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Robust control of regenerative chatter in uncertain milling process with weak nonlinear cutting forces: A comparison with linear model
9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control Available online at www.sciencedirect.com 9th IFAC Modelling, Management and 9th IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control Berlin, Germany, August 28-30, 2019 Control Control Control Berlin, Germany, August 28-30, 2019 Berlin, Berlin, Germany, Germany, August August 28-30, 28-30, 2019 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 1102–1107
Robust control of regenerative chatter in uncertain milling process with weak Robust control of regenerative chatter in uncertain milling process with weak Robust of regenerative chatter in uncertain milling weak nonlinear cutting forces: A comparison with linearprocess model with Robust control control of regenerative chatter in uncertain milling process with weak nonlinear cutting forces: A comparison with linear model nonlinear cutting forces: A comparison with linear model nonlinear cutting forces: A comparison with linear model Hamed Moradi*, Ali Nouriani,
Hamed Moradi*, Ali Nouriani, Gholamreza Hamed Moradi*, Ali Hamed Moradi*,Vossoughi Ali Nouriani, Nouriani, Hamed Moradi*, Ali Nouriani, Gholamreza Vossoughi Gholamreza Vossoughi Gholamreza Vossoughi Gholamreza Vossoughi * Centre of Excellence in Design, Robotics & Automation, School of Mechanical Engineering, Sharif University of Technology, * Centre CentreIran, of Excellence Excellence in21Design, Design, Robotics & Automation, Automation, School School of Mechanical Mechanical Engineering, Sharif Sharif University University of of Technology, Technology, Tehran, (Tel: +98 66165545, e-mail:[email protected]; [email protected]) * of in Robotics & Engineering, * Centre Centre of of Excellence Excellence in in Design, Design, Robotics Robotics & & Automation, Automation, School School of of Mechanical Mechanical Engineering, Sharif Sharif University University of of Technology, Technology, *Tehran, of Engineering, Iran, (Tel: +98 21 66165545, e-mail:[email protected]; [email protected]) Tehran, Tehran, Iran, Iran, (Tel: (Tel: +98 +98 21 21 66165545, 66165545, e-mail:[email protected]; e-mail:[email protected]; [email protected]) [email protected]) Tehran, Iran, (Tel: +98 21 66165545, e-mail:[email protected]; [email protected]) Abstract: For various types of materials, milling process is extensively used to generate complex shapes Abstract: For various various types of materials, milling process is extensively extensively used to to generate generate complex shapes with high quality. During theof process and to achieve high removal rate, precision and bettercomplex surface shapes finish, Abstract: For types materials, milling process is used Abstract: For various various types of materials, milling process is extensively extensively used to to generate generate complex shapes Abstract: For types of materials, milling process is used complex shapes with high quality. During the process and to achieve high removal rate, precision and better surface finish, chatter suppression is of great importance. An extended model of the milling process isbetter presented in finish, which with high quality. During the process and to achieve high removal rate, precision and surface with high quality. During the process and to achieve high removal rate, precision and better surface finish, with high quality. During the process and toAn achieve high removal rate, precision andisbetter surface finish, chatter suppression is of great importance. extended model of the milling process presented in which the cutting forces are described as a third-order nonlinear function of chip thickness. Uncertainties chatter suppression is of great importance. An extended model of the milling process is presented in chatter suppression is of great importance. An extended model of the milling process is presented in which chatter suppression is ofdescribed great importance. An extended model of the milling process is presented in which which the cutting forces are as a third-order nonlinear function of chip thickness. Uncertainties associated the are process and tool arenonlinear also included to achieve a more realistic model. To the forces described as aaa third-order function of thickness. Uncertainties the cutting cuttingwith forces are described as parameters third-order nonlinear function of chip chip thickness. Uncertainties the cutting forces described as third-order function of chip thickness. Uncertainties associated with the are process andantool tool parameters arenonlinear also included to achieve achieve a more more realistic model. To robust control is designed based on µ-synthesis with DK-iteration suppress regenerative chatter, H∞ parameters associated with the process and are also included to a realistic model. associated with the process and tool parameters are also included to achieve a more realistic model. To To associated with the process andantool parameters are also included to achieve a more realistic model. To robust control is designed based on µ-synthesis with DK-iteration suppress regenerative chatter, H ∞ the robust stability and performance of both linear and nonlinear algorithm.regenerative The controller guarantees control is designed based on µ-synthesis with DK-iteration suppress chatter, an H ∞ robust robust control is designed based on µ-synthesis with DK-iteration suppress regenerative chatter, an H ∞ control is designed based on µ-synthesis withand DK-iteration suppress regenerative chatter, an H∞ robust algorithm. The controller the robust performance both linear nonlinear models in the presence ofguarantees uncertainties. In bothstability models, and as the amount ofof uncertainty increases, more algorithm. The controller the robust stability and performance of both and algorithm. The controller guarantees the robust stability and performance of both linear and nonlinear algorithm. Thepresence controllerofguarantees guarantees the In robust stability and performance ofuncertainty both linear linearincreases, and nonlinear nonlinear models in the uncertainties. both models, as the amount of more actuator efforts are required to suppress the chatter. The linear model needs less actuation and time to models in the presence of uncertainties. In both models, as the amount of uncertainty increases, models in in the the presence presence of of uncertainties. uncertainties. In In both both models, models, as as the the amount amount of of uncertainty uncertainty increases, increases, more more models more actuator efforts are required to suppress the chatter. The linear model needs less actuation and time to suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed. actuator efforts are required to suppress the chatter. The linear model needs less actuation and time actuator efforts efforts are are required required to to suppress suppress the the chatter. chatter. The The linear linear model model needs needs less less actuation actuation and and time time to to actuator to suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed. suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed. suppress the However, as effects are must re-designed. © 2019, IFAC (International Federation of Automatic Hostingthe bycontroller Elsevier Ltd. Allbe reserved. suppress the chatter. chatter. However, as the the nonlinear nonlinear effectsControl) are increased, increased, the controller must berights re-designed. © 2019, IFAC (International (InternationalFederation Federationofof ofAutomatic AutomaticControl) Control)Hosting Hosting by Elsevier Ltd. All rights reserved. © 2019, 2019, IFAC byby Elsevier Ltd. AllAll rights reserved. © 2019, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. rights reserved. © (International Federation of Control) Elsevier Ltd. reserved. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. © 2019, IFAC IFAC (International Federation of Automatic Automatic Control) Hosting Hosting by Elsevier Ltd. All All rights rights reserved. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. control. The nonlinear models of the cutting process includes the 1. INTRODUCTION The nonlinear nonlinear modelsa combination of the cutting cutting process and includes the material nonlinearity, of material structural The models of process includes the 1. INTRODUCTION The nonlinear models of the process includes the The nonlinear modelsa combination of the the cutting cutting process and includes the 1. material nonlinearity, of material structural 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) nonlinearity, aaa combination of and material nonlinearity, combination of material and structural Self-excited vibrations or chatter is an undesirable material material nonlinearity, combination of material material and structural structural nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) Self-excited vibrations or chatter is an undesirable and high order nonlinear terms in cutting forces (Moradi, nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) phenomenon in machining processes which results in Self-excited vibrations or chatter an nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) Self-excited vibrations or chatter is an undesirable and high order nonlinear terms in cutting forces (Moradi, Self-excited vibrations or processes chatter is is which an undesirable undesirable phenomenon in machining results in 2012b). In some recent researches, the effects of tool wear and and nonlinear terms cutting (Moradi, and high high order nonlinear terms in cutting forces forces (Moradi, reduction of material removal rate (MRR), poor surface finish phenomenon in processes which results in highIn order order nonlinear terms in in forces (Moradi, phenomenon in machining processes which results in 2012b). some recent recent researches, thecutting effects of toolsources wear and phenomenon in machining machining processes whichsurface results in and reduction of material removal rate (MRR), poor finish process damping have been included beside the of 2012b). In some researches, the effects of tool wear and In some recent researches, the effects of tool wear and and decrease in tool life. Machine tool chatter is classified into 2012b). reduction of material removal rate (MRR), poor surface finish 2012b). In some recent researches, the effects of tool wear and reduction of material removal rate (MRR), poor surface finish process damping have been included beside the sources of reduction of material removal ratetool (MRR), poor surface finish and decrease in tool life. Machine chatter is classified into structural/cutting force nonlinearities, e.g., (Moradi, 2013a, process damping have been included beside the sources of damping have included the of regenerative and and tool life. chatter is into process damping force have been been included beside beside the sources sources of and decrease decrease in in toolnon-regenerative life. Machine Machine tool tool(mode chatter coupling) is classified classifiedtypes into process structural/cutting nonlinearities, e.g., (Moradi, 2013a, and decrease in tool life. Machine tool chatter is classified into regenerative and non-regenerative (mode coupling) types structural/cutting 2012b). Control strategies like adaptive control used to achieve force e.g., 2013a, structural/cutting force nonlinearities, nonlinearities, e.g., (Moradi, (Moradi, 2013a, (Koeinsberger and non-regenerative Tlusty, 1970). In (mode the former mechanism, regenerative coupling) types force nonlinearities, e.g., (Moradi, 2013a, regenerative and and non-regenerative (mode coupling) types structural/cutting 2012b). Control strategies like adaptive control usedtime-delay to achieve achieve regenerative and non-regenerative (mode coupling) types (Koeinsberger and Tlusty, 1970). In the former mechanism, constant cutting force and improved surface finish, 2012b). Control strategies like adaptive control used to 2012b). Control strategies like adaptive control used to achieve the cut produced at time leaves small waves in the material (Koeinsberger and Tlusty, 1970). In the former mechanism, 2012b). Control strategies like adaptive control used to achieve (Koeinsberger and and Tlusty, Tlusty, 1970). 1970). In In the former former mechanism, mechanism, constant cutting force and improved surface finish, time-delay (Koeinsberger the cut produced produced at time time leaves small smallthe waves in thetool. material feedback controlforce (Nayfeh, 2012) surface and optimal control of cutting and finish, constant cutting force and improved surface finish, time-delay regenerated during each subsequent pass of in thethe This constant the cut at leaves waves material constant cutting force and improved improved surface finish, time-delay time-delay the cut produced at time leaves small waves in the material feedback control (Nayfeh, 2012) and optimal control of of the cut produced at time leaves small waves in the material regenerated during each subsequent pass of the tool. This regenerative chatter (Moradi, 2013b) have been developed. In feedback control (Nayfeh, 2012) and optimal control feedback control (Nayfeh, 2012) and optimal control interaction leads to thesubsequent chip thickness variation and feedback regenerated during each pass of the tool. This control (Nayfeh, 2012) and optimal control of of regenerated during each subsequent pass of the tool. This regenerative chatter (Moradi, 2013b) have been developed. In regenerated during each subsequent pass of the tool. This interaction leads to the chip thickness variation and addition, piezoelectric active vibration control systems (Parus regenerative chatter (Moradi, 2013b) have been developed. In regenerative chatter (Moradi, 2013b) have been developed. In consequently cutting force variation and increase in vibration interaction leads to the chip variation and chatter (Moradi, 2013b) have beensystems developed. In interaction leads to the chip thickness variation and addition, piezoelectric active vibration control (Parus interaction leads toforce the variation chip thickness thickness variation and regenerative consequently cutting and increase in vibration et al., 2013), sliding mode control (Park et al., 1998) and addition, piezoelectric piezoelectric active active vibration vibration control control systems systems (Parus (Parus amplitude of the tool.force In the mode coupling mechanism, the addition, consequently cutting variation and in addition, piezoelectric active vibration control (Parus consequently cutting force variation and increase in vibration et al., al., 2013), 2013), sliding mode control (Park etHsystems al.,and 1998) and consequently cutting force variation and increase increase in vibration vibration adaptive amplitude of the tool. In the the mode coupling mechanism, the various robust sliding control mode approaches including et sliding mode control (Park et al., 1998) and ∞ et al., 2013), control (Park et al., 1998) and tool traces out an elliptic path that varies the depth of cut in amplitude of the tool. In mode coupling mechanism, the et al., 2013), sliding mode control (Park et al., 1998) and amplitude of the tool. In the mode coupling mechanism, the and adaptive various robust control approaches including H ∞ amplitude of the tool. In the mode coupling mechanism, (Carrillo et al., 1999) have been implemented either tool traces out an elliptic path that varies the depth of cut cutthe in techniques various robust robust control approaches including H∞∞ and and adaptive adaptive various control approaches including H such a wayout to an bolster the path coupled modes the of vibrations. tool elliptic that depth in various robust control approaches including H tool traces traces out an elliptic path that varies varies the depth of of cut cutThe in techniques ∞ and adaptive (Carrillo et al., 1999) have been implemented either tool traces out an elliptic path that varies the depth of in for chatter suppression or achievement of the constant cutting such a way to bolster the coupled modes of vibrations. The techniques (Carrillo et al., 1999) have been implemented techniques (Carrillo (Carrillo et et al., al., 1999) 1999) have have been been implemented implemented either either regenerative type is found to be themodes most of detrimental the techniques such bolster the vibrations. either such aaa way way to to bolster the coupled coupled modes of vibrations.to The The for chatter suppression or achievement of the constant cutting such way to bolster the coupled modes of vibrations. The forces. Also, control strategies based onof regenerativerate type is found found to be be the the most detrimental detrimental to to the the for suppression or the cutting for chatter chatter suppression or achievement achievement offuzzy-logic the constant constantmethods cutting production in most machining processes. regenerative type is to most for chatter suppression or achievement of the constant cutting regenerative