Investigation of Micro Grinding via Kinematic Simulations

Abstract

With increasing demand for micro structured surfaces in hard and brittle materials, the importance of micro grinding increases. The application of micro pencil grinding tools (MPGTs) in combination with ultra precision multi axes machine tools allow an increased freedom of shaping. However, small dimensions of the grinding tools below 500 µm substantiate high rotational speeds and low feed rates to enable the machining process. Besides, the abrasive grits of the tool can be large in comparison to the tool dimensions. All factors will influence the resulting surface topography of the workpiece. But some of the topography properties are no longer accessible for optical measurements, making process evaluations and improvements difficult.

In the present contribution the measurement results are supplemented by the results of a kinematic simulation model. The built up of such a kinematic simulation is described, which considers real process and tool properties. The results received by the simulation are compared to measurements to validate the model and point out the advantages of the simulations. In a further step, a principle is shown how the simulation can be used to make the undeformed chip thickness accessible, a process result which cannot be measured within the real machining process.

1 Introduction

Micro grinding is an abrasive process which is suitable to machine brittle materials such as hardened steel, silicon [1], or glass [2]. Micro structuring such materials can be used to create special surface properties and structures which are necessary for the application in optical or electronic industry. Tools for this process are MPGTs which usually have a diameter below 500 µm [3]. However, such tools have many statistical characteristics concerning the grit geometry and the location of the grits on the tool surface. For this reason, the distribution of the abrasive grits of each tool is unique. In peripheral grinding this property is also transferred to the resulting workpiece topography leading to unique scratch-structures on the surface [4]. Especially when using larger sizes for the abrasive grits, which are beneficial concerning tool life and cutting characteristics [5], the scratches significantly influence the resulting surface properties. For the investigation of the micro grinding process, this major influence of the scratches leads to a disadvantage: The topography is no longer mainly influenced by the process parameters. Instead, the tools and their grit distribution become a prominent factor influencing the final surface. However, the real tools cannot be used for multiple experiments since they are affected by wear. This hinders the comparison between two surfaces, received by different process parameters and different tools: The influence of the process parameter cannot be separated from the influence of the tool statistics.

At this point, using kinematic simulations can simplify the investigation of the micro grinding process. In its basic idea, a kinematic simulation model does not consider material behavior, such as material deformations or machine stiffness [6]. Instead, the focus of the simulation is on the influence of kinematic process parameters such as feed rate or the tilt angle of the spindle on the resulting workpiece shape and surface topography.

In general, a kinematic simulation model consists of a tool model and a workpiece model [7]. Such simulations are time discrete, which means that the state of the model is calculated after discrete time steps [8]. Hence, the relative motion between both models is determined for discrete time steps evaluating kinematic equations [9]. For each time step, the intersection between the virtual workpiece and the tool model is calculated [9]. The intersecting volumes are removed from the workpiece model to update the workpiece shape and to represent the virtual machining process.

For kinematic simulation, the workpiece model is spatially discretized. One suitable method is the dexel model, established by van Hook [10]. In the model, usually two dimensions of the workpiece are divided into equidistant calculation points. In the perpendicular third dimension, the surface height is stored on each calculation point as a numerical value. Several authors already used comparable approaches to visualize surfaces received by grinding [6, 11, 12].

The tool model is assumed to be a rigid body which means that it does not change during the simulation. Hence, tool wear is usually not considered [13]. In grinding, different approaches exist to model the tool with focus on the abrasive grits. The challenge is that each grinding tool is unique regarding the arrangement and shape of the abrasive grits. One approach which is based on digitization of real abrasive grits was developed by Klocke et al. [14]. They used a micro computer tomograph to receive three-dimensional volume models of the abrasive grits [14]. However, such digitization techniques go along with high measurement and data processing effort [14].

As an alternative, statistical properties of the grits on the grinding tool are used as basis for modeling [15]. In this case, simplified grit geometries, based on standard geometric bodies are used to approximate the real grit shape. In literature, spheres [16], cuboids, octahedrons [17], tetrahedrons [12], cones [18], ellipsoids, or general polyhedrons [19] were used as basic geometries for the virtual grit representations. To improve the compliance of the basic geometries with the real grits, further adaptations were applied: The intersection of hexahedrons with tetrahedrons in different size ratios leads to grit shapes fitting to the ideal crystal morphology of cubic boron nitride (cBN) grits [14, 17]. Further adaptions of the resulting virtual grit representations were done by Warnecke et al. by fitting planes with different orientations, randomly cutting areas of the grit volume [17]. Compared to this, Chen et al. used tetrahedrons to model the grits, of which the corner points were randomly cut off, resulting in further triangular facet on virtual grit representation [12]. In general, the choice of the virtual grit representation depends not only on the material of the abrasive grits, but also on the manufacturing process, since the grit shapes of, for instance, cBN depends on the temperature and pressure during the synthetic production [20]. Due to the different shapes, the size of the abrasive grits is not the same for all grits within a sample and they differ from the nominal value. Koshy et al. took the sieving process of the grits as basis: The lower and upper mesh size of the sieves were used to calculate the mean value and standard deviation [21], assuming a normal distribution of the grit sizes [22]. Another approach is based on measurements of the abrasive grits. Warnecke et al. used scanning electron microscopy (SEM) images of the grits to measure the length of the short and long axis of the grits [17]. This approach was also adopted by Chen et al. who measured the edge length of the grits via SEM images [12]. For analyzing powders in general, laser diffraction is a suitable method using the light scattering properties of the powders and solving the Maxwell equations [23]. This method was already used for the characterization of abrasive grits [24]. Two theories to evaluate the light diffraction and calculate the particle size exist: The Fraunhofer approximation and the Mie theory [23, 25]. For the Fraunhofer approximation, no values for the optical properties of the particles are necessary, however it is only suitable for larger diameters and does not consider light transmission through the particles [23]. For the Mie theory, the theoretical fundamentals were set by Gustav Mie [25]. This theory does not only use light diffraction, but also transmission through the particles [26] leading to more reliable results for particles below 10 µm [27]. However, the Mie theory requires the value of the complex refraction index of the analyzed powder [23]. Concerning the results, the particle size distribution of real powders is often elongated to larger dimensions [28]. This effect is not well described by the normal distribution, but by the lognormal distribution [28].

