AnEfficientThird-OrderFull-DiscretizationMethodfor...

Research ArticleAn Efficient Third-Order Full-Discretization Method forPrediction of Regenerative Chatter Stability in Milling

Chao Huang Wen-An Yang Xulin Cai Weichao Liu and YouPeng You

National Key Laboratory of Science and Technology on Helicopter TransmissionNanjing University of Aeronautics and Astronautics Nanjing 210016 China

Correspondence should be addressed to Wen-An Yang dreamflownuaaeducn

Received 6 July 2019 Revised 22 October 2019 Accepted 8 January 2020 Published 20 June 2020

Academic Editor Huu-Tai +ai

Copyright copy 2020 Chao Huang et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+e prediction of regenerative chatter stability has long been recognized as an important issue of concern in the field of machiningcommunity because it limits metal removal rate below the machinersquos capacity and hence reduces the productivity of the machineVarious full-discretization methods have been designed for predicting regenerative chatter stability +e main problem of suchmethods is that they can predict the regenerative chatter stability but do not efficiently determine stability lobe diagrams (SLDs)Using third-order Newton interpolation and third-order Hermite interpolation techniques this study proposes a straightforwardand effective third-order full-discretization method (called NI-HI-3rdFDM) to predict the regenerative chatter stability in millingoperations Experimental results using simulation show that the proposed NI-HI-3rdFDM can not only efficiently predict theregenerative chatter stability but also accurately identify the SLD +e comparison results also indicate that the proposed NI-HI-3rdFDM is very muchmore accurate than that of other existing methods for predicting the regenerative chatter stability in millingoperations A demonstrative experimental verification is provided to illustrate the usage of the proposed NI-HI-3rdFDM toregenerative chatter stability prediction +e feature of accurate computing makes the proposed NI-HI-3rdFDM more adaptableto a dynamic milling scenario in which a computationally efficient and accurate chatter stability method is required

1 Introduction

+e control of cutting vibration and machining instabilityplays a significant role in high quality and high-efficiencymachining of a workpiece such as titanium alloy and high-temperature alloy thin-walled components +e vibrationsexisting in the machining processes consist of free vibrationforced vibration and self-excited vibration where the lastone can be further divided into three main categories re-generative chatter model coupling chatter and frictionalchatter [1 2] With respect to milling operations it has beenproved that the main factor that influences the machiningstability is regenerative chatter [3] which may limit metalremoval rate below the machinersquos capacity and hence reducethe productivity of the machine +erefore accurate andefficient regenerative chatter stability prediction is a criticaltask in milling operations for determining the potentialregions of stable machining

+emain advance in the study of machining dynamics inthe past few years has come in recognizing and interpreting avariety of different delay-differential equations (DDEs) thatarise in the milling processes when considering the regen-erative effects [4 5] Stability lobe diagrams (SLDs) obtainedwith DDEs [6 7] are the most widely applied tools for re-vealing the boundary between stable (ie chatter-free) andunstable (ie with chatter) machining and locating theirpossible optimal chatter-free parameters (eg spindle speedand depth of cut) +e theoretical advances in numericalanalysis methods and the use of on-line computers forsolving DDEs in machining process modeling have led toincreased interest in creating SLDs for stable machiningparameter selection +ese techniques are referred to asfrequency-time methods (eg zero order analytical (ZOA)[8] andmultifrequency (MF) [9]) and time-domainmethods(eg temporal finite element analysis (TFEA) [10] first-order semidiscretization method (1stSDM) [11] updated

HindawiShock and VibrationVolume 2020 Article ID 9071451 16 pageshttpsdoiorg10115520209071451

semidiscretization method (USDM) [12] first-order full-discretization method (1stFDM) [13] second-order full-discretization method (2ndFDM) [14] third-order full-discretization method (3rdFDM) [15] third-order leastsquares approximated full-discretization method(3rdLSAFDM) [16] hyper-third-order full-discretizationmethod (HTOFDM) [17] numerical integration method(NIM) [18] improved numerical integration method(INIM) [19] complete discretization scheme (CDS) [20]RungendashKutta methods (RKM) [21] RungendashKutta-basedcomplete discretization method (RK-CDM) [22] third-or-der Hermite approximation method (3rdHAM) [23] third-order HermitendashNewton approximation method (3rdH-NAM) [24] high-order full-discretization method (HFDM)[25] and third-order updated full-discretization method(3rdUFDM) [26]) ZOA methods that expand time varyingcoefficients of dynamic milling equations into Fourier seriesand truncate the series to include only the average com-ponent have been applied in regenerative chatter stabilityprediction tasks Although it was proved that the ZOA canperform well in regenerative chatter stability predictiontasks not readily applicable to low radial immersion millingis a common problem for ZOAs in regenerative chatterstability prediction +e MF method extended the study ofZOA to consider the harmonics of the tooth-passing fre-quencies for low radial immersion machining processesGenerally however this method is limited to harmonicallyexcited systems and as the harmonics in the expansion of thenonlinear force becomes increasingly high there aregrowing concerns that time cost could become a threat

In recent years intensive research efforts have beenmade in the use of time-domain techniques as an effectiveapproach to regenerative chatter stability prediction Baylyet al [10] proposed a TFEA method to predicting millingstability A comparative study indicated that the perfor-mance of their proposed method was superior to that oftraditional frequency-domain methods with regard tocomputation accuracy Insperger et al [11] proposedthe1stSDM where the latter had higher computation effi-ciency and computation accuracy Long and Balachandran[12] developed the USDM for amilling dynamics model withconsideration of the loss-of-contact and time-delay effectsand indicated that period-doubling bifurcations of periodicorbits can occur in low-immersion operations Ding et al[13] proposed the 1stFDM for milling stability prediction Acomparative study indicated that the proposed 1stFDM wassuperior to the 1stSDM with regard to computation effi-ciency Owning to its wonderful computation accuracyvarious variants of the 1stFDM [14ndash17] such as the 2ndFDM3rdFDM 3rdLSAFDM and HTOFDM were subsequentlydeveloped for milling stability prediction Ding et al [18]employed NIM to recognize milling stability boundaries andshowed that their NIM outperformed the traditional SDMand TFEA method with regard to convergence rate andcomputation efficiency Liang et al [19] extended the studyof the NIM to develop an INIM for creating SLD for millingoperations with time-varying delay Li et al [20] proposedthe CDS for milling stability prediction by discretizing thedelay time term state term and periodic-coefficient matrices

simultaneously A comparative study indicated that theirCDS could outperform the SDM and FDM with regard tocomputation efficiency Niu et al [21] applied the fourth-order RungendashKutta method to handle the Duhamel term ofDDEs for milling stability prediction +eir approach wassuperior to 1stSDM in terms of convergence rate andcomputation efficiency Li et al [22] presented the RK-CDMto predict the chatter stability for milling processes Anextensive comparison based on convergence rate andcomputation efficiency as the performance index showedthat the proposed RK-CDM could outperform the SDM andCDS in predicting milling stability However nearly all theabovementioned methods in this research area have used thecoordinates of each node as interpolation conditions thecurve was maintained relatively flat when interpolating withlower-order functions while it would not be maintainedsmooth at the joints of the piecewise curves+e result mightas well not provide the near-optimum SLD On the contrarythese methods might lead to a poor overall machining pa-rameters selection

