The Design of an Infeed Cylindrical Grinding Cycle

inventions

Article The Design of an Infeed Cylindrical Grinding Cycle

Fukuo Hashimoto

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

Received: 6 August 2020; Accepted: 21 August 2020; Published: 28 August 2020

Abstract: This paper synthesizes the design of an infeed cylindrical grinding system into a total system composed of the grinding mechanism and the grinding machine characteristics. The causalities between the grinding parameters and the machine structures are discussed, and the infeed grinding processes are analyzed as outputs that represent responses to the inputs. These relationships are integrated into a block diagram with closed-loop feedback. A novel model exhibiting practical parameters such as grinding speed, infeed rate and MRR (Material Removal Rate) is proposed. The analysis of the grinding system derived a critical factor, the “grinding time contact,” which governs the transient behaviors of process parameters such as forces and machine deﬂection. The process parameters during the infeed cycle including spark-out grinding were investigated, and the formulas required for the cycle design are presented. Furthermore, to improve accuracy and productivity, the features of the cycle design are described and procedures for controlling size error and roundness are discussed. Finally, the model was veriﬁed with infeed grinding tests applied to both the chuck-type cylindrical and centerless grinding methods.

Keywords: grinding; cylindrical grinding; grinding machine; cycle time

1. Introduction In general, the term “cylindrical grinding” describes a group of grinding processes, the common characteristic of which is the rotation of the workpiece around a fixed axis [1]. However, this paper extends the field of cylindrical grinding to the centerless-support grinding methods, which do not use a fixed axis for workpiece rotation. This paper discusses the infeed (plunge) grinding of the outer diameter of the cylindrical and ring workpieces; the relevant methods to be addressed are chuck- center type cylindrical grinding, centerless grinding and shoe-centerless grinding. In infeed cylindrical grinding, the grinding cycle must be designed based on quality and productivity demands. Usually the cycle includes grinding processes for roughing, semi-finishing and finishing, and at the end of the cycle, spark-out grinding is performed to release the machine elastic deflection caused by the grinding forces. Production engineers must make decisions about the infeed rate, the grinding stock for each process and the spark-out time; they need to estimate what the productivity based on the cycle time should be, and must determine grinding quality standards for factors such as size error and roundness. It is crucial for engineers to fully understand the behaviors of grinding parameters during the cycle and the influence of the grinding machine on the grinding processes. Early research on grinding was conducted by Schlesinger in 1936 [2]. Since then, many researchers have performed extensive studies and today, grinding fundamentals are well established [3,4]. The relationships among the fundamental parameters in infeed cylindrical grinding have been presented by Malkin, Levin, et al. and others [5–8]. Further, many kinds of grinding models have been proposed and the simulations are making great contributions to the production industry [9–11]. Chiu et al. has presented a simulation of the grinding parameters specific to infeed cylindrical grinding [12].

Inventions 2020, 5, 46; doi:10.3390/inventions5030046 www.mdpi.com/journal/inventions Inventions 2020, 5, 46 2 of 22

The influences of grinding machine characteristics on the grinding processes have been investigated as well [13–16]. Tobias revealed the dynamic characteristics of the machine tools [17], and the design methods of machine tools have also been presented [18–20]. In particular, the effects of the dynamics of machine structures on self-excited chatter vibrations have been discussed, and the theory known as “regenerative chatter vibration” has been well developed [21–23]. In the production engineering societies, both grinding fundamentals and machine tool characteristics are well understood, and many analytical models have been presented [24,25]. However, it is common that industrial engineers who understand both areas well still have difficulty applying their knowledge to the design of the grinding cycle and the prediction of grinding performance. They also lack practical tools to optimize infeed grinding processes based on the characteristics of the grinding machine being used. These issues may be arising from the fact that infeed cylindrical grinding is not treated as a total grinding system. Therefore, the grinding fundamentals and the machine characteristics are not holistically integrated, and the causalities among the system parameters are not made clear. Another reason could be the disconnection between the scientific parameters used in past proposed models and the practical parameters used in real-world grinding operations. In order to optimize infeed grinding processes with the required grinding performance and productivity, it is essential to synthesize the infeed cylindrical grinding process from the viewpoints of both the grinding and machine tool technologies. Industrial engineers should be equipped with practical tools for designing the grinding cycle and determining operational conditions such as feed rates, stock assignments, specific material removal rate (SMRR) and spark-out times. The objective of this study was to treat the various components of infeed cylindrical grinding as an overall system. Thus, this proposed system contains the grinding mechanism, the grinding fundamental parameters and the characteristics of the grinding machine structures. The causalities between the grinding parameters and the machine tools are discussed, and the infeed grinding processes are analyzed as output responses to the operational inputs from the grinding machine. These relationships are integrated into a block diagram with closed-loop feedback. A novel model exhibiting practical parameters like grinding speed, infeed rate and MRR is presented. The proposed model is directly applicable for the practical setup and grinding operations in both cylindrical and centerless grinding. The analysis of the grinding system derived a critical factor, the “grinding time constant,” which governs the transient behaviors of process parameters, such as forces and machine deflection. The process parameters used during the infeed cycle (including spark-out grinding) were investigated, and the formulas required for the cycle design are shown. Furthermore, to improve accuracy and productivity, the features of the cycle design are described and procedures for controlling size error and roundness are discussed. Finally, the model was verified with infeed grinding tests applied to both grinding methods: chuck type cylindrical grinding and centerless grinding.

2. Cylindrical Grinding System

2.1. Models of Grinding Systems A general grinding system can be represented as a closed-loop system, as shown in Figure1. The system is composed of two functions. One is the feed-force function based on the grinding mechanism, and the other is the compliance function of the grinding machine. In a grinding machine, an infeed slide with a grinding wheel head is fed into the workpiece and an amount of interference between the wheel and the workpiece is specified. The grinding wheel removes materials from the workpiece as per the actual interference amount. The grinding and subsequent material removal result in the generation of grinding forces and cause the deflection of the machine structures in the infeed direction. Therefore, the actual amount of the interference becomes the difference between the command infeed and the machine deflection. This principle can be applied to most grinding Inventions 2020, 5, x FOR PEER REVIEW 3 of 23

To investigate the regenerative self-excited vibrations in cylindrical grinding applications, including centerless grinding [11,14,21–23,25], several grinding models have been developed to date. Stability criteria to suppress the chatter vibrations have also been discussed. These models focus on the depth of cut per revolution of the workpiece, which is proportional to the grinding forces under given conditions. When the structure of the grinding machine is excited with the dynamic components of the grinding forces, the self-excited chatter vibrations occur. Thus, the models capture the dynamic characteristics of the grinding machine such as the natural frequencies and the damping factors. From this point of view, these models can be classified within the dynamic grinding model shown in Figure 2. However, they are not suitable for addressing the fundamental parameters of the grinding processes because the grinding forces are determined by the material removal rates, not Inventionsonly by 20the20 , depth5, x FOR of PEER cut. REVIEW In addition, the dynamic model does not directly present practical 3setup of 23 conditions, such as workpiece sizes, grinding speed and infeed rates. To investigate the regenerative self-excited vibrations in cylindrical grinding applications, Figure 3 shows a novel grinding model that accounts for the various setup parameters of Inventionsincluding2020 centerless, 5, 46 grinding [11,14,21–23,25], several grinding models have been developed to 3date. of 22 cylindrical grinding operations, including the material removal rate, workpiece sizes and grinding Stability criteria to suppress the chatter vibrations have also been discussed. These models focus on speed. In this model, only the static compliance of the grinding machine is considered for the grinding the depth of cut per revolution of the workpiece, which is proportional to the grinding forces under systems,process analysis. including The surface dynamic grinding, process cylindrical behaviors grinding, should centerless be analyzed grinding, by using shoe-centerless the dynamic grinding, model given conditions. When the structure of the grinding machine is excited with the dynamic double(Figuredisk 2). From grinding, this perspective, etc. the model can be considered a static grinding model. components of the grinding forces, the self-excited chatter vibrations occur. Thus, the models capture the dynamic characteristics of the grinding machine such as the natural frequencies and the damping factors. From this point of view, these models can be classified within the dynamic grinding model shown in Figure 2. However, they are not suitable for addressing the fundamental parameters of the grinding processes because the grinding forces are determined by the material removal rates, not only by the depth of cut. In addition, the dynamic model does not directly present practical setup conditions, such as workpiece sizes, grinding speed and infeed rates. Figure 3 shows a novel grinding model that accounts for the various setup parameters of cylindrical grinding operations, including the material removal rate, workpiece sizes and grinding speed. In this model, only the staticFigureFigure compliance 1.1. GeneralizedGeneralized of the grindinggrinding model.model. machine is considered for the grinding process analysis. The dynamic process behaviors should be analyzed by using the dynamic model (FigureTo 2). investigate From this theperspective, regenerative the model self-excited can be vibrationsconsidered ina static cylindrical grinding grinding model. applications, including centerless grinding [11,14,21–23,25], several grinding models have been developed to date. Stability criteria to suppress the chatter vibrations have also been discussed. These models focus on the depth of cut per revolution of the workpiece, which is proportional to the grinding forces under given conditions. When the structure of the grinding machine is excited with the dynamic components of the grinding forces, the self-excited chatter vibrations occur. Thus, the models capture the dynamic characteristics of the grinding machine such as the natural frequencies and the damping factors. From this point of view, these models can be classiﬁed within the dynamic grinding model shown in Figure2. However, they are not suitable for addressing the fundamental parameters of the grinding processes because the grinding forces are determined by the material removal rates, not only by the depth of cut. In addition, the dynamic model does not directly present practical setup conditions, Figure 1. Generalized grinding model. such as workpiece sizes,Figure grinding 2. Dynamic speed grinding and infeed model rates.for roundness stability criterion.

