3D ground surface topography modeling considering dressing and wear effects in grinding process

International Journal of Machine Tools & Manufacture 74 (2013) 29–40

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International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool

2D/3D ground surface topography modeling considering dressing and wear effects in grinding process J.L. Jiang a, P.Q. Ge a,b,n, W.B. Bi a, L. Zhang a, D.X. Wang a, Y. Zhang a a b

School of Mechanical Engineering, Shandong University, Jinan 250061, China Key Laboratory of High-efficiency and Clean Mechanical Manufacture at Shandong University, Ministry of Education, Jinan 250061, China

art ic l e i nf o

a b s t r a c t

Article history: Received 9 May 2013 Received in revised form 8 July 2013 Accepted 11 July 2013 Available online 26 July 2013

Roughness is usually regarded as one of the most important factors to evaluate the quality of grinding process and ground surface. Many grinding parameters are affecting ground surface roughness with different extents, however, the most influential factors are wheel dressing and wear effects which were unfortunately not get seriously attention in the previous researches. On the other hand, as a most common indicator, roughness is only a statistical evaluation which is not enough to describe the topography characteristics of a surface, especially under higher demands on grinding process and functional ground surface quality. Thus in this work, a 2D and 3D ground surface topography models were established based on the microscopic interaction mechanism model between grains and workpiece in grinding contact zone. In this study, besides grinding parameters, the wheel dressing and wear effects were taken into consideration, including dressing depth, dressing lead, geometry of diamond dressing tool and wear effects of both wheel and diamond dressing tool. A dressing and wear profile line, Ldw, which will describe how the grains’ shapes are changed, was established and added into a former 2D ground surface roughness prediction model. In order to obtain a better visual effect, a 3D topography model was established which is based on the interaction situations in real grinding process. Both 2D and 3D models will predict ground surface roughness more precisely and stably than traditional models by comparing with a dressing lead single-factor experiment. Results also showed that the selection of dressing parameters and dressing tools can refer to the formed shape of Ldw by comparing with grinding depth, ae, and the dressing lead should be carefully chosen which will greatly influence ground surface topography the most. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Grinding Dressing Ground surface roughness Ground surface topography

1. Introduction As one of precision manufacturing technologies, grinding process is often used for the final finishing for its ability of achieving high surface quality to satisfy severe requirements. The parameters of grinding process are directly affecting ground surface quality with different extent. However, it is confusing that, there are many factors for example, wheel type and topography, wheel dressing and conditioning, sizes and velocities of wheel and workpiece, cooling and lubricating conditions etc., which are interdependent and even non-linearly. Meanwhile, the material removal process is actually a synthesized result by thousands of grains in grinding contact zone with different sizes, locations and geometry shapes. Thus, the stochastic nature of grain distribution must be taken into consideration in the research of grinding process. All the above n Corresponding author at: Shandong University, School of Mechanical Engineering, No. 17923, Jingshi Road, Ji’nan, Shandong Province 250061, China. Tel./fax: +86 531 88399277. E-mail address: [email protected] (P.Q. Ge).

0890-6955/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmachtools.2013.07.002

aspects prevent people from understanding and modeling the grinding process with large difficulties and obstacles. Surface roughness is one of the most crucial factors of assessing the quality of grinding process and ground surface, its modeling and prediction is always a research hotspot in the previous decades. Many methods of predicting ground surface roughness have been proposed with relatively good accuracy. These models can be divided into two categories: empirical and analytical models. The empirical models are normally developed as a function of kinematic conditions [1], and they have the advantages of simple forming, high accuracy and being successfully industrial used. However, the determination of the empirical coefficients in these models is based on large amount of experiment data which are exhausting and time consuming. Thus the empirical models cannot consider all of the affecting factors mentioned above; otherwise the experimental workload will be greatly increased. Furthermore, the empirical models have a limit on that they will be invalidated when the grinding condition is changed. Another type of models, analytical models, is based on the people’s understanding of the grinding process. Many of these

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Nomenclature undeformed chip cross-sectional area dressing depth per dressing pass grinding depth grinding width wheel diameter grain diameter total dressing depth maximum grain diameter average grain diameter minimum grain diameter maximum grain penetration depth, also undeformed chip thickness hcuz grain penetration depth hcuz,max maximum grain penetration depth l the grain location at the grinding contact length lc grinding contact length lcon real contact length of a grain lcut the location when a grain starts cutting lm moving distance of grinding wheel lslid the location when a grain starts sliding lw workpiece length LA, LA(i),LAn simulated line to describe the ground surface profile Ax,y ad ae bw de dgx dtol dmax dmean dmin hcu,max

models are very complicated and have no fixed formulas, and an analytical model usually needs a large calculation amount, so they have poor applicability in industrial practice. However, the analytical model considering most of the affecting factors and can help people more in-depth understanding of grinding process. Generally, some assumptions have to be made in an analytical model to reduce the complexity. With the development of computer technology, larger computer memory and greater computing speed will support more complex models. Yet the most critical issue is still on how to understand and describe the microscopic interacting situation between grains and workpiece material including the mechanisms of material removal and chip formation. Undeformed chip thickness is usually regarded as an important variable in the researches of grinding mechanisms. Rowe et al. [2] showed that the energy distribution in the grinding contact zone should be assumed to be triangular or square law in shape by comparing the temperature distribution predicted with the measured one. The reason is that the undeformed chip thickness is not uniform and the heat generation progressively varies between zero and the maximum chip thickness. In Gopal and Rao′s research [3], chip formation and material removal depends to a large extent on the microstructure of the grinding wheel, the quantities of motion and the geometric parameters. Soneys and Peters [4] simulated the grinding processes based on the calculation of the equivalent chip thickness. This simulation combines the chip thickness, force and surface roughness. A relationship between chip thickness and surface roughness was proposed; however, this model includes an empirical factor and does not describe the microstructure of the grinding wheel [5]. Currently, one of the most popular formulas of maximum undeformed chip thickness and grain number per unit area were proposed by Malkin [1]: " hm ¼



