A review of chatter vibration research in milling

Cixu YUE,Hining GAO,Xinli LIU,Steven Y.LIANG,Lihui WANG

aHarbin University of Science and Technology,Harbin 150080,China

bGeorgia Institute of Technology,Atlanta,GA 30332-0405,USA

cKTH Royal Institute of Technology,Stockholm,Sweden

Abstract Chatter is a self-excited vibration of parts in machining systems.It is widely present across a range of cutting processes,and has an impact upon both efficiency and quality in production processing.A great deal of research has been dedicated to the development of technologies that are able to predict and detect chatter.The purpose of these technologies is to facilitate the avoidance of chatter during cutting processes,which leads to better surface precision,higher productivity,and longer tool life.This paper summarizes the current state of the art in research regarding the problems of how to arrive at stable chatter prediction,chatter identification,and chatter control/-suppression,with a focus on milling processes.Particular focus is placed on the theoretical relationship between cutting chatter and process damping,tool run out,and gyroscopic effect,as well as the importance of this for chatter prediction.The paper concludes with some reflections regarding possible directions for future research in this field.

KEYWORDS Chatter;Gyroscopic effects;Milling;Process damping;Tool runout

1.Introduction

Chatter is a form of cutting vibration that is caused by the characteristics of a machining system under the continuous action of aperiodic external exciting force.It usually shows the strong relative vibration between a tool and a workpiece in a metal cutting process.As early as 1907,the chatter problem was studied by Taylor.1Up to now,chatter research has been performed for more than a century,but it is still a major obstacle to automate machining processes such as turning,milling,and drilling(as shown in Fig.1,whereNis the workpiece speed andFis the feed force).2Chatter in cutting processes will bring many unfavorable factors.For example,it will affect the quality of the workpiece surface,reduce the accuracy of the workpiece,produce harsh noise,accelerate tool wear,and so on.Therefore,cutting chatter has been a hot topic in academia and industry.

Depending on the lack of dynamic stiffness in one or more components of a system including the machine tool,the tool holder,the cutting tool,or the workpiece,there are three kinds of mechanical vibration in metal cutting processes:free vibration,forced vibration,and self-excited vibration.Free vibration is caused by impact.Forced vibration is caused by an imbalance between gears,bearings,spindles,and other machine tools.When the causes of these two kinds of vibration have been determined,it is relatively straightforward to avoid,reduce,or eliminate them.Self-excited vibration is an unattenuated vibration caused by the alternating force produced by the interaction between the tool and the workpiece.This type of vibration increases the instability and uncontrollability of a system.Depending on the specific self-excitation mechanism present in the vibration,chatter can be divided into three types:frictional chatter,mode coupling chatter,and regenerative chatter.3Frictional chatter is caused by mutual friction between the tool and the workpiece in the same direction as that of the cutting speed.Mode coupling chatter is caused by the coupling of two natural modes of vibration because of the slight difference between the rigidity of vibrating elements in two different directions.Here,a stable condition for the sys-tem can be obtained by using experimental modelling and analysis.Regenerative chatter is caused by differences in chip thickness resulting from the phase difference between the vibration pattern formed during a prior cutting process and the vibration displacement of a subsequent cutting process(as shown in Fig.2,wherecandkare the damping and stiffness of the cutting system,respectively;aeis the radial cutting width;ftis the feed rate,Ft,jandFr,jare the tangential and radial cutting forces,respectively;Φjis the instantaneous angular immersion of toothj;VjandVj-1are the dynamic displacements of teethjandj-1,respectively).Chatter not only reduces processing efficiency,machining precision,and cutting tool and machine tool lives,but also results in excess waste of materials and sound pollution.

Fig.1 Chatter problem in cutting processes.2

The various issues pointed out above have led to the development of a specific body of research around chatter and its resolution.In the Web of Science database,the total number of relevant journal articles published from 1967 to May 30,2018 related to milling chatter can be obtained by searching using keywords ‘chatter” and ‘milling” (see Fig.3).As illustrated in Fig.3,there has been an increasing interest in researching the chatter problem in milling processes.An analysis of the principal publications for this topic is shown in Table 1.

Quintana and Ciurana4already examined the research status for chatter in cutting processes,and also classified methods used to ensure that there is no chatter.Siddhpura and Paurobally5similarly looked at the research status of chatter in turning processes,and discussed the theoretical relationship between chatter and tool wear.Munoa et al.2presented a critical review of different chatter suppression techniques,and summarized the applications and challenges of each.Chatter analysis,chatter stability prediction,and chatter detection are very complex processes.Each requires independent research on different types of cutting,e.g.,turning,milling,and drilling.Compared to other machining methods,milling has the advantage of high production efficiency and a wide range of machining processes.It is particularly suitable for machining of complex composite parts,most notably those with various special-shaped surfaces,thus offering a number of advantages over other machining methods.As a result,milling plays an important role in the mechanical manufacturing industry.By consulting the relevant literature,we have found that there is no overall summary of the chatter problem in milling processes,so in this paper,various specific issues regarding chatter in milling processes are systematically studied(i.e.,chatter prediction,chatter detection,and chatter suppression/control).We also examine and discuss the theoretical relationship between cutting chatter and process damping,tool runout,and gyroscopic effects.At the end of the paper,we suggest ways in which future research on these topics might be directed.

Fig.2 Dynamic model of a milling process.

Fig.3 Number of papers published regarding milling chatter.

Table 1 Publications according to journals.

2.State of art in stability prediction

On the premise of not wanting to change the structures and characteristics of machine tools and tool shanks,a usual approach to examining cutting chatter issues is to construct a milling Stability Lobe Diagram(SLD).This aims to identify the best cutting parameters in a stable region.Chatter does not usually occur at low speeds because of the effect of process damping,but,as the speed increases,the process damping effect is reduced.Optimization of the removal rate of materials therefore necessitates the study of an SLD.By obtaining four sets of input parameters-the cutting force coefficient,system dynamic parameters,process parameters,and tool geometry,a multiple-degree-of-freedom dynamic equation for milling processes can be established.A time-frequency method can be used to get the SLD for a milling process,and the cutting process can then be optimized,as shown in Fig.4,whereM，C，K，Kc（t），q（t），˙q（t），q（t），［q（t）-q（t-T）］are the mass,damping,stiffness,cutting force coefficient,acceleration,velocity,displacement,dynamic displacement of the cutting system,respectively;ap，Z，Dare the cutting depth,the number of tool teeth,and the tool diameter,respectively;acr,b,andnare the axial limit cutting depth,radial cutting width,and spindle speed,respectively.

2.1.Cutting force coefficient

Accurately obtaining the cutting force coefficient for a cutting process is the key to predicting the relationship between the cutting force and the stability in milling.6Cutting coefficients represent the material's yield strength,friction between the tool and the work material,and tool geometry.There are two approaches to determining cutting force coefficients:(1)the cutting force coefficient for bevel milling is obtained from an orthogonal cutting database;(2)the cutting force coefficient for bevel milling can be calibrated quickly using mechanical methods.In the first approach,a right-angle-based method for oblique cutting transformations is used,which was firstly proposed by Budak et al.7By extracting orthogonal test data,cutting force coefficients for different cutting tools and different cutting processes can be obtained.The second method for obtain cutting force coefficients can be further subdivided into the average cutting force method8and the instantaneous cutting force method.9,10The average cutting force method puts the measured average cutting force into a linear function for feed per tooth.Cutting force coefficients in different directions can then be obtained using fitting functions in Matlab software.The instantaneous cutting force method is based on the minimum objective function to fit a simulated cutting force and an experimentally measured force,with the instantaneous cutting force coefficient being obtained by inversion.

The laws whereby cutting force coefficients vary according to cutting parameters have also been studied in the literature.11-15Campatelli et al.,11,12for instance,studied the law for the cutting force coefficient according to the spindle speed,and pointed out that there was a trend towards this becoming stronger in association with the tangential force.Their results showed that the cutting force coefficient was larger in lowspeed regions,while it firstly decreased and then increased in high-speed regions.Grossi et al.13used the traditional mean force method and an instantaneous estimation method based on a genetic algorithm to get the milling force coefficient.Comparisons of simulation results showed that the instantaneous cutting force estimation method was more accurate and effective.The change rule for the cutting force coefficient in the speed range for the spindle was the same as that proposed elsewhere in the literature.11,12Using the instantaneous cutting force method and a nonlinear optimization method,the influences of different cutting parameters(feed per tooth,spindle speed,and radial cutting width)on the cutting force coefficient were studied by Rubeo and Schmitz.14Results showed that the cutting force coefficient was nonlinear in relation to the cutting parameters.This therefore implied that the cutting force coefficient was influenced by the cutting parameters.However,Wang et al.15studied the effects of cutting parameters(spindle speed and cutting depth)on the cutting coefficient,and concluded that cutting parameters did not affect the cutting force coefficient in the spindle speed range of 500-1500 r/min.

Fig.4 Procedure to obtain SLDs.

