Model Development for Optimum Setup Conditions That Satisfy Three Stability Criteria of Centerless Grinding Systems

inventions

Article Model Development for Optimum Setup Conditions that Satisfy Three Stability Criteria of Centerless Grinding Systems

Fukuo Hashimoto ID

Advanced Finishing Technology Ltd., Akron, OH 44319, USA; [email protected]

Received: 5 September 2017; Accepted: 16 September 2017; Published: 21 September 2017

Abstract: The centerless grinding process demonstrates superior grinding accuracy with extremely high productivity, but only if the setup conditions are properly set up. Otherwise, various unfavorable phenomena manifest during the grinding processes and become serious obstacles to achieving that high quality and productivity. These phenomena are associated with the fundamental stabilities of the centerless grinding system, so it is essential to keep the system stable by setting up the appropriate grinding conditions. This paper describes the development of a model for finding the setup conditions that simultaneously satisfy the three stability criteria of centerless grinding systems: (1) work rotation stability for safe operations; (2) geometrical rounding stability for better roundness; and (3) dynamic system stability for chatter-free grinding. The objective of the model development is to produce combinations of optimal setup conditions as the outputs of the model, and to rank the priority of the outputs using PI (performance index) functions based on the process aims (productivity or accuracy). The paper demonstrates that the developed model, named Opt-Setup Master, can generate the optimum setup conditions to ensure safe operations, better roundness and chatter-free grinding. It provides practical setup conditions as well as scientific parameters and fundamental grinding parameters. Finally, the paper verifies that the Opt-Setup Master provides the setup conditions that simultaneously satisfy all three stability criteria of the centerless grinding system.

Keywords: grinding; centerless grinding; process optimization; safe operation; quality; productivity

1. Introduction The centerless grinding method has been extensively applied for the production of cylindrical components such as rings, rollers, and pins. It is estimated that a single car has more than 2000 parts ﬁnished by centerless grinding processes. The centerless grinding process demonstrates extremely high productivity with very high grinding accuracy in OD size, roundness and surface integrity. However, its superior performance compared to other grinding methods can be achieved only if the grinding conditions are properly set up; otherwise, various unfavorable phenomena, such as slippages in work rotation, deformed roundness and chatter vibrations, appear during the grinding process and lead to deterioration in grinding performance [1]. These huge advantages and disadvantages come from the unique work-holding features of the centerless grinding system: (1) a loose hold on the workpiece without any mechanical constraints; (2) the work friction brake/drive mechanism of the work rotation; and (3) a self-centering mechanism called “regenerative centering”. The grinding process is very sensitive to these unique centerless setup conditions, so it is essential to secure the grinding system’s stability by setting it up appropriately. This requires controlling the three fundamental stability issues caused by the work-holding features of centerless grinding. These are: (1) work rotation stability; (2) geometrical rounding stability; and (3) dynamic system stability.

Inventions 2017, 2, 26; doi:10.3390/inventions2040026 www.mdpi.com/journal/inventions Inventions 2017, 2, 26 2 of 18

Work rotation stability is related to the work friction brake/drive mechanism of the centerless grinding system. The regulating wheel is in rolling-sliding contact with the workpiece, and provides the friction force to the workpiece that drives or brakes the work rotation. In this unique mechanism, the work rotates with almost the same peripheral velocity of the regulating wheel during the stable grinding process, in which the torque created by the grinding force balances with the torque from the friction forces acting on the regulating wheel and the blade top surface. However, under heavy grinding with excessive grinding force, control over the work rotation speed is lost due to the broken torque equilibrium, and it increases toward the grinding wheel speed. This phenomenon, called “spinners”, can cause dangerous accidents and should be avoided in order to maintain safe operations. The author is a pioneer of the study of work rotation stability and has shown that there exists an absolute safe zone where spinners do not develop [2]. The setup guidelines for safe operations are well established in the literature, and the means of satisfying the work rotation stability criterion have been demonstrated [3]. Geometrical rounding stability is related to the work-holding conditions and regenerative centering effects. Although centerless grinding technology has been around for 100 years since the method was patented by L.R. Heim in 1917 [1], a great deal of effort was exerted by early research pioneers to understand its rounding mechanism, and significant papers have been published [4–9]. The theory of the rounding mechanism has been well established, the setup guidelines for achieving better roundness have been described, and the means of satisfying the stability criteria have been clarified. The stability criteria assume that the grinding system consists of solid bodies and is dynamically stable. Under certain work-holding conditions, a specific number of lobes on the roundness of the workpiece appear or cannot be removed. It is crucial to minimize roundness errors by selecting the proper setup conditions. Dynamic system stability is related to the work-regenerative chatter vibration caused by the instability of the centerless grinding system, including the machine dynamics. The chatter vibration in centerless grinding is very severe and builds up very fast. In general, the amplitude growth rate is 10 to 100 times greater than that of center-type grinding processes, and is caused by the wheel-regenerative chatter vibration. Significant investigations have been carried out by many researchers [10–13] to understand dynamic system stability and suppress the chatter vibration. The system stability criterion has been well established, and the setup condition guidelines for chatter-free grinding are available in the literature. As mentioned above, the setup guidelines for satisfying each stable criterion have been established. However, the setup operations of centerless grinding still rely on experimental skill and the trial-and-error method. Even though each stable criterion can be satisfied individually by carefully choosing the setup conditions, it is almost impossible to simultaneously satisfy all three of the centerless grinding system’s stability criteria. Therefore, a special analytical tool for finding the optimum combination of setup conditions is greatly needed [14]. The objective of this paper is to describe the development of an analytical model capable of finding the optimum combination of setup conditions that satisfies all three stability criteria at the same time. This paper describes the structure of the developed model, which consists of the input-information session, the data bank that stores all the parameters required for the model calculations, the PI (performance index) functions for assessing the setup conditions based on the process aim (productivity or accuracy), and the output-information session. Further, this paper explains the algorithm of the model and shows how to find the setup conditions that meet the three stability criteria simultaneously. The developed model, named Opt-Setup Master, is verified through case studies in which workpieces with various sizes are ground with different grinding machines. Finally, the Opt-Setup Master demonstrates its capability to generate the optimum setup conditions that satisfy all three stability criteria, and to provide the grinding conditions that will provide safe grinding operations and chatter-free grinding with improved grinding accuracy. Inventions 2017, 2, 26 3 of 18 Inventions 2017, 2, 26 3 of 18

2. Basic2. Basic Setup Setup Conditions Conditions in in Centerless Centerless Grinding Grinding The basicThe basic setup setup parameters parameters in in centerless centerless grinding grinding areare the blade angle angle θθ, ,the the center center height height angle angle γ γ and the work rotational speed nw, as shown in Figure 1. and the work rotational speed nw, as shown in Figure1.

Figure 1. Setup conditions in centerless grinding. Figure 1. Setup conditions in centerless grinding.

The set (θ, γ, nw) of these parameters is called the “setup condition” in this paper, and it Thesignificantly set (θ, affectsγ, nw )centerless of these grinding parameters perfor ismance. called In thepractice, “setup the work condition” center height in this CH paper, (instead and it signiﬁcantlyof γ) and affects the RW centerless (regulating grinding wheel) rotation performance. speed N Inr (rpm) practice, are used the because work center these heightparametersCH can(instead be directly set up on the machine. The center height CH (mm) has the following relationship with the of γ) and the RW (regulating wheel) rotation speed Nr (rpm) are used because these parameters can be directlycenter set height up on angle the machine.γ (°) when Theangles center α and height β are small.CH (mm) has the following relationship with the ◦ center height angle γ ( ) when angles α and βγare=+α small. β (1)

-1 γ22CH= α + β CH (1) α =≅sin (2) ()DD++() DD 2rwCH rw2CH α = sin−1 =∼ (2) (Dr + Dw) (Dr + Dw) −1 22CH CH β =≅sin (3) −1 ()2CH++∼ ()2CH β = sin DDg wgw = DD (3) Dg + Dw Dg + Dw 3.14 Dg +++Dw ()(Dr + Dw) 3.14()DDDDgwrw◦ CHCH( mm()mm) ==° γγ()( ) (4) (4) 360 Dg +++Dr + 2Dw 360()DDgr 2 D w The work rotation speed nw is controlled by the RW friction drive/brake mechanism. Figure2 shows testThe results work of rotation normal speed grinding nw is forcecontrolledFn, the by friction the RW coefﬁcientfriction drive/brakeµr and the mechanism. rolling-sliding Figure velocity 2 shows test results of normal grinding force Fn, the friction coefficient μr and the rolling-sliding between RW and the workpiece during an infeed centerless grinding process [15]. In steady state velocity between RW and the workpiece during an infeed centerless grinding process [15]. In steady grinding, the sliding velocity ∆V, deﬁned as (Vw − Vr), is about +0.008 m/s, and the slippage ratio state grinding, the sliding velocity ΔV, defined as (Vw − Vr), is about +0.008 m/s, and the slippage ratio ∆V/V is about 2%, where V and V are the work and RW peripheral velocities, respectively. Since the Δr V/Vr is about 2%, wherew Vw and Vrr are the work and RW peripheral velocities, respectively. Since the slidingsliding velocity velocity is very is very small, small, the the work work rotation rotation speedspeed nww cancan be be represented represented by: by: DN ()rpm ()∼≅ Drrr Nr(rpm) nnww(rpsrps) = (5) (5) 60Dw

In theIn modelthe model development, development, the the scientiﬁc scientific parametersparameters (θ (θ, ,γγ, n, wn)w are) are used used for forthe theanalysis analysis of the of the optimumoptimum setup setup condition, condition, and and the the practical practical parameters parameters ( (θθ,, CHCH, ,NNr)r are) are the the outputs outputs of the of themodel. model. Inventions 2017, 2, 26 4 of 18 Inventions 2017, 2, 26 4 of 18