Recently the Hermite interpolation polynomial is one ofmany polynomial interpolation algorithms that have drawnincreasing attention for their attractive properties and ex-cellent error estimates on a wide range of numerical sum-mation problems Hermite interpolation polynomials notonly define the coordinate value of each node but also specifythe derivative of the curve at each node which has beenshown to be superior to the Newton interpolation andLagrange interpolation in keeping the stability of the curveWhile the Hermite interpolation polynomial solution maybe the globally stable optimum current FDM methods tendto fall into a local unstable solution +e unsmooth con-nection problem is unlikely to occur with Hermite inter-polation polynomial +erefore Hermite interpolationpolynomial can provide excellent performance for millingstability prediction problems

To the best of the authorsrsquo knowledge only three studiesin the literature have focused on the Hermite interpolationpolynomial-based machining stability prediction for millingoperations Liu et al [23] predicted the machining stabilityin milling operations using the 3rdHAM and compared theirresults with those obtained using the 1stSDM and 2ndFDMIn the efficient FDM linear interpolation and Hermite in-terpolation were approximate to the state item and time-delay item respectively Compared with the results obtainedusing the 1stSDM and 2ndFDM their FDM reducedcomputation time by almost 53 and 10 respectively Jiet al [24] used the 3rdH-NAM to resolve the problem ofmachining stability prediction for milling processes In the3rdH-NAM third-order Hermite interpolation and third-order Newton interpolation were used to approximate thestate item and time-delay item respectively Simulationsshowed that their approach was superior to Guorsquos 3rdFDMand Liursquos 3rdHAM in terms of the accuracy of identifyingthe milling stability boundaries Liu et al [25] proposed aHFDM to resolve the problem of machining stability pre-diction for turning processes and showed that their HFDMimproved calculation efficiency by about 50 compared tothe very efficient 2ndFDM In the HFDM Lagrange

2 Shock and Vibration

interpolation and Hermite interpolation were used to ap-proximate the state values and delayed state values re-spectively Although the results of these studies werepromising the problem of stability prediction accuracy hasstill remained unsolved and more accurate trials are needed

In the present research based on third-order Newtoninterpolation polynomial and third-order Hermite inter-polation polynomial techniques this study proposes astraightforward and effective third-order full-discretizationMethod (called NI-HI-3rdFDM) to predict the regenerativechatter stability in milling operations Experimental resultsusing simulation show that the proposed NI-HI-3rdFDMcan not only efficiently predict the regenerative chatterstability but also accurately identify the SLD +e compar-ison results also indicate that the proposed NI-HI-3rdFDMis very much more accurate than that of other existingmethods for predicting the regenerative chatter stability inmilling operations A demonstrative experimental verifica-tion is provided to illustrate the usage of the proposed NI-HI-3rdFDM approach to regenerative chatter stabilityprediction +e feature of accurate computing makes theproposed NI-HI-3rdFDM more adaptable to a dynamicmilling scenario in which a computationally efficient andaccurate method is required

+e study is organized as follows Section 2 describes themathematical model of milling dynamics adopted in thisstudy +e proposed NI-HI-3rdFDM for solving the math-ematical model of milling dynamics for regenerative chatterstability prediction is described in Section 3 Section 4presents the performance comparisons of the proposed NI-HI-3rdFDM with those of existing methods in the literatureExperimental results are given in Section 5 and these indicatethat the proposed NI-HI-3rdFDM is capable of efficientlycreating superior SLDs for accurate and efficient regenerativechatter stability prediction Finally Section 6 is devoted toconclusions and directions for future research

2 Mathematical Model of Milling Dynamics

It is generally accepted that the behavior characteristic of amilling operation is just as much the product of DDE as themorphology +us by considering regenerative effects thebehavior which can be adequately explained by the classicalmathematical theory of milling dynamics is described instate-space form by the equation as follows

_x(t) A0x(t) + B(t)[x(t) minus x(t minus T)] (1)

where A0 is a constant matrix and represents the time-in-variant nature of the system and B(t) is one periodic-co-efficient matrix and satisfies B(t + T) B(t)

+e general response of equation (1) can thus be ob-tained via direct numerical integration as follows

x(t) eA0 tminus t0( )x t0( 1113857 + 1113946

t

t0

eA0(tminus ξ)B(ξ)[x(ξ) minus x(ξ minus T)]dξ

(2)

In order to solve equation (2) by introducing a timeinterval τ the time period T is equally divided into m small

time intervals and therefore T mτ On each time intervalt isin [kτ (k + 1)τ](k 0 1 m) equation (2) can be re-written as the following equation

x(t) eA0(tminus kτ)x(kτ) + 1113946

t

kτeA0(tminus ξ)B(ξ)[x(ξ) minus x(ξ minus T)]dξ

(3)

Let xk denote x(kτ) then xk+1 can be got from equation(3) as the following equation

xk+1 eA0τxk + 1113946

τ

0eA0(τminus δ)B(δ)[x(δ) minus x(δ minus T)]dδ (4)

where δ ξ minus kτ δ isin [0 τ] In earlier investigations it hasbeen demonstrated that the performance of several ap-proaches including NIM CDS RK-CDM FDM and itsvariants are very sensitive to interpolation polynomial forapproximating the state item and time-delay item In thecurrent study the state term x(δ) is interpolated by third-order Newton interpolation polynomial while the time-delay term x(δ minus T) is approximated by third-order Hermiteinterpolation polynomial Specifically on the time interval[kτ (k + 1)τ] the nodal values x(kτ + τ) x(kτ) x(kτ minus τ)and x(kτ minus 2τ)(denoted as xk+1 xk xkminus1 and xkminus2 respec-tively) are used to approximate x(δ) and the nodal valuesx(kτ minus T) x(kτ + τ minus T) and x(kτ + 2τ minus T) (denoted asxkminusm xk+1minusm and xk+2minusm respectively) are used to approx-imate x(δ minus T) In addition because high order interpola-tion of B(δ) has no apparent effect on improving thecomputation efficiency and computation accuracy of thedynamic milling system [27] linear interpolation isemployed to approximate B(δ) with the boundaries[B(kτ)B(kτ + τ)] which is denoted here as [BkBk+1]

+e state term x(δ) is determined by third-order Newtoninterpolation polynomial as follows

x(δ) a1xk+1 + b1xk + c1xkminus1 + d1xkminus2 (5)

where

a1 δ3 + 3τδ2 + 2τ2δ

6τ3

b1 δ2 + 3τδ + 2τ2

2τ2minusδ3 + 3τδ2 + 2τ2δ

2τ3

c1 δ + 2ττ

minusδ2 + 3τδ + 2τ2

τ2+δ3 + 3τδ2 + 2τ2δ

2τ3

d1 1 +δ + 2ττ

+δ2 + 3τδ + 2τ2


6τ3

(6)

Using the derivative formula the first time derivative ofthe x(t) at t tkminusm and t tk+1minusm could be got from

_xkminusm xk+1minusm minus xkminusm

τ

_xk+1minusm xk+2minusm minus xk+1minusm

τ

(7)

Shock and Vibration 3

+en the time-delay term x(δ minus T) is obtained by third-order Hermite interpolation polynomial at the time span[tkminusm tk+1minusm] and expressed as

x(δ minus T) a2xkminusm + b2xk+1minusm + c2xk+2minusm (8)

where

a2 τ3 minus τ2δ minus τδ2 + δ3

τ3

b2 τ2δ + 2τδ2 minus 2δ3

τ3

c2 δ3 minus τδ2

τ3

(9)

Similarly the periodic-coefficient term is achieved bylinear interpolation at the time span [kτ (k + 1)τ] resultingin

B(δ) δτBk+1 +

τ minus δτ

Bk (10)