For placing the virtual grit representations on the virtual tool surface, the single grits are randomly rotated around their axis first [29]. The protrusion height of the grits on the tool surface is modeled using a normal distribution with a mean value equal to the mean value of the grit size [30]. The position of each virtual grit representation on the virtual tool surface can be set with a uniform distribution, until a given grit density on the tool is reached [6]. Another method was introduced by Chen et al. [31]. In their model, all grit representations were initially located on an equidistant mesh on the virtual tool surface to predefine the number of grits within the model [31]. Then, a slight deviation of the position of each grit was added to randomize the grit positions [31]. Liu et al. denoted the method as “shaking” of the virtual grit representations [18].

However, with this principle, overlapping of the grits with each other is possible. To avoid this, the placing on the tool surface was restricted: When spheres are used as virtual grit representations, Koshy et al. formulated a mathematical condition to assure that the center points of two grits have at least a distance equal to the sum of both grit radii [21]. If more complex grit shapes are used, the condition can also be applied by calculating the minimum enclosing sphere of each virtual grit representation [12].

In previous works we already performed kinematic simulations for grinding processes. Adjustments to the principle were made to determine forces and chip thickness in the conventional grinding process [32]. With the help of kinematic simulations, we investigated the undeformed chip thickness of single grit scratch tests in combination with the pile-up effect [33]. We also performed investigations concerning micro grinding with MPGTs. The generation of the bottom surface of ground micro channels were investigated via kinematic simulations [13]. We found that the radial position of the abrasive grit on the tool face causes a step shaped workpiece topography [13].

The investigation of peripheral grinding using kinematic simulations for electroless plated MPGTs was not yet performed. To fill this research gap, in the present contribution the kinematic simulation is applied to micro grinding and the built up of the simulation model is described. One advantage of the kinematic simulation is, that the same tool model can be used for an arbitrary number of virtual experiments. Furthermore, errors appearing due to the digitization of the real ground surface topographies can be avoided within simulations. The simulations were based on real grinding experiments with MPGTs and implemented in Matlab¹. The work originated within the IRTG 2057 on Physical Modeling for Virtual Manufacturing Systems and Processes.

2 Properties of the MPGTs

The MPGTs considered within this contribution were self-manufactured. The tool blanks were made of high speed steel and a cylindrical shape with a nominal height of 400 µm and a diameter of 360 µm was manufactured on the tool tip. This cylindrical shape was the base for the abrasive body of the MPGT. The abrasive body was produced via an electroless plating process, a solely chemical process we developed previously [34]. The abrasive grits were made of cBN with a nominal grit size between 20 µm and 30 µm. For the plating process, the abrasive grits were added to the plating solution in the form of loose powder. During the plating process a nickel-phosphorous compound was deposited, which embedded the abrasive grits in one layer on the cylindrical shape of the tool blank [35]. The final nominal diameter of the tool was 400 µm, which is composed of the blank diameter (360 µm) and twice the smallest nominal grit size (20 µm). The properties of the tools researched within this contribution are depicted in Fig. 1.

3 Model of the MPGT for Kinematic Simulations

For modeling the tool, geometric properties of the real MPGTs are necessary. A digitization of the complete micro grinding tool was not applicable due to the size and curvature of the tools. As an alternative, statistical methods were applied to describe the properties of the tools and to get the geometric information. For this purpose, values for the statistics of grit size and grit shape distribution were used for modeling. Both statistics are received by analysis of the real cBN grits and the real MPGTs. In the following sections it will be shown how the results of the analyses can be used to generate a virtual representation of the MPGT.

Fig. 1.

Geometric properties of the MPGTs

3.1 Analysis of the Grit Size Distribution

The cBN particles used as abrasive grits for the tools were purchased in the form of powder with a nominal size between 20 µm and 30 µm. However, solely the range is not suitable to be used in the simulations, since within this range a uniform distribution of the grit sizes would be assumed. Instead, the distribution of the grit sizes was needed to receive realistic sizes of the virtual grit representation.

A method was necessary which delivered repeatable results, without subjective evaluation. This can efficiently be done via particle analysis. The 3P instruments Bettersizer S3 plus (See Footnote 1) was applied which uses the principle of laser diffraction. A large amount of grits can be analyzed within a short period of time and in a repeatable way. The original cBN powder was used to measure the grit size distribution. For a measurement of the grits, which were actually embedded on the MPGTs, hundreds of tools would have had to be chemically dissolved to receive a suitable quantity of grits.

