Dynamic performance of a long-stroke fast tool servo system

Zheng Gong, Dehong Huo,,a)Zengyuan Niu, Wanqun Chen, and Kai Cheng

ABSTRACT A fast tool servo(FTS)system can be used to efficientl manufacture optical freeform surfaces.This paper investigates the dynamic performance of an FTS system driven by a voice coil motor and guided by air bearings.A simulation model and testing platform are developed to evaluate the load capacity and stiffness of the air bearings.The mechanical dynamic performance of the designed FTS,including modal and harmonic analyses,is assessed using finit element analysis.A nonlinear relation between air-bearing stiffness and mechanical bandwidth is obtained.The working dynamic performance is tested through system runout,tracking performance,and closed-loop tests.Quantitative relations between air-bearing stiffness and the mechanical and working bandwidths are established and analyzed.Machining experiments verify the feasibility of the FTS system with 31.05 N/μm stiffness air-bearings.©2022 Author(s).All article content,except where otherwise noted,is licensed under a Creative Commons Attribution(CC BY)license(http://creativecommons.org/licenses/by/4.0/).https://doi.org/10.1063/10.0011434

KEYWORDS FTS,Dynamic performance,Air bearing,Stiffness

I.INTRODUCTION

Optical freeform surfaces are widely used in lighting systems,laser scanners,fibe connectors,and other applications.1These surfaces either do not rotate or have microstructures.According to the surface topography,a slow tool servo or a fast tool servo(FTS)systemcan be used to manufacture such surfaces.2,3Compared to a slow tool servo,an FTS is more efficien for manufacturing surfaces with a complex geometry.4According to the driving source,FTS systems can be categorized as piezoelectric FTS(PZT-FTS)or Lorentz-force FTS systems.

PZT-FTS uses piezoelectric ceramics as the driving source.Guided by a flexibl hinge mechanism,it can achieve a highfrequency performance of up to a few thousand hertz.Yanget al.5used finit element analysis to develop a PZT-FTS with a dominant natural frequency of 2191 Hz and a static stiffness of 33.62 N/μm.Altintas and Woronko6designed an FTS driven by piezoactuators with flexur hinges.With a stroke of 36μm and a natural frequency of 3200 Hz,the system had a stiffness of 370 N/μm in the direction of motion.A sliding mode controller helped it to achieve a tracking error of±10 nm.To increase the stroke of a PZT-FTS,the flexur can be designed as an amplifier Wang and Yang7developed an FTS for use in noncircular piston turning.The system reached a stroke of 540μm with repetitive control.Zhuet al.8presented a multi-objective optimum design of an FTS system.Dynamic modelling and analysis were applied to the flexure-base mechanism,and differential evolution was found to be the optimal design algorithm.The designed FTS had a stoke of up to 10.25μm and a frequency of at least 2 kHz.Ganet al.9integrated a cost-effective position sensitivity detector into an FTS system.With a proportional-integral feedback controller and by collecting the global straightness error,the surface roughness of aluminum and brass was improved.

Lorentz-force FTS uses a voice coil motor(VCM)or linear motor as the driving source.With well-designed support components,it can achieve a long stroke of up to a millimeter and a working frequency of a few hundred hertz.Rakuff and Cuttino10designed a flexur mechanism driven by a VCMthat had a 25 mm stroke.The flexur acted like a guide system due to its stiffness.The control system combined a proportional-integral-derivative(PID)feedback controller and a linear current amplifier The maximum stroke was 2 mm,and the bandwidth was up to 140 Hz.Liuet al.11also implemented a flexur hinge as the guide system.Final closed-loop performance tests showed that the system had a 1 mm stroke and a working frequency of around 30 Hz.Liuet al.12designed a long-stroke FTS system that combined a triangular slideway,air bearings,and a VCM.Six air bearings were installed around the slideway to lift it.A finit element analysis of the slideway found that the largest deformation was 0.13μm.Similar structures were implemented by Tianet al.13The slideway had a T-shape and the stroke was up to 30 mm.The dynamic performance of the system was assessed with a modal analysis,which provided information about the firs and second natural frequencies.Gutierrez and Ro14designed an FTS and control system using parametric modelling.A closed-loop controller with a linear quadratic regulator was implemented,and a simulation and physical models showed that it had good tracking performance for sine and ramp signals.