Figure 2. Dynamic grinding model for roundness stability criterion. Figure 2. Dynamic grinding model for roundness stability criterion. Figure3 shows a novel grinding model that accounts for the various setup parameters of cylindrical grinding operations, including the material removal rate, workpiece sizes and grinding speed. In this model, only the static compliance of the grinding machine is considered for the grinding process analysis. The dynamic process behaviors should be analyzed by using the dynamic model (Figure2). From this perspective, the model can be considered a static grinding model. Inventions 2020, 5, x FOR PEER REVIEW 4 of 23

Inventions 2020, 5, 46 4 of 22 Inventions 2020, 5, x FOR PEER REVIEW 4 of 23

Figure 3. Static grinding model for practical grinding operations.

2.2. The Modeling of the Infeed Cylindrical Grinding System

2.2.1. The FundamentalsFigure of 3.InfeedStatic Cylindrical grinding model Grinding for practical grinding operations. Figure 3. Static grinding model for practical grinding operations. 2.2. TheFigure Modeling 4a illustrates of the Infeed the Cylindricalcenter-chuck Grinding type method System of cylindrical grinding. With this method, the 2.2.amount The Modeling of the infeed of the Infeedbecomes Cylindrical the radial Grinding reduction System of the workpiece, which is represented as the 2.2.1.method The parameter Fundamentals c = 1.0. of Infeed Cylindrical Grinding 2.2.1.Figure TheFigure Fundamentals4 a4b,c illustrates illustrates of the theInfeed center-chuck centerless Cylindrical grinding type Grinding method method of and cylindrical the shoe grinding.-centerless Withgrinding this method, therespec amountFiguretively. 4a of Withillustrates the infeed these the becomesmethods, center the -thechuck radial amount type reduction methodof the infeed ofof thecylindrical becomes workpiece, grinding. the which diameter With is represented reduction this method, of as the the methodamountworkpiece. parameterof the The infeed methodc = becomes1.0. parameter the cradial is assumed reduction to be of c =the 0.5 workpiece,in centerless which methods. is represented as the method parameter c = 1.0. Figure 4b,c illustrates the centerless grinding method and the shoe-centerless grinding method, respectively. With these methods, the amount of the infeed becomes the diameter reduction of the workpiece. The method parameter c is assumed to be c = 0.5 in centerless methods.

Figure 4. Cylindrical grinding methods (WP: workpiece, GW: grinding wheel, RW: regulating wheel). Figure 4. Cylindrical grinding methods (WP: workpiece, GW: grinding wheel, RW: regulating wheel). (a) Center-chuck type grinding c = 1.0; (b) Centerless grinding c = 0.5; (c) Shoe centerless grinding (a) Center-chuck type grinding c = 1.0; (b) Centerless grinding c = 0.5; (c) Shoe centerless grinding c = c = 0.5. 0.5. Figure4b,c illustrates the centerless grinding method and the shoe-centerless grinding method, respectively.The SMRR With (specific these methods,material removal the amount rate) ofQw the’ can infeed be represented becomes theby: diameter reduction of the Figure 4. Cylindrical grinding methods (WP: workpiece, GW: grinding wheel, RW: regulating wheel). workpiece. The method parameter c is assumed to be c = 0.5 in centerless methods. (a) Center-chuck type grinding c = 1.0; (b) Centerless = ∙ ∙grinding ∙ c = 0.5; (c) Shoe centerless grinding c = (1) The SMRR (speciﬁc material removal rate) Q ’ can be represented by: 0.5. w where c is the grinding method parameter, is the diameter of the workpiece and is the infeed rate. The ordinary unit of SMRR is /Q(w0 =·c)π. d Thatw fi means represents the removed chip(1) The SMRR (specific material removal rate) Qw·’ can· ·be represented by: volume per unit width in unit time. The MRR (material removal rate) is: where c is the grinding method parameter, dw is the diameter of the workpiece and fi is the infeed rate. 3 = =∙∙∙ ∙ (2(1) ) The ordinary unit of SMRR is mm /(mm s). That means Qw represents the removed chip volume per · 0 unitwherewhere width c b is is the inthe unitgrinding width time. of methodthe The workpiece. MRR parameter, (material The tangential removal is the rate) diametergrindingQw is: forceof the workpiece is obtained and by: is the infeed rate. The ordinary unit of SMRR is /( · ). That means represents the removed chip volume per unit width in unit time. The MRRQ w(material = b ∙Qw0 removal rate) is: (3)(2) · = ∙ 3 (2) wherewhereb uis is the the width specific of the grinding workpiece. energy The with tangential the unit grinding of (J/mm force) andF tis obtainedis the grinding by: speed. The wherenormal b isgrinding the width force of the isworkpiece. written by: The tangential grinding force is obtained by: Qw Ft ==u ∙ (4)(3) · vs = ∙ (3) where u is the speciﬁc grinding energy with the unit of (J/mm3) and v is the grinding speed. The normal 3 s where u is the specific grinding energy with the unit of (J/mm ) and is the grinding speed. The grinding force Fn is written by: normal grinding force is written by: Fn = η Ft (4) · = ∙ (4) Inventions 2020, 5, 46 5 of 22

Inventions 2020, 5, x FOR PEER REVIEW 5 of 23 where η is the force ratio deﬁned as Fn/Ft. The equivalent chip thickness heq is deﬁned as: where η is the force ratio defined as /. The equivalent chip thickness ℎ is defined as: Q w0 heqℎ= = (5)(5) vs

heqℎis oneis one of of the the critical critical parameters, parameters, such such as as surface surface roughness roughness and and residual residual stresses, stresses, that that govern govern grindinggrinding quality quality [3 ].[3]. As As shown shown in in the the above above equations, equations, the the speciﬁc specific tangential tangential grinding grinding force forceFt0 has thehas followingthe following relationship relationship with hwitheq : ℎ: Ft0 = u heq (6) = · ∙ ℎ (6) Figure5 is a graphical representation of the infeed cylindrical grinding test results showing the Figure 5 is a graphical representation of the infeed cylindrical grinding test results showing the linear relationship between heq and Ft0 under the various grinding speeds. The grinding power Pg linear relationship between ℎ under the various grinding speeds. The grinding power with the ordinary unit of (kW) can be expressed by: with the ordinary unit of (kW) can be expressed by:

Pg==uQ∙w (7) · (7)

