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National Key Laboratory of Science and Technology on Helicopter Transmission，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract: Optimization of machining parameters is of great importance for multi-pass end milling because machining parameters adversely or positively affect the time and quality of production. This paper develops a second-order fulldiscretization method（2ndFDM）-based 3-D stability prediction model for simultaneous optimization of spindle speed，axial cutting depth and radial cutting depth. The optimal machining parameters in each pass are obtained to achieve the minimum production time comprehensive considering constraints of 3-D stability，machine tool performance，tool life and machining requirements. A cloud drop-enabled particle swarm optimization（CDPSO）algorithm is proposed to solve the developed machining parameter optimization，and 13 benchmark problems are used to evaluate CDPSO algorithm. Numerical results show that CDPSO algorithm has a certain advantage in computational cost as well as comparable search quality and robustness.A demonstrative example is provided.

Key words：machining parameter；multi-pass end milling；chatter stability；particle swarm optimization（PSO）；cloud model

0 Introduction

Optimization of machining parameters that ad⁃versely or positively affect the machining process be⁃havior is of great importance to shorten the produc⁃tion time，explore the production potential and im⁃prove the efficiency of multi-pass end milling opera⁃tion. Over the past few decades，a number of re⁃searchers have contributed significant research in the formulation of machining parameter optimization models and in the design of their solution algo⁃rithms，claiming to have made headway in their re⁃spective machining operations［1-3］. However，most current researches are focused on how to solve the parameter optimization model efficiently，while little attention is paid to improve the practicality of optimi⁃zation model. Taking milling operations for exam⁃ple，most of the related literatures simplified the mill⁃ing operation to symmetrical milling，and most opti⁃mization models did not consider the constraint of machining stability. For large，complex，thin-walled structures，such as helicopter rotor blades and air⁃craft envelope，however，chatter easily occurs and results in poor surface quality and machining inaccu⁃racy because of low rigidity of this kind of structure.

How to avoid chatter occurrence in the machin⁃ing process is a challenging issue and has caught the attention of researchers and technicians in the field of machining. The most common method is to pre⁃dict the machining stability lobe diagrams（SLD），and many methods have been proposed［4-5］. Unfortu⁃nately，all of these researches are devoted to ma⁃chining parameter selection in the stability domain without considering production time and quality. Bu⁃dak et al.［6］and Chen et al.［7］built mathematical model of optimization on machining parameters with the constraint of machining stability to seek the opti⁃mal combination of axial cutting depth and radial cut⁃ting depth without chatter for the maximum material removal rate，but only two variables were consid⁃ered in these studies. The three variables related to machining stability，i.e.，axial cutting depth，radial cutting depth，and spindle speed are equally impor⁃tant，as they all directly affect the rationality of the parameters obtained. Thus，an accurate three-di⁃mensional（3-D）stability model is established in this paper for the optimization of multi-pass end ma⁃chining parameters.

In recent years，particle swarm optimization（PSO）and its variants have been successfully devel⁃oped to tackle different optimization problems［8-9］.Although PSO appears to be fast in finding solu⁃tions near optimal，it is weak in subsequent exploita⁃tion. This study proposed a cloud drop-enabled PSO（CDPSO）algorithm inspired by the excellent properties of the normal cloud model［10］，which can effectively improve the randomness and fuzziness of the particle evolution. Numerical results obtained us⁃ing the proposed CDPSO algorithm to solve bench⁃mark problems indicate that it performs better than other four existing algorithms. And a demonstrative example shows that optimal parameters obtained by the proposed optimization model performs better than empirical parameters by 34% with respect to production time.

1 3‑D Milling Stability Model

Considering that the degree of freedom most concerned in machining is only the one perpendicu⁃lar to the machined surface. Therefore，the milling dynamic model used in this study is based on the dy⁃namic equation described as follows

whereζis the relative damping；wnthe angular natu⁃ral frequency；dthe axial cutting depth；mtthe modal mass of the tool；Tthe regenerative delay andh(t)a specific cutting force coefficient expressed as

hereZis the number of the cutter teeth；KtandKnare the tangential and normal cutting force coeffi⁃cients，respectively；g(φj(t)) is a window function involving the angular position of thejth tooth

whereφstandφexare the start and exit angles，re⁃spectively； for up-milling，φst=0 andφex=arccosfor down-milling，φst=arccosandφex=π，wherea Dis the radial immersion ratio.

