Analysis of grinding mechanics and improved grinding force model based on randomized grain geometric characteristics

Mingzhng LIU, Chngh LI,*, Ynin ZHANG, Min YANG, Tng GAO,Xin CUI, Xioming WANG, Wnho XU, Zongming ZHOU, Bo LIU,Zr SAID, Runz LI, Shuhm SHARMA

a School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao 266520, China

b Hanergy (Qingdao) Lubrication Technology Co., LTD, Qingdao 266520, China

c Sichuan Future Aerospace Industry LLC, Shifang 618400, China

d Department of Sustainable and Renewable Energy Engineering, University of Sharjah, Sharjah 27272, United Arab Emirates

e Department of Biomedical Engineering, University of Southern California, Los Angeles, CA 90089–1111, USA

f Department of Mechanical Engineering, I.K.Gujral Punjab Technical University, Jalandhar 144603, India

KEYWORDS Effective action area;Grinding force;Improved model;Mechanical behaviour;Randomized grain

Abstract Too high grinding force will lead to a large increase in specific grinding energy,resulting in high temperature in grinding zone, especially for the aerospace difficult cutting metal materials,seriously affecting the surface quality and accuracy.At present, the theoretical models of grinding force are mostly based on the assumption of uniform or simplified morphological characteristics of grains, which is inconsistent with the actual grains.Especially for non-engineering grinding wheel,most geometric characteristics of grains are ignored,resulting in the calculation accuracy that cannot guide practical production.Based on this,an improved grinding force model based on random grain geometric characteristics is proposed in this paper.Firstly, the surface topography model of CBN grinding wheel is established, and the effective grain determination mechanism in grinding zone is revealed.Based on the known grinding force model and mechanical behavior of interaction between grains and workpiece in different stages, the concept of grain effective action area is proposed.The variation mechanism of effective action area under the influence of grain geometric and spatial characteristics is deeply analyzed,and the calculation method under random combination of five influencing parameters is obtained.The numerical simulation is carried out to reveal the dynamic variation process of grinding force in grinding zone.In order to verify the theoretical model,the experiments of dry grinding Ti-6Al-4 V are designed.The experimental results show that under different machining parameters, the results of numerical calculation and experimental measurement are in good agreement, and the minimum error value is only 2.1 %, which indicates that the calculation accuracy of grinding force model meets the requirements and is feasible.This study will provide a theoretical basis for optimizing the wheel structure, effectively controlling the grinding force range, adjusting the grinding zone temperature and improving the workpiece machining quality in the industrial grinding process.

1.Introduction

Grinding is the most widely used high-efficiency and low-cost finishing process in the manufacturing industry.1During grinding,the interaction between the grinding wheel and workpiece generates grinding force at sliding, elastic/plastic deformation,and chip-forming stages.2–3Grinding force is an important parameter to measure grinding performance,and it has a direct impact on wheel durability, grinding specific energy, heat source temperature field, workpiece surface quality, and shape-position accuracy.4However, due to the random distribution of grain pose on the grinding wheel surface and the characteristic of grain cutting at a negative rake angle, the mechanical behavior of the grinding wheel in the process of removing the workpiece material is more complex than that in other machining methods(such as turning and milling).This phenomenon is mainly attributed to the following points.(i)The geometric characteristics(mainly tilted angle)of each grain vary irregularly, resulting in different interference processes between the grain and workpiece.(ii) The grinding wheel rotates at a high speed,and the grains(effective grain number)change rapidly and dynamically in the grinding zone during the workpiece feeding process.In addition,the grinding process is accompanied with intense friction and elastic/plastic deformation of the workpiece material.As a result, the grinding force measured through experiments cannot accurately explain the complex mechanical behavior of the material removal process,thereby hindering the disclosure of the scientific origin.Therefore, a prediction model must be established to reveal the dynamic evolution mechanism of grinding force in the process of removing workpiece materials,the grinding process must be optimized, and the structure of the machine tool and wheel needs to be improved to ensure grinding quality.This objective has always been a fundamental part of grinding theory.

The establishment and numerical analysis of grinding force prediction models have consistently been a research hotspot.Modeling methods are mainly divided into two categories:empirical and theoretical modeling.5At the early stage, due to randomized grain characteristics on the grinding wheel surface,theoretical research on grinding force has focused on the establishment of empirical models.Empirical models are result-oriented prediction models that can predict the grinding force by establishing a mathematical regression equation between the input parameters and output results on the premise of a specific machine tool,grinding wheel,and workpiece and process conditions without considering the cutting mechanism.6Grinding force models established through this method have a certain engineering value, but they are highly dependent on the specific research condition and machining technology.Consequently, their reference value is low.

In the process of theoretical modeling, important factors,such as tribology, material science, and dynamics, should be considered comprehensively, and the factors that need to be considered in modeling should be improved as much as possible from the physical sense.Malkin and Cook7divided grinding force into friction force and cutting force in accordance with the action mechanism and established a corresponding theoretical model on the basis of the influence of the contact area between the grains and workpiece after wearing.After their study, the research on grinding force theoretical models entered a new stage.Younis and Alawi8extended the grinding force model proposed by Malkin and divided the grinding process into three contact stages: friction,ploughing, and cutting.The relationship between the energy required for chip formation and cutting force was analyzed from the perspective of energy accumulation and force, and the calculation method of cutting force was improved.Tang et al.9proposed a new grinding force model from the perspective of energy distribution; the model divides cutting force into static and dynamic parts in accordance with the difference in shear strain and temperature in the grinding zone.Then,the researchers conducted grinding experiments to verify the feasibility and accuracy of the model.Basing on workpiece material characteristics, Li et al.10comprehensively considered the cooperative relationship among grain radius distribution, cutting strain rate, and plastic–brittle deformation transition in the grinding process for the first time and established a grinding force model with high prediction accuracy.Mishra and Salonitis11used multiple regression analysis to identify the dynamic behavior of grinding force ratio and proposed a modified grinding force model and a simplified new method to estimate the empirical factor.A grinding process experiment was conducted to verify the precision of the model.Zhou et al.12established a grinding force model for titanium alloy by combining the backpropagation (BP) neural network and optimization genetic algorithm(GA).Compared with the prediction results of the traditional regression equation, the grinding force model established by GA–BP showed higher prediction accuracy (error value less than 5 %).

Among numerous modeling methods, the theoretical method based on the surface topography of the grinding wheel is close to the real grinding condition.On the basis of the contact form between the grains and workpiece, the generation and action mechanism of grinding force in the grinding process have been revealed in previous studies.A mathematical model of grinding force has been established by analyzing the shape/distribution of grains, the interaction between grains and the workpiece, the mechanical properties of materials, and the material removal mechanism.Relevant studies have also pointed out that the randomness of cutting edges is positively correlated with the complexity of grain shape.13–16The distribution of grains is an important part of establishing the surface topography of the virtual grinding wheel and revealing the mechanism of grinding force.At present, two main methods based on device scanning and statistics are commonly utilized.Butlert et al.17used 3D measurement technology to observe and simulate the morphology of a real grinding wheel.Zhang et al.18employed an ultra-depth-of-field camera to photograph a grinding wheel and counted the number of grains per unit area.Liu et al.19used statistical methods to establish the basic morphological parameters of grains and two new indicators to characterize the surface morphology of the grinding wheel;then, a 3D model of grain distribution was established.Hou et al.20believed that the diameter of grains conforms to the normal distribution.Wang et al.21and Chen et al.22proposed a calculation method of average grain diameter on the basis of the rule that particle size conforms to the normal distribution.Huang et al.23established a grinding force theoretical model with two-factor variables by considering the randomness of grain protrusion height on the grinding wheel.The model can be adjusted for different grinding methods by changing the variables.Li et al.24considered the random distribution mechanism of grains and constructed a grinding force model of single grains under the action of sliding,ploughing,and cutting on the basis of the difference in cutting depth.Then, a macroscopic grinding force model was established using the discrete integral, and the dynamic grinding force predicted by the model was in good agreement with experimental results.Zhang et al.25simplified grains into cones,divided the grinding force into ploughing and cutting forces from the perspectiszve of the material removal principle of a single grain,and refined the modeling process of grinding force.Jamshidi and Budak26proposed a new grain–workpiece interference mechanism.An analytical grinding force model based on the kinematics and geometry of grains was established in consideration of the grinding wheel surface wear,and it revealed the inherent characteristics of grinding force under high specific grinding energy, low feed speed, and small cutting depth.Sun et al.27combined the processing characteristics of self-rotating grinding with the motion equation of grains and divided the material removal method into friction slip and plastic flow.A theoretical model of self-rotating grinding force was established.Wang et al.28presented a grinding force prediction model based on the material removal mechanism of random grain distribution and specific grinding energy under different undeformed chip thicknesses.Ni et al.29reported that grinding force is formed by the joint action of friction and cutting forces and established friction and cutting force models by comprehensively considering the shape of a single grain, average grinding depth, effective grain number,and contact arc length between the grain and workpiece.