Maxwell normal-stress motor-driven FTS(MNM-FTS)is similar to Lorentz-force FTS,since for both the driving sources are motors.However,MNM-FTS relies on the Maxwell electromagnetic normal stress as the driving force.Lu and Trumper15designed an electromagnetically driven FTS with two flexures which guided the tool tip to move along theZdirection.The experimental results showed that a full stroke of 50μm and working frequencies of up to 1 kHz were achieved.A prototype of an MNM-FTS was designed based on a flexur mechanism by Nieet al.16The fina system had a stroke of 11.3μm and a closed-loop bandwidth of 3 kHz.Zhuet al.17built a new MNM-FTS,which realized a stroke of 260μm and a working frequency of 1121 Hz.

Apart from the mechanism,an appropriate control algorithm is also important for working performance.Five different control algorithms are widely used in FTS control:PID control,10repetitive control,18sliding mode control,19active disturbance rejection control,20and zero-phase error-tracking control.21The control algorithms are not necessarily independent,since two or more control algorithms can work together to help the system to achieve good tracking performance.

Based on this literature review,we choose motors as the driving source and air bearings as the guide system to achieve a long stroke.Most studies have focused on the air bearings.However,air bearings offer stiffness only in the direction perpendicular to the motion of the system,so that it is undamped in the direction of motion.There has been little research into the dynamic performance of an FTS system with air bearings,even though their good dynamic performance can offer good tracking performance during operation.The relation between air-bearing stiffness and system dynamic performance is unknown,yet when designing air bearings,it is important to consider the dynamic performance of the system.

This paper describes the design of a long-stroke FTS system with a VCM and air bearings.Models were built to simulate the mechanical dynamic performance of the system and for modal and harmonic analyses.Two different air bearings with different stiffnesses were designed and manufactured for the system,and a testing platform was built.With a well-designed control system,the system tracking performance and working bandwidth were determined.From a comparison of the results for the systems with different air bearings,the nonlinear relation between air-bearing stiffness and dynamic performance of the system was determined.Machining experiments verifie the feasibility of the system with the designed air bearing.

II.MECHANICAL SUBSYSTEM DESIGN

A long-stroke FTS system based on a VCM was designed to track a sine signal from 1 mm at 30 Hz to 0.1 mm at 100 Hz.The motion of the system is described by these equations:

whereXis the travel distance of the system(m),ais the acceleration of the system(m/s2),tis time(s),Ais the maximum stroke(m),fis the frequency(Hz),ωis angular frequency(rad/s),Fis the required driving force(N),andmis the mass of all the moving components(kg).The mechanical design requirements can be define according to Eqs.(1)-(3),as listed in Table I.

Figure 1 is a schematic diagram of the designed system.The VCM is connected to a hollow square slide.Four air bearings are arranged outside the slide and connected to the outer plate with ballheaded screws.

The fina moving mass of the system is 995.1 g.Because there are four air bearings,the moving parts(motor coil+slide+connection plate)have negligible friction.The air-fil surface of each air bearing is 40×50 mm2.The air-bearing stiffness is the only uncertain value.It depends on the performance of the air bearing and the system as a whole.

A.Simulation of air bearings

With a fixe air-fil surface area,the air bearing can be designed as an orific air bearing.An orific plug is used to form an air fil with a certain stiffness.Figure 2 is a schematic diagram of an air bearing.