The process usually involves using a time-frequency domain method to solve a delay differential equation,on the basis of which an SLD can then be obtained,so an accurate cutting force coefficient can increase the robustness of the prediction.Assuming that the cutting force coefficient varies with speed,Grossi et al.16proposed a stability analysis method that considers the continuous variation of the cutting force coefficient according to the spindle speed.This method is more robust than traditional stability prediction methods,as shown in Fig.5,where ωcis the chatter frequency.A dynamic milling force coefficient model for milling processes has now been established by Yue et al.17,18Here,a dynamic milling force coefficient is combined with an improved semi-discrete method to obtain an SLD for a process.Compared to the results of predictions based on a constant coefficient,the prediction accuracy is greatly improved.

2.2.Dynamic characteristics of a system

In order to obtain a stability boundary,the dynamic behavior of a system has to be predicted or modeled by simulating the responses of the machine tool,tool shank,and tool system.The transfer function for an MDOF system can be identified using structural dynamic tests.The structure is excited with an impact hammer that is instrumented with a piezoelectric force transducer.Resulting vibrations are then measured using displacement,velocity,or acceleration sensors.However,a sensor installed on the system has an effect on the frequency response function of the system structure.19Due to transmission quality effects,systematic errors are introduced into the measured frequency response function,resulting in model parameters for the structure(e.g.,natural frequencies)shifting away from their correct values.A number of studies examined how to get rid of the effect of the sensor quality on frequency response functions.20,21For example,two accelerometers of different masses were tested to see if they can remove the effect of sensor mass.22Experiments were also conducted with and without virtual mass to assess the mass load effect.23The common method nowadays is to use the Sherman-Morrison formula,which draws upon an initial data matrix and modified data to effectively invert the modified data matrix directly.24The main application of this formula was originally the analysis of the structural modification problem in structural dynamics.The deformation formula for inverse matrix inversion has also been applied to some reanalysis problems.25,26At the same time,another new method was proposed to eliminate the sensor mass effect on the test frequency response function.27Subsequently,a structural modification method which can be used to compensate for the quality effect of an accelerometer was proposed,and a more accurate SLD was obtained.28

Fig.5 SLDs for experimental validation.16

For thin-walled components,the dynamic characteristics of a system are changed with the removal of material and changes in the position of machining.This effect increases the complexity of describing the dynamic characteristics of the system.As a result,dynamic modeling and analysis of changes in the positions of the workpiece and the tool become central to predicting the stability of the system.The dynamic behavior of a workpiece can be obtained by using either the finite element method(FEM)29-32or the structural dynamic modification method(SDM).33-36

In the FEM,a cutting process is divided into multiple cutting steps over the whole length of a workpiece.For each cutting step,a finite element model of the work piece is established,and model analysis is carried out to obtain the dynamic characteristics for this step.Alongside this,experimental model analysis is performed to identify the damping information,or else small constant damping is assumed.In this way,the FEM can be used to obtain the dynamic behavior of the whole machining process.Adetoro et al.29improved the stability model by considering nonlinear variations of the cutting force coefficient and the axial immersion angle,as well as the influences of these upon the axial depth of cut.The dynamic characteristics of a workpiece were then analyzed using the FEM and a Fourier transform,and the SLD of the milling process was thus obtained.Experimental results verified that the proposed method was consistent.Song et al.30used the FEM to obtain the dynamic properties of a thinwalled part when the tool position was changing and material being removed.The curve equation for the model's characteristics that changed with the tool position was regressed.A dynamic SLD with time variance was then obtained using a semi-discrete method.Campa et al.31also used the FEM to calculate the model parameters at each discrete step of a tool-path during milling thin floors using bull-nose end cutters.Ding and Zhu32proposed two FEMs to obtain the model parameters of a workpiece after each machining step.It was concluded that the accuracy of the model parameters obtained using a 3D finite element model was higher.By using a frequency domain method,three-dimensional stability lobes can be obtained for processing of thin-walled parts.The correctness of the proposed method and model has been verified by a series of milling experiments.The advantage of the FEM is that a higher accuracy can be obtained theoretically when the cutting step size is reduced.However,more cutting steps means that more FEMs and longer time are needed to reconstruct the model and to re-do the analysis,making it less efficient.For capturing the radial depth of cutting,the accuracy and efficiency of this approach are greatly reduced,making it more or less unusable.

Alan et al.33used an SDM prediction workpiece dynamic frequency function combined with a zero-order frequency domain method to obtain an SLD with different processing positions and material removal.Here,it was concluded that removal of material did not obviously change the dynamic characteristics of a workpiece,but the location of the lobe had a significant effect.With the processing position close to the fixed end,the rigidity of the workpiece increased,and the stability was enhanced.Budak et al.34obtained an initial frequency response function for a workpiece based on an initial finite element mesh,and then used the SDM to obtain the frequency response function as material was removed along the tool path.Based on the updated frequency response function and a frequency domain method,an SLD was generated.It was concluded that the natural frequency of the workpiece varied with the removal of material and the processing position,which affected the processing stability limit.The acoustic signal and surface texture features of the process are gathered together,and their verification is shown in Fig.6,in whichuis the feed direction,andvis the cross feed direction.Song et al.35,36presented a new method for prediction of dynamic SLDs in thin-wall milling using the dynamic correction of an equal-mass structure.The input data for this method is just two sets of dynamic characteristics for original(unmachined)and final(machined)structures.It uses a Sherman-Morrison-Woodbury formula to estimate the frequency response function after changes in the workpiece structure.Here,it is not necessary to remodel and re-grid the FEM to improve efficiency.The probability of a singularity occurring is very low compared to those of other existing methods.Once the dynamic characteristics of the tool position are determined,these can be used along the direction of machining to predict specific dynamic SLDs.

Fig.6 Simulation and experimental study on milling stability.34

2.3.Process parameters

The feed per tooth/feedrate of a cutting process,the workpiece's curvature radius,and the lead angle of the milling cutter are constantly changing in complex machining processes.Because time delay plays a key role in the determination of chatter,it is necessary to study the effect of feed speed on time delay and then assess its effect on stability.Balachandran et al.37-39considered the feedrate effect on time delay,along with loss-of-contact effects,and presented a model in which time delay along the feed direction differs from time delay in the opposite direction.The associated model is a nonlinear,non-homogeneous,and non-autonomous delay differential system with periodic coefficients and two constant time delays.Based on the above research,Long et al.40studied the effects of the feedrate on the entry cutting angle,the exit cutting angle,and the amplitude of the feed mark.It was concluded that the critical axial cutting depth decreases as the feedrate increases and as the feedrate is increased,the difference between the stability chart obtained for the system with a constant delay and that obtained for the system with variable delays becomes more pronounced.When there is tool runout,the influence of the feed per tooth on the SLD for milling cannot be ignored.41A small feed per tooth has a great influence on stability,but the effect reduces as the feed per tooth is increased.The study also found that the flip bifurcation zone not only occurs at a small radial cut,but also at a large radial cut if the feed is sufficiently small.Sun et al.42,43thus proposed a form of non-chatter tool orientation algorithm,which can optimize the tool feed direction and produce stable milling.

The surface curvature of a convex die mold can directly affect the contact area between a tool and a workpiece,and thereby has an effect upon milling stability.Study results have shown that the limit for the axial cutting depth decreases gradually as the radius of curvature increases.44Different machining positions for a die workpiece with a convex surface can cause the lead angle of a milling cutter to change,thus impacting upon stability.It has been found that,as the lead angle of a milling cutter increases,the stability lobe is shifted downward,and the limit for the axial cutting depth is reduced.45

2.4.Tool geometry

The geometrical parameters of a tool influence the cutting force coefficient,the kinetics of the machining process,and the directional coefficients of the dynamic force.Cutting tools with different shapes are widely used in milling processes.The consequent discontinuity between machining processes makes stability analysis more complicated.Altintas et al.46-50used an average directional factor that was independent of the helix angle to construct a milling SLD where a large radial cutting depth was needed.Simulation and experiments to test their approach showed good consistency.The indication was that the effect of the helix angle on stability can be set aside when there is a large radial cutting depth.Wan et al.41studied the influence of the helix angle on stability where there was low radial immersion and a low feed rate.The study found that the helix angle had a significant influence on stability,but as the radial depth and feed rate were increased,the influence gradually decreased.Variable pitch milling cutters have been widely studied because they can dispose of any regeneration effects and alter or modify system behaviors,effectively suppressing milling chatter and improving machining process stability.Research relating to this will be examined more closely in Section 4.1.

2.5.Dynamic modeling

When the rigidity of a workpiece is far greater than that of a tool and the helix angle of a milling cutter is zero,the tool is usually simplified to a two-degree-of-freedom spring-dampermass system model in the direction of the feed and the lead,46-48as shown in Fig.7(a).k，care the stiffness and damping of the tool/workpiece,respectively.However,when the helix angle of a milling cutter is not zero,the milling force in the machining process will stimulate a three-directional vibration mode in the cutter in the feed direction,the lead direction,and the tool axis direction.Altintas49therefore proposed a three-degree-of-freedom milling model that takes into account the axial vibration mode.In this chatter model,a workpiece is regarded as a rigid body,and its vibration is disregarded.These aforementioned models have been widely used to predict cutter vibration.However,when the rigidity of a workpiece and that of a tool are similar,the effect of the workpiece dynamics is not at all negligible.This makes models of the stability of a system using the preceding chatter models inaccurate.Budak and Altintas50and Tang et al.51established new four-degree-of-freedom models that take into consideration the vibrations of both a tool and a workpiece,as shown in Fig.7(b).In these four-degree-of-freedom models,each of the milling cutter and the workpiece has a two-degree-offreedom spring-mass-damping system.On the basis of this model,Liu et al.52used a relative transfer function to study the effect of damping on the chatter stability of thin-walled parts during a milling process.Results showed that process damping can improve stability in the low-speed zone.

