Topography Modeling of Surface Grinding Based on Random Abrasives and Performance Evaluation

Abstract

The surface morphology and roughness of a workpiece are crucial parameters in grinding processes. Accurate prediction of these parameters is essential for maintaining the workpiece’s surface integrity. However, the randomness of abrasive grain shapes and workpiece surface formation behaviors poses significant challenges, and accuracy in current physical mechanism-based predictive models is needed. To address this problem, by using the random plane method and accounting for the random morphology and distribution of abrasive grains, this paper proposes a novel method to model CBN grinding wheels and predict workpiece surface roughness. First, a kinematic model of a single abrasive grain is developed to accurately capture the three-dimensional morphology of the grinding wheel. Next, by formulating an elastic deformation and formation model of the workpiece surface based on Hertz theory, the variation in grinding arc length at different grinding depths is revealed. Subsequently, a predictive model for the surface morphology of the workpiece ground by a single abrasive grain is devised. This model integrates the normal distribution model of abrasive grain size and the spatial distribution model of abrasive grain positions, to elucidate how the circumferential and axial distribution of abrasive grains influences workpiece surface formation. Lastly, by integrating the dynamic effective abrasive grain model, a predictive model for the surface morphology and roughness of the grinding wheel is established. To examine the impact of changing the grit size of the grinding wheel and grinding depth on workpiece surface roughness, and to validate the accuracy of the model, experiments are conducted. Results indicate that the predicted three-dimensional morphology of the grinding wheel and workpiece surfaces closely matches the actual grinding wheel and ground workpiece surfaces, with surface roughness prediction deviations as small as 2.3%.

1 Introduction

Superalloys are used to manufacture turbine discs and blades for engine hot-end components due to their high strength and oxidation resistance at high temperatures, accounting for more than 50% of the materials used in aircraft engines [1,2,3]. High-temperature nickel-based alloys (e.g., GH4169, GH4049, etc.) have been developed to meet the demands for high-temperature resistance and stress-bearing capacity [4,5,6]. However, the low thermal conductivity of GH4169 leads to heat buildup during grinding, causing deterioration in the surface quality of the workpiece [5, 7,8,9]. To improve surface integrity and achieve green manufacturing, nano-lubricant minimum quantity lubrication (NMQL) has been applied to nickel-based alloy cutting, representing a significant improvement over traditional metal grinding fluids [10,11,12,13]. Furthermore, predicting the surface morphology and roughness in nickel-based alloy grinding with NMQL has become a research focus, which is key to advancing aerospace parts manufacturing from green precision methods to intelligent and efficient processes [14,15,16,17].

The abrasives on the grinding wheel remove material from the workpiece by cutting. Predicting workpiece surface morphology and roughness remains challenging due to the complex shapes and distribution of abrasives [18,19,20]. Modeling a single abrasive, as the basic unit on the wheel, is crucial [21,22,23,24]. Liu et al. [25] developed a prediction model for workpiece surface roughness using a 3D grinding wheel model (spherical, truncated cone, and cone abrasives) based on a 2D cutting edge profile. Zhang et al. [26] established a 3D morphology and surface roughness prediction model for axial ultrasonic-assisted cross grinding using spherical abrasive modeling. Dong et al. [27] simulated the machining process and predicted the appearance of the finished surface using a mathematical model of the distribution of cone abrasives in the sand belt, accounting for abrasive wear in their study [28]. In the study by Yan et al. [29], the surface morphology of the wheel was measured using a white-light interferometer, and they proposed that abrasives in grinding could be simplified as cones. Hegeman et al. [30] established the 3D surface morphology of the wheel based on ellipsoidal abrasives. Additionally, considering the influence of abrasive position, a random vibration function was added to the ellipsoidal abrasives. Xiao et al. [31] developed a surface morphology model for the specific stacking effect in gear grinding, elucidating the abrasive grain removal mechanism applicable to workpieces with complex geometries. Zhang et al. [32] created a surface morphology model for grinding hard alloys and analyzed the influence of wheel morphology on the maximum material removal rate. Eder et al. [33] investigated the evolution process of surface morphology in nano-grinding and deduced the governing evolution laws. Kang et al. [34] established a three-dimensional morphology model for circular abrasive grains in slider seat rings, exploring the effect of abrasive grain overlap rate on workpiece morphology and removal rate. Wu et al. [35] developed a surface roughness model for ductile-brittle materials, investigating the effect of brittle fracture on surface morphology and the influence of processing parameters on surface roughness.

Additionally, the spatial distribution of abrasives on the wheel surface significantly affects surface roughness. Scholars have used analytical models based on probability analysis to describe the randomness of the grinding process. Hou and Komanduri [36] found that the particle size of abrasives on the wheel follows a normal distribution [37]. Chen and Rowe [38,39,40] proposed a method to determine the position of abrasives on the wheel by randomly distributing them based on the spatial position distribution of abrasives [41]. Zhang et al. [42] established an abrasive position model with a normal distribution through the random movement of abrasives in the X, Y, and Z directions during wheel modeling.

However, there are some shortcomings in the existing research on 3D models of the wheel and the surface morphology of the workpiece. Firstly, the modeling of abrasive morphology on the wheel is based on known shape models such as spherical, conical, or hexahedral, which differ from the actual shape of abrasives [43, 44]. Additionally, in the process of modeling workpiece surface morphology, the grinding is based on the geometry of known abrasives, making it inconsistent with the actual workpiece surface morphology [45, 46]. Therefore, establishing a random geometry model for actual abrasives is key to accurately modeling the wheel and workpiece surface morphology [47].

