A novel stability prediction approach for thin-walled component milling considering material removing process

Jiahao SHI,Qinghua SONG,Zhanqiang LIU,Xing AI

Key Laboratory of High Efficiency and Clean Mechanical Manufacture,Ministry of Education,Shandong University,Ji’nan 250061,China

School of Mechanical Engineering,Shandong University,Ji’nan 250061,China

A novel stability prediction approach for thin-walled component milling considering material removing process

Jiahao SHI,Qinghua SONG*,Zhanqiang LIU,Xing AI

Key Laboratory of High Efficiency and Clean Mechanical Manufacture,Ministry of Education,Shandong University,Ji’nan 250061,China

School of Mechanical Engineering,Shandong University,Ji’nan 250061,China

The milling stability of thin-walled components is an important issue in the aviation manufacturing industry,which greatly limits the removal rate of a workpiece.However,for a thin-walled workpiece,the dynamic characteristics vary at different positions.In addition,the removed part also has influence on determining the modal parameters of the workpiece.Thus,the milling stability is also time-variant.In this work,in order to investigate the time variation of a workpiece’s dynamic characteristics,a new computational model is firstly derived by dividing the workpiece into a removed part and a remaining part with the Ritz method.Then,an updated frequency response function is obtained by Lagrange’s equation and the corresponding modal parameters are extracted.Finally,multi-mode stability lobes are plotted by the different quadrature method and its accuracy is verified by experiments.The proposed method improves the computational efficiency to predict the time-varying characteristics of a thin-walled workpiece.

1.Introduction

Due to its wide applications in military,energy,and aerospace industries,milling of thin-walled components is of great significance.However,the high flexibility of a cutting toolworkpiece system and improper cutting parameters contribute to severe vibration,and the most important source of the problems that emerge during thin-walled components milling is regenerative chatter.

Regenerative chatter is a kind of self-excited vibrations,and the cutting amount simultaneously depends on the paths left by the tooth tips of the current tooth and the previous tooth,so the cutting force varies as well as the chip thickness.There are many methods proposed to predict chatter stability,including frequency-domain methods1–3and time-domain methods.4–8However,all the se studies assumed that the dynamic characteristics of a system did not change during the whole machining process.

As for a thin-walled workpiece,due to the variation of the dynamic characteristic with the motion of the cutter position,the stability also changes during the machining process.Bravo et al.9and Sheng et al.10divided the milling process into different stages and obtained frequency responsefunctions(FRFs)through an impact test.Although this method would be more accurate,it was almost impractical to conduct enough impact tests to obtain a workpiece’s dynamic characteristic during milling.In order to avoid repeated impact tests,a workpiece’s FRF was updated by the finite element method(FEM).11–15However,this method is of low efficiency since a workpiece should be remodeled and correspondingly,modal analysis should also be re-conducted.In this condition,Budak et al.16and Song et al.17utilized a structural dynamic modification scheme to improve the FEM-based method,where only a workpiece’s initial dynamic characteristic was needed.Similarly,taking into consideration the influence of material removal and the change of the mode shape along the cutting path and tool axis,Yang et al.18also obtained in-process workpiece dynamics through structural dynamic modification.Neglecting the influence of material removal,Li et al.19obtained position-dependent FRFs through the effective stiffness which was dependent on mode shape.Instead of the commonly adopted FEM,the multispan plate(MSP)mode was proposed to model the varying thickness of a flexible pocketstructure.20Compared with FEM-based methods,the MSP model offers 10–20 times higher computational efficiency.Combining the high computational efficiency of the MSP model and the versatility of FEM-based methods,Ahmadi21proposed finite strip modeling(FSM)to model thin-walled structures in pocket milling operations.

It is worth noting that due to the change of the mode shape along the cutting path,some specific modes may also affect the stability of a thin-walled workpiece.Based on frequency methods,Budak and Altintas22obtained the stability of a flexible multi-mode structure.However,the dynamic characteristic was assumed as unchanged.Thvenot et al.11,12took the first several modes into consideration and determined the stability limitation by selecting the lowest envelop of each mode.According to Ref.23,this lowest envelop method(LEM)may cause some prediction errors when the modes of a workpiece are not well separated.Zhang et al.24predicted the stability of a multi-mode cantilever plate and updated the dynamic characteristic by an impact test.Song et al.25took the influences of different modes into consideration.However,material removal was neglected.