The air bearings were simulated using computational flui dynamics(CFD).The parameter values for the simulation are listedin Table II.The simulation applies the Navier-Stokes equations and the pressure disturbance of the air fil is then calculated.The load capacity and stiffness can be calculated as follows:

TABLE I.Mechanicaldesign requirements.

FIG.1.Schematic diagram of the designed system.

whereWis the load capacity(N),Pis the pressure distribution in the air fil(Pa),Pais the outlet pressure(Pa),xandyare the dimensions of the air bearing(m),Kis the air bearing stiffness(N/μm),andhis the air-fil thickness(m).

In this paper,the dimensions of the air bearing and the orific plug are fixed Therefore,the number of orifice and air-fil thickness are the variables that determine the performance of an air bearing.The simulation models and pressure distributions are shown in Fig.3.The pressure uniformly falls in moving from the position of the orific plug.

The load capacity and stiffness can be calculated from the pressure distribution,as plotted in Fig.4.With an increase in the air-fil thickness,the load capacity decreases.There is an optimal air-fil thickness where the stiffness reaches a maximum.The maximum values of stiffness for the three different air bearings are 32,23,and 7 N/μm,respectively,for an air-fil thickness of 14μm.Increasing the number of orifice can increase the stiffness of the air film but the improvement in going from fiv to seven[k1 in Fig.4(a)]is smaller than in going from one to fiv[k2 in Fig.4(b)].More orifice may increase the air-bearing stiffness.However,despite the higher manufacturing cost and air flowrate seven orifice were chosen for the air bearing.

FIG.2.Schematic diagram ofthe designed air bearing.

B.Control subsystem design

Figure 5 is a block diagram for the designed system control.A controller(Omron CK3M)with a serving frequency of 16 kHz was chosen,as it allows user-define control algorithms to be implemented.The motor drive(Elmo Gold Hornet)amplifie the current to drive the motor and works as the current loop.The encoder(Renishaw Tonic Ti2000)measures the actual displacement signal and sends it back to the controller to form a closed-loop control system.

The current loop uses a PI control algorithm.A bandwidth of 3 kHz is achieved after tuning,which is sufficien for the designed FTS system.The position loop uses a standard servo algorithm(advanced PID control22)from CK3M,as shown in Fig.6.

The system bandwidth is an important factor affecting the dynamic performance of the system.It depends on the mechanical bandwidth,control system,and control algorithms.The control algorithm here was the same in all tests.

TABLE II.Simulation parameters.

FIG.3.CFD simulation models(left)and pressure distributions(right).

FIG.4.Influenc of number of orifice on(a)load capacity and(b)stiffness.

FIG.5.Controlblock diagram of designed FTS system.

III.DYNAMIC PERFORMANCE OF THE MECHANICAL SYSTEM MODEL

In the mechanical system design,the requirement for the airbearing stiffness is unknown.Modal and harmonic analyses were applied to analyze the dynamic performance of the system to propose an air-bearing stiffness design requirement.Both analyses were conducted in Ansys Workbench.

A.Modal analysis

FIG.6.Standard servo algorithm.

FIG.7.Simplifie working model.

The model of the system was simplified as shown in Fig.7.The mounting screws for the air bearing are connected to the baseplate,and the ball heads are connected to ball sockets on the air bearings.Therefore,the fixe supports and frictional connections are the screw threads and the surfaces connecting the ball head screws and air bearings.Each air fil is modelled as eight spring elements with the same stiffness.Figure 8 shows the positions of the springs.The cutting resistance is the resultant of three equal forces(10 N)in theX,Y,andZdirections.

FIG.8.Schematic diagram of spring positions.

FIG.9.Modalanalysis results.

FIG.10.Relation between air-bearing stiffness and first-orde naturalfrequency.

FIG.12.Enlarged view of harmonic response in the Y direction.

A modal analysis of the simplifie model gave the vibration modes and the natural frequencies.Taking an air-bearing stiffness of 30 N/μm as an example,the results of the modal analysis for the system are shown in Fig.9.