To address the randomness of abrasive shapes and the complexity of material removal in grinding, a single abrasive geometry model based on the random plane method is established. Then, the kinematic model of a single abrasive and the interference geometry model of abrasive/workpiece interaction were established to predict surface morphology [48].

2 Geometric Modeling of Wheel Based on Random Plane

2.1 Random Plane Abrasive Modeling

Both natural and artificial abrasives must undergo crushing to determine particle size. The geometric characteristics and sizes of abrasives after crushing are highly random due to the disordered crushing process. As shown in Figure 1, the morphology of abrasives was observed using laser confocal microscopy. During abrasive crushing, internal defects in the material cause dislocation and slip between internal grains under external pressure, resulting in brittle fracture [49,50,51,52]. Therefore, the abrasives have the following characteristics: they are irregular polyhedra with a random number of faces. The size of the abrasives varies randomly within a certain range, and the mesh number of the wheel characterizes the particle size of the abrasives.

Figure 1

Abrasive morphology and binary image

A random plane abrasive modeling method is proposed: multiple random planes are established, and the shape enclosed by these planes is defined as the surface of random abrasives. The specific method is as follows:

First, a spherical equation limiting the minimum particle size of abrasives is established with the origin of coordinates as the center:

$$x^{2} + y^{2} + z^{2} = r^{2} .$$

(1)

Random points on the sphere are randomly selected as the base point to establish the section plane of the sphere (that is, a random plane). The coordinates of random points conform to the following equation:

$$\left\{ {\begin{array}{*{20}ll} {x_{i} = \pm r \cdot rand,} \\ {y_{j} = \pm r \cdot rand,} \\ {x_{i}^{2} + y_{j}^{2} + z_{k}^{2} = r^{2} .} \\ \end{array} } \right.$$

(2)

Then a plane tangent to the sphere is established by random points (x_i, y_j, z_k) on the sphere. The equation of the tangent plane is as follows:

$$x_{i} x + y_{j} y + z_{k} z = r^{2} .$$

(3)

A single abrasive is formed by establishing multiple random planes, as shown in Figure 2. Processing the obtained MATLAB random planar graph data. A closed 3D abrasive shape can be obtained by removing the excess between the intersecting planes and selecting the minimum volume figure under multiple plane equations [53].

Figure 2

Stochastic abrasive modeling method

For a given mesh number of wheels, the particle size varies within a certain range [D_min, D_max] and follows a normal distribution. Therefore, the side length of the cuboid domain generated by abrasives is set to D_max to limit the maximum particle size, and the sphere diameter is set to D_min to limit the minimum particle size. When the initial number of planes is set to 30 or 70, the resulting abrasive grains do not match those scanned by SEM (as shown in Figure 2). Through measurement of actual abrasives, it was found that when the initial number of planes is 40 to 50, the abrasive morphology is closest to the real morphology.

2.2 Random Distribution of Abrasives and Modeling of Wheel

A portion of the area involved in grinding on the wheel was selected for modeling, as shown in Figure 3(a). The sampling width and length of the wheel are denoted as W and L, respectively. Since abrasive grains are randomly distributed on the grinding wheel surface, the grain size distribution is determined by randomly selecting the sampling range. To facilitate the analysis of abrasive grain morphology, half of the sampling length is chosen to be equal to the sampling width.

Figure 3

Wheel modeling area and schematic diagram of abrasive position: (a) modeling of local abrasive grains on the grinding wheel surface, (b) distribution of abrasive grain morphology in the grinding wheel model)

Hou and Komanduri [36] found that the abrasives on the wheel obey the normal distribution, and according to the 3σ principle of the normal distribution, it can be expressed as:

$$\Phi \left( {d_{g} } \right) = \frac{1}{{\sqrt {2\uppi } \sigma }}\exp \left[ { - \frac{{\left( {d_{g} - D_{ave} } \right)}}{{2\sigma^{2} }}} \right],$$

(4)

where d_g is the abrasive diameter, σ is the standard deviation of abrasive diameter, and D_ave is the mean abrasive diameter.

The wheel is composed of abrasive, binder, and porosity. The volume fraction of abrasive can be calculated by the wheel tissue number S. Thus, the abrasive spacing can be obtained:

$$L_{r} = D_{ave} \left( {\sqrt {\frac{\uppi }{{8\left( {32 - S} \right)}}} - 1} \right).$$

(5)

Therefore, the number of abrasives in the axial and circumferential direction of the wheel is:

$$N_{x} = \frac{L}{{D_{ave} { + }L_{r} }},$$

(6)

$$N_{y} = \frac{W}{{D_{ave} + L_{r} }}.$$

(7)

The surface of the wheel is dispersed into several small cuboids, as shown in Figure 3(b) The initial coordinates of the abrasive are in the center of the small cube, and the side length of the cube is D_ave + L_r.

Further, the position of abrasives is arranged on the surface of the wheel according to the distance between abrasives. A randomly transformed position coordinate is applied to each abrasive to realize the random distribution of abrasive positions in axial, circumferential and protruding heights. The central position coordinate matrix of abrasives is as follows:

$$\left( {x_{ij}^{\prime} ,{y_{ij}^{\prime}}, {z_{ij}^{\prime} }} \right)^{{\text{T}}} = \left( {j \cdot L_{g} \cdot i \cdot L_{g} \cdot 0} \right)^{{\text{T}}} ,\;1 \le i \le N_{x} ,1 \le i \le N_{y} ,$$

(8)

where x_ij′, y_ij′, and z_ij′ are the initial coordinates of the abrasives in row i and column j, L_g = L_r + D_ave.