Fig.7 Different chatter prediction spring-damper-mass system models.

In side-milling processes for thin-walled structures,a workpiece has greater flexibility than a milling cutter.As a result,chatter stability is mainly affected by the dynamic characteristics of the workpiece.In this case,researchers have treated the tool as a rigid body and ignored its vibrational mode.For this reason,thin-walled workpieces can be simplified as singledegree-of-freedom spring-damper-mass systems for modeling,as shown in Fig.7(c).Wang et al.53,for example,predicted regenerative chatter for an I-shaped flexible workpiece in a vertical precision milling phase.Tang and Liu54also considered the effect of the radial cutting depth on stability for these kinds of cases.When milling frame-type thin-walled parts from the bottom,the workpiece is more flexible in the axial direction than the tool,and the relationship between the two is opposite in the radial direction.Fei et al.55therefore established a threedegree-of-freedom dynamic model of the milling process,and used a semi-discrete method to achieve stability,as shown in Fig.7(d).The effectiveness of this model was proven in cutting experiments.

2.6.A method for solving chatter models

Mathematically,the cutting dynamics for regenerative effects are controlled by the delay differential equation(DDE)of time-period parameters.A time-delay method can thus be used to solve DDEs and get the stability boundary for milling processes.The method is composed of critical machining parameters so that the cutting area can be divided into stable and unstable cutting zones.Appropriate cutting parameters can be selected from SLDs.In this way,chatter can be avoided,processing efficiency improved,and better processing quality achieved.Frequency domain methods,discrete methods,and numerical methods are all common approaches here.

Frequency domain methods are generally used to express DDEs for chatter models in the frequency domain using a Fourier transform,and the stability boundary of milling is then obtained using the control theory.The zero-order frequency domain(ZOA)method was firstly proposed in 1995 by Altintas and Budak46.Examining the regenerative effect,this method only considers two degrees of freedom for a tool in its milling chatter model.The time-variable directional dynamic cutting force coefficient matrix is approximated using a zero-order harmonic component undergoing a Fourier transform,and the axial cutting depth and cutting speed without chatter are directly obtained from the linear analytical expression.Without any numerical iteration,the prediction accuracy is the same as that using a time-domain solution or other numerical methods,and the computational efficiency is improved.This method has been expanded upon so that it can be applied to variable-pitch cutters,47ball-end milling,48face milling with an inclination angle,49inserted cutter milling,56helical milling,57and bull-nose end milling.58The zero-order frequency domain method is the most efficient and widely applied method for solving chatter stability so far adopted.However,this method is not able to predict the additional stable area under low radial immersion.As the ratio of cutting width/tool diameter decreases,there is a large deviation in the prediction of local stability,59as shown in Fig.8,where circles represent stable cutting,crosses quasi-periodic chatter,and diamonds periodic chatter;stability boundaries(lines):experimental(black),predicted by the SD(grey solid)and ZOA(grey dashed)methods;aeis the cutting width,andDis the tool diameter.The reason for this phenomenon is that,when it is calculating the dynamic cutting coefficient,the algorithm uses zero-order averaging in the tooth period without considering the influence of higher-order terms.Low radial immersion is regarded as periodic impulse excitation,so higher-order terms have much greater influences on the cutting coefficient.The stability method for low radial immersion is therefore more complex than that for ordinary radial immersion,and cannot predict double cycles.In order to avoid the above problems,Merdol and Altintas60proposed a multifrequency method,which takes into account the higher harmonics of directional factors.Here,the algorithm iteratively searches for the chatter frequency in the calculation process,and needs to solve multiple eigenvalues.Bachrathy and Stepan61further extended the multi-frequency method to predict stability for all tool structures,including complex ones where there can be distributed delays.By combining the extended multi-frequency method with a multidimensional dichotomy,the computational efficiency can be greatly improved.The extended multi-frequency method has been proven to still obtain reliable stability prediction results when the quality of the frequency response function is poor.In a further refinement,Otto et al.62applied the multi-frequency domain method to solving milling stability with non-uniformpitch and variable-helix tools.

Fig.8 Experimental and predicted stability boundaries for upmilling at various radial immersions.59

Regenerative chatter models are generally a system of DDEs.The stability is determined by the eigenvalue of the single-valued system operator,but the single-valued operator is represented by an infinite dimension matrix,which makes it difficult to solve closed-form stability prediction results.Discrete methods usually use a finite-dimensional transformation matrix to approximate a single-valued operator of infinite dimensions,thus reducing the difficulty of solving and the calculation time.The main kinds of discrete methods are:the semi-discrete method,the fully-discrete method,and the time finite element method.Insperger et al.63,64were the first to propose a semi-discrete method(SD)and apply it to stability predictions for a single-degree-of-freedom milling system and a two-degree-of-freedom one in 2002.This method uses weighting of two adjacent hysteresis values in each time segment,approximates the time lag term in the time segment,and performs zero-order averaging processing on the periodic coefficient term to produce a discrete dynamic system that approximates the original differential dynamic system over a single cycle.Stability is then judged in relation to the size of the spectral radius of the state transition matrix.In order to further improve the convergence of stability,a first-order semi-discrete method was proposed.65Altintas et al.66compared frequency domain and semi-discrete methods,and concluded that the accuracy and speed of the semi-discrete method would depend on the discrete time interval and the number of nodes used.This method can take into account complicated working conditions,such as tools having variable pitches and/or spindles with variable speeds.Wan et al.41,67also considered the influences of the runout of a cutter and variabletooth spacing and multi-modal cutting dynamics.On the basis of this,they improved the traditional semi-discrete method.Their improved method has then been further extended to cover thread milling68and multifunctional tools.69Recent research has focused on the use of different methods to improve calculation speed and convergence.Dong et al.70reconstructed the semi-discrete method to predict the stability of a milling process on a Shannon standard orthogonal basis.Compared with conventional semi-discrete methods,the computational speed was increased by 5 times and 2.6 times respectively for single-degree-of-freedom models and two-degree-offreedom models that otherwise possessed the same level of precision.

In order to improve the efficiency of the time-domain state transition matrix,Bayly et al.71proposed a method using time domain finite element analysis in 2003 that could predict the dynamic stability of an intermittent milling process,based on the assumption that time delay is equal to the passage period of the cutter tooth.This method first of all demanded precise separation of the free vibration of the cutting tool from the forced movement process.It then used three piecewise Hermite polynomials to approximate the vibration displacement.After this,dynamic mapping of the discrete system was constructed using a weighted residual method to approximate the original system,and a system state transition matrix of the cutting cycle was obtained using a polynomial integral.Finally,the stability of milling was assessed using the Floquet theory.As the free vibration of a tool does not need to be densely time dispersed,this method offers significantly higher efficiency for milling stability problems where there are high discontinuity and low radial immersion.Garg et al.72extended the method for stability analysis in time-delay systems by incorporating parameters for excitation of periodic coefficients.They then compared the advantages and disadvantages of using two types of single-type units and multi-type units,increasing the number of units to approximate the exact solution.Patel et al.73used a time-based finite element method to solve continuous differential equations,and obtained stability trends according to different radial cut-off and helix angles in down-milling and up-milling,together with the difference between chatter islanding and secondary Hopf bifurcation lobes.Mann and Patel74extended the method to a system that could be applied to all available space state model representations.Under conditions of low radial immersion,Sims et al.75used a time-based FEM to predict the stability of a variablepitch cutter with the same helix angle,and concluded that the method had a higher computational efficiency than those of semi-discrete methods.However,this approach is only suitable for predicting the stability of variable-pitch milling cutters with low radial immersion.

In 2010,Ding et al.76proposed a fully-discrete method based on a direct integral approach towards obtaining the dynamic response of a milling process in order to facilitate milling stability prediction.As the matrix exponent function involved in the computation is only dependent on speed and has no relation to the axial depth of cut,it has a higher computational efficiency than those of semi-discrete methods.There are three significant differences between fully-and semi-discrete approaches77:(1)semi-discrete methods are only discretized for the delay term,while fully-discrete methods are used to discretize the state term and the delay term using a linear-difference approach;(2)the periodic direction coefficients in semi-discrete methods are obtained by using very small time periods,whilst in fully-discrete methods,they are obtained by inserting the boundary value of the time period;(3)to obtain a stable lobe map,semi-discrete methods require a double loop for the spindle speed and axial depth to calculate the transfer matrix,whereas fully-discrete methods only require one cycle for the spindle speed to calculate the transfer matrix.Yue et al.78proposed a predictive model for chatter stability that took into consideration the characteristics of the contact between a tool and a workpiece,and then used a fully-discrete method to solve it and obtain the stability of a milling process.In order to improve the accuracy of fully discrete prediction methods,a corresponding second-order fully-discrete method,79third-order fully-discrete method,80and hyper-third-order fully-discrete method81have all been proposed.However,as the order increases,so does the calculation time.In order to solve this tension between calculation time and prediction accuracy,Tang et al.82proposed an improved fully-discrete method that is based on high-order interpolation of state and delay terms to predict milling stability.The computational efficiency was improved by establishing a state transition matrix to directly compensate for the higher order difference value in each time period.Li et al.83proposed another kind of fully-discrete method where all time correlation terms have been dispensed and numerical iteration used instead to get an iterative formula,and thus a Floquet transfer matrix.However,this method does not have a discrete periodic coefficient matrix.For this reason,Xie84proposed yet another improved fully-discrete method that has a higher computational efficiency than that of Li's method83.After this,Li et al.85proposed a Runge-Kutta method,that is also a kind of fully-discrete method,to predict milling process stability.Here,an iterative binary search approach is used to determine the stability boundary instead of an iterative sequential search,which greatly reduces the time spent in computation.

