A material removal model for nonconstant-contact flexible grinding

Contact calculation is of great importance in predicting the material removal (MR) of flexible grinding process (FGP). The contact is mostly considered approximately constant in the existing MR models, while the situations that contact varies a lot after FGP are ignored. Therefore, a novel model is proposed in this paper to take those situations into consideration. Firstly, the nonconstant-contact situation is introduced. Then, an equivalent method is developed to convert the nonconstant-contact grinding process into the accumulation of several quasi-constant-contact grinding processes. Based on the equivalent method, a MR model is established, and the procedure to obtain the model parameters by the finite element analysis (FEA) is introduced. In the end, the equivalent method and the MR model are tested by a series experiments of different process parameters. Results show that the proposed MR model can predict the material removal effectively for the nonconstant-contact situations.


Introduction
Grinding is an important processing method of finish machining. To achieve higher quality of grinding, much effort has been made in studying the grinding process. In order to predict the undeformed chip thickness (UCT) of grinding, the relation between UCT and the specific grinding energy was investigated by Jin and Stephenson, which indicated an exponential relation between each other [1]. Zhang et al. investigated the maximum UCT in the face grinding of soft-brittle HgCdTe films [2], and Zhang et al. developed an integrated model with a workpiece model, a kinematic model, and a calculation model to describe and simulate the distribution of the UCT [3]. In terms of grinding force, Younis et al. proposed a model by taking the three stages (rubbing, plowing, and cutting) into consideration for the cylindrical grinding [4], which was improved by Durgumahanti et al. through further investigation on the plowing effect [5]. Dai et al. investigated the influence of the types of abrasive wear on grinding force and grinding ratio, and found different effect for three typical grinding stages [6]. Jiang et al. held the opinion that better understanding would be achieved by establishing a multi-factor model, and an integrated model was thus developed by taken the tool characteristics, grinding force, grinding temperature, and other grinding factors into consideration [7]. Now that rigid grinding tool and workpiece system is mainly adopted in conventional grinding, the removal depth is almost equal to the geometric interference depth, so that higher stiffness requirement is raised to reduce the residual stock on workpiece caused by the elastic deformation of the grinding system [8]. However, with the growth of stiffness, the interaction between grinding tool and workpiece would become more intense, which could introduce surface defects including scratches, micro-cracks, and even grinding burn [9]. Besides, it is difficult for conventional grinding to obtain a smooth surface for complex surfaces [10]. As a result, the flexible grinding process (FGP) was developed to reduce the possible damage of conventional rigid grinding by increasing the flexibility of the system. The researches of flexible grinding were reported to achieve a more moderate abrasive-workpiece interaction, a better surface integrity, and a more stable grinding process [11][12][13], indicating the potential wide application in ultra-precision machining.
Since the grinding tool is much softer than workpiece, the contact is different with that in the conventional grinding, resulting the invalidity of conventional material removal (MR) model. To develop the removal model suitable for FGP, some simplifications are mostly made including rigid workpieces and steady and constant contact. At the same time, the Preston equation or the Archard wear equation is mostly adopted for removal calculation, while it is necessary for the two equations to obtain the contact pressure between the tool and workpiece. For example, Rao et al. treated the grinding tool as a simple Winkle elastic foundation and the workpiece as a rigid base [14]. Zhu et al. assumed that the contact between the tool and workpiece obeyed the Hertz contact theory, and the original profiles of grinding tool and workpiece were chosen for calculation [15]. Considering the specific contact state, the finite element method (FEM) was also selected for contact calculation [16].
With the establishment of removal model, FGP was reported to be applied in the finishing stage of many precision parts such as aero-engine blades, optical lenses, and medical instruments [17][18][19]. Meanwhile, higher requirements were put forward to FGP for more situations, which made the mentioned simplifications irrational sometimes. For example, the deformation of low-rigidity parts may also influence the calculation of contact [20] and the contact may not be steady due to the grinding vibration [21]. Another typical situation is the correction of local error on the workpiece. Due to the tool wear and other reasons, it is inevitable that local error exceeds the limits of tolerance, such as the local protrusion caused by tool wear and the nonuniform allowance distribution of the precision-forged aero-engine blades, which is shown in Figs. 1 and 2 [22,23], respectively. In those cases, it is difficult to reprocess since the allowance of most area of workpiece has reached the tolerance limit, and it is also hard for local machining processes to maintain a smoothly transition between the processing area and the other area. Since the ability of FGP to achieve a smoothly transition, it was thus reported for local error correction. However, it can be seen that the profile of local area varies a lot after local processing, so that the assumption that the contact maintains constant is not suitable in this situation, introducing the failure of the exiting FGP removal models. In the above literatures 22 and 23, the removal model was developed through the empirical approach, which was considered unable to reveal the physics of the process and unbeneficial for the control of processing period and cost [15]. As a result, a novel MR model is proposed in this paper to take the nonconstantcontact situations into account.
The rest of this paper is arranged as follows. The nonconstant-contact situations and the equivalent method in FGP are explained in Sect. 2. Then, the MR model is developed in Sect. 3. In Sect. 4, a series experiments are carried out for verification. And in Sect. 5, results are analyzed and discussed. Conclusions are summarized in the last section.