type is found to be the most detrimental to the forces. Also, strategies based on fuzzy-logic methods regenerative type is found to be the most detrimental to the (Zuperl et al.,control 2012) are the other techniques used in this area. production rate rate in most machining processes. forces. Also, control strategies based on fuzzy-logic methods forces. control strategies based on methods production in machining processes. forces. Also, Also, control strategies based on fuzzy-logic fuzzy-logic methods production in most machining processes. et al., 2012) are the other techniques used in this area. In the earlyrate works of this area, modelling and analysis of the (Zuperl production rate in most most machining processes. (Zuperl et al., 2012) are the other techniques used in this area. (Zuperl et al., 2012) are the other techniques used in this area. processes high dimensional accuracy In the the early early works of ofand thisrelative area, modelling modelling and analysis analysis of the the (Zuperl etmilling al., 2012) are the require other techniques used in this area. complex geometry work-piece/tool motion in Precision In works this area, and of In the early works of this area, modelling and analysis of the Precision milling processes require high dimensional accuracy In the early works of this area, modelling and analysis of the in spite of the variations in cutting conditions such as spindle complex geometry and relative work-piece/tool motion in Precision milling processes require high dimensional accuracy Precision milling processes require high dimensional accuracy milling process wasand carried out. work-piece/tool Mechanistic approach was complex geometry relative motion in milling processes require high dimensional accuracy complex geometry and relative work-piece/tool motion in in spite of the variations in cutting such spindle complex geometry and relative work-piece/tool motionwas in Precision speed and depth of cut as well as conditions variations in theas dynamic milling process process was carried out. Mechanistic approach in spite of the variations in cutting conditions such as spindle in spite of the variations in cutting conditions such as spindle extensively used was for governing dynamic equations of various milling carried out. Mechanistic approach was in spite of the variations in cutting conditions such as spindle milling process was carried out. Mechanistic approach was speed and depth of cut as well as variations in the dynamic milling process was carried out. Mechanistic approach was properties of the machine tool system. However, most of the extensively used for governing dynamic equations of various speed and depth of cut as well as variations in the dynamic speed and depth of cut as well as variations in the dynamic milling processes and prediction of cutting forces, deflection extensively used for governing dynamic equations of various speed and depth of cut as well as variations in the dynamic extensively used for governing dynamic equations of various properties of the machine tool system. However, most of the extensively used for dynamic equations of various researches considered a relatively simplified model of the milling processes andgoverning prediction of cutting forces, deflection properties of the machine tool system. However, most of properties of the machine tool system. However, most of the of machine components and form errors (Budak, 2006). In this milling processes and prediction of cutting forces, deflection properties of the machine tool system. However, most of the milling processes and prediction of cutting forces, deflection researches considered a relatively simplified model of the milling processes and prediction of cutting forces, deflection process. Such models are commonly based on idealized of machine components and form errors (Budak, 2006). In this researches considered a relatively simplified model of the researches considered aa relatively simplified model of the approach, by using and experimental data, cutting force of machine components form errors (Budak, 2006). In this researches considered relatively simplified model of the of machine components and form errors (Budak, 2006). In this process. Such models are commonly based on idealized of machine components and form errors (Budak, 2006). Inforce this process. models of machining process and imperfections or approach, by using experimental data, cutting Such models are commonly based on idealized process. Such models are commonly based on idealized coefficients are calibrated for certain cutting conditions. After approach, by using experimental data, cutting force process. Such models are commonly based on idealized approach, by by using using experimental experimental data, data, cutting cutting force force models of machining process and imperfections or approach, uncertainties in the model or cutting conditions are not taken coefficients are calibrated for certain cutting conditions. After models of machining process and imperfections models of of machining machining process process and and imperfections imperfections or or determination the flankfor plowing and shear coefficients certain cutting conditions. After or coefficients are areofcalibrated calibrated for certainforces, cuttingfriction conditions. After models uncertainties in the model or cutting conditions are not taken coefficients are calibrated for certain cutting conditions. After into account. These sources of uncertainty alongside with the determination of the flank plowing forces, friction and shear uncertainties in the model or cutting conditions taken uncertainties in in the the model model or or cutting cutting conditions conditions are are not not taken angles from theof orthogonal cutting test and transforming them into determination the flank plowing forces, friction and shear uncertainties are not taken determination of the flank plowing forces, friction and shear account. These sources of uncertainty alongside with the determination the flank plowing forces, friction and shear effects of cutting forces nonlinearity and regenerative chatter; angles from theoforthogonal orthogonal cutting test and transforming them into into account. These sources of uncertainty alongside with the account. These sources of alongside the to the from oblique conditions, cutting forces are computed angles the cutting test and transforming them into account. These sources of uncertainty uncertainty alongside with with the angles from the orthogonal cutting test and transforming them effects of cutting forces nonlinearity and regenerative chatter; angles from the orthogonal cutting test and transforming them lead to inaccurate modelling. to the oblique conditions, cutting forces are computed effects of cutting forces nonlinearity and regenerative chatter; effects of cutting forces nonlinearity and regenerative chatter; (Armarego and Deshpande, 1991). Also, closedare formcomputed solution effects to the oblique conditions, cutting forces of cutting forces nonlinearity and regenerative chatter; to the oblique conditions, cutting forces are computed to inaccurate modelling. to the oblique conditions,1991). cutting forces are computed (Armarego and Deshpande, Also, closed form solution lead lead to inaccurate modelling. lead to inaccurate modelling. for the evaluation of linear (Bayoumi et al.closed 1994) form and nonlinear (Armarego and 1991). solution In this unlike the previous researches which discussed to paper, inaccurate modelling. (Armarego and Deshpande, 1991). Also, closed form solution (Armarego and Deshpande, Deshpande, 1991). Also, Also, closed form solution lead for the evaluation of linear (Bayoumi et al. 1994) and nonlinear In this paper, unlike the previous researches which models (Moradi, 2012a, 2013a) of cutting forces in end milling for the evaluation of linear (Bayoumi et al. 1994) and nonlinear either SDOF turning processes (Park et al., 1998) ordiscussed a linear In this paper, unlike the previous researches which discussed for the evaluation of linear (Bayoumi et al. 1994) and nonlinear In this paper, unlike previous researches which for the evaluation of linear (Bayoumi et al.forces 1994) in and nonlinear thisSDOF paper, turning unlike the the previous researches whichordiscussed discussed models (Moradi, 2012a, 2013a) of cutting cutting end milling In either processes (Park et al., 1998) a et linear process have been presented. models (Moradi, 2012a, 2013a) of forces in end milling model of cutting forces in milling process (Van Dijk al., either SDOF turning processes (Park et al., 1998) or linear models (Moradi, 2012a, 2013a) of cutting forces in end milling either SDOF turning processes (Park et al., 1998) or linear models 2013a) of cutting forces in end milling either SDOF turning processes (Park et al., (Van 1998)Dijk or aaa et linear process (Moradi, have been been2012a, presented. model of cutting forces in milling process al., process have presented. model of cutting forces in milling process (Van Dijk et process have have been presented. model cutting forces milling process al., process been presented. model of of cutting Ltd. forces in milling process (Van (Van Dijk Dijk et et al., al., 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier All in rights reserved.
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Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
2012), an extended model of the milling process is considered in which the cutting forces are described as a third-order nonlinear function of chip thickness, through coupled nonlinear delay differential equations. The proposed modelling is essential for the machining of materials in which the cutting forces are inherently nonlinear functions of chip thickness. Uncertainties associated with the machine-tools parameters and cutting parameters are also included to achieve a more realistic model. To suppress regenerative chatter, an H∞ robust control strategy is developed based on µ-synthesis and DK-iteration algorithm. The control inputs are the counterbalance forces in x-y directions, implemented by actuators (e.g., piezo-electric actuators). The robust performance and robust stability of the control system is investigated and compared for the linear and nonlinear models, in the presence of uncertainties. Moreover, the effect of uncertainty amount on the behaviour of tool tip vibrations and required actuator forces is studied. Also, it has been shown that as the nonlinear effects are strengthened, the robust controller must be re-designed.
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d𝐹𝐹𝑥𝑥,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = −d𝐹𝐹𝑡𝑡,𝑗𝑗 cos 𝜙𝜙𝑗𝑗 − d𝐹𝐹𝑟𝑟,𝑗𝑗 sin 𝜙𝜙𝑗𝑗 ; d𝐹𝐹𝑦𝑦,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = +d𝐹𝐹𝑡𝑡,𝑗𝑗 sin 𝜙𝜙𝑗𝑗 − d𝐹𝐹𝑟𝑟,𝑗𝑗 cos 𝜙𝜙𝑗𝑗
(1)
Using Eq. (1) and integrating analytically along the cut portion of the flute 𝑗𝑗, yields the closed form expressions for cutting forces. Aligning the flute 𝑗𝑗 = 0 at 𝜙𝜙 = 0 in the beginning of the algorithm, the cutting forces contributed by all flutes are calculated and integrated digitally to obtain the total instantaneous forces on the cutter at immersion 𝜙𝜙 as (Moradi, 2013b): 𝐹𝐹𝑥𝑥 (𝜙𝜙) = ∑𝑁𝑁−1 𝑗𝑗=0 𝐹𝐹𝑥𝑥𝑥𝑥 ;
𝐹𝐹𝑦𝑦 (𝜙𝜙) = ∑𝑁𝑁−1 𝑗𝑗=0 𝐹𝐹𝑦𝑦𝑦𝑦
(2)
2.2 Nonlinear Modelling of Cutting Forces with Regenerative Chatter Effect Since stability theories cannot predict interesting phenomena in milling process with linear dynamics, nonlinear modelling of milling process is of great importance. In this paper, to achieve a more realistic model, cutting forces are expressed as a complete third-order polynomial function of chip thickness as: 𝑑𝑑𝑑𝑑𝑡𝑡,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [𝜁𝜁1 ℎ𝑗𝑗 3 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜁𝜁2 ℎ𝑗𝑗 2 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) +
𝜁𝜁3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜁𝜁4 ] 𝑑𝑑𝑑𝑑
(3-1)
𝑑𝑑𝑑𝑑𝑟𝑟,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [𝜂𝜂1 ℎ𝑗𝑗 3 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜂𝜂2 ℎ𝑗𝑗 2 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) +
𝜂𝜂3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜂𝜂4 ] 𝑑𝑑𝑑𝑑
Fig. 1. (a) Dynamics of the milling process (b) dynamic chip thickness in regenerative chatter mechanism Unlike the previous works, a nonlinear model of cutting forces in milling process and its equivalent linear model are presented. Nonlinear dynamic modelling is essential for milling processes in which cutting forces are inherently nonlinear functions of chip thickness. On the other hand, linear modelling of the problem makes the design of robust control strategy to be more convenient. Therefore, after design of control strategy for the linear model, it is implemented on the more realistic nonlinear model. Results approve that the designed controller has an appropriate robust performance for both of the linear and nonlinear models.
(3-2)
where cutting force coefficients ζi , ηi , i = 1. .4 are found directly from experimental force signals, as presented in (Moradi, 2013a). In the presence of regenerative chatter, the chip thickness is expressed as: h (ϕj ) = [cf sinϕj + vj,0 − vj ] g (ϕj )
(4)
ϕst < ϕj < ϕex ϕst > ϕj or ϕex < ϕj
(5)
As presented by (Moradi, 2013b), g(ϕj ) is a unit step function determining whether the tooth is in or out of cut and is described in terms of start (ϕst ) and exit immersion (ϕex ) angles of the cutter as: g(ϕj ) = {
1 0
Since static part of the chip thickness has no effect on regeneration mechanism, and according to coordinates shown in Fig. 1, Eq. (4) is reduced to: h (ϕj ) = [Δx sinϕj + Δy cosϕj ] g (ϕj )
2. LINEAR/NONLINEAR MODELLING OF THE CUTTING FORCES IN MILLING PROCESS
(6)
where 𝛥𝛥𝛥𝛥 = 𝑥𝑥(𝑡𝑡) − 𝑥𝑥(𝑡𝑡 − 𝜏𝜏); 𝛥𝛥𝛥𝛥 = 𝑦𝑦(𝑡𝑡) − 𝑦𝑦(𝑡𝑡 − 𝜏𝜏);
2.1 Calculation of Cutting Forces in Classical Approach Dynamics of the milling process as a 2DOF model and the variation of chip thickness is shown in Fig. 1. The immersion angle is measured clockwise from the y-axis and the axial (𝑎𝑎) and radial (𝑤𝑤) depths of cut are constant. Assuming the bottom end of one flute as the reference immersion angle 𝜙𝜙. According to coordinates in Fig. 1, elemental forces in feed (x) and normal (y) directions are expressed in terms of tangential and radial forces as (Moradi, 2013b):
𝜏𝜏 = 2𝜋𝜋⁄(𝑁𝑁 𝛺𝛺)
(7)
[x(t), y(t)] and [x(t − τ), y(t − τ)] represent dynamic displacements of the cutter at the present and previous tooth periods and τ is the delay time; where Ω is the spindle speed in (rad⁄s). The extended nonlinear modelling of cutting forces and development of closed form expression through Fourier series expansion have been accomplished in the previous researches.
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2019 IFAC MIM 1104 Berlin, Germany, August 28-30, 2019
Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
Development of the proposed extended nonlinear model has been experimentally accomplished in (Moradi, 2013a). Dynamics of regenerative chatter and internal resonance phenomenon , nonlinear dynamics and bifurcation analysis of the process in the presence of tool wear, process damping and cutting force nonlinearities and using a tunable vibration absorber suppress regenerative chatter have been investigated for the proposed extended nonlinear model. According to the formulation presented in (Moradi, 2012a, b), closed form expressions for the nonlinear cutting forces in x-y directions are explained as: 𝑁𝑁 {𝛼𝛼 ′1 𝛥𝛥𝑥𝑥 3 + 𝛽𝛽 ′1 𝛥𝛥𝑦𝑦 3 + 𝛼𝛼 ′ 2 𝛥𝛥𝑥𝑥 2 + 𝛽𝛽 ′ 2 𝛥𝛥𝑦𝑦 2 + 𝛼𝛼 ′ 3 𝛥𝛥𝛥𝛥 + 2𝜋𝜋 𝛽𝛽 ′ 3 𝛥𝛥𝛥𝛥 + 3𝛾𝛾 ′1 𝛥𝛥𝑥𝑥 2 𝛥𝛥𝛥𝛥 + 3𝛾𝛾 ′ 2 𝛥𝛥𝛥𝛥𝛥𝛥𝑦𝑦 2 + 2𝛾𝛾 ′ 3 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 + 𝛾𝛾 ′ 4 } 𝑁𝑁 {𝛼𝛼 𝛥𝛥𝑥𝑥 3 + 𝛽𝛽1 𝛥𝛥𝑦𝑦 3 + 𝛼𝛼2 𝛥𝛥𝑥𝑥 2 + 𝛽𝛽2 𝛥𝛥𝑦𝑦 2 + 𝛼𝛼3 𝛥𝛥𝛥𝛥 + 𝐹𝐹𝑥𝑥 = − 2𝜋𝜋 1 𝛽𝛽3 𝛥𝛥𝛥𝛥 + 3𝛾𝛾1 𝛥𝛥𝑥𝑥 2 𝛥𝛥𝛥𝛥 + 3𝛾𝛾2 𝛥𝛥𝛥𝛥𝛥𝛥𝑦𝑦 2 + 2𝛾𝛾3 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥+𝛾𝛾4 } (8)
𝐹𝐹𝑦𝑦 =
where for half immersion up-milling with 𝜙𝜙𝑠𝑠𝑠𝑠 = 0 and 𝜙𝜙𝑒𝑒𝑒𝑒 = 𝜋𝜋⁄2 (as for the case study here), coefficients of Eq. (8) are determined in terms of 𝜁𝜁𝑖𝑖 , 𝜂𝜂𝑖𝑖 𝑖𝑖 = 1. .4. Similarly, as the cutting force signals are measured, this approach can be extended to any other machining condition (for more details, see the work by Moradi et al. 2013a). Especially, the proposed modelling is essential for machining of materials in which cutting forces are inherently nonlinear functions of chip thickness (e.g., Titanium alloys), (as in the works by the Altintas, 2000; Wiercigroch & Budak, 2001; Stepan et al., 2005; Vela-Martinez, 2009; Moradi et al., 2013a). After determination of nonlinear cutting forces including delay regenerative chatter, nonlinear dynamics of the process is described as: 𝑚𝑚𝑥𝑥 𝑥𝑥̈ + 𝑐𝑐𝑥𝑥 𝑥𝑥̇ + 𝑘𝑘𝑥𝑥 𝑥𝑥 = 𝐹𝐹𝑥𝑥 (𝛥𝛥𝑥𝑥 𝑖𝑖 , 𝛥𝛥𝑦𝑦 𝑖𝑖 ) + 𝑢𝑢𝑥𝑥 𝑚𝑚𝑦𝑦 𝑦𝑦̈ + 𝑐𝑐𝑦𝑦 𝑦𝑦̇ + 𝑘𝑘𝑦𝑦 𝑦𝑦 = 𝐹𝐹𝑦𝑦 (𝛥𝛥𝑥𝑥 𝑖𝑖 , 𝛥𝛥𝑦𝑦 𝑖𝑖 ) + 𝑢𝑢𝑦𝑦
𝑖𝑖 = 1,2,3
(9)
where cutting forces Fx and Fy , as functions of time delay, are given by Eq. (8); ux and uy are the actuators forces (control inputs) implemented on the cutting tool structure. Also, it is assumed that there is only one effective direction for chatter vibration in x-y directions.