1=2 #1=2 4 vw ae ; Cr vs de

C¼

1 LB

ð1Þ

where C is the grain number per unit area, L and B are the lateral and longitudinal distance between two adjacent grains, respectively; r is the chip width-to-thickness ratio. However, the formulas above are

Lsd N Nlc Nt Nv nd Qw ′ Ra ttotal Vcut,xy vs vw x xp y y′comp y′drw,min Y′CL Δt δ s φ

standard line to describe ideal ground surface wheel structure number number of grains in grinding contact zone total number of grains passing through line LA number of grains per unit grinding wheel volume dressing passes in a dressing process specific remove rate roughness, Ra the time of LA changing into LAn removed workpiece material volume by grain Gx,y wheel speed work speed variable x expressing grain diameter the x-coordinate value of the intersecting point of line y¼ ymax and line y¼ g(x) protrude height of a grain compensation amount lowest point location of Ldw center line of LA time interval of two adjacent grains coefficient, equals to dmax  dmin standard deviation of normal distribution volume percentage of grains

obtained under these assumptions: (1) all the grains are spherical and share the same size; (2) the locations of all the grains follow uniform distribution and the protrusion heights are the same. These assumptions do not reflect the characters of stochastic nature and the values of L, B and r are difficult to determine. Hou [6] used probability statistics method to analyze the mechanics of the grinding process. The numbers of contacting and cutting grains are determined for a given depth of wheel indentation and the undeformed chip thickness was described by the minimum grain diameter. In Hou′s research, it is assumed that the diameter of grains is under normal distribution and using only one variable x to express both grain size and grain protrusion, which means that the biggest grain has the highest protrusion and vice versa. However, this assumption is not corresponding with the actual interacting situation very well. The grains are random located within the wheel, so the size and protrusion height of a grain are independent with each other and they should be represented by two variables. Younis and Alawi [7] developed an undeformed chip thickness model which is described by Rayleigh′s probability density function (p.d.f.) and this model has been used extensively in the following research papers: [8–11]. However, the Rayleigh′s p.d.f. is defined by only one paremeter, β, which is hard to determine and has no clear physical meanings, and this model also cannot give a clear relationship between grinding conditions and undeformed chip thickness. Jiang and Ge [12] have been developed a comprehensive model which can successfully describe the microinteracting mechanism between workpiece material and grains with different size, location and protrude height. The model is based on a new method of calculating grain number per unit grinding wheel volume and undeformed chip thickness. Their work gives a deeper insight into contacting situation in grinding zone which is hard to achieve using experimental methods. Although these models and simulation methods mentioned above can relatively well predict ground surface roughness with different extent, unfortunately, none of them can able to take the grinding wheel dressing and wear effects into consideration which actually have great influence on ground surface quality. The parameters of dressing process, such as dressing depth, dressing lead, dressing passes number and the geometry of dressing tool,

J.L. Jiang et al. / International Journal of Machine Tools & Manufacture 74 (2013) 29–40

will have great influence on the contacting situation between grains and workpiece material. In the research of Oliveira et al. [13], the combination of the dressing depth, dressing velocity and the dressing tool tip geometry determine the wheel topography characteristic. It is proved that the patterns on grinding wheel surface produced by dressing operation can be transferred to the workpiece surface during the grinding process. In the research of Brinksmeier and Cinar [14], a collision number id is defined considering dressing parameters and it is found that the active surface topography of the grinding wheel after dressing strongly depends on the collision number. Saad et al. [15] has developed an empirical model of predicting ground workpiece surface roughness incorporating single-point dressing parameters and diamond-roll dressing parameters. It is found that the coefficients have a linear relationship with the inverse of the overlap ratio for single-point dressing. However, the method was developed based on Soneys′ [4] equivalent chip thickness formula whose simple form cannot able to describe the complex chip formation process and cannot reflect the stochastic nature of grinding process. Shanawaz [19] showed that the ground surface roughness from electrolytic in-process dressing (ELID) grinding process (Ra ¼0.81 μm) was much lower than which was obtained from conventional grinding (Ra ¼1.72 μm) in their experiments of grinding of aluminum silicon carbide metal matrix composite materials. In Young′s research [20], the ground surface roughness decreases with dressing cross feed and dressing speed ratio decreasing and decreases with the dressing pass to about 3–5 and then comes to a constant value. In other research papers it was showed that, different truing and dressing conditions will achieve different cutting asperity densities [21]; finer dressings (fine truing lead, slow dressing feed and small truing/dressing depth) will produce higher densities of cutting asperities [22] and experiments in [23] showed that smaller dressing depth, dressing speed ratio and dressing cross-feed rate will produce smoother ground surface. At the same time, the wheel wear will also affect grinding process. Although the ground surface roughness will be reduced when the cutting edges are worn flat, however their ability of removing material is weakened, thus in grinding contact zone, sliding and plowing between grains and workpiece material are mainly occurred. Here much energy of heat will be generated and majoring of the heat will transferred into workpiece material and cause thermal damage and other unexpected defects. Additionally, the roughness parameter cannot fully express the characteristics of surface topography. Roughness is essentially a statistical indicator of peaks and valleys on the surface, so the surfaces with different topographies may have the same roughness values. Therefore, when higher requirements on surface quality is put forward, for example some surfaces of optical elements or precision bearing raceways, the roughness indicator is not enough for expressing the topography of the surfaces. Towards to this problem, many researchers have established 2D/3D ground surface topography models which have better visual effects. Chen and Rowe [16–18] have discussed comprehensively the impact of single-point dressing on grinding process. Simulated grinding wheel topography was established taking account of the motion of the dressing tool, grain size, grain spacing, grain fracture and break-out. Then a 2D ground surface topography was obtained and contains features which bear a resemblance to the experimental surface. However, the method of calculating undeformed chip thickness has not given, and the influences of wear effect of grinding wheel and dressing tools was not taken into consideration. In the research of Gong et al. [24], the virtual reality technology was applied to simulate ground surface, a virtual grinding wheel has been created and a 3D images were obtained and the dressing and wear effects were considered. However, in this model, only one parameter (maximal remaining