Within the particle analysis, water was used as dispersant medium and the dispersant stirrer had a rotational speed of 2500 rpm. According to the standard measurement procedure, cBN grits were added to the dispersant medium until an obscuration of 12% within the measurement system was displayed. Before the grit size analysis was started, ultrasound was activated for one minute to remove bubbles within the dispersant medium. During the measurement a total amount of 10,000 grits were analyzed. The evaluation of the measurement was done using the Mie theory since grit sizes partly below 10 µm appeared in the cBN powder sample and the transmission of the light through the particles is possible. For using this evaluation method, the complex refraction index was necessary, a parameter which was not exactly known for cBN. Hence, the option of the measurement system to calculate its value was used, leading to a value of 2.12-0.001i. Using the calculated value of the complex refraction index, a measurement residual of 6.945% for the particle size analysis was reached. The histogram of the grit sizes is given in Fig. 2. It can be seen that the distribution of the grit sizes has no symmetrical shape, but the left flank is steeper than the right flank. Besides, ten percent of the grits have a size below the d₁₀ value of 12.57 µm and 90 percent of the grits are below the d₉₀ value of 27.77 µm. However, grit sizes below 12.57 µm and above 27.77 µm also occurred in the abrasive grit sample: The smallest grit size identified by the measuring system was about 3.55 µm and the largest grit size about 51.82 µm.

Fig. 2.

Histogram of the grit size distribution received by the particle size analysis.

The finite number of analyzed grits cause that no continuous distribution function of the grit sizes can be determined by the particle analysis. As a consequence, values between the discrete measured values are also possible values for the grit size with a certain probability. For this reason, a distribution function was fitted into the discrete data using the Matlab (See Footnote 1) function “fitdist”. Due to the asymmetric shape of the grit size distribution, the normal distribution did not fit well. A better correlation was received using the log-normal distribution function. Fitting a log-normal distribution function into the measured data of the particle size analysis led to a distribution which is described by a mean value of 2.9375 and a standard deviation of 0.3133. Both values were used as input parameters for the simulation.

3.2 Analysis of the Grit Shape

In case of the grit shape, it cannot be assumed that all grit shapes included in the original powder of cBN are also found on the tool. Grits with an unfavorable geometry which does not allow a form locked join within the bond are not embedded. For instance, this can be grits which only have a small volume fraction inside the bond leading to a weak connection between grit and bond. Hence, the shape of the grit had to be evaluated by analyzing the abrasive body of the manufactured tools as it is shown in Fig. 3. Since the protrusion height of the grits, which is the distance between the outermost point of the grit and the bond material, is not evaluated within this contribution, three dimensional topography measurements to receive the height information were not necessary. Hence, SEM images of the peripheral surface of real MPGTs were analyzed which have a higher lateral resolution in comparison to optical topography measurements. The procedure for the evaluation of the SEM image is depicted in Fig. 3 and was applied to ten different MPGTs for statistical validation. In detail, an area of 200 µm times 200 µm in the middle of the depicted tool surface was considered and a uniform number of 20 grits for each tool were analyzed. The number of 20 grits was available within the evaluation area for all analyzed tools. Furthermore, using a constant value of measured grits instead of all available grits avoided overrating of individual MPGTs with larger grit density.

Each grit shape was manually allocated to the best fitting basic grit shape. For the allocation, two classes were defined for grits with the following properties:

• Elongated grits with large aspect ratios or predominantly sharp corner points and acute angles.

• Bulky grits with aspect ratios similar to cuboids or predominantly obtuse angles.

Fig. 3.

Area for evaluation of the grit shapes on a SEM image of the MPGT.

Subsequently to the allocation, the ratios of elongated and bulky shape were calculated in relation to the total number of evaluated grits in order to estimate the probability of the grit classes. The result of the evaluation is depicted in Fig. 4. It can be seen that both grit classes appear approximately with the same probability. The probability values are used as input parameters for the simulation, what will be described in later sections.

Fig. 4.

Definition of the two grit shape classes

3.3 Requirements and Assumptions for the Tool Model

For the application within a kinematic simulation of the peripheral micro grinding process, the following requirements and assumptions for the tool model were defined:

• 1) Due to the comparatively large abrasive grits for micro grinding in combination with the associated low grit density, each single grit has a high impact on the final surface topography. Hence, the grit shape is represented in the resulting surface topography. Therefore, detailed virtual grit representations are necessary for the simulation.

• 2) The exact number of grits on the real tool is not known. Hence, a predefined number of grits on the tool model is not suitable. Instead, the goal is to implement a virtual saturation process of virtual grit representations on the virtual tool surface.

• 3) Input data for tool modeling are supposed to be determined with less effort in a repeatable way and the procedure should be transferable to other tool diameters and abrasive grit sizes.

• 4) As for the real tool, overlapping grits should be avoided within the tool model.

The superordinated procedure which is used to model the tool is depicted in Fig. 5. In the next sections, the single steps are described in detail.

Fig. 5.

Subordinated procedure to model the MPGT.

3.4 Modeling of the Virtual Bond of the Tool Model

The bond is the material in which the grits are embedded with a certain volume fraction. The embedded volume fraction depends on the thickness of the bond material.

For the tool model, the bond is considered to be an ideal cylinder. Within this cylinder, the bond is limited by the blank material, which is also considered to be an ideal cylinder with a smaller radius. The radius of the blank is set by the manufacturing process to 180 µm. The difference between the blank radius and the radius of the bond surface is the thickness of the bond as it is depicted in Fig. 6. In the real manufacturing process of the MPGTs, this value depends on the plating time. In previous work, we found a deposition rate of about 21 µm/h [35]. For the tools, a total plating time of 35 min was applied leading to a bond thickness of 12.25 µm. Adding the value to the radius of the tool blank provides the radius of the bond. Both, the radius of the blank and the radius of the bond are used for the tool model.

Fig. 6.

Definition of the bond thickness for MPGTs.

