The Effect of Torsional Vibrations on Metal Cutting Dynamics

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ScienceDirect ScienceDirect Procedia CIRP 00 00 (2018) 000–000 000–000 Procedia Procedia CIRP CIRP 00 (2018) (2018) 000–000

Procedia CIRP 00 000–000 Procedia CIRP 00 (2017) 000–000 Procedia CIRP 00 (2018) (2018) 000–000 Procedia CIRP 77 (2018) 461–464

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8th CIRP CIRP Conference on on High Performance Performance Cutting (HPC (HPC 2018) 8th 8th CIRP Conference Conference on High High Performance Cutting Cutting (HPC 2018) 2018) 8th CIRP Conference on High Performance Cutting 8th CIRP Conference on High Performance Cutting (HPC (HPC 2018) 2018)

The Effect of Torsional Vibrations on Metal Cutting Dynamics The Effect of Torsional Vibrations on Metal Cutting Dynamics The Effect of Torsional Vibrations on Metal Cutting Dynamics The Effect 28th of Torsional Vibrations on Metal Cutting Dynamics a,* Conference, May b a,* aa CIRP Otto Design Nantes, France Andreas , Martin Kolouchb2018, , G¨unter Radons ∗∗ ∗ ∗∗ ∗

Andreas , Martin Kolouchb , G¨u Andreas Otto Ottoa,* unter nter Radons Radonsaa a,* , Martin Kolouchb , G¨ Andreas Otto , Martin Kolouch , G¨ u nter Andreas Otto , Martin Kolouch , G¨unter Radons Radons

a,* b a aa Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany a Institute Institute of of Physics, Physics, Chemnitz Chemnitz University University of of Technology, Technology, 09107 09107 Chemnitz, Chemnitz, Germany Germany Instituteaa Institute for Machine Machine Tools and and Forming Technology IWU, Reichenhainer Reichenhainer Straße 88, 88, 09126 Chemnitz, Chemnitz, Institute for Tools Forming Technology IWU, Straße 09126 of Chemnitz University of 09107 Germany of Physics, Physics, Chemnitz University of Technology, Technology, 09107 Chemnitz, Chemnitz, Germany Institutea Institute for Machine Tools and Forming Technology IWU, Reichenhainer Straße 88, 09126 Chemnitz,

Fraunhofer Germany Fraunhofer Germany A new methodology to analyze the functional and physical architecture of Fraunhofer Germany Corresponding author. Tel.: +49-371-531-37717; fax: +49-371-5318-37717. E-mail address: [email protected] Fraunhofer Institute for Machine Tools and Forming Technology IWU, Reichenhainer Straße 88, 09126 Chemnitz, Germany Corresponding author. Tel.: fax: +49-371-5318-37717. E-mail address: [email protected] Fraunhofer Institute for Machine Tools and Forming Technology Reichenhainer Straße 88, 09126 Chemnitz, Germany Corresponding author. Tel.: +49-371-531-37717; +49-371-531-37717; fax: +49-371-5318-37717. E-mailIWU, address: [email protected] existing products for an assembly oriented product family identification Corresponding Corresponding author. author. Tel.: Tel.: +49-371-531-37717; +49-371-531-37717; fax: fax: +49-371-5318-37717. +49-371-5318-37717. E-mail E-mail address: address: [email protected] [email protected] bb b bb b

Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat Abstract Abstract Abstract Abstract Abstract École Nationale Supérieure d’Arts et Métiers, Arts et cutting Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metzmodel 57078, is France The effect of torsional vibrations on the dynamics of metal processes is studied. An extended dynamic process presented, where The effect torsional vibrations on dynamics of cutting is extended dynamic process is presented, where The effect of ofangle torsional vibrations on the the dynamics of metal metal resulting cutting processes processes is studied. studied. An An extended dynamic process model model isStability presented, where the spindle and the time delay are dynamic variables in delay differential equations with state-dependent delay. lobes are The effect of torsional vibrations on the dynamics of metal cutting processes is studied. An extended dynamic process model is presented, where the spindle angle and the time delay are dynamic variables resulting in delay differential equations with state-dependent delay. Stability lobes are The effect ofangle torsional vibrations on the dynamics of metal resulting cutting processes is studied. An extended dynamic process model isStability presented, where the spindle and the time delay are dynamic variables in delay differential equations with state-dependent delay. lobes are calculated based on the new extended process model and compared with results from the classical chatter theory with constant spindle speed and the spindlebased angle andthe thenew time delay are dynamic variables resulting with in delay delay differential equations with state-dependent state-dependent delay. Stability Stability lobesand are calculated on extended process model and compared results from the classical chatter theory with constant spindle speed *the Corresponding author. Tel.: +33 3delay 87 37are 54 30; E-mail address: [email protected] spindle angle and the time dynamic variables resulting in differential equations with delay. lobes are calculated based on the new extended process model and compared with results from the classical chatter theory with constant spindle speed and delay. For For based an ideal ideal symmetric tool, stable stable cutting is and characterized by periodic spindle speed but butchatter still aatheory constant delay. In this this case,speed the main main calculated on the process model compared with results the with constant spindle and delay. an tool, cutting is characterized aaa periodic spindle speed still constant delay. In case, the calculated based on symmetric the new new extended extended process model and compared by with results from from the classical classical chatter theory with constant spindle speed and delay. For an ideal symmetric tool, stable cutting is characterized by periodic spindle speed but still a constant delay. In this case, the main difference between the new extended and thecutting classical model is a different periodic solution to periodic torsional tool displacements, which results delay. For an symmetric tool, is by spindle speed but aa constant delay. In the main difference new and classical model is different periodic solution to periodic torsional tool displacements, which delay. Forbetween an ideal idealthe symmetric tool, stable stable cutting is characterized characterized by aa periodic periodic spindle speed but still still constant delay. In this this case, case, theresults main difference between the new extended extended and the thelobes. classical modelthe is aanew different periodic solution to periodic torsional tool displacements, which results only in slight modifications of the stability However, theory shows that it is important to consider tool asymmetries (e.g. runout), difference between the new extended and the classical model is a different periodic solution to periodic torsional tool displacements, which results only in slight modifications of the stability lobes. However, the new theory shows that it is important to consider tool asymmetries (e.g. runout), difference between the new extended and the classical model is a different periodic solution to periodic torsional tool displacements, which results only in slight modifications of the stability lobes. However, the new theory shows that it is important to consider tool asymmetries (e.g. runout), because this would would result in in of periodically varying delays during stable cuttingshows and larger larger stability changes due to to torsional vibrations.(e.g. runout), Abstract only in modifications the lobes. However, the new that is to tool because this result periodically varying during cutting and changes due vibrations. only in slight slight modifications the stability stability lobes.delays However, thestable new theory theory that it itstability is important important to consider consider tool asymmetries asymmetries because this would result in of periodically varying during stable cuttingshows and larger stability changes due to torsional torsional vibrations.(e.g. runout), c 2018  2018 The The Authors. Published by Elsevier Elsevier Ltd. delays c because this would result in periodically varying delays during stable cutting and larger stability changes due to torsional  Authors. Published by Ltd. because this would result in periodically varying delays during stable cutting and larger stability changes due to torsional vibrations. vibrations. c 2018 The  Authors. Ltd. © Published by Elsevier Peer-review under environment, thePublished responsibility of the the International Scientific Committee of the the 8th Conference on High Performance Cutting  2018 The Authors. by Elsevier Ltd. Peer-review under the responsibility of International Scientific Committee of 8th CIRP Conference Performance Cutting In business theElsevier trend towards morelicense product variety and customization isCIRP unbroken. Due to on thisHigh development, the need of cc today’s  2018 The Authors. Published by Ltd. Peer-review under the responsibility of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting This is an open access article under the CC BY-NC-ND (https://creativecommons.org/licenses/by-nc-nd/4.0/) (HPCand 2018). Peer-review under the responsibility of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting (HPC 2018). agile reconfigurable production systems emerged to cope with various products and product families. To design and optimize production Peer-review under the responsibility of the International Scientific Scientific CommitteeCommittee of the 8thofCIRP Conference on Highon Performance Cutting (HPC 2018). Selection and peer-review under responsibility of the International the 8th CIRP Conference High Performance (HPC 2018). systems as well asCutting; to choose the Chatter; optimalVibrations; product matches, product analysis methods are needed. Indeed, most of the known methods aim to (HPC 2018). Cutting (HPC 2018). Metal Sawing; Stability; State-dependent Keywords: Metal Cutting; Sawing; Chatter; Vibrations; Stability; State-dependent delay; delay; Keywords: Metal or Cutting; Sawing;family Chatter;on Vibrations; Stability; delay;families, however, may differ largely in terms of the number and Keywords: analyze a product one product the physical level.State-dependent Different product Metal Cutting; Sawing; Chatter; Vibrations; Stability; State-dependent delay; Keywords: Metal Cutting; Sawing; Chatter; Vibrations; Stability; State-dependent delay;of appropriate product family combinations for the production Keywords: nature of components. This fact impedes an efficient comparison and choice system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster hand, an spindle speed is possible, which 1. these products in new assembly oriented product families for the optimization of existing assembly lines and variation the creation future reconfigurable hand, an active active spindle speed variation is of possible, which leads leads 1. Introduction Introduction hand, an active spindle speed variation is possible, which leads 1. Introduction to a continuous time-dependent variation of the delay and can assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and hand, an active spindle speed variation is possible, which 1. Introduction to aa continuous continuous time-dependent variation of the the delay delay andleads can hand, an active spindle speed variation is possible, which leads 1. Introduction to time-dependent variation of and can a functional analysis is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the be used to stabilize cutting processes [5–7]. Whereas in the latIncreasing quality requirements and decreasing product life to a continuous time-dependent variation of the delay and be aused used to stabilize stabilize cutting processes processes [5–7].ofWhereas Whereas inand the can latIncreasing quality quality requirements requirements and and decreasing decreasing product product life life to continuous time-dependent variation the delayin can be to cutting [5–7]. the latIncreasing similarity between product families by providing design support to both, production system planners and product designers. An illustrative ter two cases the time delay is determined a priori by one or cycles demand flexible metal cutting processes at the perforbe used to stabilize cutting processes [5–7]. Whereas in the latIncreasing quality requirements and decreasing product life ter used two to cases the time time delay is determined determined priori by by onelator cycles demandquality flexible metal cutting cutting processes at product the perforperforbe stabilize cutting processes [5–7]. Whereas in the Increasing requirements andprocesses decreasingat life two cases delay is aasteering priori one or cycles demand flexible metal the example of a nail-clipper ismachine used tocutting explain the proposed methodology. Anter industrial case the study on two product families of columns of more ’external’ parameters of the system, the situation is commance limit of modern tools. From a dynamical point ter two cases the time delay is determined a priori by one or cycles demand flexible metal processes at the performore ’external’ parameters of the system, the situation is commance limit of modern machine tools. From aa dynamical point ter two cases the time delay is determined a priori by one or cycles demand flexible metal cutting processes at the performore ’external’ parameters of the system, the situation is commance limit of modern machine tools. From dynamical point thyssenkrupp Presta Francemachine is then carried out to give astable first industrial evaluation ofdifferent the proposed approach. pletely if passive spindle speed variations are taken of view, an ideal cutting process, which we call cutting, more ’external’ parameters of the system, the situation is commance limit of modern tools. From a dynamical point pletely different if passive spindle speed variations are taken of view, an ideal cutting process, which we call stable cutting, more ’external’ parameters of the system, the situation is commance limit of modern machine tools. From a dynamical point pletely different if passive spindle speed variations are taken of view, an ideal cutting process, which we call stable cutting, ©is2017 The Authors. Published by Elsevier B.V. into account. We passive speed variations as characterized periodic motion with awesmall ampletely different if passive spindle speed are of view, ideal cutting process, which call stable into account. We define spindle speed as is by periodic motion with small stationary pletely different if define passive spindlespindle speed variations variations are taken taken of view, an anunder idealby cutting process, call stationary stableofcutting, cutting, intoDesign account. We define passive spindle speed variations variations as is characterized characterized by periodic motion with aawe small stationary amPeer-review responsibility of thewhich scientific committee theam28th CIRP Conference 2018. passive

