The Effect of Grinding Wheel Contact Stiffness on Plunge Grinding Cycle

inventions

Article The Eﬀect of Grinding Wheel Contact Stiﬀness on Plunge Grinding Cycle

Fukuo Hashimoto 1,* and Hiroto Iwashita 2 1 Advanced Finishing Technology Ltd., Akron, OH 44319, USA 2 Toba Engineering, Saku, Nagano 384-2101, Japan; [email protected] * Correspondence: [email protected]

Received: 16 November 2020; Accepted: 16 December 2020; Published: 16 December 2020

Abstract: This paper presents the effect of grinding wheel contact stiffness on the plunge grinding cycle. First, it proposes a novel model of the generalized plunge grinding system. The model is applicable to all plunge grinding operations including cylindrical, centerless, shoe-centerless, internal, and shoe-internal grinding. The analysis of the model explicitly describes transient behaviors during the ramp infeed and the spark-out in the plunge grinding cycle. Clarification is provided regarding the premise that the system stiffness is composed of machine stiffness and wheel contact stiffness, and these stiffnesses significantly affect productivity and grinding accuracy. The elastic deflection of the grinding wheel is accurately measured and formulas for representing the deflection nature under various contact loads are derived. The deflection model allows us to find the non-linear contact stiffness with respect to the normal load. The contact stiffnesses of four kinds of grinding wheels with different grades and bond materials are presented. Both cylindrical grinding and centerless grinding tests are carried out, and it is experimentally revealed that the time constant at ramp infeed and spark-out is significantly prolonged by reducing the grinding force. It is verified that a simulation of the grinding tests using the proposed model can accurately predict critical parameters like forces and machine deflection during plunge grinding operations. Finally, this paper provides a guideline for grinding cycle design in order to achieve the required productivity and grinding accuracy.

Keywords: grinding; grinding wheel; contact stiﬀness; plunge grinding; grinding machine

1. Introduction In the plunge grinding of cylindrical components, the grinding performance is assessed by productivity and grinding accuracy. The productivity in the number of finished parts per unit time is determined by the grinding cycle time consisting of the plunge-infeed time and the spark-out time. As the actual infeed to the CNC command has a time delay, the required cycle time is largely affected by the system stiffness. On the other hand, the grinding accuracy is significantly governed by the system stiffness. In fact, the actual depth of cut always lower than the command value due to the elastic deflections of the grinding system. These deflections become the root cause of geometrical errors such as size error, profile error, roundness error, etc., in the ground products. The system stiffness is composed of the grinding machine stiffness and the contact stiffness of grinding wheel. The machine stiffness has a linear characteristic under general grinding conditions, and it is relatively easy to integrate it to the grinding system. However, it is well known that the contact deflection in wheel-workpiece interactions shows a non-linear characteristic. As the result, the process parameters like grinding forces in the grinding cycle exhibit complicated behaviors and the phenomena make it difficult to design a proper grinding cycle. Thus, it is crucial to take the influence of contact stiffness on the grinding processes into account in order to achieve the required productivity and accuracy.

Inventions 2020, 5, 62; doi:10.3390/inventions5040062 www.mdpi.com/journal/inventions Inventions 2020, 5, 62 2 of 27

There are many publications related to the wheel contact stiffness. However, no published paper has described the effect of wheel contact stiffness on the plunge grinding cycle. Much research has been done in the area of grinding wheel deflection. Hahn [1] first suggested that the deflection of the wheel surface could explain the differences in material removal rates arising from the conformity difference between internal and external grinding. Peklenik [2] proposed the use of the thermocouple method to measure grinding temperature and contact length. He showed a difference in the effective number of cutting points with the wheel grade which attributed to differences in wheel elasticity. Snoeys and Wang [3] carried out static loading tests to verify their model of grain mounted on springs and concluded that the model provided more realistic values for the deflection of the wheel. Brown, Saito, and Shaw [4] analyzed the influence of elastic deflection on contact length and established a contact length model using Hertzian theory. Kumar and Shaw [5] analyzed the deflection of the grinding wheel and workpiece separately and concluded that the predominant deflection was due to the grinding wheel, not the workpiece. Krug and Honcia [6] estimated the amount of local wheel deflection and found that it would be on the order of 1 µm for a vitrified wheel and 2.5 µm for a resinoid wheel. Nakayama, Brecker, and Shaw [7] conducted experiments to measure the deflection associated with an individual grain and showed that the local elastic deformation of the grain was the same order of magnitude as the undeformed chip thickness. Nakayama [8] observed deflection of the grains that behaved as if the grains on the wheel were supported by non-linear springs. Zhou and Lutterwelt [9] presented a measuring method to identify the maximum contact length and local length at the wheel-work contact zone. Rowe et al. [10] proposed a model based on the theory of cylinders in contact, including the effect of grains, and explained why measured contact length can be 50% to 200% greater than geometric contact length. Hucker, Farris, and Chandrasekar [11] conducted grinding tests to measure machine and wheel deflection under various grinding loads and concluded that the contact stiffness of CBN wheels is about three times greater than that of Al2O3 wheels. Yamada et al. [12,13] proposed a model of a wheel consisting of rigid body and spring elements and measured the contact stiffness of the wheel under both stationary conditions and grinding operations. Papanikolaou and Salonitis [14] applied a three-dimensional molecular dynamics simulation to investigate the effect of contact stiffness on grinding processes under various grinding speeds. Self-excited regenerative chatter vibrations are another aspect of the contact stiffness of the wheel that havet been extensively investigated. Chatter vibrations are generally associated with the dynamic compliances of the grinding machine, and a great deal of research has been devoted to understanding these instabilities [15–19]. Snoeys and Brown [20] proposed a model representing work-regenerative and wheel-regenerative chatter vibrations in cylindrical grinding. The model was represented by a block diagram with feedback loops of machine compliances, including the contact stiffness of the wheel. Inasaki [21] experimentally investigated wheel contact stiffness with hard-spring characteristics and pointed out that it changed with grinding conditions such as speed ratio, depth of cut, etc. He discussed the effects of contact stiffness on the stability of the chatter vibrations. Miyashita, Hashimoto, and Kanai [22] analyzed the dynamic stability of a centerless grinding system and proposed a new setup index for suppressing work-regenerative chatter vibrations. Their model indicated that the contact stiffnesses of both the grinding wheel and regulating wheel had significant influence on chatter vibrations. Hashimoto, Kanai, and Miyashita [23] investigated the growing mechanism of chatter vibrations in cylindrical grinding and proposed a stabilization index related to the wheel contact stiffness for selecting the grinding wheel specifications. A parameter related to the dynamic time constant was introduced and discussed the in-process measurement of the time constant to predict the stability of the chatter vibrations. In the field of grinding cycle investigations, King and Hahn [24] analyzed a cylindrical grinding system and introduced a concept of time constant which was applicable to the cylindrical grinding, but not to centerless grinding. Malkin and Koren [25] investigated the plunge grinding cycle and proposed Inventions 2020, 5, 62 3 of 27 accelerated spark-out to reduce the cycle time. Chiu and Malkin [26] presented a simulation of the grinding parameters specific to infeed cylindrical grinding. The aforementioned research concentrated primarily on the modeling and measurement of real contact length and the deflection characteristics at the local contact zone of the wheel. This literature revealed the nature of wheel elastic deflection and provided valuable evidence about contact stiffness with hard-spring characteristics. However, the extensive knowledge gleaned about wheel deflection has been little used in practical applications. Current practices for determining the operational grinding conditions of the grinding cycle still largely rely on a cut-and-try method. One reason for this may be the disconnection between the scientific parameters used in past research and the practical parameters required for real-world grinding operations. It is necessary to establish practical procedures for setting up the proper grinding conditions including grinding stocks, infeed rates, spark-out time, and the selection of the grinding wheel. Furthermore, clear guidelines for designing the grinding cycle that take the effect of contact stiffness on grinding performance into account are indispensable. This paper presents the effect of grinding wheel contact stiffness on the plunge grinding cycle. It also provides guidelines for the design of the grinding cycle. First, the generalized plunge grinding system including cylindrical, centerless, shoe-centerless, internal, and shoe-internal grinding is explicitly described and analyzed in terms of system outputs corresponding to the inputs of the plunge grinding conditions. The analysis describes the influence of system stiffness on the time constant that governs the transient behaviors of the process parameters such as forces, power, and machine elastic deflection. Additionally, it clarifies that system stiffness is composed of machine stiffness and wheel contact stiffness. These stiffnesses significantly affect the grinding performance indicators such as productivity and grinding accuracy. The elastic deflection of the grinding wheel is accurately measured and a function for representing the deflection nature under various contact loads is derived. The function allows us to find the non-linear contact stiffness with respect to the normal load. The contact stiffnesses of four kinds of grinding wheels with different grades and bond materials are presented, and the changes in contact stiffness after dressing are discussed. Experimental tests for both cylindrical grinding and centerless grinding are conducted, and it is experimentally confirmed that the time constant at ramp infeed and spark-out is significantly prolonged by reducing the grinding force. It is verified that a simulation of the grinding tests using the proposed model can accurately predict critical parameters like forces, machine deflection, and system stiffness during plunge grinding operations. Finally, this paper emphasizes that it is vital to take the effect of wheel contact stiffness into consideration during the design of the plunge grinding cycle.