3.5 Validation of the Bond Thickness

Measurements on real tools were used to check whether the analytical value for the bond thickness is suitable for modeling. For the analysis, the faces of the tools were considered, and the diameter of the bond was measured using SEM images of the tools. Before the SEM images were performed, the tools were cut to make the bond visible and improve the identification of the bond diameter. For cutting, the MPGT was rotated, and a dicing blade was used. The feed rate of the dicing blade was set to a value of 1 mm/min for a gentle cutting process. Dough mixed with cBN powder was used to remove the emerging burr.

The electroless plating process does not produce ideally smooth surfaces and minor variations in bond thickness may occur. Therefore, the determined value for the diameter also varies depending on the measuring position. For this reason, the measurement of the bond diameter was statistically validated. Within the measurement procedure, rectangles were set around the tool with each side touching the bond of the tool image. The rectangle led to two perpendicular measurements for the diameter of the bond, using both dimensions of the fitted rectangle. For the statistical validation, the procedure was performed three times using rectangles with an angle of 0°, +120° and −120° as it is depicted in Fig. 7. Thus, for each tool six values for the bond diameter were available. The measurement principle was applied to the SEM images of six different MPGTs. To get the value for the bond thickness, the measured values are halved to receive the values for the bond radius. Subtracting the radius of the tool blank leads to the value of the bond thickness. The mean value of the measurements was 13.03 µm with a standard deviation of 2.91 µm.

Even though large standard deviations appear, which can be explained by material deformations due to the cutting process of the tool faces, the mean value fits well to the value received by using the deposition rate to calculate the bond thickness.

Fig. 7.

Evaluation of the bond diameter using rectangles with different tilt angles.

3.6 Modeling of the Abrasive Grits

For the virtual grit representations, the previously defined two classes of grits are approximated with basic geometries. In order to meet the requirement of realistic virtual grit representations, those basic geometries were considered which are constructed of plane facets and then fit to the real crystal shape of the grits. Thus, spheres and ellipsoids are excluded.

The implementation of the virtual grit representations into the tool model was performed considering the limited computational power: To keep the memory requirement low, only the corner coordinates of each virtual grit representation is stored. Besides the points, a list is created, in which the corner points are allocated to the grit facets.

At the beginning of the generation of a virtual grit representation, the corner points are set so that the size of the virtual grit in each dimension is equal to one. Thus, multiplying the size of the grit by the coordinates of the corner points, the virtual grit representation is stretched to its final size.

For the elongated grit class, tetrahedrons were selected as virtual grit representation as it is shown in Fig. 8. Tetrahedrons consist of spatially arranged triangular faces, which create sharp corner points as it was defined for the elongated grit class. Besides, tetrahedrons appear in the ideal crystal morphology of cBN [36].

The bulky shape is represented using a polyhedron (see Fig. 8). Starting point for modeling the polyhedron is a cuboid. Since the real grits allocated to the bulky class did not have sharp corner points, the corner points of the cuboid are cut off at a randomized position. It is done by spanning another triangular facet which substitutes the corner point of the cuboid.

To span the new facet, the corner point of the original cuboid is used. The corner point of the original cuboid is stored three times to receive three individual points at the same position. Each of the three corner points is displaced in the direction of one edge of the cuboid. The displacement is done with randomized values, leading to arbitrary triangular facets. The procedure is depicted in Fig. 8.

The decision which virtual grit representation geometry is chosen is done by the probability evaluated for the grit shape distribution. Subsequently, the virtual grit representation size was adjusted. For this purpose, random sizes are calculated according to a log-normal distribution. The distribution function is defined using the mean value and standard deviation received by the particle size analysis. In a last step, the grit is randomly rotated around its three rotational axes.

Fig. 8.

Modeling of the polyhedron (a) and allocation of the grit models to the grit classes (b)

3.7 Positioning of the Virtual Grit Representations on the Virtual Tool

In the next step of the tool model generation algorithm, the virtual grit representations are placed on the virtual tool surface. Since peripheral grinding is considered, the virtual grit representations are only located on the peripheral surface of the virtual tool.

Using cylinder coordinates, the radial position of the grit determines the protrusion height. The position is determined by a normal distribution. However, the grits are supposed to be inside the virtual bond. Thus, the values for the radial position are checked: It was avoided that the grit is located inside the blank material and outside of the bond material.

For the first virtual grit representation, all height and angular positions on the tool peripheral surface are equally probable. However, for all subsequent grits the positions which are already occupied by virtual grits are no longer available. In order to represent this effect in the simulation, a virtual saturation process of the tool model surface with grits was replicated: The randomly calculated position of the virtual grit representation is compared with the position of the grits already placed on the tool model. However, due to the randomized shape of the virtual grit representations, computationally extensive calculation would have been necessary to determine the intersection between two grits. To circumvent such calculations, the comparison between the grit positions is abstracted: Since only the height and angular positions of the grits are relevant for a possible interaction with other grits, the checking of the grit positions was reduced to two dimensions (angular and height direction). Furthermore, the grit shape is reduced to a quadrangle enclosing the virtual grit representation. Using this procedure, it is possible to calculate the intersections between the grits using two-dimensional geometry: A valid position is only present if the smallest or largest angle respectively height position of the new grit is not inside the squares of an already existing grit.

Two special cases are taken into account which could lead to overlapping grits event though the previously mentioned condition was met:

• The new grit model completely encloses the quadrangle of an already existing virtual grit representation.

• The angular position of the new grit is close or equal to 360°. Here, the maximum angular position of the grit could reach values above 360°, leading to overlapping with existing grits with minimum angular positions close to 0°.