Substituting equations (5) (8) and (10) into equation (4)leads to

Hk1 minus I1113872 1113873xk+1 + H0 +Ηk01113872 1113873xk + Hkminus1xkminus1

+ Hkminus2xkminus2 Hkmxkminusm + Hkmminus1xk+1minusm + Hkmminus2xk+2minusm

(11)

where

H0 Φ0 eA0τ

Hk1 Φ5 + 3τΦ4 + 2τ2Φ3

6τ4Bk+1 +

minusΦ5 minus 2τΦ4 + τ2Φ3 + 2τ3Φ2

6τ4Bk

Hk0 Φ5 + 4τΦ4 + 5τ2Φ3 + 2τ3Φ2

2τ4Bk+1 +

minusΦ5 minus 3τΦ4 minus τ2Φ3 + 3τ3Φ2 + 2τ4Φ1

2τ4Bk

Hkminus1 Φ5 + τΦ4 minus 2τ2Φ3

2τ4Bk+1 +

minusΦ5 + 3τ2Φ3 minus 2τ3Φ2

2τ4Bk

Hkminus2 minusΦ5 + τ2Φ3

6τ4Bk+1 +

Φ5 minus τΦ4 minus τ2Φ3 + τ3Φ2

6τ4Bk

Hkmminus2 Φ5 minus τΦ4

τ4Bk+1 +

minusΦ5 + 2τΦ4 minus τ2Φ3

τ4Bk

Hkmminus1 minus2Φ5 + 2τΦ4 + τ2Φ3

τ4Bk+1 +

2Φ5 minus 4τΦ4 + τ2Φ3 + τ3Φ2

τ4Bk

Hkm Φ5 minus τΦ4 minus τ2Φ3 + τ3Φ2

τ4Bk+1 +

minusΦ5 + 2τΦ4 minus 2τ3Φ2 + τ4Φ1

τ4Bk

Φ1 1113946τ

0e

minusA0δdδ

Φ2 1113946τ

0e

minusA0δδdδ

Φ3 1113946τ

0e

minusA0δδ2dδ

Φ4 1113946τ

0e

minusA0δδ3dδ

Φ5 1113946τ

0e

minusA0δδ4dδ

(12)

+ematricesΦ1ndashΦ5 could be calculated byΦ0 and A0 asΦ1 Aminus1

0 Φ0 minus I( 1113857 (13)

Φ2 Aminus10 Φ1 minus τI( 1113857 (14)


Φ3 Aminus10 2Φ2 minus τ2I1113872 1113873 (15)



Note that it is thus highly desirable to find a one-to-onediscrete mapping between the current and previous period[27] all the current period variables x (ie xk+1 xk xkminus1 xkminus2)should be located in the left hand side of equation (11) whileall the previous period variables x (ie xk+2minusm xk+1minusm xkminusm)in the right hand side of equation (11) If not correspondingterms need to be converted into a required range In equation(11) when k 1 the left hand side variables xkminus1 and xkminus2 ofequation (11) are equal to x0 and xminus1 which are also presentedas xmminusm and xmminus1minusm in the previous period +ereforeequation (11) is rewritten as

H11 minus I1113872 1113873x2 + H0 +Η101113872 1113873x1 H1mx1minusm + H1mminus1x2minusm

+ H1mminus2x3minusm minus H1minus1xmminusm minus H1minus2xmminus1minusm

(18)

When k 2 the left hand side variable xkminus2 of equation(11) is equal to x0 which is also equal to xmminusm in the previousperiod +en equation (11) can be rewritten as

H21 minus I1113872 1113873x3 + H0 +Η201113872 1113873x2 + H2minus1x1 H2mx2minusm

+ H2mminus1x3minusm + H2mminus2x4minusm minus H2minus2xmminusm(19)

When k m the right hand side variable xk+2minusm ofequation (11) is equal to x2 which is in the current periodEquation (11) can be rewritten as

Hm1 minus I1113872 1113873xm+1 + H0 +Ηm01113872 1113873xm + Hmminus1xmminus1

+ Hmminus2xmminus2 minus Hmmminus2x2 Hmmxmminusm + Hmmminus1xm+1minusm

(20)

According to equations (11) and (18)ndash(20) the discretemapping is obtained as

D1

x1x2⋮xm

xm+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2

x1minusm

x2minusm

⋮xmminusm

xm+1minusm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where

D1

I 0 0 0 middot middot middot 0 0 0 0 0

H0 + H10 H11 minus I 0 0 middot middot middot 0 0 0 0 0

H2minus1 H0 + H20 H21 minus I 0 middot middot middot 0 0 0 0 0

H3minus2

⋮

H3minus1

⋮

H0 + H30

⋮

H31 minus I

⋮

middot middot middot

⋱

0

⋮

0

⋮

0

⋮

0

⋮

0

⋮0 0 0 0 middot middot middot Hmminus1minus2 Hmminus1minus1 H0 + Hmminus10 Hmminus11 minus I 0

0 minusHmmminus2 0 0 middot middot middot 0 Hmminus2 Hmminus1 H0 + Hmminus10 Hmminus11 minus I

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2

0 0 0 0 0 middot middot middot 0 0 I

H1m H1mminus1 H1mminus2 0 0 middot middot middot minusH1minus2 minusH1minus1 0

0 H2m H2mminus1 H2mminus2 0 middot middot middot 0 minusH2minus2 0

0 0 H3m H3mminus1 H3mminus2 middot middot middot 0 0 0

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

0 0 0 0 0 middot middot middot Hmminus1m Hmminus1mminus1 Hmminus1mminus2

0 0 0 0 0 middot middot middot 0 Hmm Hmmminus1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)

+e transition matrix of dynamic system Φ is definedover one period T as

Φ D1( 1113857minus 1D2 (23)

Finally if the maximum modulus for the eigenvalue ofthe transition matrix V is the at most one then concludethat the dynamicmilling system is in stable (ie chatter-free)machining in accordance with Floquet theory [28] other-wise conclude that the dynamic milling system is in unstable

(ie with chatter) machining in accordance with Floquettheory [28]

3 The Proposed Method for MillingStability Analysis

+e motion of a single DOF milling system with singlediscrete time delay can be expressed as a functional dif-ferential equation as follows


euroq(t) + 2ζωn _q(t) + ω2nq(t)

minuswh(t)

m[q(t) minus q(t minus T)]

(24)

where ζ ωn mt w and q(t) denote the relative dampingnatural frequency modal mass depth of cut and dis-placement vector of the cutter respectively Tooth-passingperiod T is equal to 60(NΩ) whereN is the number of teethand Ω is the spindle speed +e directional cutting forcecoefficient h(t) is one periodic function with period T and isdefined as

h(t) 1113944N

j1g ϕj(t)1113872 1113873sin ϕj(t)1113872 1113873 Ktcos ϕj(t)1113872 1113873 + Knsin ϕj(t)1113872 11138731113960 1113961

(25)

where Kt and Kn are the tangential and normal cutting forcecoefficients and ϕj(t) denotes the jth toothrsquos angular po-sition and is expressed as

ϕj(t) 2πΩ60

t +2π(j minus 1)

N (26)

+e window function g(ϕj(t)) is defined by

g ϕj(t)1113872 1113873 1 ϕst ltϕj(t)lt ϕex

0 otherwise1113896 (27)

where ϕst and ϕex are the start and exit angles respectivelyAs for up-milling ϕst 0 and ϕex arcos(1 minus 2aD) as fordown-milling ϕst arcos(2aD minus 1) and ϕex π whereaD is the radial depth of the cut ratio [5]