Further, the position of the abrasives is randomly transformed, and the transformed coordinates are:

$$\left( {x_{ij} ,y_{ij} ,z_{ij} } \right)^{{\text{T}}} = \left( {x_{{_{ij} }}^{\prime} + \Delta x_{ij} ,y_{{_{ij} }}^{\prime} + \Delta y_{ij} ,z_{{_{ij} }}^{\prime} + \Delta z_{ij} } \right)^{{\text{T}}} ,$$

(9)

where x_ij, y_ij, and z_ij are the position coordinates of the corresponding abrasive after random transformation. Δx_ij, Δy_ij, and Δz_ij are the coordinate values transformed by each abrasive in three directions, respectively, and their sizes are set according to the method studied by Zhang et al. [54, 55].

In addition, to ensure that the positions of abrasives will not overlap and collide after random movement, the motion relationship between abrasives should also meet the following requirements:

$$\sqrt {\left( {x_{ij} - x_{i + + j + + } } \right)^{2} + \left( {y_{ij} - y_{i + + j + + } } \right)^{2} } \ge \frac{{D_{ij} + D_{i + + j + + } }}{2}.$$

(10)

The calculation process and results of the geometric model of the wheel are illustrated in Figure 4. For the resin-based bonded corundum wheel (outer diameter 300 mm, inner diameter 76.2 mm, thickness 20 mm), the sampling area of the wheel is a rectangle with dimensions 1 mm width and 2 mm length [56]. The 3D morphology of the sampled area of the wheel with different grit sizes (80#, 160#, and 240#) is depicted.

Figure 4

Grinding wheel morphology calculation process

The 3D morphology of the simulated wheel was validated, as shown in Figure 5. The actual wheel profile can be used to analyze the prominent positions and distances of abrasives [57]. The distances and height differences of abrasive vertices can be analyzed, and the simulation results agree well with the experimental values. The effective height difference of the wheel surface is 300 μm, consistent with the height difference of the abrasives on the simulated wheel surface. Furthermore, the contour height curves of the wheel surface closely match the calculated positional characteristics of the abrasives.

Figure 5

The simulation and experimental results

3 Workpiece Surface Morphology Generative Model

3.1 Contact Arc Length of a Single Abrasive Cutting Workpiece

The coordinate system is established with the lowest point of contact between the wheel and the workpiece as the origin of the grinding process. The X, Y, and Z axes correspond to the coordinate system of the grinding machine, respectively. The motion diagram of a single abrasive is depicted in Figure 6.

Figure 6

Single abrasive trajectory and contact arc length

The contact arc length between a single abrasive and the workpiece is AC, and according to the relative motion relationship between the wheel and the workpiece, the equation of the cycloid is:

$$\left\{ \begin{aligned} x & = r_{s} \sin \psi \pm v_{\psi } , \\ y & = r_{s} \left( {1 - \cos \psi } \right), \\ \end{aligned} \right.$$

(11)

where Ψ is the angular displacement of the abrasive, v_Ψ is the horizontal movement distance of the wheel, “+” represents reverse grinding, “−” represents smooth grinding.

$$v_{\psi } = \frac{{\psi \cdot v_{0} }}{{2\uppi }},$$

(12)

where v₀ is the horizontal movement distance of the workpiece corresponding to each rotation of the wheel, and its expression is:

$$v_{\psi } = \frac{{\psi \cdot v_{0} }}{{2\uppi }} = \frac{{v_{w} }}{{60n_{s} \times 2\uppi }}\psi = \frac{{v_{w} }}{{60n_{s} \times 2\uppi }}\frac{{r_{s} }}{{r_{s} }}\psi = \frac{{r_{s} v_{w} }}{{60v_{s} }}\psi ,$$

(13)

where, n_s represents the rotational speed, v_w denotes the workpiece feed rate, r_s signifies the grinding wheel radius, and v_s indicates the peripheral speed of the grinding wheel.

Plug v_Ψ in the cycloidal equation:

$$\left\{ \begin{aligned} x & = r_{s} \left( {\sin \psi \pm \frac{{v_{w} }}{{60v_{s} }}\psi } \right), \\ y & = r_{s} \left( {1 - \cos \psi } \right). \\ \end{aligned} \right.$$

(14)

Differentiate the above equation:

$$\left\{ \begin{aligned} {\text{d}}x & = r_{s} \left( {\cos \psi \pm \frac{{v_{w} }}{{60v_{s} }}} \right){\text{d}}\psi , \\ {\text{d}}y & = r_{s} \sin \psi d\psi . \\ \end{aligned} \right.$$

(15)

The differential value of motion contact arc length l_k is:

$$\begin{aligned} {\text{d}}l_{k} & = \sqrt {{\text{d}}x^{2} + {\text{d}}y^{2} } = \left[ {r_{s}^{2} \left( {\cos \psi \pm \frac{{v_{w} }}{{60v_{s} }}} \right)^{2} + r_{s}^{2} \sin \psi^{2} } \right]^{\frac{1}{2}} {\text{d}}\psi \\ & = r_{s} \left[ {1 \pm \frac{{2v_{w} }}{{60v_{s} }}\cos \psi + \left( {\frac{{v_{w} }}{{60v_{s} }}} \right)^{2} } \right]^{\frac{1}{2}} {\text{d}}\psi . \\ \end{aligned}$$

(16)

Since the value of a_p is very small, Ψ is very small, so cosΨ = 1. The above equation can be rewritten as:

$$\begin{aligned} {\text{d}}l_{k} & = r_{s} \left[ {1 \pm \frac{{2v_{w} }}{{60v_{s} }} + \left( {\frac{{v_{w} }}{{60v_{s} }}} \right)^{2} } \right]^{\frac{1}{2}} {\text{d}}\psi \\ & = r_{s} \left( {1 \pm \frac{{v_{w} }}{{60v_{s} }}} \right){\text{d}}\psi . \\ \end{aligned}$$

