Crushing Characteristics of Sorghum Grains Subjected to Compression and Impact Loading at Different Moisture Contents

Abstract

Sorghum is an important grain crop in many countries worldwide, yet it often suffers from high levels of fragmentation during harvest due to varying maturity. To this end, a study was conducted to investigate the crushing characteristics of sorghum grains subjected to compression and impact loading at different moisture contents. By configuring sorghum kernels with varying ranges of water and determining their physical parameters, such as length, width, etc., the geometric mean diameter of sorghum kernels was 3.105–3.550 mm, and the sphericity was above 75%. Compression tests were conducted on sorghum kernels in the triaxial direction. The compression energy was calculated to be 13.409–19.229 J on the X-axis, 16.313–21.409 J on the Y-axis, and 17.609–24.741 J on the Z-axis. In contrast, the apparent contact modulus of elasticity was calculated, with the maximum modulus of elasticity up to 72 MPa in the Z-axis direction, and the variations in the X-axis and Y-axis were approximate. Finally, mechanical impact tests were conducted to measure the critical angle of seed breakage, and a mathematical model was established to predict the impact of mechanical breakage force. The error between the predicted and experimental values was within 3%. This paper conducted compression and impact mechanics tests on sorghum seeds at different moisture contents to provide a design basis for sorghum harvesting and processing and other harvesting equipment.

1. Introduction

Sorghum, also known as corn, is a critical dryland crop in China with drought and salinity tolerance, high photosynthetic capacity, and a wide range of uses. It plays an essential role in agricultural production and plantation restructuring, and is widely grown in northeastern, southwestern, and northern China [1]. Sorghum is damaged by impact during harvesting, threshing, separation, cleaning, transportation, and storage [2]. When the seed coat is damaged, saprophytic and phytopathogenic microorganisms attached to the outside of the hull can invade the seeds. The seeds will age and deteriorate, severely affecting the storage life of sorghum [3]. Cracked or broken seeds have a shortened storage life and loss of activity. The seeds fail to germinate and reduce the germination rate of the seeds [4]. Currently, sorghum is increasingly mechanized for harvesting, storage, transportation, and processing, and in each process, the compressive impact loads on sorghum seeds are involved. In this case, the breakage loss of sorghum seed grains is also severe, as shown in Figure 1. Therefore, studying the characteristics of sorghum seed grains under resistance to compression and impact crushing is essential for designing agricultural equipment for sorghum seeding, cleaning, harvesting, and threshing.

The mechanical properties of the grain seeds of soybean, corn, wheat, and rice are studied by scholars in several countries around the world [5]. The basic parameters of the seed grains were determined [6,7]. The effect of different moisture contents on the mechanical properties of the seed grains was determined [8,9,10]. The mechanical properties of the seeds were measured by mechanical tests, such as the elastic modulus of the seeds by compression and impact [11,12,13,14,15,16]. The discrete element method is applied to build mechanical models of seeds, such as rice and wheat, to simulate the stress of seed crushing [17,18,19,20,21].

However, research on sorghum’s mechanized production and mechanical properties is weak. Sui et al. analyzed the problems in sorghum production and made reasonable recommendations [22]. Zeng et al. combined the discrete unit method to model grain fragmentation [23]. Qiu et al. studied the viscoelasticity of different varieties of grains [24]. Feng et al. looked at the tensile fracture of three sorghum spike flaps [25]. Zargar et al. tested the affect of geometry and boundary conditions on the characterization of mechanical properties of sorghum [26]. Mathematical modeling of the sorghum grain drying process and analysis of its energy properties were studied by Perazzini et al. [27]. Analysis of factors affecting the mechanical properties of plant straws, such as sorghum, was performed by Maraldi et al. [28].

Therefore, to research the breaking characteristics of sorghum seeds subjected to compression and impact loading at different moisture contents, the work in this paper can be summarized in the following three points: (1) Analyze the effect of moisture content on the triaxial dimensions of sorghum seeds. (2) Analyze the effect of moisture content on the breaking energy and elastic modulus of sorghum seeds through compression tests. (3) Obtain the critical conditions for breaking sorghum seeds through impact mechanics tests and establish a critical impact force model to predict the impact mechanic properties using compression test properties.