Inventions 2017, 2, 26 4 of 18

Figure 2. Infeed centerless grinding process. Dr = 255 mm, Dw = 30 mm, Nr = 30 rpm, Vw = 0.4 m/s, Figure 2. Infeed centerless grinding process. Dr = 255 mm, Dw = 30 mm, Nr = 30 rpm, Vw = 0.4 m/s, Sliding velocity: 0.008 m/s, Slippage ratio: 2%. SlidingFigure velocity: 2. Infeed 0.008 centerless m/s,Slippage grinding ratio:process. 2%. Dr = 255 mm, Dw = 30 mm, Nr = 30 rpm, Vw = 0.4 m/s, Sliding velocity: 0.008 m/s, Slippage ratio: 2%. 3. Centerless Grinding Systems and the Characteristic Equation 3. Centerless Grinding Systems and the Characteristic Equation 3. CenterlessSince the Grinding three stability Systems criteria and influence the Characteristic each other Equation and are significantly affected by the setup Since the three stability criteria influence each other and are significantly affected by the setup conditions,Since the it threeis necessary stability tocriteria assess influence these stabilities each other as and a total are significantly system—including affected theby themachine setup conditions, it is necessary to assess these stabilities as a total system—including the machine dynamic dynamicconditions, characteristics, it is necessary the tocenterless assess these grinding stabilities mechanism as a totaland thesystem—including grinding processes. the Figuremachine 3 characteristics, the centerless grinding mechanism and the grinding processes. Figure3 shows showsdynamic a blockcharacteristics, diagram ofthe the centerless centerless grinding grinding mechanism system. The and system the grinding consists processes.of the regenerative Figure 3 a blockcenteringshows diagram a block mechanism ofdiagram the centerless[16], of thethe regenerativecenterless grinding grinding system. function Thesystem. [17], system the The relationship consistssystem consists of between the regenerative of depth-of-cutthe regenerative centering and mechanismthecentering normal [mechanism16 grinding], the regenerative force, [16], thethe regenerativecontact function stiffness [17 function], the of the relationship [17], wheels, the relationship the between wheel depth-of-cutfiltering between functions, depth-of-cut and the and normal andthe grindingmachinethe normal force, dynamics grinding the contact [12]. force, The stiffnessthe dyna contactmic of stiffness thebehavior wheels, of ofthe thethe wheels, wheelrounding the filtering wheel mechanism functions,filtering can functions, be and investigated the and machine the dynamicsbasedmachine on [ 12dynamicsthe]. characteristic The dynamic [12]. The equati behavior dynaonmic of ofthe behavior the closed rounding loopof the centerless mechanism rounding grinding mechanism can be system investigated can in Figurebe investigated based 3. on the characteristicbased on the equation characteristic of the equati closedon loop of the centerless closed loop grinding centerless system grinding in Figure system3. in Figure 3. Regenerative Regenerative Grinding where, Slide stiffness Normal ε΄= sin(ϒ)/cos(θ-β) center-function function where, feed Regenerative1 Regenerative Grinding force (1-ε) = cos(θ-β)/cos(θ-α) Slide + −2πS ⋅ ′ Normal ε΄= sin(ϒ)/cos(θ-β) f center-function−ϕ S −ϕ S z function− stiffnessb k Fn −ε′ 1 + −ε 2 cs 1 e w s = σ + jn (s: Laplace operator) 1 e zcr (1 )e feed 1 force (ϕ1-1:ε Blade) = cos( phaseθ-β)/cos( angleθ-α) +- −2πS ′ f −ϕ −ϕ z − b⋅k Fn −ε′ 1SContact+ − Complianceε 2S csof regulating1 e wheel w ϕs =2: σRegulating + jn (s: Laplace wheel operator) phase angle 1 e zcr (1 )e - + (1−ε) bϕ:1 :Grinding Blade phase width angle z Contact Compliance of regulating wheel 2 cr b⋅k′ kϕ΄w:: Regulating Specific grinding wheel stiffnessphase angle + + (1−εcr) b: Grinding width z Contact Compliance of grinding wheel k΄cr: Contact stiffness of regulating wheel cr b⋅k′ k΄w: Specific grinding stiffness + + 1 cr k΄cs : Contact stiffness of grinding wheel Contact Compliance⋅ ′ of grinding wheel νk:΄ crMode: Contact number stiffness of regulating wheel + b kcs + 1 Uk΄νcs: ν: -Contactth. mode stiffness orientation of grinding factor wheel Compliance of⋅ grinding′ machine ν: Mode number b kcs kmν: ν-th. mode machine static stiffness + uν Uν: ν-th. mode orientation factor Compliance of grindingGν (s) machine G(s): Machine dynamic transfer function ν kmν kmν: ν-th. mode machine static stiffness uν Zcs: Grinding wheel filter function  Gν (s) ZcrG(s): :Regulating Machine dynamic wheel filter transfer function function ν k mν Zcs: Grinding wheel filter function Figure 3. Block diagram of centerless grindingZcr: Regulating system. wheel filter function Figure 3. Block diagram of centerless grinding system. The characteristic functionFigure 3. isBlock represented diagram by: of centerless grinding system. The characteristic function is represented by:1 The characteristic function is represented−= by: gs() fs1() (6) −=gs() ()1 (6) − fs= g(s) (6) where f (s) where Ze()1− −2π s where −=−1 cs (7) ϕϕ −−2π s − - 12s − 2πs s fs11() 11−+−εε′ eZZecscs ()11 Z− e() e −−=−= − cr (7) (7) 0 −ϕϕϕ1s − ϕ2s f (s()) 1 −−+−εεεe′ - 12s + Zcr()(1 − ε)e s fs11 e()− ε Zcr e '  111  g()sbk=++ Gs() wm0  1 ′ (()1 −′εε) 1  (8) g(s) = bkw ' bk11+ 1 bk+ k Gm(s)  (8) g()sbk=++bk0 csbk0 crk m Gs() wm cs′ cr′ m  (8) bkcs bk cr k m  Inventions 2017, 2, 26 5 of 18

Inventions 2017, 2, 26 5 of 18 By solving the characteristic roots of Equation (6), the dynamic rounding stability can be evaluated By solving the characteristic roots of Equation (6), the dynamic rounding stability can be and the transient behavior of the waviness amplitude in work roundness can be calculated during the evaluated and the transient behavior of the waviness amplitude in work roundness can be calculated grinding process. The characteristic root can be represented by: during the grinding process. The characteristic root can be represented by:

ssjn==+σ + jn (9) (9) wherewheres is s is the the Laplace Laplace operator, operator,σ σis is thethe amplitudeamplitude growth growth rate rate per per unit unit radian, radian, and and n isn isthe the number number ofof lobes lobes in in the the work work roundness. roundness. TheThe transienttransient of the amplitude change change AA(t()t on) on roundness roundness waviness waviness during the grinding process can be expressed by: during the grinding process can be expressed by: ()= ( ) A tA0exp 2πσ ntw (10) A(t) = A0 exp(2πnwσt) (10) where A0 is the initial amplitude of the waviness, nw is the work rotation speed in rps and t is the wheregrindingA0 is time. the initialWhen amplitudeσ is positive, of the waviness,amplitude nofw nis lobes the work grows rotation with grinding speed intimerps tand andt theis the grindinggrinding time. process When can beσ isidentified positive, as thethe chatter amplitude vibration. of n lobes In case grows of σ < with0, the grindingamplitude time of n lobest and is the grindingdecreased process with cangrinding be identiﬁed time t, and as the grinding chatter vibration. process becomes In case ofstableσ < with 0, the improved amplitude roundness. of n lobes is decreasedWhen with the grinding effect of machine time t, and vibration the grinding is neglig processible, the becomes response stable of the with transfer improved function roundness. Gm(s) is degeneratedWhen the to effect a constant of machine and the vibration resulting is system negligible, is of a the kinematic response nature, of the referred transfer to function as “geometricGm(s) is degeneratedrounding stability” to a constant [12]. Then, and the the resulting characteristic system equation is of a is kinematic simplified nature, as: referred to as “geometric rounding stability” [12]. Then, the characteristic−2πs equation is simpliﬁed as: ()1− e  1 ()1− ε b  −=k ' ++ (11) −−φφssπ w   11−+−εε′′′e1 12− e−()2 s e kkk 1 (1 − ε) b − = 0 cs+ cr+ m  0 −ϕ s −ϕ s kw 0 0 (11) 1 − ε e 1 + (1 − ε)e 2 kcs kcr km

4.4. Three Three Stability Stability Criteria Criteria in in Centerless Centerless GrindingGrinding

4.1.4.1. Work Work Rotation Rotation Stability Stability Criterion Criterion FigureFigure4 shows 4 shows the the torques torques acing acing on on the the workpieceworkpiece during the the centerless centerless grinding grinding process. process. Tg Tgis is thethe grinding grinding torque torque given given by by the the tangential tangential grinding grinding force force Ft. Ft.Tb Tband andTr areTr theare frictionthe friction torques torques acting acting on the blade and the regulating wheel, respectively. Under the stable grinding process, the on the blade and the regulating wheel, respectively. Under the stable grinding process, the following following torque-quilibrium relationship is maintained. The work peripheral velocity Vw is torque-quilibrium relationship is maintained. The work peripheral velocity Vw is controlled by the controlled by the friction drive/brake mechanism of RW, and becomes almost the same as the RW friction drive/brake mechanism of RW, and becomes almost the same as the RW peripheral velocity Vr. peripheral velocity Vr.

StableStable grinding grinding: : TgTg==+ Tb+ TrTr , VwVw ≅=∼ VrVr (12) (12)

Vg Vw Vr

Workpiece

Grinding wheel Tg Tr Regulating wheel Blade FigureFigure 4. 4.Torques Torques actingacting on workpiece during during centerless centerless grinding. grinding.