Figure 5. Equivalent chip thickness vs. grinding force in cylindrical grinding. Figure 5. Equivalent chip thickness vs. grinding force in cylindrical grinding. 2.2.2. Machine Stiffness in the Grinding System 2.2.2. Machine Stiffness in the Grinding System Figure6 illustrates the sti ffnesses of major structures in a grinding machine. In Figure6a, δs is the deflectionFigure caused 6 illustrates by the normal the stiffnesses grinding of force majorFn actingstructures on the in grindinga grinding wheel machine. head. In ks Figure is the sti6a,ff ness is ofthe the deflection grinding wheelcaused support by the normal system representedgrinding force by Fn /actingδs. Figure on 6theb shows grinding the wheel work supporthead. ks head is the andstiffnessδw is the of deflection the grinding of the wheel work support head support system system represented caused by by Fn/.kw. denotesFigure 6b the shows stiffness the of work the worksupport head head support and system. is the In deflection Figure6c, ofkwo therepresents work head the support stiffness system of the caused workpiece by F itself.n. Adenotes thin, ring-shapedthe stiffness workpiece of the work sometimes head support has a low system. stiffness In Figure that cannot 6c, be ignored.represents Figure the 6 stiffnessd shows of the the stiworkpieceffness in the itself. contact A thin area, ring between-shaped the workpiece grinding sometimes wheel and has the a workpiece,low stiffness which that cannot is referred be ignored. to as “contactFigure sti6dff ness”shows and the is stiffness denoted in by thekcon contact. In general, area thebetween contact the sti ffgrindingness kcon wheelof the grindingand the wheelworkpiece, has hard-springwhich is referred characteristics to as “contact in which stiffness” the higher and contact is denoted force providesby . theIn gene higherral, stitheffness. contact stiffness of the grinding wheel has hard-spring characteristics in which the higher contact force provides the higher stiffness.

The grinding system stiffness , with the loop from the workpiece to the grinding wheel via the above stiffnesses shown in Figure 6a–d, can be represented by: 1 1 1 1 1 = + + + (8) Inventions 2020, 5, x FOR PEER REVIEW 6 of 23

InventionsThe2020 compliance, 5, 46 1/km of the grinding system is found by the summation of all compliances6 of 22 discussed in Figure 6.

Figure 6. Machine stiffness in a grinding system. (a) Grinding wheel support system; (b) Work head supportFigure system;6. Machine (c) Workpiece stiffness in sti aff grindingness; (d) system. Contact ( stia)ff Grindingness of grinding wheel support wheel. system; (b) Work head support system; (c) Workpiece stiffness; (d) Contact stiffness of grinding wheel. The grinding system stiffness km, with the loop from the workpiece to the grinding wheel via the above2.3. The stiff Blocknesses Diagram shown of in a Figure Cylindrical6a–d, Grinding can be represented System by:

Figure 7 is the schematic of a cylindrical1 1 grinding1 1 machine.1 The grinding wheel mounted on the = + + + infeed slide is fed into the workpiece as a command infeed. The actual infeed can be expressed(8) km ks kw kwo kcon by: The compliance 1/km of the grinding system is found by the summation of all compliances () = () − () (9) discussed in Figure6. where t is the grinding time and is the grinding machine deflection in the infeed direction given 2.3. The Block Diagram of a Cylindrical Grinding System by Equation (14). The actual infeed rate is:

Figure7 is the schematic of a cylindrical grinding machine.() The grinding wheel mounted on the () = (10) infeed slide is fed into the workpiece as a command infeed. The actual infeed rw can be expressed by: The SMRR is expressed by: rw(t) = I f (t) de(t) (9) − () = ∙ ∙ ∙ () (11) where t is the grinding time and de is the grinding machine deﬂection in the infeed direction given by The MRR is: Equation (14). The actual infeed rate fi is: () = ∙ () (12) drw(t) fi(t) = (10) The normal grinding force can be written asdt follows: ∙ The SMRR Qw0 is expressed by: () = ∙ () (13)

where η is the force ratio (Fn/Ft) andQ w0( tis) = thec π grindingdw f (t) speed. The elastic deflection of(11) the · · · i grinding machine in the infeed direction can be expressed by: The MRR Qw is: Inventions 2020, 5, 46 7 of 22 Inventions 2020, 5, x FOR PEER REVIEW 7 of 23

Qw(t) = b1Qw0(t) (12) () = · ∙ () (14) The normal grinding force Fn can be written as follows: The deflection given by Equation (14) is fed back to Equation (9). Thus, the infeed cylindrical grinding processes can be described as a closed-loopη u system repeating Equations (9)–(14). Fn(t) = · Qw(t) (13) By using Equations (11)–(13), Equation (14) canvs be· rewritten as follows: where η is the force ratio (Fn/Ft) and vsis∙ the∙ grinding ∙ ∙ ∙ speed. The elastic deﬂection de of the grinding () = ∙ () = ∙ () (15) machine in the infeed direction can be expressed ∙ by:

where T is the time constant of the grinding system1 in units of (s). de(t) = Fn(t) (14) ∙ ∙ km ·∙ ∙ ∙ = (16) ∙ The deﬂection de given by Equation (14) is fed back to Equation (9). Thus, the infeed cylindrical grindingThe time processes constant can T be is described one of the as most a closed-loop critical parameters system repeating in the cylindrical Equations g (9)–(14).rinding system.

Figure 7. Schematic of cylindrical grinding machine. Figure 7. Schematic of cylindrical grinding machine. By using Equations (11)–(13), Equation (14) can be rewritten as follows: The Laplace transformation of Equations (9)–(14) leads to the following Equations: c (π )dw b η(u) de(t) =· · =· · · f−i(t)=()T fi(t) (17(15)) vs km · · · () = ∙ () (18) where T is the time constant of the grinding system in units of (s). () = ∙ ∙ ∙ () (19) c π dw b ηu T=() ·=· ∙·· ·() (20(16)) vs km ∙ · () = ∙ () (21) The time constant T is one of the most critical parameters in the cylindrical grinding system. The Laplace transformation of Equations (9)–(14)1 leads to the following Equations: () = ∙ () (22) Rw(s) = I f (s) De(s) (17) − where s is the Laplace operator. Actual infeed (), actual infeed rate (), SMMR (), normal grinding force () and elastic deflection F(s)(=) ares Rw Laplace(s) functions. (18) i · Figure 8 shows the block diagram of the infeed cylindrical grinding system representing the Qw0(s) = c π dw Fi(s) (19) transfer functions given by Equations (17)–(22). · The· · infeed cylindrical grinding system can be expressed by a closed-loop feedback system.Qw( sThe) = transferb Qw0(s )function G(s) of the system is represented(20) · by: Inventions 2020, 5, 46 8 of 22

η u Fn(s) = · Qw(s) (21) vs · 1 De(s) = Fn(s) (22) km · where s is the Laplace operator. Actual infeed Rw(s), actual infeed rate Fi(s), SMMR Qw0(s), normal grinding force F (s) and elastic deﬂection D (s) are Laplace functions. Inventions 2020, 5, x FOR PEERn REVIEW e 8 of 23 Figure8 shows the block diagram of the infeed cylindrical grinding system representing the transfer functions given by Equations (17)–(22). The infeed cylindrical grinding system can be expressed 1 1 by a closed-loop feedback system.() = The( transfer) = function G(s) of the system= is represented by: ∙ ∙ ∙ ∙ ∙ + 1 (23) + 1 ∙ R 1 1 G(s) = w (s) = = (23) c π dw b η u + Thus, infeed cylindrical grindingI f can be represented· · · · · s + 1by theTs first1-order lag system with the time vs km constant T given by Equation (16). ·

Figure 8. Block diagram of cylindrical grinding system. Figure 8. Block diagram of cylindrical grinding system. Thus, infeed cylindrical grinding can be represented by the ﬁrst-order lag system with the time constant3. ResponsesT given to by Command Equation Infeed (16). in a Cylindrical Grinding System

3.3.1. Responses Responses to to Command Step Infeed Infeed(Spark- inOut a Grinding) Cylindrical Grinding System The spark-out in the infeed grinding cycle corresponds to the step response in the closed-loop 3.1. Responses to Step Infeed (Spark-Out Grinding) feedback system. In order to simulate the spark-out process, it is assumed that the step command

inThe the spark-out infeed slide in theposition infeed is grindinggiven to the cycle system. corresponds to the step response in the closed-loop feedback system. In order to simulate the spark-out process, it is assumed that the step command I0 in () = (24) the infeed slide position is given to the system.