2. Material and Methods

2.1. Material

2.1.1. Sorghum Seeds

The Chengde experimental field of the Hebei Academy of Agricultural Sciences provided the sorghum used in this experiment. The sorghum varieties were Jinza 34, Liaonuo 11, and Longza 20, widely grown in northern and northeastern China and the country. Considering that the moisture content of sorghum at storage and processing is 10% to 13%, the moisture content (wet basis) of the materials used in this experiment, Jinza 34, Liaonuo 11 and Longza 20, was 10.39%, 12.65%, and 11.39%, respectively, at the time of storage in storage; the moisture content of sorghum at harvest was about 18% to 21%.

2.1.2. Sample Preparation

The moisture content of the sorghum grain population was rationally divided and configured to 14% (in the medium moisture content zone, equivalent to damp material grains), 17% (in the medium moisture content zone, comparable to low moisture material grains), 20% (in the high moisture content zone, equivalent to freshly harvested grains), and 23% (in the high moisture content zone, slightly above the moisture content of freshly harvested sorghum) based on the initial wet basis moisture content of the sorghum grain Samples.

Sorghum involves different moisture contents in different harvest periods, different processing methods, and different storage environments. Therefore, the use of the above four moisture contents covers all possible moisture contents involved and the test results are more comprehensive.

Initial Wet Basis Moisture Content

The moisture content of sorghum at different harvest periods was measured using the thermostatic drying method [29], where sorghum seed samples were weighed to a real 0.01 g and placed in a drying oven at a temperature of 105 °C for 4.5 h. After the time was up, the drying oven was opened, and the samples were immediately covered with the box lid, removed and weighed, and the data were recorded and then placed in the drying oven for 0.5 h. After the time was up, the above operation was repeated. If the mass difference is less than 0.02 g, the drying of the grain is considered to be completed. The moisture content was calculated according to formula (1), and the average value was repeated three times.

$$H=\frac{A-B}{A}\times 100\%$$

(1)

where H is the moisture content, %; A is the initial fresh mass of sorghum seeds, g; B is the final dry mass of sorghum seeds, g.

Preparation of Materials with Different Moisture Contents

Samples of sorghum seeds with different moisture contents were prepared to study the effect of moisture content on mechanical properties. The sorghum seeds were weighed using an analytical balance with an accuracy of 0.01 g. Samples with 14%, 17%, 20%, and 21% moisture content need to be prepared in 2 batches. Uniform shaking is required every two hours during preparation. The water distribution process was done in a plastic bag, the treated samples were sealed and placed in a refrigerated box for backup, and the samples were taken out and restored to room temperature before the test to configure the moisture content according to Equation (2).

$$M=m\times \frac{H_2-H_1}{1-H_2}$$

(2)

where M is the mass of water required to be blended, g; m is the mass of sorghum seeds, g; H₁ is the initial moisture content of the grain seed population, %; H₂ is the moisture content after blending is required, %.

2.1.3. Experimental Instruments

TA.XT plus (produced by Stable Micro Systems, Godalming, UK), testing speed range 0.01–40 mm/s, testing distance precision 0.001 mm, testing force precision 0.0002%; INC Ax-act scanning electron microscope (produced by Hitachi, Tokyo, Japan); SZ680 continuous zoom body microscope (eyepiece 10X/23 mm, objective zoom range 0.68–4.7X); MP2002 electronic balance (Shanghai Precision Instrument Co./Ltd., Shanghai, China), range 300 g, precision 0.01 g; American Fowler digital vernier caliper, range 0–150 mm, resolution 0.01 mm (Canton, MA, USA); Pendulum Impact Load Tester from XinSanSi (Shanghai, China) Enterprise Development Co.

2.2. Methods

2.2.1. Measurement of Triaxial Dimensions

The kernels’ most intuitive shape and size are represented by the triaxial dimensions, which are essential in the kernel screening process. The sorghum kernels were selected one at a time from a sealed bag configured for moisture content, and their length (L), width (W), and height (H) were determined using a digital vernier caliper. The data were recorded 100 times by repeating the measurements. The triaxial scale of sorghum grains is shown in Figure 2 [30].