For simplicity, Eq. (11) is described in the simple modal form as: ̃𝑥𝑥 {𝛼𝛼3 𝛥𝛥𝛥𝛥 + 𝛽𝛽3 𝛥𝛥𝛥𝛥 + 𝛾𝛾4 } + 𝑢𝑢̂𝑥𝑥 𝑥𝑥̈ + 𝜇𝜇𝑥𝑥 𝑥𝑥̇ + 𝜔𝜔𝑥𝑥2 𝑥𝑥 = −𝑁𝑁 2 ̃𝑦𝑦 {𝛼𝛼′3 𝛥𝛥𝛥𝛥 + 𝛽𝛽′3 𝛥𝛥𝛥𝛥 + 𝛾𝛾′4 } + 𝑢𝑢̂𝑦𝑦 (12) 𝑦𝑦̈ + 𝜇𝜇𝑦𝑦 𝑦𝑦̇ + 𝜔𝜔𝑦𝑦 𝑦𝑦 = +𝑁𝑁 where, 𝑘𝑘𝑖𝑖 𝑁𝑁 ̃𝑖𝑖 = , 𝑁𝑁 , 𝑚𝑚𝑖𝑖 2𝜋𝜋 𝑚𝑚𝑖𝑖 2√𝑘𝑘𝑖𝑖 𝑚𝑚𝑖𝑖 𝑢𝑢̂𝑖𝑖 = 𝑢𝑢𝑖𝑖 ⁄𝑚𝑚𝑖𝑖 , 𝑖𝑖 = 𝑥𝑥, 𝑦𝑦
𝜇𝜇𝑖𝑖 = 2 𝜁𝜁𝑖𝑖 𝜔𝜔𝑖𝑖 , 𝜁𝜁𝑖𝑖 =
𝑐𝑐𝑖𝑖
, 𝜔𝜔𝑖𝑖2 =
To achieve an approximate dynamics of the delayed expressions in cutting forces, Eq. (12) is transferred into the Laplace domain and the first order Pade approximation as 𝑒𝑒 −𝑠𝑠𝑠𝑠 ≅ 1 − 𝑠𝑠𝑠𝑠 is used for delayed terms (i.e., a truncated Taylor series expansion for 𝑒𝑒 −𝑠𝑠𝑠𝑠 with one delayed term is used). Thereafter, applying the inverse Laplace transformation on the resulting equation, yields the approximated dynamics of the linear delayed system as: ̃𝑥𝑥 𝛼𝛼3 𝜏𝜏)𝑥𝑥̇ + 𝑁𝑁 ̃𝑥𝑥 𝛽𝛽3 𝜏𝜏𝑦𝑦̇ + 𝜔𝜔𝑥𝑥2 𝑥𝑥 = −𝑁𝑁 ̃𝑥𝑥 𝛾𝛾4 + 𝑢𝑢̂𝑥𝑥 𝑥𝑥̈ + (𝜇𝜇𝑥𝑥 + 𝑁𝑁 (13) 2 ′ ′ ̃𝑦𝑦 𝛽𝛽 𝜏𝜏)𝑦𝑦̇ − 𝑁𝑁 ̃𝑦𝑦 𝛼𝛼 3 𝜏𝜏 𝑥𝑥̇ + 𝜔𝜔𝑦𝑦 𝑦𝑦 = 𝑁𝑁 ̃𝑦𝑦 𝛾𝛾 ′ + 𝑢𝑢̂𝑦𝑦 𝑦𝑦̈ + (𝜇𝜇𝑦𝑦 − 𝑁𝑁 3 4 The validity of such approximation was investigated by Moradi et al. (2012b). According to Eq. (13), the transfer functions between the output vector [𝑥𝑥, 𝑦𝑦]𝑇𝑇 and control inputs [𝑢𝑢̂𝑥𝑥 , 𝑢𝑢̂𝑦𝑦 ]𝑇𝑇 are found as: 𝑋𝑋(𝑠𝑠) 𝐺𝐺 (𝑠𝑠) [ ] = [ 11 𝑌𝑌(𝑠𝑠) 𝐺𝐺21 (𝑠𝑠)
𝐺𝐺12 (𝑠𝑠) 𝑈𝑈𝑥𝑥∗ (𝑠𝑠) ] ][ 𝐺𝐺22 (𝑠𝑠) 𝑈𝑈𝑦𝑦∗ (𝑠𝑠)
where, 𝐺𝐺 (𝑠𝑠) 𝑮𝑮(𝒔𝒔) = [ 11 𝐺𝐺21 (𝑠𝑠)
𝛬𝛬 (𝑠𝑠) 𝐺𝐺12 (𝑠𝑠) 1 [ 4 ]= 𝛥𝛥(𝑠𝑠) −𝛬𝛬3 (𝑠𝑠) 𝐺𝐺22 (𝑠𝑠)
−𝛬𝛬2 (𝑠𝑠) ] (15-1) 𝛬𝛬1 (𝑠𝑠)
𝛬𝛬1 (𝑠𝑠) = 𝑠𝑠 2 + 𝜓𝜓𝑥𝑥 𝑠𝑠 + 𝜔𝜔𝑥𝑥2 , 𝛬𝛬2 (𝑠𝑠) = 𝜑𝜑𝑥𝑥 𝑠𝑠 𝛬𝛬3 (𝑠𝑠) = −𝜑𝜑𝑦𝑦 𝑠𝑠, 𝛬𝛬4 (𝑠𝑠) = 𝑠𝑠 2 + 𝜓𝜓𝑦𝑦 𝑠𝑠 + 𝜔𝜔𝑦𝑦2 (15-2) ̂𝑥𝑥 (𝑠𝑠) − 𝑁𝑁 ̃𝑥𝑥 𝛾𝛾4 ⁄𝑠𝑠 , 𝑈𝑈𝑦𝑦∗ (𝑠𝑠) = 𝑈𝑈 ̂𝑦𝑦 (𝑠𝑠) + 𝑁𝑁 ̃𝑦𝑦 𝛾𝛾′4 ⁄𝑠𝑠 𝑈𝑈𝑥𝑥∗ (𝑠𝑠) = 𝑈𝑈 𝛥𝛥(𝑠𝑠) = 𝑠𝑠 4 + 𝑎𝑎1 𝑠𝑠 3 + 𝑎𝑎2 𝑠𝑠 2 + 𝑎𝑎3 𝑠𝑠 + 𝑎𝑎4 , 𝑎𝑎1 = 𝜓𝜓𝑥𝑥 + 𝜓𝜓𝑦𝑦 , 𝑎𝑎2 = 𝜔𝜔𝑥𝑥2 + 𝜔𝜔𝑦𝑦2 + 𝜓𝜓𝑥𝑥 𝜓𝜓𝑦𝑦 + 𝜑𝜑𝑥𝑥 𝜑𝜑𝑦𝑦 , 𝑎𝑎3 = 𝜔𝜔𝑥𝑥2 𝜓𝜓𝑦𝑦 + 𝜔𝜔𝑦𝑦2 𝜓𝜓𝑥𝑥 , 𝑎𝑎4 = (𝜔𝜔𝑥𝑥 𝜔𝜔𝑦𝑦 )2 , ̃𝑥𝑥 , ̃𝑦𝑦 , 𝜑𝜑𝑥𝑥 = 𝛽𝛽3 𝜏𝜏𝑁𝑁 𝜑𝜑𝑦𝑦 = 𝛼𝛼 ′ 3 𝜏𝜏𝑁𝑁
2.3 Control Transfer Function for the Linear Model of Cutting Forces with Regenerative Chatter
(14)
̃𝑥𝑥 , 𝜓𝜓𝑥𝑥 = 𝜇𝜇𝑥𝑥 + 𝛼𝛼3 𝜏𝜏𝑁𝑁
(15-3)
̃𝑦𝑦 𝜓𝜓𝑦𝑦 = 𝜇𝜇𝑦𝑦 − 𝛽𝛽 ′ 3 𝜏𝜏𝑁𝑁
In the case of linear modelling of cutting forces, elemental forces in tangential and radial directions are described as (instead of Eq. (3)): 𝑑𝑑𝐹𝐹𝑡𝑡,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [ 𝜁𝜁3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜁𝜁4 ] 𝑑𝑑𝑑𝑑 𝑑𝑑𝐹𝐹𝑟𝑟,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [𝜂𝜂3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜂𝜂4 ] 𝑑𝑑𝑑𝑑
(10)
Following the same procedure explained in the previous section, closed form expressions for the linear model of cutting forces are determined. Therefore, dynamics of the milling process with linear model of cutting forces and in the presence of regenerative chatter is described as: 𝑚𝑚𝑥𝑥 𝑥𝑥̈ + 𝑐𝑐𝑥𝑥 𝑥𝑥̇ + 𝑘𝑘𝑥𝑥 𝑥𝑥 = − 𝑚𝑚𝑦𝑦 𝑦𝑦̈ + 𝑐𝑐𝑦𝑦 𝑦𝑦̇ + 𝑘𝑘𝑦𝑦 𝑦𝑦 =
𝑁𝑁
𝑁𝑁
2𝜋𝜋
2𝜋𝜋
{𝛼𝛼3 𝛥𝛥𝛥𝛥 + 𝛽𝛽3 𝛥𝛥𝛥𝛥 + 𝛾𝛾4 } + 𝑢𝑢𝑥𝑥
{𝛼𝛼′3 𝛥𝛥𝛥𝛥 + 𝛽𝛽′3 𝛥𝛥𝛥𝛥 + 𝛾𝛾′4 } + 𝑢𝑢𝑦𝑦
(11)
Fig. 2. Schematic of the closed-loop control system in the peripheral milling process 3. DEVELOPMENT OF ROBUST CONTROL STRATEGY Closed-loop control system of the milling process is shown in Fig. 2. A control system is robust if it is insensitive to differences between the actual system and the model of the system which is used to design the controller. These differences are referred to as model uncertainty. The 𝐻𝐻∞ robust control technique is used to check if the design specifications
1121
2019 IFAC MIM Berlin, Germany, August 28-30, 2019
Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
are satisfied even for the worst-case uncertainty. General components of the 𝐻𝐻∞ control strategy are shown in Fig. 3, in which the variables 𝑟𝑟, 𝑦𝑦, 𝑒𝑒, 𝑢𝑢, 𝑑𝑑 and 𝑑𝑑𝑖𝑖 are the reference input, output variables, tracking error, control input, output disturbance and input disturbance, respectively. Dynamic plant and controller are denoted by 𝐺𝐺 and 𝐾𝐾 while 𝑊𝑊𝑚𝑚 , 𝑊𝑊𝑃𝑃 and 𝑊𝑊𝑢𝑢 are the weighting functions.
Sensitivity transfer function 𝑆𝑆 = (𝐼𝐼 + 𝐺𝐺𝐺𝐺)−1 is the transfer function of (𝑦𝑦⁄𝑑𝑑 ) and (𝑒𝑒⁄𝑟𝑟 ). The complementary sensitivity function,𝑇𝑇 = 𝐺𝐺𝐺𝐺(𝐼𝐼 + 𝐺𝐺𝐺𝐺)−1 , is the transfer function of (𝑦𝑦⁄𝑟𝑟 ). In addition, the control signal is explained as 𝑢𝑢 = 𝐾𝐾𝐾𝐾(𝑟𝑟 − 𝑑𝑑 − 𝐺𝐺𝑑𝑑𝑖𝑖 ). Consequently, to achieve disturbance rejection and appropriate transient performance, ‖𝑆𝑆‖∞ must be small while for the perfect tracking, |𝑇𝑇(𝑗𝑗𝑗𝑗)| must have unit value. Due to constraint 𝑆𝑆 + 𝑇𝑇 = 𝐼𝐼 (identity), the weighted sensitivity ‖𝑊𝑊𝑃𝑃 𝑆𝑆‖∞ and complementary sensitivity ‖𝑊𝑊𝑚𝑚 𝑇𝑇‖∞ functions are minimized. Also, to restrict the magnitude of the control effort, the upper bound 1/|𝑊𝑊𝑢𝑢 | on magnitude of 𝐾𝐾𝐾𝐾 must be small. To achieve these conditions, a stacking mixed sensitivity function is used as (Skogestad, 2005): ‖𝑁𝑁‖∞ = max𝜎𝜎̅(𝑁𝑁(𝑗𝑗𝑗𝑗)) < 1;
𝑁𝑁 = ‖𝑊𝑊𝑝𝑝 𝑆𝑆
𝜔𝜔
𝑊𝑊𝑢𝑢 𝐾𝐾𝐾𝐾
𝑊𝑊𝑚𝑚 𝑇𝑇‖∞
(16)
1105
force coefficients of nonlinear Eq. (3) are found as (Moradi et al. 2013b): 𝜁𝜁1 = 5600 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚3 , 𝜁𝜁2 = −4893 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚2 , 𝜁𝜁3 = 2736 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚 , 𝜁𝜁4 = −81 𝑎𝑎𝑐𝑐 𝑁𝑁 (18) 𝜂𝜂1 = 6958 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚 𝑚𝑚3 , 𝜂𝜂2 = −5754 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚2 , 𝜂𝜂3 = 1752 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚 , 𝜂𝜂4 = −31 𝑎𝑎𝑐𝑐 𝑁𝑁 Similarly, the above procedure can be used for other machining conditions in which cutting force signals are measured experimentally. For the nonlinear model, helical cutter is divided into small discrete elements along the cutter axis, and cutting force components are calculated by digital integration of Eq. (3) and then using Eqs. (1) and (2). Total cutting force is found as: 𝐹𝐹(𝜙𝜙) = √𝐹𝐹𝑥𝑥 2 (𝜙𝜙) + 𝐹𝐹𝑦𝑦 2 (𝜙𝜙)
(19)
There is good agreement between linear and nonlinear models for a light Aluminium material (𝐴𝐴𝐴𝐴7075); which shows weakly nonlinear characteristic, However, in the case of highly nonlinear materials (such as Titanium alloys), the difference between linear and nonlinear models of cutting forces is significant. (Moradi et al. 2013b) For simulation of the problem and design of controller, nominal realistic values for model parameters of Eq. (9) are extracted from Moradi et al. (2012b). Furthermore, number of tool teeth are assumed to be 𝑁𝑁 = 4; resulting in 𝜏𝜏 = 0.005 𝑠𝑠. The force parameters for the linear and nonlinear models are given by Eqs. (17) and (18), respectively. The open-loop system of milling process (Eq. (9) with 𝑢𝑢𝑥𝑥 = 𝑢𝑢𝑦𝑦 = 0) is simulated in time domain via SIMULINK Toolbox of MATLAB. The critical depth cut in which the system changes from the stable condition to the unstable one is found as 𝑎𝑎𝑙𝑙𝑙𝑙𝑙𝑙 = 3.1 𝑚𝑚𝑚𝑚.