Numerical methods usually directly solve the delay differential equation and obtain the dynamic response for a milling process.After this,the stability of the milling process can be assessed using the stability criteria constructed according to the response or whether the amplitude of the response vibration is divergent.Smith and Tlusty86established a time domain simulation model of peak-to-peak values for cutting force,and used the variation in the peak-to-peak values as a chatter criterion.Altintas and Campoanes87also proposed an improved time-domain model for milling under conditions of low radial immersion.Here,the ratio of the dynamic cutting thickness to the static cutting thickness,obtained by simulation,is used as the dimensionless chatter discrimination coefficient.Lu et al.88and Wan et al.41applied this method to micro-milling and automobile panel die milling,respectively.In both cases,simulation and experimental results exhibited good levels of agreement.Li et al.89adopted the ratio of the maximum dynamic cutting force to the maximum static cutting force predicted by simulation as their criterion of chatter.Qu et al.90,meanwhile,made use of the statistical variance of dynamic displacement as a chatter detection criterion to obtain the SLD of a cutting process.In this work,they concluded that the feed per tooth had little effect on stability but a significant effect on the machined surface quality.

With regard to solving delay differential equations,Zatarain et al.91proposed an implicit subspace iteration method to calculate the stability of a milling process.The use of‘step matrices' replaces the transfer matrix and only iteratively calculates dominant eigenvalues,thus reducing computational time.Even in extremely-low-velocity situations,the proposed method is five times better than semi-discrete methods.Zhang et al.92proposed a numerical differential method,based on finite difference and extrapolation methods,to arrive at the stability for high-speed milling.The numerical extrapolation method can be repeatedly used to obtain stability predictions,with various orders of accuracy.Lehotzky et al.93improved spectral element(SE)methods to meet the reduction of the convergence rate found when the SE approach was used for discontinuous cutting.This resulted in a notable improvement in the convergent speed of the calculation time,the stable boundary,and the convergence rate of the maximal characteristic multiplier.Liang et al.proposed an improved numerical integration method,which modified the Floquet transition matrix by using an offset matrix to obtain a stability lobe,updated with varying time delay(VTD),as shown in Fig.9.94It can be seen from Fig.9(a)that,in the case of low radial immersion,the resonance region has some overlap with the chatter region.In Fig.9(b),notice that the SLD with VTD is in good agreement with the results obtained from the experiment.Also notice that the lower peak of the surface location error(SLE)and the larger cutting depth limit of the SLD do not possess the same spindle speed in the predicted results as they do when the VTD is taken into account.Niu et al.95presented a generalized form of the Runge-Kutta method(GRKM)that is based on a Volterra integral equation of the second kind.It has been proven that the GRKM has a higher convergence rate and computational accuracy than those of semi-discrete methods.Beyond this,Zhang et al.96divided the tooth passing period equally into a finite set of time intervals,and the Simpson method was then used in each time interval to estimate the state items.The stability of a milling process was then obtained using the Floquet theory.This method offers higher computational efficiency and precision than those of either fully-or semi-discrete methods.Karandikar et al.97used the random walk strategy to predict stability for a milling case with uncertain factors considering both theoretical and experimental results.

Fig.9 Predicted SLD with VTD.94

3.Experimental techniques for chatter detection

A time-frequency domain method can be used for solving dynamic equations in a milling process so as to obtain a corresponding stable leaf map.Appropriate cutting parameters can then be selected to avoid the occurrence of chatter vibration and improve the processing accuracy for the surface profile and the material removal rate.However,this method requires that industrial users have a basic knowledge of processing technologies and materials,and a complete mechanical dynamics analysis.It is therefore very difficult to use this method in a workshop.As machining progresses,machine tools,workpieces,and cutting tools are constantly changing.This makes it difficult to predict machining stability accurately and select suitable machining parameters.As a solution to this,researchers can detect signals in a machining process,for instance,acceleration signals,cutting force signals,or acoustic signals,and use these for the purposes of either online or offline chatter detection.They can then adjust processing parameters to achieve stable cutting.

3.1.Vibration signal acquisition and processing technology

Acceleration signals can provide an in-depth understanding of the dynamics of a cutting process,which is very useful for monitoring its state.The vibration measurement technique is one of the most commonly used ways of detecting regenerative chatter.Friedrich et al.98obtained a multivariable stability lobe diagram(MSLD)of a milling process by using an extended support vector machine(CSVC)and a neural network continuous learning algorithm(MI- flush-RANEKF),as shown in Fig.10,whereacr,b,n,andJare the axial limit cutting depth,the radial cutting width,the spindle speed,and the trust value,respectively.By comparing Fig.10(a)-(c),it can be seen that as the training data increases,the accuracy of the stability boundary obtained by the two algorithms gradually increases as well.A comparison between Fig.10(a)and(c)shows that the CSVC has no trust criteria,so we cannot judge the reliability of the output value.MI- flush-RANEKF introduces the trust valueJto determine the reliability of the output stability.This is based on the value of the trust valueJrelative to 0.If we compare all three graphs in Fig.10(c),we can see that as the training parameters increase,the negative value(blue)and the positive value(yellow)of the trust valueJare gradually getting closer to 0(green),which means that the reliability is increasing.Pe′rez-Canales et al.99used the approximate adjustment method of entropy to detect the state of a machining process.By assuming that there is a correlation between machining chatter and stochastic dynamics,we can detect the chatter of a machining process by detecting the randomness content of its associated acceleration signal.Experimental results have shown that this method is effective for chatter detection.Cao et al.100used a wavelet packet transform to decompose the acceleration signal in a cutting process,so that the chatter signal was distributed within a certain bandwidth.The wavelet packet containing the chatter signal was then reconstructed,and the reconstructed signal was analyzed using a Hilbert-Huang transform to obtain the Hilbert-Huang spectrum of the full time-energy distribution of the acceleration signal.The mean and standard deviation of the corresponding Hilbert-yellow spectrum were then calculated to detect the state of the cutting process.Fu et al.101used ensemble empirical mode decomposition(EEMD)to decompose a measured vibration signal into a series of intrinsic mode functions(IMFs).They then selected the characteristics of intrinsic mode functions according to the energy rule.After this,they used Hilbert spectral analysis to calculate the characteristics of inherent model functions,which provided them with the Hilbert time/frequency spectrum.The spectral characteristics of the system were then quantized by the normalized energy ratio(NER)and the coefficient of variation(CV).A Gauss hybrid model could now be used to automatically calculate the threshold,thereby enabling detection of different processing states.Ji et al.102used aggregation empirical mode decomposition(AEMD)to extract the characteristic information of an original vibration signal.Power spectrum entropy and fractal dimensions were then used to extract the frequency and shape of the signal.These two characteristics were now combined to identify the state of the milling process,which made the prediction result more reliable.An online chatter detection system has also been developed,and its effectiveness proven experimentally.Lamraoui et al.103performed multi-band resonance filtering and envelope analysis on the acceleration signal of a milling process to improve the signal to-noise ratio and sensitivity of generation characteristics.The extracted features were then sorted according to entropy,and divided into those indicative of stable or unstable cutting by using a radial basis function(RBF)and a multi-layer perceptron(MLP)neural network.By doing this,it was possible to detect chatter in the milling process.

