Ma Junjin,Zhang Dinghua,Wu Baohai,Luo Ming,Chen Bing

Key Laboratory of Contemporary Design and Integrated Manufacturing Technology,Ministry of Education,Northwestern Polytechnical University,Xi’an 710072,China

Vibration suppression of thin-walled workpiece machining considering external damping properties based on magnetorheological fluids flexiblefixture

Ma Junjin,Zhang Dinghua*,Wu Baohai,Luo Ming,Chen Bing

Key Laboratory of Contemporary Design and Integrated Manufacturing Technology,Ministry of Education,Northwestern Polytechnical University,Xi’an 710072,China

Milling of the thin-walled workpiece in the aerospace industry is a critical process due to the high flexibility of the workpiece.In this paper,a flexible fixture based on the magnetorheological(MR)fluids is designed to investigate the regenerative chatter suppression during the machining.Based on the analysis of typical structural components in the aerospace industry,a general complex thin-walled workpiece with fixture and damping constraint can be equivalent as a rectangular cantilever beam.On the basis of the equivalent models,natural frequency and mode shape function of the thin-walled workpiece is obtained according to the Euler–Bernoulli beam assumptions.Then,the displacement response function of the bending vibration of the beam is represented by the product of all the mode shape function and the generalized coordinate.Furthermore,a dynamic equation of the workpiece-fixture system considering the external damping factor is proposed using the Lagrangian method in terms of all the mode shape function and the generalized coordinate,and the response of system under the dynamic cutting force is calculated to evaluate the stability of the milling process under damping control.Finally,the feasibility and effectiveness of the proposed approach are validated by the impact hammer experiments and several machining tests.

1.Introduction

In aerospace industry,the machining vibration of thin-walled workpiece is an undesirable phenomenon.The analysis of the negative influence caused by machining vibration,especially for machining flexible workpiece such as aeroengine blades,casings,impellers,blisks,has received special attention due to its effects on the accuracy of the final workpiece,the production efficiency and the life of machine spindles and cutters.Milling process,which plays a critical role in manufactur-ing,has been extensively used to the finishing machining of the low rigidity and large machining de formation workpiece,and the machining vibration is a serious problem in machining.There fore,it is necessary to investigate the machining vibration dynamic response of the thin-wall workpiece to suppress vibration in machining.

For the efforts of investigating the milling vibration of thinwalled flexible workpiece,reasonable cutting parameters should be selected in machining according to chatter stability diagram.Recently,in the aspect of chatter stability prediction,numerous theoretical and experimental investigations have been made in model prediction,analysis,and various designs of fixture devices and so on.To accurately study the machining vibration,the prediction and analysis of the dynamic machining stability of the thin-walled workpiece is a fundamental problem.Many research works about this aspect have been developed.Altintas and Budak1,2presented an analytical method(ZOA method)based on the dynamic milling coefficients approximated by their Fourier series to predict the milling stability lobes.Merdol and Altintas3overcome the existence of additional stability regions and period doubling bifurcations in small radial cut depth during milling process.Zhou et al.4proposed an analytical model considering tool-workpiece engagement region and different cutter lead angles for chatter stability prediction in bull-nose end milling of aeroengine casings,and the predicted method is well agreed with experimental results.Insperger and Ste′pa′n et al.5–7proposed the semidiscretization method to efficiently conduct stability analysis to the linear milling delayed systems.Considering the calculated efficiency,Ding et al.8,9presented thefull-discretization methods to predict milling stability based on different mathematic theory.Then,this method is extended by Liu et al.10based on the Hermite interpolation and the Floquet theory.