(17)

Integrate the above equation:

$$l_{k} = r_{s} \left( {1 \pm \frac{{v_{w} }}{{60v_{s} }}} \right)\int_{0}^{\psi } {{\text{d}}\psi = r_{s} } \left( {1 \pm \frac{{v_{w} }}{{60v_{s} }}} \right)\psi .$$

(18)

Due to the value of Ψ is small, so sinΨ = Ψ, then:

$$\psi = \sqrt {1 - \cos^{2} \psi } = \left[ {1 - \left( {\frac{{r_{s} - a_{p} }}{{r_{s} }}} \right)^{2} } \right]^{\frac{1}{2}} = 2\left[ {\frac{{a_{p} }}{{d_{s} }} - \frac{{a_{p}^{2} }}{{d_{s}^{2} }}} \right]^{\frac{1}{2}} \approx 2\sqrt {\frac{{a_{p} }}{{d_{s} }}} ,$$

(19)

where, d_s represents the radius of the grinding wheel post-grinding.

Then the value of Ψ is substituted into the arc length equation (18):

$$l_{k} = 2r_{s} \left( {1 \pm \frac{{v_{w} }}{{60v_{s} }}} \right)\sqrt {\frac{{a_{p} }}{{d_{s} }}} = \left( {1 \pm \frac{{v_{w} }}{{60v_{s} }}} \right)\sqrt {a_{p} \cdot d_{s} } .$$

(20)

Further, according to Eq. (11), the motion equation of abrasives can be derived and expressed as:

$$y = \frac{{x^{2} }}{{d_{s} \left( {1 \pm \frac{{v_{w} }}{{v_{s} }}} \right)^{2} }}.$$

(21)

3.2 Furrow Depth of a Single Abrasive Cutting Workpiece

The actual cutting depth is less than the theoretical value due to material recovery after elastic deformation during grinding [58]. Therefore, the actual surface curve of the workpiece should account for the elastic recovery, as depicted in Figure 7(a).

Figure 7

Schematic diagram of elastic deformation of workpiece surface and elastic sphere geometric contact: (a) actual elastic recovery amount, (b) schematic diagram of geometric contact with elastic balls

Taking the surface of the workpiece after machining as the reference plane, the cutting depth equation of the abrasive can be expressed as:

$$h = a_{p} - \frac{{x^{2} }}{{d_{s} \left( {1 \pm \frac{{v_{w} }}{{v_{s} }}} \right)^{2} }}.$$

(22)

The height of actual workpiece surface formed during the grinding of a single abrasive is:

$$Z_{n} = z_{n} + \varepsilon_{wn} ,$$

(23)

where Z_n is the actual coordinate height value of the workpiece, z_n is the theoretical coordinate height value of the workpiece. ε_wn is the elastic recovery amount of the workpiece can be solved according to Hertz contact theory. The schematic diagram of elastic sphere geometric contact is illustrated in Figure 7(b).

When two elastic spheres come into contact with different radii, the pressure of the two spheres at r distance from the center point is:

$$p\left( r \right) = p_{0} \sqrt {1 - \left( \frac{r}{a} \right)^{2} } ,$$

(24)

where p₀ is the maximum pressure in the contact area of the two spheres and a is the radius of the contact surface.

The displacements are determined at distances r from the center of the sphere within the contact circle, as well as at points outside the contact circle at various distances r from the center.

$$\left\{ \begin{aligned} u_{z} \left( r \right) & = \frac{{1 - v^{2} }}{E}\frac{{\uppi \cdot p_{0} }}{4a}\left( {2a^{2} - r^{2} } \right),r \le a, \\ u_{z} \left( r \right) & = \frac{{1 - v^{2} }}{E}\frac{{p_{0} }}{2a}\left[ {\left( {2a^{2} - r^{2} } \right)\sin^{ - 1} \left( \frac{a}{r} \right) + a \cdot r \cdot \sqrt {a - \left( \frac{a}{r} \right)^{2} } } \right],r > a, \\ \end{aligned} \right.$$

(25)

where v and E are the Poisson’s ratio and elastic modulus of the workpiece respectively.

Due to the action of pressure, the elastic deformation of the two spheres occurs at the contact point, the displacement in the contact area is:

$$u\left( r \right) = u_{z1} + u_{z2} = \delta - \frac{{r^{2} }}{{2R^{*} }},$$

(26)

where δ is the deformation of the two elastic spheres and R^* is the comprehensive radius of curvature of the two spheres.

$$\frac{1}{{R^{ * } }} = \frac{1}{{R_{1} }} + \frac{1}{{R_{2} }}.$$

(27)

Since each point on the contact surface has the same pressure value in the two elastic spheres, then:

$$\begin{aligned} u\left( r \right) & = u_{z1} + u_{z2} = \frac{{1 - v_{1}^{2} }}{{E_{1} }}\frac{{\uppi \cdot p_{0} }}{4a}\left( {2a^{2} - r^{2} } \right) + \frac{{1 - v_{2}^{2} }}{{E_{2} }}\frac{{\uppi \cdot p_{0} }}{4a}\left( {2a^{2} - r^{2} } \right) \\ & = \frac{1}{{E^{ * } }}\frac{{\uppi \cdot p_{0} }}{4a}\left( {2a^{2} - r^{2} } \right). \\ \end{aligned}$$

(28)