However,these grinding force models still have some shortcomings, which lead to low accuracy and are mainly reflected in the following aspects.(i) The shape of grains in current grinding wheel surface topography models is mainly spherical or conic, thus ignoring the highly randomized geometric characteristics of real grains and resulting in a large deviation from actual grinding conditions.(ii) The grain geometric morphology on the grinding wheel is irregular, and most researchers considered only the protruding height and grain size without analyzing the apex angle,let alone the spatial tilted angles that considerably affect the grinding force.(iii)Most previous studies established models based on the average grain penetration depth or average chip thickness, so these models can predict only the average grinding force and cannot obtain the details of grinding force at the microscopic scale.

To improve the accuracy of known grinding force models and by taking the effective action area between grains and the workpiece as the main research content, this study improves known grinding force models by fully exploring the influence on the variation mechanism of the effective working area with randomized grain geometric characteristics.First, a surface topography model of a grinding wheel with random grains is established to reveal the decision mechanism of effective grains in the grinding zone and the micro-interaction process at different contact stages.Second, the variation mechanism of the effective action area with 3D grain rotation is revealed.Lastly,the dynamic variation law of grinding force and the distribution law of grain geometric characteristics in the grinding zone are determined.This study provides a theoretical basis for the intelligent monitoring and regulation of grinding force, optimization of the grinding wheel structure,improvement of workpiece surface quality, and reduction of energy loss in the grinding process.

2.Grinding wheel surface topography model

The machining performance of the grinding wheel is determined as abrasive, which is one of the most critical factors.Cubic boron nitride (CBN) abrasive can maintain stable cutting performance due to its merits of high hardness, strength,and thermal stability.The CBN grinding wheel has a uniform grain distribution and good thermal conductivity, which can effectively reduce the thermal damage on the workpiece surface.Therefore,the CBN grinding wheel is selected for theoretical and experimental research in this paper.

2.1.Geometrical morphology

The shape of abrasive grains plays important roles during the grinding process and profoundly affects the prediction of grinding force.Therefore, determining the shape of abrasive grains is necessary.By observing the surface morphology of an actual CBN grinding wheel with an OLS-5000 microscope,as shown in Fig.1, this study finds that the grains are mostly shaped as an irregular quadrilateral pyramid.A sufficient number of abrasive grains (more than 200 in each area) are randomly collected from six areas along the circumference of the grinding wheel for observation and statistics to show the universality of the grain shape.The grains with a quadrilateral pyramid shape in areas 1–6 account for more than 50%,with a maximum of 62.4 %.The research results show that the larger the number of cutting edges is, the shorter the sliding and plowing stages of grains are in the grinding process.This condition reduces the plastic accumulation on both sides of the cutting path and increases the material removal rate.30Therefore,by ensuring that the grinding wheel topography model is as close to the actual grinding wheel surface as possible while guaranteeing modeling feasibility, the grain shape is assumed to be a regular quadrilateral pyramid in this study for calculation and analysis convenience.

The particle size following the normal distribution directly determines the protruding height distribution, which affects the final cutting depth.31–32Therefore, according to the 6σ principle of normal distribution, the particle size D(i)of any grain i can be expressed in Eq.(1).

Fig.1 Grain shape of CBN grinding wheel and partial enlarged view of number distribution proportion with typical shape.

where Dmaxis maximum particle size, Dminis minimum particle size and randn is standard normally distributed random number.

For a regular quadrilateral pyramid grain, the cutting performance is optimal when the cutting edge located in the central longitudinal section is parallel to the forward direction of the grain.33Therefore, for analysis convenience,α(i)can be set as the apex angle in the central longitudinal section through the apex of any grain i.To obtain the distribution of α(i), grains with typical characteristics are measured with an OLS-5000 microscope, as shown in Fig.1.After data statistics, α(i)is found to range from 31.8° to 113°.Within the range, the α(i)values are randomly generated and assigned to any grain i.According to Fig.2, protruding height h(i)originalwith the original non-tilted state,outer circle diameter of the grain bottom surface d(i), and particle size D(i)exist in three situations.

The grain tilted angles(for clarity,the rotation angle will be used later), namely, θxand θy, along the x-axis and y-axis affect protrusion height h(i)original.Therefore,the final protrusion height h(i)of each grain i shown in Eq.(2).

2.2.Grain distribution

The abrasive vibration method is adopted in this study to simulate the random distribution of grain position.According to Liu et al.,34for grinding wheel 80#, the position distribution of grains in the sampling interface tends to be stable when the number of vibration times is 1000.Before vibration,assuming that the grains are uniformly distributed on the data,grain number N on the sampled grinding wheel surface can be calculated with Eq.(3).

Fig.2 Geometric characteristics of regular quadrilateral pyramidal grain.

The spatial parameters need to be set to fully display the randomness of grain tilted angle distribution.The initial values of rotation angles θx, θy, and θzare 0.After determining the grain position distribution, random increments Δθx, Δθy, and Δθzare added to θx, θy, and θz, respectively.The value range of Δθxand Δθyis set to[0,20°]due to the symmetry of grains,and the value range of Δθzis set to [0, 45°].

where Sgis organization number of grinding wheel, Lsis circumferential length of grinding zone, Wsis axial width of grinding zone, dmaxis maximum outer circle diameter of grain bottom surface and dminis minimum outer circle diameter of grain bottom surface.

After vibration, the bottom surface central coordinate of any grain i be set to (x(i), y(i), z(i)).Then, the mathematical model of CBN grinding wheel surface topography can be expressed by N × 9 matrix GNin Eq.(4).

The CBN grinding wheel surface topography based on the randomized grain geometric characteristics after the program simulation is shown in Fig.3.

2.3.Effective grains

Only the grains that interfere with the workpiece can generate heat.Given the difference in the protruding height of grains,identifying effective grains is important for improving the accuracy of the temperature field distribution model.35A thin layer whose thickness is the maximum particle size of grains is obtained from the grinding wheel surface then expanded.h(i)and h(i)mare the protruding height and final cutting depth of any grain i, respectively.

h(i)maxand h(i)mmaxare the maximum protrusion height and maximum final cut depth, respectively.When grinding wheel searches for relative zero point in actual machining process,it is required that the grain with maximum protruding height just touches workpiece surface.

When grinding depth apis given and ap≤h(i)max, as shown in Fig.4(a), on the basis of the grain with the maximum protrusion height in the grinding zone, the range of grain protrusion height that can interfere with the workpiece is shown in Eq.(5).

With the increase in ap, the number of effective grains increases.When apincreases to a certain extent (ap≥h(i)max),the number of effective grains does not increase, but not all grains are involved in the cutting process.36As shown in Fig.4(b)-I, for the grains on the same row along the wheel axial direction, taking grain with the maximum protrusion height as the datum, the range of protrusion height that interferes with the workpiece is shown in Eq.(6).

For the adjacent grains on the same row along the circumferential wheel direction, whether the latter grain interferes with the workpiece or not depends on the spacing and protrusion height difference between the former and latter grains.Therefore, the protrusion height difference of adjacent grains must be determined.Let the former grain be the reference grain i.If the protruding height of latter grain i–1 is greater than that of grain i, then grain i-1 must interfere with the workpiece.If the protruding height of grain i-1 is less than that of grain i, the critical protruding height of i–1 needs to be determined, as shown in Fig.4(b)-II.The protruding height range where grain i–1 can interfere with the workpiece is shown in Eq.(7).

Fig.3 CBN grinding wheel surface topography based on randomized grain geometric characteristics.