However, once this quilibrium condition is broken by the excessive grinding torque Tg Tg overcomingHowever, the once friction this quilibrium torques (Tb condition + Tr) during is broken grinding, by the the excessive work velocity grinding Vw torquesuddenly overcomingincreases thetoward friction the torques grinding (Tb wheel+ Tr) speed during Vg grinding,. the work velocity Vw suddenly increases toward the grinding wheel speed Vg. Unstable grinding: Tg>+ Tb Tr , Vw ≅ Vg (13) Unstable grinding : Tg > Tb + Tr, Vw =∼ Vg (13) Inventions 2017, 2, 26 6 of 18

This phenomenon, called “spinners”, can create a potentially very dangerous situation and Inventionsshould2017 be, avoided2, 26 for safe operations. 6 of 18 Figure 5a,b show the geometrical arrangement of the centerless grinding process and the forces acting on the workpiece at any cut section perpendicular to the work axis l during grinding. The This phenomenon, called “spinners”, can create a potentially very dangerous situation and should variables fT and fN represent the tangential and normal grinding forces per unit width at the cut be avoided for safe operations. section. Rb and Rr are the resultant forces, while μb and μr are the friction coefficients at the contact Figure5a,b show the geometrical arrangement of the centerless grinding process and the points with the blade and the RW, respectively. w(l) is the work weight per unit width at the cut forces acting on the workpiece at any cut section perpendicular to the work axis l during grinding. Section l. The torque equilibrium equation can be written by: The variables fT and fN represent the tangential and normal grinding forces per unit width at the cut L μμ+−+ section. Rb and Rr ared theω resultant()B forces,()lBflC while µb ()and µ()r are the () lCwl friction coefﬁcients() at the contact I = rl() 12rTr 12 dl points with the blade and the RW, respectively.()w(l)μ is() the+ work weight per unit width at the(14) cut dt0 A12r l A Section1. The torque equilibrium equation can be written by: where L Z dω (B=−−−1µμr(l) + θαB2) fT(l) − (C() θα1µr(l) + C2)w(l) I = r(l)A1 b cos( ) sin dl (15)(14) dt (A1µr(l) + A2) 0 A =−+−μ sin()()θα cos θα (16) where 2 b A1 = µb cos(θ − α) − sin(θ − α) (15) =−μγ( + γ )()()()() − + μ θ +βμ +−θ +β BA11bbbsin k cos 1 k sin k cos  (17) A2 = µb sin(θ − α) + cos(θ − α) (16)

B1 = A1 − µ (sin γ + k cos=−γ) − [(μγ1 +()kµ ) sin −(θ + γβ) + (k − µ ) cos(θ + β)] (18)(17) b BA22b cosb k sin b

B2 = A2 − µb(cos γ − k sin γ) (18) =−θ μ ()θα − (19) C1 sinb cos sin C1 = sin θ − µb(cos θ − sin α) (19) C = μ cosα C2 = µb cos α (20)(20) I andI andω ωare are the the mass mass moment moment of ofinertia inertia andand thethe angular velocity velocity of of the the workpiece. workpiece. k isk isthe the force force ratioratio (fN (/fNf/Tf).T). ForFor convenience,convenience, the plus sign sign of of μµrr isis assigned assigned to to the the downward downward friction friction force force and and the the minusminus sign sign is is assigned assigned to to the the upward upward one. one.

(a) (b)

FigureFigure 5. 5.Arrangement Arrangement of of centerless centerless grindinggrinding andand forces acting on on workpiece. workpiece. (a ()a Forces) Forces acting acting on on workpiece during centerless grinding; (b) Cylindrical workpiece. workpiece during centerless grinding; (b) Cylindrical workpiece.

The generalized motion Equation (14) is applicable to any cylindrical-shaped workpiece; for example,The generalized simple cylindrical, motion Equation tapered, (14)and ismultip applicablele stepped to any diameter cylindrical-shaped workpieces. Equation workpiece; (14) for example,indicates simple that, in cylindrical, addition to tapered, being affe andcted multiple by the primary stepped setup diameter conditions workpieces. (θ, γ), the Equation rotational (14) indicatesmotion that,of the inworkpiece addition is to affected being affectedby the grinding by the forces primary and setupthe friction conditions force on (θ ,RW.γ), the rotational motionThe of the upper-limit workpiece tangential is affected grinding by the force grinding fU under forces theand stable the grinding friction condition force onRW. is derived from EquationThe upper-limit (14). tangential grinding force fU under the stable grinding condition is derived from Equation (14). (C1µr0 + C2) fU = w (21) (B1µr0 + B2) Inventions 2017, 2, 26 7 of 18 Inventions 2017, 2, 26 7 of 18 Inventions 2017, 2, 26 7 of 18 ()CCμ + = 10r 2 fU ()μ + w (21) ()CC10μr + 2 f = BB10r 2w U ()μ + (21) where µr0 is the maximum static friction coefficientBB10r of RW. 2 When the tangential grinding force fT is where μr0 is the maximum static friction coefficient of RW. When the tangential grinding force fT is smaller than fU, the work rotation speed Vw can be controlled with the RW speed Vr. wheresmaller μ thanr0 is the fU, themaximum work rotation static friction speed Vwcoefficient can be controlled of RW. When with thethe tangentialRW speed grindingVr. force fT is Figure6 shows the results of the calculation of fU with respect to the blade angle θ with various smallerFigure than 6 f Ushows, the work the results rotation of speedthe calculation Vw can be of controlled fU with respect with theto the RW blade speed angle Vr. θ with various friction coefficients µr0. The grinding force fU is normalized with the diameter d of a simple cylindrical frictionFigure coefficients 6 shows μ ther0. The results grinding of the force calculation fU is normalized of fU with with respect the todiameter the blade d of angle a simple θ with cylindrical various workpiece made of steel. fU increases with increased θ. When θ is greater than a certain angle with frictionworkpiece coefficients made of μ steel.r0. The fU grinding increases force with fU increased is normalized θ. When with θthe is diametergreater than d of aa certainsimple cylindricalangle with µr , the fU value becomes infinite. Under this condition, there is no risk of the spinners phenomenon 0 workpieceμr0, the fU value made becomes of steel. infinite. fU increases Under with this increased condition, θ. thWhenere is θ no is riskgreater of the than spinners a certain phenomenon angle with f occurring.μoccurring.r0, the ThefU value The zone zonebecomes with with the infinite. the infinite infinite UnderU fUvalue value this iscondition,is called the the th “safeere “safe is operatio no operation risk nof zone.” the zone.” spinners For For instance, instance,phenomenon there there is is ◦ no limitoccurring.no limit on f Uon Thewhen fU when zone a blade awith blade the of of θinfinite >θ > 42 42° isf Uis value used used is withwith called an theRW “safe ofof μµr 0roperatio =0 0.25.= 0.25. n zone.” For instance, there is no limit on fU when a blade of θ > 42° is used with an RW of μr0 = 0.25.

FigureFigure 6. 6.Upper Upper limit limit tangentialtangential grinding grinding force. force. Figure 6. Upper limit tangential grinding force. Figure 7 shows the safe operation zones under various μr0 values on the (θ–γ) chart and Figure7 shows the safe operation zones under various µ values on the (θ–γ) chart and provides providesFigure guidelines 7 shows forthe satisfyingsafe operation the work zones rota undertion stabilityvariousr0 μcriterionr0 values (WRSC).on the (θStable–γ) chart grinding and guidelinesprovideswithout for anyguidelines satisfying risk of spinnersfor the satisfying work can rotation be the obtained work stability rotaby criterionselectingtion stability (WRSC).the set criterion of Stable(θ, γ(WRSC).) from grinding the Stable safe without operationgrinding any risk of spinnerswithoutzone, and canany the be risk WRSC obtained of spinners is satisfied by selecting can with be obtainedthe the setup set of byconditions (θ selecting, γ) from ( θthe the, γ ).set safe of operation(θ, γ) from zone, the safe and operation the WRSC is satisﬁedzone, with and the the WRSC setup is conditions satisfied with (θ, γ the). setup conditions (θ, γ).

Figure 7. Safe operation chart. Figure 7. Safe operation chart. Figure 7. Safe operation chart.

4.2. Geometrical Rounding Stability Criterion When the grinding system is stable and the inﬂuence of the machine dynamics on the rounding mechanism is negligible, the stability of the rounding mechanism is predominantly affected by the Inventions 2017, 2, 26 8 of 18 Inventions 2017, 2, 26 8 of 18 4.2. Geometrical Rounding Stability Criterion 4.2. Geometrical Rounding Stability Criterion InventionsWhen2017, 2the, 26 grinding system is stable and the influence of the machine dynamics on the rounding8 of 18 When the grinding system is stable and the influence of the machine dynamics on the rounding mechanism is negligible, the stability of the rounding mechanism is predominantly affected by the mechanism is negligible, the stability of the rounding mechanism is predominantly affected by the geometrical arrangement of the centerless grinding system under the solid-body machine structure. geometrical arrangement of the centerless grinding system under the solid-body machine structure. geometricalBy analyzing arrangement the characteristic of the centerless roots of Equation grinding (11), system the undereffect theof the solid-body center height machine angle structure. γ on By analyzing the characteristic roots of Equation (11), the effect of the center height angle γ on Bygeometrical analyzing rounding the characteristic stability can roots be ofassessed. Equation Figure (11), 8a theshows effect the ofcharacteristic the center root height distributions angle γ on geometrical rounding stability can be assessed. Figure 8a shows the characteristic root distributions geometricalfor the odd rounding lobes and stability the even can lobes be [14]. assessed. When Figure a lower8a center shows height the characteristic angle such as root γ < distributions3° is set up, for the odd lobes and the even lobes [14]. When a lower center height angle such as γ < 3° is◦ set up, forthe the amplitude odd lobes growth and the rates even of the lobes 3, 5, [14 and]. When 7 lobe as become lower center close to height zero, angleand the such roundness as γ < 3erroris set due up, the amplitude growth rates of the 3, 5, and 7 lobes become close to zero, and the roundness error due theto amplitude these odd growthlobes cannot rates ofbe the improved 3, 5, and during 7 lobes th becomee grinding close process. to zero, On and the the other roundness hand, errorwhen duea to these odd lobes cannot be improved during the grinding process. On the other hand, when a tohigher these odd center lobes height cannot angle be such improved as γ > during 9° is set the up, grinding the amplitude process. growth On the rates other of hand,even lobes when like a higher 18, higher center height angle such as γ > 9° is set up, the amplitude growth rates of even lobes like 18, center20, and height 22 become angle such close as toγ >zero 9◦ isand set the up, roundness the amplitude error growth cannot ratesbe improved of even lobes due to like the 18, even-lobe 20, and 22 20, and 22 become close to zero and the roundness error cannot be improved due to the even-lobe becomewaviness. close to zero and the roundness error cannot be improved due to the even-lobe waviness. waviness.