plitude. However, practice often undesired vibrations is characterized by periodic motion aa small stationary amplitude. However, in practice often undesired large vibrations is characterized by in periodic motion with smalllarge stationary amplitude. However, in practice oftenwith undesired large vibrations limit the productivity because they lead to bad surface finish, plitude. However, in practice often undesired large vibrations limit the the However, productivity because they to surface finish, undesired vibrations Keywords: Assembly; Design method;often Familylead identification plitude. in practice limit productivity because they lead to bad badlarge surface finish, increased tool wear and noise. They occur stable cutting belimit the because to surface finish, increased tool noise. They occur if stable cutting belimit the productivity productivity because they lead toif bad surface finish, increased tool wear wear and and noise. they Theylead occur ifbad stable cutting become unstable. In this case, the vibration amplitude increases increased tool wear and noise. They occur if stable cutting become In vibration increases increased tool wear andcase, noise.the occur amplitude if stable cutting become unstable. unstable. In this this case, theThey vibration amplitude increases until nonlinearities such as the loss of contact between the tool come unstable. In this case, the vibration amplitude increases until such as of between the come unstable. In this vibration amplitude until nonlinearities nonlinearities suchcase, as the thetheloss loss of contact contact betweenincreases the tool tool and the workpiece become become limiting factor. The resulting large 1.until Introduction nonlinearities such loss contact between the tool and workpiece limiting The resulting until nonlinearities such as as aaathe the loss of offactor. contact between thelarge tool and the the workpiece become limiting factor. The resulting large amplitude vibrations are called chatter [1,2]. The main reason and the workpiece become a limiting factor. The resulting large amplitude vibrations are called chatter [1,2]. The main reason and the workpiece become a limiting factor. resulting large amplitude vibrations are called chatter [1,2].The The main reason for chatter isthe the regenerative regenerative effect. Ainwavy wavy surface is left left Due tovibrations fastare the domain of amplitude called chatter The main for is A is amplitude are development called effect. chatter [1,2]. [1,2]. Thesurface main reason reason for chatter chattervibrations is the the regenerative effect. A wavy surface is left on the workpiece due to small vibrations of the structure. The communication and an ongoing trend of digitization and for chatter is the regenerative effect. A wavy surface is left on the workpiece due to small vibrations of the structure. The for chatter is the due regenerative effect. A of wavy surface is The left on the workpiece to small vibrations the structure. repetitive cutting of theto same wavy surface results under some digitalization, manufacturing are facing important on the due small vibrations of the structure. The repetitive cutting wavy surface under on the workpiece workpiece duethe smallenterprises vibrations ofresults the structure. The repetitive cutting of of thetosame same wavy surface results under some some conditions in positive net amount of mechanical energy leadchallenges in a today’s market environments: a energy continuing repetitive of same wavy surface results under some conditions in amount of mechanical leadrepetitive cutting of the the net same wavy surface results under some conditionscutting in aa positive positive net amount of mechanical energy leading to the increasing amplitude of mechanical vibrations. tendency towards reduction of product development times and conditions in a positive net amount of mechanical energy ing to the increasing amplitude of mechanical vibrations. conditions in a positive net amount of mechanical energy leadleading to the increasing amplitude of mechanical vibrations. Mechanical models for the prediction of chatter vibrations shortened product lifecycles. In addition, there is an increasing ing to the increasing amplitude of mechanical vibrations. Mechanical models for the prediction of chatter vibrations ingMechanical to the increasing amplitude of mechanical vibrations. models for the prediction of chatter vibrations areMechanical helpful during the development development of new new machine tools and demand of customization, at the same time invibrations a global models for the chatter are helpful the of tools models for being the prediction prediction ofmachine chatter vibrations areMechanical helpful during during the development of newof machine tools and and for the optimization of cutting process parameters. An essencompetition with competitors all over the world. This trend, are helpful during the development of new machine tools and for the optimization of cutting process parameters. An essenare helpful during theofdevelopment of new machineAn tools and for the optimization cutting process parameters. essential parameter of the models is the time delay of the regenerfor the optimization of cutting process parameters. An essenwhich is inducing the development from macro to micro tial parameter of the models is the time delay of the regenerfor optimization cuttingis process essential the parameter of the of models the timeparameters. delay of theAnregenerative effect, which isdiminished the time timeisbetween between two subsequent cuts at at tial of the delay of regenermarkets, results lot time sizes due to augmenting ative effect, which is the two subsequent cuts tial parameter ofinthe the models the time delay of the the regenerativeparameter effect, which is models the timeis between two subsequent cuts at the same workpiece location on the workpiece surface. The ative effect, which is the time between two subsequent cuts at product varieties (high-volume to low-volume production) [1]. the same workpiece location on the workpiece surface. The ative effect, which is location the time on between two subsequent at the same workpiece the workpiece surface.cuts The classical chatter theory based on an ideally constant delay (and the same workpiece location on the workpiece surface. The To cope with this augmenting variety as well as to be able to classical chatter theory based on an ideally constant delay (and the samechatter workpiece location onan theideally workpiece surface. The classical theory based on constant delay (and constant spindle speed) is well-developed well-developed [1,3,4]. Ondelay the other classical chatter based on ideally constant (and identify possible optimization potentials in the existing constant speed) is On the other classical chatter theory theory based on an an ideally[1,3,4]. constant (and constant spindle spindle speed) is well-developed [1,3,4]. Ondelay the other constant spindle speed) is well-developed [1,3,4]. On the other production system, it is important to have a precise knowledge constant spindle speed) is well-developed [1,3,4]. On the other c 2018 The Authors. Published by Elsevier Ltd. 2212-8271 

