# Study of the Movement of Chips during Pine Wood Milling

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Theoretical Approach

#### 2.2.1. Analysis of Chip Morphology

_{0}is the tool’s back angle, and the back angle of each cutting edge is equal. The front angle of each cutting edge is 0°, whereas d

_{0}and d

_{1}refer to the widths of the earlywood and latewood layers, respectively.

_{0}= 0° and a back angle of ɑ

_{0}= 30° on the cutting edge. The milling cutter is applied to the wood with the annual ring pattern shown in Figure 2a. The lengths of the cutting edges 1 to 5 are denoted by l

_{1}to l

_{5}, and the inclination angle λ

_{0}= 45° is applied to edges 2 and 5. Thus, the #1, #3, and #4 cutting edges are computed with single-layer properties during cutting, whereas the #2 and #5 cutting edges are calculated using multi-layer properties. For instance, if l

_{2}cosλ

_{0}< (d

_{0}|d

_{1}) for the No. 2 cutting edge, it is computed as single-layer property, and vice versa.

_{w}(i.e., the width of the cutting edge), and the single cutting depth be a′

_{c}(i.e., the depth of cutting into the workpiece by the No. 1 cutting edge). The value of a′

_{c}can be expressed as a function of the feed rate V

_{f}and spindle speed n:

_{F}represents the bending moment induced by the cutting chip.

_{c}and F

_{t}be the horizontal and vertical components of the cutting force, respectively, between the circumferential milling cutter and the wood. The equations for F

_{c}and F

_{t}are presented as (6) and (7) in reference [17]:

_{m}is the radius of chip deflection, r

_{f}is the radius of the wood fiber sieve tube, v

_{olf}is the wood fiber volume fraction, and S

_{m}is the wood shear strength in the corresponding direction. Additionally, φ denotes the rotation angle, μ is the friction coefficient, and θ represents the angle between the annulus and the horizontal plane.

_{c}, and the component force perpendicular to the annual layer pointing in the direction of the wood specimen is represented by F

_{t}. As illustrated in Figure 3, the wood chips are subjected to a force N, which can be described by the following Equation (8) [18]:

_{0}of 5° and a trailing angle of ɑ

_{0}of 30°. This cutter angle serves to enhance tool life but results in average cutting surface quality, as per previous research [17]. In Figure 3c, the wood damage mainly comprises interfiber tearing. This damage can be predicted by calculating the average bending strength σf, influenced by factors such as the percentage of earlywood and latewood layers and the gap. Equation (9) presents the prediction model for chip length L

_{1}, which is formed by the cutting edges 1, 3, and 4. Accordingly, the shape of the chip produced by these cutting edges can be approximated by a rectangular sheet of length L

_{1}, width a

_{w}, and thickness a′

_{c}, without accounting for the real-time effect of feed rate V

_{f}on a′

_{c}.

#### 2.2.2. Modeling of Chip Boundary Surface f(x,y)

_{0}, it must satisfy V

_{0}= 2πnr, where r corresponds to the radius of gyration of the knife tip. Taking the air resistance coefficient as C = 0.5 and the air density as ρ

_{a}= 1.3 g/L, with a maximum windward area of S = L

_{a}× a

_{w}and the mass of wood chips as m, we obtain the velocity V

_{t}of the wood chips at time t, as shown in Equation (10):

_{1}wood chips above the y-axis in the absence of air resistance during free fall. The M

_{1}coordinates can be described using variables, as presented in Equation (11). The highest point reached by the wood chip can be calculated using Equation (12):

_{2}in the z-direction is zero as depicted in Figure 4c. The variation laws of angles β′

_{yoz}and β′

_{xoy}for any wood chip M

_{3}on the boundary surface are represented in Equation (13) as follows:

_{yoz}and β′

_{xoy}are linearly related. That is to say, β′

_{yoz}= kβ′

_{xoy}/2 + b. The relationship between the two angles can be solved using Equation (14) presented below:

_{3′}s displacement in the xoy plane, as given in Equation (15). Furthermore, the time t is represented by Equation (16). Equation (18) accurately describes the relationship satisfied by β′

_{xoy}:

_{3}point can be described as in Equation (19):

_{1}(x,y), for the boundary surface of the wood chip in the x+, y+, and z+ space. The collapse of β′

_{xoy}is achieved by substituting Equation (18) into Equation (21), resulting in the following equation:

_{3}in the x+, y+, and z+ space can be expressed using Equation (23). Upon completing the necessary steps, the equation for the boundary surface of the wood chip in the x+, y+, and z+ space can be obtained as shown in Equation (21), and is represented by f

_{2}(x,y):

_{1}(x,y) and f

_{2}(x,y) about the yoz plane. This implies that the boundary surface for the x-directional chip can be obtained by replacing x with -x. It is worth noting that in Equations (22) and (24), V

_{t}refers to the velocity on the diffuse boundary surface at time t. By substituting Equation (16) into Equations (10), (14) and (18), we obtain the results as shown in Equation (25). The values of β

_{yoz}and β

_{xoy}are related to the spindle speed n, feed rate V

_{f}, and depth of cut a

_{c}, which shall be regressed polynomially later.

#### 2.3. Methods

_{yoz}, the projected angle along the z-axis β

_{xoy}, and the average chip size L

_{a}. During milling operations, if the spindle speed drops below 6000 r/min, the machine will produce excessive noise and vibrations. The maximum spindle speed for the machine is 11,500 r/min; therefore, the allowable range for spindle speed n is 6000 r/min to 11,000 r/min. Similarly, the upper limit for feed speed is based on the maximum feed speed allowed by the machine of 1800 mm/min; thus, the range for feed speed V

_{f}is 430 mm/min to 1770 mm/min. The depth of cut is also subject to the machinable range of the milling cutter, and the range for cutting depth a

_{c}is designated between 5 mm to 15 mm.

## 3. Results and Discussion

_{yoz}pertains to the forward-looking spreading angle, representing the projection of the chip spreading angle β along the x-axis, whereas β

_{xoy}denotes the top-looking spreading angle, representing the projection of the chip spreading angle β along the z-axis. The parameters L

_{a1}to L

_{a4}are the average chip sizes that are evenly spaced, with L

_{a}representing the overall average chip size. The numbers 1 to 20 in the table correspond to the CCD test numbers, whereas 21 to 26 are the supplementary experiment numbers that were utilized to study the chip size distribution.

_{a}< 0.2 mm). In this paper, we primarily focus on the main diffusion zone.

#### 3.1. Effect of Milling Parameters on the Orthogonal Diffusion Angle β_{yoz}

_{yoz}mentioned in this paper refers to the angle clamped by the tangent lines of the upper and lower boundaries at the initial point, as shown in Figure 5a. The analysis of the experimental data indicated that the overall model regression was highly significant (p < 0.0001). However, the depth of cut a

_{c}had almost no effect on the orthogonal diffusion angle because the p-value of a

_{c}is greater than 0.1. The analysis of variance for the regression model is presented in Table 4. The polynomial obtained from the regression has been illustrated in Equation (26). The visualization of the chip ortho-visual diffusion angle β

_{yoz}is shown in Figure 6 for better understanding.

_{f}results in an increase in the chip orthogonal diffusion angle β

_{yoz}. Consequently, the cutting force and its amplitude also increase [17], making the cutting condition more severe. As a result, the standard deviation of the generated chip size becomes larger and the range of the initial direction of chip movement becomes more random, leading to an increase in the area of the main diffusion zone. On the contrary, increasing spindle speed n and reducing feed speed V

_{f}leads to a softer cutting process, with smaller chip sizes and a reduced range of initial velocity direction. This, in turn, results in a decrease in the positive diffusion angle β

_{yoz}. Notably, β

_{yoz}is found to be independent of the depth of cut a

_{c}. This suggests that the direction of the initial chip shot on the plumb plane is largely independent of the chip shape and size.

#### 3.2. Effect of Milling Parameters on the Top View Diffusion Angle β_{xoy}

_{xoy}in the experiment. Table 5 displays the analysis of variance for the regression model. To improve the regression effect, a more complex regression polynomial of β

_{xoy}was required compared to that of β

_{yoz}, as shown in Equation (27). Figure 7 depicts the visualization of the chip top view diffusion angle β

_{xoy}.

_{xoy}generally increases with increasing depth of cut a

_{c}, and feed rate V

_{f}. The theoretical analysis described in the previous section suggests that the chip diffusion angle in the xoy direction is determined by the direction of the chip at the moment it leaves the cutting edge. The time for this release is influenced by the centripetal force provided by the cutting force and the friction coefficient between the cutting edge and the chip. As the depth of cut and feed rate increase, the average chip size also becomes larger. This is due to the fact that large chips have a larger contact area, which causes them to break away from the cutting edge more slowly. Additionally, increasing chip size results in higher air resistance and a corresponding increase in positive pressure, ultimately slowing down the release time of the chip from the cutting edge, leading to an increase in the top view diffusion angle β

_{xoy}.