The stabilizing 𝐻𝐻∞ optimal controller is determined by minimizing the function ‖𝑁𝑁(𝐾𝐾)‖∞ . The conditions of nominal stability (NS) and performance (NP); robust stability (RS) and performance (RP) are discussed by Moradi et al. (2015).
4.2 Characteristics of the Weighting Functions and Design of Robust Controller Consider the matrix transfer function 𝐆𝐆 (𝐬𝐬) given by Eq. (10) and nominal values of parameters given by Eqs. (17), (18) and (19). To study a real case of uncertain model, uncertain parameters are changed around their nominal values as:
Fig. 3. Components of the robust H∞ control design 4. IMPLEMENTATION OF ROBUST CONTROL, RESULTS AND DISCUSSIONS
̅ ≤ Θ ≤ (1 + 𝜒𝜒) Θ ̅ (1 − 𝜒𝜒) Θ
4.1 Specifications of the Milling Process and Cutting Conditions To simulate the problem and investigate the performance of robust controller, a typical case study with the experimental cutting forces in tangential and radial directions is considered. Plotting the forces 𝐹𝐹𝑡𝑡 and 𝐹𝐹𝑟𝑟 versus chip thickness ℎ; using linear regression via Curve Fitting Toolbox of MATLAB (CFTM), applied on Eq. (3) in un-differential form, yields the cutting force coefficients for an orthogonal cutting test, e.g., for more details see Moradi, et al. (2013a): 𝐾𝐾𝑡𝑡𝑡𝑡 = 1445 𝑎𝑎𝑐𝑐 , 𝐾𝐾𝑟𝑟𝑟𝑟 = 311.4 𝑎𝑎𝑐𝑐 𝑁𝑁. 𝑚𝑚𝑚𝑚−1 𝐾𝐾𝑡𝑡𝑡𝑡 = 15.6 𝑎𝑎𝑐𝑐 , 𝐾𝐾𝑟𝑟𝑟𝑟 = 72.4 𝑎𝑎𝑐𝑐 𝑁𝑁
(17)
where 𝑎𝑎𝑐𝑐 is the axial depth of cut (in 𝑚𝑚𝑚𝑚). Again plotting the forces 𝐹𝐹𝑡𝑡 and 𝐹𝐹𝑟𝑟 versus chip thickness ℎ, and using cubic regression via CFTM (with a third order polynomial), cutting
(20)
where 𝜒𝜒 is the uncertainty amount in percent and Θ can take ̅ is the any of parameters given by Eqs. (17), (18) and (19); Θ nominal value of Θ. In many practical cases, the configuration of multiplicative input uncertainty is used to represent the various sources of dynamic uncertainty (e.g., see Skogestad & Postlethwaite, 2005); where the perturbed plant 𝐆𝐆𝐩𝐩 (𝐬𝐬) is described as: 𝐺𝐺𝑝𝑝 (𝑠𝑠) = 𝐺𝐺 (𝑠𝑠) [1 + 𝑊𝑊𝑚𝑚 (𝑠𝑠) Δ𝐼𝐼 (𝑠𝑠)]; ⏟|Δ𝐼𝐼 (𝑗𝑗𝑗𝑗)| ≤ 1 ∀𝜔𝜔 (21) ‖Δ𝐼𝐼 ‖∞ ≤1
𝐺𝐺 (𝑠𝑠) is the nominal plant and Δ𝐼𝐼 (𝑠𝑠) is any stable transfer function with a magnitude less or equal to 1, at each frequency. Uncertain plant 𝐺𝐺𝑝𝑝 (𝑠𝑠) is replaced by a disc type approximation 𝐺𝐺′𝑝𝑝 (𝑠𝑠) of radius |𝑊𝑊𝑚𝑚 (𝑗𝑗𝑗𝑗) 𝐺𝐺(𝑗𝑗𝑗𝑗)| where the weight function 𝑊𝑊𝑚𝑚 (𝑠𝑠) is determined as Skogestad (2005):
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Γ𝐼𝐼 (𝜔𝜔) = max |
𝐺𝐺𝑝𝑝 (𝑗𝑗𝑗𝑗)
𝐺𝐺𝑝𝑝 ∈Π 𝐺𝐺(𝑗𝑗𝑗𝑗)
− 1| ,
Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
|𝑊𝑊𝑚𝑚 (𝑗𝑗𝑗𝑗)| ≥ Γ𝐼𝐼 (𝜔𝜔), ∀𝜔𝜔
(22)
where Π is the set of possible perturbed plants. According to Eqs. (14), (15) and (20), there are 25 cases of perturbed plants for 𝐺𝐺12 (𝑠𝑠) and 𝐺𝐺21 (𝑠𝑠) and 26 cases of perturbed plants for 𝐺𝐺11 (𝑠𝑠) and 𝐺𝐺22 (𝑠𝑠). The weight function of uncertainty (𝑊𝑊𝑚𝑚 ), upper bound on relative errors, is found as: 1 𝑊𝑊𝑚𝑚 (𝑠𝑠) = 10 [ 1
1 ] 1
(23)
At the low frequencies, the weight function 1⁄|𝑊𝑊𝑝𝑝 | as the upper bound on the sensitivity function |𝑆𝑆|, is equal to the parameter 𝐴𝐴 (typically𝐴𝐴 ≈ 0) in correlation with the steady state tracking error. At the high frequencies, it approaches to the maximum peak amplitude 𝑀𝑀 ≥ 1 ; and the bandwidth frequency (𝜔𝜔𝐵𝐵 ) indicates the speed of time response. To have a desired step response with a maximum overshoot of 10 percent, the rise time of about 0.2s, and a perfect tracking (error≈0), the appropriate performance weight 𝑊𝑊𝑝𝑝 is defined as: 𝑊𝑊𝑃𝑃 (𝑠𝑠) =
𝑠𝑠/1.1+10
𝑠𝑠+8×10−5
Fig. 4. Tool tip vibrations of the open-loop system in the presence of 30 percent uncertainty.
(24)
Using a 𝜇𝜇-synthesis code and DK-iteration algorithm developed in Robust Control Toolbox of MATLAB, 𝐻𝐻∞ robust controllers of 6-order are obtained. Using the same toolbox and plotting their Bode diagram, these controllers are approximated with reduced third order controllers as: 𝑘𝑘 (𝑠𝑠) 𝑘𝑘12 (𝑠𝑠) ] 𝐾𝐾(𝑠𝑠) = [ 11 𝑘𝑘21 (𝑠𝑠) 𝑘𝑘22 (𝑠𝑠) 𝑘𝑘11 (𝑠𝑠) = 830 (𝑠𝑠 + 2070)(𝑠𝑠 + 70)⁄Φ(𝑠𝑠) ; 𝑘𝑘12 (𝑠𝑠) = −390 (𝑠𝑠 + 2120)(𝑠𝑠 + 60)⁄Φ(𝑠𝑠) 𝑘𝑘21 (𝑠𝑠) = −5 (𝑠𝑠 + 3820)(𝑠𝑠 + 1230)⁄Φ(𝑠𝑠); 𝑘𝑘22 (𝑠𝑠) = 390⁄𝑠𝑠 ; Φ(𝑠𝑠) = 𝑠𝑠(𝑠𝑠 2 + 115𝑠𝑠 + 3.65 × 105 )
(25)
4.3 Performance of the Roust Controller in Chatter Suppression of the Linear/Non-linear Models Figure 4 shows time response of the tool tip vibrations of the open-loop system with linear/nonlinear models of cutting forces, in the presence of uncertainty (e.g., 30% uncertainty𝜒𝜒 = 0.3 in Eq. (20)). As it is observed, unstable conditions occur in the presence of uncertainties (without controller). Moreover, vibrations amplitudes are larger in the nonlinear model because the existence of high order delayed terms (Δ𝑥𝑥 𝑖𝑖 , Δ𝑦𝑦 𝑖𝑖 , Δ𝑥𝑥 Δ𝑦𝑦 𝑗𝑗 , Δ𝑥𝑥 2 Δ𝑦𝑦; 𝑖𝑖 = 2,3; 𝑗𝑗 = 1,2) induce a more unstable response. The effect of robust controller on chatter suppression of the linear/nonlinear model and in the presence of 10%, 30% and 50% uncertainty is shown in Fig. 5. As it is observed, for both uncertain models of cutting forces, robust controller acts efficiently in chatter suppression. Moreover, as the amount of uncertainty increases, chatter is suppressed in longer time. The required actuator forces for chatter suppression in linear and nonlinear models are shown in Fig. 6. According to these figures and as physically expected, as the amount of uncertainty increases, more actuator forces are required for chatter suppression.
Fig. 5. Tool tip vibrations of the milling process after implementation of the optimal robust controller in the presence of uncertainty. For a given uncertainty, according to Fig. 5, chatter is suppressed in shorter time with less vibration amplitudes in the case of linear model. Moreover, as shown in Fig. 6, less actuator forces are required to suppress chatter in the case of linear model. This is because the existence of high order delayed terms in the nonlinear model.
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Fig. 6. Required actuator force to suppress regenerative chatter vibrations of the milling process; in the presence of uncertainty. 5. CONCLUSIONS In this paper, an extended nonlinear model of the milling process is considered in which the cutting forces are described as a third-order nonlinear function of chip thickness (through coupled nonlinear delay differential equations). The proposed modelling is more essential for the machining of materials in which the cutting forces are inherently nonlinear functions of chip thickness. To achieve a more realistic model, uncertainties associated with the machine-tools and cutting parameters are also included. To suppress the regenerative chatter, an 𝐻𝐻∞ robust control strategy is developed based on 𝜇𝜇 -synthesis and DK-iteration algorithm. In the presence of uncertainties, the robust performance and robust stability of the control system is investigated and compared for the linear and nonlinear models. REFERENCES Altintas Y, Stepan G, Merdol D and Dombovari Z (2008) Chatter stability of milling in frequency and discrete time domain. CIRP Journal of Manufacturing Science and Technology 1: 35-44. Armarego EJA and Deshpande NP (1991) Computerized endmilling force predictions with cutting models allowing eccentricity and cutter deflections. Ann. CIRP 40: 25-29. Bayoumi AE, Yucesan, G and Kendall LA (1994) An analytic mechanistic cutting force model for milling operations: a theory and methodology. Trans. ASME Journal of Engineering Industry 116(8): 324-30.
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Budak E (2006) Analytical models for high performance milling, part I: cutting forces, deflections and tolerance integrity. Int. Journal of Machine Tools and Manufacture 46: 1478-88. Carrillo FJ, Rotella F and Zadshakoyan M (1999) Delta approach robust controller for constant turning force regulation. Journal of Control Engineering Practice 7: 1321-1331. Chiou R and Liang SY (1998) Chatter stability of a slender cutting tool in turning with tool wear effect. Int. Journal of Machine Tools & Manufacture 38(4): 315-327. Koeinsberger F and Tlusty J (1970) Machine Tool Structures. First ed. New Jersey: Pergamon press. Moradi H, Movahhedy MR and Vossoughi GR (2012a) Dynamics of regenerative chatter and internal resonance in milling process with structural and cutting force nonlinearities. Journal of Sound and Vibration 331(16): 3844-3865. Moradi H, Movahhedy MR and Vossoughi GR (2012b) Bifurcation analysis of milling process with tool wear and process damping: Regenerative chatter with primary resonance. Journal of Nonlinear Dynamics 70(1): 481-509 Moradi H, Vossoughi GR and Movahhedy M (2013a) Experimental dynamic modelling of milling with process damping, structural and cutting force nonlinearities. Journal of Sound and Vibration 332: 4709-4731. Moradi H, Vossoughi GR, Movahhedy MR and Salarieh H (2013b) Suppression of nonlinear regenerative chatter in milling process via robust optimal control. Journal of Process Control 23(5): 631-648. Moradi H and Vossoughi GR (2015) Robust control of the variable speed wind turbines in the presence of uncertainties: A comparison between H„ and PID controllers. Energy (90): 1508–1521. Nayfeh AH and Nayfeh NA (2012) Time-delay feedback control of lathe cutting tools. Journal of Vibration and Control 18(8): 1106-1115 Park Y, Kim TY, Woo J, Shin D, Kim J (1998) Sliding mode cutting force regulator for turning processes. Int. Journal of Machine Tools & Manufacture 38(8): 911-930. Parus A, Powałka B, Marchelek K, Domek S and Hoffmann M (2013) Active vibration control in milling flexible workpieces. Journal of Vibration and Control 19(7): 1103-1120. Skogestad S and Postlethwaite I (2005) Multivariable Feedback Control, New York: John Wiley and Sons. Van Dijk NJM, Van De Wouw N, Doppenberg EEJ, Oosterling HAJ and Nijmeijer H (2012) Robust active chatter control in the high-speed milling process. IEEE Trans. on Control Systems Technology 20(4): 901-917 Vela-Martínez L, Jáuregui-Correa JC, González-Brambila OM, Herrera-Ruiz G and Lozano-Guzmán A (2009) Instability conditions due to structural nonlinearities in regenerative chatter. Journal of Nonlinear Dynamics 56: 415–27. Zuperl U, Cus F and Reibenschuh M (2012) Modeling and adaptive force control of milling by using artificial techniques. Journal of Intelligent Manufacturing 23(5): 1805-1815.