Without loss of generality，Eq.（1）can be ex⁃pressed as state-space form

where

The response of Eq.（4）ont∈[kτ，(k+1)τ]can be expressed as

In a similar way

wheret∈[0，τ]. The following equations can be ob⁃tained by handling the Duhamel term of Eq.（7）with the method described by using second-order full-discretization method（2ndFDM）［4］.

where

where

Thereafter，the discrete map can be described

as

where

whereδ=[I-Fk+1]-1and the transition matrix within a periodic time interval can be established by using the sequence of discrete mapsDk(k=0，1，2，…，m-1)，which can be utilized to predict the chatter stability region about spindle speed and axial cutting depth via Floquet theory.

The third variable of the 3-D stability model is the radial immersion ratio，wherea∈(0，D].

2 Optimization Model

In order to obtain the ideal stability model，an infini⁃tesimal quantitysmis defined for facilitating model⁃ing. The value range is equally divided intoεsmall pieces，thus

2.1 Objective functions

In the case of milling，the production time has always been considered as the objective functions in most attempts to optimize the machining parame⁃ters. And first of all，it is necessary to construct the model about the length of tool travel.

The length of tool travel in rough pass can be expressed as

At this point，the formulation of 2ndFDMbased 3-D stability model of spindle speed，axial cutting depth and radial cutting depth has been com⁃pleted.

wherearis the radial cutting depth in rough machin⁃ing；apthe approach distance；andean arbitrary set distance to avoid possible accidents and damage，which is taken as 2 mm in this study；GInt(⋅) and SInt(⋅) denote the greatest and the smallest integer operators，respectively.

The approach distance can be expressed as

whereDis the diameter of the tool.

The travel length of the finish pass is measured from the contact of tool and workpiece to separation

whereasis the radial cutting depth in finish pass.

The production time can be expressed as

whereTp（minute）is the preparation time，TLthe clamping time，Tathe adjustment time，Tmthe ma⁃chining time，andTrthe tool changing time. SinceTpandTLare always fixed and have no effect on the total production time，they can be ignored.

Considering that machining operation of multipass end milling consistsnrough passes and one fin⁃ish pass，it can be obtained that

Substituting Eq.（18）into Eq.（17），the objec⁃tive function can be expressed as

where

whereh1（minutes per millimeter）andh2（minutes）are constants related to tool travel and approach/de⁃part time，which are taken as 7E—4 and 0.3 in this study；Ωsis the spindle speed（revolution per min⁃ute），ftsthe feed rate per tooth（millimeters per tooth），Zthe number of cutter teeth；andTtcthe tool changing time（minutes per tooth）required for each edge and taken as 1.5 in this study；triandtsare the tool life［11］in rough and finish pass，respec⁃tively，which can be calculated by

whereCv，Kv，xv，yv，sv，qv，pvandlare constants and exponents associated with the tool and workpiece material.

2.2 Constraint functions

To ensure the safety of machining process and the quality of product，including spindle speed，feed rate，axial cutting depth，radial cutting depth，ma⁃chining force，machining torque，machining power，surface roughness and tool life must be selected within a predetermined interval. The description of the above constraints can be found［11］，and it will not be repeated in this paper. In addition to the above constraints，the optimization model construct⁃ed in this study also contains three-dimensional ma⁃chining stability region constraint on spindle speed，axial cutting depth and radial cutting depth，which is described in the previous section.

2.3 Optimization model

Objective function

Constraint functions

whereFri，FsandFmaxare the machining forces of theith rough pass，finish pass and available maximum value，respectively；Tmri，TmsandTmmaxare the ma⁃chining torques of theith rough pass，finish pass，and available maximum value，respectively；pri，psandpmaxthe machining powers of theith rough pass，finish pass，and available maximum value，respec⁃tively；Rri，Rs，RrmaxandRsmaxthe surface roughnesses of theith rough pass，finish pass，required value in rough pass and finish pass，respectively；tri，tsandTrthe tool lifes of theith rough pass，finish pass and required value，respectively；anddtdrianddsthe to⁃tal axial cutting depth，axial cutting depths of theith rough pass and finish pass，respectively.