31

height) was used to describe dressing effect and several coefficients were introduced to describe grain diameter changing, grain geometrical shape, grain roughness and grain wear which value are lacking of theoretical basis. Aurich et al. [25] have developed a kinematic simulation of the grinding process model (KSIM). Rely on KSIM, a series of important terms in grinding process can be obtained such as undeformed chip thickness and cross sectional area, grinding forces and ground surface topography. KSIM also considered the dressing effects however in a quite simple way that was just simply realized by cutting down the grain protrusion height of every single grain to a specified limit (dressing height) which leads to a larger number of grains with the same maximum grain protrusion height. Actually, this approach is more like a grinding wheel wear process. In the work of this paper, the interaction mechanism between grains and workpiece material are firstly modeled and discussed in Section 2, which give a basis of the following simulations. In Section 3, a ground surface topography model considering dressing and wear effects was established which is a modified model from the author′s former research which already took all of the grinding parameters included. In Section 4, additionally, a 3D model of ground surface topography, which is more close to the grinding process, also considering dressing and wear and has better visual effects, has been developed. The 3D simulated surface was compared and in good accordance with the measured workpiece surface under the same grinding conditions. It is found that both analytically and experimentally, the value of dressing leads has a greater impact compared with the other dressing parameters.

2. Interaction mechanisms between grains and workpiece material in grinding contact zone The detailed introduction of microscopic interaction mechanism was published in the author′s former paper [12], and the following are some basic principles. In grinding contact zone, a large amount of grains with random geometry and random distributed location contact with workpiece material with different microscopic interacting types, which will affect both grain number and single grain force calculation. Generally, it is believed that a grain will experience three stages when it goes through grinding contact zone: sliding, plowing and cutting. But for different grains, the starting points of plowing and cutting stages are different for the reason that the critical conditions of plowing and cutting are related with grain sizes and penetration depths. Based on this point, it is reasonable to suppose that when a grain is at the end of grinding contact length, it may not enter the cutting stage and still in sliding or plowing stage, or even cannot contact with workpiece material in the whole contact length, so that different grains will experience different interacting situations which means that the starting points and the lengths of sliding, plowing and cutting stages are different between grains within grinding contact zone. Furthermore, these three types of grains may exist at the same location of the grinding contact zone. As shown in Fig. 1, there are nine grains in grinding contact zone. Because of the different sizes and protrusion heights, these grains contacting with workpiece material in different situation. Fig. 1 shows that: (1) only Grain No. 7 contacts with workpiece material from the beginning of the grinding contact zone and the others do not; (2) Grains No. 1 and No. 3 share the same ‘grain′s real contact length', but the lengths of sliding, plowing and cutting stages are different between the two; (3) Grain No. 4 and No. 9 are end at plowing stage and Grain No. 2 is end at sliding stage; (4) Grain No. 5–No. 8 experience three stages, but their ‘grain′s real contact lengths′ and the starting points & lengths of three

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Fig. 1. Interaction situations between grains and workpiece material.

Fig. 2. The grains distribution and grinding wheel surface topography modeling.

stages are totally different with each other; (5) at the same location l, Grain No. 4, 8 are in sliding stage, Grain No. 1, 3, 5–6, are in plowing stage, Grain No. 7 is in cutting stage and Grain No. 2, 9 cannot contact with workpiece material. In order to describe the contacting situation of grains, two important terms should be obtained first. They are the number of grains per unit grinding wheel volume (Nv), and the maximum undeformed chip thickness (hcu,max). The grain′s shape is assumed to be spherical. Under a specified particle size number of abrasive grains, the grain′s diameter is assumed to follow normal distribution, the maximum grain diameter is dmax, the minimum grain diameter is dmin, and the average grain diameter is dmean. Thus, dmean ¼(dmax+dmin)/2. The value of dmax and dmin can be determined from the nominal grain size of a particle size number. Table 1 shows the values of sieve openings and the corresponding dmax, dmin, dmean for different particle sizes [6]. Using dgx to describe the diameter of a grain and define variable x: dgx ¼ dmean þ x; x∈½δ=2; δ=2; δ ¼ dmax dmin

ð2Þ

For more precise calculation, the standard deviation, s, is set as s ¼(δ/2)/4.4. In the neighborhood of value x, the probability of grain number can be expressed as: 2 8:8 P g ðxÞ ¼ pffiffiffiffiffiffi eð1=2Þðð4:4=δ=2ÞxÞ dx δ 2π

ð3Þ

The volume fraction of the grains in unit grinding wheel volume is:
3 4 dgx π N v P g ðxÞ 2 δ=2 3