Fig.10 Multivariable stability lobe diagram(MSLD)of a milling process.98

3.2.Cutting force signal acquisition and processing technology

Cutting force is widely used in cutting chatter recognition,because it deforms the machined material and makes chips,has high sensitivity,and is quick to respond to changes in a cutting state.Peng et al.104used the cutting force signal for detection in a real machining process.A support vector machine was built using the Matlab LIBSVM toolbox.Feature vectors were selected from the wavelet energy entropy theory and used to train the support vector machine.The chatter recognition accuracy was as high as 98.333%.A training vector machine was then used to classify the features of a simulated cutting force signal obtained from a dynamic cutting force model to determine the machining status.The critical axial cutting depth corresponding to each spindle speed and the radial cutting depth were found,giving a 3-dimensional SLD.This was in good agreement with the stability of the lobe images obtained using both zero-order frequency domain and semi-discrete methods.The cutting force signal in a milling process was used by Ren et al.105to develop a chatter detection method that was based on multi-scale permutation entropy.0.8078 and 0.7842 were selected as the thresholds for judging the processing state.First of all,the entropy of a single-scale arrangement of data segments was calculated.Whether the cutting tool was involved in cutting was determined according to whether the entropy was more than 0.8078.If the entropy value was lower than 0.8078,then the permutation entropy at scale 4 was calculated.If the entropy was lower than 0.7842,cutting was stable.Higher values indicated that there was milling chatter.The model aliasing problem in empirical mode decomposition(EMD)of cutting force signals was investigated,and a chatter recognition method combining EMD and wavelet packet decomposition(WPD)was proposed as a solution.106Here,EMD decomposition of a cutting force signal was carried out first of all to reconstruct an intrinsic mode function containing the chatter frequency.Then,based on the energy distribution of the wavelet packet node,the node with the largest energy distribution was selected to reconstruct the signal.The signals were analyzed using a Hilbert-Huang transform(HHT)to obtain the mean and standard deviation of the Hilbert-Huang spectrum as a chatter identification index.Experimental results showed that the method is effective at identifying chatter characteristics in a milling process.Subsequently,a milling chatter recognition method combining VMD and energy entropy was proposed,which improved the sensitivity to chatter,as shown in Fig.11,107whereKis the number of modes and α is the quadratic penalty.Three groups of experiments on three cutting conditions were carried out.Simulation and experimental results showed that the method could detect chatter effectively.Zhang et al.108used variational mode decomposition and wavelet packet decomposition to decompose the cutting force signal in a process to get two groups of sub-signals.Energy features were extracted from the two sets of sub-signals to obtain the energy entropy.The current state of the process was detected according to changes in the energy entropy,and the feasibility of the method was proven experimentally.

3.3.Sound signal acquisition and processing technology

Acoustic signals or acoustic emission signals generated by mechanical vibrations in a cutting area can be used to detect the presence of chatter.Simple non-contact measuring devices can be used for this,making signal collection easy and reliable,so chatter detection technology based on sound signals have been widely used.109-114Delio et al.109compared microphones with acceleration and displacement sensors,and concluded that,in many cases,acoustic signals are more effective for chatter detection.Altintas and Chan110carried out spectrum analysis on the sound signal of a process,and adopted the maximum amplitude in the spectrum as an indicator of chatter,which can be used for online chatter detection.Schmitz et al.111used the statistical variance in audio signals sampled at each rotation as an indicator of chatter.

Based on the fact that stable cutting is synchronous and unstable cutting asynchronous,it is concluded that stable cutting will have a low variance and unstable cutting a high statistical variance.Quintana et al.112collected sound signals from a milling process by installing a microphone in the shell of the machine tool.Offline analysis of time-based sound signals was used to identify chatter frequencies.A three-dimensional sound map of the milling test was set up to obtain the SLD for the milling process,as shown in Fig.12.Tsai et al.113proposed an acoustic chatter signal index(ACSI)to quantify sound signals in a milling process.A spindle rotational speed compensation strategy(SSCS)was used to compensate for the spindle speed,which is effective at controlling vibration in cutting processes.The effectiveness of a real-time chatter prevention strategy using acoustic signal feedback was proven by detecting the surface quality of a workpiece after testing.Cao et al.114used a synchronous compression transform to analyze sound signals obtained from a microphone during a cutting process,and obtained the time-frequency domain characteristic of the signals.A filtering technique was used in the time-frequency domain to remove the spindle rotation frequency,the passing frequency of the cutter tooth,and its high order harmonics,with the goal of highlighting the chatter frequency.A singular value decomposition(SVD)method was used to concentrate the TF matrix,and then the first-order singular value was calculated to give a chatter detection index.The effectiveness of this method was verified through cutting experiments.

Fig.11 Flow diagram of a chatter detection method based on VMD and energy entropy.107

Fig.12 An SLD based on sound signals.112

3.4.Visual signal acquisition and processing technology

Identifying the stability of machining processes based on machined surface quality is another method of chatter identification.There is not much research on the use of visual technology for chatter detection and evaluation.Szyd?owski and Powa?ka115proposed a surface detection algorithm based on machine vision signals.Their algorithm is based on local gradient estimation and able to use milling parameters.The hypothesis is that the local gradient direction could be used to calculate the ridge/valley orientation of the machined surface.The average level of error is obtained by calculating the surface error map according to the direction of the local ridge.By comparing a fast Fourier transform of the cutting force signal with the average error level of the machined surface,one can get a threshold to assess whether chatter is occurring in a machining process.The advantage of this method is that it is based on processing parameter image analysis,which is effective for improving analysis efficiency.The disadvantage is that the initial threshold has to be set for each process.Lei and Soshi116proposed a new method for using visual signals to detect the status of a cutting process.Here,the installation of a polarizing filter in front of the light source first of all eliminates the effect of specular reflection on the accuracy of the Polarization Vision System,so that accurate image information about the machined surface can be obtained.Then,the image information is transformed into gray levels to extract the pixel intensity.The spatial frequency and wavelength of a workpiece are obtained using a fast Fourier transform of the pixel intensity.The correlation between the wavelength and the cutting speed is then used to obtain the vibration frequency of the corresponding surface.When compared to the acoustic emission detection technology,the spectrum calculated using visual systems has a more obvious dominant peak frequency.This makes it easier to judge the status of processing.

Table 2 Characteristics and limitations of chatter recognition methods.

Therefore,we have seen that various kinds of processing signal(acceleration signals,cutting force signals,sound signals,and visual signals)can be used for online and of fline detection of chatter in milling processes.The characteristics and limitations of each of these methods are summarized in Table 2.

4.Chatter control/suppression technology

The control/suppression of chatter is a challenging problem in milling processes.In the last few decades,machining technology has made great progress.As other things improve,including better efficiency,higher accuracy,lower rejection rates,and lower production costs,there is a greater pressure to develop more effective chatter control/suppression technology as well.Broadly speaking,chatter control/suppression techniques can be divided into passive and active controls.

4.1.Passive control technology

Passive control technology aims to improve the stability of machining processes by improving machine design or using equipment that can absorb extra energy or get rid of the regeneration effect,thereby changing or modifying a system's behaviors(see Fig.13,whererandsare the dynamic displacements of these absorbers,andkandcare the damping and stiffness of the cutting system,respectively).Passive control techniques usually use dampers or special tools to suppress chatter(see Fig.14,whereM,k1,andc1are the mass,damping,stiffness of the original system,respectively;m,k2,andc2are the mass,damping,stiffness of the damper,respectively;x/x0is the ratio of the dynamic displacement to the static displacement).Ziegert et al.117explored the possibility of placing a mechanical damper directly into a rotating tool.A cylindrical shock absorber was inserted into a matched shaft hole along the center line of a milling cutter.As the tool rotated during milling,the centrifugal force between the multi- fingered shock absorber and the tool body created pressure,which caused energy dissipation through friction when there were bending vibrations.In the best case,this damper could increase the processing efficiency by up to 53%.Moradi et al.118created a twodegree-of-freedom system for milling processes,and designed an optimal adjustable damper in theXandYdirections to suppress regenerative chatter during flank milling.The optimal parameters of the shock absorber were obtained by a Matlab optimization algorithm.The study concluded that adjustable shock absorbers could improve the stability of the process effectively and have good robustness in dynamic models with uncertain parameters.Adjustable shock absorbers were subsequently applied to a milling process for cantilever plates,where they also worked effectively to inhibit processing chatter.119

Fig.13 Impact of a passive damper on stability.

Fig.14 Passive control technology.

Saadabad et al.120designed the best tuned vibration absorber in the X-Y directions for flank milling processes.Here,a multi-loop complex algorithm is used to determine the optimal rigidity and mass for a vibration absorber so as to improve global stability in milling processes.Adjustable shock absorbers can also reduce chatter caused by the effect of workpieces bending during milling processes.121Tunable mass dampers(TMDs)are commonly used in passive systems.Several combinations of TMDs can be used simultaneously in a production process.Rashid A et al.122fixed four TMDs to a workpiece resulting in a 98%reduction in the vibration amplitude during processing.Studies by Yang et al.123have shown that using multiple identical TMDs has a better damping effect than that of a single TMD.Burtscher et al.124presented an adaptive tuning mass damper(ATMD)with a variable mass,and gave the design program of the ATMD,as shown in Fig.15.The variable meaning in Fig.15 was described in reference.124The real amplitude of the frequency response function of the spindle system was increased by nearly 52%with the ATMD optimized by a genetic algorithm.The experiment showed that the difference between the dominant frequency of the spindle system with or without ATMD was nearly 17%,which proved the effectiveness of the AMTD in reducing cutting vibration.Recently,new methods have been developed to suppress chatter in milling processes.Zhang et al.125,126submerged processing parts in viscous liquid,which reduced the milling force coefficient and the natural frequency of the system and also improved the system damping,thus realizing milling process chatter suppression.Butt et al.127designed a non-contact eddy current damping device to suppress chatter.The advantage of the device is that there is no additional physical addition,and it is suitable for different sizes of workpieces.Wan et al.128proposed an effective method to improve the milling process stability by attaching appropriate additional masses to thin walled parts.