Decision variables：n，dri，ari，Ωri，fri，ds，as，Ωs，fs，wherenis the number of rough passes；ariandasare the radial cutting depths of theith rough pass and finish pass，respectively，ΩriandΩsthe spindle speeds of theith rough pass and finish pass，respec⁃tively，andfriandfsthe feed rates of theith rough pass and finish pass，respectively.

Before searching for optimal values of decision parameters and minimization production time，the first issue is to obtain the optimal number of passes and the optimal distribution of total stock. Thus，a methodology is implemented to accomplish this task.

If the difference between total axial cutting depth and axial cutting depth of finish pass is divisi⁃ble by axial cutting depth of rough pass，nequals the corresponding integer quotient；Otherwise，rounds the quotient to the nearest integer in the direction of positive infinity asn，and the axial cutting depth of the last rough machining is equal to the total cutting depth minus the depth of the finishing and the depth of the first(n-1) rough passes.

3 Solution Method and Its Perfor‑mance Evaluation

3.1 Solution method

Fig.1 is an overall flowchart of the proposed CDPSO algorithm for parameter optimization.First，initialize personal best solution（pbest）of each particle and global best solution（gbest）of the whole swarm. Second，perform cloud mutation op⁃erator and two-point crossover operator on particles to generate new particles，and update swarm based on PSO and niching-gene-algorithm-based tourna⁃ment selection （NGATS）［12］strategy. Finally，perform simplex crossover（SPX）［13］operator on particles to generate new particles，and update swarm based on dominant particle replacement mechanism.

Fig.1 Framework of the proposed CDPSO algorithm

The proposed CDPSO algorithm is based on an improved version of PSO，which can be de⁃scribed as

whereis the gbest of the whole swarm in thekth iteration，the pbest of theith particle in thekth it⁃eration，and rand() the random numbers uniformly distributed between 0 and 1.wdenotes inertia weight factor，and decreases gradually with the in⁃crease of the square of iteration numbers. IterNum is the current number of iteration and IterNumMax the maximum number of iterations. In this study，wmax=0.9，wmin=0.4.

3.1.1 Constraints handing

Since the optimization of machining parameters involves lots of constraints while PSO is not a con⁃strained optimization algorithm，a simple and effi⁃cient constraint processing scheme is implemented in this study，that is，converting multi-constraint and single-objective optimization problem into bi-ob⁃jective optimization problem to minimize the initial objectivef(pi) and the degree of constraint viola⁃tionG(pi) simultaneously. For the sake of clarity，letf(pi) =(f(pi)，G(pi)). If there only exist in⁃equality constraints，G(pi)can be expressed as

wherecis the total number of constraints in a speci⁃fied problem andgj(pi) thejth constraint. Hence，is considered as the global optimal（minimum）solution if and only ifsuch thatAdditionally，if there exist equality constraints，convert them into inequality constraints as|h(x)|-δ≤0，whereδ=1.0E-4.

3.1.2 Genetic manipulation

The two-point crossover operator and SPX op⁃erator are utilized to process particles in global search and local search，respectively，to increase the diver⁃sity of swarm.Besides，a cloud mutation operation is

proposed to process the pbestand the gbestin this study，because it can unify the fuzziness and ran⁃domness，and can transform between qualitative con⁃cepts and quantitative data，thus，the cloud mutation operation can efficiently improve the uncertainty of evolution.The detail is expressed as

an d（thelth gene locus of）are reined by cloud drop generated by normal cloud model de⁃scribed in Eqs.（28—30）. En and He indicate entro⁃py and hyper entropy，respectively. They decrease with the increase of iteration number as described in Eqs.（26—27），leading to the decrease of En'.Thus，the explorative ability is maintained at a high level in the early stage and kept a global conver⁃gence in the later. The operation object of mutation onis the whole chromosome and that onis the part of gene locus，as shown in Eqs.（29—31）. Tak⁃ing thelth gene locus as an example，the probability of mutation is，wherenis the total number of design variables. In this study，Enmax=5.0E-3，Enmin=5.0E-7， Hemax=3.0E-3， Hemin=3.0E-7，and the number of generated cloud drop in each iteration is 200.