In machining vibration suppression,many different passive and active control methods are investigated.Zhang et al.11presented a piezoelectric active vibration control method that increases the damping of the thin-walled flexible workpiece to mitigate the milling vibration.Rashid et al.12adopted an active controlled palletised workholding system,which is different from traditional approach in machining systems to dampen the unwanted vibration.However,many considerable piezoelectric devices and external system in their method,which take much time and is not suitable for complex parts.Sathianarayanan et al.13and Mei et al.14,15proposed the method of using magnetorheological fluids to adjust the stiffness of boring bar,which can efficiently improve the stability of boring bar and reduce machining vibration.However,they only investigated the effects of the damping material on the boring bar and the properties of the wholefixture are not considered.Shamoto et al.16presented a new method of simultaneous double-sided milling to machineflexible plates and the chatter vibration is efficiently suppressed by the experimental validation.Kolluru et al.17presented a novel surface damping solution which took distributed discrete masses as viscoelastic layer to large thin-walled casings for reducing the machining vibration obviously.Zhang et al.18proposed a component synthesis active vibration suppression method based on zeroplace-ment technique to suppress the vibration of flexible systems.Yang et al.19designed a vibration suppression device for the thin-walled flexible workpiece based on the electromagnetic induction principle,and the excitation tests and machining tests are carried out to verify the efficiency of the device.Zeng et al.20proposed a novel fixture design approach to suppress the machining vibration of flexible workpiece.In this work,appropriate fixture layout scheme concerned with suppressing the machining vibration of the flexible workpiece is designed based on the proposed dynamic modelof workpiece- fixture-cutter system.Later,Wan et al.21,22presented a new fixture layout optimization method for the milling of thin-walled multi-framed workpiece to obtain better fixture supports scheme,which took the machining quality and suppression of machining vibration into consideration,in addition,the predicted and experimental results show that the reasonable fixture design can efficiently improve the machining accuracy and significantly suppress machining vibration of flexible workpiece.However,those works only focus on the properties of the fixture support and the effect of the damping properties of the whole fixture on the dynamic machinability is ignored.

Notwithstanding many valuable results aforementioned,current researches on chatter stability prediction and vibration control are mainly concentrated on the effect of the cutting process parameters,while the actual dynamic response properties of the machining system depended on the whole damping properties of the system according to the dynamic differential equation to some extent.However,the dynamic damping properties are not considered in these studies.To address this issue,a semi-active control flexible fixture based on MR fluids is proposed for the complex thin-walled workpiece machining in this paper,which takes the aforementioned missing damping factor into consideration.

Hence force the paper is organized as follows:Section 2 describes the problem formulation of the thin-walled workpiece in machining considering the equivalent model of the thin-walled workpiece.Then,natural frequency and mode shapefunction of the thin-walled workpiece based on the Euler–Bernoulli beam assumptions is calculated in Section3 3.Section 4 constructsthe dynamic equation of the workpiece- fixture system considering the external damping factor,and the response of system under the dynamic cutting force is calculated to evaluate the stability of the workpiecefixture system.Milling experimental tests are carried out for validating the feasibility and efficiency of the proposed chatter theory and the flexible fixture,and the experimental results is discussed in Section 5.Following the discussion,some conclusions and future works are summarized in the end.

2.Problem formulation

The machining of thin-walled workpiece is critical to obtain high precision and good quality of the workpiece in aerospace industry.Due to the typical structure of the thin-walled workpiece used in the aerospace industry,the parts with fixture constraint can be approximately equivalent as a rectangular cantilever beam as shown in Fig.1.Firstly,a solid workpiece with lower precision and larger scattered errors is fixed on the working table of machine tools by the fixture,which can keep the correct position between the cutting tools and the workpiece,and an amount of materials is removed for the required final structure in machining process.In this process,the two stages are divided.One is rough machining,the workpiece with big cutting allowance and the cutter with shank of tool can be regarded as rigid,the cutter with short cutting tool can be regarded as rigid.There fore,the large spindle speed and depth of cut are used to improve the material removal efficiency.The other stage is finishing machining.At this stage,the desired geometrical and surface integrity of the thin-walled workpiece can be obtained.During the finishing process,the short cutting tool can be regarded as rigid and the workpiece has the thinwalled properties and is considered as flexible,there fore,the machining vibration is a dominant factor in fluencing the dynamic machinability.

Fig.1 A rectangular plate representing thin-walled workpiece.