2. Plunge Grinding System and the Grinding Cycle

2.1. Plunge Grinding System In order to investigate the effect of the grinding wheel’s elastic contact deflection on the plunge grinding process, it is necessary to clarify the relationship between the operational setup parameters and the machine characteristics (including the abrasive tools). There are several methods of plunge internal and external grinding, as shown in Figure1. The grinding method parameter c is used to identify the influence of the amount of infeed on the size reduction of the workpieces. In the centerless grinding and shoe centerless grinding methods (Figure1d,e), the amount of infeed becomes the diameter reduction of the workpieces and c = 0.5 is denoted. On the other hand, in Figure1a–c,f, the amount of infeed gives the radial reduction on the external or internal ground surface of the workpieces, and c = 1.0 is assigned for these methods. Inventions 2020, 5, 62 4 of 27 Inventions 2020, 5, 62 4 of 29

Figure 1. Plunge grinding operations and the method parameter c.(a) Center type cylindrical grinding; Figure 1. Plunge grinding operations and the method parameter c. (a) Center type cylindrical (b) Chuck type cylindrical grinding; (c) Internal grinding; (d) Centerless grinding; (e) Shoe centerless grinding; (b) Chuck type cylindrical grinding; (c) Internal grinding; (d) Centerless grinding; (e) Shoe grinding; (f) Shoe internal grinding. centerless grinding; (f) Shoe internal grinding. In plunge grinding, as shown in Figure1, the grinding wheel mounted on the infeed slide is fed In plunge grinding, as shown in Figure 1, the grinding wheel mounted on the infeed slide is fed into the workpiece as a command infeed If. The actual infeed rw can be expressed by: into the workpiece as a command infeed If. The actual infeed 𝑟 can be expressed by:

r𝑟w((𝑡)t) = I =f 𝐼 ((𝑡)t) d −e( 𝑑 t(𝑡)) (1) − (1) wherewheret tis is the the grinding grinding time time and andde is 𝑑 the is elastic the elastic deﬂection deflection of the of grinding the grinding system system including including the contact the 𝑓 deﬂectioncontact deflection of grinding of grinding wheel. The wheel. actual The infeed actual rate infeedfi is: rate is: () 𝑓(𝑡) = . (2) drw(t) fi(t) = . (2) dt The SMRR (specific material removal rate) 𝑄 is expressed by:

The SMRR (specific material removal rate) Qw is expressed by: 𝑄 (𝑡) = 𝑐0 ∙ 𝜋 ∙𝑑 ∙ 𝑓(𝑡) (3) where 𝑑 is the internal or external diameterQw0(t) of= thceπ workpiece.dw fi(t) The MRR (material removal rate) (3)𝑄 is: · · · where dw is the internal or external diameter of the workpiece. The MRR (material removal rate) Qw is: 𝑄(𝑡) =𝑏∙𝑄 (𝑡) (4) where b is the grinding width. The normalQ wgrinding(t) = b Q forcew0(t) 𝐹 can be written as follows: (4) · 𝜂∙𝑢 where b is the grinding width. The normal𝐹 grinding(𝑡) = force∙𝑄F(𝑡)n can be written as follows: (5) 𝑣 η u where η is the force ratio of Fn to tangential force Ft, 𝑣 is the grinding speed, and u is the specific Fn(t) = · Qw(t) (5) vs · energy. The elastic deflection 𝑑 of the grinding system in the infeed direction can be expressed by: where η is the force ratio of Fn to tangential force Ft1 , vs is the grinding speed, and u is the specific 𝑑(𝑡) = ∙𝐹(𝑡) (6) energy. The elastic deflection de of the grinding system𝑘 in the infeed direction can be expressed by: where 𝑘 is the equivalent system stiffness. 1 Equivalent system stiffness 𝑘 consistsde(t) of= the Fstiffnessn(t) of each machine structure involved(6) in k · the grinding system and the stiffness of the grindingeq wheel. The main components of equivalent stiffness 𝑘 are shown in Figure 2. Material removal by grinding generates a normal grinding force where keq is the equivalent system stiffness. 𝐹, as described in Equation (5). The force 𝐹 gives the elastic deflection 𝑑, which is determined by the system stiffness 𝑘 . Figure 2a,b illustrate the stiffnesses 𝑘 𝑎𝑛𝑑 𝑘 of the grinding wheel

Inventions 2020, 5, 62 5 of 29 support system and the workpiece support system, respectively. Figure 2c shows the stiffness 𝑘 of workpiece itself, which cannot be ignored when 𝑘 is low (as in the case of thin-wall rings). Figure 2d shows the local deflection 𝛿 at the contact area between the grinding wheel and the workpiece, and 𝑘 is the contact stiffness of the grinding wheel. The equivalent system stiffness 𝑘 is expressed by: 1 1 1 = (7) 𝑘 𝑘 𝑘 where 𝑘 is the mechanical stiffness represented by the stiffness of each mechanical structure related toInventions the grinding2020, 5, 62system: 5 of 27 1 1 1 1 1 = ⋯= (8) Equivalent system stiffness keq𝑘consists𝑘 of𝑘 the stiffness𝑘 of each𝑘 machine structure involved in the grinding system and the stiffness of the grinding wheel. The main components of equivalent stiffness wherekeq are 𝑘 shown is the in Figurei-th stiffness2. Material of the removal mechanical by grinding structure generates relevant a to normal the grinding grinding system force F andn, as n is thedescribed degrees in Equationof freedom. (5). TheSince force eachFn givescomponent the elastic of deflection mechanicalde, which stiffness is determined 𝑘 has bya thelinear system spring characteristic,stiffness keq. Figure 𝑘 remains2a,b illustrate a constant the stiffnesses during kthes and plunge kw of the gr grindinginding process. wheel support On the system other andhand, the the wheelworkpiece contact support stiffness system, 𝑘 has respectively. the non-linear Figure characteristics2c shows the stiffnessreferred ktowo asof “hard-spring.” workpiece itself, As which a result, thecannot equivalent be ignored system when stiffnesskwo is low 𝑘 (as behaves in the case as of athin-wall non-linear rings). spring, Figure which2d shows changes the local in stiffness deflection value dependingδc at the contact on the area grinding between conditions. the grinding As wheel such, andit is the crucial workpiece, to take and thek cspringis the contactcharacteristics stiffness at of the contactthe grinding area of wheel. the grinding wheel into account in the design of the plunge grinding cycle.

Figure 2. Components of equivalent system stiffness in plunge grinding system. (a) Stiffness of grinding Figure 2. Components of equivalent system stiffness in plunge grinding system. (a) Stiffness of wheel head; (b) Stiffness of work head; (c) Stiffness of workpiece; (d) Contact stiffness of grinding wheel. grinding wheel head; (b) Stiffness of work head; (c) Stiffness of workpiece; (d) Contact stiffness of

grindingThe equivalent wheel. system stiﬀness keq is expressed by:

The deflection 𝑑 given by Equation (6)1 is fed1 back1 to Equation (1). Thus, the plunge grinding = + (7) process can be described as a closed-loop systemkeq krepeatingm kc Equations (1)–(6). By using Equations (3)– (5), Equation (6) can be rewritten as follows: where km is the mechanical stiﬀness represented by the stiﬀness of each mechanical structure related to the grinding system: n 1 1 1 1 X 1 = + + + = (8) km ks kw kwo ··· kmi i=1

where kmi is the i-th stiffness of the mechanical structure relevant to the grinding system and n is the degrees of freedom. Since each component of mechanical stiffness km has a linear spring characteristic, km remains a constant during the plunge grinding process. On the other hand, the wheel contact stiffness kc has the non-linear characteristics referred to as “hard-spring.” As a result, the equivalent system stiffness keq behaves as a non-linear spring, which changes in stiffness value depending on the Inventions 2020, 5, 62 6 of 29 Inventions 2020, 5, 62 6 of 27

𝑐∙𝜋∙𝑑 ∙𝑏∙𝜂∙𝑢 𝑑(𝑡) = ∙ 𝑓(𝑡) = 𝑇 ∙ 𝑓(𝑡) (9) grinding conditions. As such, it is crucial to𝑣 take∙𝑘 the spring characteristics at the contact area of the grinding wheel into account in the design of the plunge grinding cycle. where T is the time constant of the grinding system. The deflection de given by Equation (6) is fed back to Equation (1). Thus, the plunge grinding ∙∙∙∙∙ process can be described as a closed-loop system𝑇= repeating Equations. (1)–(6). By using Equations (3)–(5),(10) ∙ Equation (6) can be rewritten as follows: The time constant T is one of the most critical parameters in the plunge grinding system. The Laplace transformation of Equationsc π dw (1)–(6)b η u leads to the following equations: de(t) = · · · · · fi(t) = T fi(t) (9) vs keq · · 𝑅(𝑠)·=𝐼(𝑠) −𝐷(𝑠) (11) where T is the time constant of the grinding system.

𝐹(𝑠) =𝑠∙𝑅(𝑠) (12) c π dw b η u T = · · · · · . (10) vs keq 𝑄 (𝑠) =𝑐∙𝜋∙𝑑· ∙𝐹(𝑠) (13) The time constant T is one of the most critical parameters in the plunge grinding system. 𝑄 (𝑠) =𝑏∙𝑄 (𝑠) (14) The Laplace transformation of Equations (1)–(6) leads to the following equations: 𝜂∙𝑢 Rw𝐹((s𝑠))==I f (s) ∙𝑄De((s𝑠)) (11)(15) 𝑣 −

F (s) = s Rw(s) (12) i ·1 𝐷(𝑠) = ∙𝐹(𝑠) (16) Qw0(s) = c𝑘π dw F (s) (13) · · · i where s is the Laplace operator. Actual infeedQw(s )𝑅=(𝑠)b Q, wactual0(s) infeed rate 𝐹 (𝑠), SMMR 𝑄 (𝑠), normal(14) · grinding force 𝐹(𝑠), and elastic deflection 𝐷(𝑠)η areu Laplace functions. Fn(s) = · Qw(s) (15) Figure 3 shows a block diagram of the plungevs · grinding system representing the transfer functions given by Equations (11)–(16). The plunge1 grinding system can be expressed by a closed- De(s) = Fn(s) (16) loop feedback system. The transfer function G(s) ofkeq the· system is represented by:

s 𝑅 R (s) 1 F1(s) Q (s) where is the Laplace operator.𝐺(𝑠) = Actual(𝑠) infeed= w , actual infeed rate= i ,. SMMR w0 , normal 𝑐∙𝜋∙𝑑 ∙𝑏∙𝜂∙𝑢 (17) grinding force Fn(s), and elastic deﬂection𝐼 De(s) are Laplace functions.𝑠1 𝑇𝑠 1 Figure3 shows a block diagram of the plunge𝑣 grinding ∙𝑘 system representing the transfer functions givenThus, by Equations plunge grinding (11)–(16). can The be plunge represented grinding by system a first-order can be expressed lag system by with a closed-loop the time feedback constant T system. The transfer function G(s) of the system is represented by: given by Equation (10). The cut-off frequency 𝑓 (Hz) of the plunge grinding system is represented by: R 1 1 G(s) = w (s) = = . (17) I c π dw b η u Ts + f 𝑓· ·=· · · .s + 1 1 (18) vs keq ·

Figure 3. Block diagram of the plunge grinding system. Figure 3. Block diagram of the plunge grinding system.