The actual infeed rate () can be expressed as follows (see Appendix A): I (t) = I f 0 (24) () = (25) The actual infeed rate fi(t) can be expressed as follows (see AppendixA): The actual infeed position (), the normal grinding force () and the deflection () I0 t during spark-out grinding are obtained by:f (t) = e T (25) i T − () = (1 − ) (26) The actual infeed position rw(t), the normal grinding force Fn(t) and the deﬂection de(t) during spark-out grinding are obtained by: () = ∙ (27) t rw(t) = I0 1 e− T (26) − () = (28)

Simulations of the changing processes of (), () and () during spark-out grinding are shown in Figure 9a–c. Inventions 2020, 5, 46 9 of 22

t Fn(t) = I kme− T (27) 0· t de(t) = I0e− T (28)

Simulations of the changing processes of rw(t), Fn(t) and de(t) during spark-out grinding are shownInventions in 20 Figure20, 5, x FOR9a–c. PEER REVIEW 9 of 23

Inventions 2020, 5, x FOR PEER REVIEW 9 of 23

Figure 9. Responses to step infeed in cylindrical infeed grinding (spark-out grinding). (a) Actual infeed Figure 9. Responses to step infeed in cylindrical infeed grinding (spark-out grinding). (a) Actual I0 = 5 µm; (b) Normal grinding force Fn(0) = 99.8 N; (c) Machine deﬂection de (0) = 5 µm. infeedFigure I0 = 5 μm; 9. Responses (b) Normal to step grinding infeed force in cylindrical Fn (0) = 99.8 infeed N; grinding(c) Machine (spark deflection-out grinding). de (0) =(a 5μm.) Actual 3.2. Responsesinfeed to I0Ramp = 5 μm; Infeed (b) Normal (Plunge grinding Grinding) force Fn (0) = 99.8 N; (c) Machine deflection de (0) = 5μm. 3.2. Responses to Ramp Infeed (Plunge Grinding) Infeed3.2. Responses grinding to Ramp with Infeed a constant (Plunge rateGrinding) in the grinding cycle corresponds to the ramp response in a feedbackInfeed system.grinding To with simulate a constant the processesrate in the of grinding infeedor cycle plunge corresponds grinding to with the anramp infeed response rate, itin is a feedbackInfeed system. grinding To simulate with a constant the processes rate in the of grindinginfeed or cycle plunge corresponds grinding to withthe ramp an infeed response rate, in it is assumeda feedback that the system. command To simulate of a ramp the processes infeed as of the infeed input or isplunge given grinding to the grindingwith an infeed system. rate, The it is infeed assumed that the command of a ramp infeed as the input is given to the grinding system. The infeed commandassumedI f (t that) is shownthe command as: of a ramp infeed as the input is given to the grinding system. The infeed command () is shown as: command () is shown as: I (t) = Ip t (29) f · () = ∙ () = ∙ (29) (29) where Ip is a constant infeed rate. The actual feed rate fi(t) can be expressed as follows (see AppendixB):

wherewhere is a is constant a constant infeed infeed rate. rate. The The actual actual feedfeed rate (()) can can be be expressed expressed as follows as follows (see (see t AppendixAppendix B): B): f (t) = Ip 1 e− T . (30) i − () = 1 − . (30) The actual infeed position rw(t), the( normal) = grinding1 − force. Fn(t) and the deﬂection de(t) during(30) infeed (plunge)The actual grinding infeed can position be obtained (), the by: normal grinding force () and the deflection () The actual infeed position (), the normal grinding force () and the deflection () during infeed (plunge) grinding can be obtained by: during infeed (plunge) grinding can be obtainedh by: t i r (t) = I t T e T w p 1 − (31) () = [ −− 1 −− ] (31) () = [ − 1 − t ] (31) ( ) = T Fn( )t Ip km T 1 e− (32) (32) = ∙ · ∙·(1 − ) () = ∙ ∙(1 − t ) (32) ( ) = T de(t) = Ip∙ T1 − e− (33) (33) · − () = r (∙t) F1 −(t) d (t) (33) SimulationsSimulations of the of changing the changing processes processes of ofw (,),n (and) ande (during) during the the infeed infeed grinding grinding with a constantwith rate a constant are shown rate are in Figureshown in10 Figurea–c. 10a–c. Simulations of the changing processes of (), () and () during the infeed grinding with a constant rate are shown in Figure 10a–c.

Figure 10. a FigureResponses 10. Responses to ramp to ramp infeed infeed in cylindrical in cylindrical infeed infeed grinding grinding (plunge (plunge grinding). grinding). ( ()a) Actual Actual infeed; (b) Normalinfeed; grinding(b) Normal force; grinding (c) Machine force; (c) Machine deﬂection. deflection.

Figure3.3. The 10. Effects Responses of the Ti tome rampConstant infeed on Infeed in cylindrical Grinding Processes infeed grinding (plunge grinding). (a) Actual infeed; (b) Normal grinding force; (c) Machine deflection.

3.3. The Effects of the Time Constant on Infeed Grinding Processes Inventions 2020, 5, x FOR PEER REVIEW 10 of 23

Figure 11 shows the effects of the time constant T on the infeed grinding processes. Figure 11a is the response to the ramp infeed command. The actual infeed position delays the command and gradually catches up. Eventually it attains the same slope as the command infeed. The asymptote of the actual infeed curve intersects at the time T of the horizontal axis. Figure 11b shows the response of the normal grinding force and deflection to the ramp infeed. These outputs are gradually increased according to the response of the first-order lag system and Inventionsfinally reach2020, 5a, 46constant command input. The time constant T is defined as the time when the output10 of 22 reaches 63%. At three times the time constant (3T), the output reaches 95% of the command input. Inventions 2020, 5, x FOR PEER REVIEW 10 of 23 Figure 11c is the spark-out response to the step infeed command. The output of the force or the 3.3. The Effects of the Time Constant on Infeed Grinding Processes deflection is gradually reduced and finally converges to zero. The time constant T is the time when Figure 11 shows the effects of the time constant T on the infeed grinding processes. Figure 11a the outputFigure is11 reduced shows by the 63%. effects Again, of the 3Ttime gives constant the 95% reduction.T on the infeed grinding processes. Figure 11a is the response to the ramp infeed command. The actual infeed position delays the command and is theFigure response 12 presents to the ramp the relationship infeed command. between Thethe system actual infeedstiffness position km and the delays time the constant command T. As and gradually catches up. Eventually it attains the same slope as the command infeed. The asymptote of graduallyEquation (16) catches indicates, up. Eventually the higher itthe attains system the stiffness, same slope the shorter as the the command time constant. infeed. The The design asymptote with of the actual infeed curve intersects at the time T of the horizontal axis. thethe actual more rigid infeed grinding curve intersectsmachine provides at the time a shorterT of the time horizontal constant axis.T and reduces the spark-out time. Figure 11b shows the response of the normal grinding force and deflection to the ramp infeed. These outputs are gradually increased according to the response of the first-order lag system and finally reach a constant command input. The time constant T is defined as the time when the output reaches 63%. At three times the time constant (3T), the output reaches 95% of the command input. Figure 11c is the spark-out response to the step infeed command. The output of the force or the deflection is gradually reduced and finally converges to zero. The time constant T is the time when the output is reduced by 63%. Again, 3T gives the 95% reduction. Figure 12 presents the relationship between the system stiffness km and the time constant T. As Equation (16) indicates, the higher the system stiffness, the shorter the time constant. The design with the more rigid grinding machine provides a shorter time constant T and reduces the spark-out time. Figure 11. Effect of time constant on infeed grinding processes. (a) Slide position; (b) Forces, power Figure 11. Effect of time constant on infeed grinding processes. (a) Slide position; (b) Forces, power and machine deflection; (c) Spark-out in forces, power and deflection. and machine deflection; (c) Spark-out in forces, power and deflection. Figure 11b shows the response of the normal grinding force and deflection to the ramp infeed. These outputs are gradually increased according to the response of the first-order lag system and finally reach a constant command input. The time constant T is defined as the time when the output reaches 63%. At three times the time constant (3T), the output reaches 95% of the command input. Figure 11c is the spark-out response to the step infeed command. The output of the force or the deflection is gradually reduced and finally converges to zero. The time constant T is the time when the output is reduced by 63%. Again, 3T gives the 95% reduction. Figure 12 presents the relationship between the system stiffness km and the time constant T. As EquationFigure 11. (16) Effect indicates, of time the constant higher on the infeed system grinding stiffness, processes. the shorter (a) S thelide time position; constant. (b) Forces, The design power with the moreand machine rigid grinding deflection; machine (c) Spark provides-out in forces, a shorter power time and constant deflection.T and reduces the spark-out time.

