Calibration and Testing of Discrete Element Simulation Parameters for Ultrasonic Vibration-Cutter-Soil Interaction Model

Abstract

This paper established an accurate discrete element for ultrasonic vibration-cutter-soil interaction model to study the interaction mechanism between the soil-engaging component and the soil. In order to reduce the interaction between calibration parameters and improve the calibration accuracy, it is proposed that the soil constitutive, contact parameters, and bonding parameters be calibrated by combining the soil repose angle experiment and the soil resistance experiment of ultrasonic vibration cutting. The study adopts the Hertz-Mindlin (no slip) contact model used in EDEM, to explore soil particle interactions. The central composite design is used to achieve systematic investigation. 3-factor 3-level orthogonal design experiment was employed using the coefficient of restitution, the coefficient of static friction, and the coefficient of rolling friction as key test factors and soil’s repose angle as the response index. Based on the Hertz-Mindlin with bonding contact model, Design-Expert 13.0 software was used to design the Plackett-Burman experiment, the steepest ascent, and the Box-Behnken experiment. With the maximum soil cutting resistance in ultrasonic vibration cutting experiment used as the response value, the adhesion parameters were optimized, and the optimal solution combination was obtained as: Normal Stiffness = 4.635 × 10⁶ N/m, Shear Stiffness = 3.401 × 10⁶ N/m, and Bonded Disk Radius = 2.57 mm. The optimal parameter combinations obtained from the calibration experiments were verified in two ways: ultrasonic vibration cutting and non-ultrasonic vibration cutting. The results showed that the errors between the simulation values and the actual values of the two comparative experiments were less than 5%, and the model calibrated for the three parameters can be used to study the drag reduction mechanism of ultrasonic vibration cutting in soil.

1. Introduction

In agricultural production, agricultural machinery’s soil-engaging components are subject to high resistance, thus leading to high energy consumption during operation. Therefore, it is vital to reduce the resistance of soil-engaging component [1,2,3,4]. Currently, the main ways to reduce resistance include the bionic method, coating of soil-engaging components, vibration method, ultrasonic method, etc. Among them, the ultrasonic method has a better resistance reduction performance, about 30% [5]. To further improve the effect of ultrasonic incision and drag reduction, it is necessary to explore the ultrasonic drag reduction mechanism. However, current research on ultrasonic incision and drag reduction is primarily experimental, and it is difficult to understand the drag reduction mechanism in depth. The discrete element method can accurately simulate soil’s physical and mechanical properties and is widely used in soil researches [6,7,8,9,10,11]. Therefore, this article intends to use this method to analyze the drag reduction mechanism of ultrasonic cutting to improve the drag reduction effect. Constructing a specific, accurate discrete element for ultrasonic vibration-cutter-soil interaction model is necessary for using the discrete element method to study the ultrasonic drag reduction mechanism. Therefore, the establishment process of this model is studied in this paper.

The discrete element method has become widely used by researchers to simulate and analyze particle-related agricultural machinery used for cultivating, sowing, and harvesting [12,13,14,15]. An appropriate contact model must be selected and calibrated before discrete element simulation. Generally speaking, the cohesion between soil particles causes the soil to form blocks. When calibrating, the bonding parameters between discrete element particles must be added. Therefore, the Hertz-Mindlin model combined with the JKR model or the Hertz-Mindlin model combined with the Bonding model is generally used. The soil constitutive parameters, contact parameters and bonding parameters need to be calibrated during the experimental design.

Makang et al. [16] used the discrete element method to calibrate soil through the soil repose angle experiment, established a soil-tillage cutter interaction model, and studied the effects of horizontal and vertical forces and soil disturbance on bulk density at different speeds and depths. Shi et al. [17] used the repose angle as an indicator to calibrate the soil parameters of farmland in Northwest China. Then, they verified it through soil direct shear tests and established a discrete element model of farmland soil in the Northwest China’s arid area. Wu et al. [18] calibrated the parameters of cohesive soil through the soil repose angle experiment, established a discrete element model of cohesive soil, and revealed the movement rules of cohesive soil disturbed by soil-engaging components. When calibrating the soil parameters above, soil repose angle experiment was carried out on the soil to calibrate the soil’s constitutive, contact, and bonding parameters. However, the cohesive force between the particles of cohesive soil and hard soil and the other two groups of parameters will mutually influence, each other, thereby reducing the calibration accuracy. Calibrating these parameters separately will decrease the interaction between parameters, enhancing calibration accuracy. Therefore, this paper proposes that drying, crushing, and sieving of the hard soil particles shall be performed before the experiment to eliminate the influence of cohesive force on the calibration process of constitutive parameters and contact parameters. Subsequently, intact soil samples calibrate the cohesive force, thus improving the overall calibration accuracy.