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Robust control of regenerative chatter in uncertain milling process with weak Robust control of regenerative chatter in uncertain milling process with weak Robust of regenerative chatter in uncertain milling weak nonlinear cutting forces: A comparison with linearprocess model with Robust control control of regenerative chatter in uncertain milling process with weak nonlinear cutting forces: A comparison with linear model nonlinear cutting forces: A comparison with linear model nonlinear cutting forces: A comparison with linear model Hamed Moradi*, Ali Nouriani,
Hamed Moradi*, Ali Nouriani, Gholamreza Hamed Moradi*, Ali Hamed Moradi*,Vossoughi Ali Nouriani, Nouriani, Hamed Moradi*, Ali Nouriani, Gholamreza Vossoughi Gholamreza Vossoughi Gholamreza Vossoughi Gholamreza Vossoughi * Centre of Excellence in Design, Robotics & Automation, School of Mechanical Engineering, Sharif University of Technology, * Centre CentreIran, of Excellence Excellence in21Design, Design, Robotics & Automation, Automation, School School of Mechanical Mechanical Engineering, Sharif Sharif University University of of Technology, Technology, Tehran, (Tel: +98 66165545, e-mail:[email protected]; [email protected]) * of in Robotics & Engineering, * Centre Centre of of Excellence Excellence in in Design, Design, Robotics Robotics & & Automation, Automation, School School of of Mechanical Mechanical Engineering, Sharif Sharif University University of of Technology, Technology, *Tehran, of Engineering, Iran, (Tel: +98 21 66165545, e-mail:[email protected]; [email protected]) Tehran, Tehran, Iran, Iran, (Tel: (Tel: +98 +98 21 21 66165545, 66165545, e-mail:[email protected]; e-mail:[email protected]; [email protected]) [email protected]) Tehran, Iran, (Tel: +98 21 66165545, e-mail:[email protected]; [email protected]) Abstract: For various types of materials, milling process is extensively used to generate complex shapes Abstract: For various various types of materials, milling process is extensively extensively used to to generate generate complex shapes with high quality. During theof process and to achieve high removal rate, precision and bettercomplex surface shapes finish, Abstract: For types materials, milling process is used Abstract: For various various types of materials, milling process is extensively extensively used to to generate generate complex shapes Abstract: For types of materials, milling process is used complex shapes with high quality. During the process and to achieve high removal rate, precision and better surface finish, chatter suppression is of great importance. An extended model of the milling process isbetter presented in finish, which with high quality. During the process and to achieve high removal rate, precision and surface with high quality. During the process and to achieve high removal rate, precision and better surface finish, with high quality. During the process and toAn achieve high removal rate, precision andisbetter surface finish, chatter suppression is of great importance. extended model of the milling process presented in which the cutting forces are described as a third-order nonlinear function of chip thickness. Uncertainties chatter suppression is of great importance. An extended model of the milling process is presented in chatter suppression is of great importance. An extended model of the milling process is presented in which chatter suppression is ofdescribed great importance. An extended model of the milling process is presented in which which the cutting forces are as a third-order nonlinear function of chip thickness. Uncertainties associated the are process and tool arenonlinear also included to achieve a more realistic model. To the forces described as aaa third-order function of thickness. Uncertainties the cutting cuttingwith forces are described as parameters third-order nonlinear function of chip chip thickness. Uncertainties the cutting forces described as third-order function of chip thickness. Uncertainties associated with the are process andantool tool parameters arenonlinear also included to achieve achieve a more more realistic model. To robust control is designed based on µ-synthesis with DK-iteration suppress regenerative chatter, H∞ parameters associated with the process and are also included to a realistic model. associated with the process and tool parameters are also included to achieve a more realistic model. To To associated with the process andantool parameters are also included to achieve a more realistic model. To robust control is designed based on µ-synthesis with DK-iteration suppress regenerative chatter, H ∞ the robust stability and performance of both linear and nonlinear algorithm.regenerative The controller guarantees control is designed based on µ-synthesis with DK-iteration suppress chatter, an H ∞ robust robust control is designed based on µ-synthesis with DK-iteration suppress regenerative chatter, an H ∞ control is designed based on µ-synthesis withand DK-iteration suppress regenerative chatter, an H∞ robust algorithm. The controller the robust performance both linear nonlinear models in the presence ofguarantees uncertainties. In bothstability models, and as the amount ofof uncertainty increases, more algorithm. The controller the robust stability and performance of both and algorithm. The controller guarantees the robust stability and performance of both linear and nonlinear algorithm. Thepresence controllerofguarantees guarantees the In robust stability and performance ofuncertainty both linear linearincreases, and nonlinear nonlinear models in the uncertainties. both models, as the amount of more actuator efforts are required to suppress the chatter. The linear model needs less actuation and time to models in the presence of uncertainties. In both models, as the amount of uncertainty increases, models in in the the presence presence of of uncertainties. uncertainties. In In both both models, models, as as the the amount amount of of uncertainty uncertainty increases, increases, more more models more actuator efforts are required to suppress the chatter. The linear model needs less actuation and time to suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed. actuator efforts are required to suppress the chatter. The linear model needs less actuation and time actuator efforts efforts are are required required to to suppress suppress the the chatter. chatter. The The linear linear model model needs needs less less actuation actuation and and time time to to actuator to suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed. suppress the chatter. However, as the nonlinear effects are increased, the controller must be re-designed. suppress the However, as effects are must re-designed. © 2019, IFAC (International Federation of Automatic Hostingthe bycontroller Elsevier Ltd. Allbe reserved. suppress the chatter. chatter. However, as the the nonlinear nonlinear effectsControl) are increased, increased, the controller must berights re-designed. © 2019, IFAC (International (InternationalFederation Federationofof ofAutomatic AutomaticControl) Control)Hosting Hosting by Elsevier Ltd. All rights reserved. © 2019, 2019, IFAC byby Elsevier Ltd. AllAll rights reserved. © 2019, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. rights reserved. © (International Federation of Control) Elsevier Ltd. reserved. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. © 2019, IFAC IFAC (International Federation of Automatic Automatic Control) Hosting Hosting by Elsevier Ltd. All All rights rights reserved. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust Keywords: Milling process; chatter suppression; linear/nonlinear cutting forces; uncertainty, robust control. control. The nonlinear models of the cutting process includes the 1. INTRODUCTION The nonlinear nonlinear modelsa combination of the cutting cutting process and includes the material nonlinearity, of material structural The models of process includes the 1. INTRODUCTION The nonlinear models of the process includes the The nonlinear modelsa combination of the the cutting cutting process and includes the 1. material nonlinearity, of material structural 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) nonlinearity, aaa combination of and material nonlinearity, combination of material and structural Self-excited vibrations or chatter is an undesirable material material nonlinearity, combination of material material and structural structural nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) Self-excited vibrations or chatter is an undesirable and high order nonlinear terms in cutting forces (Moradi, nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) phenomenon in machining processes which results in Self-excited vibrations or chatter an nonlinearities (Moradi, 2012a, 2013a, Vela-Martínez, 2009) Self-excited vibrations or chatter is an undesirable and high order nonlinear terms in cutting forces (Moradi, Self-excited vibrations or processes chatter is is which an undesirable undesirable phenomenon in machining results in 2012b). In some recent researches, the effects of tool wear and and nonlinear terms cutting (Moradi, and high high order nonlinear terms in cutting forces forces (Moradi, reduction of material removal rate (MRR), poor surface finish phenomenon in processes which results in highIn order order nonlinear terms in in forces (Moradi, phenomenon in machining processes which results in 2012b). some recent recent researches, thecutting effects of toolsources wear and phenomenon in machining machining processes whichsurface results in and reduction of material removal rate (MRR), poor finish process damping have been included beside the of 2012b). In some researches, the effects of tool wear and In some recent researches, the effects of tool wear and and decrease in tool life. Machine tool chatter is classified into 2012b). reduction of material removal rate (MRR), poor surface finish 2012b). In some recent researches, the effects of tool wear and reduction of material removal rate (MRR), poor surface finish process damping have been included beside the sources of reduction of material removal ratetool (MRR), poor surface finish and decrease in tool life. Machine chatter is classified into structural/cutting force nonlinearities, e.g., (Moradi, 2013a, process damping have been included beside the sources of damping have included the of regenerative and and tool life. chatter is into process damping force have been been included beside beside the sources sources of and decrease decrease in in toolnon-regenerative life. Machine Machine tool tool(mode chatter coupling) is classified classifiedtypes into process structural/cutting nonlinearities, e.g., (Moradi, 2013a, and decrease in tool life. Machine tool chatter is classified into regenerative and non-regenerative (mode coupling) types structural/cutting 2012b). Control strategies like adaptive control used to achieve force e.g., 2013a, structural/cutting force nonlinearities, nonlinearities, e.g., (Moradi, (Moradi, 2013a, (Koeinsberger and non-regenerative Tlusty, 1970). In (mode the former mechanism, regenerative coupling) types force nonlinearities, e.g., (Moradi, 2013a, regenerative and and non-regenerative (mode coupling) types structural/cutting 2012b). Control strategies like adaptive control usedtime-delay to achieve achieve regenerative and non-regenerative (mode coupling) types (Koeinsberger and Tlusty, 1970). In the former mechanism, constant cutting force and improved surface finish, 2012b). Control strategies like adaptive control used to 2012b). Control strategies like adaptive control used to achieve the cut produced at time leaves small waves in the material (Koeinsberger and Tlusty, 1970). In the former mechanism, 2012b). Control strategies like adaptive control used to achieve (Koeinsberger and and Tlusty, Tlusty, 1970). 1970). In In the former former mechanism, mechanism, constant cutting force and improved surface finish, time-delay (Koeinsberger the cut produced produced at time time leaves small smallthe waves in thetool. material feedback controlforce (Nayfeh, 2012) surface and optimal control of cutting and finish, constant cutting force and improved surface finish, time-delay regenerated during each subsequent pass of in thethe This constant the cut at leaves waves material constant cutting force and improved improved surface finish, time-delay time-delay the cut produced at time leaves small waves in the material feedback control (Nayfeh, 2012) and optimal control of of the cut produced at time leaves small waves in the material regenerated during each subsequent pass of the tool. This regenerative chatter (Moradi, 2013b) have been developed. In feedback control (Nayfeh, 2012) and optimal control feedback control (Nayfeh, 2012) and optimal control interaction leads to thesubsequent chip thickness variation and feedback regenerated during each pass of the tool. This control (Nayfeh, 2012) and optimal control of of regenerated during each subsequent pass of the tool. This regenerative chatter (Moradi, 2013b) have been developed. In regenerated during each subsequent pass of the tool. This interaction leads to the chip thickness variation and addition, piezoelectric active vibration control systems (Parus regenerative chatter (Moradi, 2013b) have been developed. In regenerative chatter (Moradi, 2013b) have been developed. In consequently cutting force variation and increase in vibration interaction leads to the chip variation and chatter (Moradi, 2013b) have beensystems developed. In interaction leads to the chip thickness variation and addition, piezoelectric active vibration control (Parus interaction leads toforce the variation chip thickness thickness variation and regenerative consequently cutting and increase in vibration et al., 2013), sliding mode control (Park et al., 1998) and addition, piezoelectric piezoelectric active active vibration vibration control control systems systems (Parus (Parus amplitude of the tool.force In the mode coupling mechanism, the addition, consequently cutting variation and in addition, piezoelectric active vibration control (Parus consequently cutting force variation and increase in vibration et al., al., 2013), 2013), sliding mode control (Park etHsystems al.,and 1998) and consequently cutting force variation and increase increase in vibration vibration adaptive amplitude of the tool. In the the mode coupling mechanism, the various robust sliding control mode approaches including et sliding mode control (Park et al., 1998) and ∞ et al., 2013), control (Park et al., 1998) and tool traces out an elliptic path that varies the depth of cut in amplitude of the tool. In mode coupling mechanism, the et al., 2013), sliding mode control (Park et al., 1998) and amplitude of the tool. In the mode coupling mechanism, the and adaptive various robust control approaches including H ∞ amplitude of the tool. In the mode coupling mechanism, (Carrillo et al., 1999) have been implemented either tool traces out an elliptic path that varies the depth of cut cutthe in techniques various robust robust control approaches including H∞∞ and and adaptive adaptive various control approaches including H such a wayout to an bolster the path coupled modes the of vibrations. tool elliptic that depth in various robust control approaches including H tool traces traces out an elliptic path that varies varies the depth of of cut cutThe in techniques ∞ and adaptive (Carrillo et al., 1999) have been implemented either tool traces out an elliptic path that varies the depth of in for chatter suppression or achievement of the constant cutting such a way to bolster the coupled modes of vibrations. The techniques (Carrillo et al., 1999) have been implemented techniques (Carrillo (Carrillo et et al., al., 1999) 1999) have have been been implemented implemented either either regenerative type is found to be themodes most of detrimental the techniques such bolster the vibrations. either such aaa way way to to bolster the coupled coupled modes of vibrations.to The The for chatter suppression or achievement of the constant cutting such way to bolster the coupled modes of vibrations. The forces. Also, control strategies based onof regenerativerate type is found found to be be the the most detrimental detrimental to to the the for suppression or the cutting for chatter chatter suppression or achievement achievement offuzzy-logic the constant constantmethods cutting production in most machining processes. regenerative type is to most for chatter suppression or achievement of the constant cutting regenerative type is found to be the most detrimental to the forces. Also, strategies based on fuzzy-logic methods regenerative type is found to be the most