If an overlapping of the virtual grit representations is detected, the calculated position of the grit is discarded. Consequently, the algorithm skips back to the calculation of the grit position and a new position is generated. Especially with increasing amount of virtual grit representations which are already placed on the virtual tool, the procedure can end in an infinite program loop. To avoid this and to limit the total computing time, the maximum number of loops was limited to 100 repetitions. When no valid position for the virtual grit representation was found within this period, the virtual grit representation is discarded and a new one is generated. With increasing computing time, the grit density on the virtual tool is enlarged. At the same time, the probability of a new grit to be located on the tool model decreases which is comparable to the saturation process of the real tool.

For the tool models used within the present contribution, a number of 1,000 virtual grit representations were predefined. After generating the tool model, between 580 and 680 virtual grit representations were actually integrated into the final tool model depending on the grit sizes. However, the amount of virtual grit representations on the virtual tool is not a static value but it can change due to the statistical characteristics of the grinding tool. The process fits to the real electroless plating process: Only if enough space is left on the tool surface, a further grit can be embedded. Hence, also for the real process a saturation point for the amount of grits on the tool is reached. A comparison between the tool model and the real MPGT will be given in the next section.

3.8 Evaluation of the Grit Size on the Real Tool

During the plating process, the chemical solution is set into rotation to swirl up the cBN grits. However, the rotation of the solution, including the cBN causes centrifugal forces acting on the abrasive grits. Due to the centrifugal forces, especially larger and therefore heavier cBN grits tend to move away from the middle of the plating solution. Such grits cannot reach the tool surface and are therefore not embedded. Thus, the coating process operates as grit size filter, what means, that not all grit sizes of the original cBN powder are actually present on the final tool.

For the evaluation of this effect, the values of the particle size analysis were compared to values of grit measurements of the real tool. Optical measurements using a confocal microscope were not suitable due to the same reasons mentioned in the section on the grit shape analysis. Thus, the analyses were done using SEM images of tool peripheral surface. To avoid the influence of the curvature of the tool, only the grits which were located inside a 200 µm × 200 µm square in the middle of the tool image were considered as it was already done for the grit shape analysis. The measurement of each grit was done two times: One measurement in the longest and one in the shortest expansion of the grit. The dimension was measured using two parallel lines enclosing the grit. It was applied to the SEM images of ten different tools and for each tool, the size of 20 grits were analyzed. A log-normal distribution was fitted into the resulting grit sizes, as it was also done for the results of the particle size analysis. The corresponding mean value and the standard deviation are given in Table 1 in comparison to the values received by the particle size analysis.

Table 1. Mean value and the standard deviation of the fitted distribution functions for the grit sizes received via particle size analysis in comparison to the statistical values received by grit size measurements in SEM images.

The values show that the measurements via the SEM images have a lower mean value and a lower standard deviation. This supports the previously mentioned filtering effect of the coating process of the MPGTs. To make the difference between both statistics visible, both are used to generate a tool model. The comparison is depicted in Fig. 9. On the left, the tool model using the complete data of the particle size analysis is shown (tool model 1 (TM-1)). When comparing it to the image of the real tool in the middle, it is again recognizable that the grit sizes are too large. On the right side, a tool model calculated with the grit sizes (comparison tool model (TM-C)) measured via the SEM images is shown, which fits much better to the real tool. For further analysis, both tool models are compared. With the virtual tools, parameters are available which are not directly accessible by measurements of the real tool. The first one is the total amount of virtual grit representations. The tool modeled via the data received by the SEM image (TM-C) has 680 grits on its peripheral surface, which is 100 grits more than for the other tool model. This fact also corresponds to the smaller grit sizes of the tool model on the right in Fig. 9: Since the virtual grit representations are smaller, more models can be placed on the surface until the virtual saturation is reached. Furthermore, the protrusion heights of the virtual grit representations were calculated virtual grit representation. For the tool modelled via the particle size analysis (TM-1), the mean value of the protrusion heights is 10.86 µm with a standard deviation of 6.89 µm. For the tool models created via the SEM measured values (TM-C), the mean value of 7.80 µm is significantly smaller which is also true for the standard deviation of 5.72 µm.

Fig. 9.

Comparison of the tool model 1 (TM-1) calculated via the data of the particle size analysis, the tool model received by the data of the grit size measurement (TM-C) and the SEM image of the real tool.

3.9 Adaption of the Grit Sizes for the Tool Model

In the previous section it was shown that the grit sizes of the real tool differ from the grit sizes measured with the particle analysis. To feasibly use the data from the particle size analysis within the modelling procedure for the MPGT, the statistical distribution received via the particle size analysis was adjusted in the following way: As larger grit sizes do not appear on the final MPGT, values of larger grit sizes were excluded from the distribution function. For this purpose, the threshold value was set to the d₉₀ value of the particle size distribution. It is a standard parameter [23], which is used to characterize the particle size distributions. Its value is directly exported by the particle measurement system. The implementation in the algorithm for the tool model is done by a testing loop of the grit size: According to the determined distribution function of the particle size analysis, a value for the size of the virtual grit representation is calculated. If the value is above d₉₀ it is discarded, and a new value is generated. The resulting tool model, which will be called TM-2, is depicted in Fig. 10 in comparison to the tool model using the original particle size distribution (TM-1), and to the model generated with the statistical values received via the measurement of the SEM images (TM-C). Comparing TM-2 to TM-1, the amount of grits are increased as expected due to the decreasing average grit sizes. The amount of modelled grits is also closer to the number of grits in TM-C. Furthermore, the mean value and standard deviation of the grit protrusion heights of TM-2 are much closer to the corresponding values of TM-C. An exact match does not have to be achieved, since the parameters mentioned are subject to statistical fluctuations even in real tools. In a last step, the grits were given an identification number to allocate the corner points to the corresponding virtual grit representation.

Fig. 10.