Figure 12. System stiffness and time constant.

4. Grinding Cycle Design

4.1. Grinding Cycle Figure 13 diagrams a typical infeed grinding cycle. At the home position of the infeed slide, a workpiece is loaded and starts to rotate. First, the infeed slide travels toward the workpiece with a rapid forward speed. Then, the slide approaches the workpiece with a fast feed rate appropriate for rough grinding. During the fast feed, at point O, the grinding wheel contacts the workpiece and the rough grinding starts. At point A, the infeed rate is reduced for semi-finish grinding. The infeed rate is further reduced at point B for finish grinding. At point C, the infeed slide stops and maintains its Figure 12. System stiﬀness and time constant. Figure 12. System stiffness and time constant. 4. Grinding Cycle Design 4. Grinding Cycle Design 4.1. Grinding Cycle 4.1. Grinding Cycle Figure 13 diagrams a typical infeed grinding cycle. At the home position of the infeed slide, a workpieceFigure is 13 loaded diagrams and startsa typical to rotate.infeed First,grinding the infeedcycle. At slide the travels home towardposition the of workpiecethe infeed slide, with aa rapidworkpiece forward is loaded speed. and Then, starts the to slide rotate. approaches First, the the infeed workpiece slide travels with a toward fast feed the rate workpiece appropriate with for a roughrapid forward grinding. speed. During Then, the fastthe slide feed, approaches at point O, thethe grindingworkpiece wheel with contactsa fast feed the rate workpiece appropriate and thefor rough grinding. During the fast feed, at point O, the grinding wheel contacts the workpiece and the rough grinding starts. At point A, the infeed rate is reduced for semi-finish grinding. The infeed rate is further reduced at point B for finish grinding. At point C, the infeed slide stops and maintains its Inventions 2020, 5, 46 11 of 22

roughInventions grinding 2020, 5, x FOR starts. PEER At REVIEW point A, the infeed rate is reduced for semi-ﬁnish grinding. The infeed11 of 23 rate is further reduced at point B for ﬁnish grinding. At point C, the infeed slide stops and maintains itsposition position for for spark spark-out-out grinding. grinding. During During the the spark spark-out,-out, the the infeed infeed slide slide retr retractsacts toward toward the home home position withwith aa rapidrapid backwardbackward speedspeed atat pointpoint D.D.

Figure 13. Diagram for infeed grinding cycle.

It isis criticalcritical forfor infeedinfeed grindinggrinding operationsoperations to determinedetermine the infeed positions O to D;D; thethe infeedinfeed rates and spark-outspark-out time based on the hard stock conditions; and and the the requirements requirements for for productivity. productivity. To designdesign thethe grindinggrinding cycle,cycle, thethe followingfollowing parametersparameters shouldshould bebe consideredconsidered and bebe quantitativelyquantitatively analyzed during the infeed grindinggrinding processes.processes. i. Actual relative approaches between the grinding wheel and workpiece. i. Actual relative approaches between the grinding wheel and workpiece. ii. Actual infeed rates. ii. Actual infeed rates. iii. Grinding forces. iii. Grinding forces. iv. Elastic deflection of grinding machine. iv. Elastic deflection of grinding machine. The cycle time (CT) is defined as the total time from the home position to the home position via pointsThe O, cycleA, B, timeC and (CT) D. is defined as the total time from the home position to the home position via points O, A, B, C and D. 4.2. Analysis of Infeed Grinding Processes 4.2. Analysis of Infeed Grinding Processes The grinding cycle setup for the analysis is shown in Figure 14. , and are the infeed ratesThe for grinding the rough, cycle semi setup-finish for theand analysis finish grinding is shown operations, in Figure 14 respectively.. Ip1, Ip2 and I p The3 are infeed the infeed rate ratesfi (t) forduring the rough,the cycle semi-finish can be expressed and finish by grinding the following operations, Equations. respectively. The infeed rate fi (t) during the cycle can be expressed by the following Equations. 0 ≤ ≤ : () = 1 − e (34) t 0 t t : f (t) = I 1 e T (34) a i p1 ( )− ( ) ≤ ≤ − < ≤ : () = 1 − e + f(t)e (35) (t ta) (t ta) −T −T ta < t tb: fi(t) = Ip2 1 e− + fi(ta)e− ( ) (35) ≤ − < ≤ : () = 1 − e + ()e (36) (t t ) (t t ) − b − b T T tb < t tc: fi(t) = Ip3 1 e− + fi((tb)e)− (36) ≤ − < ≤ : () = ()e (37) (t tc) − tc < t t : f (t) = f (tc)e− T (37) ≤ d i i The actual slide position rw(t) during the cycle can be represented by:

h t i 0 t ta: rw(t) = I t T 1 e− T . (38) ≤ ≤ p1 − −

(t ta) (t ta) − h i − ta < t t :rw(t) = Ip (t ta) T 1 e− T + I ta rw(ta) 1 e− T + rw(ta). (39) ≤ b 2 − − − p1 − −

Figure 14. Setup of infeed grinding cycle. Inventions 2020, 5, x FOR PEER REVIEW 11 of 23

position for spark-out grinding. During the spark-out, the infeed slide retracts toward the home position with a rapid backward speed at point D.

Inventions 2020, 5, 46 12 of 22

Figure 13. Diagram for infeed grinding cycle. (t t ) (t t ) − b h i − b t < t tc: rw(t) = Ip (t t ) T 1 e− T + I ta + Ip (t ta) rw(t ) 1 e− T + rw(t ). (40) b ≤ 3 − b − − p1 2 b − − b − b It is critical for infeed grinding operations to determine the infeed positions O to D; the infeed (t tc) h i − rates and sparktc < t -outtd :timerw(t )based= Ip 1onta +theIp 2hard(tb stockta) + Iconditions;p3(tc tb) andrw( tthec) 1requirementse− T + forrw( tproductivity.c) (41) To design the grinding≤ cycle, the following− parameters− should− be considered− and be quantitatively analyzedThe normal during grindingthe infeed force grindingFn(t) canprocesses. be expressed by: i. Actual relative approaches between the grinding wheel and tworkpiece. 0 t ta: Fn(t) = Ip1 km T 1 e− T . (42) ii. Actual infeed rates. ≤ ≤ · · −

(t ta) (t ta) iii. Grinding forces. − h i − ta < t t : Fn(t) = Ip km T 1 e− T + I ta rw(ta) kme− T (43) iv. Elastic deflection of≤ grindingb machine.2· · − p1· − (t t ) (t t ) The cycle time (CT) is defined as the total− btime from the home position to the home− b position via tb < t tc: Fn(t) = Ip3 km T 1 e− T + [Ip1 ta + Ip2(tb ta) rw(tb)]kme− T (44) points O, A, B, C≤ and D. · · − · − − (t tc) h i − tc < t t : Fn(t) = I ta + Ip (t ta) + Ip (tc t ) rw(tc) kme− T (45) 4.2. Analysis of Infeed Grinding≤ d Processesp1· 2 b − 3 − b − The deﬂection de(t) can be expressed by: The grinding cycle setup for the analysis is shown in Figure 14. , and are the infeed rates for the rough, semi-finish and finish grinding operations, t respectively. The infeed rate fi (t) 0 t ta: de(t) = Ip1 T 1 e− T (46) during the cycle can be expressed by≤ the≤ following Equations.· −

(t ta) (t ta) − − ta < t 0t :≤de(≤t)=: I p T1()e=− T 1+−dee(ta)e − T ((47)34) ≤ b 2· − ( ) ( ) (t t ) (t t ) < ≤ : () = 1 − e − b + f (t )e − b t < t tc : de(t) = Ip T 1 e− T + de(t )e− T ((48)35) b ≤ 3· − b ( ) (t tc) − < ≤ : t c < t()t =: de(t)1=−dee(tc)e− T+ ()e ((49)36) ≤ d () The simulation results of fi(t), rw(t), Fn(t) and de(t) during the infeed grinding cycle (see Figure 14) < ≤ : () = ()e (37) are shown in Figure 15a–d.