When calibrating bonding parameters, in order to find the suitable parameters to simulate the interaction between cinnamon soil and soil-engaging components, Qiu et al. [19] employed the Hertz-Mindlin with JKR model to calibrate the parameters of cinnamon soil via uniaxial compression experiment, and finally validated results through soil trench-cutting experiment, and found a error of 10.32%. Zhang et al. [20] used the Hertz-Mindlin with JKR model to establish a maize root-soil mixture (MRSM) DEM model and calibrated the root-soil adhesion parameters through the rotating drum test. Song et al. [21] used the Hertz-Mindlin with bonding model to calibrate the parameters for the mulberry soil through a soil repose angle experiment and sliding friction angle experiment, providing theoretical support for the structural optimization of soil-engaging parts of mulberry orchard farming machinery. To sum up, the currently used methods are soil repose angle experiment, direct shear test, rotating drum test, and sliding friction angle test, mostly calibrating soil particles under static or low-speed flow [22,23,24]. However, in high-frequency vibration scenario, like ultrasonic vibration, it is not easy to achieve accurate calibration using traditional methods. Therefore, this article introduces high-frequency vibration cutting experiments to calibrate bonding parameters using cutter-cutting experiments with and without ultrasonic vibration. During the calibration process, since that small particle size makes full-scale simulation difficult. To reduce the amount of calculation, researchers generally choose to enlarge the particle diameter [25,26]. Huang et al. [27] selected a particle size of 2 mm to calibrate the ultrasonic-soil discrete element model. Mostafa Bahrami et al. [28] established a discrete element model of the plate load test, and the particle size selected for calibration was 10 mm. Shi et al. [17] selected a particle size of 5 mm when calibrating the discrete element model of soil particles in farmland in the arid area of Northwest China. However, when other parameters remain consistent, different particle size amplification result in inconsistent calibration results. Therefore, the parameters must be recalibrated to achieve the calibration effect. The calibration objective of this paper is to provide an accurate discrete element model for the study of drag reduction mechanism of ultrasonic vibration cutting, so a specific discrete element model is developed with ultrasonic vibration-soil-cutting cutter as the research object.

The soil calibrated in this study is sandy loam with low water content. Once the adhesive bonding between soil particles are destroyed during farming, they cannot be restored, so the Hertz-Mindlin bonding model was selected. Unlike other studies, this paper innovatively proposes soil parameter calibration by combining the soil repose angle and ultrasonic cutting soil resistance experiments. First, the soil repose angle is used as the index, and the dried and crushed soil is used as the calibration object. An 3-factor 3-level orthogonal design experiment was performed to calibrate the restitution, static, and rolling friction coefficients; using the maximum ultrasonic cutting resistance as the experimental index, Plackett-Burman experiments were carried out to screen out the three most significant factors among the five factors affecting the Hertz-Mindlin with bonding model, whose calibration range was further narrowed through climbing experiments; and finally, the Box-Behnken experiments performed to calibrate the three significant factors to obtain the optimal parameter combination. Compared with the measured values, it provides reliable soil particle parameters for establishing an accurate discrete element model of ultrasonic vibration-cutter-soil interaction.

2. Materials and Methods

2.1. Measurement of Basic Soil Parameters

The trial was conducted on 8 October 2023 using sandy loam soil with low moisture content, at the on-campus experimental field of Qingdao Agricultural University (120.3959843° N, 36.3192722° E). The five-point sampling method was used to take soil from the experiment field, 500 g of soil was weighed at the sampling points using an electronic balance, and then soil sieves were used to grade the soil particles; particles with diameter < 0.25 mm account for 20.16%, particles with diameter in the range of 0.25 mm–1 mm account for 41.20%, and particles with diameter >1 mm account for 38.64%. The density of soil particles was 2670 kg/m³ using the specific gravity bottle method.

Soil moisture content was measured using the drying method. To ensure that the soil samples obtained represent the soil in the experimental field, the cutting ring method was used to sample soil at depths of 5 cm, 10 cm, and 15 cm, ensuring that the cut soil samples were as neat as possible without compaction or damage. Then, the soil samples were placed in sealed bags to measure the mass using a balance and record. After that, the soil samples were dried using far-infrared rapid, constant temperature drying oven (YHG-300-BS Shanghai Yuejin Medical Equipment Co., Ltd., Shanghai, China), and the moisture content was measured to be 9%.

Undisturbed soil was prepared by using a cylindrical iron sheet with openings at the top and bottom to cut vertically into the soil in the test field, then the surrounding soil was cleaned, and the cylindrical soil specimen was obtained by inserting a steel plate at the bottom of the sheet, finally, the specimen was placed on a wooden board.

2.2. Soil Repose Angle Experiment

Soil repose angle experiments can reveal the friction and interaction between soil particles. The experimental devices used include an electronic balance, a funnel, a bracket, and a Mitutoyo high-accuracy digital micrometer (measuring range: 4–90°, accuracy: 0.05°), and a grade 65 steel plate of 210 mm × 160 mm.

During the experiment (Figure 1), first, the steel plate was carpeted with a layer of soil particles; the bottom of the funnel was blocked with a baffle plate; then 500 g of sieved soil was poured into the funnel; when the soil came to a complete standstill, the baffle plate was withdrawn to allow the soil particles fall freely. Finally, the soil repose angle was measured after the soil particles came to a complete standstill.