Let p(t) m _q(t) + mζωnq(t) and x(t) q(t) p(t)1113858 1113859then equation (16) can be expressed as


where

A0 minusζωn mminus1

t

mt ζωn( 11138572

minus mtω2n minusζωn

⎡⎣ ⎤⎦

B(t) 0 0

minuswh(t) 01113890 1113891

(29)

+emodal parameters relating to the dominant mode ofthe milling system are given in Table 1 For more infor-mation about periodic DDEs one can refer to the literature[5]

31 Convergence Rate In order to evaluate the convergencerate of different time-domain methods such as the NIM3rdHAM 3rdFDM and the proposed NI-HI-3rdFDM thelocal error ||u|minus |u0|| is expressed as the function of thediscrete number Noting that |u| is the maximummodulus ofthe eigenvalue of the transitionmatrix and |u0| is obtained bythe 1stSDM [11] atm 200 All the other parameters for themilling process are summarized in Table 2 All of the cal-culation programs are written in MATLAB 92 and run on apersonal computer Intelreg Coretrade i5-7400 CPU and 8GBmemory

Figure 1 presents the convergence rates of the NIM3rdHAM 3rdFDM and the proposed NI-HI-3rdFDM inpredicting the milling stability Figure 1 indicates that theproposed NI-HI-3rdFDM can converge sooner than theNIM 3rdHAM and 3rdFDM can Specifically the localerrors ||u|minus |u0|| of the NIM 3rdHAM 3rdFDM and theproposed NI-HI-3rdFDM were 00155 00084 00233 and06272times10minus3 respectively when Ω 5000 rpm w 05mmand m 40 +e local error of the proposed NI-HI-3rdFDMdecreased by 90 925 and 973 in comparison to that ofthe NIM 3rdHAM and 3rdFDM respectively Although theproposed NI-HI-3rdFDM converged a little slower than3rdHAM in case of w 05mm and |u0| 10726 for otherthree cases the proposed NI-HI-3rdFDM was fairly fasterthan 3rdHAM +is slow convergence rate can be explainedby the fact the derivability of the state item was utilized toenhance the calculation precision in 3rdHAM

32 Stability Lobe Diagrams In order to demonstrate thecomputation accuracy and computation time performancesof the proposed NI-HI-3rdFDM the SLDs obtained with theNIM 3rdHAM 3rdFDM and the proposed NI-HI-3rdFDMfor m 20 30 and 40 are summarized in Table 3 Note thatthe machining operation is up-milling operation the ratio ofradial cutting depth and tool diameter is aD 1 the spindlespeed Ω and depth of cut w is set to be 5000 rpm to10000 rpm and 0mm to 4mm over a 200 times 100 sized grid ofparameters respectively [14] It is also worth noting that inTable 3 the stability charts obtained with the 1stSDM form 200 in red color were regarded as the criterion-refer-enced SLDs As can be seen the stability charts of theproposed NI-HI-3rdFDMwere much closer to the criterion-referenced lobe diagrams than those of the NIM 3rdHAMand 3rdFDM Table 3 indicates that the proposed NI-HI-3rdFDM can also achieve better or at least comparableperformance with existing methods such as the NIM3rdHAM and 3rdFDM with respect to the computationtime Specifically in the case ofm 30 the computation timeof the NIM 3rdHAM 3rdFDM and the proposed NI-HI-3rdFDM were 10 s 92 s 38 s and 23 s respectively In thiscase the computation efficiency of the proposed NI-HI-3rdFDM increased by 75 and 395 compared with that ofthe 3rdHAM and 3rdFDM respectively except for the NIMAlthough the computation time of the proposed NI-HI-3rdFDM was slightly greater than that of the NIM thecomputation accuracy obtained with the proposed NI-HI-3rdFDM (corresponding computation time 23 s) was evenequal to that of the NIM when m 60 (correspondingcomputation time 37 s) Moreover in the case of m 40the stability charts of the proposed NI-HI-3rdFDM were ingood agreement with the criterion-referenced lobe A visualexamination of Table 3 also revealed that the proposed NI-HI-3rdFDM not only obtained successfully accurate millingstability prediction but also saved a lot of computation time

In addition the proposed NI-HI-3rdFDM can also beextended to a multifreedom milling system A stabilityprediction of a two-DOF milling system was employed toevaluate the accurate and efficient performances of NI-HI-


3rdFDM in predicting milling stability Note that the samesystem parameters taken from the literature [5] were used+e SLDs calculated from up-milling operations withm 40and aD 1 05 01 and 005 are shown in Table 4 As can beseen in Table 4 the SLD of the proposed NI-HI-3rdFDMwas

relatively identical to that of the 1stFDM and the 1stSDMbut the computation time of the proposed NI-HI-3rdFDMwas much less than that of the 1stFDM and the 1stSDMunder the same milling parameters +e proposed NI-HI-3rdFDM achieved better and comparable efficiency

Table 1 Modal parameters

Number of teeth (N) Frequency f (Hz) Damping (Ζ) Mass mt (kg)Tangential cutting forcecoefficient Kt (Nm2)

Normal cutting forcecoefficient Kn (Nm2)

2 922 0011 003993 6times108 2times108

Table 2 Parameters for the milling process

Milling type Radial depth of cut ratio aD Spindle speed Ω (rpm) Depth of cut w (mm)Up-milling 1 5000 02 05 07 1

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]

3rdFDM [15]The proposed NI-HI-3rdFDM

m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)

Figure 1 Convergence rate for the NIM 3rdHAM 3rdFDM and the proposed NI-HI-3rdFDM (a) w 02mm |u0| 08192 (b)w 05mm |u0| 10726 (c) w 07mm |u0| 12197 (d) w 1mm |u0| 14040


Table 3 SLD for the single DOF milling model using the NIM 3rdHAM 3rd FDM and the proposed NI-HI-3rdFDM

m 20

NIM [18]Criterion-referenced lobe

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)

3rdHAM [23]Criterion-referenced lobe

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)

Computation time 6 s Computation time 55 s

3rdFDM [15]Criterion-referenced lobe

Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000

e proposed NI-HI-3rdFDMCriterion-referenced lobe

Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000



compared with the 1stFDM and the 1stSDM Especially foraD 005 the runtime of the proposed NI-HI-3rdFDMdecreased from 200 s to 160 s when compared with the1stFDM and from 1197 s to 160 s when compared with the1stSDM

4 Comparison and Discussion

Performances of the proposed NI-HI-3rdFDMwere comparedwith those of existing methods in the literature In order toillustrate the fact that specific interpolations of state and time-delay items can be associated independently with differentproblems when relevant process knowledge is accessible theother candidate method called 3rdH-HAM for chatter stability

prediction that uses two three-order Hermite approximationalgorithms to simultaneously approximate the state item andtime-delay item respectively is also provided in the com-parison between the proposed NI-HI-3rdFDM and otherexisting methods Figure 2 compares the convergence rate forchatter stability prediction of the proposed NI-HI-3rdFDMwith that of the 3rdH-HAM 3rdUFDM and 3rdH-NAM Asseen in Figure 2 the proposed NI-HI-3rdFDM outperformedall the compared methods in converging local discretizationdifferences between the approximate and the exact solutionover the discretization interval except in the case ofw=02mm and |u0| = 08192 In this case there existed manyweak points for the proposed NI-HI-3rdFDMwhen comparedwith the 3rdH-NAM +is relatively slow convergence can be

Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 4 SLD for the two-DOF milling model using the 1stSDM 1stFDM and the proposed NI-HI-3rdFDM

Radial depthof cut ratio 1stSDM [11] 1stFDM [13] +e proposed NI-HI-3rdFDM

aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25

Computation time 1247 s Computation time 282 s Computation time 256 s

aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)



explained by the fact that the derivative of x(δ) in the 3rdH-NAMwas utilized to approximate the state term with low axialdepth of cut which resulted in the increase of accumulativeprecision It can also be concluded from Figure 2 especiallyfrom this case that the method using third-order Newtoninterpolation algorithm and third-order Hermite interpolationalgorithm simultaneously approximated the state item andtime-delay item respectively (ie NI-HI-3rdFDM) and out-performed the method using two three-order Hermite inter-polation algorithms to simultaneously approximate the stateitem and time-delay item respectively (ie 3rdH-HAM) +ismeans that three-order Hermite interpolation algorithm wasideal only when it was a time-delay item interpolator and failedto be ideal for the interpolation of state item

Table 5 presents a comparison of the SLDs of the 3rdH-HAM 3rdH-NAM 3rdUFDM and the proposed NI-HI-3rdFDM in creating lobe diagrams Table 5 indicates that theproposed NI-HI-3rdFDM can create lobe diagrams moreaccurately than the 3rdH-HAM 3rdH-NAM and 3rdUFDMcan+e proposedNI-HI-3rdFDMwas also capable of creatinglobe diagrams with smaller value of m but without any loss incomputation accuracy the 3rdH-HAM 3rdH-NAM and3rdUFDM had no such capability As can be seen using thethird-order Hermite interpolation algorithm to approximatethe time-delay term enhanced notably performances of theproposed NI-HI-3rdFDM especially the computation accu-racy For a certain accuracy of SLD the third-order Hermiteinterpolation algorithmwasmore effective than the third-order

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m

3rdH-HAM3rdH-NAM [24]

3rdUFDM [26]The proposed NI-HI-3rdFDM

(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)

Figure 2 Comparison of convergence rates between the proposed NI-HI-3rdFDM and existing methods (a) w 02mm |u0| 08192 (b)w 05mm |u0| 10726 (c) w 07mm |u0| 12197 (d) w 1mm |u0| 14040


Newton interpolation algorithm in approximating the time-delay term +e abovementioned characterization of idealinterpolation implies that when special interpolation algo-rithms existed approximating the state item and time-delayitem simultaneously in creating SLD for chatter stabilityprediction it was more reasonable to combine the state-iteminterpolation and time-delay-item interpolation informationinto one scheme and analyzed their behavior jointly +e si-multaneous analysis of state and time-delay items was veryhelpful in the approximation search for each discretization step+e third-order Newton interpolation algorithmwas obviouslynot good enough for the proposed NI-HI-3rdFDM to find theoptimal interpolation solution of the state itemHowever whenthe third-order Newton interpolation algorithm approximatedthe state item the proposed NI-HI-3rdFDM increased theefficiency by 848 (1-37243) compared with the 3rdH-NAMand by 178 (1-3745) compared with the 3rdUFDM underthe similar accuracy condition +erefore it can be concluded

that a third-order Newton interpolation of state item and athird-order Hermite interpolation of time-delay item were agood compromise between the performance and the executiontime of the proposed NI-HI-3rdFDM

5 Experimental Verification

In this section a two-DOF milling system with m 40 wasutilized to verify how the proposed NI-HI-3rdFDM canfunction effectively in the chatter stability prediction +esame experimental verification was also carried out byGradisek et al [29] +e modal parameters and cuttingexperiment data applied to the proposed NI-HI-3rdFDMwere the same as those used in the study of Gradisek et al[29] thereby ensuring unbiased verification Figure 3 showsthe SLDs for up-milling operations with aD 1 05 01 and005 +e results of the investigation indicate that for aD 1and 05 the chatter stability boundaries of the proposed NI-

Table 5 Comparison of SLDs between the proposed NI-HI-3rdFDM and existing methods

3rdH-HAMCriterion-referenced lobe

w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)

3rdH-NAM [24]Criterion-referenced lobe

w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)

Computation time 129 s m 40 Computation time 243 s m 62

w (m

m)

3rdUFDM [26]Criterion-referenced lobe

Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)

The proposed NI-HI-3rdFDMCriterion-referenced lobe

Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000



HI-3rdFDM absolutely agreed with Gradisekrsquos experimentalobservations For aD 01 and 005 the proposed NI-HI-3rdFDM also possessed good stability prediction capabilitybut not without weak points for some specific cutting re-gimes It was relatively difficult for the proposed NI-HI-3rdFDM to predict the stability boundaries as radial im-mersion was decreased +e abovementioned weak points ofthe chatter stability prediction capability could be tackled bymeans of additional consideration of certain nonlinearfactors such as tool deflection and effects of edge forcesresiding in the low radial immersion milling +e variationof the radial immersion during cutting due to the tooldeflection and the effects of edge forces could be attached tothe milling dynamics modeling to improve the chatter

stability prediction performance of the proposed NI-HI-3rdFDM with regard to aD 01 and 005

6 Conclusions

Regenerative chatter can be related to different assignablecauses that affect adversely a machining operation such aslimiting metal removal rate below the machinersquos capacityand reducing the productivity of the machine +erefore theprediction of regenerative chatter stability is very helpful foroptimally stable cutting parameter selection in ensuing ahigh-performance machining In past years various full-discretization methods have been designed for predictingregenerative chatter stability +e main problem of such

aD = 1 m = 40w

(mm

)

Stability boundariesStable cuttingUnstable cutting

0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)

Figure 3 Experimental and predicted stability boundaries


methods is that they can predict the regenerative chatterstability but do not efficiently determine SLDs In this studyusing third-order Newton interpolation and third-orderHermite interpolation techniques a straightforward and ef-fective NI-HI-3rdFDM was developed for the prediction ofregenerative chatter stability +e experimental results ob-tained using simulation indicate that the proposed NI-HI-3rdFDM can not only efficiently predict the regenerativechatter stability but also accurately identify the SLD Em-pirical comparisons showed that the proposed NI-HI-3rdFDM outperformed the existing approaches in the liter-ature in predicting the regenerative chatter stability while alsoproviding the capability of faster and more accurate com-putation that facilitates dynamic milling scenario in which acomputationally efficient and accurate chatter stability pre-diction method is required+is study also demonstrated thatthe chatter stability boundaries of the proposed NI-HI-3rdFDM agree with experimental observations

In this study only a linear milling dynamics model wasadopted thus the predictions and experiments agree onlyqualitatively (ie with respect to the structure of the stabilityboundary) for lower radial immersions Hence for applyingthe proposed NI-HI-3rdFDM in a practical chatter stabilityprediction scenario other common types of nonlinearfactors (eg tool deflection and effects of edge forces) shouldalso be included in the milling dynamics modeling toovercome the usage limitation of the proposed method

Data Availability

No data were used to support this study But the authorswould like to share their MATLAB codes to defend thesubmitted results

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was funded partially by the National ScienceFoundation of China (51775279 and 51775277) NationalDefense Basic Scientific Research Program of China(JCKY201605B006) Jiangsu Industry Foresight and CommonKey Technology (SBE2018030858) and Open Foundation forGraduate Innovation Base of NUAA (KFJJ20180518)

References

[1] M Wiercigroch and E Budak ldquoSources of nonlinearitieschatter generation and suppression in metal cuttingrdquo Phil-osophical Transactions of the Royal Society of London Series AMathematical Physical and Engineering Sciences vol 359no 1781 pp 663ndash693 2001

