Introduction

Leymus chinensis is a Gramineae Leymus plant, often referred to as the ‘King of Grasses’ and also known as Elymus. It is one of the essential forage grasses in the central and northeastern regions of Inner Mongolia Autonomous Region of China1,2. Leymus chinensis is characterized by high yield, rich nutrition content, good palatability, and high feeding value. Its root system is well-developed, with complex transverse rhizome that easily form a dense root network, which helps bind and stabilize the soil. It plays a crucial role in windbreaks and sand fixation, water conservation, soil and water conservation, and soil improvement2. Currently, the research on the Leymus chinensis industry in China primarily focuses on breeding superior varieties, increasing reproductive capacity, expanding planting areas, enhancing seed harvesting capabilities, and ensuring forage yields3. However, there is limited specialized machinery for specific grass species, and the avaliable types are few. In particular, the harvesting machinery for Leymus chinensis seeds is not yet sufficiently developed. There are problems such as significant harvest losses and low seed cleaning rate4. These problems are hindering the development of the Leymus chinensis seeds industry, so it is necessary to study the efficient production equipment of Leymus chinensis seeds.

The research on the interaction between crops and mechanical equipment and the force and motion characteristics of crops in equipment is very important for the equipment’s design and optimization. However, due to the small size and complex shape of the Leymus chinensis seeds, it is difficult in accurately analyzing the force and motion of the seeds in the production machinery through physical tests and the complex forces acting on the seeds during the harvest process5,6,7. P. A. Cundall et al.8 proposed the discrete element method (DEM), which can accurately analyze the stress state and motion behavior of agricultural materials and track their mechanical behavior and motion trajectory in real-time. The accuracy of its simulation model and parameters is an essential factor affecting the simulation results9,10. However, the three-dimensional model is divergent from the real model, and the direct application of physical parameters to the simulation model is not accurate enough. Therefore, calibrating the material’s discrete element simulation parameters to obtain a more accurate discrete element model is imperative11,12.

In recent years, an increasing number of researchers have expand the study of agricultural materials’ discrete element simulation parameters calibration. This supplies a theoretical rationale for the materials’ complex motion and force analysis, simulation study on the interactive mechanism between materials and mechanical equipment, and the optimization design of related mechanical equipment5,13. Considering the characteristics of flax grains determined through physical test, Shi et al.6 calibrated the rolling friction coefficient of the flax model through the simulation stacking test and verified it through the seeding test. Using the EDEM rapid filling method, Wu et al.14 established a discrete element model of Peucedanum seeds. Then they optimized the significant factors, using the repose angle as the response variable. Applying the Hertz–Mindlin contact model with JKR for particles interactions, Xing et al.15 calibrated the discrete element parameters for soil particles’. Then they verified the model and parameters’ accuracy by stacking and soil-breaking resistance tests. Employing the repose angle as the optimization target value, Shi et al.16 acquired the farmland soil particles’ parameters in northwest China’s arid area. With the penetration resistance of duckbill structure as the response value, they carried out simulation and actual comparison tests to verify the accuracy of the calibrated parameters. Based on the principle of particle contact scaling, Li et al.17 performed dimensional analysis, enlarged wheat flour particles, calibrated simulation parameters by stacking angle and verified the model’s accuracy. Chiho et al.18 proposed a calibration method for non-spherical viscous particles with reference to the JKR theory and surface energy. Then the powder’s rolling friction coefficient was calibrated. Based on the fish pellet feed’s discrete element contact parameters that was calibrated, Niu et al.19 confirmed the feed’s parameters of the bonding model through the uniaxial compression test and stacking test. Combining physical and simulation test methods, Ma et al.20 calibrated the required discrete element parameters for the compression process of alfalfa straw. Then they extracted the optimized simulation parameters using the physical stacking angle as the target value. Taking cotton straw as the main object, Zhang et al.21 executed the stacking simulation experiment relying on the Hertz–Mindlin (no slip) model and the bending simulation experiment using the bonding model. Combined with physical experiments, the cotton straw’s discrete element parameters were obtained. Li et al.22 calibrated the required parameters for the simulation of wheat straw through compression, and shear and three-point bending tests. K. G. Santos et al.23 measured the stacking angle of dried cherries by rotating drum method. Then the contact parameters between dried cherries particles was confirmed with the angle as an evaluation index. In summary, in the current research of discrete element simulation parameters calibration, it mainly uses the repose angle as the evaluation index, and the optimization or verification is usually carried out through the stacking test. A small amount of research employs compression tests, bending tests, soil-breaking resistance tests, etc. It provides a reference for the paper’s research. However, most research objects are large seeds, soil, powder, straw, etc. For Leymus chinensis seeds, there are no calibrated discrete element simulation parameters.