Based on the above literature review,it can be seen that often the dynamic characteristic of a workpiece was obtained by the FEM.Apart from the MSP method,the re is no other independent numerical method which is not involved with the FEM.In addition,Refs.11,12considering both multi modes and material removal did not give a clear explanation about the relationship between the position,mode shape,and stability limitation.This paper focuses on the mathematic model of a thin-walled flexible part,and tries to conduct the oretical analysis of the time-variant modal parameters of the workpiece.

The contribution of this paper lies in that the influences of the cutter position and material removal is investigated by the oretical analysis.Since a system does not need to be remodeled during the machining process,compared with the finite element method,the calculation efficiency can be improved a lot.The remainder of this paper is summarized as follows.The equation of motion of the flexible thin-walled part and modal analysis are given in Section 2.Then,the different quadrature method is briefly introduced to calculate stability lobes in Section 3,and the milling stability prediction offlexible part milling and experimental results are presented in Section 4.Conclusions are drawn in the last section.

2.Time-varying model of thin-walled plate

As shown in Fig.1,during the milling process,the material is continuously removed along the trajectory of the moving cutter.Correspondingly,the workpiece’s dynamic characteristics,such as natural frequency and stiffness,also change during this process.In this section,different from Refs.16–18,where the FEM is used to conduct modal analysis,a more efficient method is proposed to obtain the dynamic characteristics of the workpiece.

2.1.Energy formula

In Fig.1,a cantilevered thin-walled rectangular plate under milling machining,with a length of L,a width of W,and a thickness of h,is modeled.The spindle speed is Ω and the feed rate is c.The classical plate the ory,which is suitable for thinwalled plate analysis,is used,and only the bending effect of the thin-walled plate is considered.Since,during the milling process,the material is removed as the cutter moves along the cutting path,the shape of the plate is varying during this period.However,it is lucky that the shape of the plate is predictable,and the material removal volume is relatively small compared to that of the unprocessed plate.Therefore,it is reasonable to assume that the plate’s modal shape is unchanged and correspondingly,the kinetic energy,T,of the machined plate can be expressed as

where ρ is the density,hris the thickness of the removed part,namely the radial cutting depth,w is the displacement along z direction,and the over-dot denotes differentiation with respect to time.It should be mentioned that A represents the surface area of the unprocessed plate while Ar(t)stands for that of the removed part,which is a function of the time.

Fig.1 Model of a cantilevered plate.

Similarly,the potential energy V*of the machined plate can be calculated as

wheref(x,y)is a function of x and y,and D*and Dr*are the complex flexural rigidities of the unprocessed plate and removed part,respectively,which are given by

where μ is the poisson ratio,E is the elastic module,i is an imaginary unit,and γ is the loss factor of the w*orkpiecefixture system.To distinguish from a real number,‘’is added in the upper right corner of the complex number.

In this paper,the Rayleigh-Ritz method(RRM)is used to approximately express the transverse displacement w.The Rayleigh-Ritz solutions for this problem are of the following form

where N is the number of Rayleigh-Ritz terms,qij(t)is the Rayleigh-Ritz coefficients,and ξi(x)and ηj(y)are admissible functions. ξi(x)and ηj(y)arefound by solving the clampedfree beam function and free-free beam function,respectively,which are expressed as26

where αiand βjare roots of the characteristic equation,and satisfy coshαicosαj=-1 and coshβicosβj=1,respectively.

Since the main aim is to extract the modal parameters,the external work done by the milling force is neglected.Therefore,the Lagrange of this system without a load is La=T-V*,where T and V*are the kinetic energy and strain energy obtained by Eqs.(1)and(2).The equation of motion of the system obtained by Lagrange’s equation is shown as

where the double-over-dot denotes second-order differentiation with respect to time,0 is the column vector of N2×1,q(t)=[q11(t),q12(t),...,qNN(t)]T.For the machining condition shown in Fig.1,where the cutter moves from the left to the right,the radial cutting depth is ar,and the axial cutting depth is aa,so matrices S and U*can be calculated by

and

In Eqs.(8)and(9),u=m+(i-1)×N and v=n+(j-1)× N,where i,j,m,and n=1,2,...,N.(·)i,xrepresents first-order differentiation with respect to x,which is similar to(·)i,xx,(·)j,y,and(·)j,yy.

It should be mentioned that Eqs.(1)–(10)show the process to obtain the governing equation of the cantilevered rectangular plate.For more complex workpieces,such as thin-walled plates and shells with different shapes,boundary conditions,and varying thickness,some mathematic skills are needed to deal with the m(e.g.,skills mentioned in Ref.27).Of course,for some quite complicated shapes,some reasonable simplifications are also needed.