process-induced speed variations, due tospeed torsional displaceinto account. define passive spindle variations as process-induced speed variations, e.g. due torsional displaceinto account. We We define passivee.g. spindle variations as process-induced speed variations, e.g. due to tospeed torsional displacements of of the the tool. tool.speed Significant torsional displacements occur, for process-induced variations, e.g. due to torsional displacements Significant torsional displacements occur, for process-induced variations, e.g. due to torsionaloccur, displacements of the tool.speed Significant torsional displacements for example, in face face milling and sawing sawing operations [8,9]. Their Their oriments of tool. Significant torsional displacements occur, for example, in and operations [8,9]. ments of the the tool.milling Significant torsional displacements occur,orifor example, in face milling and sawing operations [8,9]. Their origins are, one the one hand, torsional flexibility in the tool/tool example, in face milling and sawing operations [8,9]. Their origins are, one the one hand, torsional flexibility in the tool/tool example, in face andtorsional sawing operations origins are, one themilling one hand, flexibility[8,9]. in theTheir tool/tool holder/spindle system and, torsional on the the other hand, limiting limiting torque gins are, one flexibility in holder/spindle system and, on hand, torque gins are, one one the the one hand, hand, flexibility in the the tool/tool tool/tool holder/spindle system and, torsional on the other other hand, limiting torque of spindle drive for compensating the process torque via the of the product range and characteristics manufactured and/or holder/spindle system and, on the other hand, limiting torque of the spindle drive for compensating the process torque via the holder/spindle system on the other hand, limiting torque of the spindle drive for and, compensating the process torque via the spindle speed control. In other words, passive spindle speed assembled in this system. In this context, the main challenge in of the spindle drive for compensating the process torque via the spindle speed control. In other words, passive spindle speed of the spindle for compensating the passive process spindle torque via the spindle speed drive control. In other words, speed variations are a variation of the cutting speed even if no modelling and analysis is now not only to cope with single spindle speed control. In other words, passive spindle speed variations are a variation of the cutting speed even if no spindle speed words,speed passive spindle variations are acontrol. variationInofother the cutting even if no speed speed variation is programmed in the numerical controller (CNC). products, aare limited product range or existing product families, variations aa variation of the cutting speed even no speed variation programmed in controller variations variation ofthe thenumerical cutting speed even if if(CNC). no speed variation is isare programmed in the numerical controller (CNC). The consequences of passive spindle speed variations for the but also to be able to analyze and to compare products to define variation is programmed in the numerical controller (CNC). The consequences of passive spindle speed variations for variation is programmed in the numerical controller (CNC). The consequences of passive spindle speed variations for the the process dynamics are nearly unexplored, but obviously, they new product families. It can be observed that classical existing The consequences of passive spindle speed variations for the process dynamics are nearly unexplored, but obviously, they The consequences passive spindle speed the process dynamics areofnearly unexplored, butvariations obviously,forthey lead to state-dependent delays. The effect of state-dependent product families are regrouped in function of clients or features. process dynamics are nearly unexplored, but obviously, they lead to state-dependent delays. The effect of state-dependent process dynamics are nearly obviously, they lead to state-dependent delays.unexplored, The effectbut of state-dependent delays in metal cutting cutting is only rarely studied in the literature. literature. However, assembly oriented product hardly to find. lead to state-dependent delays. The effect of state-dependent delays metal is only rarely studied in the lead to in state-dependent delays. Thefamilies effect ofare state-dependent delays in metal cutting is only rarely studied in the literature. In Refs. [8,10,11] the so-called tangential regenerative effect delays in metal cutting is only rarely studied in the literature. On the product family level, products differ mainly ineffect two In Refs. [8,10,11] the so-called tangential regenerative delays metal cutting is only rarely studied in the literature. In Refs.in [8,10,11] the so-called tangential regenerative effect was studied for sawing, turning and drilling operations. In In Refs. [8,10,11] the so-called tangential regenerative effect main characteristics: (i) the number of components and (ii) the was studied for sawing, turning and drilling operations. In the In [8,10,11] the so-called tangential effect wasRefs. studied for sawing, turning and drilling regenerative operations. In the considered processes stable cutting is associated with a constant was studied for sawing, turning and drilling operations. In type of components (e.g. mechanical, electrical, electronical). considered processes stable cutting is associated with a constant was studiedprocesses for sawing, turning andisdrilling operations. In the the considered stable cutting associated with a constant delay and state-dependent state-dependent delays arise only in case case ofaaperturbaperturbaconsidered processes stable cutting is with constant Classical methodologies considering mainly single products delay and only in of considered processes stabledelays cuttingarise is associated associated with constant delay and state-dependent delays arise only in case of perturbations and the influenceexisting on the the stability behavior is small. small. In Ref. Ref. delay and state-dependent delays arise only in of or solitary, already product families analyze the tions and influence stability is In delay andthe state-dependent delays arisebehavior only in case case of perturbaperturbations and the influence on on the stability behavior is small. In Ref. [12] the effect of state-dependent delays is studied for milling tions and the influence on the stability behavior is small. In Ref. product structure on a physical level (components level) which [12] the effect of state-dependent delays is studied for milling tions andeffect the influence on the stability behavior is small. In Ref. [12] the of state-dependent delays is studied for milling processes, where state-dependent delays occur due tofor lateral vi[12] the effect of state-dependent delays is milling causes regarding andelays efficient definition and processes, where occur due to lateral vi[12] thedifficulties effect of state-dependent state-dependent delays is studied studied milling processes, where state-dependent delays occur due tofor lateral viprocesses, where state-dependent delays occur due to lateral comparison of different product families. Addressing this processes, where state-dependent delays occur due to lateral vivi-