10 10 Conditions σ Conditions 8 σ Blade angle θ=30° 8 BladeStock angleremoved θ=30° in dia. =35-40μm m μ ｍ 4 m 6

μ Stock removed in dia. =35-40μm μ ｍ 4 Work size (dia. x width) 6 μ WorkX: 9 x size 30 mm (dia. x width) X:Δ: 149 x x 30 30 mm mm 4 2 4 Δo:: 14 20 x x 30 30 mm mm 2 Roundness o: 20 x 30 mm 2 Roundness Roundness 2 Roundness Amplitude growth rate ３ ϒopt ９ Roundness 0 Amplitude growth rate ３ ϒopt ９ 0 5 10 0 0246810 Center-height angle ° 0 0246810 Center-height angle ° Center-height5 angle °10 Center-height angle ° Center-height angle ° Center-height angle ° (a)(b)(c) (a)(b)(c) Figure 8. Geometrical rounding stability. (a) Center-height angle vs. amplitude growth rates [14]; (b) FigureFigure 8. 8.Geometrical Geometrical roundingrounding stability. (a (a) )Center-height Center-height angle angle vs. vs.amplitude amplitude growth growth rates rates[14]; ( [b14) ]; Effect of γ on roundness [6]; (c) Effect of γ on roundness [16]. (b)Effect Effect of of γ γonon roundness roundness [6]; [6 (];c) (Effectc) Effect of γ of onγ onroundness roundness [16]. [ 16]. These results suggest the existence of an optimum center height angle γopt that yields a These results suggest the existence of an optimum center height angle γopt that yields a minimumThese results roundness suggest error, the and existence γopt = of6.7° an has optimum been proposed center height as that angle optimumγopt thatangle yields [12]. Miyashita a minimum minimum roundness error, and ◦γopt = 6.7° has been proposed as that optimum angle [12]. Miyashita roundnesset al. reported error, the and experimentalγopt = 6.7 resultshas been shown proposed in Figure as 8b, that and optimum indicated angle that [the12]. optimum Miyashita center et al. et al. reported the experimental results shown in Figure 8b, and indicated that the optimum center reportedheight angle the experimental exists around results 7° [6]. Rowe shown et in al. Figure reported8b, on and theoretical indicated and that experimental the optimum analysis center of height the height angle exists around◦ 7° [6]. Rowe et al. reported on theoretical and experimental analysis of the anglerounding exists mechanism around 7 of[6 ].a workpiece Rowe et al.with reported a flat face. on Figure theoretical 8c shows and the experimental effect of the analysiscenter height of the rounding mechanism of a workpiece with a flat face. Figure 8c shows the effect of the center height roundingangle on mechanism the roundness of aerror. workpiece The grinding with a test ﬂat resu face.lts Figure verified8c that shows the theoptimum effect ofcenter the centerheight angle height angle on the roundness error. The grinding test results verified that the optimum center height angle angleγopt on exists the around roundness 6°–8° error. [8]. The grinding test results veriﬁed that the optimum center height angle γopt exists around 6°–8° [8]. γopt existsIt is aroundwell known 6◦–8 ◦that[8]. an odd number of lobes appears under a lower center height condition It is well known that an odd number of lobes appears under a lower center height condition suchIt isas well γ < 3°, known as shown that in an Figure odd number 9a. To minimize of lobes appears the roundness under aerror lower with center odd heightnumbers condition of lobes, such it such as γ < 3°, as shown in Figure 9a. To minimize the roundness error with odd numbers of lobes, it asisγ recommended< 3◦, as shown that in Figurelower center9a. To height minimize angles the be roundness avoided as error much with as possible. odd numbers Under of a relatively lobes, it is is recommended that lower center height angles be avoided as much as possible. Under a relatively higher center height condition, a specific even number of lobes appears, as shown in Figure 9b. recommendedhigher center that height lower condition, center height a specific angles even be avoided number as of much lobes as appears, possible. as Under shown a relativelyin Figure higher 9b. centerWhere height the center condition, height a specific angle is even known, number the ofeven lobes number appears, of lobes as shown ne that in Figurewill appear9b. Where can be the found center Where the center height angle is known, the even number of lobes ne that will appear can be found by 180/γ [6]. heightby 180/ angleγ [6]. is known, the even number of lobes ne that will appear can be found by 180/γ [6].

研研研研調調調RW 調調調 GW 調調調研研研研GW RW 研研研研GW RW 研研研研 γ 調調調RW GW γ

受受 Blade

7 lobes γ＝5º γ＝7.5º γ＝9º ＝＝ 3 lobes 5 lobes 7 lobes 36γ＝ lobes5º 24γ lobes7.5º 20γ lobes9º 3 lobes 5 lobes 36 lobes 24 lobes 20 lobes

ne=180/5=36 ne=180/7.5=24 ne=180/9=20 ne=180/5=36 ne=180/7.5=24 ne=180/9=20 (a) (b) (a) (b) Figure 9. Effect of center-height angle on rounding mechanism. (a) Odd lobe appearance at lower γ; Figure 9. Effect of center-height angle on rounding mechanism. (a) Odd lobe appearance at lower γ; Figure(b) Even 9. Effect lobe appearance of center-height at a specific angle onγ. rounding mechanism. (a) Odd lobe appearance at lower γ; (b) Even lobe appearance at a specific γ. (b) Even lobe appearance at a speciﬁc γ. Inventions 2017, 2, 26 9 of 18 Inventions 2017, 2, 26 9 of 18 These even numbers of lobes are the result of the geometrical rounding instability caused by the geometrical arrangement of the regulating wheel or the blade. Figures 10 and 11 show the geometricalThese even rounding numbers stability of lobes criteria are the of resultthe regulating of the geometrical wheel and rounding the blade, instability respectively. caused by the geometricalFigure arrangement10a is an example of the of regulating regulating wheel wheel or thegeometrical blade. Figures rounding 10 and instability. 11 show When the geometrical a peak of roundingwaviness stabilityon the work criteria roundness of the regulating contacts wheel with the and regulating the blade, respectively.wheel and the waviness becomes a valleyFigure at the 10 agrinding is an example point (a ofnd regulating vice versa), wheel the geometricalwaviness erro roundingr cannot instability. be removed When during a peak the ofgrinding waviness process. on the Conversely, work roundness Figure contacts 10b is withan example the regulating of regulating wheel wheel and the geometrical waviness becomesstability. aWhen valley a atpeak the contacts grinding with point the (and regulating vice versa), wheel the and waviness the waviness error cannotbecomes be a removedpeak at the during grinding the grindingpoint (and process. vice versa), Conversely, the waviness Figure 10 errorb is ancan example be removed of regulating during wheelthe process. geometrical Therefore, stability. the Whenregulating a peak wheelcontacts geometrical with the regulatingrounding stability wheel and criterion the waviness (RW − becomesGRSC) can a peak be atsummarized the grinding as pointfollows: (and vice versa), the waviness error can be removed during the process. Therefore, the regulating wheel geometrical rounding stability criterion (RW − GRSC) can be summarized as follows: 180 ()RW−= GRSC Unstable: Even int eger 180γ (22) (RW − GRSC) Unstable : = Even integer (22) γ 180 ()RW−= GRSC Stable:180 Odd int eger (23) (RW − GRSC) Stable : γ = Odd integer (23) γ

GW RW GW RW n=16 n=16 ϒ ϒ

[180/ϒ]=16: Even [180/Unstableϒ]=16: Even [180°/ϒ°]=Even integer

(a)

GW RW GW RW n=15 n=15 ϒ ϒ

[180/Stableϒ]=15: Odd [180°/ϒ°]=Odd[180/ integerϒ]=15: Odd

(b)

FigureFigure 10.10.Regulating Regulating wheel wheel (RW) (RW) geometrical geometrical rounding rounding stability stability criteria. criteria. (a) RW geometrical (a) RW geometrical rounding unstablerounding conditions; unstable conditions; (b) RW geometrical (b) RW geometrical rounding stable rounding conditions. stable conditions.

Likewise,Likewise, FigureFigure 1111aa isis anan exampleexample ofof bladeblade geometricalgeometrical roundingrounding instability.instability. WhenWhen aa peakpeak ofof wavinesswaviness contactscontacts withwith thethe toptop surfacesurface ofof thethe bladeblade andand thethe wavinesswaviness becomesbecomes aa peakpeak atat thethe grindinggrinding pointpoint (and(and vicevice versa),versa), the wavine wavinessss error error cannot cannot be be removed. removed. In In addition, addition, again, again, Figure Figure 11b 11 isb an is anexample example of blade of blade geometrical geometrical roundi roundingng stability. stability. When When a valley a valley contacts contacts with withthe top the surface top surface of the ofblade the bladeand the and waviness the waviness becomes becomes a peak a peakat the at grin theding grinding point point(and (andvice ve vicersa), versa), the waviness the waviness error B − GRSC errorcan be can removed. be removed. Similarly, Similarly, the the blade blade geometrical geometrical rounding rounding stability stability criterion criterion ( B − GRSC)) cancan bebe summarizedsummarized asas follows:follows: 90 − θ − β ( − ) 90 −−θ β= ()BB−=GRSCGRSC UnstableUnstable ::Even Evenint integer eger (24) γγ (24) 90 − θ − β (B − GRSC) Stable : 90 −−θ β= Odd integer (25) ()B −=GRSC Stable:γ Odd int eger γ (25) Inventions 2017, 2, 26 10 of 18 Inventions 2017, 2, 26 10 of 18

GW n=16 GW n=16 ββ

θ θ

Unstable [(90°-θ°-β°)/ϒ°]=Even integer

(a)

GW GW n=15 n=15 β β

θ θ

[(90°-θ°-β°)/ϒ°]=Odd integer Stable (b)

Figure 11.11. BladeBlade geometricalgeometrical roundingrounding stabilitystability criteria.criteria. (a) BladeBlade geometricalgeometrical roundingrounding unstableunstable conditions; ((b)) BladeBlade geometricalgeometrical roundingrounding stablestable conditions.conditions.