FigureFigure 14.14. SetupSetup ofof infeedinfeed grindinggrinding cycle.cycle.

4.3. The Influence of the Time Constant on the Grinding Cycle Figure 16 shows the influence of the time constant on the behaviors of the process parameters during infeed cylindrical grinding, assuming the typical cycle shown in Figure 14. The time delay becomes greater with the increased time constant T and the response to the command infeed becomes slower with lower system stiffness km, as shown in Figure 16a. When the time constant T = 0.5 s as in Figure 16b, the deflection reaches a steady state at each operation from rough to finish grinding, and the final deflection at the end of the spark-out grinding becomes almost zero. This indicates that the grinding cycle is properly designed. On the other hand, when the time constant T = 1.5 s or 2.6 s, the deflection does not reach a steady state at each grinding operation and is still in a transient stage. The final deflections at the end of the spark-out grinding do not converge. It means that the size error of the workpiece is greater than that of T = 0.5 s, due to the remaining deflection. Additionally, it indicates that the stiffer machine provides the reduced time constant T and the smaller size error. Inventions 2020, 5, 46 13 of 22 Inventions 2020, 5, x FOR PEER REVIEW 13 of 23

Figure 15. Process parameters during the infeed grinding cycle. (a) Infeed rate; (b) Command and Figure 15. Process parameters during the infeed grinding cycle. (a) Infeed rate; (b) Command and Inventionsactual 2020 slide, 5, x position; FOR PEER (c REVIEW) Normal grinding force; (d) Deﬂection of grinding machine. 14 of 23 actual slide position; (c) Normal grinding force; (d) Deflection of grinding machine.

4.3. The Influence of the Time Constant on the Grinding Cycle Figure 16 shows the influence of the time constant on the behaviors of the process parameters during infeed cylindrical grinding, assuming the typical cycle shown in Figure 14. The time delay becomes greater with the increased time constant T and the response to the command infeed becomes slower with lower system stiffness km, as shown in Figure 16a. When the time constant T = 0.5 s as in Figure 16b, the deflection reaches a steady state at each operation from rough to finish grinding, and the final deflection at the end of the spark-out grinding becomes almost zero. This indicates that the grinding cycle is properly designed. On the other hand, when the time constant T = 1.5 s or 2.6 s, the deflection does not reach a steady state at each grinding operation and is still in a transient stage. The final deflections at the end of the spark-out grinding do not converge. It means that the size error of the workpiece is greater than that of T = 0.5 s, due to the remaining deflection. Additionally, it indicates that the stiffer machine provides the reduced time constaFigurent T 16.andInfluences the smaller of thesize time error. constant on the behaviors of process parameters. (a) Actual infeed Figure 16. Influences of the time constant on the behaviors of process parameters. (a) Actual infeed position; (b) Deflection of grinding machine. position; (b) Deflection of grinding machine. 5. Grinding Accuracy and Stock Assignments 5. Grinding Accuracy and Stock Assignments 5.1. Size Error 5.1. Size Error In the primary grinding cycle shown in Figure 17, the deflection de at the grinding time tp can be obtainedIn the by: primary grinding cycle shown in Figure 17, the deflection at the grinding time can be obtained by: Tp de tp = Ip T 1 e− T . (50) · − = ∙ 1 − . (50) where Ip is the infeed rate and Tp is the infeed grinding time. When Tp is sufficiently long enough compared to the time constant T, the deflection de can be approximately represented by: where is the infeed rate and is the infeed grinding time. When is sufficiently long enough

compared to the time constant T, the deflection can be approximately represented by:

() ≅ ∙ . (51)

The elastic deflection of the grinding system at the final grinding time becomes the size error of the ground workpiece. Therefore, the size error in diameter ∆ can be found by:

∆() = 2 ∙ ∙ ∙ (1 − ) ≅ 2 ∙ ∙ ∙ (52)

where is the spark-out time. When the size error in diameter is required to be within ∆, the spark-out time for achieving the objective can be obtained by:

Figure 17. Primary grinding cycle.

∆ = − ∙ (53) 2 ∙ ∙ ∙ Inventions 2020, 5, x FOR PEER REVIEW 14 of 23

Figure 16. Influences of the time constant on the behaviors of process parameters. (a) Actual infeed position; (b) Deflection of grinding machine.

5. Grinding Accuracy and Stock Assignments

5.1. Size Error

In the primary grinding cycle shown in Figure 17, the deflection at the grinding time can be obtained by:

= ∙ 1 − . (50)

where is the infeed rate and is the infeed grinding time. When is sufficiently long enough

compared to the time constant T, the deflection can be approximately represented by:

() ≅ ∙ . (51)

The elastic deflection of the grinding system at the final grinding time becomes the size error of the ground workpiece. Therefore, the size error in diameter ∆ can be found by:

Inventions 2020, 5, 46 14 of 22 ∆() = 2 ∙ ∙ ∙ (1 − ) ≅ 2 ∙ ∙ ∙ (52)

where is the spark-out time. When the size error in diameter is required to be within ∆, the de tp T Ip. (51) spark-out time for achieving the objective can be ·obtained by:

Figure 17. Primary grinding cycle. Figure 17. Primary grinding cycle.

The elastic deﬂection de of the grinding system at the ﬁnal grinding time ts becomes the size error of the ground workpiece. Therefore, the size error in diameter∆ ∆Sd can be found by: = − ∙ (53) 2 ∙ ∙ ∙ Tp Tsp Tsp ∆S (ts) = 2 c Ip T 1 e− T e− T 2 c Ip Te− T (52) d · · · − · · · where Tsp is the spark-out time. When the size error in diameter is required to be within ∆Sd, theInventions spark-out 2020,time 5, x FORTsp PEERfor achievingREVIEW the objective can be obtained by: 15 of 23

5.2. Roundness ∆Sd Tsp = T ln (53) − · 2 c Ip T In infeed grinding, the depth of cut Δ per revolution· · of· the workpiece is written by:

5.2. Roundness ∙ () ∙ ∆() = = (1 − ) (54) In infeed grinding, the depth of cut ∆ per revolution of the workpiece is written by: where is the rotational speed of the workpiece in rps. The depth of cut Δ is observed as a step on the roundness of the workpiece during infeedc fi (grinding,t) c Ip as shownt in Figure 18. The step becomes the ∆(t) = · = · 1 e− T (54) roundness error. The step-roundness Δ can benw reducednw by spark− -out grinding.

∙ ∙ where nw is the rotational speed∆( of) = the workpiece1 − in rps. The≅ depth of cut ∆ is observed as a( step55) on the roundness of the workpiece during infeed grinding, as shown in Figure 18. The step becomes the roundnessIn order error. to Thesuppress step-roundness the step-roundness∆ can be to reduced less than by ∆ spark-out, the required grinding. spark-out time can be calculated by: c I Tp Tsp c I Tsp p ∙ ∆ p ∆(ts) = · 1 e− T e− T · e− T (55) nw =− − ∙ nw (56) ∙

Figure 18. Roundness error caused by the residual depth of cut. (a) Work rotation during infeed Figure 18. Roundness error caused by the residual depth of cut. (a) Work rotation during infeed grinding; (b) Step-roundness. grinding; (b) Step-roundness.

5.3. Grinding Stock Assignments and SMRR Figure 19 shows a grinding cycle diagram with SMRRs and grinding stocks for rough, semi- finish and finish grinding. The total stocks in diameter S consist of S1 for rough, S2 for semi-finish and S3 for finish grinding.