The soil repose angle was measured as 34.40°. As shown in Figure 2, the image was processed using matlab 2020, and the fitted equation along the slope is:

(1)$y=0.69008x+3.517$

The slope in the equation is converted into an angle, then the accumulation repose angle was obtained, i.e., 34.61°.

2.3. Soil Resistance Experiment of Ultrasonic Vibration Cutting

A soil resistance experiment platform of ultrasonic vibration cutting was built with a cutter carrier, ultrasonic generator, control box, ultrasonic cutting knife, computer, and signal acquisition software (Figure 3). The cutter carrier was equipped with a limit sensor and a force sensor. The ultrasonic generator frequency is 20 kHz. The cutting speed, depth, and data collection interval can be set via the control software.

Then, a soil-cutting experiment was conducted on the cutting cutter using ultrasonic vibration. The cutting depth was set to 0.1 m, the cutting speed was 0.05 m/s, and the data collection interval was 100 ms. The experiment index was the resistance of the cutting cutter when the soil cutting depth was 0.1 m, at this point, the resistance was the maximum resistance encountered by the cutting blade when cutting the soil.

2.4. Soil Resistance Validation Experiment of Non-Vibration Cutting

To verify the accuracy of the proposed discrete element model of soil-ultrasonic vibration-cutter interaction, cutting resistance validation experiments were carried out using a cutter without ultrasonic vibration. The maximum resistance measured in the experiment was 85.1 N.

2.5. Establishment of Ultrasonic Vibration-Soil-Cutter Discrete Element Model

2.5.1. DEM Contact Model Selection

EDEM is a numerical method used to simulate interactions and motions between vibration. In EDEM simulation and analysis, choosing a suitable contact model to describe the contact behavior and interaction between soil particles is extremely important [29,30,31,32,33,34]. The Hertz-Mindlin (no slipping) contact model is based on the Hertz-Mindlin theory, which considers the elastic deformation and sliding friction between particles and is mainly used to simulate the elastic and frictional interactions between particles [35,36]. As shown in Figure 4, the Hertz-Mindlin bonding model considers the adhesion effect between particles based on the Hertz-Mindlin model. Since the bonding between soil particles is destroyed during soil sieving, the Hertz-Mindlin (no slipping) contact model is used to simulate the soil repose angle, which does not consider the adsorption effect after the destruction of the bonding between soil particles.

2.5.2. The Hertz-Mindlin with Bonding Model

(2) $\delta F_n=-V_n{S}_{n}A\partial t$

(3) $\delta F_t=-V_t{S}_{t}A\partial t$

(4) $\delta M_n=-{\omega }_n{S}_{t}J\partial t$

(5) $\delta M_t=-{\omega }_n{S}_t{\displaystyle \frac{J}{2}}\partial t$

(6) $A=\pi R_b^2$

(7) $J={\displaystyle \frac{1}{2}}\pi R_b^4$

The bond breaks when the normal and tangential shear stress exceeds a particular predefined value:

(8)${\sigma }_{\max }<{\displaystyle \frac{-F_n}{A}}+{\displaystyle \frac{2M_t}{J}}{R}_b$

(9)${\tau }_{\max }<{\displaystyle \frac{-F_t}{A}}+{\displaystyle \frac{2M_n}{J}}{R}_b$

F_n and F_t are the normal and tangential stress. R_b is the bonding radius, S_n and S_t are the normal and tangential stiffness, respectively, and t is the time step. V_n and V_t are the normal and tangential velocities of the particles, respectively, and ${\omega }_n$ and ${\omega }_t$ are the normal and tangential angular velocities, respectively. A is the contact area, J is the moment of inertia. M_n and M_t are the normal and tangential torque.

These bonding forces/moments are additionally added to the standard Hertz-Middlin forces [37,38,39]. Since this model can work when the particles are not in contact, the contact radius should be set to be larger than the actual contact radius. This model can only be used for particles and interparticle contacts.

2.5.3. EDEM Model Building

(1) Soil repose angle simulation experiment

A funnel model is built using SolidWorks 2018 (Figure 5), specifying dimensions such as a height (h) of 20 cm, upper length (l₁) of 15 cm, and lower length (l₂) of 2.5 cm; then the model is exported in IGS format and then imported into EDEM 2020. First, a particle factory is built above the funnel, and then two layers of soil particles are spread on the bottom steel plate. Second, let the particle factory generate soil particles, then let the particles fall to the ground through the funnel, and wait for the particle pile to be utterly stationary before measuring the soil repose angle.

(2) ultrasonic vibration cutting soil simulation experiment

As shown in Figure 6, the DEM model was built based on the ultrasonic cutting experiment platform, and only the cutter, soil, and box required for soil cutting were kept. The particle factory with radius R = 0.18 m and H = 0.14 m was established in EDEM software after filling 170,000 particles. The cutter diagram is drawn in solidworks 2018, exported in IGS format and then imported into discrete element software. The cutter is 0.2 m in length, 0.13 m in height and 0.025 m in width.

