In the context of non-standard blade geometries of knife-like tools with tapered cutting edges where the width of the blade surface varies with feed, there is limited research on predicting grinding forces considering the changing contact line. To enhance the accuracy of predicting grinding forces during the blade surface grinding of knife-like tools, a novel analytical-regression correction method is proposed. This method employs an analytical approach to analyze the varying contact line between the grinding wheel and the tool during grinding, enabling the determination of irregularly shaped grinding contact zones. By introducing exponential coefficients related to the grinding contact line, a regression analysis is employed to refine a variable edge-width grinding force model. In comparison to the conventional constant contact line blade surface grinding force prediction, this model is better suited for non-standard blade geometries of knife-like tools in grinding processes. Results indicate that the average relative error between the predicted values from the variable edge-width grinding force model and the actual measurements remains within 9 %, thereby validating the model's effectiveness in predicting grinding forces.

High-precision blade grinding of tools is a crucial step in the modern tool manufacturing process chain. Accurate prediction of grinding forces can enhance the prediction of blade surface deformation during the grinding of dagger-shaped tools, thereby improving subsequent deformation compensation accuracy. High-precision blade grinding can enhance tool lifespan, reduce cutting forces, and minimize residual surface stresses during machining, effectively boosting the cutting performance of tools. To this end, the field has conducted impactful research, such as the ultrasonic dagger blade (Wang et al., 2017), which is widely used in the aerospace composite material manufacturing field and involves a series of complex processes in its manufacturing, including heat treatment, wire cutting, edge grinding, and coating. Among these processes, the edge-grinding process is the core of the entire process chain, and the precision of the edge formation is related to the cutting force state and the axial vibration performance of the tool (Sun et al., 2023). Due to the cantilever clamping scheme used during the grinding of dagger blade edges, the tool exhibits a slender and thin geometry, resulting in a relatively small bending stiffness of the tool. Additionally, the thickness of the blade gradually decreases during the grinding process. Therefore, during grinding, the blade blank is highly susceptible to elastic deformation caused by grinding forces, and after grinding, it elastically recovers, leaving some material residues on the blade edge. As a result, the blade edge takes on a convex shape, affecting the precision of the edge shape and consequently impacting the service life and ultrasonic cutting performance of the dagger blade.

The grinding force model serves as a representation of the relationship between various physical quantities in the grinding process. It enables the optimization of grinding conditions and parameters to achieve the desired grinding outcomes (Fu et al., 2015). In their work, Sun et al. (2015) conducted indentation experiments and divided the material into two stages: a ductile stage and a simultaneous ductile–brittle stage based on the depth of the tough–brittle transition. They first derived the grinding force per unit width of a single abrasive grain separately for each stage. Subsequently, by considering the grinding contact arc length and the total number of abrasive grains, they integrated the expressions for the total grinding force in both stages (Sun et al., 2015). Zhang et al. (2018) considered the plastic build-up mechanism of material removal and various lubrication conditions. They simulated the depth of cut for each dynamically effective grinding grain and introduced the plastic build-up rate parameter. Based on these considerations, they developed a single-grain grinding force model. Furthermore, they synthesized the grinding forces generated by individual grains to calculate the total grinding force (Zhang et al., 2018). Similarly, Li et al. (2019) considered the effects of the strain rate, the random distribution of the abrasive radius, and the depth of the elastic–plastic transition in a comprehensive manner. They used a genetic algorithm to train key model parameters using experimental force data, thereby establishing a grinding force model. The modeled force agrees well with the measured force. This model enabled an in-depth understanding of the deformation mechanism of a crystal solid involved in ultra-precision grinding and the effect of the strain rate on its material removal (Li et al., 2019). Jamshidi et al. (2019) discretized the grinding wheel into cells along the axial direction according to the width of the cutting edge of the grinding grain, established the equations for the dynamic radius of the grinding grain on the cell and the total number of grinding grains in each cell, established the equations for the displacement of a single grinding grain in the

Therefore, a grinding force model based on the variable grinding contact area has been established to effectively overcome the limitations of pure theoretical modeling or empirical modeling. This model takes into account the influence of practical factors and can rapidly and efficiently obtain a high-precision grinding force model with only a small number of samples. The subsequent sections of this paper are organized as follows.