In this paper, the Leymus chinensis seed discrete element simulation model was established with the Hertz–Mindlin (no slip) model, and its shape and dimensions were determined by the physical experiment. In view of the limitations of its shape characteristics, the static stacking anglewas used as the response factor24. A Plackett–Burman Design (PBD) test, steepest slope climbing (SSC) test and Response Surface Methodology (RSM) analyst test based on Box–Behnken Design (BBD) were performed to calibrate the Leymus chinensis seeds’ discrete element simulation parameters. By comparing the physical and simulation test results, the reliability of the simulation parameters and model was verified25. This study provides a reference for the correlation studies, including the interaction mechanism between Leymus chinensis seeds and mechanical equipment and the force and motion behavior of Leymus chinensis seeds in mechanical equipment based on the discrete element method.

Materials and methods

Determination of physical property parameters

The materials were gathered from the ‘Zhongke No.1’ Leymus chinensis seeds produced in July 2023, from the Third Company of the 136th Regiment of the Eighth Division of Xinjiang Production and Construction Corps (as shown in Figs. 1, 2). The seeds belong to granular material, the Hertz–Mindlin model can be used to establish its discrete element model26. The required parameters include the three axis dimensions, density, thousand seed weight, moisture content, Poisson ratio, shear modulus, coefficient of static friction (COS), coefficient of rolling friction (COR), and coefficient of collision recovery (COCR) of Leymus chinensis seeds27. The angle of repose (AOR) was used as the evaluation index in the simulation test.

Fig. 1
Fig. 1
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Leymus chinensis plant area.

Fig. 2
Fig. 2
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Leymus chinensis seeds (with bran).

Measurement of basic physical property parameters

A CD-30C digital vernier caliper (Precision: 0.01 mm, Dongguan Sanliang Precision Instrument Co., Ltd., China) was used to measure the seeds’ three-axis size. With reference to the Chinese national standard GB/T 2930.9-2017 ‘Weight determination of grass grain inspection procedures’28 and GB/T 5519-2018 ‘Determination of thousand seed weight of grains and beans’29, the seeds’ density and thousand seed weight were measured by WKT-120A electronic density meter (Quality precision: 0.001 g, Density precision: 0.0001 g/cm3, Jiangsu VicoMeter Instrument Co., Ltd., China). According to GB/T 2930.8-2017 ‘Determination of moisture content in grass grain inspection procedure’30, the seeds’ moisture content was measured using low-temperature drying method. The test instrument is the SO-6F drying test chamber (Strayfield, UK). Each group was repeated 5 times. Then the basic physical property parameters’ average values of Leymus chinensis seeds were measured. The results is shown in Table 1.

Table 1 The basic physical property parameters of Leymus chinensis seeds.
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Determination of Poisson’s ratio

The Leymus chinensis seed particles are too small, and there are significant differences in length, thickness and width, so it is intricate in measuring the Poisson’s ratio. In this paper, the Poisson’s ratio was determined using the definition method. That means conducting a compression test on Leymus chinensis seeds to measure the variable quantity in the length and thickness directions, then the ratio of strain in the length direction to that in the thickness direction was calculated31. Randomly selected 20 Leymus chinensis seeds, and their original length and thickness were respectively recorded. The compression test of Leymus chinensis seeds (as shown in Fig. 3) was carried out by the TA.XT Plus physical property tester (Measurement range: 0~500 KN, Precision: 0.001 N, Shanghai Rui Fen Intelligent Technology Co., Ltd., China). In the control system of the tester, set the loading speed to 0.5 mm/s, and select a stainless steel cylinder with a diameter of 5 mm as the probe, the probe was adjusted to a suitable position, and 1 mm was loaded along the thickness direction of Leymus chinensis seed.

Fig. 3
Fig. 3
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Compression test of Leymus chinensis seeds. 1. Computer 2. Leymus chinensis seed 3. Physical property tester.