_{xoy}demonstrates inconsistencies with the theory in the range of a

_{c}= 7–9 mm and V

_{f}= 900–1500 mm/min. This may be due to the fact that the depth of cut is too small for all five cutting edges of the circumferential milling cutter to be fully involved in cutting. Furthermore, at higher feed rates, chips experience greater air resistance and less collision, resulting in a deviation from the expected pattern. However, as spindle speed n increases, the chip has a greater initial velocity, which tends to eliminate the observed deviation.

#### 3.3. Effect of Milling Parameters on the Average Chip Size L_{a}

_{f}, and depth of cut a

_{c}, revealed that all three contributed significantly to the average chip size La (p-values less than 0.0004 for each factor). The analysis of variance for the regression model is presented in Table 6, and the regression polynomial for La can be found in Equation (28). A visualization of the mean chip size L

_{a}is displayed in Figure 8.

_{a}increases as the depth of cut a

_{c}increases, the spindle speed n decreases, or the feed rate V

_{f}increases. These results align closely with previous research in the field of wood processing [17] and are consistent with the theoretical analysis presented earlier. Notably, the depth of cut has the strongest impact on the average chip size when compared to the other variables.

#### 3.4. Distribution State of Chip Size

_{j}. The R

^{2}value even reached 1 when performing the second-order exponential autoregression on the data. To avoid overfitting, an exponential model without constant terms was utilized in the data analysis. The regression results remained effective.

_{j}in the sampling direction, as shown in Figure 9a. This indicates that the cutting stability increases with an increase in spindle speed n. This finding is generally consistent with the analytical results demonstrated in Figure 6 and the trend observed in the Ogun PS study [22] in terms of spindle speed and chip size. Moreover, the mean value of the sampled chip size L

_{j}data increased as the spindle speed n increased, which is consistent with the analytical findings in Figure 8.

_{f}led to a more uneven distribution of the sampled chip size L

_{j}in the sampling direction, as demonstrated in Figure 9b. This indicates that the cutting stability decreases as the feed rate V

_{f}increases, which is consistent with the analytical findings presented in Figure 6 and also in alignment with results from Ispas.M on cutting stability aspects [23]. Additionally, the mean value of the sampled chip size L

_{j}data increased as the feed rate V

_{f}was increased, which is in agreement with the analytical results demonstrated in Figure 8.

_{c}led to a more uneven distribution of the sampled chip size L

_{j}in the sampling direction. The distribution increased dramatically between a

_{c}= 10 mm and a

_{c}= 13 mm, as illustrated in Figure 9c. This indicates a decrease in cutting stability with an increase in cutting depth a

_{c}, which is consistent with the analysis results presented in Figure 7 and the findings of Ogun PS [22]. Furthermore, the mean value of the sampled chip size L

_{j}data increased as the depth of cut a

_{c}was increased, which is consistent with the analytical findings demonstrated in Figure 8.

#### 3.5. Solution of Chip Boundary Surface Model f(x,y)

_{1}(x,y) and f

_{2}(x,y). Moreover, in situations where the distance between the milling plane and the table is 100 mm, the theoretical chip dispersal area on the table corresponds to the area enclosed by the two surfaces of f

_{1}(x,y) and f

_{2}(x,y), as well as the plane of z = −100 mm. As illustrated in Figure 10, the simulated chip boundary surfaces closely resemble the actual spread of chips generated during cutting. The area between the two surfaces effectively corresponds to the genuine chip dispersal area.

_{yoz}or β′

_{xoy}tends towards its maximum value on one axis when the other axis approaches zero. In this study, a linear relationship was assumed between the two, as presented in Equation (14). Although this method does not affect the limited position of each axis of the wood chip diffusion boundary, the surfaces are not smoothly connected. Therefore, future research can explore the nonlinear relationship between β′

_{yoz}and β′

_{xoy}. On the other hand, as the movement process of very fine chips is difficult to observe, the vortex region caused by the wind force of the milling cutter was not analyzed in this paper. As such, future observations and studies can be carried out to investigate the movement of extremely small chips near the milling cutter. Furthermore, if we further investigate the impact of tool angle on chip diffusion boundary based on the findings of this study, it would facilitate extending the methodology proposed in this paper to a wider scope.