Inventions 2020, 5, 62 7 of 27

Thus, plunge grinding can be represented by a ﬁrst-order lag system with the time constant T given by Equation (10). The cut-oﬀ frequency fc (Hz) of the plunge grinding system is represented by:

1 Inventions 2020, 5, 62 fc = .7 (18)of 29 2πT

2.2.2.2. Analysis of Plunge Grinding Cycle TheThe plungeplunge grindinggrinding cyclecycle starts with a rapid forwardforward infeed of the slide from the home position. position. Typically,Typically, thethe slideslide feedfeed raterate isis thenthen changedchanged forfor rough,rough, semi-finishsemi-finish oror finishfinish grinding.grinding. AfterAfter infeedinfeed grinding,grinding, spark-out grinding is performed in order to reducereduce thethe elasticelastic deflectiondeflection ofof thethe grindinggrinding system,system, followedfollowed by the rapidrapid retractionretraction of thethe slideslide toto thethe homehome position.position. To study the eeffectffect ofof thethe wheel’swheel’s contactcontact stistiffnessffness onon thethe plungeplunge grindinggrinding process,process, thisthis paperpaper focusesfocuses onon aa primaryprimary grindinggrinding cyclecycle consistingconsisting ofof plungeplunge grindinggrinding withwith aa constantconstant infeedinfeed raterate followedfollowed byby spark-outspark-out grinding.grinding. TheThe diagramdiagram inin FigureFigure4 4aa depicts depicts the the plunge plunge grinding grinding operation. operation. TheThe infeedinfeed slideslide is is rapidly rapidly fed fed forwardforward toto positionposition PP fromfrom thethe homehome positionposition H.H. Then,Then, thethe infeedinfeed raterate isis changedchanged toto aa constantconstant infeedinfeed rate 𝐼 at point P. At some point O, which is determined by the size of the incoming workpiece, the rate Ipat point P. At some point O, which is determined by the size of the incoming workpiece, the 𝐼 grindinggrinding wheelwheel mounted mounted on on the the slide slide contacts contacts the workpiecethe workpiece at the at infeed the infeed rate Ip andrate plunge and grinding plunge starts.grinding At starts. position At A, position the slide A, infeed the slide is stopped infeed is and stopped spark-out and grindingspark-out starts. grinding At point starts. B, At the point slide B, is rapidlythe slide retracted is rapidly to retracted the home to position the home H. position Figure4 H.b shows Figure the 4b primaryshows the cycle primary with cycle plunge with grinding plunge 𝑡 (𝑡 −𝑡 ). timegrindingtp and time spark-out and timespark-out(ts t ptime). −

Figure 4. Plunge grinding cycle. (a) Diagram of plunge grinding operations; (b) Primary cycle of Figure 4. Plunge grinding cycle. (a) Diagram of plunge grinding operations; (b) Primary cycle of plunge grinding. plunge grinding. During plunge grinding 0 t tp , the actual feed rate f (t) can be obtained by the following During plunge grinding (0𝑡𝑡≤ ≤ ), the actual feed rate 𝑓i (𝑡) can be obtained by the following equations [27]: equations [27]: t 0 t tp : fi(t) = Ip 1 e− T . (19) ≤ ≤ − 0𝑡𝑡: 𝑓(𝑡) =𝐼 1 −tp 𝑒 (.t tp) (19) − tp t ts : f (t) = Ip 1 e− T e− T . (20) ≤ ≤ i − · () ( ) The actual infeed position𝑡rw𝑡𝑡t is: : 𝑓(𝑡) =𝐼 1 − 𝑒 ∙𝑒 . (20)

h t i The actual infeed position 0𝑟(t𝑡) is:tp : rw(t) = Ip t T 1 e− T . (21) ≤ ≤ − − " !# 0𝑡𝑡: 𝑟(𝑡) =𝐼𝑡 − 𝑇(t tp 1) −𝑒 tp. (21) − tp t ts : rw(t) = Ip tp T e− T e− T . (22) ≤ ≤ − − () The normal grinding force𝑡 𝑡𝑡Fn(t)is:: 𝑟(𝑡) =𝐼 𝑡 −𝑇𝑒 −𝑒 . (22)

t The normal grinding force 0𝐹(𝑡)t is:tp : Fn(t) = Ip keq T 1 e− T . (23) ≤ ≤ · · − 0𝑡𝑡: 𝐹(𝑡) =𝐼 ∙𝑘 ∙𝑇(1−𝑒 ). (23)

𝑡 𝑡𝑡: 𝐹(𝑡) =𝐼 ∙𝑘 ∙𝑇1−𝑒 ∙𝑒 . (24)

The grinding power 𝑃(𝑡) is:

Inventions 2020, 5, 62 8 of 27

Inventions 2020, 5, 62 tp (t tp) 8 of 29 − tp t ts : Fn(t) = Ip keq T 1 e− T e− T . (24) ≤ ≤ · · − · 𝑣 P (t) The grinding power g 0𝑡𝑡is: : 𝑃(𝑡) = ∙𝐼 ∙𝑘 ∙𝑇1−𝑒 . (25) 𝜂 vs t 0 t t : P (t) = I k T 1 e T . (25) p g p eq − ≤ ≤ 𝑣 η · · · − 𝑡 𝑡𝑡: 𝑃 (𝑡) = ∙𝐼 ∙𝑘 ∙𝑇1−𝑒 ∙𝑒 . (26) 𝜂 tp (t tp) vs − tp t ts : Pg(t) = Ip keq T 1 e− T e− T . (26) The deflection 𝑑(𝑡) causing≤ ≤ the size and ηroundn· · ·ess error− is ·obtained by: The deﬂection de(t) causing the0𝑡𝑡 size and: 𝑑 roundness(𝑡) =𝐼 ∙𝑇1−𝑒 error is obtained. by: (27)

t T 0 t tp : de(t) = Ip T 1 e− .() (27) ≤ ≤ · − 𝑡 𝑡𝑡: 𝑑(𝑡) =𝐼 ∙𝑇1−𝑒 ∙𝑒 . (28) tp (t tp) − tp t ts : de(t) = Ip T 1 e− T e− T . (28) When plunge grinding time≤ ≤𝑡 is long enough· − compared· to time constant T, the above Equations (20), (22), (24), and (26) for spark-out grinding can be approximately represented by the When plunge grinding time tp is long enough compared to time constant T, the above Equations (20), following equations: (22), (24), and (26) for spark-out grinding can be approximately represented by the following equations: () 𝑡 𝑡𝑡: 𝑓 (𝑡) ≅𝐼 ∙𝑒(t tp) . (29) − tp t ts : f (t) Ip e− T . (29) ≤ ≤ i · () ! ( ) (t tp) (30) 𝑡 𝑡𝑡: 𝑟 𝑡 ≅𝐼(𝑡 −𝑇∙𝑒 − ). tp t ts : rw(t) Ip tp T e− T . (30) ≤ ≤ − · (t tp) (31) 𝑡 𝑡𝑡: 𝐹(𝑡) ≅𝐼 ∙𝑘 ∙𝑇∙𝑒− . tp t ts : Fn(t) Ip keq T e− T . (31) ≤ ≤ · · · (t tp) 𝑣 vs − 𝑡 tp𝑡𝑡t ts: 𝑃: P(𝑡g)(t≅) ∙𝐼Ip∙𝑘keq T∙𝑇∙𝑒e− T . . (32) (32) ≤ ≤ 𝜂 η · · · ·

(t tp) −() tp t ts : de(t) Ip T e− T .(33) (33) 𝑡 𝑡𝑡≤ ≤ : 𝑑(𝑡) ≅𝐼 ·∙𝑇∙𝑒· . Figure5 is an example simulation of the plunge grinding process. The actual slide position rw(t) Figure 5 is an example simulation of the plunge grinding process. The actual slide position 𝑟(𝑡) (Figure5a) is calculated by Equations (21) and (22). The actual slide position delays the command and (Figure 5a) is calculated by Equations (21) and (22). The actual slide position delays the command gradually catches up. Eventually, it attains the same slope as the command feed. The asymptote of and gradually catches up. Eventually, it attains the same slope as the command feed. The asymptote the actual infeed curve intersects at time T on the horizontal axis. The position difference between the of the actual infeed curve intersects at time T on the horizontal axis. The position difference between command feed and the actual feed represents the system deflection and is also the size error. The deflection the command feed and the actual feed represents the system deflection and is also the size error. The is reduced with increased spark-out time. The grinding power Pg(t) found by Equations (25) and deflection is reduced with increased spark-out time. The grinding power 𝑃(𝑡) found by Equations (26) is the response to the ramp infeed command with a constant rate and the no-infeed command for (25) and (26) is the response to the ramp infeed command with a constant rate and the no-infeed the spark-out. command for the spark-out.

Figure 5. Simulation of plunge grinding processes (conditions: c = 1.0, dw = 177.8 mm, b = 30 mm, vs. Figure 5. Simulation of plunge grinding processes (conditions: c = 1.0, dw = 177.8 mm, b = 30 mm, vs. = 45 m/s, η = 2.0, keq = 2421 N/mm, u = 26.7 J/mm3, S = 0.3 mm, Qw = 2.0 mm3/mm s, Ip = 3.58 µm/s, = 45 m/s, η = 2.0, keq = 2421 N/mm, u = 26.7 J/mm³, S = 0.3 mm, Qw’ =0 2.0 mm³/mm·s,· Ip = 3.58 μm/s, tp tp = 41.9 s, ts = 56.9 s). (a) Infeed slide position; (b) Grinding power. = 41.9 s, ts = 56.9 s). (a) Infeed slide position; (b) Grinding power.

The power is gradually increased and finally reaches a constant value. The time constant T is defined as the time when the power reaches 63.2% (1−𝑒 = 0.632) of the final value. The definition

Inventions 2020, 5, 62 9 of 27 Inventions 2020, 5, 62 9 of 29 of T can Thebe applied power is to gradually the spark-out increased grinding and finally shown reaches in aFigure constant 5b. value. The transient The time constantcurves of T isother 1 defined as the time when the power reaches 63.2% (1 e− = 0.632) of the final value. The definition parameters, including the actual infeed rate 𝑓(𝑡), the −normal grinding force 𝐹(𝑡), and the deflection of T can be applied to the spark-out grinding shown in Figure5b. The transient curves of other 𝑑(𝑡), follow the curve of Figure 5b. parameters, including the actual infeed rate fi(t), the normal grinding force Fn(t), and the deflection d (t), follow the curve of Figure5b. 2.3. Thee Effect of Grinding Wheel Contact Stiffness on the Plunge Grinding Process 2.3. The Effect of Grinding Wheel Contact Stiffness on the Plunge Grinding Process As mentioned above, system stiffness 𝑘 consists of a constant mechanical stiffness 𝑘 and contact stiffnessAs mentioned 𝑘 of above, the grinding system sti wheelffness withkeq consists non-linear of a constant charac mechanicalteristics. In sti thisffness section,km and the contact effect of systemstiff stiffnessness kc of (particularly the grinding wheelwheel with contact non-linear stiffness characteristics. 𝑘) on the plunge In this section,grinding the process effect of is system discussed. stiFigureffness 6 (particularly shows the wheeleffect of contact system stiff stiffnessness kc) on on the the plunge plunge grinding grinding process process. is discussed. The response of the Figure6 shows the e ffect of system stiffness on the plunge grinding process. The response of the grinding power 𝑃 is increased with increased system stiffness. Accordingly, the greater the system grinding power Pg is increased with increased system stiffness. Accordingly, the greater the system stiffness 𝑘, the shorter the time constant T. In the design of a plunge grinding cycle, the spark-out stiffness keq, the shorter the time constant T. In the design of a plunge grinding cycle, the spark-out time can be reduced by increasing system stiffness. time can be reduced by increasing system stiffness.