Z φ¼

δ=2

ð4Þ

The term φ, which can get from wheel structure number, N, as follows [29]: φ¼

3 ð37NÞ; % 2

ð5Þ

Thus the total number of grains per unit grinding wheel volume, Nv, can be obtained: pffiffiffiffiffiffi 3φδ 2π Nv ¼ ð6Þ R δ=2 2 3 4:4π δ=2 dgx eð1=2Þðð4:4=δ=2ÞxÞ dx The protrude height of a grain can be expressed by variable y. In this paper, it is assumed that the maximum grain diameter equals with the maximum protrusion height, dmax ¼ymax, which has been confirmed by experimental data from measuring instruments and SEM micrographs [26–28]. The maximum undeformed chip thickness, hcu,max, means the largest penetration depth of all over the grains in grinding contact zone, so within the range of y∈½dmax hcu;max ; dmax , the contacting interaction may occur between grains and workpiece material.

J.L. Jiang et al. / International Journal of Machine Tools & Manufacture 74 (2013) 29–40

33

Table 1 Sizes of sieve openings, dmax, dmin and dmean [6]. Particle size number (#)

24

30

36

46

54

60

70

80

90

100

Sieve opening (mm) dmax (mm) dmin (mm) dmean (mm)

0.762 0.762 0.589 0.850

0.589 0.589 0.476 0.676

0.476 0.476 0.354 0.532

0.354 0.354 0.291 0.415

0.291 0.291 0.255 0.323

0.255 0.255 0.211 0.273

0.211 0.211 0.178 0.233

0.178 0.178 0.152 0.194

0.152 0.152 0.142 0.165

0.142 0.142 0.114 0.128

location in grinding contact zone: ( 0 l∈½0; lc lcon Þ hcuz ¼ l h d þ y l∈½l cu;max max c lcon ; lc Þ lc

ð10Þ

The undeformed chip cross sectional area, Ax,y, can be expressed using hcuz and dgx:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2hcuz 1 2 dgx hcuz  hcuz ðdgx hcuz Þ Ax;y ¼ dgx arccos 1 2 2 dgx ð11Þ Fig. 3. Four stages while a grain passing through grinding contact length.

Now the size and the location of a grain can be expressed using two variables x, y, and using Gx,y to express a grain, Gx,y ¼G(dgx, y). Pick a thin layer from the outer surface of a grinding wheel with thickness equals to dmax, and the layer then is stretched. The grains distribution and grinding wheel surface topography are established as shown in Fig. 2: The interaction situation between a grain and workpiece material can be divided into three types: sliding, plowing and cutting. The type of interaction is related with grain penetration depth, hcuz, and grain diameter, dgx. The critical condition of plowing and cutting can be defined by two parameters, ξplow, ξcut. So a grain will be in sliding stage when hcuz oξplowdgx; or in plowing stage when ξplowdgx ohcuz oξcutdgx; or in cutting stage when hcuz 4 ξcutdgx. A grain may not contact with workpiece material from the beginning of grinding contact zone, so it will experience four stages as like Fig. 3 shows: uncontact, sliding, plowing and cutting stages. In Fig. 3, a conception of ‘grain′s real contact length' is defined: lcon which is no longer than grinding contact length, lc. For a single grain with a protrude height, y, the following relationship can be obtained: hcuz;max ¼ hcu;max ðdmax yÞ

Z V cut;xy ¼

lc

then in the real contact length of a grain, there are three stages existing: sliding, plowing and cutting. It is assumed that the penetration depth of a grain, hcuz, will change proportionally with its location, l, in grinding contact zone, thus the following relationship can be obtained: ð9Þ

Substituting Eqs. (7) and (8) into Eq. (9), the penetration depth of a grain, hcuz, is a function of the grain′s protrude height and

ð12Þ

Ax;y dl lcut

here, the establishing condition of Eq. (12) is that the cutting interaction will occur on grain Gx,y. Thus, the maximum penetration depth of grain Gx,y must be larger than critical cutting condition, hcuz,max 4ξcutdgx, which is equivalent with: y≥gðxÞ ¼ dmax hcu;max þ ξcut ðx þ dmean Þ

ð13Þ

Meanwhile, the value ranges of x, y is: (

x∈½δ=2; δ=2

ð14Þ

y∈½dmax hcu;max ; dmax 

thus, the Eqs. (13) and (14) must be satisfied simultaneously and the value ranges of x, y is shown in Fig. 4: In Fig. 4, the dashed area is the value ranges of x, y. There are two situations: hcu,max 4ξcutdmax and hcu,max o ξcutdmax exist. Thus the probability function Pcut can be expressed as: P cut ¼ ∬Ω P g ðxÞ Z

ð7Þ

Using variablep l ffiffiffiffiffiffiffiffiffi to express the location in the grinding contact zone,l∈½0; lc ; lc ¼ ae de , and it is obvious that hcuz varies with l. If the protrude height of a grain equals to the maximum value, y¼ymax ¼dmax, its real contact length equals to lc, lcon ¼lc, if the protrude height equals to the minimum value, y ¼ymin ¼dmax  hcu,max, its real contact length lcon ¼0. So the real contact length of a grain only relates with the grain's protrude height:
 dmax y ð8Þ lcon ¼ lc 1 hcu;max

hcuz;max lcon ¼ hcuz lcon lc þ l

Integrate Eq. (11) from lcut to lc, the removed workpiece material volume by grain Gx,y, Vcut,xy, can be derived as:

P g ðyÞdydx ¼

where xmax ¼

(

xmax δ=2

Z

dmax

gðxÞ

δ=2 xp ¼

2 8:8 1 pffiffiffiffiffiffi eð1=2Þð4:4=ðδ=2ÞxÞ dydx hcu;max δ 2π ð15Þ

; hcu;max 4 ξcut dmax ξcut hcu;max dmean 1

; hcu;max o ξcut dmax

The total number of grain in the grinding wheel volume with thickness equals to hcu,max per unit time and unit width is: Ntotal ¼ vs hcu;max N v

ð16Þ

Thus, the total volume removed by all cutting grains per unit time and unit width, Vcut, is: V cut ¼ N total P cut V cut;xy

ð17Þ

Obviously, Vcut shares the same meaning with specific remove rate, Qw′(Qw′¼ aevw): Q w ′ ¼ ae vw ¼ V cut

ð18Þ

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J.L. Jiang et al. / International Journal of Machine Tools & Manufacture 74 (2013) 29–40

Fig. 4. Value ranges of x, y in two situations. (a) hcu,max 4xcutdmax and (b) hcu,max o xcutdmax.

Substitute Eq. (6), (12), (15) and (17) into Eq. (18), and get the following equation: ae vw ¼ vs hcu;max Nv Z xmax Z dmax 2 8:8 1 pffiffiffiffiffiffi eð1=2Þðð4:4=ðδ=2ÞÞxÞ  V cut;xy dydx hcu;max δ=2 gðxÞ δ 2π

ð19Þ

In Eq. (19), there is only one unknown quantity, hcu,max, the value can be obtained by repeated iteration method. Based on the grain–workpiece microscopic interaction mechanism model, the contacting situation of every single grain can be expressed, and then the numbers and distributions of each type of grains can be obtained. By combining the single grain force model, the distributions of grinding forces and even grinding heat flux can be deduced, instead of the assumed rectangular, triangular shaped heat fluxes by former researches. Also since the contacting situation of each grain can be described, in the author′s former research [12], a ground surface topography model was developed and the calculation results were compared with experimental data in [8,26]. However, the dressing and wear effects were cannot take into considered, so its practical application was limited. Hence in this work, this calculation method was improved and new 2D & 3D models were proposed considering wheel dressing and wear effects.

Fig. 5. Ground surface roughness model.

can be approximately considered that these Nt grains pass through LA successively, change LA into LAn and the time interval of two adjacent grains will be Δt: Δt ¼

t total 1 ¼ vs  bw  hcu;max  Nv Nt

ð22Þ

The profile of LA can be regarded as a set of many points: 3. Establishment of 2D ground surface topography modeling considering dressing and wear effects Generally, the value of roughness is obtained by measuring at the direction perpendicular to work speed. Thus, a model can be generated as shown in Fig. 5. In Fig. 5, a coordinate system, b  y′, is established, and the origin of coordinates is at ground workpiece surface. A line LA is defined to express workpiece surface profile which is perpendicular to work speed. In grinding process, the locations of all points on LA will be changed by contact grains and line LA will become to be curve LAn after grinding. The variance of all points on curve LAn can be considered as roughness Ra. Let ttotal to express the time of LA changing into LAn: t total ¼

lc vw

ð20Þ

The total number of grains passing through LA is Nt: N t ¼ t total  vs  bw  hcu;max  N v

ð21Þ

when a grain passes by, the profile of LA will be changed. The changing situation depends on the size and location of the grain and the profile of LA before contacting with the grain. Therefore, it

LA ¼ fA1 ; A2 ; :::; Aj1 ; Aj ; Ajþ1 ; :::; An g

ð23Þ

The distance of adjacent points on LA is a quite small value: 0.5%  dmean and the profile of LA can be enough accurately expressed. The number of points on LA is n ¼Nt/(dmean  0.5%). Let LA(i) to be the profile of LA before the ith grains, Gi, passing by: LA ðiÞ ¼ fA1 ðiÞ ; A2 ðiÞ ; :::; Aj1 ðiÞ ; Aj ðiÞ ; Ajþ1 ðiÞ ; :::; An ðiÞ g; 1≤i≤n

ð24Þ

LA(i),

the y′-coordinate of some If the grain, Gi, contacts with points on LA(i) will be changed. In this case, the grain's diameter, penetration depth and its location in the coordinate system b–y′ need to be determined. 1. Grain diameter. The diameter of a grain is dgx ¼dmean+x, the value of x can be obtained by using a Matlab function ‘randn()’, which can get a random number in the range [dmin,dmax] and follows normal distribution. 2. The location of grain Gi at y′ direction and its penetration depth. In order to determine the penetration depth, a ‘standard line’, Lsd, is defined to describe the ideal ground surface. Before grinding, Lsd coincides with LA, y′sd ¼ae, and after grinding, y′sd ¼0. Lsd moves a fixed distance at the -y′ direction each time

J.L. Jiang et al. / International Journal of Machine Tools & Manufacture 74 (2013) 29–40

when a grain passes by. This fixed distance is: ae Δy′sd ¼ Nt

ð25Þ

The time when grain Gi passes is ti, ti ¼i  Δt, and its penetration depth can be obtained relative to the location of Lsd(i) at y′ direction, y′sd(i): ¼ ae y′ðiÞ sd

t total t i t total

ð26Þ

At t¼ti, the location of grain Gi in grinding contact length is: li ¼ lc

t total t i t total

ð27Þ

The protrude height of grain Gi is yi, yi ∈½dmax hcu;max ; dmax , and follows random distribution, its value can be obtained by using another Matlab function ‘rand()’. As expressed in Eq. (29), the penetration depth of grain Gi relativing to Lsd(i) is hi: hi ¼

li hcu;max dmax þ yi lc

ð28Þ

It should be noticed that the value of hi is allowed to be negative. The standard line Lsd(i) means ideal ground surface, so LA(i) may be on the upper side to Lsd(i), therefore, when a grain on the upper side Lsd(i), it can still be contacting with LA(i). Based on hi, the y′-coordinate of grain Gi is y′G(i): ðiÞ