2.2.Modal analysis

Eq.(7)describes the governing motion of the plate’s free vibration,and can be used to extract the natural frequencies and loss factors of different modes.Assuming q(t)in the form ofˉq exp （iλt）,Eq.(7)can be rewritten as

Since the stiffness matrix U*is a complex matrix,the generalized eigenvalue λ2is also a complex number.As a result,the derived complex eigenvalue problem,Eq.(11),gives the rorder natural frequency ωrand loss factor γras follows

where Re[·]and Im[·]mean the real and imaginary parts,respectively.For a small loss factor,it has the following relationship with the damping ratio,ζr

It should be mentioned that the loss factor of the workpiece-fixture system is different from that of a specific mode and should befirstly determined by a modal test.According to Eq.(13),the loss factor of a specific mode can be obtained by a modal experiment and the n based on Eq.(12),the loss factor of the workpiece-fixture system can be determined by the iteration method.

Apart from the natural frequency and loss factor of each mode,the FRF of the treated plate is now of primary interest.Assuming a unit harmonic point load applied at point(xin,yin),the FRF with a response at point(xre,yre)can be calculated as

where W(x,y)=[ξ1(x)η1(y),ξ2(x)η2(y),...,ξN(x)ηN(y)]T,and S and U*are calculated by Eqs.(8)–(10),respectively.After the calculation of the FRF,the rational fraction polynomials algorithm is adopted to fit the FRF curves to obtain the modal parameters.28

3.Milling stability with multiple modes

As shown in Fig.2,since the workpiece is flexible as compared to the tool,here,only the traverse vibration and cutting force in y-direction are considered.When considering n modes of the workpiece,the motion of equation can be expressed as

where M=diag(m1,m2,...,mn),C=diag(c1,c2,...,cn),K=diag(k1,k2,...,kn),and Z(t)=aahz(t)[φ1,φ2,...,φn]T×[φ1, φ2,..., φn]which is periodic with Tp.mi,ci,and kirepresent the ith-order modal mass,damping,and stiffness,respectively.The term φistands for the ith-order mode shape of the cutter position and

In Eq.(16),Ntis the number of cutter teeth,and Ktand Knare the tangential and normal linearized cutting force coefficients,respectively.φj(t)is the angular position of tooth j,and g(φj(t))is the switching function,which can be defined as

where φstand φexare the start and exit angles of tooth j.

Fig.2 Single degree-of-freedom milling system.

Here,in this paper,the different quadrature method(DQM)is used to approximate the original delay differential equation(DDE)(Eq.(15))as a set of algebraic equations and the n construct a Floquet transition matrix for milling stability analysis.The DQM was originally introduced by Bellman to solve partial differential equations(PDEs).29Then it has been widely used in engineering and physics problems.The basic idea of the DQM is that a partial derivative of a function with respect to a coordinate direction at a sampling grid point within an interval along that direction can be approximated as a linear weighted sum of all the function values at the sampling grid points within the whole interval.Compared with other well-known approximation methods,such as the semi-discretized technique,the advantages of this semianalytical time-domain method are its simplicity and high computational efficiency.More details about this method are described in Ref.30

Thefirst step of the DQM is re-expressing Eq.(15)as the following state-spaceform

Considering the characteristic of the milling process,one can assume that the time interval[0,Tp]is composed of a free vibration period,[0,tf],and a forced vibration period,[tf,Tp],during which the cutter contacts with the workpiece.For the first time interval,the state of the system at the end of the free vibration duration t=tfcan be analytically expressed as

Denoting the time span of the forced vibration duration Tc=Tp-tf,and letting τ=(t-tf)/Tcas the nondimensional time,Eq.(17)can be re-expressed as

For the normalized second time interval[0,1],it can be discretized by the sampling grid points τi,i=0,1,...,m.According to the idea of the DQM,each component of the derivative,˙~x（τ）,can be expressed as the weighted linear sums of~x（τ）at all the sampling grid points.Therefore,for each component of˙~x(τi),i=1,2,...,m,it can be rewritten in the following discreteform

where?is the Kronecker product defined for two matrices A and B so that A ? B=[AijB],I2n×2nis the 2n×2n identity matrix,and D is a matrix with an order of m×(m+1)and calculated by the following mth-order Lagrange polynomial

In this paper,Chebyshev-Gauss-Lobatto sampling grid points are used to determine the value of each τi.

Finally,by noting that ti=tf+τiTcand combining Eqs.(19)–(21),the state transition relationship can be obtained as follows

The stability of Eq.(15)can be approximated by the Floquet transition matrix Φ = Ψ-1Γ.Note that the integer m determines the dimension of matrices in Eq.(23)and the precision of the results.