Nonconstant contact in FGP
In order to explain the change of contact, an experiment was conducted with different feed rates. Figure 3a shows  the structure of the tool, and the actual tool in this experiment is shown in Fig. 3b and denoted as TOOL_1. The flexible tool used is composed of tool core, flexible substrate, and grinding film, the material of which is steel, nitrile rubber (NBR), and CBN sand belt, respectively. The workpiece is made from steel and pre-processed on a surface grinding machine. The grinding process is shown in Fig. 4a where the direction of feed is perpendicular to that of the grinding speed. A plate workpiece (W_P) is used and the surface after grinding is shown in Fig. 4b, where the grinding mark is changing gradually with feed rate, and the contact profile tends to be the geometric tool-workpiece interference when the total dwelling time increases.
To calculate the MR, a section S is taken for illustration which is perpendicular to the workpiece surface and parallel with the feed direction. CA is an arbitrary point on the intersection line of contact area and section S. The contact is shown in Fig. 5, where MN is the contact line. Before grinding, the contact is symmetric as described in most literatures shown in Fig. 5a. Due to MR, partial material at CA has been removed before M reached there, causing the decrease of actual tool offset within MN. Specifically, the amount of the decrease becomes larger from N to M, causing an unusual and asymmetric contact as shown in Fig. 5b. Suppose the contact pressure distribution of MN before grinding is P(x) for a given tool offset 0 . With the growing MR, the real contact pressure distribution within MN is not P(x) but P � (x) , P �� (x) , and so on as shown in Fig. 6, which is much different with the P(x) and can hardly be calculated directly.
To solve this problem, an equivalent method is developed to convert the unusual and asymmetric contact process into the accumulation of several quasi-constantcontact processes. The implementation method is to convert a one-pathed grinding with large MR into the accumulation of several repeated one-pathed grindings with small MR. Since the MR of each one-pathed grinding in the repeated grinding processes is small, the contact of each grinding process can be approximately regarded constant. As a result, a large MR can be calculated.

