling process is 20 kHz. The thrust force was measured by the monitoring software in the NC system and its sampling frequency is 1 kHz.

4.2. The Result of the Cutting Force

The results of the cutting force in the three directions are shown in Figure 4.

It can be seen from Figure 4 that there are obvious fluctuations in the cutting force. The amplitude of the cutting force in y direction is the largest and that in z direction is the smallest. The main harmonic components of the cutting force in the three directions are the three times harmonics of the rotation frequency, which are generated by the process of discontinuous contact between the cutter and the material. In addition, there are other harmonics in cutting force due to the tool eccentricity, spindle vibration and other factors.

4.3. The Result of the Thrust Force in PMLM

The spectral analysis of the thrust force in the machining process is shown in Figure 5. The original main thrust force harmonics of PMLM are obtained as shown in

Table 3. The original main thrust harmonics.

Order	Source	Frequency (Hz)	Amplitude(μm)
1	End effect	1.08	20.58
2	Slotting effect	1.91	6.11
3	Drive circuit	3.42	6.84
4	Drive circuit	6.74	1.53
5	Slotting effect, drive circuit	7.40	4.62

through the frequency spectrum analysis of the thrust force with the constant velocity.

It can be seen from Figure 5 and Table 3 that there are lots of thrust harmonics in addition to the cutting force and original thrust harmonics. There are many new thrust harmonics on both sides of the adjusted thrust caused by the cutting force. The new thrust force harmonics caused by the couplings between the cutting force and the original thrust harmonics are shown in

Table 4. The coupling thrust force caused by the first-order cutting force.

Order	Source	Test Results (Hz)	Theoretical Calculations (Hz)	Deviation (%)
1	Air-gap Fluctuation	35.41	35.90	1.36
2	Air-gap Fluctuation	39.24	39.32	0.20
3	Air-gap Fluctuation	42.07	41.94	0.31
4	Full-loop Control	42.07	42.18	0.26
5	Air-gap Fluctuation	43.24	42.74	1.17
6	Full-loop Control	42.90	42.84	0.14
7	Air-gap Fluctuation	44.74	43.85	2.03
8	Air-gap Fluctuation	45.57	45.76	0.41
9	Full-loop Control	46.08	46.16	0.17
10	Full-loop Control	47.41	47.66	0.52
11	Full-loop Control	51.57	51.49	0.15
12	Full-loop Control	53.07	52.99	0.15
13	Air-gap Fluctuation	53.07	53.4	0.62
14	Air-gap Fluctuation	55.07	55.31	0.43
15	Full-loop Control	57.24	56.32	1.63
16	Air-gap Fluctuation	56.24	56.42	0.32
17	Full-loop Control	56.24	56.98	1.30
18	Air-gap Fluctuation	57.07	57.22	0.26
19	Air-gap Fluctuation	59.57	59.84	0.45
20	Air-gap Fluctuation	63.41	63.26	0.24

,

Table 5. The coupling thrust force caused by the second-order cutting force.

Order	Source	Test Results (Hz)	Theoretical Calculations (Hz)	Deviation (%)
1	Air-gap Fluctuation	85.23	85.55	0.37
2	Air-gap Fluctuation	88.90	88.97	0.08
3	Air-gap Fluctuation	91.73	91.59	0.15
4	Full-loop Control	91.73	91.83	0.11
5	Air-gap Fluctuation	92.57	92.39	0.19
6	Full-loop Control	92.57	92.49	0.09
7	Air-gap Fluctuation	93.57	93.5	0.07
8	Air-gap Fluctuation	95.23	95.41	0.19
9	Full-loop Control	95.56	95.81	0.26
10	Full-loop Control	97.06	97.32	0.27
11	Full-loop Control	101.23	101.14	0.09
12	Full-loop Control	102.73	102.65	0.08
13	Air-gap Fluctuation	103.23	103.05	0.17
14	Air-gap Fluctuation	104.73	104.96	0.22
15	Full-loop Control	105.57	105.97	0.38
16	Air-gap Fluctuation	106.06	106.07	0.01
17	Full-loop Control	106.73	106.63	0.09
18	Air-gap Fluctuation	106.73	106.87	0.13
19	Air-gap Fluctuation	109.73	109.49	0.22
20	Air-gap Fluctuation	113.06	112.91	0.13

and

Table 6. The coupling thrust force caused by the third-order cutting force.

Order	Source	Test Results (Hz)	Theoretical Calculations (Hz)	Deviation (%)
1	Air-gap Fluctuation	134.89	135.05	0.12
2	Air-gap Fluctuation	138.72	138.47	0.18
3	Air-gap Fluctuation	141.39	141.09	0.21
4	Full-loop Control	141.39	141.33	0.04
5	Air-gap Fluctuation	142.22	141.89	0.23
6	Full-loop Control	142.22	141.99	0.16
7	Air-gap Fluctuation	142.89	143.0	0.08
8	Air-gap Fluctuation	144.73	144.91	0.12
9	Full-loop Control	145.39	145.31	0.06
10	Full-loop Control	146.72	146.82	0.07
11	Full-loop Control	150.89	150.64	0.17
12	Full-loop Control	152.22	152.15	0.05
13	Air-gap Fluctuation	152.22	152.55	0.22
14	Air-gap Fluctuation	154.55	154.46	0.06
15	Full-loop Control	155.39	155.47	0.05
16	Air-gap Fluctuation	155.39	155.57	0.12
17	Full-loop Control	156.22	156.13	0.06
18	Air-gap Fluctuation	156.22	156.37	0.10
19	Air-gap Fluctuation	158.89	158.99	0.06
20	Air-gap Fluctuation	162.89	162.41	0.30

. Due to the limitations of experimental conditions, only the frequency is experimentally verified. In Table 4, Table 5 and Table 6, the coupling thrust harmonics of different components are separated by the frequency characteristics from experimental data.