2212-8271  2018 The TheAuthors. Authors. Published by Elsevier Ltd. cc© 2018 2212-8271 Published by Ltd. 2212-8271  2018 The Authors. Published by Elsevier Elsevier Ltd. Peer-review under the responsibility of the the Scientific Committee of the the 8th CIRP CIRP Conference on on High Performance Performance Cutting (HPC (HPC 2018). This is an copen access article under theInternational CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review of Scientific 2212-8271 2018 The Authors. by cunder 2212-8271  2018the Theresponsibility Authors. Published Published by Elsevier Elsevier Ltd. Ltd. Peer-review under the responsibility of the International International Scientific Committee Committee of of the 8th 8th CIRP Conference Conference on High High Performance Cutting Cutting (HPC 2018). 2018). Selection © and peer-review under responsibility of the International Scientific Committee of the 8th CIRP Conference Cutting on High(HPC Performance Cutting 2212-8271 2017 The Authors. Published by Elsevier B.V. Peer-review under the responsibility of the International Scientific Committee of the 8th CIRP Conference on High Performance Peer-review under the responsibility of the International Scientific Committee of the 8th CIRP Conference on High Performance Cutting (HPC 2018). 2018). (HPC 2018). Peer-review under responsibility of the scientific committee of the 28th CIRP Design Conference 2018. 10.1016/j.procir.2018.08.273

