Predictive Modeling and Parameter Optimization of Cutting Forces During Orbital Drilling＊

Shan Yicai（單以才），Li Liang（李亮），He Ning（何寧），Qin Xiaojie（秦曉杰），Zhang Ting（章婷）

1.Nanjing College of Information Technology，Nanjing，210046，P.R.China；2.College of Mechanical and Electrical Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing，210016，P.R.China；3.School of Mechanical Engineering，Nanjing Institute of Technology，Nanjing，211167，P.R.China

1 Introduction

With the widely use of titanium alloy，carbon fiber reinforced plastics（CFRP）and aluminum alloy in aviation parts，highly efficient and precise hole－making technology has become a research hotspot［1－2］.As a new hole－making technology，orbital drilling has some advantages including small axial force，minor tool wear，convenient chip removal，high dimension accuracy，etc.And compared with conventional drilling，it can process various holes with one tool.Therefore，orbital drilling has a broad application prospect in aviation assembly［3－5］.

As radial and axial cutting edges both participate in cutting，it is more difficult to control cutting forces in orbital drilling than in conventional drilling.Research on cutting forces and parameter optimization of orbital drilling has great values on machining accuracy enhancement and processing quality improvement.By now，many scholars have conducted various experimental and theoretical researches on cutting forces of orbital drilling.

On the aspect of theoretical modeling，Bayoumi，et al.analyzed the relation between cutting parameters and tool rotation angle，then established an analytic model of cutting forces.Further，the effect of radial and axial engagements and cutting speed on the cutting parameters was investigated［6］.Qin Xuda，et al.simplified the influence of cutting forces on axial cutting edge into a z－direction single force，then presented a novel analytical model of cutting force［7］.Under certain time domain，Liu Changyi established cutting force models with inputs as helical feeding，spindle rotation speed，axial and radial cutting depth and tool geometric parameters［8］.As orbital drilling process is effected by various factors（impact load，tool wear，chip deformation，etc.），large deviation is inevitable between theoretical and measured cutting forces.

On the aspect of experimental studies，Ni Wangyang analyzed orbital drilling kinetics，and investigated the influence of revolution speed，axial feeding rate and revolution radius on cutting forces［9］.Iyer et al.carried out contrast experiments between orbital drilling and conventional drilling on die steel AISI D2.The research confirmed the superiority of orbital drilling in the aspects of cutting forces，hole quality and tool wear，ect［10］.Denkena，et al.analyzed the impact of axial and tangential feed on milling forces through undeformed chip model［11］.Hiroyuki Sasahara studied the effect of minimum quantity lubrication on chip temperature，chip deformation，cutting temperature，cutting forces and shape error，and finally pointed out that the cutting force of orbital drilling was smaller than that of conventional drilling［12］.Wang Ben pointed out that there was little difference between x－direction cutting force in orbital drilling and that in conventional drilling，while z－direction cutting force in orbital drilling was smaller than that in conventional drilling［13］.Though the influence of cutting parameters on cutting forces has been analyzed，prediction model of cutting force has not been established in above references.

In China，Yuan Zhixing built up cutting force models of x－and z－direction through orbital drilling in die steel.These models neglected the effect of revolution radius on cutting forces［14］.Wang Haiyan，et al.built a multi－objective optimization model with two objectives including tool life and material removal rate.Pareto genetic algorithm was introduced to optimize cutting parameters［15］.

If conventional cutting parameters are used as inputs in establishing cutting force models of orbital drilling，forces of axial and radial edges need to be built respectively.We attempt to avoid the complexity of modeling with two edges，and chose four cutting control parameters as inputs.To realize predicting and controlling cutting forces during orbital drilling，we focus on establishing models of cutting forces when machining different holes with one tool，and optimizing cutting control parameters.