ðiÞ y′ðiÞ G ¼ y′sd hi þ dgx =2

ð29Þ

3. The location of grain Gi at b direction. The location of Gi at b direction is bG(i) which can be obtained by using Matlab function ‘rand()’ within the range [0,bw]. From Eqs. (1)–(3), the interaction situation between Gi and LA(i) is shown in Fig. 6: From Fig. 6, when grain Gi passes, the locations of some points on LA(i) are changed and the new profile LA(i+1) is performed. For example, as shown in Fig. 6, the points Aj(i) and Aj+k(i) on LA(i) are changed to be Aj(i+1) and Aj+k(i+1) after the grain passed by. At the end, the ultimate ground surface profile, LAn, will be performed. The variance of LAn is the roughness Ra. In this model, the dressing and wear effects cannot be taken into consideration, thus the calculated ground surface roughness value are higher than experimental data [8,12]. Dressing operation will greatly change the topography of grinding wheel surface by different dressing parameters as well as wear effect. So it is necessary to analyze on how dressing and wear will

Fig. 6. Interaction situation between Gi and LA(i).

influence grinding process. !1=2 n 1 2 Ra ¼ ∑ ðy′ n ΔÞ n j ¼ 1 Aj

35

ð30Þ

In order to establish a modified model based on this method considering dressing and wear effects by developing a dressing and wear profile line, Ldw. The line Ldw will change the grain′s shape and contains all the dressing parameters and wear factors of wheel and dressing tools. Then the shape-changed grains contact with workpiece material in grinding process and the ground surface topography considering dressing and wear effects will obtained. Only single point dressing tool will be discussed and the following are some assumptions: 1. In dressing process, the dressing tool is considered to be a rigid body. The grain′s material overlapping with dressing tool will be removed completely. 2. In grinding process, the grains are considered to be a rigid body. The workpiece material overlapping with grains will be removed completely. The dressing tool′s tip is simplified to be a cone shape and its apex angle is 60–901. However, the tip is not an ideal cone shape, so it is more reasonable to be a truncated cone shape. Thus distance from the top surface to the tip point is the wear amount of the dressing tool, wd. As shown in Fig. 7(a), a dressing line (Ld) is established with only one pass dressing. The dressing direction is from left to right, ad is the dressing depth per pass, fd is the dressing lead which is the distance of the two adjacent tips, wd is the wear amount of the dressing tool, and φd is the semi-apex angle of dressing tool tip. Actually, there are several dressing passes in a dressing operation. In a dressing operation, each dressing pass has the same dressing depth, ad. As shown in Fig. 7(b), at b direction, the dressing lines of each pass are random located, and the bold line is a composite dressing line of a 3 passes dressing process. The total dressing depth is dtol, and the number of dressing passes in a dressing process is nd: dtol ¼ nd ad

ð31Þ

The wheel wear effect can also be added in the dressing line. In grinding operations, the upper layer of grinding wheel will be worn out, and the tip of the cutting edges will be flattened. If it is assumed that the top surface of the flattened cutting edges have the same height, then the wheel wear effect, using wheel wear amount ws, can be added into dressing line and the dressing and wear profile line will be developed, Ldw, as shown in Fig. 7(c). The upper surface's location of the grinding wheel will be changed by dressing process, so the compensation amount of dressing should be considered. Using term y′comp to express compensation amount and using y′drw,min to express the lowest point location of the line Ldw, so y′comp ¼y′drw,min. However, there is no mathematical formula of compensation amount for the reason that the final shape of the line Ldw will be very complex by many passes dressing and the value of y′comp can only be obtained from the computer memory when calculating. In the grinding process simulation, first put the lowest point of line Ldw at the grinding depth's location, y′drw,min ¼ae, as shown in Fig.7(d), and Ldw will change at y′ direction together with standard line Lsd. Meanwhile, Ldw will move a random distance at b direction per each grain. Thus, the grain′s shape will be changed by line Ldw first, then it will cut the workpiece material and change line LA. All the grain's y′-coordinate value should be

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Fig. 7. The establishment and usage of dressing and wear line, Ldw.

changed by minus compensation value: y′ðiÞ sd

t t ¼ ae total i y′comp t total

ð32Þ

Fig. 7(f) shows the interacting situation between a dressed grain and workpiece surface profile LA, and Fig. 8 is a comparison between calculation results based on the original methods in [12] and in this work with and without dressing and wear effects. It can be seen that the final shape of line LAn are quite different, and experimental result shows that the modified method will give more accurate results.