In Sect. 2, an analytical regression correction method is proposed. Currently, there is no grinding force model specifically addressing the widening of the grinding width for dagger blade conical edges. This makes it difficult to accurately predict the grinding forces when irregular material removal volumes occur. To address the characteristics of the widening of the grinding width along the feed direction for dagger blade edges, an equivalent mapping method for the grinding contact area is introduced to obtain the removal area of irregular shapes. In Sect. 3, an experimental setup is designed and constructed for experiments. Using regression analysis with the introduction of exponential factors related to practical factors, the grinding force model from Sect. 2 is validated. Finally, a comprehensive summary of the contributions made in this paper is provided in Sect. 4.

Currently, most grinding force modeling is directed towards regular components to investigate the influence of various process parameters on grinding forces. However, when grinding dagger blade edges, the shape of material removal changes with different cutting depths and feed lengths, making it challenging to establish a grinding force model for the entire blade surface grinding process.

Schematic diagram of ultrasonic dagger grinding.

The diagram shown in Fig. 1 illustrates the schematic of grinding the oblique blade surface of an ultrasonic dagger blade. It represents the linear reciprocating grinding of the oblique blade surface at a specific grinding depth through the end face of a bowl-shaped cubic boron nitride (iCBN) grinding wheel. In the diagram,

The grinding force generated by a single abrasive grain is related to the depth of cut of the abrasive grain and the thickness of the abrasive chip (Li et al., 2017), which can be expressed as the grinding force per unit grinding area, and the expression for the unit grinding force is as follows:

The material of the workpiece is primarily removed by the abrasive grains at the outermost periphery of the grinding wheel's end face. In this paper, the equivalent mapping of the grinding arc at different feed positions of the grinding wheel is carried out, and it is mapped to the grinding arc of the grinding wheel when grinding square workpieces, as shown in Fig. 1. This includes the equivalent grinding arcs in Zone I and Zone II. Here,

Assuming that the quadrilateral cone abrasive grain is ground using a tapered surface, the abrasive chips are produced by extrusion of the workpiece material, the grinding force on the front tool surface of the abrasive grain is the contact area of the abrasive grain and the abrasive chips multiplied by the grinding force per unit area (Younis et al., 1987), and the grinding force is established in this cutting attitude:

Based on the average undeformed grinding thickness and grinding depth, find the contact area between the front surface of the abrasive grain and the abrasive chip (Younis and Alawi, 1984):

The average cross-sectional area of the abrasive chips can be obtained as

When grinding, the abrasive grain produces a circular surface or a small plane because of the abrasive tip, which slips and rubs against the workpiece surface, causing elastic deformation of the workpiece surface and generating frictional forces. The normal and tangential friction forces generated by the slip friction of a single abrasive grain (Tang et al., 2009) are

The average contact pressure is based on the study of Malkin (1989), which gives

According to the undeformed average grinding thickness and Eq. (9), the following are obtained.

Relative motion occurs between the abrasive chip formation and the front tool surface of the abrasive grain, generating normal friction and tangential friction.

In this paper, the single abrasive grinding force is divided into the chip formation force and friction force, where the friction force is divided into the friction force when the abrasive chips flow out and the friction force generated by the slip friction of the abrasive particles:

The following are the total normal force and total tangential force during grinding.

The unknown coefficients in Eqs. (20) and (21) are calculated based on the regression solution of the grinding force data measured by the grinding experiment, and Eqs. (20) and (21) calculate the total normal grinding force and the total tangential grinding force on the grinding wheel, while the grinding force on the workpiece in the

Decomposition of the grinding force on the workpiece.