The triaxial arithmetic means diameter D_a and geometric mean diameter D_g of sorghum seeds are calculated as follows:

$$D_a=\frac{H+W+L}{3}$$

(3)

$$D_g={\left(L\times W\times H\right)}^{1/3}$$

(4)

where D_a is the arithmetic mean diameter of the seeds, mm; D_g is the geometric mean diameter of the seeds, mm.

Because of the irregular shape of sorghum seeds and the significant variation in morphology, sphericity is usually used to represent the shape of sorghum. The formula for sphericity S_p is

$$S_p=\frac{{\left(L\times W\times H\right)}^{1/3}}{L}$$

(5)

where: S_p is the sphericity of the seeds, %; L, W, and H are the length, width, and height of the seeds, mm.

2.2.2. Compression Test

The physical property analyzer was selected to operate in compression mode. The SMS P/36R cylindrical compression probe and 100 × 90 mm compression base was used as the compression device. The sorghum grain was simulated to be subjected to compression loads in different directions during harvesting, transportation, and mechanized processing. The sorghum seeds were compressed in X, Y, and Z directions for three axes, as shown in Figure 3. The arrow indicates the direction in which the compression force F is applied.

During the test, the compression speed was adjusted to 0.02 mm/s, the pre-test speed was 0.4 mm/s, the trigger force was 0.049 N, and the return speed after the test was 2.0 mm/s. Before the test, the sorghum seeds were placed on the compression base for the compression test. After compression, the sorghum grains were carefully placed under a body microscope and photographed with pointed tweezers.

There is a clear biological yield point in the sorghum grain during compression deformation. When the applied load does not reach the yield point, the relationship between force and deformation at this stage is approximately linear. When the applied load reaches the biological yield point, damage is caused to the microstructure of the sorghum material. Therefore, the force corresponding to the yield point is the sorghum grain’s maximum compression force F (N), which is the first peak point on the force–displacement curve during the compression deformation of the grain. The area enclosed by the curve before the maximum compression force and the horizontal coordinate (shaded area in the figure) is the corresponding compression damage energy W (MJ). As the load increases, local tissue damage occurs in the sorghum seeds and enters the plastic zone. Finally, as the load increases, the maximum peak point is reached, the rupture point of which is shown in Figure 4. At this point, the material undergoes microstructural damage under the applied load. The compression damage energy was calculated according to Equation (6).

$$W={{\displaystyle \int }}_0^{D_F}FdD$$

(6)

where W is the compression damage energy, MJ; F is the compression damage force, N; and D is the value of the horizontal coordinate, mm; D_F is the biological yield distance, mm.

According to the requirements of ASAE S3684 DEC2000 (R2017) standard (American Society of Agricultural and Biological Engineers, 2017), the apparent contact modulus of the elasticity of sorghum can be calculated using the Hertz formula, as shown in Equation (7).

$$E=\frac{0.338K_U^{\frac{3}{2}}F\left(1-{\mu }^2\right)}{D^{\frac{3}{2}}}\left(\frac{1}{R_U}+\frac{1}{R_U^{\prime }}\right)$$

(7)

where E is the apparent contact modulus of the elasticity of sorghum, M pa; F is the load, N; D is the deformation, mm; μ is the Poisson’s ratio of sorghum seeds concerning other crops and a value of 0.4; and R_U, R’_U are the principal radii of curvature at the contact points on the upper surface of sorghum during compression, mm.

Two parallel rigid plates compress sorghum seeds in the height direction, and the principal radii of curvature at the contact points can be expressed as Equations (8) and (9).

$$R_U=\frac{\frac{W^2}{4}+H^2}{2H}$$

(8)

$$R_U^{\prime }=\frac{\frac{L^2}{4}+H^2}{2H}$$

(9)

where L, W, and H are the length, width, and height of sorghum seeds, respectively, mm; K_U is a constant determined by the radius of principal curvature, which can be obtained from the ASAE S368.4 DEC2000 (R2017) standard by calculating the cosine of the principal plane angle cos θ. According to the Hertzian contact theory, Equation (10) is obtained.

$$cos\theta =\frac{1/R_U-1/R_U^{\prime }}{1/R_U+1/R_U^{\prime }}$$

(10)

2.2.3. Impact Mechanics Test

A single sorghum grain was fixed to a homemade baffle fixture, and a pendulum dynamic load tester was used to perform impact tests on sorghum grains. Fifty repetitions of each sample were performed. After the impact, the impacted grains were carefully and quickly placed under a body microscope with pointed forceps to observe the degree of breakage of sorghum grains and to record the angle of the pendulum (β). The schematic diagram of sorghum seed impact is shown in Figure 5.