detrimental to the (Zuperl et al.,control 2012) are the other techniques used in this area. production rate rate in most machining processes. forces. Also, control strategies based on fuzzy-logic methods forces. control strategies based on methods production in machining processes. forces. Also, Also, control strategies based on fuzzy-logic fuzzy-logic methods production in most machining processes. et al., 2012) are the other techniques used in this area. In the earlyrate works of this area, modelling and analysis of the (Zuperl production rate in most most machining processes. (Zuperl et al., 2012) are the other techniques used in this area. (Zuperl et al., 2012) are the other techniques used in this area. processes high dimensional accuracy In the the early early works of ofand thisrelative area, modelling modelling and analysis analysis of the the (Zuperl etmilling al., 2012) are the require other techniques used in this area. complex geometry work-piece/tool motion in Precision In works this area, and of In the early works of this area, modelling and analysis of the Precision milling processes require high dimensional accuracy In the early works of this area, modelling and analysis of the in spite of the variations in cutting conditions such as spindle complex geometry and relative work-piece/tool motion in Precision milling processes require high dimensional accuracy Precision milling processes require high dimensional accuracy milling process wasand carried out. work-piece/tool Mechanistic approach was complex geometry relative motion in milling processes require high dimensional accuracy complex geometry and relative work-piece/tool motion in in spite of the variations in cutting such spindle complex geometry and relative work-piece/tool motionwas in Precision speed and depth of cut as well as conditions variations in theas dynamic milling process process was carried out. Mechanistic approach in spite of the variations in cutting conditions such as spindle in spite of the variations in cutting conditions such as spindle extensively used was for governing dynamic equations of various milling carried out. Mechanistic approach was in spite of the variations in cutting conditions such as spindle milling process was carried out. Mechanistic approach was speed and depth of cut as well as variations in the dynamic milling process was carried out. Mechanistic approach was properties of the machine tool system. However, most of the extensively used for governing dynamic equations of various speed and depth of cut as well as variations in the dynamic speed and depth of cut as well as variations in the dynamic milling processes and prediction of cutting forces, deflection extensively used for governing dynamic equations of various speed and depth of cut as well as variations in the dynamic extensively used for governing dynamic equations of various properties of the machine tool system. However, most of the extensively used for dynamic equations of various researches considered a relatively simplified model of the milling processes andgoverning prediction of cutting forces, deflection properties of the machine tool system. However, most of properties of the machine tool system. However, most of the of machine components and form errors (Budak, 2006). In this milling processes and prediction of cutting forces, deflection properties of the machine tool system. However, most of the milling processes and prediction of cutting forces, deflection researches considered a relatively simplified model of the milling processes and prediction of cutting forces, deflection process. Such models are commonly based on idealized of machine components and form errors (Budak, 2006). In this researches considered a relatively simplified model of the researches considered aa relatively simplified model of the approach, by using and experimental data, cutting force of machine components form errors (Budak, 2006). In this researches considered relatively simplified model of the of machine components and form errors (Budak, 2006). In this process. Such models are commonly based on idealized of machine components and form errors (Budak, 2006). Inforce this process. models of machining process and imperfections or approach, by using experimental data, cutting Such models are commonly based on idealized process. Such models are commonly based on idealized coefficients are calibrated for certain cutting conditions. After approach, by using experimental data, cutting force process. Such models are commonly based on idealized approach, by by using using experimental experimental data, data, cutting cutting force force models of machining process and imperfections or approach, uncertainties in the model or cutting conditions are not taken coefficients are calibrated for certain cutting conditions. After models of machining process and imperfections models of of machining machining process process and and imperfections imperfections or or determination the flankfor plowing and shear coefficients certain cutting conditions. After or coefficients are areofcalibrated calibrated for certainforces, cuttingfriction conditions. After models uncertainties in the model or cutting conditions are not taken coefficients are calibrated for certain cutting conditions. After into account. These sources of uncertainty alongside with the determination of the flank plowing forces, friction and shear uncertainties in the model or cutting conditions taken uncertainties in in the the model model or or cutting cutting conditions conditions are are not not taken angles from theof orthogonal cutting test and transforming them into determination the flank plowing forces, friction and shear uncertainties are not taken determination of the flank plowing forces, friction and shear account. These sources of uncertainty alongside with the determination the flank plowing forces, friction and shear effects of cutting forces nonlinearity and regenerative chatter; angles from theoforthogonal orthogonal cutting test and transforming them into into account. These sources of uncertainty alongside with the account. These sources of alongside the to the from oblique conditions, cutting forces are computed angles the cutting test and transforming them into account. These sources of uncertainty uncertainty alongside with with the angles from the orthogonal cutting test and transforming them effects of cutting forces nonlinearity and regenerative chatter; angles from the orthogonal cutting test and transforming them lead to inaccurate modelling. to the oblique conditions, cutting forces are computed effects of cutting forces nonlinearity and regenerative chatter; effects of cutting forces nonlinearity and regenerative chatter; (Armarego and Deshpande, 1991). Also, closedare formcomputed solution effects to the oblique conditions, cutting forces of cutting forces nonlinearity and regenerative chatter; to the oblique conditions, cutting forces are computed to inaccurate modelling. to the oblique conditions,1991). cutting forces are computed (Armarego and Deshpande, Also, closed form solution lead lead to inaccurate modelling. lead to inaccurate modelling. for the evaluation of linear (Bayoumi et al.closed 1994) form and nonlinear (Armarego and 1991). solution In this unlike the previous researches which discussed to paper, inaccurate modelling. (Armarego and Deshpande, 1991). Also, closed form solution (Armarego and Deshpande, Deshpande, 1991). Also, Also, closed form solution lead for the evaluation of linear (Bayoumi et al. 1994) and nonlinear In this paper, unlike the previous researches which models (Moradi, 2012a, 2013a) of cutting forces in end milling for the evaluation of linear (Bayoumi et al. 1994) and nonlinear either SDOF turning processes (Park et al., 1998) ordiscussed a linear In this paper, unlike the previous researches which discussed for the evaluation of linear (Bayoumi et al. 1994) and nonlinear In this paper, unlike previous researches which for the evaluation of linear (Bayoumi et al.forces 1994) in and nonlinear thisSDOF paper, turning unlike the the previous researches whichordiscussed discussed models (Moradi, 2012a, 2013a) of cutting cutting end milling In either processes (Park et al., 1998) a et linear process have been presented. models (Moradi, 2012a, 2013a) of forces in end milling model of cutting forces in milling process (Van Dijk al., either SDOF turning processes (Park et al., 1998) or linear models (Moradi, 2012a, 2013a) of cutting forces in end milling either SDOF turning processes (Park et al., 1998) or linear models 2013a) of cutting forces in end milling either SDOF turning processes (Park et al., (Van 1998)Dijk or aaa et linear process (Moradi, have been been2012a, presented. model of cutting forces in milling process al., process have presented. model of cutting forces in milling process (Van Dijk et process have have been presented. model cutting forces milling process al., process been presented. model of of cutting Ltd. forces in milling process (Van (Van Dijk Dijk et et al., al., 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier All in rights reserved.
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Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
2012), an extended model of the milling process is considered in which the cutting forces are described as a third-order nonlinear function of chip thickness, through coupled nonlinear delay differential equations. The proposed modelling is essential for the machining of materials in which the cutting forces are inherently nonlinear functions of chip thickness. Uncertainties associated with the machine-tools parameters and cutting parameters are also included to achieve a more realistic model. To suppress regenerative chatter, an H∞ robust control strategy is developed based on µ-synthesis and DK-iteration algorithm. The control inputs are the counterbalance forces in x-y directions, implemented by actuators (e.g., piezo-electric actuators). The robust performance and robust stability of the control system is investigated and compared for the linear and nonlinear models, in the presence of uncertainties. Moreover, the effect of uncertainty amount on the behaviour of tool tip vibrations and required actuator forces is studied. Also, it has been shown that as the nonlinear effects are strengthened, the robust controller must be re-designed.
1103
d𝐹𝐹𝑥𝑥,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = −d𝐹𝐹𝑡𝑡,𝑗𝑗 cos 𝜙𝜙𝑗𝑗 − d𝐹𝐹𝑟𝑟,𝑗𝑗 sin 𝜙𝜙𝑗𝑗 ; d𝐹𝐹𝑦𝑦,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = +d𝐹𝐹𝑡𝑡,𝑗𝑗 sin 𝜙𝜙𝑗𝑗 − d𝐹𝐹𝑟𝑟,𝑗𝑗 cos 𝜙𝜙𝑗𝑗
(1)
Using Eq. (1) and integrating analytically along the cut portion of the flute 𝑗𝑗, yields the closed form expressions for cutting forces. Aligning the flute 𝑗𝑗 = 0 at 𝜙𝜙 = 0 in the beginning of the algorithm, the cutting forces contributed by all flutes are calculated and integrated digitally to obtain the total instantaneous forces on the cutter at immersion 𝜙𝜙 as (Moradi, 2013b): 𝐹𝐹𝑥𝑥 (𝜙𝜙) = ∑𝑁𝑁−1 𝑗𝑗=0 𝐹𝐹𝑥𝑥𝑥𝑥 ;
𝐹𝐹𝑦𝑦 (𝜙𝜙) = ∑𝑁𝑁−1 𝑗𝑗=0 𝐹𝐹𝑦𝑦𝑦𝑦
(2)
2.2 Nonlinear Modelling of Cutting Forces with Regenerative Chatter Effect Since stability theories cannot predict interesting phenomena in milling process with linear dynamics, nonlinear modelling of milling process is of great importance. In this paper, to achieve a more realistic model, cutting forces are expressed as a complete third-order polynomial function of chip thickness as: 𝑑𝑑𝑑𝑑𝑡𝑡,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [𝜁𝜁1 ℎ𝑗𝑗 3 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜁𝜁2 ℎ𝑗𝑗 2 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) +
𝜁𝜁3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜁𝜁4 ] 𝑑𝑑𝑑𝑑
(3-1)
𝑑𝑑𝑑𝑑𝑟𝑟,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [𝜂𝜂1 ℎ𝑗𝑗 3 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜂𝜂2 ℎ𝑗𝑗 2 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) +
𝜂𝜂3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜂𝜂4 ] 𝑑𝑑𝑑𝑑
Fig. 1. (a) Dynamics of the milling process (b) dynamic chip thickness in regenerative chatter mechanism Unlike the previous works, a nonlinear model of cutting forces in milling process and its equivalent linear model are presented. Nonlinear dynamic modelling is essential for milling processes in which cutting forces are inherently nonlinear functions of chip thickness. On the other hand, linear modelling of the problem makes the design of robust control strategy to be more convenient. Therefore, after design of control strategy for the linear model, it is implemented on the more realistic nonlinear model. Results approve that the designed controller has an appropriate robust performance for both of the linear and nonlinear models.
(3-2)
where cutting force coefficients ζi , ηi , i = 1. .4 are found directly from experimental force signals, as presented in (Moradi, 2013a). In the presence of regenerative chatter, the chip thickness is expressed as: h (ϕj ) = [cf sinϕj + vj,0 − vj ] g (ϕj )
(4)
ϕst < ϕj < ϕex ϕst > ϕj or ϕex < ϕj
(5)
As presented by (Moradi, 2013b), g(ϕj ) is a unit step function determining whether the tooth is in or out of cut and is described in terms of start (ϕst ) and exit immersion (ϕex ) angles of the cutter as: g(ϕj ) = {
1 0
Since static part of the chip thickness has no effect on regeneration mechanism, and according to coordinates shown in Fig. 1, Eq. (4) is reduced to: h (ϕj ) = [Δx sinϕj + Δy cosϕj ] g (ϕj )
2. LINEAR/NONLINEAR MODELLING OF THE CUTTING FORCES IN MILLING PROCESS
(6)
where 𝛥𝛥𝛥𝛥 = 𝑥𝑥(𝑡𝑡) − 𝑥𝑥(𝑡𝑡 − 𝜏𝜏); 𝛥𝛥𝛥𝛥 = 𝑦𝑦(𝑡𝑡) − 𝑦𝑦(𝑡𝑡 − 𝜏𝜏);
2.1 Calculation of Cutting Forces in Classical Approach Dynamics of the milling process as a 2DOF model and the variation of chip thickness is shown in Fig. 1. The immersion angle is measured clockwise from the y-axis and the axial (𝑎𝑎) and radial (𝑤𝑤) depths of cut are constant. Assuming the bottom end of one flute as the reference immersion angle 𝜙𝜙. According to coordinates in Fig. 1, elemental forces in feed (x) and normal (y) directions are expressed in terms of tangential and radial forces as (Moradi, 2013b):
𝜏𝜏 = 2𝜋𝜋⁄(𝑁𝑁 𝛺𝛺)
(7)
[x(t), y(t)] and [x(t − τ), y(t − τ)] represent dynamic displacements of the cutter at the present and previous tooth periods and τ is the delay time; where Ω is the spindle speed in (rad⁄s). The extended nonlinear modelling of cutting forces and development of closed form expression through Fourier series expansion have been accomplished in the previous researches.
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Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
Development of the proposed extended nonlinear model has been experimentally accomplished in (Moradi, 2013a). Dynamics of regenerative chatter and internal resonance phenomenon , nonlinear dynamics and bifurcation analysis of the process in the presence of tool wear, process damping and cutting force nonlinearities and using a tunable vibration absorber suppress regenerative chatter have been investigated for the proposed extended nonlinear model. According to the formulation presented in (Moradi, 2012a, b), closed form expressions for the nonlinear cutting forces in x-y directions are explained as: 𝑁𝑁 {𝛼𝛼 ′1 𝛥𝛥𝑥𝑥 3 + 𝛽𝛽 ′1 𝛥𝛥𝑦𝑦 3 + 𝛼𝛼 ′ 2 𝛥𝛥𝑥𝑥 2 + 𝛽𝛽 ′ 2 𝛥𝛥𝑦𝑦 2 + 𝛼𝛼 ′ 3 𝛥𝛥𝛥𝛥 + 2𝜋𝜋 𝛽𝛽 ′ 3 𝛥𝛥𝛥𝛥 + 3𝛾𝛾 ′1 𝛥𝛥𝑥𝑥 2 𝛥𝛥𝛥𝛥 + 3𝛾𝛾 ′ 2 𝛥𝛥𝛥𝛥𝛥𝛥𝑦𝑦 2 + 2𝛾𝛾 ′ 3 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 + 𝛾𝛾 ′ 4 } 𝑁𝑁 {𝛼𝛼 𝛥𝛥𝑥𝑥 3 + 𝛽𝛽1 𝛥𝛥𝑦𝑦 3 + 𝛼𝛼2 𝛥𝛥𝑥𝑥 2 + 𝛽𝛽2 𝛥𝛥𝑦𝑦 2 + 𝛼𝛼3 𝛥𝛥𝛥𝛥 + 𝐹𝐹𝑥𝑥 = − 2𝜋𝜋 1 𝛽𝛽3 𝛥𝛥𝛥𝛥 + 3𝛾𝛾1 𝛥𝛥𝑥𝑥 2 𝛥𝛥𝛥𝛥 + 3𝛾𝛾2 𝛥𝛥𝛥𝛥𝛥𝛥𝑦𝑦 2 + 2𝛾𝛾3 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥+𝛾𝛾4 } (8)
𝐹𝐹𝑦𝑦 =
where for half immersion up-milling with 𝜙𝜙𝑠𝑠𝑠𝑠 = 0 and 𝜙𝜙𝑒𝑒𝑒𝑒 = 𝜋𝜋⁄2 (as for the case study here), coefficients of Eq. (8) are determined in terms of 𝜁𝜁𝑖𝑖 , 𝜂𝜂𝑖𝑖 𝑖𝑖 = 1. .4. Similarly, as the cutting force signals are measured, this approach can be extended to any other machining condition (for more details, see the work by Moradi et al. 2013a). Especially, the proposed modelling is essential for machining of materials in which cutting forces are inherently nonlinear functions of chip thickness (e.g., Titanium alloys), (as in the works by the Altintas, 2000; Wiercigroch & Budak, 2001; Stepan et al., 2005; Vela-Martinez, 2009; Moradi et al., 2013a). After determination of nonlinear cutting forces including delay regenerative chatter, nonlinear dynamics of the process is described as: 𝑚𝑚𝑥𝑥 𝑥𝑥̈ + 𝑐𝑐𝑥𝑥 𝑥𝑥̇ + 𝑘𝑘𝑥𝑥 𝑥𝑥 = 𝐹𝐹𝑥𝑥 (𝛥𝛥𝑥𝑥 𝑖𝑖 , 𝛥𝛥𝑦𝑦 𝑖𝑖 ) + 𝑢𝑢𝑥𝑥 𝑚𝑚𝑦𝑦 𝑦𝑦̈ + 𝑐𝑐𝑦𝑦 𝑦𝑦̇ + 𝑘𝑘𝑦𝑦 𝑦𝑦 = 𝐹𝐹𝑦𝑦 (𝛥𝛥𝑥𝑥 𝑖𝑖 , 𝛥𝛥𝑦𝑦 𝑖𝑖 ) + 𝑢𝑢𝑦𝑦
𝑖𝑖 = 1,2,3
(9)
where cutting forces Fx and Fy , as functions of time delay, are given by Eq. (8); ux and uy are the actuators forces (control inputs) implemented on the cutting tool structure. Also, it is assumed that there is only one effective direction for chatter vibration in x-y directions.