Figure 6. The effect of system stiffness on the plunge grinding process. Figure 6. The effect of system stiffness on the plunge grinding process. Figure7 presents the e ffect of system stiffness keq on the time constant T. T becomes longer where kFigureeq becomes 7 presents smaller. Bythe using effect Equations of system (7) stiffness and (10), the𝑘 time on constantthe timeT constantcan be rewritten T. T becomes as follows: longer where 𝑘 becomes smaller. By using Equations (7) and (10), the time constant T can be rewritten as 1 1 follows: T = F + = Tm + Tc. (34) km kc 1 1 F 𝑇=𝐹 = n=𝑇 𝑇. (34) 𝑘F 𝑘 (35) fi Inventions 2020, 5, 62 10 of 29 where F is the ratio of normal grinding force Fn to infeed𝐹 rate fi. 𝐹= (35) 𝑓 where F is the ratio of normal grinding force 𝐹 to infeed rate fi. Given that machine stiffness 𝑘 is a constant, the machine time constant 𝑇 also becomes a constant determined by the grinding machine. On the other hand, wheel contact stiffness 𝑘 has a non-linear characteristic and is dependent on the applied load. In a grinder with a constant machine stiffness 𝑘, the time constant T is composed of machine time constant 𝑇 and contact time constant 𝑇, as shown in Figure 7.

Figure 7. The eﬀect of system stiﬀness on the time constant. Figure 7. The effect of system stiffness on the time constant.

3. Measurement of Contact Stiffness of Grinding Wheel

3.1. Experimental Setup In order to measure the contact stiffness, various grinding wheels were mounted on a center- type cylindrical grinder. A workpiece was supported by the centers of the main work head and tailstock head. The machine and workpiece specifications are shown in Table 1. The specifications of four kinds of grinding wheels are shown in Table 2.

Table 1. Specifications of applied grinding machine and workpiece.

Center type cylindrical grinder: Tsugami T-PG350 Grinding machine Special feature: Main shaft and tailstock shaft are supported by hydrostatic bearings Center distance 350 mm Work swing 280 mm Diameter 405 mm Grinding wheel Width 50 mm Grinding wheel motor 11 kW Machine net weight 29 kN Chrome molybdenum steel SCM435, non-hardened Shape: Ground portion diameter 100 mm Workpiece Width 20 mm Both side shafts: Diameter 20 mm; length 87.5 mm Radial stiffness with center supports at the middle: 10.4 kN/mm

Table 2. Specifications of grinding wheels tested.

Abrasive Material Abrasive Type Grain Size # Grade Structure Bond Type

White alundum 𝑨𝒍𝟐𝑶𝟑 WA 60 J 8 V: Vitrified White alundum 𝑨𝒍𝟐𝑶𝟑 WA 60 L 8 V: Vitrified White alundum 𝑨𝒍𝟐𝑶𝟑 WA 60 M 8 V: Vitrified White alundum 𝑨𝒍𝟐𝑶𝟑 WA 60 L 8 B: Resinoid

Inventions 2020, 5, 62 10 of 27

Given that machine stiffness km is a constant, the machine time constant Tm also becomes a constant determined by the grinding machine. On the other hand, wheel contact stiffness kc has a non-linear characteristic and is dependent on the applied load. In a grinder with a constant machine stiffness km, the time constant T is composed of machine time constant Tm and contact time constant Tc, as shown in Figure7.

3. Measurement of Contact Stiﬀness of Grinding Wheel

3.1. Experimental Setup In order to measure the contact stiffness, various grinding wheels were mounted on a center-type cylindrical grinder. A workpiece was supported by the centers of the main work head and tailstock head. The machine and workpiece specifications are shown in Table1. The specifications of four kinds of grinding wheels are shown in Table2.

Table 1. Speciﬁcations of applied grinding machine and workpiece.

Center type cylindrical grinder: Tsugami T-PG350 Grinding machine Special feature: Main shaft and tailstock shaft are supported by hydrostatic bearings Center distance 350 mm Work swing 280 mm Diameter 405 mm Grinding wheel Width 50 mm Grinding wheel motor 11 kW Machine net weight 29 kN Chrome molybdenum steel SCM435, non-hardened Shape: Ground portion diameter 100 mm Workpiece Width 20 mm Both side shafts: Diameter 20 mm; length 87.5 mm Radial stiﬀness with center supports at the middle: 10.4 kN/mm

Table 2. Speciﬁcations of grinding wheels tested.

Abrasive Material Abrasive Type Grain Size # Grade Structure Bond Type

White alundum Al2O3 WA 60 J 8 V: Vitriﬁed

White alundum Al2O3 WA 60 L 8 V: Vitriﬁed

White alundum Al2O3 WA 60 M 8 V: Vitriﬁed

White alundum Al2O3 WA 60 L 8 B: Resinoid

The experimental setup is illustrated in Figure8. The main shaft and the tailstock shaft were supported by hydrostatic bearings. Pressure gauges were installed to measure the pocket pressures of the hydrostatic bearings. So, the normal load L acting on the contact area between the workpiece and the wheel can be measured by detecting the pressure differences of the pockets in the hydrostatic bearings. Also, a load compensator [28] was attached to the machine. The compensator provided the same force as the detected load L to the workpiece. Therefore, the effective radial stiffness of the workpiece was significantly increased from 10.4 kN/mm to 200 kN/mm by the load compensator. The contact deflection δc of the grinding wheel can be found by measuring the deflection δs at the flange of the grinding wheel and the workpiece deflection δw. The deflections δs and δw are measured with cantilever type electric micrometers. The estimated measurement error is 0.2 µm which is accurate ± enough compared with the amounts of deflections δs and δw. The micrometers are mounted on the base of the grinding machine. The difference (δs δw) corresponds to the wheel deflection δc. − Inventions 2020, 5, 62 11 of 29

same force as the detected load L to the workpiece. Therefore, the effective radial stiffness of the workpiece was significantly increased from 10.4 kN/mm to 200 kN/mm by the load compensator. The

contact deflection 𝛿 of the grinding wheel can be found by measuring the deflection 𝛿 at the flange of the grinding wheel and the workpiece deflection 𝛿. The deflections 𝛿 and 𝛿 are measured with cantilever type electric micrometers. The estimated measurement error is 0.2 μm which is

accurate enough compared with the amounts of deflections 𝛿 and 𝛿 . The micrometers are Inventionsmounted2020 on, 5 ,the 62 base of the grinding machine. The difference (𝛿 − 𝛿) corresponds to the11 wheel of 27 deflection 𝛿.

FigureFigure 8. 8.Experimental Experimental setup. setup.

3.2.3.2. FootprintFootprint MethodMethod TheThe workpieces workpieces painted painted black black were were pressed pressed with with a a normal normal load load L L to to the the grinding grinding wheel wheel as as shown shown inin Figure Figure8 .8. Both Both shafts shafts of of the the workpiece workpiece and and the the wheel wheel were were locked locked in in rotation. rotation. The The contact contact footprints footprints of the wheel were left on the surface of the workpiece, and the contact length l was measured with a of the wheel were left on the surface of the workpiece, and the contact lengthc 𝑙 was measured with microscope.a microscope. This This paper paper names names the the method method “footprint “footprint method”. method”. Figure9a,b describe the contact length and the contact deﬂection δ for external grinding and Figure 9a,b describe the contact length and the contact deflection 𝛿c for external grinding and internal grinding, respectively. Contact deﬂection δ is approximately expressed by: internal grinding, respectively. Contact deflection c𝛿 is approximately expressed by:

2 lc𝑙 δ𝛿c≅ .. (36) (36) 44∙𝑑deq · where 𝑑 is the equivalent wheel diameter represented by: where deqis the equivalent wheel diameter represented by: 𝑑𝑑 𝑑 = dwds (37) deq = 𝑑 𝑑 (37) dw ds ± where 𝑑 and 𝑑 are the diameters of grinding wheel and workpiece, respectively. The + sign is whereused fords andexternaldw are grinding the diameters and the of − grindingsign for internal wheel and grinding. workpiece, respectively. The + sign is used for external grinding and the sign for internal grinding. − In the external grinding shown in Figure9a, deq is always less than both ds and dw. In the internal grinding shown in Figure9b, deq is always bigger than ds and dw. Typically, the contact length lc for internal grinding is much longer than that of external grinding. The contact situation can be considered equivalent to one in which a ﬂat-plate workpiece is ground by a surface grinder with the grinding wheel (diameter deq), as shown in Figure9c. Figure 10 shows microscopic images of the footprints when the grinding wheel (WA60J8V) was pressed to the workpiece with a varying speciﬁc load L’. Increasing load L’ increases not only the band width corresponding to contact length lc, but also the number of contact marks corresponding to the Inventions 2020, 5, 62 12 of 29

In the external grinding shown in Figure 9a, 𝑑 is always less than both 𝑑 and 𝑑. In the internal grinding shown in Figure 9b, 𝑑 is always bigger than 𝑑 and 𝑑. Typically, the contact length 𝑙 for internal grinding is much longer than that of external grinding. The contact situation can be considered equivalent to one in which a flat-plate workpiece is ground by a surface grinder with the grinding wheel (diameter 𝑑), as shown in Figure 9c.

Inventions 2020, 5, 62 12 of 29

Inventions 2020, 5, 62 12 of 27 In the external grinding shown in Figure 9a, 𝑑 is always less than both 𝑑 and 𝑑. In the internal grinding shown in Figure 9b, 𝑑 is always bigger than 𝑑 and 𝑑. Typically, the contact numberlength 𝑙 of for abrasive internal grains. grinding As theis much contact longer boundary than th wasat of unclear external sometimes, grinding. theThe measurementcontact situation of thecan contact be considered length by equivalent the microscope to one doesin which not havea flat-plate high accuracy. workpiece The is ground estimated by errora surface is 0.1 grindermm. ± Thewith measurement the grinding resultswheel of(diameter the contact 𝑑 length), as shown are discussed in Figurein 9c. the next section.

Figure 9. Contact length and deflection. (a) External grinding; (b) Internal grinding; (c) Equivalent wheel diameter.

Figure 10 shows microscopic images of the footprints when the grinding wheel (WA60J8V) was pressed to the workpiece with a varying specific load L’. Increasing load L’ increases not only the band width corresponding to contact length 𝑙, but also the number of contact marks corresponding to the number of abrasive grains. As the contact boundary was unclear sometimes, the measurement of the contactFigure length 9. Contact by length the microscope and deﬂection. does (a) External not have grinding; high (accuracy.b) Internal grinding;The estimated (c) Equivalent error is 0.1 Figure 9. Contact length and deflection. (a) External grinding; (b) Internal grinding; (c) Equivalent mm. Thewheel measurement diameter. results of the contact length are discussed in the next section wheel diameter.

Figure 10 shows microscopic images of the footprints when the grinding wheel (WA60J8V) was pressed to the workpiece with a varying specific load L’. Increasing load L’ increases not only the

band width corresponding to contact length 𝑙, but also the number of contact marks corresponding to the number of abrasive grains. As the contact boundary was unclear sometimes, the measurement of the contact length by the microscope does not have high accuracy. The estimated error is 0.1 mm. The measurement results of the contact length are discussed in the next section

Figure 10. Contact footprint of grinding wheel WA60J8V, ds = 195 mm, dw = 98 mm, b = 20 mm. (a) 1 N/mm; (b) 2.5 N/mm; (c) 5 N/mm; (d) 10 N/mm.