## 4. Conclusions

_{a}and chip diffusion boundary surface model f(x,y) and used CCD testing to analyze the key parameters β

_{yoz}and β

_{xoy}of f(x,y), influenced by the feed rate V

_{f}, spindle speed n, and cutting depth a

_{c}. The accuracy of the chip boundary surface f(x,y) was confirmed. The chip diffusion area was divided into the main diffusion area, random diffusion area, and vortex area based on the chip motion state. The spindle speed n and feed rate V

_{f}had the greatest impact on the orthogonal diffusion angle of the main diffusion zone, whereas the cutting depth a

_{c}had the greatest effect on the top view diffusion angle of the main diffusion zone. Furthermore, the authors noticed that the chip size increased exponentially with the percentage of the sampling spacing, and the fitting results were highly accurate. The chip boundary surface model f(x,y) was relatively effective in predicting the morphology and diffusion state of chips generated during circumferential milling of pine wood, which is significant for chip control in the wood milling process. Lastly, this study provides guidance for future research.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Wood circumferential milling and wood chip diffusion state. (

**a**) Pine specimen clamped on the machine. (

**b**) Milling cutter fixed on the electric spindle and pine specimen fixed on the fixture. (

**c**) Schematic diagram depicting the chip generation process.

**Figure 2.**Analysis of the forces during circumferential milling of wood. (

**a**) The state when the cutting edge is parallel to the annual wheel of the wood. (

**b**) The state when the cutting edge is perpendicular to the annual wheel of the wood. Numbers 1 to 5 refer to the serial numbers of the five cutting edges of the circumferential milling tool. (

**c**) The state of cutting edge No. 1 when it is at an angle of 45° to the annual wheel of the wood and the parameters utilized for the calculation of the tool and the layer of the annual wheel of the wood are shown.

**Figure 3.**Top view of wood circumferential milling and force analysis. (

**a**) The actual circumferential milling process of the machine. (

**b**) A theoretical illustration of the wood circumferential milling process. (

**c**) A local enlarged view of the wood chip formation area. The wood chip length denoted by L

_{1}and the cutting depth represented by a′

_{c}have been amplified for labeling ease.

**Figure 4.**Boundary of the wood chip diffusion surface. (

**a**) The boundary trajectory after the chip is detached from the milling cutter. (

**b**) The boundary in the yoz plane. M

_{1}is the wood chip on the boundary line of this plane. β

_{yoz}is the initial angle of the wood chip boundary in the yoz plane and the maximum angle. (

**c**) The boundary in the xoy plane. M

_{2}is the wood chip on the boundary line of this plane. β

_{xoy}is the initial angle of the wood chip boundary in the xoy plane and the maximum angle. (

**d**) For any chip M

_{3}located on the boundary surface, the angles β′

_{yoz}and β′

_{xoy}represent the inclination of the initial velocity with respect to the xoy plane and the current plane with respect to the yoz plane, respectively.

**Figure 5.**Milling test results and observed chip sizes. (

**a**) Front view of test No. 6. (

**b**) Top view of test No. 9. (

**c**) Test No. 6 top view of the scattered state of chip distribution after machining. Four equally spaced areas of d were taken and then photographed using a portable microscope. (

**d**) The four white square areas in Figure (

**c**) were photographed by a 1000× portable microscope. L

_{a1}to L

_{a4}are the average chip sizes in these four areas, respectively.

**Figure 6.**Feed rate V

_{f}and spindle speed n versus orthogonal diffusion angle β

_{yoz}at depth of cut a

_{c}= 10 mm. (

**a**) Plot of the response surface of the orthogonal diffusion angle β

_{yoz}with respect to spindle speed n and feed rate V

_{f}. (

**b**) Contour plot of the response surface.

**Figure 7.**Depth of cut ac and feed rate V

_{f}at spindle speed n = 8500 r/min versus the pitch diffusion angle β

_{xoy}. (

**a**) Plot of the response surface of the overhead diffusion angle β

_{xoy}with respect to the depth of cut ac and feed rate V

_{f}. (

**b**) Contour plot of the response surface.

**Figure 8.**Relationship between the average chip size L

_{a}and feed rate V

_{f}, spindle speed n, and depth of cut a

_{c}. (

**a**) Response surface plot of average chip size L

_{a}as a function of depth of cut a

_{c}and spindle speed n. (

**b**) Contour plot of the response surface (

**a**). (

**c**) Response surface plot of average chip size L

_{a}as a function of depth of cut a

_{c}and feed rate V

_{f}. (

**d**) Contour plot of the response surface (

**c**).

**Figure 9.**The relationship between sampling relative position D and the sampled chip size L

_{j}at different cutting parameters is depicted as L

_{j}(D), where d represents the sample spacing, j denotes the current sampling sequence, and k represents the total number of samples. The sampling relative position is expressed by (j/k)d × 100% within the range of 0% to 100%. (