In machining,due to the bad rigidity and weak intensity of the thin-walled plate,when the cutting tool passes through the surface to be machined,the large machining vibration can in fluence the machining efficiency and deteriorate the surface accuracy.To address this problem,a semi-active control flexible fixture based on MR fluids is used for the thin-walled plate machining considering damping factor,and the workpiecefixture-cutter system is shown in Fig.2.To investigate an approximate dynamic response analysis solution,the complex thin-walled workpiece is simplified as a uniform rectangular thin-walled plate(the length is 0＜x＜l,the width w?l,and the thickness of the plate z?w),which is shown in Fig.1.

Fig.2 Workpiece-fixture-cutter system based on magnetorheological fluids.

In Fig.2,a flexible fixture based on the magnetorheological(MR) fluids could be designed by adding the additional damping property to the machining system.Under the tunable magnetic field intensity H,the suspended magnetic particles in the container align into chains along the direction of external magnetic field.Then,the MR fluids transmit from a liquid state to a nearly solid state in millisecond and change back to the liquid state when the external magnetic field is removed,which is defined as magnetorheological effect.During this process,the shear stress of MR fluids sharply increases,which can provide adequate damping support force for the thin-walled workpiece contacted with the MR fluids in machining.There fore,the rigidity of the thin-walled workpiece is largely improved.In the machining process,due to the flexibility of the thinwalled workpiece,the workpiece isproneto produce vibrations and de formation as shown in Fig.3,the dynamic displacement of the workpiece is equal to the de formation of the magnetic particle chains,and the thin-walled workpiece in the MR fluids is subjected to the action of the evenly distributed damping force,which improve the stiffness of the thin-walled workpiece and effectively suppress machining vibration.

3.Natural frequency and mode shape functions of thin-walled workpiece

The thin-walled workpiece is modeled as a continuous,orthotropic rectangular thin-walled plate with the length l,width w,uniform thickness z(Fig.1).Due to the width of the rectangular plate workpiece w?l,the uniform rectangular thin-walled plate can be regarded as an Euler–Bernoulli beam to approximately obtain its natural frequency and mode shapefunctions.Assumed the length of the beam is l,ρ is the mass unit length and EI is the flexural rigidity of the cantilever beam.There fore,the coordinate system is shown in Fig.4,then,the infinitesimal section dx is selected for analysis,x1is the distance from bottom to the infinitesimal section dx,and Q,M represent the moment and torque on the infinitesimal section dx.Based on the Euler–Bernoulli beam assumptions, for the infinitesimal section dx in force analysis,the motion equation of infinitesimal section dx in free vibration along y direction can be expressed as

In moment analysis,the influence of the moment of inertia is neglected,so the moment equilibrium method is used to analyze the infinitesimal section dx.

By inserting Eq.(2)into Eq.(1),the motion equation of the bending vibration of the beam in free vibration can be obtained as

Considering the boundary conditions of cantilever beam,the natural frequency ωiof the cantilever beam can be derived from Eq.(3)

And all the mode shape functions corresponding to its natural frequency ωiare represented as

Fig.3 Damping properties of MR fluids generated by current and machining vibration.

Fig.4 Bending vibration of beam.

4.Dynamic response of thin-walled workpiece with MR fluids’damping constraint

When the dynamic cutting force F（t）and the damping force P（x，t）generated by MR fluids are applied on the cantilever beam as shown in Fig.5,the motion equation of the bending vibration of the beam can be written as

For simplification of expression in the later Section,the solution of Eq.(6)can be expressed as

Fig.5 Bending vibration of beam under dynamic cutting and damping force.

where Xi（x）contained in Xi= ［X1，X2，X3，···，Xi，···，Xn］represents mode shapefunction of the bending vibration of the beam,and qi（t）contained in q= ［q1，q2，···，qi，···，qn］Tis the generalized coordinate of the system.Using Eq.(7),the kinetic energy Ekof the dynamic cutting force vibration of the thin-walled workpiece can be obtained as

where each element of the generalized velocity˙qiis˙qi=?y/?t,and each element of the generalized mass matrix M corresponding to the generalized coordinate is given by