3.3. Deﬂection Method

The deflections of δs and δw were directly measured under loading and unloading conditions. This paper names this method “deflection method”. Figure 11a shows the measurement results without the load compensator. When the specific load L’ was increased from zero to 22 N/mm, deflection δs

Inventions 2020, 5, 62 13 of 29

Figure 10. Contact footprint of grinding wheel WA60J8V, 𝑑 = 195 mm, 𝑑 = 98 mm, 𝑏 = 20 .mm (a) 1 N/mm; (b) 2.5 N/mm; (c) 5 N/mm; (d) 10 N/mm.

3.3. Deflection Method

The deflections of 𝛿 and 𝛿 were directly measured under loading and unloading conditions. This paper names this method “deflection method”. Figure 11a shows the measurement results without the load compensator. When the specific load L’ was increased from zero to 22 N/mm, deflection 𝛿 increased from zero to point B via point A. At point B, L’ gradually reduced to zero via C. During measurement, the deflection 𝛿 changes O-A’-B’-C’-O. There are small differences in the deflection values between loading and unloading. Figure 11b is the result of the deflection measurements after implementing the load compensator. The deflection 𝛿 of the workpiece was drastically reduced by the compensator, which also created an improvement in measuring accuracy. In both cases (Figure 11a,b), the hard- Inventions 2020, 5, 62 13 of 27 spring characteristics were clearly observed in the changes to 𝛿, especially at the lower load L’, while 𝛿 linearly changed with load L’. The contact deflection 𝛿 at load L’ is found by (𝛿 −𝛿) underincreased loading from or unloading zero to point conditions. B via point It A. was At point also B,observed L’ gradually that reduced the grinding to zero viawheel C. During elastically deflectedmeasurement, with the theload; deflection it behavedδw changes like a hard O-A’-B’-C’-O. spring at Therethe lower are small load di andfferences like a inlinear the deflection spring at the highervalues load. between loading and unloading.

Figure 11. Deﬂections of wheel and workpiece under loading and unloading with grinding wheel FigureWA60J8V. 11. Deflections (a) Without of load wheel compensator; and workpiece (b) With under load compensator.loading and unloading with grinding wheel WA60J8V. (a) Without load compensator; (b) With load compensator. Figure 11b is the result of the deﬂection measurements after implementing the load compensator.

TheFigure deflection 12a showsδw of the workpiecerepeatability was of drastically the measurement reduced by of the 𝛿 compensator,. The specific which load also L’ was created increased an fromimprovement O to B via A, in then measuring L’ was accuracy. reduced In to both O casesvia C. (Figure Then, 11 L’a,b), was the increased hard-spring from characteristics O to B via were D again and clearlywas reduced observed to inO thevia changes E. The tests to δs, wereespecially repeat at theed four lower times. load L’, Although while δw linearlythere was changed a difference with in load L’. The contact deflection δc at load L’ is found by (δs δw) under loading or unloading conditions. 𝛿 between the first loading and unloading, there was −no difference after the second loading. The It was also observed that the grinding wheel elastically deflected with the load; it behaved like a hard

spring at the lower load and like a linear spring at the higher load. Figure 12a shows the repeatability of the measurement of δs. The specific load L’ was increased from O to B via A, then L’ was reduced to O via C. Then, L’ was increased from O to B via D again and was reduced to O via E. The tests were repeated four times. Although there was a difference in δs between the first loading and unloading, there was no difference after the second loading. The deflection patterns with the load were repeated and the good repeatability was confirmed. Figure 12b shows the difference in δs between the first loading and unloading at the different maximum loads R1, R2, R3, and R4. The difference was seen only at the first loading. After the second loading, there was no difference, as shown in Figure 11b. Inventions 2020, 5, 62 14 of 29 deflection patterns with the load were repeated and the good repeatability was confirmed. Figure

12b shows the difference in 𝛿 between the first loading and unloading at the different maximum loadsInventions R1, R2,2020 R3,, 5, and 62 R4. The difference was seen only at the first loading. After the second14 ofloading, 27 there was no difference, as shown in Figure 11b.

Figure 12. Repeatability of the measurements of wheel deflection, grinding wheel WA60J8V. (a) Same Figure 12. Repeatability of the measurements of wheel deflection, grinding wheel WA60J8V. (a) Same load cycle; (b) Incremental load cycle. load cycle; (b) Incremental load cycle. 3.4. Modeling of Grinding Wheel Contact Deflection 3.4. Modeling of Grinding Wheel Contact Deflection Figure 13 compares the measurement results of wheel contact deflection δc obtained by the deflection method with the load compensator and the footprint method. The deflections δ found by the Figure 13 compares the measurement results of wheel contact deflection 𝛿c obtained by the deflection method are greater than the footprint values at each specific load L0. Although the deflection deflection method with the load compensator and the footprint method. The deflections 𝛿 found by method has much higher measurement accuracy than the footprint method, the difference is obvious. It the deflection method are greater than the footprint values at each specific load L’. Although the can be considered that the footprint method’s results reflect only the deflection of the local contact area, deflection method has much higher measurement accuracy than the footprint method, the difference while the deflection method’s results contain the deflections of both the local contact area and the wheel is obvious.body from It can the contactbe considered point to the that bore the with footprint wheel flange. method’s Both results methods reflect reveal only that wheel the deflection deflection isof the localcomposed contact area, of non-linear while characteristicsthe deflection at themethod’s lower range results of the contain load and the linear deflections spring characteristics of both the at local contactthe higherarea and range the of wheel the load. body In thisfrom case, the the contact critical point specific to the load bore is about with 3 Nwheel/mm. flange. Both methods reveal thatFrom wheel the experimentaldeflection is observations, composed of it isnon-linear assumed that characteristics the contact sti atffness the oflower the grinding range of wheel the load and islinear composed spring of characteristics a non-linear spring at the and higher a liner range spring. of the Figure load. 14 Ina illustratesthis case, thethe wheel-workpiece critical specific load is aboutinference 3 N/mm. with the equivalent wheel diameter deq. The contact deflection δc under the normal load L is represented by a hard spring shown in Figure 14b. It is assumed that the hard spring can be replaced with the series connection of two springs shown in Figure 14c. One has the non-linear spring constant ka, which represents the local stiffness of the contact area. The other has the linear spring constant kb, which represents the body stiffness of the grinding wheel itself.

Inventions 2020, 5, 62 15 of 29

Inventions 2020, 5, 62 15 of 27 Inventions 2020, 5, 62 15 of 29

Figure 13. Comparison of wheel deflections obtained by the deflection method and the footprint method.

From the experimental observations, it is assumed that the contact stiffness of the grinding wheel is composed of a non-linear spring and a liner spring. Figure 14a illustrates the wheel-workpiece inference with the equivalent wheel diameter 𝑑. The contact deflection 𝛿 under the normal load L is represented by a hard spring shown in Figure 14b. It is assumed that the hard spring can be replaced with the series connection of two springs shown in Figure 14c. One has the non-linear spring constant 𝑘, which represents the local stiffness of the contact area. The other has the linear spring constantFigure 𝑘 13., whichComparison represents of wheel the deﬂections body stiffness obtained of bythe the grinding deﬂection wheel method itself. and the footprint method. Figure 13. Comparison of wheel deflections obtained by the deflection method and the footprint method.

From the experimental observations, it is assumed that the contact stiffness of the grinding wheel is composed of a non-linear spring and a liner spring. Figure 14a illustrates the wheel-workpiece inference with the equivalent wheel diameter 𝑑. The contact deflection 𝛿 under the normal load L is represented by a hard spring shown in Figure 14b. It is assumed that the hard spring can be replaced with the series connection of two springs shown in Figure 14c. One has the non-linear spring constant 𝑘, which represents the local stiffness of the contact area. The other has the linear spring constant 𝑘, which represents the body stiffness of the grinding wheel itself.

Figure 14. Modeling of the contact stiﬀness of the grinding wheel. (a) Contact of wheel with workpiece; Figure 14. Modeling of the contact stiffness of the grinding wheel. (a) Contact of wheel with (b) Model of contact stiﬀness; (c) Series connection of linear and non-linear springs. workpiece; (b) Model of contact stiffness; (c) Series connection of linear and non-linear springs.

In Figure 14c, contact deﬂection δc is expressed by: In Figure 14c, contact deflection 𝛿 is expressed by:

δc = δa + δb (38) 𝛿 =𝛿 𝛿 (38) where δ𝛿ais is the the deflection deflection of of the the non-linear non-linear spring spring with withka, and𝑘, δandb is the𝛿 is deflection the deflection of the linearof the springlinear withspringkb .with As δ a𝑘represents. As 𝛿 represents the non-linear the non-linear deflection deflection with load withL, it load is assumed L, it is thatassumed deflection that deflectionδa can be represented𝛿 can be represented by: by: L0 δ = A 1 e S (39) Figure 14. Modeling of the contact stiffnessa of the −grinding wheel. (a) Contact of wheel with(39) 𝛿 = 𝐴(1− − 𝑒 ) workpiece; (b) Model of contact stiffness; (c) Series connection of linear and non-linear springs. where A and S are constants and L0 is the specific load per unit contact width. It is also assumed that the body of the grinding wheel has a spring constant of kb. So, deflection δb is expressed by: In Figure 14c, contact deflection 𝛿 is expressed by:

𝛿 =𝛿L𝛿0 (38) δ b = (40) kb0 where 𝛿 is the deflection of the non-linear spring with 𝑘, and 𝛿 is the deflection of the linear wherespringk bwith0 is the 𝑘 specific. As 𝛿 spring represents stiffness the pernon-linear unit width. deflection Therefore, withcontact load L,deflection it is assumedδc can that be deflection rewritten as𝛿 follows: can be represented by: L0 L0 δc = A 1 e− S + . (41) − kb0 (39) 𝛿 = 𝐴(1 − 𝑒 )