2.5.4. Simulation Parameter Settings

(1) Soil repose angle simulation experiment

In this part, the simulation parameters in Table 1 are first input into the EDEM software. Spherical particles with a radius of 2 mm were selected for the experiment, and the particle factory are set to generate 500 g soil particles. The simulation duration is 18 s, fixed time step 20%, and gravity 9.8 m/s².

(2) Ultrasonic vibration cutting simulation experiment

The simulation duration of this experiment is 2 s in total, the time step is set to 1 × 10⁻⁸ s, the cutter falling speed is set to 0.05 m/s, and the amplitude of 20 kHz is applied. The cutter’s movement is designed to perform a uniform vertical cutting speed, ensuring a straightforward downward motion without any complex trajectories. The cutting resistance data can be readily obtained from the post-processing panel of the DEM simulation software, which allows for the direct export of results to an Excel spreadsheet. Figure 7 shows the change process of the force on the bonding key during cutter cutting. As the cutting depth increases, the number of contacts between the bonding key and the cutter gradually increases, and the stress and damage also gradually increases, which can better reflect the relationship between the cutter and the soil. Therefore, the soil parameters are calibrated based on the maximum cutting depth, at which the maximum cutting resistance occurs, as the experimental index.

2.5.5. Index Measurement

(1) Soil repose angle $\theta$

The soil repose angle reflects the friction properties (contact properties) between particles in the soil. It can be calibrated through soil repose angle experiments to obtain reliable soil restitution coefficient, static friction coefficient, and rolling friction coefficient. In the simulation experiment, the repose angle is obtained from Formula (9)

(10)$\sin \theta ={\displaystyle \frac{h}{l}}$

(2) Cutting resistance F

The cutting resistance of the cutter can reflect the physical and mechanical characteristics of the interaction between the ultrasonic cutter and the soil, so the cutting resistance of the cutter is used as an indicator. The resistance reduction effect of ultrasonic cutters can also be objectively evaluated through the cutting resistance of the cutter.

2.6. Experiment Scheme

Firstly, due to the bonding relationship between soil particles in the field, the direct use of the undisturbed soil cannot reflect the friction relationship between soil particles. Therefore, after soil screening, the soil repose angle was used as an indicator to obtain particle simulation constitutive parameters in low-speed flow and static conditions.

Secondly, a resistance calibration experiment of ultrasonic vibration cutting is performed. There is a bonding force between real soil particles that cannot be restored quickly after destruction. Therefore, undisturbed soil is used as the experimental object for bonding key calibration. In the experiment, the cutting cutter destroys the force between soil particles. The maximum resistance of the ultrasonic vibration cutting soil is used as the indicator to calibrate the parameters of interaction between soil particles.

2.6.1. Soil Repose Angle Calibration Experiment

Three parameters, including the coefficient of restitution, coefficient of static friction, and coefficient of rolling friction between particles, were calibrated to obtain the accurate constitutive parameters of soil particles in the simulation experiment of the angle of repose.

Before the experiment, two layers of soil particles were spread on the bottom steel plate. During the experiment, there was no displacement between the soil particles and the bottom plate. After screening the soil particles, a soil repose angle experiment was conducted. The soil screening process destroys the bonding force between soil particles. Therefore, when calibrating soil particles, the contact parameters between the particles and the steel plate and the bonding parameters between the particles are not considered. Only the coefficient of restitution, coefficient of static friction, and coefficient of rolling friction between particles were calibrated.

A simulation experiment using the coefficient of restitution, coefficient of static friction, and coefficient of rolling friction of soil particles as influencing factors, the soil repose angle as an indicator was carried out. This experiment involved three factors, each with three levels and a horizontal value range, as outlined in Table 2.

2.6.2. Soil Resistance Calibration Experiment of Ultrasonic Vibration Cutting

In order to simulate the bonding state of soil with low moisture content accurately, this article uses the Hertz-Mindlin with bonding model that comes with the EDEM software for simulation calibration. The model parameters include five types: Normal Stiffness, Shear Stiffness, Critical Normal Stress, Critical Shear Stress, and Bonded Disk Radius. This paper uses the soil penetration resistance of ultrasonic vibrating cutting cutters as an indicator to conduct calibration experiments on five parameters. Plackett-Burman experiments were first carried out to find the three parameters that have the greatest influence on the index from the five parameters, and then the range of values of the three parameters was further narrowed down by Steepest ascent, and finally the optimal parameters were obtained by Box-Behnken experiments.

Table 3 shows the simulation experiment with Normal Stiffness, Shear Stiffness, Critical Normal Stress, Critical Shear Stress, and Bonded Disk Radius as the influencing factors. The maximum resistance of simulated ultrasonic cutting is the experiment index; the experiment has five factors with two levels for each factor, and the range of the values of the levels is referred to in Table 1.