In variable-pitch tools, flutes are distributed irregularly along the perimeter of a tool.Hahn129was the first to propose a variable-pitch milling cutter,pointing out the benefits of breaking the phase between past and present vibrations,and thus looking at how to make the constant delay of traditional milling cutters become multiple discrete delays.Slavicek130was the first to model an unequal-tooth milling cutter and then apply an orthogonal cutting chatter theory.This study showed that the ultimate axial depth for variable-pitch tools could double the depth achieved by traditional milling cutters.Sellmeier and Denkena131and Altintas et al.47have also shown that variable-pitch milling cutters can improve the ultimate axial cutting depth for high spindle speeds.However,this has little effect at lower speeds.The key to obtaining the maximum stability is to find the optimal pitch between cutter teeth within a specific spindle speed range.This has led to many researchers attempting to quickly design variable-pitch milling cutters for different conditions by using the simplest methods possible.Some researchers have used time-frequency domain methods to acquire the best design parameters for stability models where there are different tooth angles.Altintas et al.47and Budak et al.132-134used a zero-order frequency domain method to obtain the stability of different angles,and then determined the optimal pitch angle based on actual requirements.

Olgac and Sipahi135used a method where they clustered treatment of characteristic roots to solve the dynamics of variable-pitch milling so as to obtain an SLD.Final results provided a theoretical basis for the pitch angle formation of tools and optimal cutting conditions.They conducted experimental verification for both four-and six-tooth milling cutters,and found that helical tools with non-constant or alternating helix angles introduced continuous changes in the local pitch angles along a tool's axis.This special geometry caused continuous variations in the delay between flutes,thus perturbing a regenerative effect.Turner et al.136used an equivalent variable-pitch representation of variable helix tools to establish an analytical model for them.A corresponding stability model was obtained by combining calculation of the mean helix angle of each tooth with a zero-order frequency domain method.This analytical model is only suitable for low radial immersion.In the time domain,Sims used the modern control theory to obtain the stability lobes of a milling process,then used the differential evolution algorithm to optimize the pitch angle and the helix angle of the cutter,137and validated simulation results.138Dombovari and Stepan139analyzed the effect of the helix angle on the milling stability,and pointed out the cutting behaviors of variable-helix angle milling cutters under high-and low-speed cutting conditions.Hayasaka et al.140used the design index(ap/alim)to design the difference of the helical angle of a milling cutter,and presented a generalized design method of a variable-helical angle end-milling cutter for chatter suppression.The effectiveness of the proposed method was verified by milling experiments under multiple conditions.The dynamics and stability of variable-pitch and helix tools have also been modelled by Comak and Budak.141Here,the model was solved in the frequency domain as well as by using a semi-discrete method.The optimal pitch and helix angle,along with their corresponding stability,were obtained.The absolute stability limit was increased by between 25%and 50%,when compared to those of regular tools.Otto et al.62presented a new method for identification of chatter stability lobes from a linearized system that is based on a multifrequency solution.Here,too,it was concluded that milling cutters with variable helix angles could significantly increase the ultimate axial cutting depth.

Fig.15 ATMD design procedures.124

Variable spindle speed(VSS)is another alternative method for regenerative chatter disruption.It has a similar mechanism to the use of a cutter with variable pitch or helix.However,its positive effect is due to the reason that the phase relationship between the inner and outer modulations on a machined surface is changed by modulating the traditional constant spindle speed(CSS)periodically.The limitations of this method are the responsiveness of machine tools and the load on spindle motors not designed to constantly change the spindle speed.The idea of suppressing chatter by a spindle speed variation came up firstly in the 1970s.Modeling of variable-speed milling is more complex than that of turning since the speed variation frequency and the tooth passing frequency interact,and the resulting system is typically quasiperiodic.It was not until 2002 that Sastry et al.142applied the idea of a spindle speed variation to milling,and found that chatter could be effectively suppressed at low spindle speeds.Zatarain et al.143modeled a milling process with a variable spindle speed in the frequency domain.Results showed that a variable spindle speed could effectively suppress cutting chatter at low spindle speeds,and a sinusoidal speed variation is more effective than a triangular speed variation in restraining cutting chatter.The model was validated by a semi discrete method and timedomain simulation,and its correctness was verified by experiments.Seguy et al.144,145studied the effect of a variable spindle speed on regenerative vibration of a machine tool in a highspeed milling process,and obtained an SLD under triangulation by a semi-discretization method.It was shown that periodic doubling chatter could effectively be suppressed by a spindle speed variation,although the technique was not effective for quasiperiodic chatter above the Hopf lobe.

Totis et al.146obtained the delay differential equation with a variable delay based on coordinate transformation from the time domain to the corner domain.Then the Chebyshev collocation method was used to evaluate the region discretization and the stability of the system.Compared with those of the semi-discrete method,the new method has a higher prediction accuracy and a faster calculation speed.Based on the Floquet theory of the variable-step numerical integration method,Niu et al.147obtained the stability of a variable-spindle speed milling process.It was found that compared with the semi discretization method and the constant-step numerical integration method,the proposed method had the advantages of high computational accuracy and efficiency,and sinusoidal modulation could achieve higher stable productivity while triangular modulation permitted larger modulation parameter options.Jin et al.148established a milling dynamics equation that considered the effects of variations of the cutter pitch angle and spindle speed,and linear stability analyses were carried out via an updated semi-discretization method.Results showed that the combined milling process exhibited a great capability to avoid the onset of milling chatter.

4.2.Active control technology

Active control technology for chatter suppression determines the dynamic behaviors of machine tools by detecting their states,and then adjusts their working statuses by active execution of a decision.Active vibration damping systems are usually composed of monitoring,diagnosis,and execution elements.This kind of control technology is becoming more and more important because of progresses in the fields of computing,sensing,and actuation.Active control technology can bring about notable improvements in the stability of processes.Munoa et al.149,150used external acceleration sensors to provide feedback to a control loop,and then used a machine's own drive to suppress the occurrence of chatter in a milling process(see Fig.16).The variable meaning in Fig.16 was described in reference.150This led to improvements in the overall stability of heavy milling,which was verified experimentally.Van Dijk et al.151developed a methodology based on a robust control approach using μ-synthesis to proactively control chatter in a milling process.This aimed to provide chatter-free cutting operations within a range of predetermined technological parameters(e.g.,cutting depth and speed).Gourc et al.152modelled milling stability with active magnetic bearings,and provided experimental validation of this approach.They noted in particular that strong forced vibrations could also limit the depth of cut,and integrated this limitation in their stability diagram.Monnin et al.153,154proposed using a mechatronic system integrated in a spindle unit that was combined with two different optimal control strategies that could suppress the occurrence of milling chatter(see Fig.17,wherew:disturbance generated by the milling process;u:reference signal delivered to the actuating system;y:information provided by the sensing system;z:tool center point deviations).Compared to that of traditional milling,the minimum limit for the axial cutting depth was increased by 55%,and efficiency by 91%.Wang et al.155also integrated a piezoelectric actuator into a spindle,which made the stiffness of the cutting system time-varying and ultimately inhibited milling chatter.Dohner et al.156used an approach where an electrostrictive actuator was placed on the spindle tip.Depending on the process vibration signal detected by a strain gauge placed at the tool's root,the output voltage can be controlled by a controller,so as to change the output force of the actuator,thus suppressing any occurrence of chatter.Experimental results have shown that active control systems can generally increase the metal removal rate by an order of magnitude.

Fig.16 Active chatter suppression using machine drives.150

Fig.17 Concept of an active spindle.TCP:tool center point.154

An active fixture system can control the chatter process by affecting the relative vibration between a workpiece and a tool.To this end,Rashid and Nicolescu157constructed a form of active clamping system that uses a force sensor to pick up changes in the cutting force in a machining process.Combined with an adaptive filtering algorithm,this can control the dynamic output force of a piezoelectric actuator to improve stability.Brecher et al.158proposed a method for suppressing chatter in a process that involved installing an active clamping device on a milling machine.The designed active fixture was driven by highly dynamic axes controlled by two piezoelectric actuators.Each dynamic axis had a displacement sensor and a force sensor.The acceleration signal was used as a feedback signal,and the position of the workpiece was dynamically adjusted using a closed-loop position control method,thus suppressing chatter.Fig.18 provides a schematic active fixture diagram and an image of an experimental site.Sallese et al.159designed a special controller that combined an intelligent fixture with the closed-loop control strategy of low-frequency sinusoidal excitation to bring about chatter suppression.Using this approach,the ultimate axial cutting depth was increased by 43%in a slope cutting test,proving its effectiveness.In order to tackle the problem of chatter in flexible parts,Parus et al.160used a linear quadratic Gauss(LQG)algorithm and a piezoelectric actuator to form an active control system that could improve stability.Once again,the effectiveness of this control system was proven using experiments.