Andreas Otto et al. / Procedia CIRP (2018) 461–464 A. Otto et al. / Procedia CIRP 00 77 (2018) 000–000

2462 a)

Φ(t)

b) τ0 Delay

Ω0t

Delay calculation τ(t)

Φ(t) x(t-τ0)

x(t)

Cutting force

Structural dynamics

Delay

F(t)

Spindle angle

x(t-τ(t))

x(t)

F(t)

Cutting force

Structural dynamics

F(t)

Fig. 1. (a) Conventional model for the dynamics of metal cutting processes with constant delay τ0 ; (b) Extended dynamic model for metal cutting processes. The spindle angle Φ(t) and also delay τ(t) are dynamic variables and depend on the state of the system.

brations of the tool. The authors did focus on the nonlinear behavior of the process dynamics but not on the effect of passive spindle speed variations. In this paper, the effect of torsional vibrations on the dynamics and stability of metal cutting processes is studied. In Sec. 2 an extended model for the process dynamics is constructed, which includes a model for torsional tool displacements. In the new extended model the spindle angle and the time delay are state-dependent dynamic variables of the system. The differences between the new extended model with state-dependent delay and the well-known conventional model with constant delay are analyzed for a concrete example in Sec. 3. The numerical results and further required improvements of the model, especially regarding the effect of runout, are discussed in Sec. 4. 2. Extended dynamic model for metal cutting processes We model the structural dynamics by a system of N damped harmonic oscillators. The displacements of each oscillator caused by the cutting force F ∈ RN are given by the vector q ∈ RN . The cutting force depends on the chip thickness, whose (quasi-)static part is proportional to the product v f τ, where v f and τ denotes the feed velocity and the time delay, respectively. In milling or sawing processes, for example, the chip thickness varies with the spindle rotation angle Φ(t). Moreover, the difference q(t − τ) − q(t) between dynamic displacements at the previous and the present cut lead to dynamic variations of the chip thickness. Details on classical process models can be found in the literature [1,3,4]. In conventional models the angular velocity of the spindle is a constant parameter Ω0 . Consequently, the spindle angle is 2π is a constant pagiven by Φ(t) = Ω0 t and the delay τ0 = ZΩ 0 rameter of the system, where Z denotes the number of cutting teeth. In this case the cutting force is a function of time t, instantaneous as well as delayed displacements q(t) and q(t − τ) and the process dynamics is given by ¨ + Dq(t) ˙ + Kq(t) = F (Ω0 t, q(t − τ0 ) − q(t)) , Mq(t)

the tooth passing period τ0 . The corresponding τ0 -periodic solution qp (t) with qp (t) = q p (t + τ0 ) is the stable cutting solution specified by Mq¨ p (t) + Dq˙ p (t) + Kqp (t) = F (Ω0 t, 0) .

(2)

The stability of stable cutting is determined by the exponential behavior of small perturbations x(t), whose dynamics are obtained by substituting q(t) = qp (t) + x(t) in Eq. (1). After linearization one obtains the periodic linear DDE M¨x(t) + D˙x(t) + Kx(t) = B(t) (x(t − τ0 ) − x(t)) ,

(3)

  where B(t) = D2 F Ω0 t, qp (t − τ0 ) − qp (t) is the derivative of F with respect to its second argument and contains the directional factors [1,3,4]. The stability of Eq. (3) can be determined, for example, via the multi frequency solution or via time domain methods (see Ref. [13] and Refs. therein). If significant torsional vibrations exist, the spindle angle Φ(t) may no longer increase linearly. Thus, as an extension to conventional models we consider the spindle angle Φ at the tool tip as a dynamic variable

Φ(t, ϕ) = Ω0 t +

M

i=1

ϕi (t),

(4)

where the elements ϕi of the vector ϕ ∈ R M are M additional degrees of freedom specifying torsional displacements of the ˙ ϕ) of Eq. (4) specifies the tool tip. The time derivative Φ(t, spindle speed. Passive spindle speed variations may occur for ϕ˙ i (t)  0. As a consequence, also the time delay is not necessarily constant, but rather a dynamic variable in the extended model τ = τ(t, ϕ). It is implicitly defined by [11,14]