To satisfy the RW geometrical rounding stability criterion (RW − GRSC), the center height angle To satisfy the RW geometrical rounding stability criterion (RW − GRSC), the center height angle γ is determined by: γ is determined by: 180180 RWRW −−= GRSC Stable Stable: γ: γ= (26) Iodd (26) Iodd where Iodd is an odd integer. Also, the blade geometrical stability criterion (B − GRSC) can be satisﬁed bywhere setting Iodd up is the ancenter odd integer. height angle Also,γ thethat blade can be geometrical calculated bystability the following criterion equation. (B − GRSC) can be satisfied by setting up the center height angle γ that can be calculated by the following equation. (90 − θ) B − GRSC Stable : γ = h ()90 −θ i (27) −=γ + (Dr+Dw) B GRSC Stable : Iodd (Dg+Dr+2Dw)  ()Dr+ Dw  (27) Iodd +  ++  4.3. Dynamic System Stability Criterion  ()Dg Dr2 Dw  As mentioned, the chatter vibration in the centerless grinding system is very severe and builds up4.3. fast Dynamic because System the work-regenerativeStability Criterion self-excited vibration has such a high amplitude growth rate, as shown in Figure 12a. This phenomenon not only deteriorates grinding accuracy and productivity, but alsoAs threatensmentioned, safe the operations. chatter vibration To achieve in the a stable centerless grinding grinding process system with highis very grinding severe accuracyand builds as shownup fast inbecause Figure the 12b, work-regenerative satisfying the dynamic self-excited system vibration stability criterionhas such isa high imperative. amplitude growth rate, as shownStudy in of Figure the characteristic 12a. This phenomenon root distributions not only of Equationdeteriorates (6) clariﬁesgrinding the accuracy generation andmechanism productivity, of but also threatens safe operations. To achieve a stable grinding process with high grinding accuracy the chatter vibration [12], and the chatter generation zones are revealed on the (n·γ–n·nw) diagram for theas shown dynamic in Figure system 12b, stability satisfying criterion the dynamic (DSSC). Figuresystem 13 stability plots the criterion 3D positive is imperative. growth rates σ of the Study of the characteristic root distributions of Equation (6) clarifies the generation mechanism characteristic roots on the (n·γ–n·nw) diagram. The chatter zones are shown as “mountains” located of the chatter vibration [12], and the chatter generation zones are revealed on the (n·γ–n·nw) diagram near the natural frequencies in the (n·nw) axis. The higher the height of the mountain, the more severe thefor chatterthe dynamic vibration. system Since stability the chatter criterion mountains (DSSC). are Figure in 0 < 13 (n ·plotγ)

Out-of-roundness of Out-of-roundnessground workpiece of ground- Mag. workpiece x 5,000 -- Mag. Filter x 1-50 5,000 - Filter 1-50 0.2 μm n=12 lobes Dynamic component of grinding force 0.2 μm n=12 lobes Dynamic component of grinding force

nw=8.9 rps, ϒ=9°, nw=2.1 rps, ϒ=9°, nw/ϒ=0.23 nnww=8.9/ϒ=0.99 rps, ϒ=9°, nw=2.1 rps, ϒ=9°, nw/ϒ=0.23 nw/ϒ=0.99 (a) (b) (a) (b) Figure 12.12. DynamicDynamic stability stability of centerlessof centerless grinding grinding system system (experiment). (experiment). (a) Unstable (a) Unstable grinding grinding process Figure(chatter);process 12.(chatter); (b Dynamic) Stable (b grinding) Stablestability grinding process. of centerless process. grinding system (experiment). (a) Unstable grinding process (chatter); (b) Stable grinding process.

Figure 13. Dynamic system stability diagram for Machine A. Conditions: b = 70 mm, k’w = 2 FigureFigurekN/mm·mm, 13.13. DynamicDynamic k’cr = 0.3 system system kN/mm·mm, stability stability diagram k’cs diagram = 1 forkN/mm·mm, Machine for Machine A. kmr Conditions: =A. 0.1 Conditions: kN/bμ=m, 70 kms mm, b =k’w= 0.15 70= 2kN/mm, kN/mmμ m,k’w km ·mm,=1 2= kN/mm·mm,k’cr0.3 kN/= 0.3μm, kN/mm fnr k’cr = 100=·mm, 0.3 Hz, kN/mm·mm,k’cs fns= = 1 200 kN/mm Hz, k’cs fn· mm,1= =1 150kN/mm·mm,kmr Hz,= 0.1ζr = kN/ 0.05, kmrµm, ζ =s kms0.1= 0.05, kN/= 0.15 ζμ1m, = kN/ 0.05.kmsµ m,= 0.15km 1kN/ = 0.3μm, kN/ kmµ1m, = 0.3fnr kN/= 100μm, Hz, fnrfns = =100 200 Hz, Hz, fnsfn 1= =200 150 Hz, Hz, fnζ1r = 150 0.05, Hz,ζs =ζr 0.05, = 0.05,ζ1 =ζs 0.05. = 0.05, ζ1 = 0.05. Each grinding machine has its own dynamic characteristics with different natural frequencies, and a (n·γ–n·nw) diagram can be plotted that shows its unique chatter zones. The chatter zones are EachEach grinding grinding machine machine has has its its own own dynamic characteristics characteristics with with different different natural natural frequencies, frequencies, andidentified a (n·γ –n·nby conductingw) diagram systematiccan be plotted grinding that shows tests. Figureits unique 14a chattershows thezones. chatter The zoneschatter of zones grinding are and a (n·γ–n·nw) diagram can be plotted that shows its unique chatter zones. The chatter zones are machine A. The chatter zones located near the natural frequencies of 100 Hz, 150 Hz and 200 Hz are identifiedidentiﬁed by by conducting conducting systematic systematic grinding grinding test tests.s. Figure Figure 14a14a shows shows the the chatter chatter zones zones of of grinding shown on the vertical axis (n·nw) Hz. These chatter zones are located in 0 < (n·γ) < 180° and generate machinemachine A. A. The The chatter chatter zones zones located located near near the the natu naturalral frequencies frequencies of of 100 100 Hz, Hz, 150 150 Hz Hz and and 200 200 Hz Hz are are showneven numbers on the vertical of lobes, axis while (n·n thew) Hz. zones These located chatter in 180°zones < (aren·γ located) < 360° ingenerate 0 < (n·γ odd) < 180° numbers◦ and generate of lobes shown on the vertical axis (n·nw) Hz. These chatter zones are located in 0 < (n·γ) < 180 and generate where γ > 6.7°. Figure 14b shows the chatter zones for grinding machine B. Machine B is designed eveneven numbers of lobes, while the zones located inin 180180°◦ < ( n·n·γ)) < < 360° 360◦ generategenerate odd odd numbers numbers of of lobes lobes with high, rigid structures and possesses spindles with very high stiffness. The first chatter zone where γ >> 6.76.7°.◦. FigureFigure 1414bb shows shows the the chatter chatter zones zones for for grinding grinding machine machine B. Machine B. Machine B is designedB is designed with appears at the natural frequency of 430 Hz, which is very high, and machine B creates wider withhigh, high, rigid structuresrigid structures and possesses and possesses spindles spindles with very with high very stiffness. high stiffness. The ﬁrst chatterThe first zone chatter appears zone at chatter-free regions than conventional machine A. appearsthe natural at frequencythe natural of frequency 430 Hz, which of 430 is very Hz, high,which and is machinevery high, B creates and machine wider chatter-free B creates regionswider Figure 15a shows the chatter zones of grinding machine A plotted on a (γ–nw) chart. The chatter-freethan conventional regions machine than conventional A. machine A. practicalFigure (γ –n15aw) showschart canthe chatterdescribe zones chatter of grindingzones, but machine cannot A describe plotted onthe a chatter(γ–nw) chart.generation The Figure 15a shows the chatter zones of grinding machine A plotted on a (γ–nw) chart. The practical practicalmechanism, (γ–n thew) chartchatter can zones’ describe various chatter vibration zones, modes, but cannotor the areasdescribe where the stablechatter grinding generation can (γ–nw) chart can describe chatter zones, but cannot describe the chatter generation mechanism, occur. However, the (γ–nw) chart is very effective in setup operations when used along with the mechanism,the chatter zones’ the chatter various zones’ vibration various modes, vibration or the modes, areas where or the stable areas grinding where canstable occur. grinding However, can occur.analytical However, (n·γ–n·n thew) diagram.(γ–nw) chart The isranges very ofeffective chatter inzones setup in operations(nw/γ) are explicitly when used given, along as shownwith the in analyticalFigure 15b. (n· γ–n·nw) diagram. The ranges of chatter zones in (nw/γ) are explicitly given, as shown in Figure 15b. Inventions 2017, 2, 26 12 of 18

the (γ–nw) chart is very effective in setup operations when used along with the analytical (n·γ–n·nw) Inventions 2017, 2, 26 12 of 18 Inventionsdiagram. 2017 The, 2, ranges26 of chatter zones in (nw/γ) are explicitly given, as shown in Figure 15b. 12 of 18

~~250 600 250 600 Chatter Hz~~

~~Hz Chatter Hz Hz zone w 200 w zone w 200 Chatter w n·n Chatter n·n n·n zone n·n 400 150 zone 400 150~~

100 100 200 Work size 200 Work size 50 O dia. 10 x L 70 mm workWork size size 50 O dia. 10 x L 70 mm workWork size size •dia. 300 x L 70 mm Dia.30•dia. 30 x LL70 70 mmmm •dia. 300 x L 70 mm Dia.30•dia. 30 x LL70 70 mmmm (lobes)x(work speed) speed) (lobes)x(work (lobes)x(work speed) (lobes)x(work speed) speed) (lobes)x(work (lobes)x(work speed) 0 90 180 270 360 0 90 180 270 360 0 90 180 270 360 0 90 180 270 360 (lobes)x(center-height angle) n·ϒ° (lobes)x(center-height angle) n·ϒ° (lobes)x(center-height(lobes)x(center-height angle) angle) n· n·ϒϒ°° (a) (b) (a) (b)

Figure 14. Chatter zones on (n·γ–n·nw) diagram. (a) Grinding machine A; (b) Grinding machine B. Figure 14. Chatter zones on (nn··γ–n·n–n·nww)) diagram. diagram. (a (a)) Grinding Grinding machine machine A; A; ( (bb)) Grinding Grinding machine machine B. B.