3. Results and Analysis

3.1. The Results and Analysis of Soil Repose Angle Calibration Experiment

A soil repose angle simulation experiment on Table 4 was performed, and regression variance analysis was performed on the experimental results using design-expert 13.0 software to obtain the soil repose angle regression model:

(11)$y=27.22+5.15X_1+5.65X_2-1.34X_1{X}_2+1.28X_1{X}_3+1.15X_2{X}_3-1.88X_1^2-1.74X_2^2+2.32X_3^3$

Variance analysis and regression coefficient significance experiment were performed on each factor, with the result as shown in Table 5. It can be seen from the table that the relationship model between the experimental factor encoding and soil repose angle is highly significant with p < 0.001. At the same time, the lack-of-fit item p > 0.05 indicates that it is not significant, so the model is reasonable. R² = 0.9962 is close to 1, indicating that the model is better and can be used for prediction. Among the three experimental factors, X₁ and X₂ significantly affect soil repose angle (p < 0.01). Among the interaction effects, $X_2^1$, $X_2^3$ had highly significant effect, X₁ X₂, X₁ X₃, X₂ X₃, $X_2^2$ had significant effect on soil repose angle and all other factors had insignificant effect. From the interaction factors perspective, X₂ X₃, X₁ X₃ and X₁ X₂ are listed in descending order in terms of influence on soil repose angle.

Response surface plots were used to analyze the effect of the coefficient of restitution, coefficient of static friction, and coefficient of rolling friction on the indexes, as shown in Figure 8.

Figure 8a shows the response surface of the coefficient of restitution and coefficient of static friction on the interaction of soil repose angle when the coefficient of rolling friction is located in the horizontal center (0.2); it can be seen that the soil repose angle increases rapidly and then slowly when coefficient of restitution and coefficient of static friction increase. It can be seen from response surface that the soil repose angle is the greatest when the coefficient of restitution is 0.6, and the coefficient of static friction is 1.0. The interaction between the coefficient of restitution and the coefficient of static friction significantly affects the soil repose angle (p < 0.05).

Figure 8b shows the response surface of the interaction between the coefficient of restitution and the coefficient of rolling friction and the soil repose angle when the coefficient of static friction is at the center level (0.5). It can be seen from response surface that when the coefficient of rolling friction is constant, the soil repose angle increases with the increase of the coefficient of restitution rapidly and then slowly. The figure shows that when the coefficient of restitution is 0.6, and the coefficient of rolling friction is 0.3, the soil repose angle is the largest. Interaction between coefficient of restitution and coefficient of rolling friction has a significant effect (p < 0.05) on the soil repose angle.

Figure 8c shows the response surface of the interaction between the coefficient of static friction and the coefficient of rolling friction and the soil repose angle when the coefficient of restitution is at the center level (0.5), and it can be seen that the soil repose angle increases gradually with the increase of the coefficient of static friction, and change extent varies under different coefficient of rolling friction. The repose angle gradually increases with the increase of the coefficient of rolling friction, which is pronounced when the coefficient of static friction is 0.85–1.0. The figure shows that the soil repose angle is more significant when the coefficient of static friction is 1.0, and the coefficient of rolling friction is 0.1 and 0.3. The coefficient of static friction and the coefficient of rolling friction have a significant effect (p < 0.05) on the soil repose angle.

Combined with the actual soil repose angle of 34.61°, the optimal solution obtained through analysis is coefficient of restitution: 0.47, coefficient of static friction: 0.96, coefficient of rolling friction: 0.28.

The simulation experiment was repeated three times using optimal solution to verify. The soil repose angle results are: 33.23°, 33.02°, 33.41°. The average value is 33.22°, and the actual difference is 1.38°. The error is 4%, which is within the allowed range: 5%.

3.2. The Results and Analysis of Soil Cutting Simulation Calibration Experiment

(1) Plackett-Burman experiments.

Plackett-Burman experiments were conducted on the five factors in Table 3, and the experiment results are shown in Table 6.

Table 7 ranks the factors in terms of contribution to cutting resistance from large to small: X₅ > X₈ > X₄ > X₇ > X₆. To make the model optimization simple and feasible, Shear Stiffness (X₅), Bonded Disk Radius (X₈), and Normal Stiffness (X₄), which significantly impact the indicators, are used for parameter calibration.

The three factors that most impact the cutting resistance indicator are X₅ (Shear Stiffness), X₈ (Bonded Disk Radius), and X₄ (Normal Stiffness). The impact rates are 28.1%, 23.5%, and 19.5%, respectively. Since X₆ (Critical Normal Stress) and X₇ (Critical Shear Stress) have minimal impact on cutting resistance, in subsequent experiments, the values of X₆ and X₇ were set to the middle value of the value range, which was 5 × 10⁵ and 5 × 10⁵, respectively.