Inventions 2020, 5, 62 16 of 29 Inventions 2020, 5, 62 16 of 27 where A and S are constants and 𝐿 is the specific load per unit contact width. It is also assumed that the bodyDifferentiating of the grinding both sides wheel of has Equation a spring (41) constant provides of the 𝑘 specific. So, deflection compliance 𝛿 d isδc expressed/dL0 of the by: grinding wheel. 𝐿 dδc 𝛿A =L0 1 S (40) = e− 𝑘+ . (42) dL0 S kb0 where 𝑘 is the specific spring stiffness per unit width. Therefore, contact deflection 𝛿 can be Therefore, specific contact stiffness kc0 per unit width can be expressed by: rewritten as follows: " # A L0 1 S kc0 = 1/ e− + . (43) 𝛿 = 𝐴 1S − 𝑒 k . (41) b0 Also,Differentiating the non-linear both contact sides of sti ffEquationness of local (41) area provides per unit the width specificka0 is:compliance 𝑑𝛿⁄𝑑𝐿 of the grinding wheel. S L0 = S ka0 e . (44) A = 𝑒 . (42) In Figure 13, the best fit curves for both methods were found by using Equation (41). The solid Therefore, specific contact stiffness 𝑘 per unit width can be expressed by: line is for the deflection method and the dotted line is for the footprint method. The parameters shown 𝐴 1 in Table3 were used for the calculations of δc. kb0 represents the wheel body stiffness and parameter A 𝑘 =1 𝑒 . (43) shows the degree of local deflection δa. Parameter𝑆S expresses𝑘 the degree of raising curve at the lower specific load L . Also, the0 non-linear contact stiffness of local area per unit width 𝑘 is:

𝑆 Table 3. Parameters for representing the𝑘 contact= deflection𝑒 . of the grinding wheel WA60J8V. (44) 𝐴 Parameter Unit Deflection Method Footprint Method In Figure 13, the best fit curves for both methods were found by using Equation (41). The solid k kN/mm mm 3.5 3.5 line is for the bdeflection0 method and· the dotted line is for the footprint method. The parameters µ shown in TableA 3 were used for them calculations of 𝛿. 2.3𝑘 represents the wheel 0.8body stiffness and parameter A showsS the degreeN of/mm local deflection 𝛿. Parameter 1.1 S expresses the 1.1 degree of raising curve at the lower specific load L’. TheThe specificspecific contact contact sti stiffnessesffnesses for for both both methods methods were were calculated calculated by using by Equationusing Equation (43), as (43), shown as shown in Figure 15. At L’ > 6 N/mm·mm, both methods provide the same contact stiffness. At L’< 6 in Figure 15. At L0 > 6 N/mm mm, both methods provide the same contact stiffness. At L0 < 6 N/ N/ mm·mm, the deflection method· provides lower contact stiffness than the footprint method. The mm mm, the deflection method provides lower contact stiffness than the footprint method. The specific specific· contact stiffness 𝑘 was 3.5 kN/mm·mm at L’ > 6 kN/mm. At L’ = 2 kN/mm, 1.5 kN/mm·mm contact stiffness kc was 3.5 kN/mm mm at L > 6 kN/mm. At L = 2 kN/mm, 1.5 kN/mm mm and 0 · 0 0 · 2.5and kN 2.5/mm kN/mm·mmmm were were obtained obtained by the by deflection the deflection method method and theand footprint the footprint method, method, respectively. respectively. ·

Figure 15. Speciﬁc contact stiﬀness of grinding wheel. Figure 15. Specific contact stiffness of grinding wheel.

Based on the contact stiffness model, the specific contact stiffness kc0 is composed of two Based on the contact stiffness model, the specific contact stiffness 𝑘 is composed of two components, ka0 for local deflection and kb0 for wheel body deflection. Figure 16 shows the specific components, 𝑘 for local deflection and 𝑘 for wheel body deflection. Figure 16 shows the specific

Inventions 2020, 5, 62 17 of 29

contact stiffness 𝑘 obtained by the deflection method, where 𝑘 was found by Equation (44) and 𝑘 was set to 3.5 kN/mm·mm. The specific local area stiffness 𝑘 shows heavy hard-spring characteristics such as lower stiffness at L’< 2 kN/mm and higher stiffness at L’> 3 kN/mm. The dominant component of 𝑘 is 𝑘 at the lower load L’< 2 kN/mm and 𝑘 at the higher load L’> 4 kN/mm. Inventions 2020, 5, 62 17 of 27 Table 3. Parameters for representing the contact deflection of the grinding wheel WA60J8V. contact stiffness kParameterobtained by theUnit deflection Deflection method, where Methodk wasFootprint found byMethod Equation (44) and k c0 a0 b0 𝑘 kN/mm·mm 3.5 3.5 was set to 3.5 kN/mm mm. The specific local area stiffness ka0 shows heavy hard-spring characteristics A· μm 2.3 0.8 such as lower stiffness at L0 < 2 kN/mm and higher stiffness at L0 > 3 kN/mm. The dominant component S N/mm 1.1 1.1 of kc0 is ka0 at the lower load L0 < 2 kN/mm and kb0 at the higher load L0 > 4 kN/mm.

Figure 16. Components of specific contact stiffness. Figure 16. Components of specific contact stiffness. 3.5. Measurement Results of Grinding Wheel Contact Stiffness 3.5. Measurement Results of Grinding Wheel Contact Stiffness The contact deflections δc of four kinds of grinding wheels shown in Table2 were measured by the deflectionThe contact method deflections with the 𝛿 load of four compensator. kinds of grinding Figure17 wheels shows shown the measurement in Table 2 were results. measured The best by fitthe curves deflection were method found bywith using the load the parameters compensator. shown Figure in 17 Table shows4. As the seen measurement in Figure 17 results., all grinding The best wheelsfit curves exhibited were found heavy by hard-spring using the behaviorparameters at theshown lower in loadTable and 4. aAs liner seen spring in Figure nature 17, at all the grinding higher load.wheels The exhibited resinoid heavy bond hard-spring wheel WA60L8B behavior exhibited at the lo thewer greatest load and deflection a liner spring among nature the tested at the wheels higher andload. the The vitrified resinoid bond bond wheel wheel WA60L8V WA60L8B gave exhibited the smallest. the greatest In wheels deflection WA60M8V among and the WA60L8B, tested wheels the reductionand the vitrified of the deflection bond wheel appeared WA60L8V at L’> gave8 kN the/mm. smallest. It seems In thatwheels a structural WA60M8V change and inWA60L8B, the grinding the wheelsInventionsreduction occurred 2020 of, 5the, 62 atdeflection the higher appeared load. These at L’> phenomena 8 kN/mm. It should seems bethat investigated a structuralin change the future. in the grinding18 of 29 wheels occurred at the higher load. These phenomena should be investigated in the future.

Figure 17. Contact deﬂection of various Al2O3#60 grinding wheels. Figure 17. Contact deflection of various AlO#60 grinding wheels.

The contact stiffnesses were found by Equation (43) from the deflection curves shown in Figure 17. Figure 18 shows the specific contact stiffness 𝑘 of various 𝐴𝑙𝑂 #60 grinding wheels. The WA60L8V wheel had the highest contact stiffness, followed by the WA60J8V and WA60M8V wheels. The resinoid bond WA60J8B wheel had the lowest stiffness among them. At the specific load L’ < 2 kN/mm, all grinding wheels exhibited very low contact stiffness at the range of 0 < L’< 4 N/mm. At the higher load L’> 4 N/mm, the contact stiffness reached a constant level.

Figure 18. Contact stiffness of various AlO #60 grinding wheels.

Table 4. Parameters of various grinding wheels obtained by the deflection measurements.

Parameter Unit WA60J8V WA60L8V WA60M8V WA60L8B 𝑘 kN/mm·mm 7.7 9.1 4.5 3.6 A Μm 1 0.5 1.8 2.6 S N/mm 1.2 1.2 0.7 0.7

Inventions 2020, 5, 62 18 of 29

Inventions 2020, 5, 62 18 of 27

Table 4. Parameters of various grinding wheels obtained by the deﬂection measurements.

Parameter Unit WA60J8V WA60L8V WA60M8V WA60L8B k kN/mm mm 7.7 9.1 4.5 3.6 b0 · A Mm 1 0.5 1.8 2.6 Figure 17. Contact deflection of various AlO#60 grinding wheels. S N/mm 1.2 1.2 0.7 0.7 The contact stiffnesses were found by Equation (43) from the deflection curves shown in Figure 17. FigureThe contact 18 shows stiffnesses the specific were found contact by Equation stiffness (43) 𝑘 fromof various the deflection 𝐴𝑙𝑂 #60 curves grinding shown wheels. in Figure The 17. FigureWA60L8V 18 shows wheel the had specific the highest contact contact stiffness stiffness,kc0 of variousfollowedAl by2O the3#60 WA60J8V grinding and wheels. WA60M8V The WA60L8V wheels. wheelThe resinoid had the bond highest WA60J8B contact stiwheelffness, had followed the lowest by the stiffness WA60J8V among and them. WA60M8V At the wheels. specific The load resinoid L’ < 2 bondkN/mm, WA60J8B all grinding wheel wheels had the exhibited lowest sti veryffness low among contact them. stiffness At theat the specific range load of 0L <0 L’<< 2 4 kN N/mm./mm, allAt grindingthe higher wheels load L’> exhibited 4 N/mm, very the lowcontact contact stiffness stiffness reached at the a rangeconstant of 0level.< L0 < 4 N/mm. At the higher load L0 > 4 N/mm, the contact stiffness reached a constant level.

Figure 18. Contact stiffness of various Al2O3 #60 grinding wheels. Figure 18. Contact stiffness of various AlO #60 grinding wheels. Figure 19 shows the influence of accumulative SMR (specific material removal) on the deflection of the WA60M8VTable 4. Parameters wheel. The of dressing various grinding conditions wheels are ob singletained point by the dresser, deflection dressing measurements. lead 0.1 mm/rev., and adress depthParameter of cut 20 µm.Unit The bestWA60J8V fit curves are WA drawn60L8V in the WA60M8V figure. The deflectionWA60L8B after dressing was reduced and the deflection curve changed with increased SMR. No clear trend was observed. 𝑘 kN/mm·mm 7.7 9.1 4.5 3.6 After dressing, theA deflectionΜ curvesm gradually1 changed0.5 up and down1.8 according to2.6 the accumulative SMR progression.S The G-ratioN/mm (grinding ratio)1.2 for monitoring1.2 the wear 0.7 progress was 0.7 not investigated in this study. Figure 20 shows the changes in contact stiffness after dressing. The contact stiffness was found by applying the parameters shown in Table5 to Equation (43). The contact sti ffness at the lower range of load (L0 < 1.5 kN/mm) was very low and no corresponding change occurred. But, at the higher range (L0 > 4 kN/mm), the contact stiffness increased significantly by 20–50% after dressing.

Table 5. Parameters of wheel deﬂection taken during speciﬁc material removal (SMR) progression.