2.4.Determination mechanism of contact stages

The contact stages between the grain and workpiece are determined by the final cutting depth of the grain,namely,the maximum undeformed chip thickness.37–38On the premise that the grains in the contact zone are in an effective state, the contact stage of any grain i can be judged based on the maximum undeformed chip thickness h(i)mand can be calculated with Eq.(8).

where C is effective grain density, lCis circumferential distance between any two adjacent cutting points, bAis axial distance between any two adjacent cutting points, εais ratio of average chip width to thickness, 0.7–1.3,39taking average value of 1 here, vwis workpiece feed speed and vsis wheel cutting speed.

The contact stage of any grain i from cutting-in to cuttingout is determined by the relationship between h(i)mand critical chip thickness h(i)plowing_crior h(i)cutting_cri, as shown in Fig.5(a).In the actual grinding process, the radian of the contact area between the grinding wheel and workpiece is very small and negligible, so cutting length l(i)of grain i is proportional to cutting depth h(i)m.When the protruding height of the grain is the maximum value or hmax, the actual cutting length is equal to arc length lc.At the same time, the cutting depth reaches the maximum value hmmax.40During this process, the cutting time tcuttingneeded for the grain with the maximum protruding height to pass through the grinding zone is 100 μs (with deep depth ap= 30 μm as an example), and the values of tslidingand tplowingat the sliding and plowing stages are only 1.9 μs and 40.8 μs, respectively.Consequently,for any grain i, the time from cutting-in to cutting-out of the workpiece is less than 100 μs regardless of the contact stage.However, the macroscopic action time and the actual sampling frequency of a dynamometer (1 kHz, that is, the minimum sampling time interval is 1000 μs) are much higher than the action time between any single effective grain i and the workpiece.In other words, at the same contact stage (with the plowing stage as an example, as shown in Fig.5(b)),the difference values Δh(i-2), Δh(i-1), and Δh(i)of cutting depth corresponding to t(i-2), t(i-1), and t(i)at different times have negligible effects on the final grinding force.Therefore, in the modeling process of this study, the actual cutting depth corresponding to any grain i at any time can be approximated as the final cutting depth h(i)m.

Fig.4 Determination mechanism of effective grain of grinding wheel.

The critical chip thickness h(i)plowingand h(i)cuttingof any grain i can be calculated with Eq.(9).

Critical parameter ξplowingneeds to be solved further.At the sliding stage, assuming that the contact mode is tip micro spherical surface and plane,the contact stress is equal to yield stress σsof the workpiece,and the contact strain is equal to h(i)plowing when the grain is in a critical condition of plowing.41Hence, ξplowingcan be calculated with Eq.(10).The relevant parameters of the workpiece are introduced into Eq.(10) to obtain ξplowing= 5.6 × 10-5.

where ξcuttingis critical parameter of cutting stage, taking 0.02520,42and ξplowingis critical parameter of plowing stage.

where Ewis workpiece elastic modulus and νwhis workpiece Poisson’s ratio.

3.Improved grinding force model

On the basis of a known grinding force model and combined with the determination mechanism of effective grains and contact stages introduced in Chapter 2, the grinding force of any grain i can be calculated.The known grinding force models used in this study are described below.

Fig.5 Determination mechanism of contact state between any effective grain and workpiece.

3.1.Known grinding force model

(1) Sliding stage.

We can deduce from Section 2.4 that for any grain i, when cutting depth h(i)mis less than 5.6 × 10-5D(i), the material is in elastic deformation.If the cutting depth is greater than 5.6 × 10-5D(i), the material will undergo plastic deformation.Therefore, if the grinding wheel is rotating at a high speed,the workpiece material will enter the plastic deformation stage in an instant under the condition of an extremely shallow grain entry depth.The stress in the contact zone between the grains and workpiece is approximately equal to yield limit σs.41According to Hertz contact theory, normal force and tangential force can be calculated with Eq.(11).

(2) Plowing stage.

When the extrusion stress with which the moving grains act on the contact surface of the workpiece exceeds the yield stress of the material, the workpiece undergoes plastic deformation and slips along the rake face of the grains in a plastic flow mode.Some materials pile up on both sides of the gully under the action of the grains and friction.43Therefore, the plowing force is mainly generated by the grains overcoming the elastic–plastic deformation and friction of the material.The normal and tangential forces on the contact surface are shown in Eq.(11).

where τsis shearing strength, a is strain hardening exponent,depending on temperature and strain rate.45According to the actual working conditions in this paper, a is 0.4446,γ0is rake angle of grain, β is frictional angle, β = arctan μ, φ is shear angle and Sshis area of shear slip zone.

3.2.Actual contact area variation mechanism

On the premise that the workpiece material is determined, the contact area between the grains and workpiece is an important factor in known grinding force calculation models.However,due to the irregular geometric characteristics of grains, the variation rule of the contact area between the grains and workpiece is highly complex.Almost no studies are currently available on the synergistic influence mechanism of the geometric/spatial morphology of polyhedral grains (protrusion height,vertex angle, and three rotation angles) on grinding force.Therefore,the influence of the geometric/spatial characteristics of grains on the contact area variation mechanism needs to be investigated.

On the basis of the known grinding force model of any grain i at plowing and cutting stages given in Section 3.1, the contact area S(i) cand the area S(i) sh of the shear slip zone are unknown parameters for calculating grinding force.The geometric morphology of grains directly affects S(i) cand S(i)sh due to the random distribution of the grains’positionpose on the grinding wheel surface.The key parameters that determine the spatial morphology are rotation angles θx, θy,and θzof the grains around x,y,and z axes.Therefore,the variation law of grain geometry parameters in the contact area and area of shear slip must be explored.

The following premises need to be established before the analysis:

1) Defining the ‘‘datum plane”.As shown in Fig.7(a), the current machining surface O1O2O3O4of the workpiece is adopted as the datum plane.The interaction time between the grains and workpiece is extremely short, that is, less than 100 μs.Hence, we can assume that the datum plane of any grain i is located at final cutting depth h(i)mat different stages.The contact area between the grains and workpiece has no influence on either of the two.Therefore, the local coordinate system can be established with the reference plane for any grain i.Only the workpiece material below the datum plane belongs to the effective machining zone.

Fig.6 Stress condition of grain-workpiece contact zone and shear deformation zone.

Fig.7 Geometrical structure and spatial characteristics of any grain i.

2) Defining the ‘‘effective contact surface”.As shown in Fig.7(b), theoretically, in the process of material removal,when the actual contact surface O’AB/O’CB between the grain and workpiece with projections O’’A1B1/O’’C1B1on the plane(the green plane in Fig.7(b)) perpendicular to the movement direction of the grains and the projected area (SO′′A1B1SO′′A1B1and SO′′C1B1SO′′C1B1)is greater than 0,the actual contact surface O’AB/O’CB plays an effective pushing role in workpiece deformation and can be called the‘‘effective contact surface”.However, during actual processing, when the projected area of the contact surface is 0, the plastic flow of the workpiece material exerts an elastic coated effect on the grains,and the grains still produce extrusion pressure and friction force on the workpiece.47Therefore, the grain–workpiece interaction due to plastic material flow is not considered in the definition of the theoretical ‘‘effective contact surface”.

3) Defining the ‘‘effective area ratio”.In the process of material removal, the proportion of the projected area of any actual contact surface relative to the total projected area is called the‘‘effective area ratio”.As shown in Fig.7(b),with the actual contact surface O’CB in the initial state of any grain i as an example, the effective area ratio is shown in Eq.(14).

Therefore, in the cutting process of grain i, the effective contact area of O’CB to extrude workpiece material is shown in Eq.(15).

4)Defining the ‘‘rotation direction of grains”.The random distribution of grain position-pose results in a difference in the rotation direction of each grain around the coordinate axis.To facilitate an analysis,the initial state of any grain i is shown in Fig.7(b).On the basis of the symmetry of a normal quadrilateral pyramid grain,the following points are set.(i)When rotating around the z-axis, the positive direction of the y-axis is positive,and the negative direction is negative.The angle variation range is 0°＜θz≤45°.(ii) When rotating around the xaxis,the positive direction of the y-axis is positive,and the negative direction is negative.The angle variation range is 0°＜θx≤20°.(iii) When rotating around the y-axis, the positive direction of the x-axis is positive, and the negative direction is negative.The angle variation range is 0°＜θy≤20°, as shown in Fig.7(c).