Comparison between tool model 1 (TM-1) and tool model 2 (TM-2)

3.10 Conclusion on Tool Modeling

A principle was presented to model MPGTs using data of particle size analysis of the abrasive grits and the deposition rate of the tool bond during manufacturing.

For the tool model, polyhedrons with adapted corner points and tetrahedrons were used as basic virtual grit representations. The virtual grit representations were placed randomly on the tool surface and overlapping of the grits was avoided. The amount of virtual grit representations on the virtual tool surface was not predefined, but a virtual saturation process was used which better emulates the real manufacturing process of the MPGTs. To adjust the size of the virtual grit representations, particle size analyses were used, rapidly delivering statistically validated and repeatable results without subjective assessments. However, when comparing the values of the particle size analysis with the values received by measurements of the embedded grits in basis of SEM images of the MPGTs, it was found that the real grits on the tools are smaller. In the modelling procedure of the tool, this was considered by excluding virtual grit representation size values larger than the d₉₀ value of the particle size analysis. This resulted in virtual grit representation sizes which fitted to the real grits on the tool.

All in all, the modeling procedure of the MPGT led to a realistic virtual representation of the tool which is suitable for kinematic simulations.

4 Setup of the Simulation

For the application of the tool model within a simulation, a simulation model is necessary. The steps for the setup of this model are described in the following sections.

4.1 Workpiece Representation Within the Simulation

Besides the tool model the second part of the simulation is the virtual workpiece. In the case of ground workpiece surfaces, the contact paths of several grits overlap. This causes that perpendicular to the feed direction, the enveloping profile of the tool peripheral surface is imaged on the workpiece. In Fig. 11 this profile is depicted as a blue line. In addition, the engagement of a grit is not continuous, but periodic with each tool rotation. For this reason, an analytical description of the surface would lead to complex equation systems. As an alternative, discrete workpiece models can be applied. Such models facilitate the calculation of the tool-workpiece interaction by continuously updating the resulting virtual workpiece surface.

The discretization method applied was the dexel model. A discretization of the surface in two dimensions was used. The third dimension was the direction of the dexel which has higher accuracy. The coordinates of the simulation model were set in a way that the x-direction was the direction of the feed, the z-direction was equal to the rotational axis of the tool and the y-direction was the direction of the surface heights, corresponding to the peripheral grinding processes [37].

Since optical surface measurements were used to digitize the real ground surfaces, the dexel model was adapted to the measurement to ensure comparability: In the lateral directions of the measurement system, the resolution was 0.7 µm. This value was also used for the resolution of the dexel model in discretized x- and z-directions. The vertical resolution is much higher and was used to identify the surface heights. The principle for discretization is depicted in Fig. 11 [37]. In Matlab (See Footnote 1), the dexel grid was implemented as a matrix. Each matrix entry depicts the height of one dexel. Hence, the dexels can also be interpreted as discrete calculation points for the surface height of the virtual workpiece.

Fig. 11.

Discretization of the workpiece model [37]

4.2 Kinematics and Time Discretization

For the calculation, the relative movements between the tool model and the workpiece model are determined. The feed motion and the tool rotation are considered. Effects due to process-machine-interactions, such as material deformations, tool deflections, or limited machine stiffness were not taken into account.

The simulation is time discrete. Hence, the tool movement and its intersection with the workpiece is calculated after equal time steps. In connection with the rotational speed of the tool, the value of the time steps determines which angle of the tool rotation is included within each calculation step. Due to the small diameters, high rotational speeds are required for micro machining. Thus, a low value for the time steps was necessary. For the simulations with a constant rotational speed of 30,000 1/min, a value of 10 µs was used for the time steps to ensure, that the rotational position of the smallest virtual grit representation in two consecutive time steps are at least touching each other and a continuous engagement of the grit into the virtual workpiece is approximated. Within each time step the grit passes a distance of 6.12 µm which is equal to the dimension of the smallest grit within the tool model. On the other hand, the value of 10 µs of the time step led to an acceptable calculation time of the simulation [37].

To calculate the kinematics, each corner points of the virtual grit representation is virtually moved. For each time step, the current position is calculated in relation to the initial position of the models to reduce a chain of errors due to the numerical calculation of the trigonometry. Within the simulation, the feed motion is considered as a translatory displacement of the virtual tool model. The distance for the displacement is calculated using the feed rate multiplied by the value of the time step. The rotational motion of the virtual grit representations is realized using a rotation matrix around the z-axis. Analogous to the feed rate, the angle is calculated via the rotational speed of the tool multiplied by the value of the time step.

The complete motion data is saved for the following calculations of the tool workpiece interactions. However, a large part of the data could be excluded from further calculations because they cannot geometrically reach the virtual workpiece surface. For instance, this condition is true for grits which are turned away from the grinding zone within the current time step. Excluding such grit provides the advantage that the calculation time for the tool workpiece intersections is reduced [37].

4.3 Calculation of the Tool-Workpiece Intersection

For the calculation of the intersection between tool and workpiece model, the main task is to find the intersection point between dexels and the virtual grit representations and to determine the new height value for each dexel. The principle is depicted in Fig. 12.

Fig. 12.

Principle for the calculation of the surface heights [37]

For the calculation, a straight line is set for each dexel in the direction of the surface heights. Each of the straight lines is intersected with the facets of the virtual grit representations. To determine the intersection, a vectorial plane, spanned by the corner point vectors, is placed on the facet and the geometric intersection point is calculated. Since the points could also be outside of the facet, the facet and the intersection point are projected into the x-z-plane leading to a polygon in two dimensions. If the projected point is inside the two dimensional polygon, a valid intersection point is present. In this case, the y-coordinate is used to define the new length of the dexel on condition that the new length of the dexel is shorter than the original length [37].