For the cantilever beam,the bending potential energy Epis only considered to simplify the complexity of fixtureworkpiece system,which can be represented by

where each element of the generalized stiffness matrix K corresponding to the generalized coordinate is defined as

Due to the damping properties of the MR fluids and the beam,the flexible fixture-workpiece system can be regarded as a dissipative system.The MR fluid is damping material,which can change its rheological behavior under the different current to improve the stiffness of the thin-walled workpiece.Thus,the uniform distributed damping force per unit length to the cantilever beam can be expressed as

where c1and c2are the damping constant of the MR fluids and structural damping constant of the cantilever.There fore,when the vibration occurs,the energy loss Elis approximated by the work done of the damping force corresponding to the generalized vibration velocities.

where each element of the generalized damping matrix C corresponding to the generalized coordinate is expressed as

In machining process,using the classical virtual work principle,the work done due to dynamic cutting force is

In Eq.(15),Qiis defined as the generalized force,which is given by

In order to obtain the motion equation of the generalized coordinate qi,the Lagrangian equation of the thin-walled workpiece with MR fluids damping material constraint can be expressed as

By substituting Eq.(8),(10),(13)and(16)into Eq.(17),the equations of motion of the thin-walled workpiece with MR fluids damping material constraint can be obtained as

Considering the complexity of vibration damping system, for simplification,the damping matrix is regarded as a linear combination of mass and stiffness matrix,so a proportional damping isused to avoid cumbersome mathematical calculation:

where α and β are constants.

There fore,the Eq.(18)can be rewritten as

In order to decouple the motion differential equations(20),the solution vector q can be expressed as a linear combination of the natural modes of the undamped system,as in the case of Eq.(7),and q（x，t）= ∑ni=1Xi（x）Yi（t）=Xi（x）Y（t），i=1，2，3，...，n,where Y= ［Y1，Y2，···，Yi，···，Yn］Tis the generalized coordinates.Thus,Eq.(20)can be rewritten as

In addition,the element of matrix Xiis normalized23andis pre-multiplied by Eq.(21).Then,Eq.(21)can be decoupled as

where ωiis the ith order natural frequency of the undamped system,and 2ζiωi= α + βω2i,ζirepresents mode damping ratio corresponding to the ith order natural mode shape under the external MR fluids damping materials.According to Eq.(22),each response of the cantilever beam vibration system with the MR fluids damping constraint can be solved as a viscously damped single degree of freedom system.There fore,the response of the bending vibration of the beam under dynamic cutting and damping force is expressed as

For the cantilever beam,the initial displacement Y0（0）and velocity˙Y0（0）are zero.There fore,the 1st mode dynamic response of the workpiece-fixture system under dynamic cutting force on the top of the equivalent rectangular cantilever beam is calculated as

5.Experimental results and discussion

According to the above analysis,a flexible fixture using MR fluids is designed to exert extra damping action on the thinwalled workpiece.Without loss of generality,the thin-walled plate with the elastic edges and the MR damping support is used to represent approximately the complex thin-walled workpiece.The schematic illustration of the flexible fixture and the thin-walled workpiece studied in this research is shown in Fig.2.The purpose of the study is to design an appropriateflexible fixture considering the vibration characteristics of the combined workpiece- fixture-cutter system.To validate the effectiveness of the designed flexible fixture,both the impact experiments and the machining experiments are implemented on the three-axis milling center(YHVT8507)and the experimental tests are carried out(see Fig.6).