It can be obtained from Table 4, Table 5 and Table 6 that the max deviation between the theoretical and test results of the frequencies of the new thrust harmonics is only 2.03%, which proves the validity of the previous theoretical analysis.

4.4. The Result of the Dynamic Precision of the Direct-Driven Feed System

The displacement fluctuation of the direct drive feed system in the machining process is shown in Figure 6. The displacement with constant velocity in the machining process is extracted for the spectral analysis as shown in Figure 7.

It can be seen from Figure 6 and Figure 7 that there are more obvious displacement fluctuations for the direct drive feed system because of the effects of the machining process. The spectral distribution of displacement fluctuation is basically consistent with that of thrust harmonics.

5. Discussions

(1)

It can be seen from Figure 5 and Figure 7 that there are a large number of thrust harmonics and displacement fluctuations caused by other unknown factors. The modal analysis of the machine tool is carried out, and the modal distribution within 200Hz is shown in

Table 7. The modal distribution of the machine tool.

Order	1	2	3	4	5	6	7	8
Frequency (Hz)	17.03	36.32	50.92	72.42	97.54	113.65	161.13	193.28

. It can be obtained from Table 7, Figure 5 and Figure 7 that several mechanical vibration modes (such as 36.32 Hz, 50.92 Hz, 72.42 Hz, and 113.65 Hz) are excited during the machining process, which aggravates electromechanical couplings, leading to much more thrust harmonics and displacement fluctuations.

(2)

Thrust spectrum of PMSLM with different conditions is shown in Figure 8. Figure 8a is the thrust spectrum only considering the cutting force and thrust harmonics. Figure 8b is the thrust spectrum considering the main mechanical vibrations of the machine tool. Figure 8c is the thrust spectrum considering the couplings caused by the cutting force.

It can be seen from Figure 8 that the cutting force will be coupled with the original thrust harmonics of the direct drive feed system, leading to more new paired thrust harmonics. Furthermore, more mechanical models will be excited during the machining process, aggravating the thrust fluctuation.

(3)

In the direct drive feed system, there are dynamic electromechanical couplings between the servo system and the mechanical system. The thrust harmonics directly act on the mechanical system, leading to obvious displacement fluctuation. Meanwhile, mechanical vibrations will affect the thrust force characteristics through the air-gap fluctuation and the feedback signal. In the machining process, the electromechanical coupling will deteriorate the influence of the cutting force harmonics on the mechanical system as shown in Figure 9. Lots of paired new thrust harmonics, which are produced by the couplings between the cutting force and original thrust harmonics, will appear on both sides of the adjusted thrust due to the cutting force. The influence of the machining process on the mechanical system will be extended from the discrete frequency point caused by the cutting force to approximate frequency band caused by the thrust force, affecting the dynamic precision of the feed system and the cutting stability of the machine tool.

In the machining process of the complex parts (such as the impeller), the cutting parameters, feed velocity, and spatial position of the feed axis are constantly changing. The cutting force, thrust force, and mechanical characteristics are constantly changing. The coupled strength of different coupling loops is different. Therefore, in the high-speed machine tool with the direct drive feed system, more attention should be paid to the influence of the machining process on the internal coupling problem of the feed system to optimize the machining path and cutting parameters.

6. Conclusions

In this paper, the effects of the machining process on the thrust force and mechanical characteristics of the direct drive feed system are researched. The main conclusions can be obtained as follows.

(1)

The dynamic electromechanical couplings between the servo system and mechanical system in the direct drive feed system will aggravate the effects of the machining process on the thrust force and mechanical characteristics of the feed system. The cutting force harmonics with multi-frequencies will be coupled with the non-linearity of the drive circuit and motor structure through the mechanical vibrations. A large number of new paired thrust harmonics will be produced. The frequencies of these new coupled thrust harmonics are related to the frequencies of the original thrust force harmonics and these of the mechanical vibrations caused by the cutting force harmonics.

(2)

The influence of the machining process on the direct drive system will be extended from the discrete frequency point caused by the cutting force to approximate frequency band caused by the thrust force, which will propose higher requirements for the control system. Especially if the frequency of one of the thrust harmonics is close to a certain natural frequency of the machine tool, the resonance will be aroused, affecting the dynamic precision of the feed system and the cutting stability of the machine tool. In order to make full use of the performance advantages of the direct drive feed system in machine tools, it is necessary to consider the influence of the machining process taking account of the electromechanical couplings. Similar problems are more prominent in the feed system driven by the linear motor with a U-shaped bilateral stator.

The thrust force harmonics of the direct drive feed system in the machining process have unique frequency characteristics. The results in this paper are of great significance to the analysis and improvement of the dynamic errors for the direct drive system in practical engineering applications.