~~No chatter 100Hz chatter 150Hz chatter 200Hz chatter No chatter 100Hz chatter 150Hz chatter 200Hz chatter []γ []nw γ nw H 2020 H 1515 15 rps.~~

~~15 Chatter region rps. rps. rps.~~

~~w Chatter region w w~~

~~n 10 w n~~

~~n 10 n 1010 n  n w  wγ  γ L1 5  L1 5 5 nw  5 nw  Work speed  γ  Work speed~~

Work speed γ L2  L2 Work speed 0 0 0 33 66 99 0 33 66 99 1212 Center-heiCenter-heigghtht anangglele ϒϒ°° Center-heiCenter-heigghtht anangglele ϒϒ°° (a) (b) (a) (b) Figure 15. Dynamic system stability criterion for machine A. (a) Chatter zones confirmed by Figure 15. Dynamic system stability criterion for machine A. (a) Chatter zones confirmed by Figuregrinding 15. tests;Dynamic (b) Dynamic systemstability system stability criterion criterion. for machine A. (a) Chatter zones conﬁrmed by grinding grindingtests; (b) Dynamictests; (b) Dynamic system stability system criterion.stability criterion. Three types of stable, no-chatter zones exist for high work speed and low work speed regions. Three types of stable, no-chatter zones exist for high work speed and low work speed regions. The chatter-freeThree types conditions of stable, no-chatter for machine zones A are: exist for high work speed and low work speed regions. The chatter-free conditions for machine A are: The chatter-free conditions for machine A are: 1. (nw/γ) H > 3.0 (high-speed chatter-free zone; KH) w 1. (nn /γ/ ) H > 3.0 (high-speed chatter-free zone; KH) 1.2.( (nw/γγ)) L1 H >< 0.6 3.0 when (high-speed γ is lower chatter-free (low-speed zone; chatter-free KH) zone 1; KL1) 2. (nw/γ) L1 < 0.6 when γ is lower (low-speed chatter-free zone 1; KL1) 2.3.( (nw//γγ)) L2 L1 < < 0.28 0.6 whenwhen γγ isis lowerhigher (low-speed (low-speed chatter-free chatter-free zone zone 1; 2; KL1) KL2) 3. (nw/γ) L2 < 0.28 when γ is higher (low-speed chatter-free zone 2; KL2) 3.( nw/γ) L2 < 0.28 when γ is higher (low-speed chatter-free zone 2; KL2) Figure 16a,b redraw Figure 14b to include chatter zone boundary lines. The chatter-free zones Figure 16a,b redraw Figure 14b to include chatter zone boundary lines. The chatter-free zones are: Figure 16a,b redraw Figure 14b to include chatter zone boundary lines. The chatter-free zones are: are:

4.4.( (nw//γγ) )H H > >4.24 4.24 (KH) (KH) 4. (nw/γ) H > 4.24 (KH) ◦ 5.5.( (nw//γγ) )L1 L1 < <2.15 2.15 (KL1) (KL1) for for γ γ< <6.67° 6.67 5. (nw/γ) L1 < 2.15 (KL1) for γ < 6.67° ◦ 6.6.( (nw//γγ) )L2 L2 <1.08 < 1.08 (KL2) (KL2) for for γ >γ 6.67°> 6.67 6. (nw/γ) L2 <1.08 (KL2) for γ > 6.67° FigureFigure 1616aa showsshows aa narrownarrow stablestable zonezone inin 1.53 1.53 < < ( n(nww//γγ)) << 2.15.2.15. TheThe setupsetup forfor thisthis chatter-stablechatter-stable Figure 16a shows a narrow stable zone in 1.53 < (nw/γ) < 2.15. The setup for this chatter-stable zonezone isis tootoo riskyrisky toto use,use, soso itit isis notnot consideredconsidered anan areaarea ofof stable,stable, chatter-freechatter-free conditions. conditions. zone is too risky to use, so it is not considered an area of stable, chatter-free conditions. Since the work speed nw is controlled by the regulating wheel speed Nr, it is helpful to convert Since the work speed nw is controlled by the regulating wheel speed Nr, it is helpful to convert the (γ–nw) charts (Figures 15b and 16b) into a (γ–Nr) chart, as shown in Figure 17. For practical setup the (γ–nw) charts (Figures 15b and 16b) into a (γ–Nr) chart, as shown in Figure 17. For practical setup operations, the range of the center height angle is set to γ = 3° to 9°. Also, in this case the range of the operations, the range of the center height angle is set to γ = 3° to 9°. Also, in this case the range of the speed ratio q (defined as the ratio of the work speed Vw to the grinding speed Vg) for surface speed ratio q (defined as the ratio of the work speed Vw to the grinding speed Vg) for surface Inventions 2017, 2, 26 13 of 18

Since the work speed nw is controlled by the regulating wheel speed Nr, it is helpful to convert the (γ–nw) charts (Figures 15b and 16b) into a (γ–Nr) chart, as shown in Figure 17. For practical setup operations, the range of the center height angle is set to γ = 3◦ to 9◦. Also, in this case the range of Inventions 2017, 2, 26 13 of 18 the speedInventions ratio 2017q, 2(deﬁned, 26 as the ratio of the work speed Vw to the grinding speed Vg)13 for of 18 surface roughnessroughness control control is set is set to to 1/ 1/qq= = 25 to to 150. 150. In InFigure Figure 17, three 17, three chatter-free chatter-free stable zones—KH, stable zones—KH, KL1 and KL1 and KL2—areroughnessKL2—are shown showncontrol withinwithin is set to the 1/q constrained constrained= 25 to 150. Inrange. range.Figure The 17, The dynamic three dynamic chatter-free system system stability stable stability zones—KH, criterion criterion is KL1satisfied isand satisﬁed KL2—are shown within the constrained range. The dynamic system stability criterion is satisfied when awhen set ofa set (γ ,ofNr (γ), isNr selected) is selected from from the the chatter-free chatter-free stable stable zones zones KH, KH, KL1 KL1 and and KL2. KL2. when a set of (γ, Nr) is selected from the chatter-free stable zones KH, KL1 and KL2. 15 15 Even-lobe Odd-lobe Chatter zone 430 chatterEven-lobe zone chatterOdd-lobe zone 430 Chatter zone chatter zone chatter zone rps w rps n

~~w 10 n [Hz] 10 w [Hz] w n·n n·n~~

~~Work speed 5 Work speed 5~~

0 0 101.4 180 281.4 360 0 0 101.4 n·ϒ [°] 180 281.4 360 036912 n·ϒ [°] 036912Center-height angle ϒ° Center-height angle ϒ° (a) (b) (a) (b) Figure 16. Dynamic system stability criterion for machine B. (a) Chatter zones on· (n··γ–n·nw) FigureFigure 16. Dynamic 16. Dynamic system system stability stability criterion criterion for machinefor machine B. ( aB.) Chatter (a) Chatter zones zones on (nonγ –n(n·γn–n·nw) diagram;w) diagram; (b) Chatter zones on (γ–nw) chart. (b) Chatterdiagram; zones (b) Chatter on (γ–n zonesw) chart. on (γ–nw) chart. [nw/ϒ]H [nw/ϒ]H

~~25 Nru 25 Nru KH KH Chatter zone Chatter zone~~

~~KL1 KL2 Speed1/q=Vg/Vw ratio KL1 KL2 150Speed1/q=Vg/Vw ratio~~

~~Regulating wheel speed NrRegulating rpm Nrl 150~~

Regulating wheel speed NrRegulating rpm Nrl 0 0 3° 6.67° 9° 3° Center-height6.67° angle ϒ° 9° Center-height angle ϒ° Figure 17. Chatter free zones on γ–Nr chart. KH: High speed chatter-free zone, KL1, KL2: Low FigureFigurespeed 17. Chatter chatter-free 17. Chatter free zones. zonesfree zones on γ on–Nr γ–Nrchart. chart. KH: KH: High High speed speed chatter-free chatter-free zone, zone, KL1,KL1, KL2:KL2: Low speed speed chatter-free zones. chatter-free zones. 5. Modeling to Find the Optimum Setup Conditions that Satisfy the Three Stability Criteria of 5.Centerless Modeling Grinding to Find the Optimum Setup Conditions that Satisfy the Three Stability Criteria of 5. ModelingCenterless to FindGrinding the Optimum Setup Conditions that Satisfy the Three Stability Criteria of CenterlessThe Grinding previous section discussed the determination of setup conditions that would satisfy each individualThe previous stability section criterion. discussed This section the determination discusses the of development setup conditions of a modelthat would for the satisfy optimum each Theindividualsetup previous conditions stability section that criterion. will discussed simultaneously This section the determination satisfy discusses all three the development ofstability setup criteria. conditions of a model that for wouldthe optimum satisfy each individualsetupFigure stability conditions 18 criterion.describes that will the Thissimultaneously structure section of discusses satisfythe developed all the three development model.stability As criteria. the of first a model step, forthe thesets optimum of (θ, γ) setup conditions(bladeFigure that angle will 18 θ, describescenter-height simultaneously the anglestructure satisfy γ) satisfying of allthe threedeveloped the three stability individualmodel. criteria. As stabilitythe first criteria step, the are sets determined. of (θ, γ) (blade angle θ, center-height angle γ) satisfying the three individual stability criteria are determined. FigureTo satisfy 18 describesthe work rotation the structure stability criterion, of the developed the sets of ( model.θ, γ) are selected As the from ﬁrst the step, safe the operation sets of (θ, γ) To satisfy the work rotation stability criterion, the sets of (θ, γ) are selected from the safe operation (bladezone angle shownθ, center-height in Figure 7, angleanalyzedγ) satisfyingwith the varying the three maximum individual friction stability coefficient criteria μr0 of are a determined.given zoneregulating shown wheel. in Figure Also, 7,the analyzed sets of (θ with, γ) satisfying the varying both maximum geometrical friction rounding coefficient stability μ rcriteria—0 of a givenRW To satisfyregulating the work wheel. rotation Also, stabilitythe sets of criterion, (θ, γ) satisfying the sets both of (geometricalθ, γ) are selected rounding from stability the safe criteria— operationRW zone − GRSC and B − GRSC—are calculated. Then, the sets of (nw, γ) (work speed nw and γ) that satisfy the shown− inGRSC Figure and7 B, analyzed− GRSC—are with calculated. the varying Then, maximumthe sets of (n frictionw, γ) (work coefﬁcient speed nw andµr0 γof) that a given satisfy regulating the Inventions 2017, 2, 26 14 of 18 wheel.Inventions Also, 2017, the2, 26sets of (θ, γ) satisfying both geometrical rounding stability criteria—RW − GRSC14 ofand 18 B − GRSC—are calculated. Then, the sets of (nw, γ) (work speed nw and γ) that satisfy the dynamic systemdynamicInventions stability system 2017, 2, criterion26 stability arecriter found.ion are The found. sets of The (nw ,setsγ) are of selected(nw, γ) are from selected one of from the stable one of chatter-free the14 ofstable 18 zones:chatter-free KH, KL1zones: or KH, KL2 KL1 (see Figureor KL2 17 (see). Figure 17). dynamic system stability criterion are found. The sets of (nw, γ) are selected from one of the stable chatter-free Workzones: rotation KH, KL1 stability or KL2Geometrical (see Figure rounding 17). stability Dynamic system stability (90-θ-β)/ϒ=Odd ( θ,ϒ )