5 Application of the Simulation Model to the Investigation of Micro Grinding

In the following sections the simulation model is applied to investigate the kinematic influence of two different feed rates on the resulting surface topography and to investigate the undeformed chip thickness, which is a parameter, that cannot be measured during experiments.

5.1 Influence of the Feed Rate on the Resulting Surface Topography

The simulation model was used to investigate the influence of the feed rate on the resulting surface topography when performing peripheral micro grinding. The advantage in comparison to experiments is that solely the influence of the feed rate can be considered, independent from other effects on the surface topography that appear in real experiments.

5.1.1 Experimental Setup

Micro grinding experiments were performed to compare the real surface topographies with the topographies received by the simulations. The workpiece for the experiments was made of 16MnCr5, hardened to 650 HV 30. The workpiece was tilted to 45° leading to V-shaped grooves due to the horizontal feed direction of the tool. This setup, which is depicted in Fig. 13 enabled to make the workpiece surface, generated by the peripheral tool surface, accessible for optical measurements. The experiments were performed on a LT Ultra MMC600H ultra precision machine tool with five axes. Two different feed rates v_f = 5 mm/min and v_f = 1 mm/min were applied at a constant rotational speed of 30,000 rpm and a constant axial depth of cut of 150 µm which was achieved within one single pass. For both feed rates the feed travel was 10 mm.

Fig. 13.

Experimental setup a) kinematic properties, b) setup within the machine tool [37]

For the evaluation of the experimentally generated surfaces, a three-dimensional topography measurement system Nanofocus µsurf Explorer (See Footnote 1) was used to digitize the surfaces. Processing of the data was done according to DIN EN ISO 25178-2 [38] using the MountainsMap (See Footnote 1) software. Within the software, the complete surfaces were aligned, a filter with a cutoff wavelength of 1.5 µm was applied to remove surface fractions with short wavelength and another one was applied with a cutoff wavelength of 24 µm to remove larger wavelength fractions of the surface. For each experiment, three measurements were performed: One at the start, one in the middle and one at the end of the groove. From the results, the arithmetic mean value Sa and the root mean square Sq of the surface heights were calculated. Further details on the measurement procedure can be found in [37]. The procedure for processing of the data was used for both, the measured data and the data received by the simulations to ensure comparability [37].

5.1.2 Results of the Experiments on the Feed Rate

The results of the experiments showed the expected structure of the micro ground surface: The abrasive grits of the tool led to continuous scratches in feed direction. Perpendicular to the feed direction, the enveloping profile of the grits on the tool were imaged on the surface.

The evaluated surface parameters varied within one experiment for the different measurement positions. For the experiments with a feed rate of 1 mm/min the values for both, Sa and Sq, showed a decreasing trend from the beginning of the machined groove towards the end. This can be explained by the wear of the tool. With increasing feed travel, the abrasive grits are subjected to wear. Hence, the envelope of the tool changes leading to reduced surface parameters.

For the experiments using a feed rate of 5 mm/min the surface parameters are generally larger than those for a feed rate of 1 mm/min. In the experiments using a feed rate of 5 mm/min, the surface parameters for the middle measurement decreased which was explained due to wear. However, they increased again for the last measurement position on the left. The effect can appear due to severe wear of the tools: Due to wear the abrasive grits become smaller or even break out. This also partly changes the enveloping profile of the tool especially at the edge between tool face and peripheral surface. This leads locally to less material removal and hence increasing values of the surface parameters [37].

The results of the measurements show an advantage of the simulations: Especially the large grits cause rough surface topographies perpendicular to the feed direction which are characterized by steep gradients. Hence, due to the limited resolution of the optical measurement system, parts of the surface were not digitized correctly and missing height values within the measured data appeared. Since this is a physical constraint, it cannot be avoided by repeating the measurement or adjustments to the optical measurement procedure. These challenges do not occur when using simulations. Furthermore, since tool wear is not considered within the simulations, the tool model can be used for numerous virtual experiments which helps to compare the resulting surfaces. Hence, the investigation of the kinematic influence of the feed rate on the resulting surface topography is accessible when considering the results of the kinematic simulations [37].

Figure 14 exemplarily depicts the experimental results at the end of the groove and for both feed rates in comparison to the results received by the simulations. It is evident that the surface heights of the simulations fit to the real surface, showing characteristic scratches in feed direction due to the abrasive grits.

Regarding the simulations the discretization of the model led to minimized but not completely eliminable deviations. However, it was found that the values for both surface parameters only had a divergence below 10 nm. This difference can be neglected since the surface topography created by the grits is much more prominent. Hence, regarding the described tools with a nominal diameter of 400 µm and nominal grit sizes between 20 µm and 30 µm it could be stated that both simulated surfaces for the two different feed rates are almost identical when using the spatial discretization equal to the lateral resolution of the optical measurement system. On the other hand, the experiments showed larger surface roughnesses for the larger feed rate. Comparing the experimental results with the results of the simulation it can be concluded that the differences do not occur due to the tool kinematics. Further effects such as wear and material behavior influenced the experimental results [37].

Further details on the evaluation can be found in [37].

Fig. 14.

Experimental results of the surface heights exemplarily for the end of the groove. For two different feed rates the comparison to the simulation results is given according to [37]

5.2 Calculation of the Undeformed Chip Thickness

One process parameter which cannot be measured during machining is the undeformed chip thickness. It depicts the height of the removed material perpendicular to the trajectory of the cutting edge respectively to the grit. The undeformed chip thickness is an important measure to describe the grinding process, e.g. for determining the material separation mode when grinding brittle materials according to the theory of Bifano et al. [39].