Figure 9 shows the firs six vibration modes of the designed FTS system.The slide moves freely due to the nature of the air bearings[Fig.9(a)].Therefore,the actual first-orde natural frequency is 328.03 Hz,as shown in Fig.9(b).Figures 9(b)and 9(c)show the pitch modes in theYandXdirections,respectively.They have similar natural frequencies due to the symmetrical design of the system.Figures 9(e)and 9(f)show the bouncing modes in theYandXdirections.These depend on the air-bearing stiffness.The modal analysis suggests that the displacements for all modes occur at the air film which indicates that this is the weakest element of the system.

As the first-orde natural frequency is important for the system dynamics,a modal analysis was performed using a range of airbearing stiffnesses(5-60 N/μm)to test its influenc on the first-orde natural frequency.The results are shown in Fig.10.

FIG.11.Harmonic response in(a)the X and(b)the Y directions for an air-bearing stiffness of 30 N/μm.

FIG.13.Harmonic response of the system with three different air bearings.

An increase in air-bearing stiffness results in the system having a higher first-orde natural frequency.Therefore,a higher airbearing stiffness can extend the mechanical bandwidth,but the improvement becomes slower as the stiffness increases.Overall,a system with greater bandwidth will have better dynamic performance,but this requires the air bearing to have a higher stiffness,and it is more difficul to design such an air bearing.The harmonic response analysis was next performed to quantitatively analyze the relation between the dynamic performance of the system and air-bearing stiffness.

B.Harmonic response analysis

A force with an amplitude of 10 N in each of theX,Y,andZdirections was applied to the front of the slide of the FTS system.These forces represent the cutting resistance(Fig.7).The harmonic response analysis can be considered as a frequency sweep of the system;hence,the mechanical bandwidth of the system can be determined from the results of this analysis.The frequency of the force was varied from 0 to 2000 Hz.

The results in theXandYdirections are shown in Fig.11.Two peaks can be observed.The first-orde natural frequencies in theXandYdirections are 328 and 332 Hz,respectively.The secondorder natural frequencies are both 1165 Hz forXandYdirections.These results agree with those of the modal analysis.Taking theYdirection as an example[Fig.11(b)],the amplitude indicates that the system is bouncing in theYdirection.This bouncing affects the air-fil thickness and,thus,the air-bearing stiffness.In addition,with an increase of the external excitation,the system changes from a linear state to a nonlinear state and has a resonance peak at 328 Hz.

Figure 12 is an enlarged view of Fig.11(b)for 0-300 Hz.In the range 0-105 Hz,the curve is almost a straight line with an increase in amplitude of less than 10%,and so the system is considered to be in a linear state.This indicates that within this range,the change in the displacement of the air fil is linear,and this small change does not change the air-bearing stiffness.Therefore,in the linear range,the air-fil thickness is hardly affected by the external excitation.This frequency range can be considered as the ideal working bandwidth of the system.Beyond 105 Hz,the system is considered nonlinear,since the changes in the displacement of the air fil are nonlinear.The rapid increase in displacement means that the system is no longer operating in an appropriate regime.The threshold was set at 10%as a convenient empirical value for the subsequent quantitative calculations.Overall,for the system with 30 N/μm air bearings,the working bandwidth is 105 Hz,even though the mechanical bandwidth is 328 Hz.

As in the modal analysis,the harmonic response was evaluated for air-bearing stiffnesses from 5 to 60 N/μm.The mechanical and ideal working bandwidths of the system and the corresponding frequencies were recorded.Figure 13 shows the results for a system with air-bearing stiffnesses of 15,35,or 55 N/μm in the frequency range 0-200 Hz.

TABLE III.Comparison ofidealand mechanicalbandwidths.

FIG.14.Manufactured air bearings.