**a**) The change in L

_{j}(D) after changing the spindle speed n when the feed speed V

_{f}= 1100 and the depth of cut a

_{c}= 10. (

**b**) The change in L

_{j}(D) after changing the feed rate V

_{f}when the spindle speed n = 8500 and the depth of cut a

_{c}= 10. (

**c**) The change in L

_{j}(D) after changing the depth of cut a

_{c}at spindle speed n = 8500 and feed rate V

_{f}= 1100.

**Figure 10.**The results of solving the f(x,y) model with different milling parameters. (

**a**) The surface model f(x,y) was implemented and the actual chip scattering area was calculated using data from test No. 1. The error of area is 39.59%. (

**b**) The surface model f(x,y) was implemented and the actual chip scattering area was calculated using data from test No. 11. The error of area is 27.71%. (

**c**) The surface model f(x,y) was implemented and the actual chip scattering area was calculated using data from test No. 25. The error of area is 35.08%.

Density (kg/m ^{3}) | Compressive Strength (MPa) | Bending Strength (MPa) | Tensile Strength (MPa) | Shear Strength (MPa) | Modulus of Elasticity (MPa) | Poisson’s Ratio | Coefficient of Friction |
---|---|---|---|---|---|---|---|

420 | 50 | 87 | 104 | 10 | 12,000 | 0.65 | 0.35 |

Alpha | Spindle Speed n (r/min) | Feeding Speed V_{f}(mm/min) | Cutting Depth a_{c}(mm) |
---|---|---|---|

+1.68 | 11,000 | 1770 | 15 |

+1 | 10,000 | 1500 | 13 |

0 | 8500 | 1100 | 10 |

−1 | 7000 | 700 | 7 |

−1.68 | 6000 | 430 | 5 |

No. | Factors | Indicators | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

n | V_{f} | a_{c} | β_{yoz} | β_{xoy} | L_{a1} | L_{a2} | L_{a3} | L_{a4} | L_{a} | |