(2) Steepest ascent

Table 8 Steepest ascent aims to narrow the value range of factors and find the optimal response surface point. Take Critical Normal Stress X₆ as 5 × 10⁵ and Critical Shear Stress X₇ as 5 × 10⁵. The steepest ascent is performed on the other three factors. According to the measured value, the cutting resistance is 37.3 N, narrowing down the range between serial number 1-serial number 3, and ranges of the three factors are Normal Stiffness X₄: 1 × 10⁶–5 × 10⁶, Shear Stiffness X₅: 1 × 10⁶–5 × 10⁶, Bonded Disk Radius X₈: 2.2–2.6.

(3) Box-Behnken experiment scheme

Based on the steepest ascent results, the Box-Behnken experiment was performed on the significant factors X₄, X₅, and X₈. The factors and levels of the Box Behnken experiment are shown in Table 9.

According to Table 10 and Table 11, The model $p$ < 0.001 is highly significant, the lack-of-fit item $p$ = 0.2247 > 0.01, and the lack-of-fit item is not significant. The results of this variance analysis are accurate and usable. Among the single factors, X₄, X₅, and X₈ and the interaction between X₄ X₅ and X₅ X₈, $X_2^8$ significantly affect cutting resistance. Others are not significant. Therefore, the regression equation, are obtained by removing the interaction terms X₄ X₈, $X_2^4$, and $X_2^5$.

(12)${\gamma }_2=34.1+1.42X_4+2.61X_5+1.89X_8+0.825X_4{X}_5+0.65X_5{X}_8-0.75X_8^2$

It can be seen from Figure 9 that when the X₈ bonding radius is controlled at the 0 level, the contour density distribution is uneven, and the response surface diagram is steep, indicating that the X₄ X₅ interaction is significant. It can be seen from the figure that with the increase of the two factors, X₄ and X₅, the cutting resistance has been increasing and has no decreasing trend. It can also be seen from the response surface diagram that the increasing trend of X₅ when X₄ remains unchanged is greater than the increasing trend of X₄ when X₅ is fixed. Cutting resistance is maximum when Normal Stiffness is 5 × 10⁶ N, and Shear Stiffness is 5 × 10⁶ N. The interaction between Normal Stiffness and Shear Stiffness significantly impacts cutting resistance (p < 0.05).

It can be seen from Figure 10 that when the tangential stiffness of X₅ is controlled at the 0 level, the interaction of X₅ X₈ also has a significant impact on the experimental indicators. When the normal stiffness X₅ remains unchanged, the soil penetration resistance increases parabolically with the increase of the bonding radius X₈. When the bonding radius X₈ remains unchanged, the soil penetration resistance increases with the increase of X₅. As the two factors increase, there is no decreasing trend in soil penetration resistance. Cutting resistance is maximum when Normal Stiffness is 5 × 10⁶ N and Bonded Disk Radius is 2.6 mm. The interaction between Normal Stiffness and Bonded Disk Radius significantly impacts Cutting resistance (p < 0.05).

Optimal Parameter Solution and Experimental Verification

(1) Optimal parameter model verification

According to the measured soil penetration resistance 37.3 N, the optimal solution was obtained using Design-expert 13.0 software: Normal Stiffness = 4.635 × 10⁶ N/m, Shear Stiffness = 3.401 × 10⁶ N/m, and Bonded Disk Radius = 2.57 mm. Through the simulation using optimal parameter combination, the maximum cutting resistance was 38.8 N. The error between the simulated and measured values is less than 5%, which indicates that the simulated soil’s mechanical properties are basically consistent with actual soil.

(2) Comparative verification of experiments and simulations

Cutting experiments with and without ultrasonic vibration cutters were used to verify the accuracy of the soil physical parameters calibrated above in the experimental field of Qingdao Agricultural University. As shown in Figure 11, when no ultrasonic vibration is applied, the cutting process changes with time of the simulation and the experiment are relatively consistent. At the end of the cutting, the maximum resistance encountered by the cutter in the vibration-free simulation experiment is 81.9 N. Compared with the measured value of 85.1 N, the error is within 5%. The data during the verification of the optimal parameters also showed that the cutting process changes over time of the simulation and the experiment were consistent when ultrasonic vibration is applied. The change error was also within 5%; after comparison, the model was verified to be feasible.

4. Conclusions

• (1). Based on the discrete element method, using the Hertz-Mindlin (no slip) contact model, soil repose angle simulation experiments were conducted on the soil in the experimental field of Qingdao Agricultural University. Coefficient of restitution X₁, coefficient of static friction X₂ and coefficient of rolling friction X₃ between soil particles were used as experiment factors, and soil repose angle was used as experiment index to conduct a three-factor, three-level central composite experiment. The optimal solution was obtained as coefficient of restitution: 0.47, coefficient of static friction: 0.96, coefficient of rolling friction: 0.28, and the simulation experiment verification was performed, and the repose angle simulation result is 33.22°, with an error of less than 5%.