Table III compares the mechanical and ideal working bandwidths of systems with different values of the air-bearing stiffness.From the results,the relation between the ideal working bandwidth and mechanical bandwidth was found to be

wherefmis the mechanical bandwidth(Hz)andfwis the ideal working bandwidth(Hz).

Thus,the mechanical and working bandwidths have a linear relation.For the designed 100 Hz working frequency of the FTS system,the air-bearing stiffness should be at least 20-25 N/μm.In addition,for an air-bearing stiffness design in which the system structures are fixed the mechanical bandwidth can be simulated with a linear relation.The ideal working frequency can then be calculated and,depending on the design requirements,the air-bearing stiffness can be determined.

FIG.16.Photograph of the testing platform for the air bearings.

IV.EXPERIMENTS TO EVALUATE THE WORKING DYNAMIC PERFORMANCE

A.Design and testing of air bearings

A one-orific air bearing(Fig.14,left)and a seven-ori fic air bearing(Fig.14,right)were designed and manufactured to test the relation between system working bandwidth and air-bearing stiffness.

A testing platform for the air-bearing stiffness was designed.It uses an air cylinder and a force sensor to test the load capacity.A laser displacement sensor(KEYENCE LK-008 in LK-G5000 series,repeatability:0.005μm,accuracy:0.02%)was installed to test the airfil thickness.Figure 15 is a schematic of the testing platform.The experimental setup is shown in Fig.16.

FIG.15.Schematic of the testing platform for the air bearings.

FIG.17.Load capacity(a)and stiffness(b)of the designed air bearings.

FIG.18.Schematic of the runout test setup.

FIG.19.Controland working performance tests ofthe designed FTS system.

The air-bearing stiffness can be calculated from the slope of the load vs displacement curve.Two manufactured air bearings were tested on the platform,and the results are shown in Fig.17.The air bearing with seven orifice exhibits better performance than the one with a single orifice When the air fil was 12-13μm,the maximum stiffnesses were 31.05 and 12 N/μm,respectively,for seven orifice and one orifice The seven-orific air bearing meets the airbearing stiffness design requirements(20-25 N/μm),but the system with only one orific does not.The latter was also assembled as the control group to determine the influenc of air-bearing stiffness on the dynamic performance of the system.

B.Working performance tests

Four types of tests were performed to assess the dynamic performance of the system:system runout,open-loop,closed-loop,and tracking performance tests.

To test the system runout,a tool holder was designed and installed in the front of the FTS system.A laser displacement sensor(KEYENCE CL-P015,resolution:0.25μm)measured the vibration of the tool holder on the side of the system.The test schematic is shown in Fig.18.

The vibration of the tool holder represents the runout of the system,which has a substantial effect on the fina quality of the workpiece.The runout of the systems with a one-or seven-orific air bearing was tested under an amplitude of 1 mm at 20 Hz and an amplitude of 0.4 mm at 30 Hz.

The working bandwidth and tracking performance are two important metrics of the dynamic performance of the system.Openand closed-loop frequency sweep tests and sinusoidal signal tracking performance tests were performed on the two systems.For the control system of the designed FTS system shown in Fig.5,the test processes are illustrated in Fig.19.

The controller generates a frequency-varying sinusoidal signal for the frequency sweep in the open-and closed-loop tests.The open-loop test23is the basis of the closed-loop test.The closed-loop test can be conducted only when the system has successfully completed an open-loop test.In the closed-loop test,the input signal is the position command.The position loop and the current loop are both active.The closed-loop test aims to determine the closed-loop model and the working bandwidth.

For the tracking performance test,three displacement input signals were generated:0.44 mm at 30 Hz,0.25 mm at 40 Hz,and 0.16 mm at 50 Hz.The control algorithm for the two different systems is the same as shown in Fig.6.The only variable is the air-bearing stiffness.