The grinding experiments are conducted on a self-developed non-standard ultrasonic tool dedicated five-axis four-linkage tool grinder (as shown in Fig. 3). The grinder's main spindle is an electric spindle, and it can grind various non-standard ultrasonic tools through a circulating oil-cooling system. The specific specifications and parameters of the machine tool are shown in Table 1.

Grinding test platform.

Main specification parameters of grinding machines.

The workpiece material selected for this experiment is AISI M2, and the workpiece dimensions for the regression coefficient grinding force experiment are 5

AISI M2 material performance parameters.

Experimental grinding wheel parameters.

The grinding experimental platform consists of a grinding process system and a grinding force measurement system as shown in Fig. 4. The grinding process system includes a grinding wheel, workpiece, force measurement table, and fixed base plate. The force measurement table is connected to the fixed base plate and the swivel seat of the grinding machine's

Grinding test platform.

The technical specifications of ME-K6D40 and HSGD6X-EC.

The force sensor model in the experiment is ME-K6D40 manufactured by ME-measurement systems GmbH in Germany, and it is paired with a signal amplifier, the HSGD6X-EC. The specific technical parameters are shown in Table 4. The data acquisition software employed is Huasoft. After obtaining the experimental data, the grinding force values are obtained through filtering and processing in MATLAB.

Unlike conventional grinding, the material removal rate during ultrasonic scalpel blade grinding is high. According to the reference (Li et al., 2011), a gradual feed grinding process can be used to grind high-hardness materials, with a grinding depth of 0.1–30 mm, a feed rate of 5–500 mm min

By analyzing the grinding force model in Sect. 2.5, it can be understood that grinding force is influenced by process parameters. In this experiment, variations in grinding force were studied by altering the grinding wheel speed, workpiece feed rate, and grinding wheel depth of cut. A full factorial experimental design was employed, with each parameter having three levels as shown in Table 5. Grinding forces were measured for all combinations of process parameters, and each set of parameters was tested three times to reduce random errors. During the experiment, efforts were made to maintain consistent grinding conditions. Severely worn grinding wheels were promptly dressed, and pre-grinding was performed on the dressed wheels. The grinding state was monitored by measuring the grinding forces using a force measurement table to ensure stability. The experiment proceeded only when the grinding wheel was in a stable grinding state.

Main specification parameters of grinding machines.

To validate the proposed theoretical grinding force model in this study, changes in grinding forces were investigated by varying the wheel speed, workpiece feed rate, and grinding depth. A full factorial experimental design was employed, with each parameter set at three levels. Grinding forces were measured for all combinations of process parameters, and each parameter combination was repeated three times to reduce random errors. The measured grinding force values are shown in Table 6. By regression analysis of the experimental data, the grinding force coefficients in Table 7 were obtained.

Grinding force experimental data.

Grinding force coefficients.

The final obtained grinding force models are

The relationships between the test and theoretical values of the grinding force and the effect of each parameter are shown in Figs. 5 and 6. When

The reasons for errors in measurement are as follows.

The grinding wheel exhibits runout errors due to manufacturing precision, causing vibrations during grinding. When the grinding depth is small, this can affect the actual grinding force measurement.

The actual shape of the abrasive grains on the grinding wheel is complex. In this study, the abrasive grains are simplified as single tetrahedral grains, and the cutting angles are determined. This simplification may not accurately represent the material removal process of all the abrasive grains.

The modeling of grinding forces did not take into account the thermal softening effect of the workpiece material caused by grinding heat.

Due to multiple repeated experiments, the wear of the grinding wheel can also affect the measurement of grinding forces before and after each set of experiments.

In summary, when the workpiece width is 5 mm, the average relative error between the predicted values from the grinding force model established in this paper and the measured values is within 10 %, indicating that, within the range of experimental process parameters, it is effective at predicting grinding forces.