• (2). Five factors: Normal Stiffness X₄, Shear Stiffness X₅, Critical Normal Stress X₆, Critical Shear Stress X₇, and Bonded Disk Radius X₈ were further calibrated in the Hertz-Mindlin with bonding model. First, a Plackett-Burman experiment was performed to obtain the three factors significantly impacting the experimental indicators: Shear Stiffness X₅ > Bonded Disk Radius X₈ > Normal Stiffness X₄. The climbing experiment narrowed the value range of the three significant factors. Finally, the Box-Behnken experiment was performed to obtain the final optimal solution combination of X₄ = 4.635 × 10⁶ N/m, X₅ = 3.401 × 10⁶ N/m, and X₈ = 2.57 mm. Simulation experiments were conducted to verify that the maximum soil penetration resistance was 38.8 N and its error between the actual experimental result of 37.3 N was less than 5%.

• (3). A method of separately calibrating soil repose angle experiments and ultrasonic vibration cutting soil resistance experiment was proposed to obtain the optimal parameter combination. A discrete element model of soil ultrasonic cutting simulation resistance experiment was established. Experiments were conducted with and without ultrasonic vibration. The results show that the error between the simulation value and the actual value of the two comparative experiments is less than 5%. The resistance change trend of the simulation and the actual test, when the cutter is cutting into the soil, is consistent, indicating that the model calibrated by this method can be used for research on ultrasonic vibration cutting of soil.

Author Contributions

Conceptualization, Y.Q. and S.H.; methodology, Y.Q.; software, Y.Q.; validation, Y.Q.; formal analysis, Y.Q.; resources, J.W. and S.H.; data curation, Y.Q. and Y.L.; writing original draft preparation, Y.Q.; writing review and editing, Y.Q.; visualization, Y.Q. and C.Y.; supervision, S.L. and K.W.; project administration, J.W. and S.H. funding acquisition, J.W. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Data Availability Statement

Available upon request from the corresponding author. The data are not publicly available due to copyright implications.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. Soil repose angle experiment (a) Before experiment (b) After the experiment.

Figure 2. Corner point processing for edge contouring (a) Soil unilateral stacking image. (b) Contour Curve Image. (c) De-noise. (d) Fitted line.

Figure 3. Soil resistance experiment platform of ultrasonic vibration cutting.

Figure 4. Diagram of bond key force.

Figure 5. Soil repose angle simulation experiment.

Figure 6. Cutting experiment simulation model.

Figure 7. Change process of cutting force on soil bonding bond.

Figure 8. (a). Contour plots of interaction X1 X2 on soil repose angle (b) Contour plots of interaction X1 X3 on soil repose angle. (c) Contour plots of interaction X2 X3 on soil repose angle.

Figure 9. Response surface diagram of X4 X5 interaction to cutting resistance.

Figure 10. Response surface diagram of X5 X8 interaction to cutting resistance.

Figure 11. Simulated cutting resistance diagram.

Table 1

Simulation parameter.

Object	Parameter	Setting Values	Source
Soil	Poisson’s Ratio	0.3	[21]
	Density/(kg∙m3)	2640	Measure
	Shear modulus/Pa	1.3 × 106	[40]
steel	Poisson’s ratio	0.3	[28]
	Density/(kg∙m3)	7861	[40]
	Shear modulus/Pa	7.9 × 1010	[40]
Soil-soil	Coefficient of restitution	0.2–0.6	To be calibrated
	Coefficient of static friction	0.3–1	To be calibrated
	Coefficient of rolling friction	0.1–0.3	To be calibrated
Soil-steel	Coefficient of restitution	0.6	[41]
	Coefficient of static friction	0.57	[41]
	Coefficient of rolling friction	0.21	[41]

Table 2

Factors and levels of soil repose angle calibration experiment.

Level	Coefficient of RestitutionX1	Coefficient of Static FrictionX2	Coefficient of Rolling FrictionX3
−1	0.2	0.3	0.1
0	0.4	0.65	0.2
1	0.6	1	0.3

Table 3

Factors and levels of soil resistance calibration experiment.

Factors	Low Level (−1)	High Level (1)
Normal Stiffness X4	1 × 106	9 × 106
Shear Stiffness X5	1 × 106	9 × 106
Critical Normal Stress X6	1 × 105	9 × 105
Critical Shear Stress X7	1 × 105	9 × 105
Bonded Disk Radius X8	2.2	3.0

Table 4

Design and results of orthogonal combination experiment.

No.	Coefficient of RestitutionX1	Coefficient of Static FrictionX2	Coefficient of Rolling FrictionX3	$\mathbf{Angle}\mathbf{of}\mathbf{Repose}{\mathit{\gamma }}_1$ (°)
1	0.4	1	0.3	34.38
2	0.6	0.3	0.2	24.64
3	0.4	0.65	0.2	26.93
4	0.4	1	0.1	32.08
5	0.4	0.3	0.3	21.2
6	0.2	0.65	0.1	24.06
7	0.4	0.3	0.1	23.49
8	0.6	0.65	0.3	33.8
9	0.2	0.65	0.3	21.8
10	0.6	1	0.2	33.7
11	0.4	0.65	0.2	27.5
12	0.2	1	0.2	25.21
13	0.2	0.3	0.2	10.8
14	0.6	0.65	0.1	30.94