V.PERFORMANCE TEST RESULTS

A.System runout

The test results for system runout when the amplitude was 1 mm at 20 Hz are shown in Fig.20(a).System 1 has a one-orific air bearing(12 N/μm),and system 7 has a seven-orific air bearing(31.05 N/μm).System 1 has a runout of±6μm and system 7 has a runout of±4μm.The frequency of the runout is the same as the test frequency of the system motion.It is clear that the higherstiffness air bearing has decreased the vibration of the cutting tool and increased the stability of the system.

The system runout results for 0.4 mm at 30 Hz are shown in Fig.20(b).When the test stroke was decreased and the test frequency increased,the runout of both systems decreased but the frequency was still the test frequency.System 1 has a runout of±2μm and system 7 has a runout of less than±1μm.In the negative direction of system motion,and especially at the limit of the stroke,vibrations with different frequencies appear(the green circle in Fig.20).This is due to the surface roughness of the lateral surface of the tool holder.In the test,the surface roughness(Ra)of this lateral surface is 1μm.When the system runs at 0.4 mm at 30 Hz,the runout amplitude is close to 1μm.This indicates that the test results are due to the superposition of the surface roughness and system runout signals.

FIG.20.System runout for(a)amplitude of 1 mm at20 Hz and(b)amplitude of 0.4 mm at 30 Hz.

FIG.21.System tracking performance with an amplitude of 0.16 mm at 50 Hz.

Overall,increasing the air-bearing stiffness can improve the stability and decrease the runout of the cutting tool during manufacturing to enhance the dynamic performance of the system.

B.Tracking performance and Bode diagrams

Three different input signals were tested for two different systems,and the results for an amplitude of 0.16 mm at 50 Hz are shown in Fig.21.With the same controller and control algorithm,system 7 shows better tracking performance than system 1.With a one-orific air bearing,system 1 has larger differences in amplitude and phase.

The tracking errors are plotted in Fig.22.System 7 has a tracking error of±0.012 mm(7.5%)whereas system 1 has a tracking error of±0.032 mm(20%).For precision machining with an FTS system,a tracking error of 20%is unacceptable.It can be seen that increasing the air-bearing stiffness up to or larger than the required level can increase the tracking performance of the system.

FIG.22.System tracking error with an amplitude of 0.16 mm.at50 Hz.

FIG.23.Originaldata of closed-loop frequency sweep results.

To determine the bandwidth of the two systems,the closedloop frequency sweep results can be used to plot the Bode diagrams.The original data from the closed-loop frequency sweep are shown in Fig.23.

The state-space model is used to fi the closed-loop frequency sweep results:

The model order is 5,and the fi to estimated data can be up to 99%.The Bode diagrams of both systems are plotted in Fig.24.

FIG.24.Bode diagrams of systems 1 and 7.

Normally,the bandwidth is the corresponding frequency of a system when the magnitude reaches-3 dB,which means that the output signal has decreased to 0.707 of the input signal.This can be seen as a limitation of the dynamic performance.For an FTS system,high-positional accuracy requires a smaller tracking error,which restricts the limitation to±10%.This value also corresponds to the 10%displacement of the mechanical system below which it is considered to be linear.Therefore,the output signal should be limited to 0.90-1.10 of the input signal:

Referring to the magnitude in the Bode diagram,if the system operates between-0.9151 and 0.8279 dB,good dynamic tracking performance will have been achieved.

Figure 24 shows that the bandwidths of systems 1 and 7 are 61.35 and 149.6 Hz,respectively.Therefore,system 1 does not comply with the design requirement(>100 Hz).Increasing the airbearing stiffness from 12 N/μm(system 1)to 31.05 N/μm(system 7)means that the system has a higher bandwidth.From the Bode diagrams,Table III,and Fig.10,it can be seen that an increase of the air-bearing stiffness increases the mechanical bandwidth,leading to a higher working bandwidth and more stable working performance in the designed working frequency range.In other words,the airbearing stiffness has a positive effect on the tracking performance.Overall,the results demonstrate that system bandwidth has a strong dependency on air-bearing stiffness.