3.2.1.Rotating around z-axis

As indicated in Fig.7(a), when any grain i rotates around the z-axis,the areas of actual contact surfaces O’CB and O’AB do not change, and their sizes are equal.O’CB is taken as an example for calculation, as shown in Fig.7(b), and the calculation formula of relevant side length is shown in Eq.(16).Therefore,the area Szof O’CB and O’AB is shown in Eq.(17).

3.2.2.Rotating around x-axis

When the grain rotates around the x-axis, the actual contact area on both sides has synchronous but different direction increase/decrease changes, that is, when one side increases,the other side decreases.The negative rotation of any grain i along the x-axis can be adopted as an example.When the rotation angle is θx, the grain changes from initial state O’-ABCD to O’’-A1BC1D, and the datum plane changes from ABCD to BE1DF1, as shown in Fig.8.The actual contact areas on the increasing and decreasing sides of the grain change to O’’BE1and O’’BF1, respectively.As can be seen from Fig.8,compared with the initial contact area, O’’BE1is smaller than O’AB, and O’’BF1is larger than O’BC.Therefore, the decrease/increase areas of O’AB/O’BC, which are SA1BE1SA1BE1(ΔSx_1) and SC1BF1SC1BF1(ΔSx_2), need to be solved.Then, the actual contact area between the grains and workpiece can be determined.

Fig.8 Spatial form of any grain i after rotating around the xaxis..

3.2.3.Rotating around y-axis

Different from the rotation of the grain around the x-axis,the actual contact area on both sides increases/decreases in the same direction synchronously when the grain rotates around the yaxis, that is, both sides increase or decrease simultaneously.The negative rotation of any grain i along the y-axis can be taken as an example.When the rotation angle is θy,the grain changes from initial state O’-ABCD to O’’-AB1CD1,and the datum plane changes from ABCD to AE2CF2,as shown in Fig.9.The actual contact areas between the grain lower side and the workpiece become O”AF2and O”CF2and increase at the same time.The area increments of O’AB/O’BC, namely, SAB2F2SAB2F2(ΔSy_2)and SCB2F2SCB2F2(ΔSy_1), are calculated in the same way as SC1BF1SC1BF1in Section 3.2.2.Therefore,the calculation of increment ΔSy_1is omitted in this section.

The increment ΔSy_1of the actual contact surface O’’AF2/O’’CF2when the grain rotates around the y-axis negatively is shown in Eq.(26).Similarly,the decrement ΔSy_2of the actual contact surface when the grain rotates around the y-axis positively is shown in Eq.(27).

3.2.4.Actual contact area of grain

The analysis and calculation shown in Sections 3.2.1–3.2.3 indicate that the rotation of the grain around the x-axis and y-axis can change the actual contact area, but the rotation around the z-axis has no effect.The actual contact area of grain i is shown in Table 1.

Fig.9 Spatial form of any grain i after rotating around y-axis.

3.3.Effective contact area variation mechanism

3.3.1.Rotating around z-axis

When the grain rotates around the z-axis, although the actual contact areas of SO’CB/SO’ABdo not change, the projected areas change.The variation law of the projected area when grain i rotates negatively is shown in Fig.10.As the rotation angle of grain i increases,the projected area of O’CB increases,whereas that of O’AB decreases.When θz=- 45°, the actual contact surface becomes O’CB/O’CD/O’AB,and the projected area of O’CD and O’AB is 0.

For O’CB/O’AB, the projected lengths of edge BC/BA are shown in Eq.(28).Therefore,the projected areas of O’CB and O’AB are shown in Eq.(29).After grain rotates θz, the total projected area is shown in Eq.(30).Since BC=BA,the effective area ratio of O’CB and O’AB is shown in Eq.(31).

According to Eq.(31), effective area rate ηe_zis a function of rotation angle θz.In the rotation angle range of 0°–45°,the effective area rate of the increasing side is 0.5–1,and the effective area rate of the decreasing side is 0–0.5 regardless of whether the grain rotates positively or negatively.

3.3.2.Rotating around x-axis

When the grain rotates around the x-axis, the actual contact surface and its corresponding projected area change synchronously.The variation law of the projected area when grain i rotates θxnegatively is shown in Fig.11.As the rotation angle of grain i increases,SO′′OF1SO′′OF1increases and SO′′OE1SO′′OE1decreases.

For the actual contact surfaces O’’BE1and O’’BF1,the projections of edges are BE1/BF1and OE1/OF1, respectively.The lengths of OE1and OF1are solved in Section 3.2.2.The projected areas of O’’BE1and O’’BF1are shown in Eq.(32).After the grain rotates θx, the total projected area is shown in Eq.(33).Therefore,the effective area ratio of O’’BE1and O’’BF1is shown in Eq.(34).

Table 1 Actual contact area of any grain i after rotating around z-axis.

Fig.10 Spatial form of any grain i after rotating around the z-axis.

Similarly, when grain i rotates θxpositively, SO′′OF1decreases and SO′′OE1increases,corresponding to the same calculation method of the effective area rate.Eq.(34) indicates that effective area rate ηe_xis a function of rotation angle θxand apex angle α.In the rotation angle range of 0°–20° and apex angle range of 30°–100°, the effective area rate of the increasing side is 0.5–0.58, and the effective area rate of the decreasing side is 0.45–0.5 regardless of whether the grain rotates positively or negatively.Therefore, when the actual contact surface is fixed, the rotation of grain around x-axis has only a small effect on effective area ratio.

3.3.3.Rotating around y-axis

When the grain rotates around the y-axis, the actual contact surface and its corresponding projected area change synchronously.The variation law of the projected area when grain i rotates θynegatively is shown in Fig.12.As the rotation angle of grain i increases,SO’’OAand SO’’OCdecrease synchronously.

Fig.11 Projection area variation of grain i with negative rotation θx.

For the actual contact surfaces O’’AF2and O’’CF2,the projections of edges are AF2/CF2and OA/OC, respectively.The lengths of OA and OC are unaffected by rotation angle θy.The projected areas of O’’AF2and O’’CF2are shown in Eq.(35).

Similarly, when grain i rotates θypositively, the effective area ratio of SO’’OAand SO’’OCis 0.5.Therefore, when the actual contact surface is fixed,the rotation of the grain around the x-axis does not affect the effective area ratio.

3.3.4.Effective contact area of grain

The analysis and calculation in Sections 3.3.1–3.3.3 indicate that in a certain angle range, the grain rotation around the x-axis only has a small effect, the rotation around the y-axis has no effect, and the rotation around the z-axis plays a decisive role in the effective area rate.Therefore, to facilitate an analysis and simplify the calculation, the effective area ratio of the grain can be set to ηe_z.The effect of any grain i rotation around the coordinate axis on the effective contact area is shown in Table 2.

Given the close confined space in the grinding wheel–workpiece contact zone and the random distribution of micro-fine grains, existing observation equipment cannot accurately capture the dynamic evolution process of chip morphology when grains remove workpiece materials.To verify the effect of rotation angle θzaround the z-axis on the effective contact area,the material removal process of a single grain is simulated with the finite element simulation software, which is the most suitable software for nonlinear cutting processes with large deformation.The chip morphology and stress variation law of the deformation extrusion area are observed.High strain, high strain rate, and high temperature often occur in the abrasive cutting process.Therefore, the used constitutive equation and its parameters must be determined before simulation.Compared with other models, the Johnson–Cook model considers the thermal softening effect, strain strengthening effect,and strain rate hardening; it is suitable for describing the stress–strain relationship of metal materials at large strain rates.Therefore, the Johnson–Cook model expressed in Eq.(38) is adopted as the constitutive model of Ti-6Al-4 V, and the relevant parameters are shown in Table 3.

Fig.12 Projection area variation of grain i with negative rotation θy.

Table 2 Effective contact area of any grain i after three-dimensional rotation.

Table 3 Johnson-Cook parameters for Ti-6Al-4V47.