5.1.Impact tests

The impact hammer experiment is a general way of experimental analysis for the vibration characteristics of the combined workpiece- fixture system.The test system are model hammer(Kistler,500 N),acceleration sensors(Dytran,3325F1-16847,Ref.sensitivity 10.25 mV/g.),SIRIUS ACC data acquisition instrument(DEWESof t-SIRIUS i series),and computer.The impact hamper is used as an excitation source for the measurement of frequency response function,and an accelerometer is selected to measure the response of the workpiece- fixture system.Then,the measured vibration response modes can be used to compare and analyze the vibration characteristics of different dampers acted on the workpiece.The dimension of the flexible thin-walled workpiece made from 45 steelis 130 mm × 60 mm with the thickness of 5 mm.The Young’s modulus of the material is E=2.10 × 1011Pa.The Poisson’s ratio υ is 0.269,and the density of the material is 7850 kg/m3.The bottom of the thin-walled workpiece is fixed at the bottom of the MR fluids container,and the distributed points A,B,C,D,E,F on the thin-walled workpiece as shown in Fig.7 are chosen for impact tests,where point A and B are symmetric with respect to C,and D and F are symmetric with respect to D.

Fig.7 Distributed points A,B,C,D,E,F on thin-walled workpiece for impact tests.

As you know,different impact points can obtain different response of the thin-walled workpiece.There fore, for verifying the effectiveness and feasibility of the MR fluids flexible fixture,the impact responses on the distributed points A,B,C,D,E,F on the thin-walled workpiece are all obtained to analyze the vibration properties.The response at point A is approximately equal to that at point C due to the symmetric distribution,and the response at point D is similar to the response at point F for the same reason.However,the points B and E are located in the middle of the thin-walled plate and the response are different from points A,C and D,F on the collinear,respectively.In order to analyze the impact response problems of MR fluids flexiblefixture easily,point A is selected as the typical position for the vibration response because it is located near the cutter entry point,which is sensitive to dynamic cutting force.Thus the modal parameters of workpiece on the point A are listed in Table 1.

In order to exactly analyze the difference dynamic properties of the workpiece-fixture system with and without damping control,the frequency response functions at different damping parameters are plotted in Fig.8.It is observed that both the amplitudes of the frequency responses functions and the values of the frequency of the workpiece-fixture system are similar with and without MR fluids damping material under 1 A current in first vibration mode.Moreover,the amplitude values of the system with damper under 2 A and 3 A current reduce obviously,and the frequency response under MR fluids damping material under 2 A and 3 A current are about 390.6 Hz.In addition,all the higher vibration modes are obviously suppressed under the action of the MR fluids damping material.The whole experimental results validate that the MR fluids flexible fixture can improve the rigidity of thinwalled workpiece and suppress the machining vibration than the traditional fixture without MR fluids damping material.

Fig.6 Machining tests.

Table 1 Modal parameters of thin-walled workpiece at point A.

Fig.8 Frequency response functions of workpiece-fixture system with and without damper.

Fig.9 Comparison of the vibration properties tests with and without damper.

Furthermore,Fig.9 shows the experimental comparisons of the impact vibration characteristics of the workpiece-fixture system with and without the damping control,and Table 2 lists the peaks of the impact modal parameters for the thin-walled workpiece measured using the impact tests with and withoutdamper.In Fig.9,it is noted that the amplitude values of peaks with damper(A1=362.8 g)are smaller than that without damper(A2=832.4 g),and the system decrement with and without damper is approximately consistent with exponential function y=Ae-ζωntas the time changes.Furthermore,the decrement timefrom external excitation to stable state with the MR fluids damping materials is t1=0.02 s,which is smaller than that t2=0.045 s without the action of the MR fluids damping materials.In addition,under the hysteresis damping,the motion of the excitation can be considered to be nearly harmonic(the energy loss per cycle is small),and the decrease in amplitude per cycle can be determined using energy balance principle.There fore,the amplitude ratio of two adjacent wave peaksi,i+1 isa constant and can bederived as Ai/Ai+1= （2+ πα）/（2- πα） (α =p/k is a dimensionless damping constant,p is hysteresis damping constant.),and the amplitude ratios without and with damper are about 1.5 and 1.6 according to Table 2,respectively.

Table 2 Modal parameters of workpiece system.