Explanation and deduction of the equivalent method
Let the MR in CA after a one-pathed grinding with feed rate F 0 be h F 0 , and call this process the single-pathed grinding (SPG). Then, for the several repeated one-pathed grindings, which is called multi-pathed grinding (MPG), the other parameters except for feed rate remain the same. Suppose the number of one-pathed grindings is N g . For the first time of one-pathed grinding in the MPG, let the feed rate be F 1 , then for the second grinding is F 2 , and so on for the last grinding is The equivalent method means that when the total dwelling time spent on the same grinding distance remains the same, the total MR is approximately equal for the SPG and MPG. That is, if then A discrete method is utilized to calculate the MR of the two grinding processes. The contact line MN is divided into several equidistant grinding segments, the number of which is N c . The segment is denoted as S j , and the tool offset at one grinding segment is considered equal and denoted as j , where 1 ≤ j ≤ N c . Some assumptions are made in the calculation as follows.
(a) The MR process is relatively slow in FGP, and the material is removed bit by bit. Therefore, a minimum unit dwelling time Δt m for each segment can be defined to calculate the MR, and when the dwelling time of each segment is Δt m , the grinding process is denoted as G * in this paper. It is clear that for any total dwelling time, a multi-pathed grinding can always be found through a superposition of several G * s. The number of G * is recorded as N m , and can be calculated as where T is total dwelling time. As a result, if the MR of the SPG and the MPG consisted of several G * is approximately equal for any N m , then the equivalent method can be seen correct. (b) For each segment on the flexible tool, the relation between the contact pressure P and the tool offset is the same, written as P = K( ) , where K is a certain function. (c) The MR can be calculated by Preston's equation. (d) The MR of any segment during a single Δt m is small compared with the tool offset at that segment, which can be calculated according to Preston's equation [24], written as where k pr is a constant related to the material properties and the unit of each parameter. As a result, Δh∕ is Feed direction considered of the same order as Δt m . For any two neighboring unit dwelling time of any segment, the MR of the former is a little bit larger than that of the latter, which can be obtained using Taylor expansion as Similarly, k pr K � ( )Δt m is also considered of the same order as Δt m .
(e) The shape of flexible tool is smooth, and the tool offset of each segment is symmetrical with respect to the perpendicular bisector of MN, which is reported in most literatures. Suppose N c is even, so that the tool offset is The MR can be calculated by accumulating the MR of each segment during each unit dwelling time Δt m . Given a total dwelling time, N m can then be calculated according to Eq. (3), and the total grinding number of all segments is N c N m . Some symbols are defined for simplification. The ratio of the MR of any segment during a single Δt m to the tool offset is written as Denote the temporary total grinding number of all segments as k, where 1 ≤ k ≤ N m N c . The MR of kth grinding is Δh k , and the total MR after kth grinding is h k . F( ) is defined as For the SPG, the process is that S 1 removes the material with N m times, then S 2 with N m times and so on until S N c with N m times. So where the superscript "S" represents the SPG. Since the removal of the first grinding is small, then the removal of the second grinding can be obtained using Taylor expansion as where the third and the higher orders of Δt m are ignored in this paper. Accordingly, for 1 ≤ k ≤ N c N m , the removal can be obtained as (5) the former: Δh former = k pr K( )Δt m the latter: where h S 0 is defined to be zero. For the MPG, the process is that all the grinding segments remove the material once successively from S 1 to S N c , then for the second time and so on until for the N m th time. For the first grinding with S 1 , the removal is where the superscript "M" represents the MPG. For the second grinding with S 2 , the removal is 0 is defined to be zero. By comparing Eqs. (10) and (13), it can be seen that the total removal after N c N m th grinding is consisted of the sum of several terms with different orders of Δt m , and the terms of the first order of Δt m are the same for the two grinding processes. As for the terms of the second order of Δt m , the sum of those can be written using the last assumption as where B l is the coefficient and can be described using Eqs. (10) and (13) as where b l,j,k is the coefficient. It can be verified that for any 1 ≤ l ≤ N c ∕2 , coefficients B S l and B M l are approximately equal for the two grinding processes. For example, coefficient B S 1 can be calculated as and coefficient B M 1 can be calculated as According to the fourth assumption, the subtraction of F( − h k ) and F( − h k * ) is considered of the second order of Δt m , where 1 ≤ k, k * ≤ N m N c . Since the number of the terms with F( l − h k ) is equal for any 1 ≤ l ≤ N c ∕2 in Eqs. (16) and (17), the subtraction of B S 1 and B M 1 is also of the second order of Δt m . As a result, the total MR is approximately equal for the two grinding processes, and the equivalent method can thus be supposed correct.

Derivation process of the model
Considering the MPG consisted of several G * , the number of G * is N m . For the ith unit grinding, G * denote the total cutting number of abrasives acting on CA as n i . Let P i (j) be the contact pressure in the jth segment. Since the distribution of abrasives is uniform, the MR can then be written according to Preston of contact pressure to the contact length in MN, denoted as p * i and called equivalent contact pressure in this paper. Considering the equivalent method is found, the MR in the SPG with feed rate F 0 can be calculated as As a result, the general differential form of the relation between MR h in SPG and the reciprocal of feed rate 1∕F is It can be seen from Eq. (22) that the relation between p * and 1∕F or that between p * and h is needed to calculate h . The latter relation is chosen in this paper, which only depends on the contact characteristic between the tool and workpiece. For the convenience of measurement, the position of the maximum MR, denoted as h 0 , is taken for example to explain and verify of the model. The equivalent contact pressure at that position is recorded as p * 0 . However, it is not easy to measure p * 0 in practice, so an assumption is made that the larger p * 0 is, the faster it decreases as h 0 decreases. This idea comes from some natural phenomena such as the heat conduction becomes faster when the difference in temperature of the two contact bodies enlarges, and can be written as where a and b are the coefficients. Integrate Eqs. (23) and (24) can be obtained, where c1 , c2 , and c 3 are the coefficients to be determined. The boundary condition is that the equivalent contact pressure approaches zero when the MR is close to initial tool offset, that is And it can thus be determined that c 3 equals 1. Coefficients c1 and c2 can be obtained by using the derivative of p * 0 to h 0 . Denote the derivative as Ks when h 0 is close to zero and as Ke when h 0 is close to initial tool offset, then they can be written as and (21) respectively. After c1 and c2 are determined, h 0 can be obtained by substituting Eqs. (24) to (22) and integral calculation. Meanwhile, the boundary conditions should be met, which are the initial MR is zero and the final MR is equal to tool offset, written as and Consequently, once Ks and Ke are obtained, the MR can then be calculated.