Li et al.161managed online suppression of milling chatter by using an open controller.The signals of a dynamometer and an acceleration sensor were extracted and synchronized to analyze the frequency domain characteristics of the sampled data.This was then used to assess whether chatter was present in the cutting process.A relationship model between the chatter frequency and the spindle speed was established,which provided a theoretical basis for restraining self-excited chatter by means of a variable spindle speed.An open,fully modular software-based milling controller was designed to integrate online parameter acquisition and feedback control,and a relevant variable-spindle speed algorithm for online suppression of chatter was embedded in the controller.An online continuous variable-cutting depth chatter suppression test for an aluminum alloy workpiece was carried out.This experiment confirmed the scope for using online intelligent milling controllers to suppress chatter.In a different approach,based on feedback from sound signals generated during a cutting process,Tsai et al.113,162used an adaptive spindle speed adjustment algorithm to suppress chatter vibration in real time.The effectiveness of this approach has also been verified by experiments.

5.Effects of other factors on SLDs

In milling process stability prediction,outside of cutting force coefficients and a cutting system's inherent attributes that can directly affect the prediction of stability,process damping,the gyroscopic effect,and the tool runout can also have an effect on stability prediction.We will therefore now examine the impacts of these other factors on SLDs.

5.1.Effect of process damping on SLDs

Sisson and Kegg163have been experimentally studying how system damping can increase as a result of contact between machined surfaces and the tool rake face during low-speed cutting,and how worn tools can improve the stability of a system.Other researchers164have carried out a large number of experiments to prove Sisson and Kegg's discovery,by introducing process damping into the coefficient for dynamic cutting force and analyzing its impact on stability.The relationship between surface ripples and process damping was established by Tlusty and Ismai.165This is the most practical method for measuring the impact of damping on a process.Wu166proposed an indentation force model to describe process damping when using a tool-tool interference ploughing force.Cllagaddi167used Wu's method in a two-degree-of-freedom dynamic milling model to simulate the ploughing force,and studied the effect of process damping on stability.Their simulation results were consistent with experimental results.Ahmadi and Ismail168also used Wu's indentation force model as the basis of a linear viscous damper.They used this instead of the damping coefficient method to examine milling stability under conditions of process damping by using both multi-frequency and semidiscrete methods.Their results showed that the semi-discrete method provided accurate prediction results across the whole cutting speed range,while the multi-frequency method required higher harmonics at low speeds to achieve the same level of accuracy.169Altintas et al.170presented a new dynamic cutting force model whose coefficients could be identified from controlled oscillating tests with the aid of a fast tool servo.When the oscillating frequency and spindle speed were synchronized to achieve in-phase inner and outer modulations,the regenerative effect was eliminated,and process damping coefficients could be identified.With the increase of tool wear,damping coefficients for the process also increase,which improves processing stability.Based on the equivalent viscous damping method proposed by Budak and Tunc,171Gurdal et al.172modeled process damping as a function of the surface wavelength,and introduced it into the stability solution process.The conclusion here was that the model for process damping was strongly dependent on the vibration amplitude,as shown in Fig.19,where red dots:experimentally chatter,green dots:experimentally stable,andAcis the vibration amplitude of cutting.

Fig.18 Active fixture structure diagram and experimental site.158

Budak and Tunc173obtained an indentation coefficient by conducting an energy analysis,and then used the coefficient to predict process damping and stability limits under different conditions.Both time-domain simulation and experimental results achieved good consistency.Tunc?and Budak174studied the effects of cutting conditions and tool geometry on process damping using a series of experiments and simulations,as shown in Fig.20,whereandare the average specific process damping coefficients inXandYdirections,respectively,Vis the cutting speed,and Av.Spec.Pro.Damp is the abbreviation of the average specific process damping.This study found that,as the cutting speed decreased,the vibration frequency increased.The higher the hone radius was,the greater the total contact length between a tool and a workpiece was,with damping of the cutting process increasing gradually,and stability increasing correspondingly.Wan et al.175established a general model to assess the influences of static and dynamic ploughing mechanisms on milling system stability.This research showed that process damping caused by a ploughing effect could greatly improve the stability of a milling system.Later on,a model analysis method was proposed to identify damping in milling processes.176At a constant axial depth,Tyler et al.177described a method that can produce the analytical radial depth for cutting SLDs that includes process damping.The viability of the approach was proven experimentally.Recently,Feng et al.178systematically studied the mechanism of damping in a thin-walled milling process,and concluded that the relative vibration of the tool-workpiece system was the main source of machining damping.It was subsequently integrated into the process's governing equation to estimate the stability lobe diagrams(SLDs).

Fig.19 Effect of the vibration amplitude on milling stability.172

5.2.Effect of tool runout on SLDs

In actual multi-tooth milling processes,because a tool is never perfectly symmetrical,there is always a non-zero deviation along the axis of rotation of the spindle and the geometrical axis of the tool,which constitutes what is called‘tool runout'.The occurrence of tool runout can greatly affect the actual cutting radius of the tool.Tool runout results in an uneven distribution of cutting forces on the cutting edge,causing the frequency of the cutting force/vibration signal to change between the tooth frequency and the spindle rotation frequency,as shown in Fig.21,where ρ is the eccentricity, λis the eccentricity angle,Ris the tool radius,Ocis the tool manufacturing center,andOris the center of tool rotation.As a result,it also affects the stability of the machining process.A lot of importance has been attached to studying the relationship between tool runout and the stability of cutting processes.Insperger et al.179studied how tool runout could affect stability by introducing eccentricity into the directional force coefficient.The conclusion drawn here was that tool runout does not change the stability boundary,but rather the type of chatter frequency.

Fig.20 Effects of tool parameters on process damping in milling.174

Fig.21 Tool geometry with cutter runout.

Under the two working conditions of slot milling and 25%axial immersion milling,Otto et al.62looked at the influence of cutter runout on the stability of milling cutters using a conventional tool and a variable-helix tool.Results showed that tool runout had no effect on the stability of conventional tools.It also had less influence on the stability of an end milling cutter with variable pitch engaged in slot milling.Its greatest influence was upon the stability of an end milling cutter with variable pitch engaged in 25%axial immersion milling.With regard to the influence of tool runout on conventional tools,it appears that the eccentricity of a tool does not affect the weight of the delay distribution,making it of no moment in actual situations of use.Wan et al.41established a multidelay dynamic equation that took into account cutter runout,and studied the influences of radial immersions,feed directions,feeds per tooth,and helix angles on milling stability.The conclusion drawn in this case was that a low radial depth of cut,a small feed per tooth,and a small helix angle could all have a great impact on stability,but the influence decreased as each of these parameters was increased.For a two-degree-offreedom milling system,the relative level of stability limits for down-milling and up-milling is dominated by the model parameters in theX-direction and theY-direction.If the natural frequency of theX-mode is larger than that of theY-mode,the stability limits for up-milling will be higher than those for down-milling.In the contrary case,an opposite effect is observed.

Zhang et al.180proposed a varying-time delay model based on the trochoid for teeth during cutter/workpiece engagement,which took cutter runout into account.They then derived a state transition matrix in one cutter rotation period by using the Cotes numerical integration formula.It was concluded that tool runout effects could actually improve milling stability,as shown in Fig.22,where ρ and λ are the cutter runout value and angle,respectively,andA,B,C,andDare four different cutting conditions.The validity of this approach was proven by there being good consistency between simulated and experimental data.Ma et al.181demonstrated the effect of tool runout on the stability of multi-modal milling.Here,simulated and measured results confirmed that the occurrence of cutter runout could locally increase the stable region,and the lower the feed rate was,the larger the stability boundaries became.Thus,the influence of the helical angle on stability can be set aside when considering cutter runout and feedrate.

5.3.Impact of the gyroscopic effect on SLDs

Fig.22 Effect of tool runout on an SLD.180

The impact on stability of the gyroscopic effect of a rotating spindle-tool's rotating speed is not negligible as it approaches high speeds and above.This means that the stability of cutting systems is speed-dependent.182Movahhedy and Mosaddegh183presented a finite element-based model of a spindle-based holder-cutting tool.This model uses Timoshenko's beam theory to obtain the frequency response of a system when gyros copicterms are included.This response provides for investigation of the influence of the gyroscopic effect on the stability of high-speed spindle systems.The presence of a gyroscopic effect in the dynamics of a system can cause a splitting of each symmetrical natural frequency of the system into two backward and forward frequencies.The amplitude of the backward frequency is the determining factor in defining the stability border.It has been shown that the gyroscopic effect can lower the critical depth of cut in high-speed milling.Shi et al.184established a finite element(FE)model of a highspeed spindle system for micro-milling.Gyroscopic and mode interaction effects on the dynamics and chatter stability of the system were studied using transfer functions and the mode shapes of the FE models.Due to its much smaller mass and radius compared to those of tools used in conventional highspeed machining,the mode-splitting phenomenon resulting from the rotation was less significant.Thus,the gyroscopic effect on micro-end mill dynamics and their chatter stability is not of great account.However,the finite element approach has some limitations when analyzing a structure's vibration and dynamics.For high-frequency vibration structures,the result can be unreliable.In view of this problem,Tajalli et al.185used a dynamic stiffness(DS)method to analyze the vibration and dynamics of a structure.Based on the extended Hamilton principle,the dynamic model for a micro-end milling cutter was derived from the coupling effect within the gyroscopic effect.The impact of the gyroscopic effect on an SLD is shown in Fig.23.It can be concluded that gyroscopic effect shifts the stability lobe to the right,and the offset gradually increases as the spindle speed is increased.