By combining Eqs. (26) and (28), it can be got:

$$\frac{{\uppi \cdot p_{0} }}{{4aE^{ * } }}\left( {2a^{2} - r^{2} } \right) = \delta - \frac{{r^{2} }}{{2R^{*} }}.$$

(29)

When r = a, the radius of the contact circle can be found as:

$$a = \frac{{\uppi \cdot p_{0} \cdot R^{*} }}{{2E^{ * } }}.$$

(30)

Therefore, the sum of the deformations of two elastic spheres can be found when r = 0:

$$\delta = \frac{{\uppi \cdot a \cdot p_{0} }}{{2E^{ * } }}.$$

(31)

According to Hertz contact theory, the contact model between the abrasive and the workpiece during grinding can be described as contact between a rigid abrasive and an elastic workpiece. When solving this model, the curvature radius R of the contact point between the abrasive and the workpiece is considered, where the workpiece is treated as having an infinite radius and the abrasive as infinitely stiff relative to the workpiece.

$$\frac{1}{{R^{ * } }} = \frac{1}{{R_{1} }} + \frac{1}{{R_{2} }} = \frac{1}{R} + 0 = \frac{1}{R}.$$

(32)

Then R^* = R.

$$\frac{1}{{E^{ * } }} = \frac{{1 + v_{1}^{2} }}{{E_{1} }} + \frac{{1 + v_{2}^{2} }}{{E_{2} }} = 0 + \frac{{1 + v^{2} }}{E} = \frac{{1 + v^{2} }}{E}.$$

(33)

Substitute the above equation into Eqs. (30) and (31) to obtain the contact circle radius and elastic deformation of the workpiece ε_wn of the contact area between the abrasive and the workpiece surface and:

$$a = \sqrt[3]{{\frac{3P}{4} \cdot \frac{{R \cdot \left( {1 + v^{2} } \right)}}{E}}},$$

(34)

$$\varepsilon_{wn} = \frac{{a^{2} }}{R} = \sqrt[3]{{\frac{{9P^{2} }}{{16R\left( {\frac{{1 + v^{2} }}{E}} \right)^{2} }}}}.$$

(35)

3.3 Modelling of Single Abrasive Cutting Furrow Morphology

When calculating the material removal of a single abrasive during grinding, it is necessary to determine the interference area between the abrasive and the workpiece. This involves the maximum cross-section profile curve of random abrasives engaged in cutting during grinding, as depicted in Figure 8(a).

Figure 8

Calculation process in workpiece grinding with single abrasive: (a) maximum cross-sectional profile curve, (b) surface profile of individual abrasive grain randomly generated, (c) surface morphology diagram of individual abrasive grain grinding workpiece

Before calculating the grinding of a single abrasive, the coordinate matrix of the upper surface of the workpiece is defined first, and let the z = 0 plane as the initial surface of the workpiece. Then the workpiece surface matrix is:

$${\varvec{Z}} = \left[ {\begin{array}{*{20}c} {z_{11} } & {z_{12} } & \cdots & {z_{1n} } \\ {z_{21} } & {z_{22} } & \cdots & {z_{2n} } \\ \vdots & \vdots & \ddots & \vdots \\ {z_{m1} } & {z_{m2} } & \cdots & {z_{mn} } \\ \end{array} } \right],$$

(36)

where m and n are respectively the number of points after discretizing the workpiece in x and y direction.

Initially, the size range of abrasives and the maximum and minimum particle sizes are determined, and the random surface morphology of a single abrasive is generated, as shown in Figure 8(b). The profile matrix of a single abrasive in the YOZ plane was extracted, and the workpiece surface matrix after grinding was calculated by integrating the kinematic model of a single abrasive. Finally, the workpiece surface matrix and 3D image were generated.

The grinding parameters are shown in Table 1. The image resulting from grinding the workpiece with a single abrasive is depicted in Figure 8(c). The trajectory generated by a single abrasive on the workpiece surface forms a cycloid. Under the condition of a grinding depth set to 20 μm, numerical analysis shows that the lowest point of the workpiece surface is 16 μm. Due to the elastic deformation of the wheel and the workpiece, as well as variations in particle size, the actual cutting depth is less than the set depth.

Table 1 Grinding parameters

3.4 Modeling the Workpiece Surface in Grinding

During grinding, a series of random interference interactions occur between numerous abrasives and the workpiece surface, shaping the workpiece surface morphology through the collective effects of multiple abrasives [59,60,61,62,63]. The grinding mechanism involving multiple abrasives is depicted in Figure 9. The positioning of abrasives on the wheel also influences the formation of the workpiece surface morphology, a topic analyzed from two perspectives.

Figure 9

Mechanism of multi-abrasive grinding

3.4.1 Influence of the Axial Distribution Mode of Abrasives

The axial distribution position of abrasives and the grinding mechanism were shown in Figure 10(a). When the axial distribution mode of adjacent abrasives is shown in Figure 10(b), the axial position of abrasives overlaps, resulting in furrow overlap on the workpiece surface. When the distribution mode of abrasives is shown in Figure 10(c), the scratch distance generated by abrasive particles is d_max, and the furrow distance generated by adjacent abrasives on the workpiece surface is large [64]. When the distribution mode is shown in Figure 10(d), the distance of abrasives is between the first and the second case. Moreover, the movement trajectory between the adjacent two abrasives partially interferes, and the scratch distance generated by abrasives on the workpiece is 0 < d < d_max.

$$d_{\max } = \frac{{2d_{g} }}{{\sqrt[3]{{v_{g} }}}}.$$

(37)

Figure 10

Axial distribution of abrasive particles and grinding mechanism

3.4.2 Influence of Circumferential Distribution of Abrasives

The difference in the circumferential distance of adjacent abrasives on the wheel results in varying grinding orders on the workpiece. According to the circumferential distribution mode among any three adjacent abrasives on the wheel, the furrow order generated by different abrasives on the workpiece surface varies. The circumferential distribution position of abrasives and the grinding mechanism is shown in Figure 11(a). The sequence of scratches on the surface of the workpiece results from the different order of scratches produced by adjacent abrasives, as shown in Figure 11(b) and (c).