~~° 180/ϒ=Odd~~

~~ϒ ( ϒ, nw ) Work rotationAbsolute stability Geometrical rounding stability Dynamic system stability~~

~~(rps) Chatter zone Safe zone (90-θ-β)/ϒ=Odd w ( θ,ϒ ) γ~~

~~° 180/ϒ=Odd~~

~~ϒ ( ϒ, nw ) ( θ,ϒAbsolute)~~

(rps) Chatter zone Safe zone w wheel γ Grinding 研研研研 ( θ,ϒ ) Work speed n ( ϒ, nw ) Center-height angle θ 調調調受受 Regulating Regulating wheel Blade Blade angle θ° wheel Center-height angle ϒ Grinding 研研研研

Work speed n ( ϒ, nw ) Center-height angle θ 調調調受受 Regulating Regulating wheel Blade angle θ° Blade Center-height angle ϒ Find the sets of (θ, ϒ, nw) that satisfy three stability criteria

~~Find the sets of (θ, ϒ, nw) that satisfy three stability criteria Determine the optimum set of (θ, ϒ, nw)~~

~~Determine the optimum set of (θ, ϒ, nw) Provide the optimum setup condition (θ, CH, Nr)~~

Provide the optimum setup condition (θ, CH, Nr) Figure 18. Structure of the model for determining the optimum setup condition Figure 18. Structure of the model for determining the optimum setup condition The second step is to find the sets of (θ, γ, nw) satisfying all three stability criteria. The third step The second step is to find the sets of (θ, γ, nw) satisfying all three stability criteria. The third is to determineThe second the step optimum is to find set the from sets among of (θ, γ the, nw population) satisfying allof threethe (θ stability, γ, nw) sets criteria. by calculating The third step the PI step is to determine the optimum set from among the population of the (θ, γ, nw) sets by calculating (performanceis to determine index) the optimum function set based from amongon the the proces populations aim of(accuracy the (θ, γ ,or nw )produc sets by tivity).calculating Finally, the PI the the PI (performance index) function based on the process aim (accuracy or productivity). Finally, optimum(performance set (θ ,index) γ, nw) isfunction converted based into on practical the proces setups aim conditions (accuracy (blade or produc angletivity). θ, center Finally, height the CH , the optimum set (θ, γ, nw) is converted into practical setup conditions (blade angle θ, center height CH, RWoptimum speed Nr set) as(θ ,the γ, noutputsw) is converted of the developed into practical model. setup conditions (blade angle θ, center height CH, RW speed Nr) as the outputs of the developed model. RWFigure speed 19Nr )is as the the flow outputs chart of of the “Opt-Setup developed model.Master”, the developed model. The model requires a Figure 19 is the flow chart of “Opt-Setup Master”, the developed model. The model requires machineFigure operator 19 is tothe provide flow chart some of “Opt-Setupinput information, Master”, as the shown developed in Table model. 1. All The parameters model requires required a a machine operator to provide some input information, as shown in Table1. All parameters required formachine the calculation operator of to the provide Opt-Setup some Masterinput information, are referenced as shown from thein Table data 1.bank, All parameterswhich stores required machine for the calculation of the Opt-Setup Master are referenced from the data bank, which stores machine specifications,for the calculation machine of the dynamic Opt-Setup characteristics, Master are refe workrenced part from numbers the data with bank, dimensions, which stores RW machine friction specifications, machine dynamic characteristics, work part numbers with dimensions, RW friction characteristics,specifications, blade machine availability, dynamic etc.characteristics, The constrai worknts ofpart the numbers setup parameters with dimensions, are also RW stored friction in the characteristics, blade availability, etc. The constraints of the setup parameters are also stored in the datacharacteristics, bank, as shown blade in availability, Table 2. etc. The constraints of the setup parameters are also stored in the datadata bank, bank, as as shown shown in in Table Table2. 2.

Inputs of operational Process priority Inputs of operational DATA BANK information Process• Grinding priority accuracy information • MachineDATA specifications BANK • •GrindingProductivity accuracy • • Productivity • MachineMachine dynamics specifications • Machine dynamics • Setup constrains • Setup constrains • Wheel and blade info. • Wheel and blade info. • Work• Work PN PN and and dimensions dimensions BladeBlade availability availability θii GeometricalGeometrical stability stability criteria criteria ϒi ϒi

SymbolsSymbols θ:θ Blade: Blade angle angle WorkWork rotationrotation stability criteria criteria ( θ(θi,i, ϒ ϒi)i) CH:CH: Center-height Center-height No ϒ: ϒCH: CH angle angle Safe YesYes nwn:w Work: Work speed speed Nr:Nr: RW RW speed speed DynamicDynamic stabilitystability criteria ((nnwi,wi, ϒ ϒi)i) PerformancePerformance IndexIndex functionsfunctions PriorityPriority ranking ranking of of setupsetup condition sets sets ( (θθi,i, ϒ ϒi,i, n nwiwi) )

~~OutputsOutputs of of optimum optimum setup conditions ( (θθi,i, CHi, CHi, Nr Nri)i )~~

~~FigureFigure 19. 19. Flow Flow chartchart of of developed model model “Opt-Setup “Opt-Setup Master”. Master”. Inventions 2017, 2, 26 15 of 18~~

~~Table 1. Input information and parameters referred to data bank.~~

Inputs Action Parameters Referred to Data Bank Machine specifications, Machine dynamic characteristics (Natural Machine name Select frequencies, damping ratios) Workpiece shape Select Cylindrical (CYD), Tapered (TPD), Spherical (SRL), Multi-stepped (STD) Workpiece part-number Select Dimensions (diameter, length, etc.), profile GW diameter Measure New, worn, measured RW diameter Measure New, worn, measured RW dresser type and dress lead Select Single point dress, Rotary dress, Friction coefficient of RW Blade availability Select Blade angle θ, blade thickness t

~~Table 2. Constraints of setup parameters.~~

Setup Parameters Symbol Unit. Min. Max. Typical Range of speed ratio 1/q = Vg/Vw - 25 150 100 Range of blade angle θ ◦ 15 45 30 Range of Center-height angle γ ◦ 3 9 6.67 Range of regulating wheel speed Nr rpm 15 100 50 Range of GW diameter Dg mm 375 455 450 Range of RW diameter Dr mm 275 350 345 Range of Workpiece diameter Dw mm 5 100 40 Grinding wheel speed in revolution Ng rpm 1260 2300 1890 Grinding speed Vg m/s 30 45 45

The Opt Setup Master calculates the boundary line of the safe operation zone on the safe operation chart by using the following relationship between the dressing lead leadr and the maximum friction coefﬁcient µr0 of the rubber bonded regulating wheel [15].

µr0 = a · leadr + 0.33 (28) where a is a constant. a = 0.14 and a = 0.02 for a SPD (single-point dresser) and an RD (rotary dresser), respectively. The boundary line of the safe operation zone on the (θ, γ) chart can be expressed by:

~~γc = m · θc + b (29)~~

where m = 3.036 · a · leadr + 1.168 (30) 2 b = 77.06 · (a · leadr) − 27.55 · (a · leadr) − 21.23 (31) When θ1 of a point (θ1, γ1) is greater than (γ1–b)/m, the point is located at the right side of the safe operation zone boundary line and meets the work rotation stability criterion. The first calculation of the Opt-Setup Master is to find the sets of (θg, γg) that meet the geometrical rounding stability criteria RW − GRSC and B − GRSC. The next calculation is to determine if the sets (θg, γg) are located within the safe operation zone under the given dressing conditions of the rubber bonded regulating wheel. If they are (answer “yes”), the sets (θgw, γgw) satisfy both the geometrical rounding and the work rotation stability criteria. The final calculation is to find the work speed nw by using γgw and the chatter stability boundary lines of (nw/γgw). From these calculations, the optimum sets of (θ, γ, nw) are discovered. Then, the performance index (PI) functions that were prepared based on the process aims are applied to assess the optimum sets of (θ, γ, nw). The PI functions are summarized in Table3. The weighting factors of the PI functions are determined by applying theoretical knowledge, experimental knowledge and operational skills. PI functions can be updated with newly gained knowledge and skills. For each process aim (accuracy or productivity), the values of the PI function for all setup conditions are calculated and these sets are ranked in ascending order from minimum to Inventions 2017, 2, 26 16 of 18 those with greater values. The smaller the PI values, the more suitable the setup conditions. The setup conditions with the smallest PI function values are defined as the optimum setup conditions.