5.2.Machining tests

In order to further verify the rationality and feasibility of the MR fluids flexible fixture,machining experiments are carried out on the machining center with and without the MR damping control.Be fore machining,milling stability is used to predict the effectiveness of the MR fluids’damping fixture.Because the natural frequency of the workpiece-fixture system under the action of MR fluids damping is not equal to the tooth passing frequency,the milling stability lobe diagram is used to select the different cutting parameters to validate the milling stability.Then,the stability lobe diagram for the thin-walled workpiece in three-axis milling with different damping ratios can be calculated using the proposed model as shown in Fig.10.

Fig.10 Stability lobe diagrams at different damping ratio(Point M and N are selected for machining tests).

Fig.11 Machining vibration acceleration response of thin-walled plate(S=5000 r/min,F=500 mm/min,ae=0.3 mm).

In Fig.10,it can be observed that stability lobe charts show remarkable variation with respect to the spindle speeds and the depth of cut with and without damping control,and the milling stability region is obviously enlarged under the MR fluids damping control in vertical direction at different damping ratios and does not change in the horizontal direction with the damping ratio increasing,which is used to validate the feasibility of the MR fluids flexible fixture in milling vibration suppression.In the fixed milling system,the larger the damping ratio under the action of the MR fluids on the thin-walled workpiece is,the more obviously stable machining is.

In machining,the dimension of the flexible workpiece is 130 mm×60 mm with the thickness of 5 mm.The cylindrical helical milling cutter with diameter 10 mm and the number of teeth N=2 is selected according to the principle of minimizing the effects of the cutter vibration on tool path.Then,spindle speed S=5000 r/min,radial cutting depth ae=0.3 mm and depth of cut in axial direction ap=2 mm(point M,and the stability lobes is shown in Fig.10.)are selected for validating the damping properties of the MR fluids under different currents.When the MR damping materials does not acted on the thin-walled workpiece,the maximum instantaneous acceleration is about 150 g as shown in Fig.11(a),and the fast Fourier trans formation(FFT)method is used to analyze the time domain singles.In Fig.11(a),it is found that different harmonic frequency ωcsuch as 403.4 Hz,820.1 Hz and 1156 Hz generated during the milling besides the tooth passing frequency ωt166.7 Hz together with its integer multiples(i.e.,333.4,516.5,500,666.7,833.4,1000 Hz).There fore,the milling process is unstable and the waviness caused by chatter is printed on the machined surface.Later,when the MR fluids damping materials contacting the workpiece is under the magnetic field,the maximum acceleration reduces to about 60 g with the damping material action under I=2 A as shown in Fig.11(b).It is noted that the vibration acceleration of thin-walled plate reduces by nearly 30%compared with that without damping control.The reason for this is that the damping force generated by the magnetorheological effect of MR fluids at different electric currents can significantly improve the dynamic machinability of the thin-walled workpiece,and the milling process is stable with the electric current increasing.In addition,the frequency spectrum also denotes the stable machining process as the tooth passing frequency and its harmonics dominate the whole machining under I=2 A.In Fig.11(c),the milling process still keeps stable under I=3.5 A.However,the value of acceleration amplitude raised to about 120 g and the quality of the machined surface deteriorate as the electric current increases,which reaches the chatter frequency ωc(i.e.,571.5,655.9,738.3,822.7 Hz.)as shown in Fig.11(d).The main reason for this is that the magnetic particles of the MR fluids have the limit of saturation magnetization,and when the magnetic field strength exceeds the critical value of saturation magnetization,the shear thinning phenomenon of MR fluids works gradually and the damping effect of MR fluids reduces sharply.

Fig.12 Comparison of predicted and measured displacement responses of selected response point A with and without damper(Cutting parameters: S=3600 r/min, F=500 mm/min,ae=0.3 mm,ap=2 mm).

Fig.13 Machining vibration of whole thin-walled sheet milling process(S=3600 r/min,F=500 mm/min,ae=0.3 mm).