Item Unit Obtained Parameter SMR mm3/mm 0 8 160 320 700 k kN/mm mm 5.9 9.1 6.7 10 12.5 b0 · A µm 1.9 1.4 1.6 1.4 1.3 S N/mm 0.5 0.5 0.5 0.5 0.5 Inventions 2020, 5, 62 19 of 29

Figure 19 shows the influence of accumulative SMR (specific material removal) on the deflection of the WA60M8V wheel. The dressing conditions are single point dresser, dressing lead 0.1 mm/rev., and adress depth of cut 20 μm. The best fit curves are drawn in the figure. The deflection after dressing was reduced and the deflection curve changed with increased SMR. No clear trend was observed.Inventions 2020 After, 5, 62 dressing, the deflection curves gradually changed up and down according 19to ofthe 29 accumulative SMR progression. The G-ratio (grinding ratio) for monitoring the wear progress was not investigatedFigure 19 shows in this the study. influence of accumulative SMR (specific material removal) on the deflection of the WA60M8V wheel. The dressing conditions are single point dresser, dressing lead 0.1 mm/rev., and adress depth of cut 20 μm. The best fit curves are drawn in the figure. The deflection after dressing was reduced and the deflection curve changed with increased SMR. No clear trend was observed. After dressing, the deflection curves gradually changed up and down according to the accumulative SMR progression. The G-ratio (grinding ratio) for monitoring the wear progress was Inventions 2020, 5, 62 19 of 27 not investigated in this study.

Figure 19. The effect of accumulative specific material removal on contact deflection.

Figure 20 shows the changes in contact stiffness after dressing. The contact stiffness was found by applying the parameters shown in Table 5 to Equation (43). The contact stiffness at the lower range of load (L’< 1.5 kN/mm) was very low and no corresponding change occurred. But, at the higher range (L’> 4 kN/mm), the contact stiffness increased significantly by 20–50% after dressing. Figure 19. The effect of accumulative specific material removal on contact deflection. Figure 19. The effect of accumulative specific material removal on contact deflection.

Figure 20. The effect of accumulative specific material removal on contact stiffness. Figure 20. The effect of accumulative specific material removal on contact stiffness. 4. Results of Plunge Cylindrical and Centerless Grinding Tests

4.1. Test Results of Plunge Cylindrical Grinding Given that the contact stiffness k of the grinding wheel largely relies on the specific normal load c L0, it is expected that the normal grinding force Fn affects the time constant T in the plunge grinding Figure 20. The effect of accumulative specific material removal on contact stiffness. system because T is a function of kc (see Equation (34)). Plunge cylindrical grinding (c = 1.0) tests were carried out with various infeed rates fi, and grinding power Pg was measured during the grinding process. The grinding conditions are shown in Table6. Figure 21 shows the ramping up of the grinding power from start to steady state during the 3 plunge grinding process. At SMRR Qw = 0.25 mm /mm s, which was very low (Figure 21a), the 0 · grinding power slowly built up and finally reached a steady state power of 0.31 kW. From the power curve, the time constant T of 17 s was measured. The time constant became shorter with increased 3 SMRR Qw . At Qw = 2.0 mm /mm s, the time constant T was shortened and T = 8.2 s was obtained. 0 0 · The reason the time constant gets shorter with increased Qw0 is that contact stiffness kc becomes greater at the higher normal grinding force Fn. The estimated range of the specific normal grinding force Fn0 is 0.46~2.36 N/mm, which can be considered as the lower specific load L0. In the load range, contact stiffness kc clearly exhibits a hard-spring nature and the contact time constant Tc of Equation (34) significantly changes. Inventions 2020, 5, 62 20 of 27

Table 6. Grinding conditions of plunge cylindrical grinding.

Item Conditions Grinding machine Universal cylindrical grinder: Heald 2EF Cinternal Grinding method Chuck type cylindrical grinding Material: Through-hardened 52100 (HRC58) Workpiece Diameter: 177.8 mm Width: 30 mm Speciﬁcation: A70KmV Grinding wheel Diameter: 127 mm Width: 86.4 mm

Inventions 2020Grinding, 5, 62 speed, ratio Grinding speed: 45 m/s, Speed ratio: 1/100 21 of 29

FigureFigure 21. ExperimentalExperimental results results of ofplunge plunge cylindric cylindricalal grinding grinding andand the time the time constant. constant. (a) 𝑄 (a )=Q 0.25w0 = 3 3 3 3 0.25mm³/mm·s; mm /mm (b)s; 𝑄 (b) Q = w0.5= mm³/mm·s;0.5 mm /mm (c) s;𝑄 (c) =Q 1.0w =mm³/mm·s;1.0 mm /mm (d) s;𝑄 (d ) =Q 2.0w mm³/mm·s.= 2.0 mm /mm s. · 0 · 0 · 0 · 4.2.4.2. Test ResultsResults of PlungePlunge Centerless Grinding TableTable7 7delineates delineates thethe grindinggrinding conditionsconditions forfor plungeplunge centerlesscenterless grinding (c == 0.5)0.5) of of glass glass workpieces.workpieces. The The regulating regulating wheel wheel spindle spindle of of the the grinding grinding machine machine was was su supportedpported by by hydrostatic hydrostatic bearingsbearings andand thethe grindinggrinding forces forces were were measured measured by by detecting detecting the the pocket pocket pressures pressures of theof the bearings bearings [29 ]. [29].

Table 7. Grinding conditions of plunge centerless grinding.

Item Conditions Grinding machine Centerless grinding machine Glass HV500 Workpiece Diameter: 12.4 mm Length: 66 mm Specification: SiC, GC 100 LmV Grinding wheel Diameter: 455 mm Width: 150 mm Regulating wheel Specification: A150RR

Inventions 2020, 5, 62 21 of 27

Table 7. Grinding conditions of plunge centerless grinding.

Item Conditions Grinding machine Centerless grinding machine Glass HV500 Workpiece Diameter: 12.4 mm Inventions 2020, 5, 62 22 of 29 Length: 66 mm Specification: SiC, GC 100 LmV Grinding wheel Diameter:Diameter: 455 mm255 mm Width:Rotational 150 mm speed: 12.7 rpm Specification:Blade angle: A150RR 30° Regulating wheel Diameter: 255 mm Grinding conditions RotationalCenter speed:height 12.7 angle: rpm 6.8° Grinding speed: 29 m/s Blade angle: 30◦ Grinding conditions Center height angle: 6.8◦ Figure 22 shows test results of plungeGrinding centerle speed:ss grinding 29 m/s with different infeed rates. The periodic fluctuations observed on the force curves were due to the radial runout of the regulation wheel rotation.Figure Taking 22 shows the test average results of of plunge the fluctuat centerlessions, grinding the withtime di constantfferent infeed T was rates. measured The periodic for each fluctuations observed on the force curves were due to the radial runout of the regulation wheel rotation. plunge grinding operation. The time constant was prolonged when the specific material removal rate Taking the average of the fluctuations, the time constant T was measured for each plunge grinding was reduced.operation. The time constant was prolonged when the specific material removal rate was reduced.

= Figure 22. Experimental results of plunge centerless grinding and the time constant. (a) Qw𝑄0 =0.01 Figure 22. Experimental3 results of plunge3 centerless grinding and3 the time constant. (a) 0.01 mm /mms; (b) Qw0 = 0.02 mm /mm s; (c) Qw0 = 0.0325 mm /mm s. mm³/mm·s; (b) 𝑄· =0.02 mm³/mm·s; (c) · 𝑄 = 0.0325 mm³/mm·s.· The grinding tests were repeated many times with a specific normal load range of L0 = 0.06~0.7 N/mm The(which grinding is considered tests were a very repeated light load). many The resultstimes arewith summarized a specific innormal Figure 23load. The range time constantof 𝐿 = 0.06~0.7 T 3 converges at about 4 s at SMRR Qw > 0.03 mm /mm s. The converged time (about 4 s) corresponds N/mm (which is considered a very light0 load). The ·results are summarized in Figure 23. The time constant T converges at about 4 s at SMRR 𝑄 > 0.03 mm³/mm·s. The converged time (about 4 s) corresponds to the machine time constant 𝑇 described in Figure 7. By contrast, the time constant was rapidly prolonged where the SMRR was reduced. It reaches 28 s at SMRR 𝑄 = 0.002 mm³/mm·s. In this case, the contact time constant 𝑇 becomes 24 s. The increased 𝑇 comes from the reduced contact stiffness created by the reduced contact load. This result clearly reveals that the time constant T is significantly increased with reduced SMRR 𝑄 .

Inventions 2020, 5, 62 22 of 27

to the machine time constant Tm described in Figure7. By contrast, the time constant was rapidly 3 prolonged where the SMRR was reduced. It reaches 28 s at SMRR Qw = 0.002 mm /mm s. In this case, 0 · the contact time constant Tc becomes 24 s. The increased Tc comes from the reduced contact stiffness created by the reduced contact load. This result clearly reveals that the time constant T is significantly Inventionsincreased 2020 with, 5, 62 reduced SMRR Qw0. 23 of 29

Figure 23. The eﬀect of speciﬁc material removal rate on the time constant in plunge centerless grinding. Figure 23. The effect of specific material removal rate on the time constant in plunge centerless 5. Validationgrinding. and Discussion

5. ValidationThe plunge and cylindricalDiscussion grinding (c = 1.0) test results shown in Figure 21a–d were simulated using Equations (19)–(28) for the primary grinding cycle shown in Figure4b. The simulation results are shownThe plunge in Figure cylindrical 24a–d. Also, grinding Figure (c 25 = a–c1.0) showtest results the simulations shown in Figure of experimental 21a–d were results simulated obtained using Equationsby plunge (19) centerless–(28) for grinding the primary (c = 0.5) grinding shown incycle Figure shown 22a–c. in InFigure all simulations 4b. The simulation of plunge grindingresults are shownoperations, in Figure the transient24a–d. Also, behaviors Figure in 25a build-up–c show during the simulations ramp infeed andof experimental the reduction results during obtained spark-out by plungeare almost centerless identical grinding to the experimental (c = 0.5) shown results, in andFigure the levels22a–c. in In steady all simulations state and the of time plunge constant grinding are operations,excellent matchesthe transient to the behaviors experimental in build-up results. Theseduring correlations ramp infeed indicate and the that reduction the proposed during model spark- outcan are accurately almost identical predict critical to the parameters experimental such result as forcess, andFt and theF nlevels, power inP steadyg, machine state deflection and the de,time constantequivalent are systemexcellent sti ffmatchesness keq, andto the time experimental constant T during results. both These plunge correlations cylindrical indicate grinding that and the centerless grinding. proposed model can accurately predict critical parameters such as forces 𝐹 and 𝐹 , power 𝑃 , machineTable deflection8 lists the de, critical equi parametersvalent system obtained stiffness by thekeq, simulations and time constant in Figure T24 duringa–d. The both specific plunge cylindricalenergy u forgrinding each plunge and centerless grinding grinding. operation was found, and it was observed that u increased with increased SMRR Qw0. These phenomena are well known in grinding [30]. The equivalent chip thickness heq (defined as the ratio of SMRR Qw0 to grinding speed vs) is a key parameter that affects energy u and the finished surface quality [31]. The system stiffness keq obtained by the simulation decreased with the decrease in specific normal grinding force Fn0.

Table 8. Critical parameters obtained by the simulations in Figure 23.