[2] M Kayhan and E Budak ldquoAn experimental investigation ofchatter effects on tool liferdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Man-ufacture vol 223 no 11 pp 1455ndash1463 2009

[3] Y Altintas Manufacturing Automation Metal Cutting Me-chanics Machine Tool Vibrations and CNC Design Cam-bridge University Press Cambridge UK 2000

[4] H E Merritt ldquo+eory of self-excited machine-tool chattercontribution to machine-tool chatter research-1rdquo Journal ofEngineering for Industry vol 87 no 4 pp 447ndash454 1965

[5] T Insperger and G Stepan ldquoUpdated semi-discretizationmethod for periodic delay-differential equations with discretedelayrdquo International Journal for Numerical Methods in En-gineering vol 61 no 1 pp 117ndash141 2004

[6] R Sridhar R E Hohn and GW Long ldquoA stability algorithmfor the general milling process contribution to machine toolchatter research-7rdquo Journal of Engineering for Industryvol 90 no 2 pp 330ndash334 1968

[7] H Ding Y Ding and L Zhu ldquoOn time-domain methods formilling stability analysisrdquo Chinese Science Bulletin vol 57no 33 pp 4336ndash4345 2012

[8] Y Altintas and E Budak ldquoAnalytical prediction of stabilitylobes in millingrdquo CIRP Annals-Manufacturing Technologyvol 44 no 1 pp 357ndash362 1995

[9] S D Merdol and Y Altintas ldquoMulti frequency solution ofchatter stability for low immersion millingrdquo Journal ofManufacturing Science and Engineering vol 126 no 3pp 459ndash466 2004

[10] P V Bayly J E Halley B P Mann and M A DaviesldquoStability of interrupted cutting by temporal finite elementanalysisrdquo Journal of Manufacturing Science and Engineeringvol 125 no 2 pp 220ndash225 2003

[11] T Insperger G Stepan and J Turi ldquoOn the higher-ordersemi-discretization for periodic delayed systemsrdquo Journal ofSound and Vibration vol 313 no 1ndash2 pp 334ndash341 2008

[12] X-H Long and B Balachandran ldquoStability analysis formilling processrdquo Nonlinear Dynamics vol 49 no 3pp 349ndash359 2007

[13] Y Ding L Zhu X Zhang and H Ding ldquoA full-discretizationmethod for prediction of milling stabilityrdquo InternationalJournal of Machine Tools and Manufacture vol 50 no 5pp 502ndash509 2010

[14] Y Ding L Zhu X Zhang and H Ding ldquoSecond-order full-discretization method for milling stability predictionrdquo In-ternational Journal of Machine Tools andManufacture vol 50no 10 pp 926ndash932 2010

[15] Q Quo Y Sun and Y Jiang ldquoOn the accurate calculation ofmilling stability limits using third-order full-discretizationmethodrdquo International Journal of Machine Tools and Man-ufacture vol 62 pp 61ndash66 2012

[16] C G Ozoegwu ldquoLeast squares approximated stabilityboundaries of milling processrdquo International Journal ofMachine Tools and Manufacture vol 79 pp 24ndash30 2014

[17] C G Ozoegwu S N Omenyi and S M Ofochebe ldquoHyper-third order full-discretization methods in milling stabilitypredictionrdquo International Journal of Machine Tools andManufacture vol 92 pp 1ndash9 2015

[18] Y Ding L Zhu X Zhang and H Ding ldquoNumerical inte-gration method for prediction of milling stabilityrdquo Journal ofManufacturing Science and Engineering vol 133 no 3 ArticleID 031005 2011

[19] X G Liang Z Q Yao L Luo and J Hu ldquoAn improvednumerical integration method for predicting milling stabilitywith varying time delayrdquo 9e International Journal of Ad-vanced Manufacturing Technology vol 68 no 9ndash12pp 1967ndash1976 2013

[20] M Li G Zhang and Y Huang ldquoComplete discretizationscheme for milling stability predictionrdquo Nonlinear Dynamicsvol 71 no 1-2 pp 187ndash199 2013


[21] J Niu Y Ding L Zhu and H Ding ldquoRunge-Kutta methodsfor a semi-analytical prediction of milling stabilityrdquoNonlinearDynamics vol 76 no 1 pp 289ndash304 2014

[22] Z Q Li Z K Yang Y R Peng F Zhu and X Z MingldquoPrediction of chatter stability for milling process usingRunge-Kutta-based complete discretization methodrdquo 9eInternational Journal of Advanced Manufacturing Technologyvol 86 no 1ndash4 pp 943ndash952 2016

[23] Y Liu D Zhang and B Wu ldquoAn efficient full-discretizationmethod for prediction of milling stabilityrdquo InternationalJournal of Machine Tools and Manufacture vol 63 pp 44ndash482012

[24] Y J Ji X B Wang Z B Liu H J Wang and Z H Yan ldquoAnupdated full-discretizationmilling stability predictionmethodbased on the higher-order Hermite-Newton interpolationpolynomialrdquo International Journal of AdvancedManufacturing Technology vol 95 no 5ndash8 pp 2227ndash22422018

[25] Y Liu A Fischer P Eberhard and BWu ldquoA high-order full-discretization method using Hermite interpolation for peri-odic time-delayed differential equationsrdquo Acta MechanicaSinica vol 31 no 3 pp 406ndash415 2015

[26] Z H Yan X B Wang Z B Liu D Q Wang L Jiao andY J Ji ldquo+ird-order updated full-discretization method formilling stability predictionrdquo 9e International Journal ofAdvanced Manufacturing Technology vol 92 no 5ndash8pp 2299ndash2309 2017

[27] X W Tang F Y Peng R Yan Y H Gong Y T Li andL L Jiang ldquoAccurate and efficient prediction of millingstability with updated full-discretization methodrdquo 9e In-ternational Journal of Advanced Manufacturing Technologyvol 88 no 9ndash12 pp 2357ndash2368 2017

[28] M Farkas Periodic Motions Springer Berlin Germany 1994[29] J Gradisek M Kalveram T Insperger KWeinert G Stepan

and E Govekar ldquoOn stability prediction for millingrdquo Inter-national Journal of Machine Tools and Manufacture vol 45pp 769ndash781 2005


semidiscretization method (USDM) [12] first-order full-discretization method (1stFDM) [13] second-order full-discretization method (2ndFDM) [14] third-order full-discretization method (3rdFDM) [15] third-order leastsquares approximated full-discretization method(3rdLSAFDM) [16] hyper-third-order full-discretizationmethod (HTOFDM) [17] numerical integration method(NIM) [18] improved numerical integration method(INIM) [19] complete discretization scheme (CDS) [20]RungendashKutta methods (RKM) [21] RungendashKutta-basedcomplete discretization method (RK-CDM) [22] third-or-der Hermite approximation method (3rdHAM) [23] third-order HermitendashNewton approximation method (3rdH-NAM) [24] high-order full-discretization method (HFDM)[25] and third-order updated full-discretization method(3rdUFDM) [26]) ZOA methods that expand time varyingcoefficients of dynamic milling equations into Fourier seriesand truncate the series to include only the average com-ponent have been applied in regenerative chatter stabilityprediction tasks Although it was proved that the ZOA canperform well in regenerative chatter stability predictiontasks not readily applicable to low radial immersion millingis a common problem for ZOAs in regenerative chatterstability prediction +e MF method extended the study ofZOA to consider the harmonics of the tooth-passing fre-quencies for low radial immersion machining processesGenerally however this method is limited to harmonicallyexcited systems and as the harmonics in the expansion of thenonlinear force becomes increasingly high there aregrowing concerns that time cost could become a threat