4. Establishment of 3D ground surface topography model considering dressing and wear effects

n equal parts, and lw is divided into m equal parts, then the 3D topography of the workpiece surface can be expressed by m  n points and their heights. In the original situation (before grinding), the y′-coordinates of m  n points are equal with ae. Δb and Δl are the distance between adjacent points at bw and lw, respectively. Δb is smaller than Δl in order to get more precise details and smaller calculation time: Δb ¼ 0:005dmean ; Δl ¼ 0:1dmean ;

n ¼ bw =Δb

m ¼ lw =Δl

ð33Þ ð34Þ

For grinding the workpiece with length lw, the distance of the grinding wheel should move at lw direction is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lm ¼ lw þ 2  ðds =2Þ2 ðds =2ae Þ2 ð35Þ So the total grinding time is:

The 3D ground surface topography model established in this section is based on the interaction situations in real grinding process. Thus, the maximum undeformed chip thickness and the standard line are not necessary at all with a relatively simple calculation process. 3D model gives a better visualized topography characters. Set lw is the length of workpiece, and in b  y′ coordinate system built in Section 2, the width bw is divided into

t total ¼ lm =vw

ð36Þ

The number of total grains within the time ttotal is: Ntotal ¼ t total  vs  bw  ae  N v

ð37Þ

It is similar with 2D model that, all the grains are assumed to pass the workpiece surface one by one, so the fixed distance Δs of

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37

Fig. 8. Comparison between two models with and without dressing and wear effects under the same grinding conditions: (a) Simulated ground surface topography without dressing and wear effects. (b) Simulated ground surface topography considering dressing and wear effects under the same grinding condition with (a), fd ¼0.08 mm/r.

Fig. 9. An example of a 3D ground surface topography formation at 1/3 grinding time.

the grinding wheel moving at lw direction is: Δs ¼ lm =N total

ð38Þ

At any time ti (i¼0∼Ntotal), the grain Gi revolves around the center of the grinding wheel and remove the contacting workpiece material, which means that the grain Gi will change the y′-coordinates of points of the workpiece surface. The method of building line Ldw is the same as in 2D model, however, the application is different. In 3D model, the last form Ldw is located at y′¼ 0 and moves according with the grinding wheel center at lw direction. Based on this method, the ultimate simulated topography of workpiece surface will be formed after all the grains passed, Fig. 9

is an example of grinding a workpiece with lw ¼15 mm, bw ¼ 2 mm at its 1/3 grinding time under the same dressing and grinding conditions in Fig. 8.

5. Experimental setup In order to validate the models developed in this work, a series of surface grinding experiments were conducted on MKL7120  6 CNC surface grinding-machine using a vitrified white alumina wheel (WA60L6V). The wheel was dressed before every grinding pass. The workpiece was hardened AISI 52100 bearing steel (HRC 62) with 80  12  12 mm (length  width  height). The grinding

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and dressing parameters are shown in Table 2. Since the dressing feed speed is the most important parameter influencing grinding process, not all of the dressing parameters were analyzed in experimental studies. Grinding experiments were carried out under the same grinding parameters in two groups with different dressing feed speed, fd ¼ 0.04 mm/r and fd ¼0.08 mm/r. Each group has the same grinding and dressing conditions and repeated five times. Surface roughness measurements were made using white light interferometer which not only can get the roughness value but also can get the 3D surface topography. A Wyko NT9300 white light interferometer was employed. Each sample was measured 3 times at different points on the surface. The measurement parameters are shown in Table 3. The measured data are compared with simulated results in Section 6. Using a cone point dressing tool with its included angle on diamond is 901 and radius on diamond is 0.127 mm. According to the method of this work, using a truncated cone shape to fit the shape of the dressing tool tip and get φd ¼ 451, wd ¼0.062 mm. The wheel wear amount ws was set to be zero because that the dressing operation were conducted after each grinding process.

6. Results and discussion Based on the methods above, Three Matlab programs were edited for 2D models with and without dressing and wear effects Table 2 Grinding and dressing parameters. Grinding and dressing parameters

Values

Wheel speed, vs Work speed, vw Grinding wheel diameter, ds Grinding depth, ae Dressing depth per pass, ad Dressing passes per workpiece, nd Dressing feed speed, fd

25 m/s 200 mm/min 280 mm 0.1 mm 0.02 mm 10 times 0.04 mm/r, 0.08 mm/r

1. The distance from the bottom to the top of dressing and wear line, Ldw, should not be larger than half of grinding depth or too small. If the distance is too large, which means too coarse dressing, too sharp cutting edges will be shaped. In this situation, the ground surface topography will be get worse and the grinding wheel will be more easily worn. On the other hand, if the distance is too small, meaning too fine dressing, the chip-containing capability of grinding wheel will be weakened and produce bad influence on grinding process. The main factor of determine the y′ bottom-to-top distance of Ldw is the dressing feed speed. So it is very important to choose dressing lead according to practical grinding parameters.

Table 3 Measurement parameters. Measurement parameters Values 2.54  1.90 mm  2.5 VSI, full resolution 0.1 nm 3.976 μm 480  640 pixels

Sampling area Objective Measurement mode Vertical resolution Pixel size Size

and 3D model, respectively. The comparisons between measured and simulated results are showed in Table 4. The measured and simulated 3D topographies were compared in Fig. 10. In Table 4, the Model 1 in “Simulated roughness” is the result of 2D model without dressing and wear effects, the two groups share the same grinding parameters so “Model 1” can get only one value. The results of “Model 2” are from the 2D model considering dressing and wear effects and results of “Model 3” are from the 3D model. It is showed that not only the measured and simulated results have similar topography characters but also the roughness values have a small error of less than 8%. Comparing with 2D model, 3D model has a relatively more accuracy and more stable predicting ability. In Fig. 10, the simulated 3D ground surface topography (isometric view in Fig. 10(b) and top view in Fig. 10(d)) are compared with measured ground surface topography (isometric view in Fig. 10(a) and top view in Fig. 10(c)). Table4 and Fig. 10 showed that both the simulated roughness value and surface topography of this work are in good accordance with experiments. From the above analytical and experimental analysis, dressing feed speed indeed will influence ground surface roughness and topography greatly. Grinding wheel wear will also affecting surface roughness greatly but it is rather difficult to analyze quantitatively in experiments because the wheel wear amount is hard to control. However, with the help of the method developed in this paper, the influence of grinding wheel wear can be analyzed theoretically. Fig. 11 shows the calculated ground surface roughness values under different grinding wheel wear amount, ws. The grinding wheel will become blunt after continuing grinding operations. The cutting edges will be worn flat, then large amount of heat will be generated at the interface between grain and workpiece material which may cause thermal damages. If the wear amount are becoming close to dressing depth per pass, then another dressing operation is needed. Additionally, the shape of dressing and wear line, Ldw, can be used to evaluate the quality of dressing and grinding operations. The following are some preliminary conclusions obtained from analytical studies:

Table 4 Comparison between experimental and analytical roughness values. Sample no.