The deformation in length and thickness direction in compression test of Leymus chinensis seeds was measured by the digital vernier caliper. The compression process load–displacement data was extracted through the software analysis module of the physical property tester. The experiment was repeated 5 times, and obtained the average value. The seed’s Poisson’s ratio was calculated by formula (1), and the result was \(\mu =0.071\pm 0.026\). The elastic modulus was calculated by formula (2), and the result was \(E=1.33\times {10}^{10}\text{Pa}\). The shear modulus was calculated by formula (3), and the result was \(G=7.05\times {10}^{9}\text{Pa}\).

$$\mu =\left|\frac{{\varepsilon }_{x}}{{\varepsilon }_{y}}\right|=\frac{\Delta L/L}{\Delta H/H}$$
(1)
$$E=\frac{\sigma }{\varepsilon }=\frac{F/A}{\Delta L/L}$$
(2)
$$G=\frac{E}{2(1+\mu )}$$
(3)

where \(\mu\) is Leymus chinensis seed’s Poisson’s ratio, \({\varepsilon }_{x}\) is the seed’s lateral strain, \({\varepsilon }_{y}\) is seed’s longitudinal strain, \(\Delta L\) is the deformation in length direction of seed, mm, \(L\) is the seed’s initial length, mm, \(\Delta H\) is the deformation in thickness direction, mm, \(H\) is the seed’s initial thickness, mm, \(E\) is the elastic modulus, Pa, \(\sigma\) is the maximum compressive stress, Pa, \(\varepsilon\) is the seed’s linear strain, \(F\) is the maximum compression force on the seed, N, \(A\) is cross-sectional area of the seed, considering the shape of the seed, its cross-sectional area is approximately half of the product of the thickness and height of the seed, mm2, \(G\) is the seed’s shear modulus, Pa.

Determination of connect parameter

Determination of the collision recovery coefficient

Relying on Newton’s law of collision, the coefficient of collision recovery (COCR) is defined as the ratio of the relative velocity before and after the collision of the two objects32, that is

$$e=\frac{{v}_{2}^{{{\prime}}}-{v}_{1}^{{{\prime}}}}{{v}_{1}-{v}_{2}}$$
(4)

where \({v}_{1}\), \({v}_{2}\) are the velocity before the collision, m/s, \({v}_{1}^{\prime}\), \({v}_{2}^{\prime}\) are the velocity before the collision, m/s.

Based on the definition, the collision test was carried into execution. The Leymus chinensis seeds fell freely and collided with the contact material. The COCR measurement system was established. It included a computer, a high-speed camera, a height ruler and a drop shelf, as shown in Fig. 4. The seed fell freely from the drop shelf and collided with the contact material on the horizontal plane. After the collision, the seed bounced freely. Because the contact material is statically placed on the plane, the velocity before and after the collision is 0. Ignoring the air resistance, only the gravity does work in the seed’s process of free-falling and rebound33, so the recovery coefficient can be simplified as

$$e=\frac{{v}^{\prime}}{v}=\frac{\sqrt{2\text{g}h}}{\sqrt{2\text{g}H}}=\sqrt{\frac{h}{H}}$$
(5)

where \(e\) is the COCR, \({v}^{\prime}\) is the Leymus chinensis seed’s velocity after the collision, m/s, \(v\) is the velocity before the collision, m/s, \(\text{g}\) is the gravitational acceleration, m/s2, \(h\) is the Leymus chinensis seed’s rebound height after collision, mm, \(H\) is the seed’s release height, mm.

Fig. 4
Fig. 4
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The schematic diagram of the COCR measurement system. 1. Computer, 2. Seed, 3. drop shelf, 4. height ruler, 5. high-speed camera, 6. contact material. Note: H is the seed’s release height, h is the seed’s rebound height after collision.

Considering that the commonly used contact materials in the process of seed harvest are steel and seeds, the collision recovery coefficient was determined by the collision test of falling seeds against the contact material in this study. The contact material for the test was set to two. One is a steel plate, and the other is a seed material board made of neatly arranged Leymus chinensis seeds bonded together, as shown in Fig. 5. Then the free-falling collision test of the seeds was carried out. The FASTEC-TS4 dual-mode high-speed camera (Frame rate: 510 fps, Fastec Imaging, USA) was employed for recording the movement process about the seeds’ collision and rebound. The seed’s falling height was set at 25 cm, and the distance between the lens and the test surface was set at 25 cm. The experiment was repeated 5 times. The data processing module of the high-speed camera software ProAnalyst (2023, Xcitex Inc, USA) was used to analyze the collision process video, as shown in Fig. 6. According to the above collision coefficient measurement principle, the rebound height data information of Leymus chinensis seed was extracted. The COCR is calculated by formula (5). The test results show that: the COCR between Leymus chinensis seeds is 0.209 ± 0.048, between seeds and steel plate is 0.223 ± 0.032.