4.Case study and experimental verification

4.1.Numerical study

The previous analysis provides a the oretical method to predict the stability of thin-workpiece milling considering material removal and changes of the tool position.In this section,before conducting numerical and experimental analysis,the process to compute the stability lobe diagram(SLD)is firstly described.As shown in Fig.3,firstly,the loss factor is extracted from the modal experiment,and the material removal is determined by the cutting position.Based on the the oretical analysis in Section 2.1,the energy expression of the workpiece can be calculated.Then,based on Eqs.(11)and(12),the natural frequencies and FRFs of different positions of the workpiece can also be obtained.The modal parameters are extracted from the FRFs with the method mentioned in Section 2.2,and finally,the milling stability can be predicted with the extracted modal parameters.

In this section,the geometrical parameters of the cantilever plate are listed in Table 1.The radial immersion ratio is fixed as 0.033,and the tangential and normal cutting force coefficients are Kt=6×108N/m2and Kn=2×108N/m2,respectively.A milling cutter with four edges is used in the down milling process.

In order to describe the dynamic characteristics of the plate,as shown in Fig.4(a),eleven evenly distributed points are selected along the cutter path.Fig.4(b)and(c)illustrates the first two mode shapes of the plate.It can be seen that in the first mode shape,the points along the cutter path share similar deformation,while differ from each other in the second mode shape.It should be mentioned that the re is a node in the middle area of the cutter path.This regulation is also reflected in Fig.5.

As described in Fig.3,firstly,a modal experiment is needed to obtain the loss factor of the workpiece before milling.It should be mentioned that,due to the existence of the node in the second mode,the impact point should not be selected in the middle of the plate,where the second mode cannot be excited.Therefore,as shown in Fig.4(a),the excited points are selected in the right upper corner of the plate.In this paper,due to the small difference between the damping ratios of first two modes,the loss factor of the system is assumed as the average of those of the first two modes.Based on the theoretic analysis in Sections 2.1 and 2.2,the first two natural frequencies at the se points are calculated and shown in Fig.6.The small difference between the results calculated by the presented method and the FEM verifies the accuracy of the previous the oretical analysis.It is obvious that the re is an increasing trend for the first mode(from 869 Hz to 876 Hz),while a sinuous trend for the second mode(between 1627 Hz and 1632 Hz).This may be because,in the first mode,with the remove of the material,a decrease of the mass results in a rise of the natural frequency.However,in the second mode,due to the sinuous mode shape along the se points,the natural frequency also presents a similar trend.

Fig.3 Process to predict the SLD.

Table 1 Geometrical parameters of the workpiece.

Fig.4 Cutter positions of the flexible plate and its first two mode shapes.

Fig.5 Normalized mode shapes of the cutting points of the first two modes.

Fig.7 gives the experimental and the oretical FRFs of the workpiece.The overlap between the se two kinds of curves around the peak area also shows the reliability of the presented method.It is clear that with the moving of the cutter,the first mode almost remains unchanged,while the second mode firstly decays and the n disappears at point x=50 mm.After that,from points 50 mm to 100 mm,this mode shows a reverse trend and increases gradually.This phenomenon may account for the mode shape of the second mode,where the re is a node in the middle area of the plate.Based on Section 2.2,modal parameters are extracted from the FRFs and listed in Table 2.It can be seen that the first mode’modal parameters are almost identical to each other.However,for the second mode,the se three modal parameters all show an increasing trend from 0–50 mm and decreasing from 50 mm to 100 mm.It is easy to understand that the node at x=50 mm makes this area a rigid point.

Fig.8 shows the SLDs of the se eleven points.As shown in Fig.8(a),the SLD of point x=0 is determined by the SLDs of both modes,which means that the bold solid line(SLD considering both modes)is the lowest envelope of the thin dash line(SLD of the first mode)and the thin solid line(SLD of the second mode).A similar phenomenon also exits in other positions.According to Ref.23,this is because the modes of the workpiece are well separated.However,when the se modes are close to each other and the contributions of other modes cannot be neglected,the zero-order approximation(ZOA)method can be used to calculate the SLD.Compared to the SLD of point 0,the SLD of the second mode is improved,and the bold solid line is mainly determined by the thin dash line.This trend also shows at points 2,3,4,and 5,and finally,in the middle point 5,the bold solid line totally overlaps with the thin dash line,which means that the SLD can be accurately calculated by only considering the first mode.This is because the node of the second mode can be considered as a fixed point,which means that chatter cannot occur and the limited depth approaches to an infinite value.In Fig.8,symbols‘+’, ‘×’,and ‘*’represent three different spindle speeds,namely 4200 r/min,5700 r/min,and 6500 r/min.It is easy to find that due to the variation of the SLD,stable zones and unstable zones are distributed in different regions.For symbol‘+’,the stable zone is predicted as 0–10 mm and 90–100 mm,while the unstable zone is 10–90 mm.Since symbol ‘×’is always below the SLD curve,the region is in a stable condition.As for symbol ‘*’,the regions in 0–20 mm and 80–100 mm are predicted as the stable zone,while that in 20–80 mm as the unstable zone.