Method of determining Ks and Ke using FEA
As mentioned above, the equivalent contact pressure is of great importance to determine Ks and Ke . In this paper, the FEA is adopted for contact calculation for the tool structure is complex and the influence of sand belt is nonnegligible. The shape of workpiece is needed to explain the procedure and for the subsequent verifications. And since the edge processing of aero-engine precision-forged blades is a typical situation of nonconstant contact, a semicircular cylinder whose radius R equals 0.3 mm is modeled for example. The analysis is conducted on the software ABAQUS 6.14.

Modeling of FEA model
The simulation process is shown in Fig. 7a, where the pellets on the sand belt are ignored for simplification. A quarter model is built owing to the symmetry for analysis efficiency, and to improve the accuracy, the contact region is fine-meshed as shown in Fig. 7b. The rubber substrate and the sand belt are supposed linear elastic for the tool offset is mostly small compared to the size of the tool. The elastic constants of the sand belt are measured on the universal testing machine, and the hardness of the rubber is measured using a Shore hardness tester, which is used to calculate the Young's modulus. The density of each component is also measured to include the influence of centrifugal force. Properties of the parts are listed in Table 1.

Verification of the FEA model
Since it is not easy to measure the contact pressure precisely, the contact force and marks are used to verify the FEA model. Experiments are conducted on a plate workpiece (W_P) for observation. Results are shown in Tables 2  and 3. It can be seen that the maximum error of the simulated contact force is 0.2 N, which is less than 10% of the measured force. The simulation contact marks are close to the removal marks when the removal is very small. Thus, the FEA model can be considered reliable.

Ks determination
The calculation of Ks is shown in Eq. (26), but it is difficult to get the exact value; hence, an approximate way is adopted as Ks ≈ Δp * 0 ∕Δh 0 , where Δh 0 is small enough. As a result, the contact of the initial contact and the contact after a small removal Δh 0 are needed to be analyzed. The contact profile is simplified to be an arc, and the change of contact arc length ΔW is calculated as where W 0 is the initial contact arc length and W ∞ is length of the removed boundary when h 0 equals , as shown in Fig. 8.   A typical contact pressure distribution is shown in Fig. 9, and the equivalent contact pressure can be calculated as where P j is the contact pressure in the contact segments, and L is length of the element. Denote the initial equivalent pressure and that after Δh 0 removed as P * a and P * b , respectively. Then, Ks can be calculated as

Ke determination
Ke appears when the MR is close the given tool offset. Meanwhile, the contact profile is close to that of the tool as well. Thus, the profile of workpiece in the FEA can be modeled as a nearly complementary arc to the tool with the radius slightly larger than that of the tool, and the maximum MR can be firstly set as equal as the tool offset, as illustrated in Fig. 10a. At the same time, the tool should be in full contact with the workpiece instead of partial contact, and there should not appear stress concentration at the contact boundaries for the radius of workpiece profile should remain decreased, as shown in Fig. 10b. Therefore, the tool offset needs then to be readjusted to satisfy the contact conditions mentioned above. Denote the readjusted amount as Δh � 0 , and the equivalent contact pressure here as P * c , then Ke can be calculated as So far, coefficients c1 , c2 , and c 3 can be obtained through the mentioned procedure. Although there is a constant K * in Eq. (22), it can be determined by the calibration experiments for a certain material, and the whole MR model can then be obtained.

Comparison in MR of SPG and MPG
To verify the effectiveness of equivalent method, experiments are conducted under three different conditions. Since the shape of the grinding tool and the workpiece are mainly concerned in practice, another type of grinding tool (TOOL_2) and a cylindrical workpiece (W_C) whose radius is 3 mm are introduced as shown in Fig. 11 besides TOOL_1 and the plate workpiece (W_P). The arrangement of the experiments is listed in Table 4. As for the grinding parameters, they are set the same for each condition. The tool offset and spindle rotation speed is 0.3 mm and 10,000 r/min, respectively. Three groups of feed rates are tested under each condition for the SPG, denoted as F A , F B , and F C , and for each feed rate of F A , F B , or F C , several groups of feed rates in MPG are set to remain the total dwelling time identical. The arrangement of feed rates is listed in Table 5 where the unit is mm/min. Experiments are carried out on a grinding machine QMK 50A, and each experiment is repeated three times (E1, E2, and E3) to reduce the contingency. The maximum MR h 0 is collected for comparison which is measured by the Taylor Hobson profilometer.