Cutting speed vsfeeding speed vw, and cutting depth apare set to 30 m∙s-1, 6 mm∙s-1, and 30 μm, respectively.As shown in Fig.13, with the positive rotation of the grain as an example, different rotation angles have different effects on the chip morphology and stress in the deformation zone.When θz= 0°, the two contact surfaces of the grain–workpiece correspond to their respective chips,the chip morphology is basically the same, and the stress values in the deformation zone are close to one another.At this time, the effective area ratio of the two contact surfaces is the same, namely, 0.5.The chip morphology and stress in the extrusion zone change after grain rotation.On the side where the effective area increases(Region I),the chip rotates synchronously with the grain along the contact surface, and the chip shape gradually changes from a ribbon to a nodule.According to Eq.(31), the effective area rate is a function of rotation angle θzonly and increases synchronously; the extrusion effect of the grain on the material is enhanced, resulting in a progressive increase in stress in the deformation zone.At the same motion displacement, the interlaminar slip of the chip shear zone increases, and the nodal structure is formed rapidly.Chip fracture occurs when the shear stress caused by deformation is higher than the shear limit.On the side where the effective area decreases (Region II), the chip also rotates synchronously with the grain, but the volume decreases.Fig.13 indicates that when θz= 0°–1 0°, the separation gap between the chip and workpiece gradually decreases.When θz= 15°– 45°, the chip is not generated;plastic accumulation occurs, and the accumulation height decreases step by step.This is because a reduction in the effective area leads to the weakening of the extrusion effect of the corresponding grain on the material, and the stress in the deformation zone decreases step by step.When the effective area decreases to a certain critical value,the extrusion pressure on the contact surface between the grain and workpiece fails to separate the material from the workpiece matrix, and only plastic flow occurs to form accumulation.The figure also indicates that the critical value occurs between 10°and 15°.Therefore, on the premise of ensuring computational efficiency and accuracy,the rotation angle interval Δθzis reduced to 0.5°,and the critical rotation angle θz_cri=13.5°is obtained by stepwise simulation.

3.4.Grain rotation on shear slip zone area variation mechanism

When the grain is in the initial state,the plane B1B2B3B4where the actual contact surface is located is defined as the rake face.The angle γ0between the surface of any point P on cutting edge OB (parallel to the base plane O’EF) and rake face B1B2B3B4is the rake angle, namely, the angle between O’K and O’O, as shown in Fig.14.When the grain rotates θx, θy,and θzaround the x,y and z axes, respectively, rake face B1B2B3B4rotates synchronously with base plane O’EF, so the size of rake angle γ0does not change with the rotation angle.The formula for calculating the rake angle of any actual contact surface of grain i is shown in Eq.(39).

Through the simulation and theoretical analysis, we find that the grain rotation around the coordinate axis can change the effective contact area and affect the grain–chip interaction force.Given that the resultant force on the grain–chip contact surface is equal to the resultant force on the shear surface of the slip zone, grain rotation also affects the effective area of the shear zone.When the grain rotates,the actual contact surface and the corresponding shear slip surface rotate synchronously.Assuming that the width of the shear slip surface is equal to the width of the contact surface, we can assume that the effective area ratio of the two surfaces is the same.The effect law of any grain i rotation around the coordinate axis on the effective area of the shear slip zone is shown in Table 4.

Fig.13 Effects of different rotation angles of grain i around z-axis on chip morphology and stress in deformation zone.

Fig.14 Rake angle of any grain i after three-dimensional rotation.

3.5.Grinding force based on grain geometric characteristics

According to the analysis in Sections 3.3 and 3.4,the 3D rotation of any grain i directly affects the final contact form between the grain and workpiece, resulting in a difference in grinding force in the process of material removal.The corresponding grinding force after any grain i rotates θx, θy, and θzaround the x,y, and z axes is shown in Table 5.

In the cutting process of any grain i,the grinding force distribution in unrotated state at plowing and chip-forming stages is shown in Fig.15.The normal and tangential forces of grain i at different stages are shown in Eq.(41).

After three dimensions rotation, the normal and tangential forces at different stages are shown in Eq.(41).

Table 4 Effective shear zone area of any grain i after three-dimensional rotation.

Notably, for grain i at the chip-forming stage, the grinding force includes sliding force, plowing force, and cutting force.When calculating the sliding/plowing force, the cutting depth of the grain can be regarded as the corresponding critical chip thickness, namely, h(i)plowing_cri= ξplowing∙D(i)and h(i)cut-ting_cri= ξcutting∙D(i).In summary, the normal and tangential forces generated by all effective grains in the grinding zone are shown in Eq.(42).

where j is cyclic variable of grain number in cutting stage, k is cyclic variable of grain number in plowing stage and m is cyclic variable of grain number in sliding stage.

4.Numerical analysis of grinding force

To comprehensively analyze the dynamic grinding force evolution law of the grinding zone in the material removal process,dry grinding is adopted as the research condition.The simulation flow of dynamic grinding force in the grinding zone on the basis of the random distribution of the grain position-pose is shown in Fig.16.The simulation includes five main steps: (i)modeling of the grinding wheel surface topography;(ii)obtainment of the key geometric characteristic parameters of grains;(iii) determination of the microscopic contact state between any grain i and the workpiece; (iv) calculation of the normal/-tangential force generated by grain i at the current stage on the basis of the rotation angle of the grain around x,y,and z axes;and(v)superposition the normal/tangential force generated by all grains based on the effective grain number at differentstages in the grinding zone.Before the simulation, some assumptions are determined.(i)The wear of grains in the process of material removal can be ignored.(ii) The influence of temperature on material physical characteristics during grinding is not considered.(iii) The machining process belongs to shallow grinding because the cutting depth is small.The grinding zone can also be approximated as an inclined plane.(iv)Workpiece materials are removed in the optimal cutting state of all grains.

Table 5 Normal force in plowing stage and resultant force in cutting stage after three-dimensional rotation of any grain i.

Fig.15 Grinding force component of any initial state grain i in plowing/cutting stage.

4.1.Influence of geometric characteristics on grinding force of single grain

By substituting the parameters into Eq.(11), the range of the cutting force generated by any grain i is determined to be 0.0024–0.0035 N.When combined with the determination mechanism of effective grains on the grinding wheel surface in Section 2.4,the cutting force generated in the grinding zone during the sliding stage is found to be only 3.28–5.18 N,which is much smaller than the force generated in the whole grinding process.Therefore, the grinding force of grains at the sliding stage can be ignored.The grinding force obtained by measuring equipment in grinding experiments can be regarded as a comprehensive macroscopic embodiment of the interaction between hundreds of random grains and the workpiece.However,this method cannot directly reveal the effect of grain morphology on the grinding force produced by a single grain in the micro state.In this study, the grinding force model based on the position-pose randomization of grains can reveal in detail the influence mechanism of differential geometric characteristics of a single grain on the variation trend of grinding force.According to the grinding wheel surface topography in Section 2.3 and the analysis in Sections 3.2–3.4, the geometric parameters affecting grain morphology are apex angle α and cutting depth hm, and the spatial parameters are 3D rotation angles θx,θy,and θz.The key parameter(grain–workpiece contact area)that determines the grinding force of any grain i is a function of geometric parameters under the premise that the workpiece material properties are determined.Therefore, the influence of geometric characteristics on the grinding force of a single grain must be thoroughly analyzed.

Fig.16 Simulation flow of grinding force based on random distribution of grains on wheel surface.

Assume that any grain i is at the chip-forming stage and that particle size D(i)is 200 μm.The method of controlling variables is used for analysis to clearly show the influence rules of the geometric parameters.Referring to the geometric structure of grains in Fig.2, when the particle size is fixed, apex angle α and cutting depth hmshow a law of synchronous and reverse direction variation.When α increases, hmdecreases and vice versa.As shown in Fig.17(a), in the initial state(θx, θy, and θzare all 0), for a group of randomly generated grains in the grinding zone, the ranges of α and hmare [30°,110°] and [0.8 μm, 110 μm], respectively.With the change in α and hm,the normal/tangential force of the grain increases initially and then decreases.The maximum values of plowing,cutting, and total normal forces are reached when α(hm) is 49.1° (12.9 μm), 62.8° (10.5 μm), and 53.9° (12.1 μm), respectively.This is because the effective action area (effective contact area Seand effective shear slip area Ssh) is positively correlated with α and negatively correlated with hm, leading to an inflection point in the variation of grinding force.Although α and hminfluence the effective action area, the degree of influence differs.Before the critical value, the increment in effective action area caused by the increase in α is larger than the decrement caused by the decrease in hm;that is,α plays a dominant role, so the grinding force increases.However, after the critical value is exceeded, hmplays a dominant role and causes the grinding force to continue to decrease.