3.1.3 Evolutionary strategy

In addition to the particle swarm evolution strategy，we use NGATS as a global updating strategy to maintain the balance between selection pressure and diversity of the swarm. However，when solving the problem of low proportion of feasi⁃ble solutions，such as machining parameter optimi⁃zation，NGATS always leads to the problem of slow convergence. To handle this situation，a local search model based on clustering partition mecha⁃nism is introduced into the current algorithm. The schematic diagram of this mechanism is shown in Fig.2.

Fig.2 Schematic diagram of the local search model

Under this mechanism，the swarm of size is di⁃vided intondisjoint sub-swarms based on their loca⁃tions. Then，particles in each sub-swarm of sizeMare used to generate the same number of offspring of sizeM' through genetic manipulation. Appropriate evolutionary mechanisms are also introduced to guide the direction of evolution. Besides，only domi⁃nant particles are taken account in this strategy for they represent the most important characteristics of the swarm. Hence，after dominant particles are se⁃lected from the swarm，they are used to replace par⁃ticles dominated. The illustration of the updating strategy is as follows.

（1）Initialize the size of sub-swarmM.

（3）While the maximum number of subswarms has not been reached：

Do

① FindMparticles closest to the reference point to form a sub-swarm.

②Generate offspring through SPX.

③ Select dominant particles(i=1，…，m)from offspring swarm.

（a）Let selected offspring replace the particles in parent particles dominated by them.

（b） Put the evolved sub-swarm in the new swarm.

（c） If the maximum number of sub-swarms reaches，go to the next step；else number of subswarms plus one，operate Step 2 again.

（4）End while

（5）Output the local searched swarm.

3.2 Performance evaluation

3.2.1 Benchmark problems

To examine the performance of the proposed CDPSO algorithm，it is tested on 13 most common⁃ly used benchmark problems algorithm［14］. The test results of the proposed CDPSO algorithm are com⁃pared with four existing well-known algorithms to evaluate its performance. Before evaluation，the fol⁃lowing parameters are selected after a few numbers of trial：Acceleration coefficientsc1=1.0 andc2=0.5，the maximum and the minimum inertia weightwmax=0.9 andwmin=0.4，the probability of the crossover operation in the global search model is 0.8，the probability of mutation operation for the gbest and pbest is 1.0 and 0.2，respectively. In addi⁃tion，the probability of crossover operation in the lo⁃cal search model is 1.0. The maximum number of it⁃erations，IterNumMax，sizes of swarmNand subswarmqare listed in Table 1，which depends on the different characteristics of 13 benchmark problems，respectively. And then fitness function evaluations（FFEs）of these benchmark problems can be ob⁃tained. Besides，FFEs of the algorithms in the liter⁃atures are also listed in Table 1 to evaluate the com⁃putational cost of CDPSO algorithm.

Table 1 IterNumMax,N,q and FFEs of CDPSO and other algorithms

3.2.2 Evaluation results

The parameters set above are used for all 13benchmark problems. The comparision experiments of CDPSO algorithm on each benchmark problem are independently performed 30 runs，and the best，mean，the worst and standard deviations produced by CDPSO，SACABC，M-ABC，COMDE，and LCA algorithms are listed in Table 2.