= + + (57) The SMRRs are assigned as for rough, for semi-finish and for finish grinding. The total grinding time is:

= + + + (58) The infeed rate can be written by /( ∙ ∙ ). So, the grinding time is rewritten by:

∙ ∙ = + + + (59) 2

By setting the spark-out time to three times the time constant T, the size error can be reduced

by 95% of the deflection generated during finish grinding. When the total stock S is assigned as %, %, and % for rough, semi-finish and finish grinding, respectively, the grinding time is presented by the following Equation.

∙ ∙ ∙ = + + + 3 ∙ (60) 2 ∙ 100 Inventions 2020, 5, 46 15 of 22

In order to suppress the step-roundness to less than ∆, the required spark-out time Tsp can be calculated by: nw ∆ Tsp = T ln · (56) − · c Ip · 5.3. Grinding Stock Assignments and SMRR Figure 19 shows a grinding cycle diagram with SMRRs and grinding stocks for rough, semi-finish and finish grinding. The total stocks in diameter S consist of S1 for rough, S2 for semi-finish and S3 for finish grinding. Inventions 2020, 5, x FOR PEER REVIEW S = S1 + S2 + S3 16 (57)of 23

Figure 19. SMRR (Speciﬁc(Specific Material Removal Rate) and stock assignments.

6. ExperimThe SMRRsental areTests assigned and Simulations as Qw10 for rough, Qw20 for semi-ﬁnish and Qw30 for ﬁnish grinding. The total grinding time T is: For the validationg of the proposed model, cylindrical grinding tests on workpieces of through- T = T + T + T + T (58) hardened steel materials (HRC58) wereg conducted1 2 using3 thesp chuck-center type cylindrical grinding methodThe ( infeedc = 1.0). rate TheI detailedcan be writtengrinding by conditionsQ /(c π aredw) shown. So, the in grinding Table 1. time Tg is rewritten by: pi wi0 · · ! Table 1. Experimentalc π dw Ssetup in infeedS cylindricalS grinding. T = · · 1 + 2 + 3 + T (59) g 2 Q Q Q sp Items w10 w20 w30 Conditions

By setting theGrinding spark-out machine time Tsp to three times the timeUniversal constant cylindricalT, the size grinder error can be reduced by 95% of the deflectionGrinding generated method during finish grinding.Chuck type When cylindrical the total grinding stock S c is= 1.0 assigned as PS %, PS %, and PS % for rough, semi-finish and finish grinding,Thru-hardened respectively, steel theHRC58 grinding time T is 1 2 3 Workpiece g presented by the following Equation. Diameter dw = 177.8 mm, width b = 30 mm Al! 2O3 70 K m V Grinding wheelc π dw S PS1 DiameterPS3 dPSs =3 127 mm, width Ls = 86.4 mm Tg = · · · + + + 3 T (60) 2 100 Q Qw Qw · · w10 20 Grinding30 speed vs = 45 m/s Stocks in diameter S = 0.33 mm 6. Experimental Tests and Simulations SMRR (Specific Material Removal Rate) Qw’ = 2.0 mm³/(mm·s) For the validationInfeed of therate proposed model, cylindricalfi = grinding0.216 mm tests/min on workpieces of through-hardened steelSpark materials-out time (HRC58) were conducted usingT thesp = chuck-center 5.3 s type cylindrical grinding method (c =Grinding1.0). The time detailed grinding conditions are shownTg = 51.1 in Table s 1. Specific energy u = 26.7 J/mm³ Force ratio (Fn/Ft) η = 2.0 System stiffness km = 2421 N/mm Time constant T = 8.2 s

The grinding power was measured during the infeed grinding and the results are shown in Figure 20a. The grinding power in the steady state was 1.6 kW, and a time constant T = 8.2 s was found at the time the power reached 63% of 1.6 kW. The specific energy u can be calculated by Equation (7), obtaining u = 26.7 J/mm3. Additionally, the system stiffness km can be found by Equation (16), obtaining km = 2421 N/mm. Inventions 2020, 5, 46 16 of 22

Table 1. Experimental setup in infeed cylindrical grinding.

Items Conditions Grinding machine Universal cylindrical grinder Grinding method Chuck type cylindrical grinding c = 1.0 Thru-hardened steel HRC58 Workpiece Diameter dw= 177.8 mm, width b = 30 mm

Al2O3 70 K m V Grinding wheel Diameter ds = 127 mm, width Ls = 86.4 mm Grinding speed vs = 45 m/s Stocks in diameter S = 0.33 mm SMRR (Speciﬁc Material Removal Rate) Qw’ = 2.0 mm3/(mm s) · Infeed rate fi = 0.216 mm/min

Spark-out time Tsp = 5.3 s

Grinding time Tg = 51.1 s Speciﬁc energy u = 26.7 J/mm3 Force ratio (Fn/Ft) η = 2.0

System stiﬀness km = 2421 N/mm Time constant T = 8.2 s

The grinding power was measured during the infeed grinding and the results are shown in Figure 20a. The grinding power in the steady state was 1.6 kW, and a time constant T = 8.2 s was found at the time the power reached 63% of 1.6 kW. The speciﬁc energy u can be calculated by Equation (7), obtaining u = 26.7 J/mm3. Additionally, the system stiﬀness km can be found by Equation (16), obtainingInventionskm = 2421 2020, 5 N, x FOR/mm. PEER REVIEW 17 of 23

Figure 20. Test results and simulation for infeed cylindrical grinding. (a) Power of infeed cylindrical grinding (Experimental);Figure 20. Test results (b )and Simulation simulation offor infeedinfeed cylindrical cylindrical grinding. grinding (a) Power (Theoretical). of infeed cylindrical grinding (Experimental); (b) Simulation of infeed cylindrical grinding (Theoretical).

The simulation results calculated by using the grinding conditions (Table 1) and parameters obtained from the grinding tests (Figure 20a) are shown in Figure 20b. In the simulation, the build- up behavior of the power curve corresponding to the grinding forces was almost identical to that produced by the infeed grinding tests, and the curve of the simulated reduction behavior is also almost identical to the curve derived from the spark-out grinding test results, indicating that the proposed model can accurately predict process parameters such as actual infeed rates, forces and machine deflection during the infeed grinding cycle. Table 2 delineates the experimental conditions and parameters for the infeed centerless grinding tests (c = 0.5) using workpieces made of soda-lime glass. The regulating wheel spindle of applied centerless grinding machine was supported by hydrostatic bearings. Both the normal and tangential grinding forces were measured by monitoring the pocket pressures of the bearings [26]. A grinding wheel with SiC grains lapped with a carbide chip was applied for the infeed centerless grinding of the glass cylinders. The ground cylinders were visually transparent with no macro-cracks on the ground surface [27]. The grinding time Tg was set to 57 s (including a spark-out time of 5 s).

Table 2. Experimental setup in infeed centerless grinding.

The simulation results calculated by using the grinding conditions (Table1) and parameters obtained from the grinding tests (Figure 20a) are shown in Figure 20b. In the simulation, the build-up behavior of the power curve corresponding to the grinding forces was almost identical to that produced by the infeed grinding tests, and the curve of the simulated reduction behavior is also almost identical to the curve derived from the spark-out grinding test results, indicating that the proposed model can accurately predict process parameters such as actual infeed rates, forces and machine deﬂection during the infeed grinding cycle. Table2 delineates the experimental conditions and parameters for the infeed centerless grinding tests (c = 0.5) using workpieces made of soda-lime glass. The regulating wheel spindle of applied centerless grinding machine was supported by hydrostatic bearings. Both the normal and tangential grinding forces were measured by monitoring the pocket pressures of the bearings [26]. A grinding wheel with SiC grains lapped with a carbide chip was applied for the infeed centerless grinding of the glass cylinders. The ground cylinders were visually transparent with no macro-cracks on the ground surface [27]. The grinding time Tg was set to 57 s (including a spark-out time of 5 s).

Table 2. Experimental setup in infeed centerless grinding.