6.Discussion and outlook

Fig.23 Impact of the gyroscopic effect on an SLD.185

Getting the cutting force coefficient,the model parameters for a cutting system,the dynamic characteristics of a workpiece,the damping effect of a process,the tool runout,and the gyroscopic effect are all critical for milling stability prediction.The cutting force coefficient for a process changes according to changes in process parameters.Polynomial fitting methods are often applied to get the cutting force coefficient,but the fitting accuracy in this approach is not high.Thus,there is a need to develop new methods that can accurately establish a dynamic cutting force coefficient model.Static mode parameters for cutting systems are obtained through hammer impact experiments,but there are certain errors that occur when calculating dynamic model parameters for cutting systems during actual processing.This makes the need for an alternative method to obtain dynamic model parameters for machining processes even more pressing.The dynamic characteristics of workpieces change according to their processing positions and the removal of material.This is especially the case with thin-walled parts.Finite element models are often used to predict model parameters and vibration modes.However,damping is difficult to estimate because of the influence of clamping on the different parts of a workpiece.When the cutting speed is lower than a certain value,the effect of process damping on machining stability is greatly increased.However,no consensus has been reached so far regarding the best way to approach analytic modeling of process damping effects.Future directions for development therefore include establishing an accurate model of these.

A more accurate dynamic model of a system composed of machine tool-tool-workpiece- fixture,a low-speed cutting process dynamic model for difficult-to-machine materials,a comprehensive dynamic model for multiple chatter types,a dynamic model considering the effect of mass removal and the time-varying relative position of tool-workpiece in thin walled parts processing are subjects that need further study in this field.In the aspect of stability analysis,the three kinds of mainstream methods,i.e.,frequency domain,discrete,and numerical methods,have been quite mature,but they all have their own limitations.The frequency domain method has high efficiency.Because of its simplification,many complex nonlinear effects are ignored,so the accuracy is not high,and its generality is poor in different working conditions.The generality and accuracy of the discrete method are good,but the computational efficiency is not high.The numerical method can consider many kinds of nonlinear effects,but the calculation efficiency is the lowest,and the stability criterion is not universal.The stability analysis method still needs further improvement in order to improve its versatility,accuracy,and computational efficiency.

Chatter detection technology is often based on a force signal,vibration signal,sound signal,image signal,etc.measured in a cutting process,through signal processing methods such as wavelet analysis,Hilbert-Huang,and other spectral pre-processing,combined with different feature extraction methods such as neural network,genetic algorithm,etc.,to achieve detection of the machining state in the cutting process.Milling chatter is a nonlinear and non-stationary phenomenon.Because the Fourier transform method conceals the time domain information of a milling process,it is blind to the state transition of non-stationary signals.Time-frequency domain analysis is another method to identify non-stationary process characteristics.Common methods include wavelet analysis,wavelet packet analysis,Hilbert-Huang transform(HHT),variable mode decomposition(VMD),and so on.Wavelet analysis and wavelet packet analysis are powerful time-frequency analysis methods.The selection of an appropriate wavelet and decomposition layers still depends on the current experience.Different wavelets will lead to completely different performances.For this reason,wavelet transform is still an artificial method.HHT can decompose process signals into a set of orthogonal intrinsic modal functions,and then obtain instantaneous frequency arrays using Hilbert's spectral analysis.Because it has adaptive effect and does not involve much manual intervention,it is more suitable for on-line chatter signal processing.However,due to the lack of theoretical basis,there are still some defects such as pattern mixing and endpoint effect.EEMD is an extension of EMD.It has some improvements in solving the mixed model of EMD,and is sensitive to strong background noise.Therefore,the algorithm should be carried out several times to reduce the error caused by white noise.VMD overcomes the shortcoming of lack of a theoretical basis of EMD,and the VMD method can adaptively determine the corresponding frequency band and estimate the corresponding model,so it can be used for on-line detection of milling chatter.BPNN models often have multiple local minima and over fitting problems.For the above problems,a genetic algorithm(GA)can be used to optimize the weight and threshold of a BP neural network model.A Support Vector Machine(SVM)has better performance in solving the problem of a small sample and high-dimensional pattern classification.It has strong generalization ability and can avoid dimensional traps.Time-frequency analytical methods are computationally intensive,and pattern-based methods usually require a large number of training samples and a long training time.To apply these chatter monitoring algorithms in practice,it is necessary to improve the quantification and applicable range of chatter indicators.

The collection of cutting forces and vibration signals requires expensive instruments,and the selection of installation positions is difficult.Sound signals are easily affected by external noise.Detection processes also usually involve monitoring a single aspect.In view of the complexity of cutting processes,the detection of a single signal can lead to detection errors.Multi-signal fusion technology is the focus of future development for detection.At present,detection is mostly offline,because of the complexity of algorithms.However,recent developments in computer and sensor technologies have made online detection of a process a real possibility for future research.In the process of chatter monitoring,the key is how to determine the threshold between chatter and stability.In order to improve the reliability and efficiency of detection,we cannot ignore the characteristics of a model itself,such as the geometry of a workpiece,the tool path,and so on.

Chatter control/suppression can be divided into active and passive controls.Passive control usually uses a tuned mass block,an impact damper,a variable pitch,or a variable helix angle to bring about chatter control.Overall optimization of the tool geometry at the design stage(e.g.,a saw-tooth cutting edge and variable pitch)can significantly improve process stability.In tool optimization,however,there are no clear and accurate design guidelines for maximizing process stability,although there are methods available that are based on energy criteria and minimization of regeneration effects.The influence of special cutter geometries on damping needs further study.Dynamic stiffness modeling of different joint parts presents the biggest challenge for dynamic behavior prediction at the tool design stage.Tuned mass dampers require a large space in the critical position,and the adjustment of the dynamic characteristics is limited,while a friction damper is smaller than a tuned mass damper,which can solve the space limitation problem of the tuned mass damper,but the method lacks a design criterion.Because damping of thin-walled parts is very small,the application of passive control technology to suppress process chatter is still a challenge.Active control usually involves the collection of sensor processing information and the use of mathematical algorithms to make decisions.These are then used to drive actuator performance,thus bringing about a change in the processing state.However,for any given application,it is still difficult to accurately change the behavior of an actuator.Only a few studies have considered a cutting process model and an actuator model together to analyze the force,the active force,and the best position for measurement. Almost all existing models are noncommercial,and there is an urgent need to determine the best design strategy for creating a reliable active system that is open to real-world implementation.Other active methods,including model predictive control,robust control,adaptive control,etc.,require complex control algorithms or expensive equipment,which cannot be widely used in an actual production process.

In the field of micro-milling,the stability analysis method of traditional milling is mostly used.However,micro-milling is not a simple reduction in the scale of macro-milling,which has some characteristics of mesoscale machining(such as scale effect,minimum cutting thickness,and single-tooth cutting phenomenon)and the influence of the microstructure of workpiece materials,making micro-milling have a unique processing mechanism and characteristics.A method of analyzing the machining mechanism and characteristics of micromilling is still to be studied.

7.Conclusions

Due to a lack of dynamic stiffness in machine tool-tool,holder-tool,cutter-workpiece systems,three forms of vibration can be generated during machining:free vibration,forced vibration,and self-excited vibration.If a system is well balanced,the first two kinds of vibration can be avoided,reduced,or eliminated.Self-excited vibration is undamped vibration caused by the alternating forces produced by the interaction between a tool and a workpiece.This greatly increases the instability and uncontrollability of a system.The most common form of self-excited vibration is regenerative chatter.Since the 1950s,researchers have been carrying out extensive and thorough research regarding the problem of chatter.Advances in computers,sensors,and actuators have improved understanding of these phenomena and helped with the development of strategies for overcoming instability problems.

This paper has summarized state-of-the-art chatter prediction,chatter detection,and chatter control/suppression in milling processes,and has discussed the relationship between cutting stability and process damping,tool runout,and gyroscopic effect.The main outcomes of the paper are as follows:

(1)By looking at research regarding input parameters(cutting force coefficient,process parameters,tool parameters,system dynamics),dynamics modeling(degree-offreedom models),and the solution of delay differential equations(frequency domain methods,discrete methods,numerical methods,and so on),prediction of stability in milling processes has been summarized,and the relationship between various factors and realization of stability has been clarified.

(2)By examining works related to acceleration signals,cutting force signals,acoustic signals,and machined surface signals,we have summarized how detection of chatter in milling processes can be brought about by using neural networks,support vector machines,Hilbert-Huang algorithms,and so on.

(3)We have emphasized the distinction in passive control technology between damper technology and special tool shape design technology.

(4)By looking at detection of machine processing information and use of intelligent algorithms for analysis,storage,and judgment,we have brought together ways in which researchers are currently trying to achieve active control of milling process chatter.

(5)We have also discussed and analyzed the relationship between process damping,tool runout,gyroscopic effect,and chatter and the stability of milling processes.

Acknowledgements

This project was supported by Projects of International Cooperation and Exchanges NSFC(51720105009),the National Natural Science Foundation of China(No.51575147),and the Youth Talent Support Program of Harbin University of Science and Technology(201507).