In recent years intensive research efforts have beenmade in the use of time-domain techniques as an effectiveapproach to regenerative chatter stability prediction Baylyet al [10] proposed a TFEA method to predicting millingstability A comparative study indicated that the perfor-mance of their proposed method was superior to that oftraditional frequency-domain methods with regard tocomputation accuracy Insperger et al [11] proposedthe1stSDM where the latter had higher computation effi-ciency and computation accuracy Long and Balachandran[12] developed the USDM for amilling dynamics model withconsideration of the loss-of-contact and time-delay effectsand indicated that period-doubling bifurcations of periodicorbits can occur in low-immersion operations Ding et al[13] proposed the 1stFDM for milling stability prediction Acomparative study indicated that the proposed 1stFDM wassuperior to the 1stSDM with regard to computation effi-ciency Owning to its wonderful computation accuracyvarious variants of the 1stFDM [14ndash17] such as the 2ndFDM3rdFDM 3rdLSAFDM and HTOFDM were subsequentlydeveloped for milling stability prediction Ding et al [18]employed NIM to recognize milling stability boundaries andshowed that their NIM outperformed the traditional SDMand TFEA method with regard to convergence rate andcomputation efficiency Liang et al [19] extended the studyof the NIM to develop an INIM for creating SLD for millingoperations with time-varying delay Li et al [20] proposedthe CDS for milling stability prediction by discretizing thedelay time term state term and periodic-coefficient matrices

simultaneously A comparative study indicated that theirCDS could outperform the SDM and FDM with regard tocomputation efficiency Niu et al [21] applied the fourth-order RungendashKutta method to handle the Duhamel term ofDDEs for milling stability prediction +eir approach wassuperior to 1stSDM in terms of convergence rate andcomputation efficiency Li et al [22] presented the RK-CDMto predict the chatter stability for milling processes Anextensive comparison based on convergence rate andcomputation efficiency as the performance index showedthat the proposed RK-CDM could outperform the SDM andCDS in predicting milling stability However nearly all theabovementioned methods in this research area have used thecoordinates of each node as interpolation conditions thecurve was maintained relatively flat when interpolating withlower-order functions while it would not be maintainedsmooth at the joints of the piecewise curves+e result mightas well not provide the near-optimum SLD On the contrarythese methods might lead to a poor overall machining pa-rameters selection

Recently the Hermite interpolation polynomial is one ofmany polynomial interpolation algorithms that have drawnincreasing attention for their attractive properties and ex-cellent error estimates on a wide range of numerical sum-mation problems Hermite interpolation polynomials notonly define the coordinate value of each node but also specifythe derivative of the curve at each node which has beenshown to be superior to the Newton interpolation andLagrange interpolation in keeping the stability of the curveWhile the Hermite interpolation polynomial solution maybe the globally stable optimum current FDM methods tendto fall into a local unstable solution +e unsmooth con-nection problem is unlikely to occur with Hermite inter-polation polynomial +erefore Hermite interpolationpolynomial can provide excellent performance for millingstability prediction problems

To the best of the authorsrsquo knowledge only three studiesin the literature have focused on the Hermite interpolationpolynomial-based machining stability prediction for millingoperations Liu et al [23] predicted the machining stabilityin milling operations using the 3rdHAM and compared theirresults with those obtained using the 1stSDM and 2ndFDMIn the efficient FDM linear interpolation and Hermite in-terpolation were approximate to the state item and time-delay item respectively Compared with the results obtainedusing the 1stSDM and 2ndFDM their FDM reducedcomputation time by almost 53 and 10 respectively Jiet al [24] used the 3rdH-NAM to resolve the problem ofmachining stability prediction for milling processes In the3rdH-NAM third-order Hermite interpolation and third-order Newton interpolation were used to approximate thestate item and time-delay item respectively Simulationsshowed that their approach was superior to Guorsquos 3rdFDMand Liursquos 3rdHAM in terms of the accuracy of identifyingthe milling stability boundaries Liu et al [25] proposed aHFDM to resolve the problem of machining stability pre-diction for turning processes and showed that their HFDMimproved calculation efficiency by about 50 compared tothe very efficient 2ndFDM In the HFDM Lagrange










x(t) eA0 tminus t0( )x t0( 1113857 + 1113946

t

t0


(2)




t


(3)


xk+1 eA0τxk + 1113946

τ





where

a1 δ3 + 3τδ2 + 2τ2δ

6τ3

b1 δ2 + 3τδ + 2τ2


2τ3

c1 δ + 2ττ


τ2+δ3 + 3τδ2 + 2τ2δ

2τ3

d1 1 +δ + 2ττ

+δ2 + 3τδ + 2τ2


6τ3

(6)



τ


τ

(7)




where


τ3


τ3

c2 δ3 minus τδ2

τ3

(9)


B(δ) δτBk+1 +

τ minus δτ

Bk (10)




(11)

where

H0 Φ0 eA0τ

Hk1 Φ5 + 3τΦ4 + 2τ2Φ3

6τ4Bk+1 +


6τ4Bk

Hk0 Φ5 + 4τΦ4 + 5τ2Φ3 + 2τ3Φ2

2τ4Bk+1 +


2τ4Bk


2τ4Bk+1 +


2τ4Bk


6τ4Bk+1 +


6τ4Bk


τ4Bk+1 +


τ4Bk


τ4Bk+1 +


τ4Bk


τ4Bk+1 +


τ4Bk

Φ1 1113946τ

0e

minusA0δdδ

Φ2 1113946τ

0e

minusA0δδdδ

Φ3 1113946τ

0e

minusA0δδ2dδ

Φ4 1113946τ

0e

minusA0δδ3dδ

Φ5 1113946τ

0e

minusA0δδ4dδ

(12)


0 Φ0 minus I( 1113857 (13)









(18)







(20)


D1

x1x2⋮xm

xm+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2

x1minusm

x2minusm

⋮xmminusm

xm+1minusm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where

D1




H3minus2

⋮

H3minus1

⋮

H0 + H30

⋮

H31 minus I

⋮


⋱

0

⋮

0

⋮

0

⋮

0

⋮

0



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2





⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)


Φ D1( 1113857minus 1D2 (23)







minuswh(t)


(24)


h(t) 1113944N


(25)


ϕj(t) 2πΩ60

t +2π(j minus 1)

N (26)



0 otherwise1113896 (27)




where


t

mt ζωn( 11138572


⎡⎣ ⎤⎦

B(t) 0 0

minuswh(t) 01113890 1113891

(29)












2 922 0011 003993 6times108 2times108



| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)




m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References








































x(t) eA0 tminus t0( )x t0( 1113857 + 1113946

t

t0


(2)




t


(3)


xk+1 eA0τxk + 1113946

τ





where

a1 δ3 + 3τδ2 + 2τ2δ

6τ3

b1 δ2 + 3τδ + 2τ2


2τ3

c1 δ + 2ττ


τ2+δ3 + 3τδ2 + 2τ2δ

2τ3

d1 1 +δ + 2ττ

+δ2 + 3τδ + 2τ2


6τ3

(6)



τ


τ

(7)




where


τ3


τ3

c2 δ3 minus τδ2

τ3

(9)


B(δ) δτBk+1 +

τ minus δτ

Bk (10)