Figure 11

Circumferential distribution of abrasive particles and grinding mechanism

3.4.3 Dynamic Effective Abrasive and Cutting Depth

Considering the kinematic relationship between adjacent abrasives, it is assumed that when the second abrasive inserts into the workpiece, the material is removed completely after the interference between the first abrasive and the workpiece. The motion relationship of adjacent abrasives is depicted in Figure 12(a). When the abrasives remove the material on the surface of the workpiece during movement, the generated Z coordinate falls below 0.

Figure 12

Dynamic effective grain grinding mechanism: (a) mechanism of multi-abrasive grinding, (b) static effective abrasive, (c) maximum undeformed chip thickness, (d) dynamic effective abrasive [45]

The trajectory of abrasives in the global coordinate system is represented by a function that defines the set of motion trajectories of abrasives. The area of interference between abrasive and workpiece is defined as the maximum contour of each abrasive in the direction perpendicular to the XOZ plane. This contour is discretized into a matrix of points in MATLAB. Therefore, the motion trajectory of abrasives is considered as the collection of points and motion trajectories along the abrasive contour. Due to the random distribution of the abrasive protrusion height, abrasives below the surface of the wheel will not participate in cutting. The abrasives that do participate in cutting are termed dynamic effective abrasives (as shown in Figure 12(b)).

The maximum undeformed chip thickness of two adjacent single abrasives involved in grinding is:

$$h_{\max } = 2 \cdot \lambda \cdot \frac{{v_{w} }}{{v_{s} }} \cdot \sqrt {\frac{{a_{p} }}{D}} .$$

(38)

The maximum undeformed chip thickness of abrasive is shown in Figure 12(c), h_max is the maximum undeformed chip thickness [65, 66]. The cutting depth of abrasive increases to h_max first and then decreases gradually. λ is the spacing of abrasives for continuous cutting. According to Eq. (19), 2 $\sqrt{\frac{{v}_{w}}{{v}_{s}}}$ represents the angular displacement ψ of the abrasive grain, and $\frac{{v}_{w}}{{v}_{s}}$ denotes the grinding depth, where the maximum undeformed chip thickness is directly influenced by the grinding depth.

Wherein, the abrasive spacing and the protruding height of each abrasive on the wheel with orderly arrangement of abrasive are fixed values, while the abrasive distribution position and height are not the same on the disordered wheel. Therefore, when the effective abrasive spacing is λ, the maximum undeformed chip thickness of the abrasive grain is:

$$h_{\max \left( n \right)} = 2 \cdot \lambda_{{\left( {n\sim n - 1} \right)}} \cdot \frac{{v_{w} }}{{v_{s} }} \cdot \sqrt {\frac{{a_{p} }}{D}} + \left( {a_{p\left( n \right)} - a_{{p\left( {n - 1} \right)}} } \right),$$

(39)

where h_(n)max is the maximum undeformed chip thickness of the nth dynamic effective abrasive; λ_{(n ~ n − 1)} is the distance between the nth and (n − 1)_th dynamic effective abrasives. a_p(n) is the height of the n_th dynamic effective abrasive (as shown in Figure 12(d)), and a_{p(n − 1)} is the height of the (n − 1)_th dynamic effective abrasive.

3.4.4 Modeling Process for Creating Surface Topography

Firstly, the grinding parameters are determined. Then the surface morphology of the wheel is established according to the random single abrasive geometry model and position model. Next, calculate the movement track of each abrasive on the wheel and the scratches formed after interference with the workpiece surface. Finally, the workpiece surface height matrix formed by the interference of all scratches after grinding is computed and subsequently output as images.

The calculation flow of workpiece surface morphology is illustrated in Figure 13. The wheel parameters include the dimensions, the organization, the size, the shape of the abrasive, and the distribution state of the abrasive. Machining parameters include grinding depth, wheel speed, and feed speed. Therefore, the calculation results under the comprehensive consideration of many factors are expected to be consistent with real grinding to a certain extent.

Figure 13

Workpiece surface morphology calculation process

4 Experimental Verification and Discussion

4.1 Experimental Setup

4.1.1 Experimental Equipment

This research was conducted using a surface CNC precision grinding machine (K-P36) with a maximum wheel speed of 2000 r/min and a maximum wheel peripheral speed of 50 m/s. (The detailed experimental instruments are shown in Table 2). The wheel used in the experiment was a white corundum wheel with an outer diameter of 300 mm, an inner diameter of 76.2 mm, and a width of 20 mm. The atomization parameters of the micro-lubrication supply equipment can be adjusted by the frequency generator, the compressed air flow knob, and the flow knob, as shown in Figure 14.

Table 2 Description of the instruments and machines parameters

Figure 14

Grinding experimental equipment

4.1.2 Experimental Material

The experimental material is high temperature nickel-based alloy GH4169, whose chemical composition and mechanical properties are shown in Tables 3 and 4.

Table 3 High temperature nickel base alloy chemical composition

Table 4 Mechanical properties of high temperature nickel-based alloys

In the experiment, soybean oil served as the base oil, and Al₂O₃ nanoparticles were used as the additive phase to prepare a nanofluid with a volume fraction of 2%. Sodium dialkyl sulfate dispersant was added to enhance the dispersion of the nano-lubricant. The nano-lubricant was stirred using an ultrasonic oscillator for 30 minutes. The physical properties of Al₂O₃ nanoparticles are shown in Table 5.