To give a clear comparison between the machining vibration of the thin-walled plate with and without damping control,cutting parameters(S=3600 r/min,F=500 mm/min,radialcutting depth ae=0.3 mm,axialcutting depth ap=2 mm)are chosen according to the stability lobes for machining tests.Fig.12 shows the predicted and measured vibration displacement response of the thin-walled plate at the selected points A for the two cases with and without damping control.It can be noted that the maximum values of vibration displacement of point A on the thin-walled workpiece with the length 129 mm and the width 10 mm are 0.453,0.312 0.143 mm under the different currents 0,1,2 A.Moreover,an agreement of the measured and predicted displacement response of the point A is reached in milling process with and without damper control.However,the difference existing among the displacement values with and without damper control indicates that the damping property is a crucial factor in fluencing the dynamic behavior of the thin-walled workpiece in machining.And,the experimental results validate that the proposed equivalent dynamic models can effectively predict the dynamic response of the thin-walled workpiece during machining.In addition,the surface waviness of vibration in machining can fully reflect on the machined surface,so the machining vibration can be detected by observing the quality of the machined surface as shown in Fig.13.It is found that the quality of the machined surface with damping control is better than that without damper control obviously,which further indicates that the method of using the MR fluids flexible fixture to suppress the vibration of the thin-walled workpiece in machining is feasible and effective.

6.Conclusions

(1)An equivalent model of the typical complex thin-walled structure in the aerospace industry is proposed based on the Euler–Bernoulli beam assumptions for simplification,which is convenient to consider the MR fluids damping properties.

(2)A dynamic equation of the workpiece- fixture system considering the external MR fluids damping constraint is proposed using the Lagrangian method,and the response of system under the dynamic cutting force is calculated to evaluate the stability of the milling process under damping control compared with the response without damping control.

(3)The proposed model and approach are validated by the experimental results,which are obtained by the modal and machining tests with and without the action of MR fluids damping materials.And a good agreement has been reached in machining response control.

(4)The proposed equivalent model only treats the thinwalled plate as cantilever beam structure,and the bending potential energy is considered in the model,neglecting the torsional potential energy.In machining,the MR fluids’damping material plays an important role in improving the dynamic properties of the thin-walled workpiece,which is only validated in three-axis milling of the complex thin-walled workpiece based on Euler–Bernoulli beam assumptions.There fore, for the complex thin-walled structure workpiece(i.e.,blades,casings,impellers,blisks.)or five-axis milling,the comprehensive equivalent dynamic model and the new flexible fixture are necessary in the future because the dynamic responses in all directions determine the regenerative chatter in machining.

Acknowledgements

This work is supported by the National Basic Research Program of China(Grant No.2013CB035802)and the 111 Project of China(Grant No.B13044).

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Ma Junjin received his B.S.and M.S.degrees from Henan Polytechnic University in 2009 and 2012,respectively,and is now a Ph.D.candidate from Northwestern Polytechnical University.His main research interests are multi-axis machining and high-speed machining.

Zhang Dinghua received his B.S.,M.S.and Ph.D.degrees from Northwestern Polytechnical University in 1981,1984 and 1989,respectively,and became a prof essor there in 1991.His main research interests are multi-axis machining,high-speed machining,machining surface integrity and industrial CT technology.

Wu Baohai received the B.S.,M.S.,and Ph.D.degrees from Xi’an Jiaotong University in 1997,2000,and 2005,respectively.He is currently an Associate Prof essor in Northwestern Polytechnical University,and his research interests include multi-axis NC machining,CAD/CAM,and smart machining technology.

27 November 2015;revised 22 February 2016;accepted 22 March 2016

Available online 21 June 2016

Chatter;

Machining vibration suppression;

Milling;

Stability lobe diagram;

Thin-walled workpiece

?2016 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail addresses:majunjin@gmail.com(J.Ma),dhzhang@nwpu.edu.cn(D.Zhang),wubaohai@nwpu.edu.cn(B.Wu).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2016.04.017

1000-9361?2016 Production and hosting by Elsevier Ltd.on behalf of Chinese Society of Aeronautics and Astronautics.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).