Fig.17 Effect of geometric characteristics on grinding force of single grain.

With regard to the large number of position-pose randomized grains in the grinding zone,their geometric characteristics are different.Although geometric parameters (α, hm) and spatial parameters(θx,θy,θz)alone have a significant influence on the variation trend of normal/tangential force, after all the parameters are randomly assigned to grains,the grinding force generated by any single grain shows an irregular distribution due to the different action intensities.With sampling area S in Fig.3 as an example,the positions of grains at the different stages and their corresponding normal force (resultant force)are shown in Fig.18.The position of grains in the sampling area is randomly distributed.According to the micromechanism of the grain–workpiece in Section 2.4,only 5 grains are at the sliding stage (blue point), only 11 grains are at the plowing stage (green point), and only 84 grains are at the cutting stage (pink point).With the grains at the cutting stage as an example, the figure indicates that the normal forces of the different grains exhibit great differences and a random distribution due to the random distribution of the geometric parameters and synergistic effects, resulting in considerably different influence intensities on the effective area action.The maximum value of normal force is 0.11 N,and the minimum value is only 0.0023 N.

4.2.Dynamic grinding force evolution mechanism in grinding zone

With the mathematical model of grinding force proposed in Chapter 3,the grinding force distribution curve of the grinding wheel throughout the whole workpiece surface is obtained by numerical calculation.The normal/tangential forces show a trend of high-speed fluctuation with time within a range, as shown in Fig.19.Fig.19 also reveals in detail the dynamic grinding force evolution mechanism in the grinding zone.To express the described objects clearly, the grinding wheel and workpiece are divided into sectors on the basis of arc length lcof the grinding zone,where the sector number n of the workpiece is the ratio of workpiece length L to arc length lc.In the process of the workpiece passing through the high-speed rotating wheel, at any time, the grinding zone can be viewed as the overlapping part between a certain sector in the workpiece and the corresponding sector of the grinding wheel.The red part of the grinding wheel–workpiece contact is shown in Fig.19.In the following part, Sector 3 of the workpiece surface and its normal force Fnare adopted as an example for analysis.Given that grinding wheel speed vsis much higher than workpiece feed speed vw, when the workpiece advances along the feed direction by the length lcof Sector 3, the grinding wheel has already rotated n’sectors, that is, n’grinding zones.For the grinding wheel surface,the position-pose distribution of grains in each sector is not the same, leading to great differences in the effective grain number and grain geometry parameters that affect the grinding force.When the grinding wheel and workpiece move at a relatively high speed, the grinding zone experiences n’times of dynamic changes in the grain’s morphological characteristic within time Tintervalof workpiece moving lc.This phenomenon directly results in different n’normal forces generated.For example, under the condition of vs= 30 m∙s-1, vw= 6 mm∙s-1, and ap= 30 μm, the range of n’normal forces in time Tintervalis [168.5 N, 189.4 N].

Fig.18 Grinding force distribution of random grains in sampling area of grinding wheel surface.

The sector number n’of grinding wheel circumference is much larger than the sector number n of the workpiece within a machining stroke L(time taken T)because the circumference of the grinding wheel is much larger than workpiece length L and grinding zone length lc.Therefore,a large amount of data is generated by numerical calculation, and a high-density grinding force fluctuation line is formed.

4.3.Distribution of grain geometric characteristic in grinding zone

To further explore the association rule between the dynamic evolution of grinding force and the grain characteristic distribution, in combination with the analysis results in Sections 4.1 and 4.2,normal force is used as an example of the grinding force generated by the grinding wheel on the whole workpiece surface, and the data within L’spacing are selected for a detailed analysis, as shown in Fig.20.After enlarging and refining the sampling zone, the average value of normal force Fn_aveis adopted as the boundary, and the sampling zone is divided into two regions: Region I (Fn> Fn_ave, red dot)and Region II (Fn< Fn_ave, blue dot) (Fig.20).The sampling zone has 500 data points,namely, 264 data points in Region I and 236 data points in Region II.As can be seen in Fig.20,the amplitude of numerical variation between any adjacent sampling points is irregular; it may change only in Region I or Region II with a small variation range, or it may exhibit cross-regional variation with a large variation range.In addition, multiple continuous data points change in a region.This observation indicates that with the high-speed movement of the grinding wheel and workpiece, the effective grain number and the grain morphological characteristics in the grinding zone synchronously change, and no obvious regularity is observed.However, for the data points distributed in Region I/II, the normal force value in Region I is higher than that in Region I.Therefore,in combination with the analysis results in Section 4.1, the effective grain number and the grain geometry parameters corresponding to the data points in Region I/II are bound to exhibit a certain distribution trend.

Many studies have shown that grinding force is generated by the interaction between grains and the workpiece,and the number of grains on the grinding wheel surface has a direct impact on grinding force.43,49–51In the process of material removal,the effective grain number in the grinding zone has an actual effect on grinding force.Fig.21 shows the distribution law of effective grain number Necorresponding to all normal forces in Regions I and II.The range of effective grain number in Regions I and II is [1746, 2008] and[1596, 1858], respectively, and the distribution in their respective regions is still random.Given the fact that the normal force in Region I is higher than that in Region II, the effective grain number in Region I is generally higher than that in Region II.However,the two regions are not completely independent,and an overlapping region exists;the range is[1746,1858],and the fluctuation range is 6.1%.The overlapping region has 148 groups of effective grains in the part of Region II, which is higher than that for Region I.This result indicates that the effective grain number is not the determining factor of the final grinding force;instead,the geometric parameters play the key role.

To further explore the distribution law of the geometric parameters, numerical points A (corresponding to normal force 191.2 N) and B (corresponding to normal force 176.8 N) are randomly selected from the respective average levels of Ne-I_ave(1877) in Region I and Ne-II_ave(1727) in Region II from Fig.22.Given the large number of grains corresponding to A and B, 300 grains are randomly selected for analysis from the grinding zone corresponding to A and B to improve the resolution of the final image and ensure the universality/validity of the data distribution.

Fig.19 Dynamic grinding force evolution mechanism in grinding zone.

The distribution of grain apex angle α is shown in Fig.22(a).The distribution range of red dots(No.300)corresponding to A is [41.6°, 78.2°], and the distribution range of blue dots(No.175) and green dots (No.125) corresponding to B is[30.7°, 57.9°] and [66.9°, 108.1°], respectively.Given that the normal force of A is higher than that of B, Fig.22(a) shows that the distribution law of α corresponding to A and B is basically consistent with the variation trend of the normal force of a single grain with apex angle in Fig.17(a).In other words,the α-I of A is mostly concentrated around the inflection point,and the α-II of B is mostly concentrated on both sides.α-I and α-II respectively corresponding to the average value of α-I_ave= 59.9° and α-II_ave= 44.3°/87.5° can also verify the above-mentioned rules.Notably, the error between α-I_aveand critical value α_criin Fig.17(a) is only 10 %.Similarly,the distribution ranges of α-I and α-II are not independent,and overlapping regions exist between them, totaling to 168 groups.This result indicates that although the normal force of A is higher than that of B,not all of the single grains corresponding to A have higher normal force than that of B.The final grinding force in the grinding zone is still affected by other factors.

The distributions of spatial parameters θx,θy,and θzof the grains are shown in Fig.22(b)–(d).Similarly, the distribution law of rotation angles θx, θy, and θzcorresponding to A and B are basically consistent with the variation trend in Fig.17(b)–(d).

Different from apex angle α, the normal force of a single grain in Fig.17(b)–(d) shows a monotonically varying trend with the rotation angle.Therefore, the distributions of θx, θyand θzcorresponding to A and B are roughly separated along the average level (θx_ave= 10°, θy_ave= 10°, and θz_-ave= 22.5°).However, the grinding force is affected by many factors.Therefore, the distributions of θx, θyand θzalso have overlapping regions ([8.2°, 12.7°], [6.3°, 13.2°], [16.8°, 28.2°]),with 132, 155, and 148 groups, respectively.Notably, due to the critical angle (13.5°) that causes grinding force mutation when the grains rotate around the z-axis, a certain number of red dots exists in the range of 0° to 13.5°, as shown in Fig.22(d).