Symbol Fn0 u T keq kˆm kc0 Unit N/mm J/mm3 s kN/mm kN/mm kN/mm mm · (a) 0.46 41.4 16 1.92 2.47 0.29 (b) 0.83 37.4 14 1.99 2.47 0.34 (c) 1.36 30.7 10.7 2.13 2.47 0.52 (d) 2.36 26.7 8.2 2.42 2.47 4.0

Fn0: specific normal grinding force, u: specific energy, T: time constant, keq: system stiffness, kˆm: estimated machine stiffness, kc0: specific contact stiffness.

Inventions 2020, 5, 62 23 of 27

From the system stiffness analysis, the machine stiffness of the applied grinder was estimated to kˆm = 2.47 kN/mm as a constant value. Then, the specific contact stiffness kc0 of the grinding wheel was calculated. kc = 4.0 kN/mm mm was obtained when the infeed rate was highest (Figure 24d). When 0 · the infeed rate was slowest (Figure 24a), the specific contact stiffness was significantly reduced to kc0 = 0.29 kN/mm mm and hard-spring characteristics were clearly exhibited, as expected. · As the result of the reduced system stiffness keq due to the reduction of contact stiffness kc, the time constant T drastically increased by over 200% from 8.2 s to 17 s, as predicted by Equation (10). So, when a grinding cycle is designed, the plunge infeed time and spark-out time should be determined by taking into account the influence of the infeed rate on the time constant T. It is clear that the time constant T significantly changes by 2–2.6 times, as seen in Figures 24 and 25. The amount of change depends on the contact stiffness of the grinding wheel, which relies on contact load L; that is, normal grinding force Fn in this case. When a plunge grinding cycle with a lower infeed rate fi is set up, a long-time constant will be found due to the lower contact stiffness at the lower grinding force Fn. On the other hand, the effect of contact stiffness on the time constant can be ignored when high stock removal conditions with high SMRR Qw0 and high normal grinding force Fn Inventionsare set 2020 up., 5, 62 24 of 29

3 Figure 24. Simulations of plunge cylindrical grinding processes. (a) Qw0 = 0.25 mm /mm s; (b) Qw0 = Figure 24.3 Simulations of plunge 3cylindrical grinding processes.3 (a) 𝑄 = 0.25 mm³/mm·s;· (b) 𝑄 = 0.5 mm /mm s; (c) Qw 0 = 1.0 mm /mm s; (d) Qw0 = 2.0 mm /mm s. 0.5 mm³/mm·s; ·(c) 𝑄 = 1.0 mm³/mm·s;· (d) 𝑄 = 2.0 mm³/mm·s.·

Inventions 2020, 5, 62 24 of 27 Inventions 2020, 5, 62 25 of 29

3 Figure 25. Simulations of plunge centerless grinding. (a) Qw0 = 0.01 mm /mm s; (b) Qw0 = Figure 25. Simulations3 of plunge centerless3 grinding. (a) 𝑄 =0.01 mm³/mm·s;· (b) 𝑄 =0.02 0.02 mm /mm s; (c) Qw = 0.0325 mm /mm s. · 0 · mm³/mm·s; (c) 𝑄 = 0.0325 mm³/mm·s. In general, the ramp infeed time of the plunge grinding cycle requires at least five times T (5T). This setting provides sufficient time for reaching 99.3% of the target level of the steady state Table 8 lists5 the critical parameters obtained by the simulations in Figure 24a–d. The specific ( 1 e− = 0.993). The spark-out time requires at least three times T (3T) in order to reduce the system energy u for− each plunge grinding operation was found, and it was observed that u increased with deflection de by 95% ( 1 e 3 = 0.950). Setting the spark-out time to 3T affords the grinding operation − − increasednot SMRR onlyimproved 𝑄 . These size error, phenomena but also improved are well roundness known of thein finishedgrinding products. [30]. InThe the plungeequivalent chip thickness grindingℎ (defined operations as usingthe ratioAl2O 3ofgrinding SMRR wheels, 𝑄 to the grinding time constant speedT can 𝑣 be) reasonablyis a key parameter estimated by that affects using the data of wheel contact stiffness shown in Figures 18 and 20. It allows process engineers to energy u and the finished surface quality [31]. The system stiffness 𝑘 obtained by the simulation determine the spark-out time and the proper grinding cycle for achieving the required size error and decreased with the decrease in specific normal grinding force 𝐹 . roundness. The design procedures are described in [27]. From theFor system high-e ffistiffnessciency and analysis, high-precision the machine plunge grinding, stiffness it of is vitalthe applied to take the grinder effect of was contact estimated to 𝑘 = 2.47sti kN/mmffness on theas a plunge constant grinding value. process Then, into considerationthe specific during contact the stiffness design ofthe 𝑘 grinding of the cycle. grinding wheel was calculated. 𝑘 = 4.0 kN/mm·mm was obtained when the infeed rate was highest (Figure 24d). When the infeed rate was slowest (Figure 24a), the specific contact stiffness was significantly reduced to 𝑘 = 0.29 kN/mm·mm and hard-spring characteristics were clearly exhibited, as expected. As the result of the reduced system stiffness 𝑘 due to the reduction of contact stiffness 𝑘, the time constant T drastically increased by over 200% from 8.2 s to 17 s, as predicted by Equation (10). So, when a grinding cycle is designed, the plunge infeed time and spark-out time should be determined by taking into account the influence of the infeed rate on the time constant T. It is clear that the time constant T significantly changes by 2–2.6 times, as seen in Figures 24 and 25. The amount of change depends on the contact stiffness of the grinding wheel, which relies on contact load L; that is, normal grinding force 𝐹 in this case. When a plunge grinding cycle with a lower infeed rate fi is set up, a long-time constant will be found due to the lower contact stiffness at the lower grinding force Fn. On the other hand, the effect of contact stiffness on the time constant can

Inventions 2020, 5, 62 25 of 27

6. Conclusions This paper presented the effect of grinding wheel contact stiffness on the plunge grinding cycle and provided guidelines for the design of the grinding cycle. First, the generalized plunge grinding system (including cylindrical, centerless, shoe-centerless, internal, and shoe-internal grinding) was explicitly described and analyzed in terms of system outputs corresponding to the inputs of the plunge grinding conditions. The analysis discussed the influence of system stiffness on the time constant that governs the transient behaviors of process parameters such as forces, power, and machine deflection. Also, it clarified that system stiffness is composed of machine stiffness and wheel contact stiffness, and these stiffnesses significantly affect grinding performance indicators such as productivity and grinding accuracy. The elastic deflection of the grinding wheel was accurately measured and a function for representing the deflection nature under various contact loads was derived. The function provides the non-linear contact stiffness with respect to the normal load. The contact stiffnesses of four kinds of grinding wheels with different grades and bond materials were presented, and changes in the contact stiffness after dressing were discussed. Experimental tests for both cylindrical grinding and centerless grinding were carried out, and it was experimentally confirmed that the time constant at ramp infeed and spark-out is significantly prolonged by reducing the grinding force. It was verified that a simulation of the grinding tests using the proposed model can accurately predict critical parameters like forces, machine deflection, and system stiffness during plunge operations. Finally, this paper emphasized that it is vital to take the effect of wheel contact stiffness into consideration during the design of the plunge grinding cycle. The study provides the following conclusions.

(1) The equivalent stiffness of the plunge grinding system is composed of machine stiffness and wheel contact stiffness. The contact stiffness greatly depends on the normal contact load. It is confirmed by the measurement of wheel deflection and the plunge grinding tests with various infeed rates. (2) The contact deflection of the grinding wheel behaves like a non-linear spring representing a local contact deflection at lower load and a linear spring expressing the elasticity of the wheel body itself at a higher load. Formulas representing the wheel deflection behaviors are presented. (3) The contact stiffness significantly affects the time constant T that governs the transient behaviors in ramp infeed and spark-out grinding. (4) The time constant T is drastically prolonged with reduced normal grinding force due to the reduction of contact stiffness at lower contact loads. The grinding tests revealed that the time constant T significantly changes by 2–2.6 times.

(5) All measured Al2O3 wheels exhibit very low contact stiffness at a specific load less than 3 N/mm. At a higher load greater than 4 N/mm, contact stiffness reaches a constant. (6) Plunge grinding tests applying both cylindrical grinding and centerless grinding methods with various infeed rates experimentally verified the effect of wheel contact stiffness on the time constant T. (7) As guidelines for plunge grinding cycle, the ramp infeed grinding time should be set to at least five times T and at least three times T for spark-out grinding.

Author Contributions: Conceptualization, F.H.; methodology, F.H. and H.I.; software, F.H.; validation, F.H. and H.I.; formal analysis, F.H.; investigation, F.H.; resources, F.H. and H.I.; data curation, F.H. and H.I.; writing—original draft preparation, F.H.; writing—review and editing, F.H. and H.I.; visualization, F.H. and H.I.; supervision, F.H.; project administration, F.H.; funding acquisition, F.H. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Inventions 2020, 5, 62 26 of 27

Conﬂicts of Interest: The authors declare no conﬂict of interest. F.H. is a founder of Advanced Finishing Technology Ltd., in which he holds shares. H.I. is a founder of Toba Engineering, in which he holds shares. The sponsors of both companies 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 deq Equivalent wheel diameter De(s) Deflection of machine in s-domain ds Diameter of grinding wheel Fi(s) Actual infeed rate in s-domain dw Diameter of workpiece Fn(s) Normal grinding force in s-domain fc Cut-off frequency G(s) Transfer function in s-domain fi Actual infeed rate I f (s) Command infeed in s-domain heq Equivalent chip thickness Qw(s) MRR in s-domain ka Stiffness of wheel contact area Qw0(s) SMRR in s-domain ka0 Specific stiffness of wheel contact area Rw(s) Actual infeed in s-domain kb Stiffness of wheel body A Constant kb0 Specific stiffness of wheel body F Ratio of Fn to fi kc Contact stiffness of grinding wheel Fn Normal grinding force kc0 Specific contact stiffness of grinding wheel Ft Tangential grinding force keq Equivalent system stiffness Fn0 Specific normal grinding force km Machine stiffness of grinding system Ft Specific tangential grinding force k Stiffness of i-th. mechanical structure 0 mi GW Grinding wheel kˆm Estimated machine stiffness I f Command infeed ks Stiffness of wheel support system Ip Constant infeed rate kw Stiffness of work support system L Contact load kw0 Stiffness of workpiece itself L0 Specific contact load lc Contact length MRR Material removal rate n Degrees of freedom Pg Grinding power rw Actual infeed Qw Material removal rate t Time Qw0 Specific material removal rate tp Plunge grinding time RW Regulating wheel ts Time at end of grinding S Constant u Specific energy SMR Specific material removal vs Grinding speed SMRR Specific material removal rate δ Deflection T Time constant δa Deflection of wheel contact area Tc Contact time constant δb Deflection of grinding wheel body Tm Machine time constant δc Contact deflection of grinding wheel WP Workpiece δs Deflection at flange of grinding wheel b Width of workpiece δw Deflection of workpiece c Grinding method parameter η Force ratio (Fn/Ft) de Deflection of grinding machine

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