Within the simulation, the undeformed chip thickness can be calculated using the difference in height values of the dexels for two consecutive time steps. However, the dexels are always aligned parallel to each other instead of perpendicular to the trajectory of the associated grit. Hence, an estimation of the error was performed. For the error estimation, the maximum feed rate was set to 5 mm/min and the minimum rotational speed to 30,000 rpm, depicting a parameter combination which is limiting for micro grinding with the described MPGTs. Lower feed rates or larger rotational speed would decrease the undeformed chip thickness and thus promote material separation in ductile mode. Using these parameters, estimated and approximated values for the undeformed chip thickness in the two dimensional case were calculated as follows:

First, the estimation of the undeformed chip thickness was calculated using the vertical distance between two trajectories in direction of the dexel for discrete calculation points. In Fig. 15a) the principle is marked with blue lines. The estimation of the undeformed chip thickness is depicted by the difference of the dexel heights of the two trajectories.

Secondly, the ideal consideration of the undeformed chip thickness was calculated according to its definition in the two dimensional case for one single grit. This means, that the distance between both trajectories is calculated in radial direction of the tool as it is shown as red lines in Fig. 15a). Hence, the direction is perpendicular to the second trajectory. The positions for the calculation were identical to the positions of the dexels enabling a direct comparison. The comparison between both principles is shown on the left in Fig. 15.

Fig. 15.

a) general principle for the calculation and estimation of the undeformed chip thickness (not to scale) and b) the corresponding results using the limiting process parameters

It shows the values for the undeformed chip thickness as a function of the position in cutting direction x. For x-values below approximately 60 µm, the difference between the calculation of undeformed chip thickness and the estimation via the dexel data is not visible in the diagram. For the x-values below 60 µm, undeformed chip thicknesses below 54 nm are estimated. The relative error was determined by calculating the difference between the ideal calculation of the undeformed chip thickness for the simplified two dimensional case, and the estimation via the dexels, divided by the value of the ideal two dimensional undeformed chip thickness. For the x-values smaller than 60 µm, the relative error was below 5%. Furthermore, the relevant x-positions can be limited: Since a large part of the resulting surface is machined by further grits, only the chip thicknesses that occur at small x-values are relevant for the final workpiece surface. In the extreme case, assuming that only one grit is located on the tool, the grit would penetrate the workpiece with each tool revolution. Hence, the maximum distance between two workpiece-grit-contacts would equal the feed per revolution, which has a value of 0.167 µm and can be set as maximum x-value. In this case, the relative error would be below 0.00004%.

As a conclusion, the vertical dexel data is suitable to approximate the undeformed chip thickness in micro grinding. The limiting values for the feed rate and the rotational speed represent parameters suitable for micro machining. However, for machining in ductile mode, the feed rate will be much lower and the rotational speed higher than the limiting values. Hence, the error between the approximation and the ideal two dimensional calculation of the undeformed chip thickness will be further reduced.

The determination of the undeformed chip thickness via the vertical dexel data was done by calculating the difference between the dexel heights in two consecutive time steps. Hence, for each dexel not only a single value for the undeformed chip thickness is received, but one for each time step. As a consequence, the value of the last time step which is not equal to zero was chosen. The reason is that only the final surface is considered which is generated with the last adjustment of the dexel height.

The undeformed chip thickness was analyzed using the same parameters as for the investigation of the feed rate with 5 mm/min. Figure 16 shows that the majority of the surface has a chip thickness in the expected range of nanometers. The large value for the undeformed chip thickness at low x-values appear due to run-in effects which will be avoided in further simulations.

The patterns in feed direction (x-direction) result from the necessary discretization within the simulation model. According to the simulation, the undeformed chip thickness is not equal for the complete surface: Some areas of the surface have very low undeformed chip thicknesses below 1 µm whereas they tend to be in the area of one-digit micrometers for other areas, especially for larger values of the z-direction. The reason is the grit distribution on the tool. Depending on the grit sequence and the grit protrusion height on the tool, the undeformed chip thickness increases or decreases. The effect especially appears when comparably large grit sizes are used. The conclusion of the simulation is that for a complete material separation in ductile mode, the process parameters have to be optimized.

Fig. 16.

Result for the undeformed chip thickness of the final surface estimated via the dexel data of the simulation.

6 Conclusion and Outlook

Within the IRTG 2057 a kinematic simulation model was built up to investigate micro grinding. A method was presented to virtually represent the MPGT using input data generated by particle size analyses and knowledge about the deposition rate of the electroless plating process. The tool model considered realistic grit geometries which made a detailed analysis of the resulting surface possible. Furthermore, the detailed grit geometries enabled the calculation of the undeformed chip thickness which’s values depend on the grit shape. The modeling of the tool was supported by analysis of the real MPGTs.

The kinematic model was validated by comparing the surfaces received by the simulations with topography measurements of real micro ground surfaces. Both surfaces depicted the characteristic scratches in feed direction originated by the abrasive grits. When using the lateral resolution of the measurement system as the distances of the dexels, the simulations showed that differences between the surface topography generated by the different feed rates were not based on kinematic effects.

The model also made the undeformed chip thickness accessible, a parameter that cannot be measured experimentally. It was found that for micro grinding applications with MPGTs the undeformed chip thickness can vary strongly due to a comparable wide range of abrasive grit sizes.

In future investigations, the kinematic simulation model will be used for further analysis of micro grinding regarding process results which are not available via measurements. This includes the detailed study of the influence of the feed rate on the resulting surface, as well as the study of the chip thickness regarding the machining of glass in ductile mode.