The grinding force model established was further verified by measuring the grinding forces of workpieces with different widths to effectively predict the grinding forces of dagger knives with different equivalent sharpening widths, and the experimental and predicted values of grinding forces are shown in Figs. 7 and 8. The cross-sectional dimensions of the ground workpiece are

In Figs. 7 and 8, at the same grinding depth, the experimentally measured grinding force increases with the grinding width in a nonlinear relationship, with an overall approximate exponential increase, and the theory predicts a linear relationship between the grinding force and the grinding width. The actual grinding force refers to the grinding force measured during experiments, while the predicted grinding force refers to the grinding force predicted by the grinding force model. When

When

Since the actual grinding force and the workpiece width are exponentially related and the equivalent grinding width varies along the feed position during dagger sharpening, which affects the maximum thickness of the grinding chips and the material removal mechanism and force ratio, in order to improve the prediction accuracy of the grinding force model, this paper introduces the exponential coefficient related to the grinding width, and based on the experimental data, the variable blade width grinding force model is modified to obtain

where

In order to verify the reasonableness of the introduced index, the actual grinding forces and predicted values for different process parameters and workpiece widths were verified by comparison as shown in Figs. 9 and 10.

In Fig. 9 the maximum relative error between the predicted grinding force in the

In Fig. 10 the maximum relative error between the predicted grinding force in the

Based on the aforementioned experiments, establishing the variable edge-width grinding force model through equivalent mapping and introducing the corrective coefficient for the grinding width are reasonable. This approach enhances the predictive accuracy of the model, resulting in an average relative error within 9 %. Thus, the established grinding force model is capable of predicting grinding force values for different equivalent edge widths, accurately forecasting the grinding force during dagger blade edge grinding. In comparison to traditional constant contact line blade surface grinding force prediction with an average error within 9 % (Yang et al., 2023), this further enhances the predictive accuracy.

In the context of grinding the tapered blade surface of dagger-shaped tools where the arc length of contact changes with feed, the current prevalent fixed contact arc length grinding force prediction methods fail to address the issue of changing arc length during grinding. Therefore, a grinding contact area equivalent mapping method has been proposed to rapidly and efficiently obtain a variable contact line model during the blade grinding process, enabling the acquisition of irregularly shaped material removal areas.

Since the contact arc length is approximately equal to the workpiece width and the actual grinding force exhibits an exponential relationship with the workpiece width, in order to further investigate the influence of the contact arc length on the grinding force prediction, exponential coefficients related to the grinding width are introduced. These coefficients are calculated through regression analysis and incorporated into the derivation of the variable edge-width grinding force model. The predicted values from this variable edge-width grinding force model have an average relative error within 9 %. This is beneficial for subsequent computational analysis of grinding forces and deformation during the actual grinding process of dagger blades, providing a theoretical foundation for subsequent grinding process compensation.

The grinding force model established in this paper is for static stability and does not consider the dynamic grinding processes, such as the influence of wheel deformation and vibration on grinding forces, which may deviate from real-world process systems. Further research is needed to investigate the grinding characteristics of the ultrasonic dagger blade, including enhancing the accuracy of the grinding force model for the dagger blade and considering the loading method and material removal mode. Subsequent research should also take into account the impact of material plastic deformation on grinding deformation.

Code and data are available on reasonable request.

Conceptualization: BY, TL, DC, JR, and WL; methodology: BY, TL, and DC; validation: BY; investigation: BY and TL; data curation: JR, WL, and YC; writing – original draft preparation: BY; writing – review and editing: BY and TL; visualization: JR; supervision: WL and YC. All the authors have read and agreed to the published version of the paper.

The contact author has declared that none of the authors has any competing interests.

The funding institution did not participate in and did not affect the research design. Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research was supported by the National Natural Science Foundation of China (grant no. 51975173) and the Zhejiang Public Welfare Technology Research Program/Industry Project (grant no. LGG21E050010).

This research has been supported by the National Natural Science Foundation of China (grant no. 51975173) and the Zhejiang Public Welfare Technology Research Program/Industry Project (grant no. LGG21E050010).

This paper was edited by Jeong Hoon Ko and reviewed by two anonymous referees.