Dressing speed

Measured roughness, Ra, μm

Simulated roughness, Ra, μm

Point 1

Point 2

Point 3

Avg.

Model 1

Model 2

Model 3

0.573 Error: (1) 71% to Group 1 (2) 50% to Group 2

0.245 Error: 11.2%

0.294 Error: 7.1%

0.325 Error: 5.4%

0.367 Error: 6.8%

1 2 3 4

Group 1: 0.04 mm/r

0.259 0.292 0.276 0.299

0.288 0.260 0.296 0.252

0.288 0.246 0.263 0.271

0.274

5 6 7 8

Group 2: 0.08 mm/r

0.344 0.328 0.361 0.379

0.334 0.322 0.327 0.336

0.315 0.341 0.384 0.344

0.343

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39

Fig. 10. Comparison between measured and simulated 3D ground surface topography.

operation, or on grinding wheel with large grain size, or be replaced by a sharper one.

7. Conclusions

Fig. 11. The influence of grinding wheel wear on ground surface roughness.

2. The wear effect of dressing tools will also influence the shape of line Ldw and then ground surface quality. If the dressing tool has a too large wear amount, a larger dressing lead has to be chosen and then may cause too coarse dressing process. In this situation, a worn dressing tool can be used on coarse dressing

A new mathematical method of predicting ground surface roughness and topography, considering most of the grinding parameters, dressing parameters and wear effects of wheel and dressing tool, has been developed. The work of this paper was based on the grain–workpiece microscopic interaction mechanism model in grinding contact zone. This model gives a more detailed description of contacting situation in grinding zone between grains and workpiece material than most of traditional methods. This mechanism model can describe how different grains with different sizes, protrusion heights in different locations in grinding zone will contact with workpiece material. The trajectory was divided into four stages: uncontact, sliding, plowing and cutting for each different grain, not just simply dividing grinding zone into three stages: sliding, plowing and cutting, traditionally. The grinding zone was also re-defined as a space area of “grinding width (bw)  grinding length (lw)  undeformed chip thickness (hcu,max)”, and gave a new method of calculating undeformed chip thickness which is a very important factor in the theoretical framework of grinding process researches. Because of its ability to describe grains′ contacting situation in grinding zone in detail, this mechanism model is reliable for

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predicting ground surface roughness and topography. Meanwhile, most of the previous research on ground surface roughness prediction could not consider wheel dressing and wear effects which are actually play very important roles and have great impact on grinding process quality, there are large limitations existing on analytical research as well as practical utilization. To overcome this problem, the work of this paper took all of the dressing parameters and wear effects into consideration by the establishment of dressing and wear line (Ldw) and a new modified predicting method was proposed. This method was validated by a dressing lead single-factor experiment. Results showed that, present method will largely reduce the calculation error from traditionally 50–70% to 5–10%, and at the same time, comparing with 2D model, the 3D model has more precise and more stable predicting ability and produces ground surface topographies in good agreement with measured images. Results also showed that, comparing with other dressing parameters, the dressing lead parameter has the largest influence on grinding process quality, larger dressing lead will produce larger roughness value and vice versa. The method proposed in this paper, which is closer to the real grinding, dressing and wear process, is more reliable and can be further applied for process designing, evaluating and optimization. The selection of dressing parameters and dressing tools can refer to the formed shape of dressing and wear line Ldw by comparing with grinding depth, ae. The grinding wheel wear amount will reduce the roughness value; however, larger wheel wear amount may lead to thermal damage or other unexpected defects. The value of dressing lead should be carefully chosen according to grinding conditions; too large or too small dressing lead will both produce bad influences. Acknowledgement The work is supported by the National Basic Research Program of China (973 Program, Grant No. 2011CB706605 and No. 2009CB724403) and the SDU Graduate Innovation Foundation (Grant No. 3136007163055). References [1] S. Malkin, Grinding Technology: Theory and Application of Machining with Abrasives, Ellis, Horwood Limited, 1989. [2] W.B. Rowe, S.C.E. Black, B. Mills, H.S. Qi, M.N. Morgan, Experimental investigation of heat transfer in grinding, CIRP Annals—Manufacturing Technology 44 (1995) 329–332. [3] A.V. Gopal, P.V. Rao, A new chip-thickness model for performance assessment of silicon carbide grinding, International Journal of Advanced Manufacturing Technology 24 (2004) 816–820. [4] R. Snoeys, J. Peters, A. Decneut, The significance of chip thickness in grinding, CIRP Annals—Manufacturing Technology 23 (1974) 227–237. [5] H.K. Tonshoff, J. Peters, I. Inasaki, T. Paul, Modelling and simulation of grinding processes, CIRP Annals—Manufacturing Technology 41 (1992) 677–688. [6] Z.B. Hou, R. Komanduri, On the mechanics of the grinding process—Part I. Stochastic nature of the grinding process, International Journal of Machine Tools and Manufacture 43 (2003) 1579–1593.

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