2 Kinematics Analysis of Orbital Drilling

Fig.1shows the principle of orbital drilling［16］.During orbital drilling，tool practises three movements simultaneously（rotation around own axis，feed along axial direction，and revolution around hole axis）.Several control parameters should be considered as the following：

Hole diameter：D

Tool diameter：d

Axial feeding pitch：P

Tool rotation speed：ns

Tool revolution speed：nω

Tool revolution radius：e

Fig.1 Schematic diagram of orbital drilling

Axial feeding pitch Prepresents the axial feeding after tool finishes revolution for a round.Hence，tool axial feed speed facan be expressed as

Tool revolution radius eis the radial offset of tool axis from hole axis.In revolution，tangential feed speed ftof tool center point（TCP）is

As tool screw feed is composed of axial and tangential feeds，screw feed speed fCof TCP is

If tool tooth number is Z，the TCP cutting amount per toothδCcan be described as

where fzaand fztare tool feeding per tooth in axial and tangential directions.

During orbital drilling，tool tangential cutting amount per toothδPis different fromδC.It is expressed as

As we all know，the cutting amount per tooth is directly related to cutting forces.The more cutting amount per tooth is，the bigger the resistance to overcome in chip formation is.Thus，Macro milling force is increased.From Eqs.（1－5），we know that cutting forces of orbital drilling can be determined by control parameters as ns，nω，P，e，d，Dand Z.As for a special tool，the influence of tool geometric parameters on cutting forces can be eliminated.

3 Experimental Design

3.1 Experimental conditions

Orbital drilling experiment is conducted on Mikron UCP710high speed machining center，as shown in Fig.2.In the center，spindle power is 16kW and spindle speed is 100—18 000r／min.The workpiece material is aluminum alloy 6061.The workpiece size is 200mm×100mm×15mm.The tool is H602411－10from German Walter company，which has 45°helix angle and two teeth.Dry cutting and inverse milling are adopted in the experiment.Cutting force measurement system is composed of Kistler 9265Bdynamic force measurement instrument，Kistler 5019 charge amplifier，and computer data acquisition subsystem.

Fig.2 Experimental setup for milling forces measurement

3.2 Experimental scheme

Under specific machine tool and cutting tool，the main parameters affecting cutting forces are tool rotation speed，axial feeding pitch，tool revolution speed and tool revolution radius.For different materials and workpieces，a lot of experiments should be done for analyzing the influence of cutting parameters on cutting forces.To reduce experimental cost，orthogonal experiment method is utilized in experimental design［17］.In actual production，variation of nsleads to different axial cutting depth.Oversized revolution radius degrades machinability of orbital drilling.Hence，there is no need to use too many levels in tool axial feeding pitch and tool revolution radius.With reference to the different hole diameters and the hole processing security in aviation components，experimental factors and levels are designed as Table 1.

Table 1 Cutting control parameters and levels

4 Model Establishment and Accuracy Analysis

4.1 Characteristic analysis on cutting forces in orbital drilling

As influenced by radial and axial cutting edges in processing，the cutting force of orbital drilling is obviously different from that in conven－tional drilling［18］.Fig.3（a）shows the waveform graph of milling force in x－direction，which is composed by one low frequency signal and one high frequency signal.The low frequency waveform is caused by tool revolution，while the high frequency waveform is mainly determined by tool rotation.The high frequency is also influenced by high speed vibration and various disturbances.Two types of waveform graph are shown in Figs.3（b，c）.

Fig.3 Waveform graph of cutting force in x－direction

Cutting force measurement system can detect three milling forces as Fx，F(xiàn)yand Fzsimultaneously.In workpiece reference coordinate system，the three forces compose the cutting resultant force of orbital drilling.If tool coordinate system is set as a reference，the cutting resultant force can also be decomposed into tangential force Ft，radial force Fr，and axial force Fa.Here，F(xiàn)ris the main factor affecting hole dimensional error and shape precision.Facauses quality defects as burr of metal material and delamination of CFRP.Ftis mainly used to calculate consumed power.Therefore，the research on cutting force in tool coordinate system is significant for improving machining accuracy［19］.According to Fig.2，cutting component forces in tool coordinate system can be calculated as

To avoid calculation error caused by low frequency signal，we utilize high frequency waveform to calculate measured value of cutting force.Under signal stable stage，we select 100adjacent peaks and troughs.A bigger peak and a bigger trough are selected respectively from each four adjacent peaks and troughs.Then，the average values of peak and trough are calculated respectively.The bigger absolute average value is regarded as measured value of cutting force.We attempt to study axial and radial forces for further improving hole quality by controlling cutting forces and optimizing cutting parameters in orbital drilling.In experiment，various combinations of four cutting control parameters are designed and the values of Frand Faare measured.The values are shown as Table 2.