Fig. 5
Fig. 5
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The seed material board.

Fig. 6
Fig. 6
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ProAnalyst software video processing interface.

Determination of the static friction coefficient

The established coefficient of static friction (COS) measurement system is illustrated in Fig. 7. When the seed is lied statically on the oblique plane, there is

Fig. 7
Fig. 7
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The friction coefficient measurement system. Note:\(\alpha\) is the angle of the Inclinometer, °, \(N\) is the supporting force provided by the inclined plane of Leymus chinensis seed, N, \(G\) is the gravity on Leymus chinensis seeds, N.

$$\left\{\begin{array}{l}G\text{ sin }\alpha =f\\ N=G\text{ cos }\alpha \end{array}\right.$$
(6)

Since \(f=\gamma N\), when increasing the inclined angle, the seeds on the surface will have a sliding trend, there is

$$\gamma =\text{tan }\alpha$$
(7)

where \(G\) is the gravity on Leymus chinensis seeds, N, \(\alpha\) is the angle of Inclinometer, °, \(f\) is the seed’s friction force when it is stationary, N, \(\gamma\) is the COS of Leymus chinensis seed.

At this time, the inclined angle can be regarded as the seed’s static friction angle, and the COS can be determined by this principle and method34.

The study measured the required COS with a self-made friction coefficient meter and an electronic angle ruler. The steel plate and the homemade seed material board were respectively used as contact surfaces. When measuring the friction coefficient, a single Leymus chinensis seed was put on the contact surface, and the handle was slowly rotated to make one end of the inclined plane rise so that the inclined angle increases continuously. When the Leymus chinensis seed shows a tendency to slide on the contact surface, stop the rotation and record the angle of the inclinometer at this time. The measured angle was substituted into formula (7) to calculate the COS. The experiment was replicated 5 times, and obtained the mean value. The test results show that the COS between Leymus chinensis seeds and steel plate is 0.342 ± 0.030, between seeds is 0.659 ± 0.063.

Determination of the rolling friction coefficient

The determination method of the coefficient of rolling friction (COR) is keeping with the COS. When turning the handle to lift the inclinometer, the seed rolls immediately and stops turning. The slope angle was record at this time. Then, the COR between the seed and the measured material was calculated. The experiment was repeated 5 times to get the average value. The test results show that the COR between the Leymus chinensis seeds and the steel plate is 0.236 ± 0.078, between seeds is 0.302 ± 0.037.

Determination of static repose angle

According to the national standard GB 11986-89 ‘Measurement of the angle of repose of surfactant powders and particles’35, the angle of repose (AOR) of Leymus chinensis seeds was determined using the pouring method. A funnel was the main test device. The inner diameters of the feed port and the discharge port were respectively 150 mm and 25 mm, and the height of the blanking port from the ground was 150 mm. Randomly, 20 g Leymus chinensis seeds was weighted and poured into the inlet above the funnel to make them fall freely. After the seeds come to rest on the lower plane, the seed pile front-view image was captured using a high-speed camera. The unilateral seeds’ repose figure was processed by MATLAB software. The boundary pixel points were extracted, then the boundary curve was fitted. They were applied to calculate the AOR of Leymus chinensis seeds. The unilateral repose angle image and boundary fitting curve of the seeds are presented in Figs. 8 and 9. The test was repeated 10 times to take the mean value. The AOR measured by physical test of Leymus chinensis seeds is 40.08°.

Fig. 8
Fig. 8
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Seeds unilateral repose image.

Fig. 9
Fig. 9
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Unilateral repose angle boundary fitting curve.

Experimental design and methods

On account of the simulation model is distinguish from the actual physical model, it is improper to directly use the actual physical parameters to the simulation test. Drawing from the research methods proposed by previous researchers, this study preliminarily determined the basic Leymus chinensis seeds’s physical parameters using the method of physical experiments. Following the measured values, the range of discrete element simulation parameters was determined. Focus on improving the accuracy of the simulation model and parameters, the simulated accumulation test was performed to calibrate the relevant parameters. In this paper, the calibration method is relying on the RSM.