Verification of the MR model
Fillets of three sizes R0.1, R0.2, and R0.3 are pre-milled as the workpiece to simulate the edges of the aero-engine blades as shown in Fig. 12a, grinding machine QMK 50A and TOOL_1 are used for the test, and the schematic of grinding process is shown in Fig. 12b. The tool offset for fillets of R0.1, R0.2, and R0.3 is set to 0.1 mm, 0.2 mm, and 0.3 mm, respectively, and the spindle rotation speed is set to 10,000 r/min. Denote the reciprocal of feed rate 1/F as 1 when the feed rate is 2000 mm/min, and the scope of 1/F is set from 0.125 to 128 by adjusting the feed rate. Each experiment is repeated three times to improve the reliability of the data. The maximum MR h 0 is measured on the coordinate measuring machine HRSW PONY.

Comparison experiments
Material removal of MPG and SPG under three conditions is listed in Table 6, and the average removal is shown in Fig. 13 for comparison. The fluctuation of MR in each repeated experiment is no more than 3 μm and 5% of the average MR, indicating the reliability of the experimental results. For each condition, the average MR comes to 93%, 49%, and 80% of the given tool offset, respectively. The maximum error between the two grinding processes is 5 μm for all the data, and 3.7 μm for the average MR. For the data and the average MR with maximum error, the relative error is 1.7% and 1.3%, respectively. Considering the error caused by tool wear and measurement, the results demonstrate that within a quite large range of MR, the MR of the two grinding processes remains approximately equal independent of the shape of the tool and workpiece. Consequently, the equivalent method is validated.

Verification experiments
MR results of three group radii are illustrated in Fig. 14.
The maximum error for all the data is 5 μm, the relative error of which is 5.5%. Results of the three repeated experiments show good accordance with each other, so that the MR under each 1/F is averaged for representation below. As mentioned above, a group of experiments are needed for calibration. In this paper, the results of R0.3 are selected. Different values of K * are selected for   comparison as shown in Fig. 15a. It can be seen that when the MR is small, the relation between h 0 and 1/F can be seen nearly linear as described in most existing models, while when MR becomes large, a strongly nonlinear relation appears instead. When the value of K * lies in 1 ~ 1.2, the deviation between the prediction and the measurement is relatively small, so the value of K * is set to be 1.2 in rest of experiments, then the MR for R0.1 and R0.2 can be calculated and are compared as shown in Fig. 15b. It should be pointed out that the real tool offset when R equals 0.1 mm is not 0.1 mm but 0.12 mm. This is due to the error of tool alignment, and it is adjusted when calculation. Coefficients c1 and c2 are then calculated using the mentioned method, as listed in Table 7. Fig. 15 indicate that the calculated MR is in good agreement with that of the experiments. For each measured data, the error of the proposed model is shown in Fig. 16. The maximum error is 15 μm, and the mean error is 4.6 μm for all measured data as shown in Fig. 16a. The relative error is mostly less than 10% with the percentage of 91%, the maximum relative error is 32.8%, and the mean relative error is 5% as shown in Fig. 16b. Though the maximum relative error is large, the real error there is less than 10 μm. The possible error sources include the simplification of FGP, the equivalent method, FEA model, and tool wear. In general, the prediction error of the MR model lies in the interval of (− 10, + 15) μm, and mostly less than 10 μm with the percentage of 85%, which  is almost of the same precision as the finish milling. As a result, the MR model can meet the manufacturing tolerance of most components such as the aero-engine blades whose manufacturing tolerances are usually 0.1 mm, and the model can thus be validated.

Conclusion
In order to predict the material removal of flexible grinding process for the nonconstant-contact situations, a novel MR model is established in this paper. The equivalent method is explained that the material removal of the single-pathed grinding can be converted to that of the multi-pathed grinding. The material removal model is then deduced based on the equivalent method and developed by using FEA. Experiments show that the error caused by the equivalent method is less than 3% under different grinding conditions, and the material grinding experiments show that the maximum error and the mean error of the proposed model are 15 μm and 4.6 μm for a large range of material removal, respectively, which indicates an acceptable accuracy in practice and the effectiveness of the material model. Data availability All data and materials generated or analyzed during this study are included in this manuscript.

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Competing interests
The authors declare no competing interests.