~~Table 3. Performance Index (PI) functions.~~

Weighting Factors Process Aim PI Function ABCDE Accuracy PIa = A × I θ − 27.5 I + B × I γ − 6.67 I + C × I Nr − 50 I + D × FL + E × STYP 0.2 2 0.1 0.01 2 Productivity PIr = A × I θ − 45 I + B × I γ − 5.15 I + C × I Nr − 50 I + D × FL + E × STYP 0.1 2 0.12 0.5 2 θ: Blade angle; γ: Center-height angle; Nr: RW speed in rpm; FL: KL1 = 0, KL2 = 0.5, KH = 1, STYP: (RW + B) GRSC = −1, RW − GRSC= 0, B − GRSC = +1.

~~Table4 shows examples of the Opt-Setup Master outputs. The conditions of the model simulation are:~~

~~(1) machine B is applied (2) the process aim is accuracy-oriented (3) the work shape and the size are a cylindrical workpiece with 40 mm (D) × 60 mm (L)~~

~~(4) the rubber bonded RW is dressed with leadr = 0.5 mm/rev by SPD (5) the GW diameter is 453 mm (6) the RW diameter is 350 mm (7) the available blade angles are θ = 27.5◦ and θ = 40.3◦~~

~~Table 4. Examples of outputs from Opt-Setup Master.~~

Priority Optimum Set Up Conditions Engineering Parameter Stability Parameters Blade Blade RW Work 1/q Stable RW − Blade − Ranking Center-Height CH Angle (n /γ) Angle Thickness Speed Speed Ratio w Zone GRSC GRSC Nr (90 − θ − No. θ (◦) t (mm) CH (mm) γ (◦) n (rps) Vg/Vw (1/s) KH/KL1/KL2 180/γ (rpm) w β)/γ 1 40.3 20 12.68 41.8 6.68 6.1 25 0.91 KL2 27 7 2 40.3 20 12.66 83.5 6.67 12.2 25 1.83 KL1 27 7 3 40.3 20 13.67 45.1 7.2 6.58 25 0.91 KL2 27 6.5 Conditions: Machine B, Process aim: accuracy, Work: cylindrical Dia.40 × L60 mm.

Ranking No. 1 has the optimum setup conditions, as it has the smallest PI values. The practical setup parameters are a blade angle θ = 40.3◦ with a thickness of 20 mm, a center height of CH = 12.68 mm and RW speed of Nr = 41.8 rpm. The center height angle is γ = 6.68◦, the work speed is nw = 6.1 rps and the 1/q is 25. The optimum setup condition set was selected from the safe operating zone. Therefore, it meets the work rotation stability criterion and ensures safe operations. Further, the optimum setup condition set was selected from the stable zone KL2 for chatter-free grinding, and thus meets the dynamic system stability criterion. Also, the optimum setup condition set meets the criteria of both Equations (23) and (25) (180/6.68 = 27 and (90 − 40.3 − 2.95)/6.68 = 7), so the geometrical rounding stability criteria are maintained. Through these means, it is veriﬁed that the optimum setup condition set—provided as the outputs from the Opt-Setup Master—simultaneously satisﬁes all three stability criteria for centerless grinding. Table5 shows the optimum setup conditions as calculated by the Opt-Setup Master for infeed centerless grinding of cylindrical workpieces of various sizes by two different grinding machines, A and B. In all cases, γ = 6.67◦, one of the most preferable γ angles, is chosen. All values representing RW − GRSC and B − GRSC are odd integers, indicating that all the setup conditions meet the geometrical rounding stability criteria. Machine B has a greater chatter DSSC index (nw/γ) than machine A in chatter-stable zones KH, KL1 and KL2. Machine B’s high stiffness creates more extensive chatter-stable zones than conventional machine A. Inventions 2017, 2, 26 17 of 18

~~Table 5. The optimum setup conditions provided by the Opt-Setup Master for the grinding of cylindrical workpieces with different machines A and B.~~

Work Blade C-H RW Chatter Chatter Machine RW − GRSC Blade − GRSC Case Dia.x L. Angle Angle Speed DSSC Stable Zone No. Nr mm A/B θ (◦) γ (◦) (n /γ) KH/KL1/KL2 180/γ (90 − θ − β)/γ (rpm) w 1 A 27.5 6.67 38.6 3.37 KH 27 9 10 × 20 2 B 27.5 6.67 64.1 5.61 KH 27 9 3 A 27.5 6.67 77.1 3.37 KH 27 9 20 × 30 4 B 27.5 6.67 41.8 1.83 KL1 27 9 5 A 40.3 6.67 14.6 0.43 KL1 27 7 30 × 50 6 B 40.3 6.67 62.7 1.83 KL1 27 7 7 A 40.30 6.67 19.4 0.43 KL1 27 7 40 × 60 8 B 40.30 6.68 41.8 0.91 KL2 27 7 9 A 40.30 6.67 24.3 0.43 KL1 27 7 50 × 70 10 B 40.30 6.68 52.3 0.91 KL2 27 7 11 A 40.30 6.67 29.1 0.43 KL1 27 7 60 × 80 12 B 40.30 6.68 62.7 0.91 KL2 27 7 ◦ ◦ Condition: GW φ453 mm, Rubber bonded RW φ350 mm, µr0 = 0.4, Available blade θ = 27.5 , 40.3 .

6. Conclusions Centerless grinding systems possess some unique features, including their work rotation drive, loose work holding and regenerative centering mechanisms. However, because of these features, three fundamental stability issues arise. Many researchers have investigated the issues and provided useful guidelines for solving the issues. This paper summarizes the three fundamental stability issues: (1) work rotation stability for safe operation with no spinners; (2) geometrical rounding stability for better roundness; and (3) dynamic system stability for chatter-free grinding. It emphasizes the need for an analytical tool that can provide optimal setup conditions—those conditions that will satisfy all three stability criteria simultaneously. This paper describes a newly developed analytical tool named Opt-Setup Master, and discusses how the three stability criteria can be met. The developed Opt-Setup Master has the following features:

(1) Accepts various shapes of workpiece: cylindrical, tapered, spherical and multi-stepped (2) Applicable to any centerless grinding machine (3) Data management via a data bank (4) Inputs are easy to enter and outputs are readily usable (5) Designed for operators (6) Provides scientiﬁc parameters for engineers/managers (7) Finds all setup conditions satisfying the three stability criteria of centerless grinding systems (8) Outputs the optimum condition based on process aim

~~Conﬂicts of Interest: The authors declare no conﬂict of interest.~~

~~References~~

1. Hashimoto, F.; Gallego, I.; Oliveira, J.F.G.; Barrenetxea, D.; Takahashi, M.; Sakakibara, K.; Stålfelt, H.-O.; Staadt, G.; Ogawa, K. Advances in Centerless Grinding Technology. CIRP Ann. Manuf. Technol. 2012, 61, 747–770. [CrossRef] 2. Hashimoto, F.; Suzuki, N.; Kanai, A.; Miyashita, M. Critical Range of Set-up Conditions of Centerless Grinding and Problem of Safe Machining Operation. Jpn. Soc. Precis. Eng. 1982, 48, 996–1001. [CrossRef] 3. Hashimoto, F.; Lahoti, G.; Miyashita, M. Safe Operations and Friction Characteristics of Regulating Wheel in Centerless Grinding. Ann. CIRP 1998, 47, 281–286. [CrossRef] Inventions 2017, 2, 26 18 of 18

4. Abrasive Industry. Anon, Centerless Grinding Operation; Abrasive Industry: Auckland, New Zealand, 1925; pp. 102–105. 5. Dall, A.H. Rounding Effect in Centerless Grinding. Mech. Eng. 1946, 68, 325–332. 6. Ogawa, M.; Miyashita, M. On Centerless Grinding (1)—Theoretical Treatise on Rounding Action. Jpn. Soc. Precis. Eng. 1957, 24, 89–94. [CrossRef] 7. Reeka, D. On the Relationship between the Geometry of the Grinding Gap and the Roundness Error in Centerless Grinding. Ph.D. Thesis, RWTH, Aachen University, Aachen, Germany, 1967. 8. Rowe, W.B.; Barash, M.M.; Koenigsberger, F. Some Roundness Characteristics of Centerless Grinding. Int. J. Mach. Tool Des. Res. 1965, 5, 203–215. [CrossRef] 9. Yonetsu, S. Study on Centerless Grinding—First Report. Jpn. Soc. Mech. Eng. 1953, 19, 53–59. 10. Gurney, J.P. An Analysis of Centerless Grinding. Trans. ASME J. Eng. Ind. 1964, 86, 163–174. [CrossRef] 11. Furukawa, Y.; Miyashita, M.; Shiozaki, S. Vibration Analysis and Work Rounding Mechanism in Centerless Grinding. Int. J. Mach. Tool Des. Res. 1970, 44, 145–175. [CrossRef] 12. Hashimoto, F.; Zhou, S.S.; Lahoti, G.D.; Miyashita, M. Stability Diagram for Chatter Free Centerless Grinding and Its Application in Machine Development. Ann. CIRP 2000, 49, 225–230. [CrossRef] 13. Gallego, I. Intelligent Centerless Grinding: Global Solution for Process Instabilities and Optimal Cycle Design. Ann. CIRP 2007, 56, 347–352. [CrossRef] 14. Hashimoto, F.; Lahoti, G.D. Optimization of Set-up Conditions for Stability of the Centerless Grinding Process. Ann. CIRP 2004, 53, 271–274. [CrossRef] 15. Hashimoto, F. Effects of Friction and Wear Characteristics of Regulating Wheel on Centerless Grinding. In Proceedings of the SME, Third International Machining and Grinding Conference, Cincinnati, OH, USA, 4–7 October 1999; pp. 1–18. 16. Rowe, W.B.; Koenigsberger, F. The Work-Regenerative Effect in Centerless Grinding. Int. J. Mach. Tool Des. Res. 1964, 4, 175–187. [CrossRef] 17. Snoeys, R.; Brown, D. Dominating Parameters in Grinding Wheel and Workpiece Regenerative Chatter. In Proceedings of the 10th International Machine Tool Design and Research Conference, LanPleacaster, UK, 10–14 September 1969; Pergamon Press: Oxford, UK, 1970; pp. 325–348.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).