Fig.20 Distribution of normal grinding force.

5.Experimental verification

5.1.Experimental materials and equipment

Grinding experiments are carried out to verify the proposed grinding force model.Titanium alloy is often used to manufacture the key material of important parts in aeroengines.Therefore,the workpiece material selected for the experiments is Ti-6Al-4 V, and the geometric size is 30 mm × 30 mm × 10 mm.The physical characteristics are shown in Table 6–7.Before the experiments, the workpiece surface is polished to avoid the influence of previous scratches until the surface roughness Ra of the workpiece reaches 0.8 μm.The grinding wheel adopts the CBN grinding wheel,and the parameters are shown in Table 6.

The K-P36 numerical control surface grinder produced by SCHLEIFRING (Germany) is used for the grinding experiments.The workpiece is mounted on the fixture of the YDM-III99 three-way dynamometer, and the dynamometer is fixed on the magnetic workbench of the grinder, as shown in Fig.23.Given that the proposed modeling method aims to describe the detailed information of grinding forces, the sampling rate should be as high as possible.The sampling frequency of the grinding force dynamometer is 1 kHz, and the signal is inputted into the‘‘Grinding Force Dynamic Test System”software on a PC through a charge amplifier(YE5850D)and A/D data acquisition card (USB-6001).Although the dynamometer and acquisition card are equipped with hardware filters, the raw force signal still contains much noise.Therefore,in this study,the filter parameters are adjusted several times until the force signal fluctuation range in the nongrinding period is within 1 N.

Fig.21 Distribution of effective grain number in grinding zone corresponding to grinding force in sampling zone.

The grinding parameters used in the experiment are shown in Table 8.

5.2.Model verification

Many studies have pointed out that grinding parameters are the key influencing factors that determine grinding force.54–56In this study,the grinding forces obtained by numerical calculation and experimental measurement are compared and analyzed under different machining parameters.To compare the detailed information of grinding force,the representation form must be determined.Grinding force fluctuates in a certain range under given machining conditions.Therefore, the upper limit value,lower limit value,and average value of the data are obtained by extracting the numerical simulation/experimental measurement value for analysis.According to the numerical analysis in Chapter 4, the upper limit indicates that the synergistic effect between the morphological characteristics of grains in the grinding zone achieves the strongest effect during the material removal process, that is, the distribution of effective grain number Ne, geometric parameters (α, hm), and spatial parameters (θx, θy, θz) makes the grinding force reach the highest level.In addition, according to ‘‘upper envelope”theory proposed in previous literature,24,57the grinding force distributed within a certain range of the upper limit indicates that the grains in the grinding zone are at the chip-forming stage and exhibit a strong interaction with the workpiece.Similarly,the lower limit indicates that a large part of the grains is at the plowing stage, or the interference strength between the grains playing a cutting role and the workpiece is weak.The average value indicates that the effect of grain morphological characteristics on grinding force is at an average level, which can reflect the overall grinding force distribution in the grinding zone.

Fig.24 shows the distribution of normal/tangential forces obtained by numerical calculation and experimental measurement during a randomly selected period of 0.5 s under different cutting depths.Under the conditions of 10,20,and 30 μm,for the value domain between the upper and lower limits and even the average values,the measured normal/tangential forces with a large fluctuation range are larger than those from the numerical calculation.This result may be due to the external influencing factors, such as spindle vibration, reduction of cutting performance caused by grain wear, and material plasticity enhancement caused by high temperature, in the actual experimental measurement.With the increase in cutting depth,normal/tangential force shows a significant trend of increasing synchronously.Combined with the effective grain determination mechanism in Section 2.4, the smaller the cutting depth is, the fewer the effective grains are in the grinding zone.In addition, many grains are at the sliding and plowing stages only, whereas few grains are at the cutting stage.This condition results in low interference between the grains and workpiece, so the grinding force is at a relatively low level.With the increase in cutting depth, the proportion of the sliding/-plowing grains decreases, and the proportion of the cutting grains increases greatly.Moreover,the maximum undeformed chip thickness of a single grain becomes large,which strengthens the interaction between the grain and workpiece.According to the analysis of the effective contact area in Sections 3.2–3.5, as the penetration depth of the grains increases, the contact area between the grains and workpiece gradually increases.Thus, the grinding force is significantly increased.The geometric/spatial distribution of the grains is closely related to the variation trend of grinding force.Under certain physical characteristics of the workpiece material,the increase in grinding force means that the contact area between the effective grain group and workpiece increases.Combined with the analysis in Chapter 4, the distributions of apex angle α,protrusion height h, and rotating angles θx, θyand θzaround the three axes have the tendency to increase the contact area.The effective number of grains in the grinding zone also shows an increasing trend due to the synergistic coupling effect of various factors.

Fig.22 The apex angle and three-dimensional rotation angle distribution of sampled grains in grinding zone corresponding to grinding forces A and B.

Table 6 Parameters of Ti-6Al-4 V.52.

However, when the particle size number of the grinding wheel is determined, the protrusion height of grains is limited by the particle diameter.When the grinding depth exceeds a certain critical value, the proportion of grains in the grinding zone at the stages of sliding, plowing, and cutting may be maintained at a relatively constant state.As a result, the variation amplitude of grinding force will be greatly reduced and eventually remain stable even if the cutting depth is further increased.

To quantify the grinding force obtained through numerical calculation and experimental measurement, the upper limit value, lower limit value, and average value of grinding force in Fig.24 are statistically analyzed, as shown in Table 9.According to the comparison of the 18 groups of data, the errors of normal/tangential forces obtained by the simulation and experiment are all less than 10 %.In addition, under different cutting depths, the error values of the upper and lower limits corresponding to normal/tangential forces are generally higher than the average error due to the random shape of grains, leading to the accidental distribution of upper and lower limit values.The average error of all normal and tangential forces is 5.1 % and 1.8 %, respectively.The results of thequantitative comparison prove the rationality and accuracy of the dynamic grinding force model based on the microscopic contact state between random grains and the workpiece.

Table 7 Parameters of CBN grinding wheel.53.

To further analyze the detailed information between the numerical calculation and experimental measurement values,the normal/tangential force data corresponding to the cutting depth of 30 μm and time period of 0.5–0.55 s are extracted based on Fig.24, and the distribution curve is drawn.As can be seen from Fig.25, although the value domain between the upper and lower limits of the experimentally measured normal/tangential forces is larger than that calculated numerically, a large difference is observed between the simulated and experimental values at the sampling points.In the grinding experiments, not all of the normal/tangential forces obtained in each sampling point are higher than the numerical values.The grinding force variation trend obtained by the experiment is different from that obtained by the numerical calculation due to the inevitable difference between the real grinding wheel surface and the simulated morphology.Within the sampling range, the proportion of the sampling point number with the experimental value lower than the simulation value (red solid points) in normal and tangential forces is 48 % and 46 %,respectively.In addition, some sampling points have an error greater than 10%between the experimental value and simulation value, and they account for 12 % (maximum error of 13.5 %) and 10 % (maximum error of 15.5 %), respectively.This result is obtained because the sampling points are all highly localized micro-scale values and largely affected by the real/simulated topography.In fact, a mathematical model cannot fully predict reality.However, the model considers the randomization of grain morphological characteristics and the interaction of different contact stages, thus providing a new method for modeling grinding forces with detailed information at the microscopic scale.

Fig.26 shows the influence of workpiece feed speed vwand wheel speed vson grinding force.With the increase in vsthe normal and tangential forces increase to a certain extent, but the increment is small, namely, only 4.4 % and 6.3 %, respectively, as shown in Fig.26(a) and 26(b).Considering the dynamic grinding force evolution mechanism in Section 4.2,this result may be due to the fact that vwis much smaller than vsUnder the condition that cutting depth apis unchanged,although vwincreases, the number of grains passing through the grinding zone increases only a little per unit time, and the proportion of plowing/cutting force is relatively stable.Therefore, on the premise of ensuring that the cutting force will not be greatly increased, the workpiece feeding speed can be appropriately increased to achieve a high material removal rate.In addition,the errors of the corresponding simulated/experimental values corresponding to the upper limit,lower limit, and average values are maintained at a relatively low level.