Establishment of discrete element simulation model

In light of the shape and size characteristics of Leymus chinensis seeds measured though physical experiments, a simplified geometric model36 was established using the 3D modelling software Solidworks (2022, Dassault Systèmes—SolidWorks Corporation, France). It was converted into stl file format and input to the discrete element simulation software EDEM (2021, Altair Engineering Inc, USA) as a particle template. Considering the small size of Leymus chinensis seed particles, and aim at balancing the simulation efficiency and authenticity, the research filled the Leymus chinensis seed particle template with single spherical particles of different radii in EDEM software. It selected the Hertz–Mindlin (no slip) model as the particle contact model. Then it input the corresponding simulation parameters into EDEM to automatically calculate the material properties of the simulation model of Leymus chinensis seed particles. The established seed discrete element model is featured in Fig. 10. Similarly, using Soliworks to establish the three-dimensional model of the funnel, and it was transformed into a stl-format file. Then, as the form of a geometric model, it was imported into EDEM software. A physical plane was established at 150 mm below the funnel for grain accumulation. At the entrance above the funnel, in order to generate seed particles, a particle factory was established, which was a virtual plane. The Leymus chinensis seed particles were dynamically generated. The generation rate was set to 400 g/s, the total amount was 3 g, and the particle size was fixed. The total time of the simulation accumulation test was set to be 3 s, the time step was 4.4 × 10−7 s, and the mesh size was 40 times the minimum particle radius.

Fig. 10
Fig. 10
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Leymus chinensis seed discrete element model.

The seeds were generated through the particle factory above the funnel during the simulation. The seeds fell freely under the influence of gravity. They were stationary at the bottom of the physical plate, forming a static accumulation angle, as shown in Fig. 11. After the test, the same method as the physical experiment was used. MATLAB (2020, Mathworks Inc, USA) was used to process the accumulation angle image.

Fig. 11
Fig. 11
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The repose angle determination model.

Experimental design of response surface methodology

(1) Plackett–Burman design test

In this study, the Hertz–Mindlin (no slip) model was selected as the particle contact model of the Leymus chinensis seeds’ discrete element simulation model. The required simulation-related parameters are Poisson’s ratio, shear modulus, and COS, COR, COCR between Leymus chinensis seeds and steel, and between seeds. Taking the AOR of Leymus chinensis seeds as the response value, a Plackett–Burman Design (PBD) test was developed to identify the significant influential parameters on the AOR37. According to the physical test results, the average value was the 0 level of the test parameters. According to the measured range, which was exhibited in Table 2, the low and high levels (− 1 and + 1) parameters were set. The Design-Expert (13.0, Stat-Ease Inc, USA) was applied to formulate PBD test.

Table 2 The parameter range of the Plackett–Burman test.
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(2) The steepest slope climbing test

Relying on the test results of PBD test, considering the significant parameters’ positive and negative effects, it took a certain step size to arrange the steepest slope climbing (SSC) test. Taking the relative error of the simulation accumulation angle and the actual angle as the evaluation indicator, the optimal value interval of the simulation test parameters was determined.

(3) Box–Behnken design test

On the basis of the variation trend of relative error in the SSC, it taken the peak value as the central point (the 0 level), and its adjacent points were perceived as the low (− 1) and high (+ 1) levels respectively. The required other non-significant parameters were in line with the parameters used in the SSC test. The RSM analysis experiment based on BBD was carried out. Then the second-order regression model between each significant parameter and the simulated stacking angle was constructed.

(4) Response Surface Methodology parameter optimization and verification

Using the Design-Expert 13.0 software optimization module and considering the physically measured AOR as the target value, the second-order regression equation of the simulated value was optimized. Then the parameter combination that makes the simulated value closest to the actual value was got. Using it to carry out three groups simulation tests, the accuracy of the simulation test and the constructed model was proved, taking the relative error between the measured and simulation AOR as the performance indicator.

Results and discussion

Analysis of the screening results from the Plackett–Burman design test

Considering the AOR of Leymus chinensis seeds as the output variable, the PBD test was implemented drawing from the measured parameter range. The test scheme and results are presented in Table 3. The accumulation angle of Leymus chinensis seeds was determined by MATLAB.