Items Conditions Grinding machine Centerless grinder Grinding method Infeed centerless grinding c = 0.5 Workpiece Soda-lime glass, HV500 Diameter dw = 12.4 mm, width b = 66 mm Grinding wheel SiC, GC 100 L m V Diameter ds = 455 mm, width Ls = 150 mm Grinding speed vs = 29 m/s Regulating wheel A 150 R R Diameter dr = 255 mm, width Ls = 150 mm Rotational speed Nr = 24.4 rpm Stocks in diameter S = 0.1 mm SMRR (Speciﬁc Material Removal Rate) Qw’ = 0.0325 mm3/(mm s) · Infeed rate fi = 0.10 mm/min

Spark-out time Tsp = 5.0 s

Grinding time Tg = 57 s Speciﬁc energy u = 741 J/mm3 Force ratio (Fn/Ft) η = 4.0

System stiﬀness km = 5900 N/mm Time constant T = 4.6 s

Figure 21a is a graph of the normal force Fn and tangential force Ft measurement results found during infeed centerless grinding. From the results, the time constant T = 4.6 s was obtained, and an extremely high speciﬁc energy of u = 741 J/mm3 was measured due to the ductile grinding of the brittle materials. It is noted that the ﬂuctuations observed on Fn and Ft curves were due to the runout of regulating wheel rotation and truing errors. Figure 21b graphs the infeed centerless grinding simulation results for the same glass cylinder workpieces. The raising transients of the forces were well simulated and the levels of the forces were almost identical. The spark-out behavior was also well simulated. Inventions 2020, 5, 46 18 of 22 Inventions 2020, 5, x FOR PEER REVIEW 19 of 23

Figure 21. Grinding test results and simulation for infeed centerless grinding. (a) Test results of infeed centerlessFigure 21. grinding Grinding of test glass; results (b) Simulationand simulation results for infeed of infeed centerless centerless grinding. grinding. (a) Test results of infeed centerless grinding of glass; (b) Simulation results of infeed centerless grinding. The results for both chuck type cylindrical grinding and centerless grinding indicate that the presentedThe modelresults canfor both provide chuck accurate type cylindrical predictions grinding of the and transient centerless behaviors grinding and indicate the steady-state that the valuespresented of infeed model grinding can provide processes. accurate predictions of the transient behaviors and the steady-state values of infeed grinding processes. 7. Conclusions 7. Conclusions This paper treated infeed cylindrical grinding as a system composed of the grinding mechanism and theThis grinding paper treated machine infeed characteristics. cylindrical grinding The causalities as a system between composed the of the grinding grinding parameters mechanism and theand machine the grinding structures machine were characteristics. discussed, and The the causalities infeed grinding between processes the grinding were parameters analyzed and as output the responsesmachine to structures the operational were discussed, input from and the grindingthe infeed machine. grinding These processes relationships were analyzed were integrated as output into a blockresponses diagram to the with operational closed-loop input feedback. from the Agrinding novelgrinding machine.model These exhibitingrelationships practical were integrated parameters suchinto as a work block sizes, diagram grinding with speed, closed infeed-loop feedback. rate and MRR A novel was presented.grinding model exhibiting practical parameters such as work sizes, grinding speed, infeed rate and MRR was presented. The grinding system analysis derived a factor referred to as “grinding time constant” that governs The grinding system analysis derived a factor referred to as “grinding time constant” that the transient behaviors of process parameters such as forces and machine deﬂection. The process governs the transient behaviors of process parameters such as forces and machine deflection. The parameters used during the infeed cycle (including spark-out grinding) were investigated and the process parameters used during the infeed cycle (including spark-out grinding) were investigated formulas required for the cycle design were presented. and the formulas required for the cycle design were presented. Furthermore, to improve accuracy and productivity, the features of the cycle design were described and the procedures for controlling size error and roundness were developed. Finally, the model was Inventions 2020, 5, 46 19 of 22 veriﬁed by infeed grinding tests applied to both the chuck type cylindrical grinding and centerless grinding methods. The study produced the following conclusions and results:

(1) In infeed cylindrical grinding, including the centerless methods, the causalities between the grinding fundamentals and the machine characteristics can be clarified, and from that a model of a new system represented by a block diagram with closed-loop feedback can be proposed. (2) From the characteristic equations of the proposed grinding system, a factor called the “grinding time constant” was revealed. This time constant was found to play a critical role in the infeed process and in spark-out grinding. (3) Formulas presenting process parameters such as grinding forces and machine deflection were derived, and the procedures for the grinding cycle design were created. (4) Practical exercises for improving size error, roundness and cycle time in infeed cylindrical grinding were developed and described. (5) The model was verified by performing grinding tests on both the cylindrical and centerless grinding methods.

Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest. F.H. is a founder of Advanced Finishing Company Ltd., in which he holds shares. The sponsors had no role in the design, collection, analysis, or interpretation of data, the writing of this article, or the decision to submit it for publication.

Nomenclature s Laplace operator T Grinding time constant De(s) Deflection of machine in s-domain Tg Grinding time F(s) Rounding error in s-domain Tp Infeed grinding time Fi(s) Actual infeed rate in s-domain Tsp Spark-out time Fn(s) Normal grinding force in s-domain UV Energy-grinding speed function Gm(s) Dynamic compliance in s-domain WP Workpiece I f (s) Command infeed in s-domain b Width of workpiece Qw0(s) SMRR in s-domain c Grinding method parameter Rw(s) Actual infeed in s-domain de Deflection of grinding machine Tc(s) Depth of cut in s-domain dw Diameter of workpiece C Constant fi Actual infeed rate Fn Normal grinding force heq Equivalent chip thickness Ft Tangential grinding force kcon Contact stiffness of grinding wheel Fn’ Specific normal grinding force km Stiffness of grinding system Ft Specific tangential grinding force ks Stiffness of wheel support system 0 GW Grinding wheel kw Stiffness of work support system I f Command infeed kw0 Stiffness of workpiece itself I0 Step infeed in slide position nw Rotational speed of workpiece Ip Constant infeed rate rw Actual infeed Km Static stiffness t time Kw Grinding stiffness u Specific energy MRR Material removal rate vs Grinding speed PS Percentages of stock assignment δcon Contact deflection of grinding wheel Pg Grinding power δs Deflection of wheel support system Qw Material removal rate δw0 Deflection of work itself Qw0 Specific material removal rate δw Deflection of work support system RW Regulating wheel ∆ Depth of cut per revolution S Grinding stocks in diameter η Force ratio (Fn/Ft) SMRR Specific material removal rate Inventions 2020, 5, 46 20 of 22

Appendix A

The actual infeed rw is (see Equation (9)):

rw(t) = I (t) de(t) (A1) f − where I f (t) = I0. The machine deﬂection de is expressed by (see Equation (15)):

de(t) = T fi(t) (A2)

Therefore, rw(t) = I T f (t) (A3) 0 − i After diﬀerentiating both sides of Equation (A3):

drw(t) dI d fi(t) = 0 T (A4) dt dt − dt

drw(t)/dt = fi(t), dI0/dt = 0. Therefore,

d fi(t) f (t) = T (A5) i − dt

Take the separation of variables in Equation (A5); then take the integral of both sides. Z Z d fi(t) 1 = dt (A6) fi(t) − T

Therefore, t ln f (t) = + C (A7) i − T where C is a constant. Equation (A7) can be rewritten as:

t fi(t) = Ce− T (A8)

From Equation (A2), at t = 0, fi(0) = de(0)/T = I0/T. Therefore,

I0 t f (t) = e T (A9) i T −

Appendix B

The ramp-infeed I f (t) is (see Equation (29)):

I f (t) = Ipt (A10)

The deﬂection de of the grinding machine is (see Equation (15)):

de(t) = T fi(t) (A11)

Therefore, rw(t) = Ipt T f (t) (A12) − i After diﬀerentiating both sides of Equation (A12):

drw(t) d fi(t) = Ip T (A13) dt − dt

Then rewrite Equation (A13): d fi(t) 1 Ip + f (t) = (A14) dt T i T The solution of the ﬁrst-order linear diﬀerential Equation can be expressed by: Inventions 2020, 5, 46 21 of 22

R Z R 1 dt Ip 1 dt f (t) = e T ( e T dt + C) (A15) i − T

Therefore, t fi(t) = Ip + Ce− T (A16) given that f (0) = Ip + C = 0 at t = 0, C = Ip. Therefore, i −

t f (t) = Ip 1 e− T (A17) i −

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