Table 5 Physical properties of Al₂O₃ nanoparticles

4.1.3 Experimental Scheme

The lubrication condition used in the experiment is NMQL, and the single factor variable method is employed to investigate the influence of different mesh sizes of wheels and varying grinding depths on the surface morphology of the workpiece. The experimental scheme is shown in Table 6. After the experiment, the workpiece surface was cleaned with anhydrous ethanol and dried by blowing. The 3D morphology of the workpiece was scanned using a laser confocal microscope, and the surface roughness value was calculated for different grinding parameters.

Table 6 Experimental program

4.2 Influence of Wheel Mesh on Workpiece Morphology

4.2.1 Prediction Result

The 3D morphology of the workpiece was observed under different mesh sizes (80#, 160#, and 240 #) with a grinding depth ap of 20 μm, as shown in Figure 15. With the increase in wheel mesh size, the number of abrasives on the wheel increases and the spacing between abrasives decreases. The number of grinding marks produced on the workpiece surface increases. Additionally, with the increase in the number of abrasives, the interference between scratches produced by different abrasives also increases, thereby contributing to the formation of a better surface morphology of the workpiece [67, 68]. As the mesh size increases, the height difference between the peaks and troughs of the workpiece contour curve increases, and the distance between adjacent peaks and troughs in the X-axis direction decreases significantly.

Figure 15

Morphology and roughness profile curve of workpieces ground by grinding wheels with different meshes

Based on the 3D morphology of the workpiece, the surface contour curve is generated by extracting the height matrix from a specific section. The roughness value of the workpiece surface at this point is calculated [69]. Further, the roughness values of different positions on the workpiece are calculated multiple times, and their average value is obtained. The corresponding relationship between the mesh size of wheels and the surface roughness of the workpiece is determined. As the mesh size increases, the surface quality improves, and the roughness value Ra decreases from 3.39 μm to 1.17 μm.

4.2.2 Experimental Result

The surface morphology and roughness profile curves of the workpiece after grinding with different mesh sizes are illustrated in Figure 16. With the increase in mesh size, the height difference of the workpiece surface in the measurement area decreases noticeably, and the number of peaks and troughs can be observed in the roughness profile. This is because as the mesh size increases, the particle size decreases, leading to a reduction in scratch depth on the workpiece surface. Additionally, with a decrease in mesh size, the number of abrasives increases, resulting in more scratches generated on the workpiece surface. This is reflected in the roughness profile, where the distance between adjacent peaks and troughs is significantly reduced, thereby enhancing the appearance of the workpiece surface to some extent.

Figure16

Surface morphology and roughness profile curve of workpieces ground with different wheel meshes

The workpiece surface roughness value under different mesh sizes of the wheel is shown in Figure 17. It can be seen from the figure that the Ra of the workpiece surface decreases from 3.04 μm to 0.825 μm with an increase in wheel mesh size. The average relative deviations between the experimental and theoretical values for mesh sizes 80#, 160#, and 240# are 5.0%, 7.2%, and 8.6%, respectively. The Ra obtained by experiment follows the same variation rule as those obtained by numerical calculation. However, the calculated Ra is greater than that obtained from experimental measurements. This is because nanofluids were used as a cooling and lubrication medium during the experiment, which can improve the surface morphology of the workpiece to some extent. This condition was not considered in the calculation process.

Figure 17

Surface roughness values of workpieces under different wheel meshes

4.3 Effect of Grinding Depth on Workpiece Morphology

4.3.1 Prediction Result

Under the condition where the wheel mesh is 80 and remains unchanged, the surface roughness is calculated by varying the grinding depth a_p, and the results are shown in Figure 18. When a_p = 10 μm, there may be gaps between the abrasives, resulting in areas of the workpiece that are not fully ground, and the furrows on the workpiece surface are shallow. As the grinding depth increases, more material is removed from the workpiece, resulting in an increase in the height difference of the workpiece surface profile.

Figure 18

Surface morphology of workpieces with different grinding depths

The height matrix of a section on the workpiece surface is extracted perpendicular to the XOZ plane, and the roughness value of the workpiece at different grinding depths is then calculated. The corresponding relationship between different grinding depths and surface roughness is established by repeatedly calculating the surface roughness values using numerical analysis. As the grinding depth increases, the chip size increases, resulting in an increase in the height difference between the peaks and troughs of the workpiece surface profile. As the grinding depth increases from 10 μm to 30 μm, more material is removed by the abrasives, resulting in an increase in Ra from 2.63 μm to 3.87 μm.

4.3.2 Experimental Result

The surface morphology and roughness profile curves of the workpiece with grinding depths of 10 μm, 20 μm, and 30 μm under NMQL conditions were obtained using an 80# wheel, as shown in Figure 19. As shown in the figure, the height difference of the workpiece surface in the measurement area increases significantly with increasing grinding depth. This is because as the grinding depth increases, the grinding area of each abrasive and the maximum undeformed chip thickness increase. Consequently, the difference between the peaks and valleys on the surface roughness contour curve of the workpiece increases, leading to a deterioration in the surface morphology quality and an increase in surface roughness.