The numerical results obtained by using CDP⁃SO algorithm and other four algorithms are analyzed according to their robustness，search quality and computational cost. As shown in Table 2，CDPSO algorithm is better or no worse than the other four al⁃gorithms in terms of the“best”“mean”and“worst”objective functions for all 13 benchmark problems，which means that CDPSO algorithm is better than the other four algorithms in search quality. In terms of standard deviation，since CDPSO algorithm is de⁃signed to reduce the calculation cost and improve the calculation efficiency，FFEs are set very small，so it is not common in most problems，but the order of magnitude is already small enough. Moreover，its standard deviation will be much smaller if FFEs are properly improved，thus the robustness is not poor.In terms of computational cost，it is measured by FFEs. CDPSO algorithm has the lowest cost for 10 benchmark problems，which is 2.8%—52% lower than the lowest cost algorithm in literatures. In par⁃ticular，in g03，the optimal solution and the mini⁃mum variance are obtained using only 48% of the computational cost of COMDE. Although in g02，g06 and g10，this method does not have a good ad⁃vantage over FEEs，which may be caused by strong suboptimal solution in the problem. However，the proposed algorithm achieves the best optimization effect in these three problems after slightly enlarging the FEE，which further illustrates the advantage of CDPSO algorithm in search capability.

Table 3 Machine parameters

Table 5 Workpiece parameters

Table 6 Modal parameters of machine tool‑tool system

Table 7 Value range of machining parameters

Table 8 Other parameters related to the optimization model

4 Experiment and Results

4.1 Experiment parameters

Before the experiment，machine parameters，tool parameters，workpiece parameters and other parameters related to optimization are shown in Ta⁃bles 3—8. Besides，the machining process is down milling，the total milling depthdtotoal=20 mm and the required surface roughnessRmax=6.4µm.

Material Cemented carbide Diameter/mm 10.0 Number of teeth 3 Nose radi⁃us / mm 0.2 Tool life /min 240

The 3-D SLD based on the above parameters is obtained as Fig.3. The accuracy of the machining stability prediction model based on 2ndFDM has been described［4］，and the 3-D machining stability model in this study is a 3-D extension based on the original model. Therefore，there is no need to con⁃duct redundant demonstration for its accuracy.

Fig.3 2ndFDM-based 3-D SLD

4.2 Experimental results

The milling experiments are conducted based on the empirical parameters and the parameters ob⁃tained through the optimization model of machining parameters proposed in this study. The correspond⁃ing production time is shown in Table 9.

Table 9 Comparison of parameter optimization results of multi‑pass milling

According to the optimization results，the opti⁃mal parameters are obtained at 100% radial cutting depth （slotting），and the corresponding stable boundary is shown in Fig.4. In the figure，the three points are machining parameters corresponding to rough machining，semi-finishing and finishing，re⁃spectively，which can verify that the three sets of pa⁃rameters meet the stability constraint of milling.

Fig.4 2ndFDM-based 2-D SLD

As shown in Table 9，the empirical process pa⁃rameters are too conservative，which cannot effec⁃tively give play to the performance of the machine tool，and result in longer milling time. However，on the premise of guaranteeing the machining stability and workpiece surface quality，the optimal parame⁃ters effectively generate the performance of the ma⁃chine tool，which greatly reduce the production time，effectively improve the processing efficiency，and further verify the practicability and effectiveness of the proposed optimization model and algorithm.

5 Conclusions

Optimization of machining parameters is of great importance for multi-pass end milling opera⁃tions to shorthen production time and improve effi⁃ciency. However，there are few studies to introduce chatter stability constraint into optimization model.Moreover，nearly all of chatter stability models con⁃struct the stable domain about spindle speed and axi⁃al cutting depth only，without consideration of radial cutting depth. To solve the aforementioned prob⁃lems，the main contributions of this study are as fol⁃lows.

（1）A 2ndFDM-based 3-D stability prediction model is developed for simultaneously optimization of spindle speed，axial cutting depth and radial cut⁃ting depth.

（2）A parameter optimization model of multipass end milling is developed by taking number of passes，spindle speed，axial cutting depth，radial cutting depth and feed rate as design parameters to achieve the minimum production time while consid⁃ering a large number of constraints including 3-D stability.

（3）An algorithm named CDPSO is proposed to solve the developed parameter optimization mod⁃el and the evaluation results indicate that it has a cer⁃tain advantage in the computational cost as well as comparable search quality and robustness.

（4）A demonstrative example indicates that the developed parameter optimization model and algo⁃rithm are indeed practical.