(r/min) | (mm/min) | (mm) | (°) | (°) | (mm) | (mm) | (mm) | (mm) | (mm) | |

1 | 7000 | 700 | 7 | 25 | 134 | 0.382 | 0.692 | 1.031 | 1.812 | 0.9792 |

2 | 10,000 | 700 | 13 | 31 | 107 | 0.484 | 0.541 | 0.771 | 1.396 | 0.7980 |

3 | 10,000 | 1500 | 13 | 73 | 135 | 0.322 | 0.559 | 1.82 | 1.9975 | 1.1746 |

4 * | 8500 | 1100 | 10 | 44 | 120 | 0.258 | 0.462 | 1.048 | 1.51 | 0.8195 |

5 | 8500 | 430 | 10 | 26 | 115 | 0.237 | 0.457 | 0.855 | 1.251 | 0.7000 |

6 | 11,000 | 1100 | 10 | 32 | 114 | 0.356 | 0.549 | 0.788 | 1.455 | 0.7870 |

7 * | 8500 | 1100 | 10 | 40 | 113 | 0.354 | 0.513 | 0.823 | 1.404 | 0.7735 |

8 | 7000 | 1500 | 13 | 73 | 140 | 0.514 | 1.139 | 1.793 | 2.538 | 1.4960 |

9 | 7000 | 700 | 13 | 28 | 114 | 0.381 | 0.563 | 1.168 | 2.47 | 1.1455 |

10 * | 8500 | 1100 | 10 | 43 | 115 | 0.295 | 0.488 | 0.941 | 1.455 | 0.7948 |

11 | 10,000 | 700 | 7 | 36 | 141 | 0.296 | 0.593 | 0.855 | 1.105 | 0.7123 |

12 | 8500 | 1770 | 10 | 78 | 152 | 0.542 | 0.771 | 1.357 | 2.354 | 1.2560 |

13 | 7000 | 1500 | 7 | 74 | 94 | 0.319 | 0.469 | 0.775 | 1.63 | 0.7983 |

14 | 8500 | 1100 | 15. | 45 | 137 | 0.395 | 0.668 | 0.888 | 3.11 | 1.2653 |

15 | 8500 | 1100 | 5 | 41 | 93 | 0.341 | 0.635 | 0.918 | 1.903 | 0.9493 |

16 * | 8500 | 1100 | 10 | 44 | 119 | 0.324 | 0.474 | 0.914 | 1.427 | 0.7848 |

17 * | 8500 | 1100 | 10 | 45 | 110 | 0.289 | 0.497 | 0.911 | 1.455 | 0.7880 |

18 | 10,000 | 1500 | 7 | 44 | 117 | 0.362 | 0.714 | 0.727 | 1.378 | 0.7953 |

19 * | 8500 | 1100 | 10 | 50 | 113 | 0.233 | 0.485 | 0.944 | 1.457 | 0.7798 |

20 | 6000 | 1100 | 10 | 86 | 105 | 0.291 | 0.567 | 1.4853 | 1.919 | 1.0656 |

21 | 10,000 | 1100 | 10 | 33 | 115 | 0.291 | 0.375 | 0.779 | 1.345 | 0.6975 |

22 | 7000 | 1100 | 10 | 44 | 110 | 0.597 | 0.648 | 1.008 | 1.964 | 1.0543 |

23 | 8500 | 1500 | 10 | 65 | 143 | 0.496 | 0.755 | 1.241 | 2.14 | 1.1580 |

24 | 8500 | 700 | 10 | 36 | 131 | 0.325 | 0.434 | 0.715 | 1.314 | 0.6970 |

25 | 8500 | 1100 | 13 | 48 | 142 | 0.585 | 0.618 | 1.127 | 2.285 | 1.1538 |

26 | 8500 | 1100 | 7 | 63 | 105 | 0.409 | 0.594 | 0.809 | 1.391 | 0.8008 |

**Table 4.**The ANOVA results for the orthogonal diffusion angle β

_{yoz}are as follows: R

^{2}= 0.9105; adjusted R

^{2}= 0.8692; predicted R

^{2}= 0.7014; adeq precision = 15.6975. Notably, the overall model regression is highly significant, with a probability of noise affecting the model being only 0.01 as indicated by the F-value of 22.04. Although almost all items in the table are highly significant (p < 0.05), item B

^{2}is not significant (p > 0.1).

Source | Sum of Squares | df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 5990.94 | 6 | 998.49 | 22.04 | <0.0001 |

A–n | 1458 | 1 | 1458 | 32.19 | <0.0001 |

B–V_{f} | 3922.62 | 1 | 3922.62 | 86.6 | <0.0001 |

AB | 242 | 1 | 242 | 5.34 | 0.0378 |

A^{2} | 297.09 | 1 | 297.09 | 6.56 | 0.0237 |

B^{2} | 60.76 | 1 | 60.76 | 1.34 | 0.2676 |

AB^{2} | 654.53 | 1 | 654.53 | 14.45 | 0.0022 |

Residual | 588.86 | 13 | 45.3 | - | - |

Lack of Fit | 535.53 | 8 | 66.94 | 6.28 | 0.0592 |

Pure Error | 53.33 | 5 | 10.67 | - | - |

Cor Total | 6579.8 | 19 | - | - | - |

**Table 5.**The ANOVA results for the topographic diffusion angle β

_{xoy}are as follows: R

^{2}= 0.8563; adjusted R

^{2}= 0.7518; predicted R

^{2}= 0.5842; adeq precision = 9.5991. Notably, the overall regression of the model is significant at p = 0.0011. The F-value of 8.19 indicates that the model is primarily influenced by signal rather than noise, with only a probability of 0.11 of being affected by noise. Furthermore, all items in the table are significant except for A and A

^{2}, with p < 0.05 indicating high significance, 0.1 < p < 0.05 indicating significance, and p > 0.1 signifying insignificance.