Table 3 Scheme and results of the Plackett–Burman test.
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Appling the Design-Expert 13.0 software data analysis module, it conducted the variance analysis of the PBD test results. The simulation parameters that have a significant influence on the AOR is indicated in Table 4. It is showed that the four simulation parameters’ P value is lower than 0.05. They are the COS between Leymus chinensis seeds, the COR between seeds, the COCR between seeds and steel plate, and the COS between seeds and steel plate. That means these parameters have a significant influence on the simulated AOR. The P value of other simulation parameters is greater than 0.05, and the influence on the simulation AOR is insignificant.

Table 4 The parameters’ significance evaluation of the Plackett–Burman test.
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Analysis of the steepest slope climbing test simulation results

The PBD test was executed to identify significant parameters of four. Following these parameters’ positive and negative effects, the SSC test was designed using a certain step size increase or decrease. The required non-significant parameters were set with reference to the parameters’ average values measured by the physical experiments, which are: The Leymus chinensis seeds’ Poisson’s ratio is 0.71, the shear modulus is 7050 MPa, the COCR between seeds is 0.209, and the COR between seeds and steel plate is 0.236. The relative error of the AOR between the simulation and actual test was used as the evaluation indicator, and then the more precise simulation parameters range values were determined.

The scheme and results of the SSC test are indicated in Table 5. It demonstrates that the AOR proportional error between the simulation and actual test decreases firstly and then increases according to the order of test. The relative error of the No. 2 test is the minimum value, and the optimal value range can be determined near No. 2.

Table 5 Test scheme and results of the steepest slope climbing method.
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Analysis of response surface methodology test results

The parameters of the No. 2 test group in the SSC test was used as the center point (0), the No. 1 and No. 3 were set as the low (− 1) and high (+ 1) levels. Then the Box–Behnken Design (BBD) test was executed. It set the non-significant parameters in line with the parameters used in the steepest ascent test. The RSM test based on BBD was carried out. It was designed by Design-Expert 13.0 software. It carried out 27 sets of simulation tests, of which 3 central points were set. The scheme and results are demonstrated in Table 6.

Table 6 Scheme and results of the Box–Behnken Design test.
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Through the Design-Expert 13.0 software data analysis module, the multivariable regression fitting was carried through relying on the BBD test results. It obtained the second-order regression equation between the parameters and simulation stacking angle, which as follows:

$$\begin{aligned}\theta &=43.32+0.1000D-0.8292E+0.8358F-2.6100G+3.2100DE-0.5550DF-0.4100DG-0.5450EF+3.9800EG\\ &\quad -3.6200FG-0.9925{D}^{2}+1.7100{E}^{2}+3.4800{F}^{2}-3.5300{G}^{2} \end{aligned}$$
(8)

The simulation test’s variance analysis outcomes are illustrated in Table 7. It indicates that the P value is less than 0.0001, the lack of fit P value is 0.0065, and they are all less than 0.01. The regression equation determination coefficient R2 is 0.9516, the adjusted determination coefficient Adjusted R2 is 0.8951, and the coefficient of variation CV is 3.37%. It shows that the established model fits well, is very dependable. The model can reflect the actual situation. From Table 7, it can be seen that G, DE, EG, FG, F2 and G2 affect the AOR quite significantly. The effect of E, F and E2 on the AOR is significant. However, D, DF, DG, EF and D2 have no significant effect on the AOR.

Table 7 Box–Behnken test regression model analysis of variance.
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The effects of interaction terms DE, EG and FG on the repose angle are shown in Figs. 12, 13, 14. As shown in Figs. 12 and 13, with the increase of the COR between Leymus chinensis seeds, the AOR rising simultaneously. That may be because with the increment of the COR, the sliding resistance between the seeds increases, making the formed AOR more stable. It was evident from Figs. 13 and 14, with the increase of the COS between seeds and steel plate, the AOR initially rises and then drops. That may be because the larger the COS is, the greater the seeds’ resistance at rest is, so that the AOR formed is not easy to scatter, making it gradually increase. However, when the AOR increases to greater than the friction angle, the stability of the seed’s AOR will be destroyed, and then it will decrease.

Fig. 12.
Fig. 12.
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3D response surface (DE).

Fig. 13.
Fig. 13.
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3D response surface (EG).