Fig.6 Variations of the first two natural frequencies.

Fig.7 FRFs of different cutter positions.Solid lines:the oretic FRFs;dash lines:experimental FRFs.

4.2.Experimental verification

A milling experiment is carried out by a high-speed machining center VMC0540d with the maximum spindle speed of 30000 r/min.A GRAS 40PP microphone with a sensitivity of 50 mV/Pa is used to measure the sound pressure.The machined surface topography is also checked by a laser microscope(Keyence VK-X250).The milling experimental equipment is shown in Fig.9.The material of the workpiece is aluminum alloy 7075,whose geometric parameters are listed in Table 1,and that of the tool is cemented carbide.

The surface topography on the machined workpiece and the sound signal during the milling process are shown in Fig.10.Chatter marks are clear from the surface topography,and the surface of the stable zone is smooth with a lower roughness value in a spindle speed of 4200 r/min.In addition,the sound pressure during the milling process also shows a recognizable change.In the periods of 0–0.4 s and 3.7–4 s,the magnitude of the sound pressure is getting large obviously(about 10 Pa).However,for the time from 0.4 s to 3.7 s,the sound pressure shows a significant decrease to around 3.5 Pa.Correspondingly,the unstable zone,determined by the sound pressure,is 0–10 mm and 87.5–100 mm,while the stable zone is 10–87.5 mm.In Fig.10(b),during the milling period,the sound pressure stands at a small value,approximately 5 Pa.Meanwhile,the re are also no chatter marks in the surface topography in a spindle speed of 5700 r/min.Therefore,all the regions are in a stable condition.In a spindle speed of 6500 r/min(see Fig.10(c)),the re are three different areas in the sound pressure,namely 0–0.75 s,0.75–3.25 s,and 3.25–4 s.In the first and third time periods,both magnitudes are in higher values,about 20 Pa,which is higher than that in the second period,only around 3 Pa.Therefore,the correspondingly measured unstable zone in this milling condition is 0–18.75 mm and 81.25–100 mm,while the stable zone is 18.75–81.25 mm.Chatter marks in the surface topography during the third time period also reflect the existence of chatter.In the comparisons of the ranges of the stable domain and the chatter domain between experimental and theoretical analysis in Section 4.1,two results are in good agreement.Thus,the correctness of the proposed method can be verified.

Table 2 The first two modal parameters.

Fig.8 Dynamic SLDs of the milling process with different modes.

Fig.9 Experimental setup.

5.Conclusions

(1)A highly efficient numerical method is proposed to obtain the dynamic characteristics of a thin-walled workpiece,and the extracted modal parameters are used to plot multi-mode stability lobes.

(2)During the material removal process,the first-order natural frequency shows a growing trend,increasing from 869 Hz to 876 Hz.However,the second-order natural frequency firstly increases from 1627 Hz to 1632 Hz and the n decreases to 1628 Hz.After that,this natural frequency reaches its maximum value(1633 Hz)at the end of the milling process.

Fig.10 Time waves of sound pressure and surface topography.

(3)The first two modal parameters have different influences on the milling stability at different cutting positions.In the two ends of the cutting area,the limited axial depth is mainly determined by the second modal parameters.On the contrary,the stability lobes are only affected by the first modal parameters.

(4)The computation results are in agreement with experimental data.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(No.51575319),the Young Scholars Program of Shandong University(No.2015WLJH31),the Major National Scienceand Technology Project(No.2014ZX04012-014),and the Tai Shan Scholar Foundation(No.TS20130922).

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5 December 2016;revised 1 January 2017;accepted 13 February 2017

Available online 8 June 2017

Material removal effect;

Multi-mode;

Stability;

Thin-walled workpiece;

Vibration

*Corresponding author at:Key Laboratory of High Efficiency and Clean Mechanical Manufacture,Ministry of Education,Shandong University,Ji’nan 250061,China.

E-mail address:ssinghua@sdu.edu.cn(Q.SONG).

Peer review under responsibility of Editorial Committee of CJA.

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