(1)

where M, D and K are N × N mass, damping and stiffness matrices of the model for the structural dynamics. The closed-loop representation for a conventional model is illustrated in Fig. 1a) [1,3]. Eq. (1) is a nonlinear, non-autonomous delay differential equation (DDE). From geometric properties of the cutting process and the cutting tool it follows that for an ideal symmetric tool F (Φ(t), 0) = F (Φ(t) + ∆, 0), where ∆ := 2π Z and 0 denotes a zero vector. This means that the cutting force is periodic with

∆ = Φ(t, ϕ) − Φ(t − τ(t, ϕ), ϕ),

(5)

which means that there is a constant angle ∆ between the present cut at time t and the previous cut at time t − τ. Putting Eq. (4) in Eq. (5) results in

τ(t, ϕ) = τ0 +

1 M ϕi (t − τ(t, ϕ)) − ϕi (t), i=1 Ω0

(6)



Andreas Otto et al. / Procedia CIRP 77 (2018) 461–464 A. Otto et al. / Procedia CIRP 00 (2018) 000–000

463

3

2π where τ0 = ZΩ is the constant nominal delay. The dynamics 0 for the torsional displacements ϕ is given by

Mϕ ϕ+D ¨ ˙ ϕ ϕ+K ϕ ϕ = Fϕ (Φ(t, ϕ), τ(t, ϕ), q(t − τ(t, ϕ)) − q) , (7) where Mϕ , Dϕ and Kϕ are M × M mass, damping and stiffness matrices of the torsional oscillators. For brevity, from here we omit the time argument of the state variables if it does not correspond to a delayed variable. The vector Fϕ specifies the cutting torque acting on the torsional oscillators. It depends also explicitly on the delay τ because the chip thickness is proportional to v f τ. The new Eqs. (6) and (7) for the delay calculation and the torsional displacements are coupled with the classical structural dynamics via the spindle angle and the delay, respectively. Thus, in the extended model Eq. (1) modifies to Mq¨ + Dq˙ + Kq = F (Φ(t, ϕ), τ(t, ϕ), q(t − τ(t, ϕ)) − q) .

M¨x(t) + D˙x(t) + Kx(t) =B x1 (t)xϕ (t)+   B x2 (t) xϕ (t − τ0 ) − xϕ (t) +B x3 (t) (x(t − τ0 ) − x(t)) ,

Fig. 2. Left: Test rig for sawing on a machining center. Right: Workpiece dimensions.

τ [10,15]. B x3 is similar to B from the conventional model. The stability behavior of Eq. (10) of the extended model can be analyzed similar to Eq. (3). The differences, compared to the conventional model, are an increasing number of equations by considering perturbations xϕ (t) of the spindle angle and several new mechanisms for instabilities. 3. Effects of torsional vibrations - a case study

(9)

which is different to Eq. (2) that holds only for the conventional model. Moreover, the linearized system for small perturbations xϕ = ϕ − ϕp and x = q − q p of the stable cutting solution is different to the classical chatter theory and is given by Mϕ x¨ ϕ (t) + Dϕ x˙ ϕ (t) + Kϕ xϕ (t) =Bϕ1 (t)xϕ (t)+   Bϕ2 (t) xϕ (t − τ0 ) − xϕ (t) +Bϕ3 (t) (x(t − τ0 ) − x(t)) ,

50 mm

(8)

A block diagram of the extended dynamic model is illustrated in Fig. 1b). The extended model defined by Eqs. (4), (6), (7) and (8) is a non-autonomous DDE with state-dependent delay. Similar to the conventional model for an ideal symmetric tool a periodicity condition exists for the cutting force and torque, i.e. F (Φ − ∆, τ, 0) = F (Φ, τ, 0) and Fϕ (Φ − ∆, τ, 0) = Fϕ (Φ, τ, 0). According to Eq. (5) and Eq. (6) this periodicity condition implies that the corresponding stable cutting solution is τ-periodic and the delay is constant τ(t, ϕ) = τ0 even if passive spindle speed variations may occur. However, in the extended model the stable cutting solution qp (t) = qp (t + τ0 ) and ϕp (t) = ϕp (t + τ0 ) is defined by   Mϕ ϕ¨ p (t) + Dϕ ϕ˙ p (t) + Kϕ ϕp (t) = Fϕ Φ(t, ϕp ), τ0 , 0 ,   Mq¨ p (t) + Dq˙ p (t) + Kqp (t) = F Φ(t, ϕp ), τ0 , 0 ,

40 mm

(10)

where the coefficient matrices Bϕ1 , Bϕ2 and Bϕ3 and B x1 , B x2 , B x3 are defined via the partial derivatives of the torque Fϕ and the force F with respect to its first, second and third argument, respectively. In particular, B x1 specifies changes of the cutting force due to a rotation of the process coordinates via dynamic torsional displacements xϕ (t). The matrix B x2 is attributed to the tangential regenerative effect, i.e. changes of the cutting force and the chip thickness due to dynamic variations of the delay