Table 2 Cutting control parameters and cutting forces in experiment

4.2 Establishment of prediction model

According to Table 2，models of Frand Faare built and the influence of cutting control parameters on cutting forces is studied.Since there is different dimension and boundary in cutting control parameters，the four parameters need to be normalized，where linear function conversion is used as

where xreflects the original value，ythe converted value，xminand xmaxare the minimum and the maximum of x.

After normalization process，the data conversion is shown as Table 3.

After data normalization，prediction models of Frand Faare established by four cutting control parameters.To overcome the shortcoming of direct－vision method in distinguishing conditional error from accidental error，variance analysis isintroduced in the research.The established models are described as Eqs.（8a，8b）.

Table 3 Results of data normalization

Next，reverse normalized processing is made by Eqs.（8a，8b），the relationships are calculated between ns，nω，P，e，and Fa（or Fr），multiple quadratic regression models are gained as follows Fa＝－186＋6.98＊10－2＊x1＋3.99＊x2＋

where the four input values as x1，x2，x3and x4correspond to ns，nω，Pand e，respectively.According to the four input values in Table 2，fitted values of Frand Facan be worked out through Eqs.（9a，9b）.

The point is that（9a，9b）can only be used when parameters meet certain demands

4.3 Accuracy analysis of prediction models

In variance analysis，F(xiàn)criterion is introduced in significance testing.The results are shown in Table 4.

Table 4 Adequacy of prediction models

In Table 4，DFis a degree of freedom，SSis sum of squares，MS is mean square，and R2is determination coefficient.

Under given significant levelα，regression model is credible if F＜F1－0.05（p，n－p－1）.From Table 4，the Fvalues of the two models are both less than F1－0.05.Eqs.（8a，9a）have high significance whenα＝0.05.R2of Frand Famodels are 0.996and 0.989.The adjusted R2are 0.983and 0.957.These indicate that the predicted values of Frand Faare consistent with experimental values（as shown in Fig.4）.The analysis shows that the models are fit for predicting cutting forc－es of orbital drilling in aluminium alloy 6061.

Fig.4 Comparison of predictive value with test value

5 Effect of Cutting Control Parameters on Cutting Forces

In regression model，when three input parameters are fixed at a certain level，the model will become a single－factor model.When Pand e are both set as 1mm，the relationship curves between Fa（or Fr）and nωare shown in Fig.5，where nωis set as 15.92，23.87，31.82r／min respectively.With the increase of ns，F(xiàn)aand Frincrease at first but decrease finally.In the lowspeed stage，tool tooth friction is predominant.The increase of nscauses the increase of Faand Fr.In high－speed stage，fzaand fztdecrease with the increase of ns.Consequently，F(xiàn)aand Frdecrease accordingly.In this stage，tool feeding becomes the main factor to determine milling force.Fig.5（a）indicates that Faincreases with the increase of nω.Fig.5（b）gives the relationship between nωand Fr.Obviously，the three curves are similar under three tool revolution speeds.

When Pand e are both 1mm and nsis set as 3 000，4 500，6 000r／min，respectively，the relationship curves of Fa（or Fr）and nωare shown in Fig.6.Faand Frincrease with the increase of nω.As can be seen from Eq.（4），the increase of nωenlarges the values of fzaand fzt.Fig.6（a）shows that the influence of nωon Fais obvious when nsis low.But the influence becomes small with nsincreasing.In Fig.6（b），the increase of nωhas little influence on Fr.