Figure 19

Surface morphology and roughness profile curves of workpieces at different grinding depths

The experimental Ra value increases from 2.72 μm to 3.96 μm with the increase of grinding depth. For grinding depths of 10 μm, 20 μm, and 30 μm, the relative deviations between the experimental and numerical values are 3.4%, 8.4%, and 2.3%, respectively. Here, numerical analysis was performed on five sets of grinding depths, and experiments were conducted to verify three representative sets of grinding depths. Despite simplifying the experimental groups, the results still demonstrate the accuracy of the model and the consistency of the experimental patterns. The Ra obtained by experiment follows the same variation rule as that obtained by numerical calculation. However, the calculated Ra is smaller than that obtained from experimental measurements. From the analysis, it can be concluded that machine and wheel vibration during the experiment may reduce the surface morphology quality of the workpiece. This condition was not considered in the numerical analysis.

4.4 The Fractal Dimension of the Workpiece Surface

The fractal theory analyzes the surface morphology of the workpiece based on its multi-scale, randomness, and self-affine properties resulting from grinding. Additionally, the Ra value of the workpiece varies significantly depending on the sampling area, evaluation length, and even the instrument used. The fractal dimension remains relatively stable regardless of changes in the sampling region under the same working conditions. The box dimension method calculates the 2D fractal dimension of the workpiece surface, as shown in Figure 20(a).

Figure 20

Fractal dimension calculation process and correlation coefficient of workpiece surface: (a) Calculation process of workpiece fractal dimension, (b) Measurement and fitting graph of fractal dimension

The box dimension is defined as covering the image with square boxes of size ε. Due to the characteristics of the image itself, the box can be divided into two states: with image and without image. The number of boxes with images is defined as N(ε), and then by changing the size of ε, you can get a series of N(ε) values. As ε approaches 0, N(ε) also approaches infinity. By taking the logarithm of ε and N(ε), then using log₂ε as the horizontal coordinate and log₂N(ε) as the vertical coordinate, and fitting the data through the least square method, the negative value of the slope obtained is the box dimension, the formula is as follows:

$$D = - \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{\log_{2} N\left( \varepsilon \right)}}{{\log_{2} \varepsilon }}.$$

(40)

MATLAB is used to binarize the micrograph of the workpiece surface when calculating the box dimension of the workpiece surface. When the box covers all 0 matrices, it indicates that there is no relevant image region, while when the box covers non-0 matrices, it indicates that there is a relevant region in the box. With this method, we can calculate the number of N(ε). The fractal dimension diagram of the workpiece surface with a 240# grinding wheel and a grinding depth of 20 μm after calculation and fitting by the box dimension method was shown in Figure 20(b). Under this condition, the fractal dimension of the workpiece surface is 1.87, the correlation coefficient is 1, and the maximum error is 2.8%. By calculating the fractal dimension of the workpiece surface after machining with different wheel meshes and different grinding depths for many times and taking an average value, the relationship between the processing conditions and the fractal dimension was obtained.

When using an 80# mesh, the fractal dimension of the workpiece surface increases by 0.623% and 1.17% with a grinding depth of 10 μm compared to depths of 20 μm and 30 μm, respectively. Similarly, with a grinding depth of 20 μm, the fractal dimension of the workpiece surface after grinding with a 240# wheel increases by 0.692% and 1.19% compared to 160# and 80# wheels, respectively. Comparing the fractal dimension and Ra value and grinding depth when using an 80# wheel. At a grinding depth of 20 μm, the fractal dimension of the workpiece surface shows a negative correlation with Ra value and grinding depth. When the grinding depth is 20 μm, the fractal dimension of the workpiece surface is negatively correlated with Ra value and a positive correlation with the wheel mesh number. Smaller grinding depths result in larger fractal dimensions, whereas larger wheel meshes also result in larger fractal dimensions; however, the fractal dimension does not perfectly correlate with the Ra value.

5 Conclusions

To improve the surface morphology and roughness of workpieces during the grinding of high-temperature alloys and enhance the predictability of the process, this paper proposes a CBN wheel modeling and workpiece surface roughness prediction method based on the random plane method. And the effects of different parameters on the three-dimensional morphology and roughness of the workpiece surface are explored. The main research conclusions are as follows.

• (1) A new method for geometric modeling of a single abrasive grain using the random plane method is proposed. Through fitting analysis, it is found that when the number of random planes is between 40 and 50, the calculated geometric shape of a single abrasive grain most closely resembles the actual abrasive grain. A kinematic model for a single abrasive grain is established, which accurately fits the three-dimensional morphology characteristics of the grinding wheel. The result shows that the height difference of the abrasive grains on the wheel surface is 300 µm.

• (2) A workpiece surface elastic deformation and formation model based on Hertz theory is established, to reveal the variation of grinding arc length at different grinding depths. The cycloid trajectory geometric characteristics of the workpiece surface formation are determined, and it is demonstrated that due to the elastic deformation of the grinding wheel and workpiece, as well as the influence of the random abrasive grain size, the grinding depth is less than the set depth.

• (3) Based on the spatial distribution model of abrasive grains and the normal distribution model of grain size, geometric models of grinding wheels with different grit sizes and three-dimensional surface morphology models of the workpiece are established. The influence mechanism of the circumferential and axial distribution of abrasive grains on the surface formation of the workpiece is revealed. By combining the dynamic effective abrasive grain model with the influence of circumferential and axial abrasive grain distribution on the grinding trajectory of the workpiece surface, a predictive model for the surface morphology and roughness of the grinding wheel is established.

• (4) Experiments are conducted to explore the influence of changing the grit size and grinding depth on the surface roughness of the workpiece, and verify the accuracy of the model. Results show that by varying the depth of grinding, model accuracy is in the range of 5.0 to 8.6 percent. By varying the grit size of the grinding wheel in the experiments, it is found that the actual three-dimensional morphology of the workpiece surface closely matches the model-predicted morphology, with the surface roughness prediction deviation as small as 2.3%.