Fig.23 Experimental material and equipment.

Table 8 Grinding parameter.

The influence of vson grinding force is different from that of vw.With the increase in vsthe normal/tangential force shows an obvious downward trend,as shown in Figs.26(c)and 26(d).When apand vware certain,because vsis much larger than vw,increasing vsmeans increasing the number of grains passing through the grinding zone in unit time.As a result, the maximum undeformed chip thickness (hm) is reduced, which decreases the cutting depth of a single grain.58The chip is thinner than before,and the chip cross-sectional area is reduced,so the grinding force is also reduced.Similarly, under the condition of different grinding wheel speeds, the distribution law of the simulation/experiment value is consistent with the law revealed above, and the maximum error value is only 5.1 %.

5.3.Model Comparison

Fig.24 Distribution of experimental and theoretical value of grinding force under different grinding depths.

Table 9 Comparison of experimental and theoretical values of grinding force under different grinding depths.

Fig.25 Fluctuated curve of grinding force gained by experimental measurement and numerical calculation in sample area.

Fig.26 Comparison of experimental and theoretical values of grinding force.

To comprehensively verify the accuracy of the proposed dynamic grinding force model based on the morphological characteristics of randomized grains, grinding force models with similar modeling methods presented in previous studies are selected for a comparative analysis.The principles of model selection are as follows: (i) the grinding force modeling is based on the grinding wheel surface topography; (ii) the parameters used in the modeling process are all known data and not obtained through experimental measurement;(iii)surface grinding is adopted rather than circumferential or microgrinding; and (iv) the workpiece material should be an isotropic metal and not a brittle,composite,or anisotropic biological material.By consulting many extant studies,four grinding force models that are in line with the above-mentioned principles are selected and shown in Table 9.The grains have different shapes, namely, pyramid (No.1), tetrahedron (No.2),cone(No.3),and sphere(No.4).Except for the apex angle of No.4,the apex angles of Nos.1–3 are fixed average values,and the protrusion heights are random values within the distribution range.To make the results comparable,the parameters in the models of Nos.1–4 are unified as follows.(i)The parameters of the workpiece materials are unified as Ti-6Al-4 V, as shown in Table 6.(ii) The parameters of the grinding wheel(except for the grain geometric/spatial characteristics) are unified as CBN, as shown in Table 6.(iii) The processing parameters are vs= 30 m∙s-1, vw= 6 mm∙s-1, and ap= 10, 20,30 μm.The detailed parameters of (i) and (ii) are shown in Table 10.To minimize the error,each calculation is performed three times, and the average value is calculated to ensure the stability of the results.

Fig.27 shows the comparison of the experimental and numerical values of normal force obtained with five mathematical models with unified parameters under different cutting depths.As can be seen from the figure, the normal force of No.1, No.2, and the model in this paper is lower than the experimental value,whereas that of Nos.3 and 4 is higher than the experimental value.This result is due to the large protrusion height of the grains in Nos.3 and 4, resulting in a relatively large cutting depth of each grain.Comparison of the experimental measurement results indicates that the error value of the proposed model is the smallest under the three cutting depths.The error values of Nos.1–4 gradually increase.Specifically, the error values of the model in No.4 are the largest, namely, 9.8 %, 11.7 %, and 10.6 %.In addition to the influence of protruding height, the shape of grains also affects the accuracy of the grinding force model to a certain extent.The sphere shape ignores the main geometric characteristics,resulting in a large gap with the real grain and a large error value.‘‘Although the cone shape considers the apex angle,the judgment of the cutting edge and grain–workpiece contact surface is far from reality.The characteristics of the pyramid shape are close to those of the actual grain,so the error is rel-atively small.In addition, the modeling process of Nos.1–4 does not consider the influence of spatial characteristics (3D tilted angle) on grinding force, and the apex angle values are all fixed, which reduces the calculation accuracy.Therefore,in the process of establishing a grinding force model,the grain characteristics should be fully considered to obtain a high final accuracy.

Table 10 Similar grinding force models proposed previously.

Fig.27 Comparison between five grinding force models and experimental values under different cutting depths.

6.Conclusion

The improved grinding force prediction model based on randomized grain geometric/spatial characteristics proposed in this study reveals the determination mechanism of any effective grain and workpiece under different microscopic contact states(sliding,plowing,and cutting).The influence mechanism of the geometric characteristics on the effective action area of the grain–workpiece is investigated.The grinding force model of the whole material removal process is established by separately calculating the normal/tangential forces generated by each effective grain at different stages.The following presents the contributions and innovative research/analysis methods proposed in this study.

A new method of predicting grinding force is presented by using detailed information on the interaction between the grains and workpiece (e.g., the three components of sliding force, plowing force, and cutting force).On the basis of the grain geometry parameters (protruding height, maximum undeformed chip thickness), a new method to determine the grain action stage is proposed.In addition, a new method of deducing the differentiated ‘‘effective action area”between pyramidal grains and the workpiece in the grinding process is proposed based on the randomized morphological characteristics of grains on a real non-engineered grinding wheel and the interaction between grains and the workpiece at different stages.

A grinding force prediction model is established based on the morphological characteristics of randomized grains.The following findings are obtained from the numerical analysis of the model.(i) Under the premise of the control variables,the synchronous-reversed variation characteristics of apex angle α and cutting depth hmcan make the change in the normal/tangential force of a single grain create an inflection point,that is,it increases initially and then decreases.Rotation angles θxand θyaround the x and y axes affect the monotonicity(increase or decrease) change in grinding force.However, a critical angle (13.5°) is observed for the transition between cutting force and plowing force in the process of the grain rotating around the z-axis, leading to an abrupt change in grinding force.(ii) Through sector division, the dynamic evolution mechanism of grinding force during the high-speed relative motion between the grinding wheel and workpiece is revealed.(iii) According to the normal force distribution obtained by certain machining parameters, the distribution trends of effective grain number Ne,apex angle α,and 3D rotation angles θx,θyand θzare basically consistent with the influence law of each factor on a single grain.

Different from previous studies that predicted the average force, this study proposes a method of comparing the upper limit, lower limit, and average values in the range of grinding force fluctuation.Numerical results of full-time domain grinding forces (normal and tangential forces) with detailed information are obtained and verified through experiments.The results of the numerical calculation and experimental measurement obtained under different machining parameters are in good agreement, and the error values are less than 10 %.The range between the upper and lower limits measured by the experiment is larger than that calculated by the numerical method, and the fluctuation range of grinding force is larger for the former than for the latter.However, in some cases,the experimental measurement results of the average value are smaller than the numerical ones; the maximum error is 5.1 %, and the minimum error is 2.1 %.

Grinding force models with similar modeling processes established in previous studies are used to demonstrate the improvement in the computational accuracy of the proposed grinding force prediction model.By unifying the machining and grinding wheel/workpiece parameters, the calculated normal forces of five models are compared with the experimental results.The results show that at grinding depths of 30,20,and 10 μm,the errors of the proposed model are only 2.8%,1.8%,and 2.6%.The errors of Nos.1–4 increase step by step;in particular, the errors of No.4 (spherical grain) are as high as 9.8%,11.7%,and 10.6%.Therefore,in the process of establishing a grinding force model, the grain characteristics must be fully considered to obtain a high final accuracy.

For the grinding process of real non-engineering grinding wheels, the method of establishing a grinding force prediction model presented in this paper can enhance the understanding of existing grinding force from macro to micro scales and guide the industrial grinding process to a large extent(optimize the grinding parameters, achieve machining efficiency, optimize the structure of the grinding wheel, effectively control the grinding force range,adjust the grinding zone temperature,and improve workpiece quality).Given the increasing usage of fine grinding and microgrinding, the requirements on microgrinding details and on the efficiency and precision of microgrinding tools are increasing.Therefore, the numerical calculation model of dynamic grinding force presented in this paper, which has a broad application prospect, can be further optimized and applied to the microgrinding process.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This paper was financially supported by the National Natural Science Foundation of China (Nos.51975305, 51905289,52105264), the Key Project of Shandong Province, China(No.ZR2020KE027), the Major Research Project of Shandong Province, China (Nos.2019GGX104040 and 2019GSF108236), the Natural Science Foundation of Shandong Province, China (No.ZR2021QE116).