Fig.5 Effect of nson Faand Fr

Fig.6 Effect of nω ＝15.92r／min on Faand Fr

Fig.7 Effect of Pon Faand Fr

When nωis 15.92r／min，eis 1mm，and nsis 3 000，4 500，6 000r／min，the relation curves of Fa（or Fr）and Pare shown in Fig.7.As for radial cutting edge，axial cutting depth fzaincreases with P.And，fztof axial cutting edge also enlarges.Consequently，F(xiàn)aincreases significantly under the special ns，which can be seen from Fig.7（a）.In low－stage，F(xiàn)aincreases with the increase of ns.But，F(xiàn)adecreases with the sharp increase of ns.In lownsof Fig.7（b），F(xiàn)rdecreases with the increase of P.The reason is that under selected tool and certain revolution radius，the increase of Pamplifies cutting effect similar to conventional drilling.But Frincreases under high rotation speed.The reason is that the sharp increase of nsproduces cutting effect more similar to milling.In this case，nshas greater influence on Frthan P.

We set the following parameters：P＝1mm，ns＝3 000，4 500，6 000r／min.In Fig.8（a），the increase ofe enlarges fztunder the special ns，and further increases Fa.In lowns，the fluence of the tool－workpiece friction on cutting forces is predominant in orbital drilling，which leads to the increase of Fa.In high ns，the sharp decrease of fztand fzacauses the decrease of Fa.Under lowns，the increase of e reduces Faas shown in Fig.8（b）.The reason is that the tool－workpiece space for chip removal is increased and the materials removal cut by radial cutting edge is multiplied.While in high ns，tool dynamic characteristic exerts greater influence on cutting forces，where Frincreases accordingly.

Fig.8 Effect of e on Faand Fr

6 Optimization of Cutting Control Parameters

To process a?13hole in aluminum alloy 6061，we choose tool H602411－10.If P＝1mm，response surfaces of Faand Frbased on prediction models are shown in Fig.9.

To achieve the minimum value of Fa（or Fr），we need to optimize combination of ns，nω，and P.The optimization method should satisfy the given objective function

where i＝a，r（Note：a means axial，r means radial）.

To achieve machining efficiency，the processing time of the hole is set as 30s.The constraints of cutting control parameters are

As the objective function is multivariate nonlinear equation，it is uneasy to find a combination of optimal cutting control parameters through analytic method or experimental method.To solve the problem，we combined penalty function meth－od with prediction models to optimize cutting control parameters.

Fig.9 Response surface of milling forces based on prediction model

After multiple iterative solutions of objective functions，two optimum combinations are worked out as

Under two combinations as x＊aand x＊r，theoretical and experimental values of Faand Frare obtained.The results are：

Theoretical values：Fa＝103.99N，F(xiàn)r＝83.73N.

Experimental values：Fa＝108.68N，F(xiàn)r＝85.49N.

The consistency between theoretical and experimental values indicates that the optimization method is feasible and practicable in cutting control parameter optimization of orbital drilling.In practical application，the optimization of Frcan be used to improve hole dimensional error and shape precision.If burr at hole export needs to be strictly controlled in special occasion，the optimization of Facan be selected.

7 Conclusions

（1）Upon analyzing orbital drilling kinematics，main cutting control parameters are determined as ns，nω，Pand e，which impose great effect on cutting forces when processing with one selected tool.Nonlinear regression analysis is introduced in prediction models of Faand Fr，in which four milling control parameters are used as inputs.By Fcriterion，the Fvalues are 81.16 and 31.58，which show good significance of two cutting force models.Meanwhile，the determination coefficients R2of 0.996and 0.989prove the high fitting precision of the models.

（2）The influence of cutting control forces on Faand Frhas been studied based upon prediction models.With the increase of ns，F(xiàn)aand Frincrease firstly and then decrease.The increase of nωcan increase Faand Fr.When Pand e both increase，F(xiàn)aincreases accordingly，while Frincreases first and then decreases.

（3）Under selected scope of cutting control parameters，Matlab software is used to optimize cutting control parameters with the optimization goals as minimum Faand Fr.Comparison between theoretical analysis and verification experiment shows that the prediction errors of Faand Frare both less than 5%，which proves the validity of the cutting force model during orbital drilling.But how to realize the collaborative optimization of Faand Frwill be our major research direction in future.

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