For simplicity, Eq. (11) is described in the simple modal form as: ̃𝑥𝑥 {𝛼𝛼3 𝛥𝛥𝛥𝛥 + 𝛽𝛽3 𝛥𝛥𝛥𝛥 + 𝛾𝛾4 } + 𝑢𝑢̂𝑥𝑥 𝑥𝑥̈ + 𝜇𝜇𝑥𝑥 𝑥𝑥̇ + 𝜔𝜔𝑥𝑥2 𝑥𝑥 = −𝑁𝑁 2 ̃𝑦𝑦 {𝛼𝛼′3 𝛥𝛥𝛥𝛥 + 𝛽𝛽′3 𝛥𝛥𝛥𝛥 + 𝛾𝛾′4 } + 𝑢𝑢̂𝑦𝑦 (12) 𝑦𝑦̈ + 𝜇𝜇𝑦𝑦 𝑦𝑦̇ + 𝜔𝜔𝑦𝑦 𝑦𝑦 = +𝑁𝑁 where, 𝑘𝑘𝑖𝑖 𝑁𝑁 ̃𝑖𝑖 = , 𝑁𝑁 , 𝑚𝑚𝑖𝑖 2𝜋𝜋 𝑚𝑚𝑖𝑖 2√𝑘𝑘𝑖𝑖 𝑚𝑚𝑖𝑖 𝑢𝑢̂𝑖𝑖 = 𝑢𝑢𝑖𝑖 ⁄𝑚𝑚𝑖𝑖 , 𝑖𝑖 = 𝑥𝑥, 𝑦𝑦
𝜇𝜇𝑖𝑖 = 2 𝜁𝜁𝑖𝑖 𝜔𝜔𝑖𝑖 , 𝜁𝜁𝑖𝑖 =
𝑐𝑐𝑖𝑖
, 𝜔𝜔𝑖𝑖2 =
To achieve an approximate dynamics of the delayed expressions in cutting forces, Eq. (12) is transferred into the Laplace domain and the first order Pade approximation as 𝑒𝑒 −𝑠𝑠𝑠𝑠 ≅ 1 − 𝑠𝑠𝑠𝑠 is used for delayed terms (i.e., a truncated Taylor series expansion for 𝑒𝑒 −𝑠𝑠𝑠𝑠 with one delayed term is used). Thereafter, applying the inverse Laplace transformation on the resulting equation, yields the approximated dynamics of the linear delayed system as: ̃𝑥𝑥 𝛼𝛼3 𝜏𝜏)𝑥𝑥̇ + 𝑁𝑁 ̃𝑥𝑥 𝛽𝛽3 𝜏𝜏𝑦𝑦̇ + 𝜔𝜔𝑥𝑥2 𝑥𝑥 = −𝑁𝑁 ̃𝑥𝑥 𝛾𝛾4 + 𝑢𝑢̂𝑥𝑥 𝑥𝑥̈ + (𝜇𝜇𝑥𝑥 + 𝑁𝑁 (13) 2 ′ ′ ̃𝑦𝑦 𝛽𝛽 𝜏𝜏)𝑦𝑦̇ − 𝑁𝑁 ̃𝑦𝑦 𝛼𝛼 3 𝜏𝜏 𝑥𝑥̇ + 𝜔𝜔𝑦𝑦 𝑦𝑦 = 𝑁𝑁 ̃𝑦𝑦 𝛾𝛾 ′ + 𝑢𝑢̂𝑦𝑦 𝑦𝑦̈ + (𝜇𝜇𝑦𝑦 − 𝑁𝑁 3 4 The validity of such approximation was investigated by Moradi et al. (2012b). According to Eq. (13), the transfer functions between the output vector [𝑥𝑥, 𝑦𝑦]𝑇𝑇 and control inputs [𝑢𝑢̂𝑥𝑥 , 𝑢𝑢̂𝑦𝑦 ]𝑇𝑇 are found as: 𝑋𝑋(𝑠𝑠) 𝐺𝐺 (𝑠𝑠) [ ] = [ 11 𝑌𝑌(𝑠𝑠) 𝐺𝐺21 (𝑠𝑠)
𝐺𝐺12 (𝑠𝑠) 𝑈𝑈𝑥𝑥∗ (𝑠𝑠) ] ][ 𝐺𝐺22 (𝑠𝑠) 𝑈𝑈𝑦𝑦∗ (𝑠𝑠)
where, 𝐺𝐺 (𝑠𝑠) 𝑮𝑮(𝒔𝒔) = [ 11 𝐺𝐺21 (𝑠𝑠)
𝛬𝛬 (𝑠𝑠) 𝐺𝐺12 (𝑠𝑠) 1 [ 4 ]= 𝛥𝛥(𝑠𝑠) −𝛬𝛬3 (𝑠𝑠) 𝐺𝐺22 (𝑠𝑠)
−𝛬𝛬2 (𝑠𝑠) ] (15-1) 𝛬𝛬1 (𝑠𝑠)
𝛬𝛬1 (𝑠𝑠) = 𝑠𝑠 2 + 𝜓𝜓𝑥𝑥 𝑠𝑠 + 𝜔𝜔𝑥𝑥2 , 𝛬𝛬2 (𝑠𝑠) = 𝜑𝜑𝑥𝑥 𝑠𝑠 𝛬𝛬3 (𝑠𝑠) = −𝜑𝜑𝑦𝑦 𝑠𝑠, 𝛬𝛬4 (𝑠𝑠) = 𝑠𝑠 2 + 𝜓𝜓𝑦𝑦 𝑠𝑠 + 𝜔𝜔𝑦𝑦2 (15-2) ̂𝑥𝑥 (𝑠𝑠) − 𝑁𝑁 ̃𝑥𝑥 𝛾𝛾4 ⁄𝑠𝑠 , 𝑈𝑈𝑦𝑦∗ (𝑠𝑠) = 𝑈𝑈 ̂𝑦𝑦 (𝑠𝑠) + 𝑁𝑁 ̃𝑦𝑦 𝛾𝛾′4 ⁄𝑠𝑠 𝑈𝑈𝑥𝑥∗ (𝑠𝑠) = 𝑈𝑈 𝛥𝛥(𝑠𝑠) = 𝑠𝑠 4 + 𝑎𝑎1 𝑠𝑠 3 + 𝑎𝑎2 𝑠𝑠 2 + 𝑎𝑎3 𝑠𝑠 + 𝑎𝑎4 , 𝑎𝑎1 = 𝜓𝜓𝑥𝑥 + 𝜓𝜓𝑦𝑦 , 𝑎𝑎2 = 𝜔𝜔𝑥𝑥2 + 𝜔𝜔𝑦𝑦2 + 𝜓𝜓𝑥𝑥 𝜓𝜓𝑦𝑦 + 𝜑𝜑𝑥𝑥 𝜑𝜑𝑦𝑦 , 𝑎𝑎3 = 𝜔𝜔𝑥𝑥2 𝜓𝜓𝑦𝑦 + 𝜔𝜔𝑦𝑦2 𝜓𝜓𝑥𝑥 , 𝑎𝑎4 = (𝜔𝜔𝑥𝑥 𝜔𝜔𝑦𝑦 )2 , ̃𝑥𝑥 , ̃𝑦𝑦 , 𝜑𝜑𝑥𝑥 = 𝛽𝛽3 𝜏𝜏𝑁𝑁 𝜑𝜑𝑦𝑦 = 𝛼𝛼 ′ 3 𝜏𝜏𝑁𝑁
2.3 Control Transfer Function for the Linear Model of Cutting Forces with Regenerative Chatter
(14)
̃𝑥𝑥 , 𝜓𝜓𝑥𝑥 = 𝜇𝜇𝑥𝑥 + 𝛼𝛼3 𝜏𝜏𝑁𝑁
(15-3)
̃𝑦𝑦 𝜓𝜓𝑦𝑦 = 𝜇𝜇𝑦𝑦 − 𝛽𝛽 ′ 3 𝜏𝜏𝑁𝑁
In the case of linear modelling of cutting forces, elemental forces in tangential and radial directions are described as (instead of Eq. (3)): 𝑑𝑑𝐹𝐹𝑡𝑡,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [ 𝜁𝜁3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜁𝜁4 ] 𝑑𝑑𝑑𝑑 𝑑𝑑𝐹𝐹𝑟𝑟,𝑗𝑗 (𝜙𝜙, 𝑧𝑧) = [𝜂𝜂3 ℎ𝑗𝑗 (𝜙𝜙𝑗𝑗 (𝑧𝑧)) + 𝜂𝜂4 ] 𝑑𝑑𝑑𝑑
(10)
Following the same procedure explained in the previous section, closed form expressions for the linear model of cutting forces are determined. Therefore, dynamics of the milling process with linear model of cutting forces and in the presence of regenerative chatter is described as: 𝑚𝑚𝑥𝑥 𝑥𝑥̈ + 𝑐𝑐𝑥𝑥 𝑥𝑥̇ + 𝑘𝑘𝑥𝑥 𝑥𝑥 = − 𝑚𝑚𝑦𝑦 𝑦𝑦̈ + 𝑐𝑐𝑦𝑦 𝑦𝑦̇ + 𝑘𝑘𝑦𝑦 𝑦𝑦 =
𝑁𝑁
𝑁𝑁
2𝜋𝜋
2𝜋𝜋
{𝛼𝛼3 𝛥𝛥𝛥𝛥 + 𝛽𝛽3 𝛥𝛥𝛥𝛥 + 𝛾𝛾4 } + 𝑢𝑢𝑥𝑥
{𝛼𝛼′3 𝛥𝛥𝛥𝛥 + 𝛽𝛽′3 𝛥𝛥𝛥𝛥 + 𝛾𝛾′4 } + 𝑢𝑢𝑦𝑦
(11)
Fig. 2. Schematic of the closed-loop control system in the peripheral milling process 3. DEVELOPMENT OF ROBUST CONTROL STRATEGY Closed-loop control system of the milling process is shown in Fig. 2. A control system is robust if it is insensitive to differences between the actual system and the model of the system which is used to design the controller. These differences are referred to as model uncertainty. The 𝐻𝐻∞ robust control technique is used to check if the design specifications
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are satisfied even for the worst-case uncertainty. General components of the 𝐻𝐻∞ control strategy are shown in Fig. 3, in which the variables 𝑟𝑟, 𝑦𝑦, 𝑒𝑒, 𝑢𝑢, 𝑑𝑑 and 𝑑𝑑𝑖𝑖 are the reference input, output variables, tracking error, control input, output disturbance and input disturbance, respectively. Dynamic plant and controller are denoted by 𝐺𝐺 and 𝐾𝐾 while 𝑊𝑊𝑚𝑚 , 𝑊𝑊𝑃𝑃 and 𝑊𝑊𝑢𝑢 are the weighting functions.
Sensitivity transfer function 𝑆𝑆 = (𝐼𝐼 + 𝐺𝐺𝐺𝐺)−1 is the transfer function of (𝑦𝑦⁄𝑑𝑑 ) and (𝑒𝑒⁄𝑟𝑟 ). The complementary sensitivity function,𝑇𝑇 = 𝐺𝐺𝐺𝐺(𝐼𝐼 + 𝐺𝐺𝐺𝐺)−1 , is the transfer function of (𝑦𝑦⁄𝑟𝑟 ). In addition, the control signal is explained as 𝑢𝑢 = 𝐾𝐾𝐾𝐾(𝑟𝑟 − 𝑑𝑑 − 𝐺𝐺𝑑𝑑𝑖𝑖 ). Consequently, to achieve disturbance rejection and appropriate transient performance, ‖𝑆𝑆‖∞ must be small while for the perfect tracking, |𝑇𝑇(𝑗𝑗𝑗𝑗)| must have unit value. Due to constraint 𝑆𝑆 + 𝑇𝑇 = 𝐼𝐼 (identity), the weighted sensitivity ‖𝑊𝑊𝑃𝑃 𝑆𝑆‖∞ and complementary sensitivity ‖𝑊𝑊𝑚𝑚 𝑇𝑇‖∞ functions are minimized. Also, to restrict the magnitude of the control effort, the upper bound 1/|𝑊𝑊𝑢𝑢 | on magnitude of 𝐾𝐾𝐾𝐾 must be small. To achieve these conditions, a stacking mixed sensitivity function is used as (Skogestad, 2005): ‖𝑁𝑁‖∞ = max𝜎𝜎̅(𝑁𝑁(𝑗𝑗𝑗𝑗)) < 1;
𝑁𝑁 = ‖𝑊𝑊𝑝𝑝 𝑆𝑆
𝜔𝜔
𝑊𝑊𝑢𝑢 𝐾𝐾𝐾𝐾
𝑊𝑊𝑚𝑚 𝑇𝑇‖∞
(16)
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force coefficients of nonlinear Eq. (3) are found as (Moradi et al. 2013b): 𝜁𝜁1 = 5600 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚3 , 𝜁𝜁2 = −4893 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚2 , 𝜁𝜁3 = 2736 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚 , 𝜁𝜁4 = −81 𝑎𝑎𝑐𝑐 𝑁𝑁 (18) 𝜂𝜂1 = 6958 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚 𝑚𝑚3 , 𝜂𝜂2 = −5754 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚2 , 𝜂𝜂3 = 1752 𝑎𝑎𝑐𝑐 𝑁𝑁⁄𝑚𝑚𝑚𝑚 , 𝜂𝜂4 = −31 𝑎𝑎𝑐𝑐 𝑁𝑁 Similarly, the above procedure can be used for other machining conditions in which cutting force signals are measured experimentally. For the nonlinear model, helical cutter is divided into small discrete elements along the cutter axis, and cutting force components are calculated by digital integration of Eq. (3) and then using Eqs. (1) and (2). Total cutting force is found as: 𝐹𝐹(𝜙𝜙) = √𝐹𝐹𝑥𝑥 2 (𝜙𝜙) + 𝐹𝐹𝑦𝑦 2 (𝜙𝜙)
(19)
There is good agreement between linear and nonlinear models for a light Aluminium material (𝐴𝐴𝐴𝐴7075); which shows weakly nonlinear characteristic, However, in the case of highly nonlinear materials (such as Titanium alloys), the difference between linear and nonlinear models of cutting forces is significant. (Moradi et al. 2013b) For simulation of the problem and design of controller, nominal realistic values for model parameters of Eq. (9) are extracted from Moradi et al. (2012b). Furthermore, number of tool teeth are assumed to be 𝑁𝑁 = 4; resulting in 𝜏𝜏 = 0.005 𝑠𝑠. The force parameters for the linear and nonlinear models are given by Eqs. (17) and (18), respectively. The open-loop system of milling process (Eq. (9) with 𝑢𝑢𝑥𝑥 = 𝑢𝑢𝑦𝑦 = 0) is simulated in time domain via SIMULINK Toolbox of MATLAB. The critical depth cut in which the system changes from the stable condition to the unstable one is found as 𝑎𝑎𝑙𝑙𝑙𝑙𝑙𝑙 = 3.1 𝑚𝑚𝑚𝑚.