Table 5

Analysis of variance of orthogonal combination experiment.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value
Model	534.00	9	59.33	115.49	0.0002	Significant
X 1	212.28	1	212.28	413.20	<0.0001
X 2	255.83	1	255.83	497.96	<0.0001
X 3	0.0465	1	0.0465	0.0905	0.7785
X 1 X 2	7.16	1	7.16	13.93	0.0203
X 1 X 3	6.55	1	6.55	12.76	0.0233
X 2 X 3	5.27	1	5.27	10.25	0.0328
$X_2^1$	11.34	1	11.34	22.07	0.0093
$X_2^2$	9.74	1	9.74	18.97	0.0121
$X_2^3$	17.19	1	17.19	33.45	0.0044
residual	2.06	4	0.5138
Lack of fit	1.89	3	0.6309	3.88	0.3532	Not Significant
Pure error	0.1625	1	0.1625
Cor total	536.05	13

R² = 0.9962. Adjusted R² = 0.9875.

Table 6

Design and results of Plackett-Burman experiments.

No.	Factor					Cutting Resistance${\mathit{\gamma }}_2$ (N)
	Normal StiffnessX4	Shear Stiffness X5	Critical Normal StressX6	Critical Shear StressX7	Bonded Disk RadiusX8
1	−1	1	1	−1	1	39.8
2	1	1	1	−1	−1	40.9
3	−1	−1	−1	1	−1	27.9
4	−1	1	1	1	−1	36.3
5	1	−1	1	1	−1	31.1
6	1	−1	−1	−1	1	39.4
7	1	1	−1	−1	−1	41.1
8	1	1	−1	1	1	56.5
9	−1	1	−1	1	1	41.8
10	1	−1	1	1	1	42.7
11	−1	−1	−1	−1	−1	25.3
12	−1	−1	1	−1	1	33.8

Table 7

Results of sensitivity analysis.

Factor	$\mathbf{Cutting}\mathbf{Resistance}/{\mathit{\gamma }}_2$
	Standardization Effect	Mean Sum of Square	Contribution Degree/%
X 4	7.798	182.442	19.54
X 5	9.358	262.735	28.17
X 6	−1.228	4.526	0.49
X 7	2.671	21.413	2.28
X 8	8.565	220.078	23.56

Table 8

Design and results of steepest ascent.

No.	Normal Stiffness X4 (N/m3)	Shear Stiffness X5 (N/m3)	Bonded Disk Radius X8 (mm)	$\mathbf{Cutting}\mathbf{Resistance}{\mathit{\gamma }}_2$ (N)
1	1	1	2.2	27.0
2	3	3	2.4	34.4
3	5	5	2.6	40.1
4	7	7	2.8	47.1
5	9	9	3.0	58.7

Table 9

Factors and levels of the Box-Behnken experiment.

Factor	Code	Level
		−1	0	1
Normal Stiffness	X 4	1 × 106	3 × 106	5 × 106
Shear Stiffness	X 5	1 × 106	3 × 106	5 × 106
Bonded Disk Radius	X 8	2.2	2.4	2.6

Table 10

The design scheme and results of Box-Behnken.

No.	Factor			$\mathbf{Cutting}\mathbf{Resistance}{\mathit{\gamma }}_2$ (N)
	X 4	X 5	X 8
1	0	−1	1	31.8
2	0	1	−1	33.4
3	1	0	−1	32.8
4	0	0	0	33.9
5	−1	1	0	34.2
6	0	0	0	34.2
7	0	−1	−1	29.2
8	−1	0	−1	30.6
9	1	1	0	39.1
10	−1	−1	0	30.9
11	0	1	1	38.6
12	0	0	0	33.8
13	1	−1	0	32.5
14	1	0	1	36.7
15	0	0	0	34.5
16	0	0	0	34.1
17	−1	0	1	34.0

Table 11

ANOVA of Box—Behnken design quadratic model.

Source	Sum of Squares	df	Mean Square	F-Value	p-Value
Model	106.34	9	11.82	103.07	<0.0001	Significant
X 4	16.25	1	16.25	141.70	<0.0001
X 5	54.60	1	54.60	467.27	<0.0001
X 8	28.50	1	28.50	248.61	<0.0001
X 4 X 5	2.72	1	2.72	23.75	0.0018
X 4 X 8	0.0625	1	0.0625	0.5452	0.4843
X 5 X 8	1.69	1	1.69	14.74	0.0064
$X_4^2$	0.1289	1	0.1289	1.12	0.3241
$X_5^2$	0.0421	1	0.0421	0.3673	0.5636
$X_8^2$	2.37	1	2.37	20.66	0.0027
Residual	0.8025	7	0.1146
Lack of fit	0.5025	3	0.1675	2.23	0.2267	Not significant
Pure error	0.3000	4	0.0750
Cor total	107.14	16

R² = 0.9932. Adjusted R² = 0.9845.

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