VI.MACHINING EXPERIMENTS

The working performance tests are no-load tests of performance.Amachining experiment was designed to assess the dynamic performance with actual loads.The FTS system with a seven-orific air bearing was installed on theZ-axis of an ultra-precise diamond turning machine(Nanotech 650FGv2)through a connection plate.A test evaluated the flexibilit of the designed FTS system.The system setup and machining parameters are shown in Fig.25 and Table IV.After the machining experiment,a 3Doptical surface profile(Zygo Newview 9000)was used to mesh the surface profile In addition,the zero position of the tool was set as-0.49 mm to make sure the cutting depth was 10μm while the FTS stroke was 0.5 mm.

TABLE IV.Machining parameters.

FIG.26.FTS motion profile

First,the FTS motion profil was obtained from the FTS encoder to assess the performance of the FTS(Fig.26).With a sinusoidal input signal of 0.5 mm at 40 Hz,the system had a stroke of±0.508 062 mm,a stroke error of 1.61%,and an overshoot of 8.13μm in the firs period.

FIG.25.Photographs ofthe experimentalsetup.

Second,the surface profil was measured by the 3D optical surface profiler Altogether,19 grooves were cut into the workpiece,and the overcut area was measured at the beginning of each groove.The surface profil is shown in Fig.27.Figure 28 indicates that there was an overcut of 6.08μm and an average groove depth of 10.1683μm.These values mean that in the firs period,the system moved an extra 6.08μm and that the groove depth error was 1.683%.Compared to the motion profil(8.13μm overshoot and 1.61%stroke error),these approximate values show that the system can realize the designed machining motion,and the design parameters are reflecte in surface profile Many factors can cause differences between the motion of the system and the surface profile like insufficien control stiffness in the motion direction or vibration of the FTS system.These causes are not considered here because their influenc is negligible compared to the overshoot.This experiment verifie the feasibility of the system with seven-orific air bearings.The system with one-orific air bearings was not tested due to safety concerns.It was expected that due to the weak air-bearing stiffness,the system would fail during machining.

FIG.27.Surface profile

FIG.28.Profil parameters:(a)Overshoot and(b)cutting depth.

VII.CONCLUSIONS

In this paper,a long-stroke FTS system with a VCM and air bearings was designed.The dynamic performance for the mechanical system and the control system was analyzed.A relation between dynamic performance and air-bearing stiffness was determined.The following conclusions can be drawn:

1.For the designed FTS system,the first-orde natural frequency is the system pitch in theXandYdirections(328 Hz)and the second natural frequency is the bouncing in theXandYdirections(1164 Hz).

2.The first-orde natural frequency of the designed FTS system has a positive correlation with air-bearing stiffness,since increasing the air-bearing stiffness increases the first order natural frequency.However,the improvement becomes saturated when the air-bearing stiffness reaches a certain value.

3.A reference range of air-bearing stiffnesses(20-25 N/μm)was set for the designed FTS system.The mechanical bandwidth has a linear relation with the ideal working frequency,which can be used as a reference for air-bearing stiffness design.

4.The dynamic working performance was tested for two systems:a one-orific air bearing(12 N/μm)and a seven-orific air bearing(31.05 N/μm).The results show that increasing the air-bearing stiffness can improve the stability and decrease the runout of the cutting tool.The system with the higher air-bearing stiffness had better tracking performance,with a tracking error of 7.5%at 50 Hz and a higher system bandwidth of 149.6 Hz.

5.A machining experiment was conducted to assess the FTS system design.The system can realize a stroke of 0.5 mm at 40 Hz for a cutting depth of 10μm.The fina surface profil error was 1.683%.

ACKNOWLEDGMENTS

The authors are grateful to Uptech,Jiangsu Industrial Technology Research Institute,for providing the facilities for the machining experiments.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflict to disclose.

DATA AVAILABILITY

No actual data is available for this research.