The stabilizing 𝐻𝐻∞ optimal controller is determined by minimizing the function ‖𝑁𝑁(𝐾𝐾)‖∞ . The conditions of nominal stability (NS) and performance (NP); robust stability (RS) and performance (RP) are discussed by Moradi et al. (2015).
4.2 Characteristics of the Weighting Functions and Design of Robust Controller Consider the matrix transfer function 𝐆𝐆 (𝐬𝐬) given by Eq. (10) and nominal values of parameters given by Eqs. (17), (18) and (19). To study a real case of uncertain model, uncertain parameters are changed around their nominal values as:
Fig. 3. Components of the robust H∞ control design 4. IMPLEMENTATION OF ROBUST CONTROL, RESULTS AND DISCUSSIONS
̅ ≤ Θ ≤ (1 + 𝜒𝜒) Θ ̅ (1 − 𝜒𝜒) Θ
4.1 Specifications of the Milling Process and Cutting Conditions To simulate the problem and investigate the performance of robust controller, a typical case study with the experimental cutting forces in tangential and radial directions is considered. Plotting the forces 𝐹𝐹𝑡𝑡 and 𝐹𝐹𝑟𝑟 versus chip thickness ℎ; using linear regression via Curve Fitting Toolbox of MATLAB (CFTM), applied on Eq. (3) in un-differential form, yields the cutting force coefficients for an orthogonal cutting test, e.g., for more details see Moradi, et al. (2013a): 𝐾𝐾𝑡𝑡𝑡𝑡 = 1445 𝑎𝑎𝑐𝑐 , 𝐾𝐾𝑟𝑟𝑟𝑟 = 311.4 𝑎𝑎𝑐𝑐 𝑁𝑁. 𝑚𝑚𝑚𝑚−1 𝐾𝐾𝑡𝑡𝑡𝑡 = 15.6 𝑎𝑎𝑐𝑐 , 𝐾𝐾𝑟𝑟𝑟𝑟 = 72.4 𝑎𝑎𝑐𝑐 𝑁𝑁
(17)
where 𝑎𝑎𝑐𝑐 is the axial depth of cut (in 𝑚𝑚𝑚𝑚). Again plotting the forces 𝐹𝐹𝑡𝑡 and 𝐹𝐹𝑟𝑟 versus chip thickness ℎ, and using cubic regression via CFTM (with a third order polynomial), cutting
(20)
where 𝜒𝜒 is the uncertainty amount in percent and Θ can take ̅ is the any of parameters given by Eqs. (17), (18) and (19); Θ nominal value of Θ. In many practical cases, the configuration of multiplicative input uncertainty is used to represent the various sources of dynamic uncertainty (e.g., see Skogestad & Postlethwaite, 2005); where the perturbed plant 𝐆𝐆𝐩𝐩 (𝐬𝐬) is described as: 𝐺𝐺𝑝𝑝 (𝑠𝑠) = 𝐺𝐺 (𝑠𝑠) [1 + 𝑊𝑊𝑚𝑚 (𝑠𝑠) Δ𝐼𝐼 (𝑠𝑠)]; ⏟|Δ𝐼𝐼 (𝑗𝑗𝑗𝑗)| ≤ 1 ∀𝜔𝜔 (21) ‖Δ𝐼𝐼 ‖∞ ≤1
𝐺𝐺 (𝑠𝑠) is the nominal plant and Δ𝐼𝐼 (𝑠𝑠) is any stable transfer function with a magnitude less or equal to 1, at each frequency. Uncertain plant 𝐺𝐺𝑝𝑝 (𝑠𝑠) is replaced by a disc type approximation 𝐺𝐺′𝑝𝑝 (𝑠𝑠) of radius |𝑊𝑊𝑚𝑚 (𝑗𝑗𝑗𝑗) 𝐺𝐺(𝑗𝑗𝑗𝑗)| where the weight function 𝑊𝑊𝑚𝑚 (𝑠𝑠) is determined as Skogestad (2005):
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Γ𝐼𝐼 (𝜔𝜔) = max |
𝐺𝐺𝑝𝑝 (𝑗𝑗𝑗𝑗)
𝐺𝐺𝑝𝑝 ∈Π 𝐺𝐺(𝑗𝑗𝑗𝑗)
− 1| ,
Hamed Moradi et al. / IFAC PapersOnLine 52-13 (2019) 1102–1107
|𝑊𝑊𝑚𝑚 (𝑗𝑗𝑗𝑗)| ≥ Γ𝐼𝐼 (𝜔𝜔), ∀𝜔𝜔
(22)
where Π is the set of possible perturbed plants. According to Eqs. (14), (15) and (20), there are 25 cases of perturbed plants for 𝐺𝐺12 (𝑠𝑠) and 𝐺𝐺21 (𝑠𝑠) and 26 cases of perturbed plants for 𝐺𝐺11 (𝑠𝑠) and 𝐺𝐺22 (𝑠𝑠). The weight function of uncertainty (𝑊𝑊𝑚𝑚 ), upper bound on relative errors, is found as: 1 𝑊𝑊𝑚𝑚 (𝑠𝑠) = 10 [ 1
1 ] 1
(23)
At the low frequencies, the weight function 1⁄|𝑊𝑊𝑝𝑝 | as the upper bound on the sensitivity function |𝑆𝑆|, is equal to the parameter 𝐴𝐴 (typically𝐴𝐴 ≈ 0) in correlation with the steady state tracking error. At the high frequencies, it approaches to the maximum peak amplitude 𝑀𝑀 ≥ 1 ; and the bandwidth frequency (𝜔𝜔𝐵𝐵 ) indicates the speed of time response. To have a desired step response with a maximum overshoot of 10 percent, the rise time of about 0.2s, and a perfect tracking (error≈0), the appropriate performance weight 𝑊𝑊𝑝𝑝 is defined as: 𝑊𝑊𝑃𝑃 (𝑠𝑠) =
𝑠𝑠/1.1+10
𝑠𝑠+8×10−5
Fig. 4. Tool tip vibrations of the open-loop system in the presence of 30 percent uncertainty.
(24)
Using a 𝜇𝜇-synthesis code and DK-iteration algorithm developed in Robust Control Toolbox of MATLAB, 𝐻𝐻∞ robust controllers of 6-order are obtained. Using the same toolbox and plotting their Bode diagram, these controllers are approximated with reduced third order controllers as: 𝑘𝑘 (𝑠𝑠) 𝑘𝑘12 (𝑠𝑠) ] 𝐾𝐾(𝑠𝑠) = [ 11 𝑘𝑘21 (𝑠𝑠) 𝑘𝑘22 (𝑠𝑠) 𝑘𝑘11 (𝑠𝑠) = 830 (𝑠𝑠 + 2070)(𝑠𝑠 + 70)⁄Φ(𝑠𝑠) ; 𝑘𝑘12 (𝑠𝑠) = −390 (𝑠𝑠 + 2120)(𝑠𝑠 + 60)⁄Φ(𝑠𝑠) 𝑘𝑘21 (𝑠𝑠) = −5 (𝑠𝑠 + 3820)(𝑠𝑠 + 1230)⁄Φ(𝑠𝑠); 𝑘𝑘22 (𝑠𝑠) = 390⁄𝑠𝑠 ; Φ(𝑠𝑠) = 𝑠𝑠(𝑠𝑠 2 + 115𝑠𝑠 + 3.65 × 105 )
(25)
4.3 Performance of the Roust Controller in Chatter Suppression of the Linear/Non-linear Models Figure 4 shows time response of the tool tip vibrations of the open-loop system with linear/nonlinear models of cutting forces, in the presence of uncertainty (e.g., 30% uncertainty𝜒𝜒 = 0.3 in Eq. (20)). As it is observed, unstable conditions occur in the presence of uncertainties (without controller). Moreover, vibrations amplitudes are larger in the nonlinear model because the existence of high order delayed terms (Δ𝑥𝑥 𝑖𝑖 , Δ𝑦𝑦 𝑖𝑖 , Δ𝑥𝑥 Δ𝑦𝑦 𝑗𝑗 , Δ𝑥𝑥 2 Δ𝑦𝑦; 𝑖𝑖 = 2,3; 𝑗𝑗 = 1,2) induce a more unstable response. The effect of robust controller on chatter suppression of the linear/nonlinear model and in the presence of 10%, 30% and 50% uncertainty is shown in Fig. 5. As it is observed, for both uncertain models of cutting forces, robust controller acts efficiently in chatter suppression. Moreover, as the amount of uncertainty increases, chatter is suppressed in longer time. The required actuator forces for chatter suppression in linear and nonlinear models are shown in Fig. 6. According to these figures and as physically expected, as the amount of uncertainty increases, more actuator forces are required for chatter suppression.
Fig. 5. Tool tip vibrations of the milling process after implementation of the optimal robust controller in the presence of uncertainty. For a given uncertainty, according to Fig. 5, chatter is suppressed in shorter time with less vibration amplitudes in the case of linear model. Moreover, as shown in Fig. 6, less actuator forces are required to suppress chatter in the case of linear model. This is because the existence of high order delayed terms in the nonlinear model.
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Fig. 6. Required actuator force to suppress regenerative chatter vibrations of the milling process; in the presence of uncertainty. 5. CONCLUSIONS In this paper, an extended nonlinear model of the milling process is considered in which the cutting forces are described as a third-order nonlinear function of chip thickness (through coupled nonlinear delay differential equations). The proposed modelling is more essential for the machining of materials in which the cutting forces are inherently nonlinear functions of chip thickness. To achieve a more realistic model, uncertainties associated with the machine-tools and cutting parameters are also included. To suppress the regenerative chatter, an 𝐻𝐻∞ robust control strategy is developed based on 𝜇𝜇 -synthesis and DK-iteration algorithm. In the presence of uncertainties, the robust performance and robust stability of the control system is investigated and compared for the linear and nonlinear models. REFERENCES Altintas Y, Stepan G, Merdol D and Dombovari Z (2008) Chatter stability of milling in frequency and discrete time domain. CIRP Journal of Manufacturing Science and Technology 1: 35-44. Armarego EJA and Deshpande NP (1991) Computerized endmilling force predictions with cutting models allowing eccentricity and cutter deflections. Ann. CIRP 40: 25-29. Bayoumi AE, Yucesan, G and Kendall LA (1994) An analytic mechanistic cutting force model for milling operations: a theory and methodology. Trans. ASME Journal of Engineering Industry 116(8): 324-30.
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Budak E (2006) Analytical models for high performance milling, part I: cutting forces, deflections and tolerance integrity. Int. Journal of Machine Tools and Manufacture 46: 1478-88. Carrillo FJ, Rotella F and Zadshakoyan M (1999) Delta approach robust controller for constant turning force regulation. Journal of Control Engineering Practice 7: 1321-1331. Chiou R and Liang SY (1998) Chatter stability of a slender cutting tool in turning with tool wear effect. Int. Journal of Machine Tools & Manufacture 38(4): 315-327. Koeinsberger F and Tlusty J (1970) Machine Tool Structures. First ed. New Jersey: Pergamon press. Moradi H, Movahhedy MR and Vossoughi GR (2012a) Dynamics of regenerative chatter and internal resonance in milling process with structural and cutting force nonlinearities. Journal of Sound and Vibration 331(16): 3844-3865. Moradi H, Movahhedy MR and Vossoughi GR (2012b) Bifurcation analysis of milling process with tool wear and process damping: Regenerative chatter with primary resonance. Journal of Nonlinear Dynamics 70(1): 481-509 Moradi H, Vossoughi GR and Movahhedy M (2013a) Experimental dynamic modelling of milling with process damping, structural and cutting force nonlinearities. Journal of Sound and Vibration 332: 4709-4731. Moradi H, Vossoughi GR, Movahhedy MR and Salarieh H (2013b) Suppression of nonlinear regenerative chatter in milling process via robust optimal control. Journal of Process Control 23(5): 631-648. Moradi H and Vossoughi GR (2015) Robust control of the variable speed wind turbines in the presence of uncertainties: A comparison between H„ and PID controllers. Energy (90): 1508–1521. Nayfeh AH and Nayfeh NA (2012) Time-delay feedback control of lathe cutting tools. Journal of Vibration and Control 18(8): 1106-1115 Park Y, Kim TY, Woo J, Shin D, Kim J (1998) Sliding mode cutting force regulator for turning processes. Int. Journal of Machine Tools & Manufacture 38(8): 911-930. Parus A, Powałka B, Marchelek K, Domek S and Hoffmann M (2013) Active vibration control in milling flexible workpieces. Journal of Vibration and Control 19(7): 1103-1120. Skogestad S and Postlethwaite I (2005) Multivariable Feedback Control, New York: John Wiley and Sons. Van Dijk NJM, Van De Wouw N, Doppenberg EEJ, Oosterling HAJ and Nijmeijer H (2012) Robust active chatter control in the high-speed milling process. IEEE Trans. on Control Systems Technology 20(4): 901-917 Vela-Martínez L, Jáuregui-Correa JC, González-Brambila OM, Herrera-Ruiz G and Lozano-Guzmán A (2009) Instability conditions due to structural nonlinearities in regenerative chatter. Journal of Nonlinear Dynamics 56: 415–27. Zuperl U, Cus F and Reibenschuh M (2012) Modeling and adaptive force control of milling by using artificial techniques. Journal of Intelligent Manufacturing 23(5): 1805-1815.
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