The effects of torsional vibrations are studied for a sawing process, where torsional vibrations are known to play a significant role [8]. A test rig was constructed for systematically studying sawing processes on a five-axis machining center (see Fig. 2, left) and verifying numerical results obtained from the extended dynamic process model. The aim of the project is to study if the more accurate extended model can explain some deviations between the classical theory and experimental results. The structural dynamics for lateral and torsional dynamics at the tool center point were determined by impact hammer tests. A specific tool dummy was constructed for generating torques via hammer impacts. The modal parameters of the identified dominant modes are given in Table 1. Details of the identification procedure will be given in future publications. A sawing blade with radius R = 80 mm and Z = 64 teeth will be used for cutting tests. We are interested in the dynamic stability for sawing of thin steel tubings. An exemplary process and typical workpiece dimensions can be found in Fig. 2, right. The feed velocity v f is adjusted to the spindle speed such that it results in a vertical feed of 6.4 mm per round (0.1 mm per tooth). For the stability analysis presented here the tangential and radial specific cutting force coefficient KT = 1783 N/mm2 and Kr = 732 N/mm2 , respectively, were taken from Ref. [16]. Based on the experimentally determined parameters in Table 1, the stability lobes, i.e. the stability border in a plane spanned by nominal spindle speed Ω0 and chip width a p , were calcuTable 1. Model parameters for displacements at the tool center point. structural dynamics

mass m

damping ratio ζ

frequency ω (Hz)

x-direction

ϕ-direction

60 kg 32 kg 15 kg 38 kg 105 kg 9 kg 0.12 kg m2

0.035 0.018 0.028 0.020 0.015 0.040 0.120

147 Hz 217 Hz 341 Hz 157 Hz 247 Hz 362 Hz 65 Hz

ϕ (modified)

0.012 kg m2

0.120

65 Hz

y-direction

Andreas Otto et al. / Procedia CIRP 77 (2018) 461–464 A. Otto et al. / Procedia CIRP 00 (2018) 000–000

464 4

chip width a p (mm)

5

considered in this paper. From previous theoretical work we know that already a small delay variation τ p may have a significant effect on the stability lobes [6,20]. Hence, further research is necessary to measure the runout and the passive spindle speed variations in cutting processes to calculate the resulting delay variation and its effect on the stability behavior.

Extended model Conventional model Extended (modified) Hybrid (modified)

4 3 2

Acknowledgements

1 0 50

100

150

200

nominal spindle speed

250 0

300

350

(rpm)

The authors acknowledge financial support from the German Research Foundation (DFG) under the grant no. 321138034.

Fig. 3. Stability lobes of sawing process studied in Sec. 3.

References lated for four different cases with the semidiscretization method [17]. The result is presented in Fig. 3. The thin solid (black) and thin dashed (blue) curve represents the stability lobes of the extended and the conventional model derived from Eq. (10) and Eq. (3), respectively. Only a very small difference can be found near Ω0 = 70 rpm. To exemplify the effects of torsional vibrations the modal mass of the torsional eigenmode was decreased by one order of magnitude (see modified mϕ in Table 1). The resulting stability lobes are the thick solid (green) curves in Fig. 3. It can be seen that the deviations near Ω0 = 70 rpm increase and small deviations near Ω0 = 140 rpm occur. Finally, we made calculations with a hybrid model to isolate the dominant effect for the deviations in the extended model. In particular, we have used the linearized dynamics Eq. (3) of the conventional model with the more accurate spindle angle Φ(t, ϕ p ) obtained from Eq. (9) of the extended model. In other words, we set B x1 , B x2 , Bϕ1 , Bϕ2 and Bϕ3 equal to zero in the extended model. The resulting thick dashed (red) curve is very close to the results from the full extended model, which means that the main part of the deviations in the extended model are due to a nonlinear increase of the spindle angle Φ(t, ϕ p ). 4. Discussion and conclusions A new extended dynamic model for metal cutting processes was presented, where the spindle speed and time delay are dynamic variables and not merely constant parameters. The new degrees of freedom may affect the stability in two different ways. On the one hand, periodic torsional displacements change the shape of the stable cutting solution. On the other hand, the dimension of the linearized system Eq. (10) increases, which may have an effect on the stability, for example, due to the tangential regenerative effect [15]. We have shown via numerical simulations that the main effect arise from a changing shape of the periodic stable cutting solution. However, for moderate torsional flexibility and ideally symmetric tools the effects from dynamic torsional displacements are negligible compared to noise and other sources for deviations of the stability behavior in experiments are more relevant. However, from experiments it is known that, due to runout and small asymmetries of the tool, in practice the fundamental period of stable cutting is given by the spindle rotation period (approximately Zτ0 ) instead of the tooth passing period τ0 [18]. In this case, a periodic delay variation τ p (t, ϕ p )  τ0 appears already during stable cutting [19], which is a completely different situation compared to the case with an ideally symmetric tool

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