Source | Sum of Squares | df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 4021.77 | 8 | 502.72 | 8.19 | 0.0011 |

A–n | 80.4 | 1 | 80.4 | 1.31 | 0.2767 |

B–V_{f} | 199.72 | 1 | 199.72 | 3.25 | 0.0986 |

C–a_{c} | 968 | 1 | 968 | 15.77 | 0.0022 |

AC | 220.5 | 1 | 220.5 | 3.59 | 0.0846 |

BC | 1740.5 | 1 | 1740.5 | 28.36 | 0.0002 |

A^{2} | 26.33 | 1 | 26.33 | 0.4291 | 0.5259 |

B^{2} | 742.04 | 1 | 742.04 | 12.09 | 0.0052 |

A^{2}C | 463.85 | 1 | 463.85 | 7.56 | 0.0189 |

Residual | 675.03 | 11 | 61.37 | - | - |

Lack of Fit | 601.03 | 6 | 100.17 | 6.77 | 0.0566 |

Pure Error | 74 | 5 | 14.8 | - | - |

Cor Total | 4696.8 | 19 | - | - | - |

**Table 6.**The model’s R

^{2}value, adjusted R

^{2}value, and predicted R

^{2}value were determined to be 0.9201, 0.8735, and 0.6484, respectively, with an adeq precision value of 16.9314. The overall regression of the model is considered to be highly significant, with a p-value less than 0.0001. The F-value of 19.74 indicates that the probability of the model being influenced solely by noise is only 0.01. All variables included in the table are significant, with p < 0.05 indicating high significance, 0.1 < p < 0.05 indicating significance, and p > 0.1 signifying insignificance.

Source | Sum of Squares | df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 0.903 | 7 | 0.129 | 19.74 | <0.0001 |

A–n | 0.1679 | 1 | 0.1679 | 25.69 | 0.0003 |

B–V_{f} | 0.1792 | 1 | 0.1792 | 27.42 | 0.0002 |

C–a_{c} | 0.2535 | 1 | 0.2535 | 38.79 | <0.0001 |

AC | 0.0199 | 1 | 0.0199 | 3.04 | 0.0994 |

BC | 0.0851 | 1 | 0.0851 | 13.02 | 0.0036 |

B^{2} | 0.0514 | 1 | 0.0514 | 7.87 | 0.0159 |

C^{2} | 0.1608 | 1 | 0.1608 | 24.61 | 0.0003 |

Residual | 0.0784 | 12 | 0.0065 | - | - |

Lack of Fit | 0.0771 | 7 | 0.011 | 42.32 | 0.06 |

Pure Error | 0.0013 | 5 | 0.0003 | - | - |

Cor Total | 0.9814 | 19 | - | - | - |

**Table 7.**Parameters necessary for solving the model are listed in the table below. Specifically, d

_{c}represents the chip thickness and is also equivalent to the single depth of cut a′

_{c}, which is calculated using Equation (1). Additionally, ρ

_{wood}denotes the density of pine wood [24].

No. | β_{yoz} | β_{xoy} | g | n | V_{f} | r | L_{a} | a_{w} | d_{c} | ρ_{wood} |
---|---|---|---|---|---|---|---|---|---|---|

(°) | (°) | (N/s^{2}) | (r/min) | (mm/s) | (mm) | (mm) | (mm) | (mm) | (kg/m^{3}) | |

1 | 25 | 134 | 9.8 | 7000 | 11.67 | 15 | 0.979 | 0.489 | 8.34 × 10^{−4} | 450 |

11 | 36 | 141 | 9.8 | 10,000 | 11.67 | 15 | 0.712 | 0.356 | 3.50 × 10^{−5} | 450 |

25 | 48 | 142 | 9.8 | 8500 | 18.33 | 15 | 1.154 | 2.308 | 6.47 × 10^{−5} | 450 |

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## Share and Cite

**MDPI and ACS Style**

Yang, C.; Liu, T.; Ma, Y.; Qu, W.; Ding, Y.; Zhang, T.; Song, W.
Study of the Movement of Chips during Pine Wood Milling. *Forests* **2023**, *14*, 849.
https://doi.org/10.3390/f14040849

**AMA Style**

Yang C, Liu T, Ma Y, Qu W, Ding Y, Zhang T, Song W.
Study of the Movement of Chips during Pine Wood Milling. *Forests*. 2023; 14(4):849.
https://doi.org/10.3390/f14040849

**Chicago/Turabian Style**

Yang, Chunmei, Tongbin Liu, Yaqiang Ma, Wen Qu, Yucheng Ding, Tao Zhang, and Wenlong Song.
2023. "Study of the Movement of Chips during Pine Wood Milling" *Forests* 14, no. 4: 849.
https://doi.org/10.3390/f14040849