Fig. 14.
Fig. 14.
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3D response surface (FG).

Determination of optimal parameter combination and experimental verification

Using the Design-Expert 13.0 software optimization module, it optimized the second-order regression equation between the significant parameters and simulated AOR. It used the physically measured AOR (40.08°) as the objective value. Its objective function and optimization function is as formula (9).

$$\left\{\begin{array}{l}AOR\left(D,E,F,G\right)=40.08^\circ \\ s.t. \left\{\begin{array}{l}\begin{array}{l}0.55\le D\le 0.63\\ 0.20\le E\le 0.28\end{array}\\ 0.22\le F\le 0.30\\ 0.37\le G\le 0.45\end{array}\right.\end{array}\right.$$
(9)

A set of Leymus chinensis seeds’ parameters were optimized, making the simulated AOR keeping with the actual value. That is: the COS between seeds D is 0.627, the COR between seeds E is 0.210, the COCR between seeds and steel plate F is 0.283, the COS between seeds and steel plate G is 0.421.

With the aim of verify the accuracy and reliability of the Leymus chinensis seeds’ calibrated simulation parameters, three groups of AOR simulation experiments were carried out. The results of the test are shown in Figs. 15 and 16. Among them, the significant parameter took the optimal parameter combination, and the non-significant simulation parameter value was the same as the steepest ascent method. The simulation results show that the Leymus chinensis seeds’ AOR are respectively 40.66°, 40.63° and 40.03°. The relative errors with the physical test values are 1.45%, 1.37% and 0.12%. The average AOR of the confirmatory test is 40.44°. It’s average relative error is 0.98%. There is no distinct difference between the simulation and physical test value. It indicates that the accuracy and reliability of the simulation test and the second-order regression model between the constructed significant parameters and the AOR were proved. That presents that the method, obtained parameters and model are suitable for the calibration of the contact parameters of Leymus chinensis seeds.

Fig. 15
Fig. 15
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Seeds unilateral repose image.

Fig. 16
Fig. 16
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Unilateral repose angle boundary fitting curve.

Conclusion

  1. (1)

    Implementing the physical experiments, the Leymus chinensis seeds’ essential physical parameters were measured. It included three-axis size, density, moisture content, thousand seed weight, shear modulus and Poisson’s ratio. The self-made test bench, which including A FASTEC-TS4 camera, was employed to measure the collision recovery coefficient. The outcomes evidence that the average values of the coefficient between seeds, between seeds and steel plates are respectively 0.209 ± 0.048 and 0.223 ± 0.032. A self-made friction coefficient instrument and electronic angle ruler was deployed in measuring the seeds’ friction coefficient. The results show that the mean values of the static friction coefficient and the rolling friction coefficient between seeds are respectively 0.659 ± 0.063 and 0.302 ± 0.037, between seeds and steel plate are 0.342 ± 0.030 and 0.236 ± 0.078.

  2. (2)

    According to the results of physical experiments, it determined the range of selected simulation parameters. Treating the repose angle of Leymus chinensis seeds as the target value, it implemented the Plackett–Burman Design test to identify the factors that significantly affecting the repose angle. They are the coefficient of static friction between seeds, between seeds and steel plate, the coefficient of rolling friction between seeds and the coefficient of collision recovery between seeds and steel plate. Further, the range of significant elements was narrowed with the the steepest slope climbing test. Then, taking the results into account, the Box–Behnken Design test was proceed. It established the second-order regression equation between the influential parameters and simulation repose angle. Choosing the physically measured repose angle (40.08°) as the target value, it specified the obtained model more accurately. A set of the ideal parameters, which ensure the simulated repose angle is almost equivalent to the actual value, were optimized: the coefficient of static friction between seeds D is 0.627, between seeds and steel plate G is 0.421, the coefficient of rolling friction between seeds E is 0.210, the coefficient of collision recovery friction between seeds and steel plate F is 0.283.

  3. (3)

    In accordance with the obtained optimal parameter combination, the stacking simulation test of the Leymus chinensis seeds was carried into execution. The average simulation ropose angle is 40.66°, and it’s only a relative error of 0.9% to the physical measured value. The test verified the reliability and accuracy of the simulation parameters and model. The research establishes a simulation parameter foundation for the related study including optimization design, simulation research and harvesting technology and equipment of Leymus chinensis seeds.