(11)

where

H0 Φ0 eA0τ

Hk1 Φ5 + 3τΦ4 + 2τ2Φ3

6τ4Bk+1 +


6τ4Bk

Hk0 Φ5 + 4τΦ4 + 5τ2Φ3 + 2τ3Φ2

2τ4Bk+1 +


2τ4Bk


2τ4Bk+1 +


2τ4Bk


6τ4Bk+1 +


6τ4Bk


τ4Bk+1 +


τ4Bk


τ4Bk+1 +


τ4Bk


τ4Bk+1 +


τ4Bk

Φ1 1113946τ

0e

minusA0δdδ

Φ2 1113946τ

0e

minusA0δδdδ

Φ3 1113946τ

0e

minusA0δδ2dδ

Φ4 1113946τ

0e

minusA0δδ3dδ

Φ5 1113946τ

0e

minusA0δδ4dδ

(12)


0 Φ0 minus I( 1113857 (13)









(18)







(20)


D1

x1x2⋮xm

xm+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2

x1minusm

x2minusm

⋮xmminusm

xm+1minusm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where

D1




H3minus2

⋮

H3minus1

⋮

H0 + H30

⋮

H31 minus I

⋮


⋱

0

⋮

0

⋮

0

⋮

0

⋮

0



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2





⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)


Φ D1( 1113857minus 1D2 (23)







minuswh(t)


(24)


h(t) 1113944N


(25)


ϕj(t) 2πΩ60

t +2π(j minus 1)

N (26)



0 otherwise1113896 (27)




where


t

mt ζωn( 11138572


⎡⎣ ⎤⎦

B(t) 0 0

minuswh(t) 01113890 1113891

(29)












2 922 0011 003993 6times108 2times108



| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)




m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References


































where


τ3


τ3

c2 δ3 minus τδ2

τ3

(9)


B(δ) δτBk+1 +

τ minus δτ

Bk (10)




(11)

where

H0 Φ0 eA0τ

Hk1 Φ5 + 3τΦ4 + 2τ2Φ3

6τ4Bk+1 +


6τ4Bk

Hk0 Φ5 + 4τΦ4 + 5τ2Φ3 + 2τ3Φ2

2τ4Bk+1 +


2τ4Bk


2τ4Bk+1 +


2τ4Bk


6τ4Bk+1 +


6τ4Bk


τ4Bk+1 +


τ4Bk


τ4Bk+1 +


τ4Bk


τ4Bk+1 +


τ4Bk

Φ1 1113946τ

0e

minusA0δdδ

Φ2 1113946τ

0e

minusA0δδdδ

Φ3 1113946τ

0e

minusA0δδ2dδ

Φ4 1113946τ

0e

minusA0δδ3dδ

Φ5 1113946τ

0e

minusA0δδ4dδ

(12)


0 Φ0 minus I( 1113857 (13)









(18)







(20)


D1

x1x2⋮xm

xm+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2

x1minusm

x2minusm

⋮xmminusm

xm+1minusm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where

D1




H3minus2

⋮

H3minus1

⋮

H0 + H30

⋮

H31 minus I

⋮


⋱

0

⋮

0

⋮

0

⋮

0

⋮

0



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2





⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)


Φ D1( 1113857minus 1D2 (23)







minuswh(t)


(24)


h(t) 1113944N


(25)


ϕj(t) 2πΩ60

t +2π(j minus 1)

N (26)



0 otherwise1113896 (27)




where


t

mt ζωn( 11138572


⎡⎣ ⎤⎦

B(t) 0 0

minuswh(t) 01113890 1113891

(29)












2 922 0011 003993 6times108 2times108



| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)




m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References






































(18)







(20)


D1

x1x2⋮xm

xm+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2

x1minusm

x2minusm

⋮xmminusm

xm+1minusm

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

where

D1




H3minus2

⋮

H3minus1

⋮

H0 + H30

⋮

H31 minus I

⋮


⋱

0

⋮

0

⋮

0

⋮

0

⋮

0



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

D2





⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)


Φ D1( 1113857minus 1D2 (23)







minuswh(t)


(24)


h(t) 1113944N


(25)


ϕj(t) 2πΩ60

t +2π(j minus 1)

N (26)



0 otherwise1113896 (27)




where


t

mt ζωn( 11138572


⎡⎣ ⎤⎦

B(t) 0 0

minuswh(t) 01113890 1113891

(29)












2 922 0011 003993 6times108 2times108



| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)




m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References

































minuswh(t)


(24)


h(t) 1113944N


(25)


ϕj(t) 2πΩ60

t +2π(j minus 1)

N (26)



0 otherwise1113896 (27)




where


t

mt ζωn( 11138572


⎡⎣ ⎤⎦

B(t) 0 0

minuswh(t) 01113890 1113891

(29)












2 922 0011 003993 6times108 2times108



| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)




m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References





































2 922 0011 003993 6times108 2times108



| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

001

002

003

004

005

(a)

| |u|

ndash |u

0| |

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

0

002

004

006

008

01

(b)

| |u|

ndash |u

0| |

0

002

004

006

008

01

012

014

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(c)

| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020

NIM [18]3rdHAM [23]


m

(d)




m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References

































m 20


0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000Ω (rpm)


0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000Ω (rpm)



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References
































Table 3 Continued

m 30


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 1000050000

1

2

3

4w

(mm

)


Ω (rpm)6000 7000 8000 9000 100005000







Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References




































Table 3 Continued

m 40


Ω (rpm)

0

1

2

3

4w

(mm

)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000



Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000


Ω (rpm)

0

1

2

3

4

w (m

m)

6000 7000 8000 9000 100005000





aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References


































aD 1

times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm) times104

0

02

04

06

08

1

w (m

m)

15 2505 1 2Ω (rpm)

05 1 15 2times104

0

02

04

06

08

1

w (m

m)

Ω (rpm)25


aD 05

times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

15 2505 1 2Ω (rpm) times104

0

05

1

15

2

25

w (m

m)

05 1 15 2 25Ω (rpm)


aD 01

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)


aD 005

times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

15 2505 1 2Ω (rpm) times104

0

2

4

6

8

10

w (m

m)

05 1 15 2 25Ω (rpm)





| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References


































| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020m



(a)

| |u|

ndash |u

0| |

0

001

002

003

004

70 9050 6030 8040 10020



m

(b)



| |u|

ndash |u

0| |

0

001

002

003

004

005

006

70 9050 6030 8040 10020m

(c)



| |u|

ndash |u

0| |

0

005

01

015

02

025

70 9050 6030 8040 10020m

(d)









w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References






































w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)

0

1

2

3

4

7000 900080006000 100005000Ω (rpm)


w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000

w (m

m)


Ω (rpm)

0

1

2

3

4

7000 900080006000 100005000





6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References


































6 Conclusions


aD = 1 m = 40w

(mm

)


0

01

02

03

04

05

06

07

08

09

1

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(a)


aD = 05 m = 40

0

01

02

03

04

05

06

07

08

09

1

w (m

m)

191713 1815 16 20 2310 12 21 2211 2414Ω (krpm)

(b)

aD = 01 m = 40

0

03

06

09

12

15

18

21

24

w (m

m)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(c)

aD = 005 m = 40

0

03

06

09

12

15

18

21

24w

(mm

)


Ω (krpm)191713 1815 16 20 2310 12 21 2211 2414

(d)





Data Availability




Acknowledgments


References


































Data Availability




Acknowledgments


References
































AnEfficientThird-OrderFull-DiscretizationMethodfor...

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