3. Results and Discussion

3.1. Sorghum Seed Triaxial Size

The triaxial dimensions of sorghum seeds of different varieties and different moisture contents were compared, as shown in Figure 6. As we can see from the figure: with the increase in moisture content, the length, width, and height of the three sorghum seeds became more extensive, with the sorghum seeds of Liaonuo 11 being flatter and the sorghum seeds of Longzhu 20 being more rounded.

The sphericity, arithmetic mean diameter, and geometric mean diameter were calculated from the triaxial dimensions, as shown in

Table 1. Arithmetic means diameter, geometric mean diameter and sphericity of different varieties at different moisture contents.

Species	Moisture Content (%)	Triaxial Arithmetic Means Diameter Da (mm)	Geometric Mean Diameter Dg (mm)	Sphericity (%)
Jinza 34	10.3 ± 0.148	3.605 ± 0.036	3.492 ± 0.028	0.781 ± 0.041
	14.0 ± 0.022	3.611 ± 0.023	3.498 ± 0.023	0.785 ± 0.042
	17 ± 0.023	3.619 ± 0.025	3.506 ± 0.024	0.792 ± 0.047
	20 ± 0.012	3.628 ± 0.027	3.516 ± 0.027	0.782 ± 0.048
	23 ± 0.032	3.636 ± 0.028	3.523 ± 0.027	0.782 ± 0.045
Liaonuo 11	12.65 ± 0.023	3.540 ± 0.035	3.398 ± 0.028	0.768 ± 0.046
	14 ± 0.045	3.546 ± 0.021	3.405 ± 0.035	0.765 ± 0.048
	17 ± 0.012	3.557 ± 0.038	3.415 ± 0.036	0.758 ± 0.025
	20 ± 0.025	3.564 ± 0.034	3.421 ± 0.039	0.774 ± 0.036
	23 ± 0.029	3.575 ± 0.023	3.433 ± 0.024	0.764 ± 0.034
Longza 20	11.33 ± 0.023	3.581 ± 0.031	3.498 ± 0.031	0.776 ± 0.032
	14 ± 0.025	3.582 ± 0.035	3.508 ± 0.038	0.775 ± 0.035
	17 ± 0.036	3.604 ± 0.034	3.518 ± 0.037	0.772 ± 0.034
	20 ± 0.025	3.617 ± 0.036	3.530 ± 0.032	0.773 ± 0.038
	23 ± 0.031	3.636 ± 0.037	3.550 ± 0.034	0.778 ± 0.039

. The table shows that all three sorghum seeds have a sphericity of 75% or more. Multiple continuous sphere models can be created to act as sorghum seeds when performing discrete meta-simulations of sorghum seeds.

3.2. Compression Test

3.2.1. Triaxial Compression Force, Compression Energy, Modulus of Elasticity

When sorghum seeds were compressed along the X-axis direction, the grain skin and endosperm at the tip of the seeds cracked, as shown in Figure 7a. The rupture was not apparent when compressed along the Y-axis, as shown in Figure 7b. When compressed along the Z-axis direction, the seeds were most severely broken, and the endosperm showed multiple cracks opening, as shown in Figure 7c.

After collecting the data of XYZ’s three-axis damage force, a binary function with the independent variables of moisture content and X (Y or Z) axis dimensions is established, and the surface of the opposing force is obtained, as shown in Figure 8.

It can be seen from Figure 8 that the magnitude of the injury force in the X and Z-axis directions is negatively correlated with the moisture content and size. In the Y-axis direction, the magnitude of the damaging force is negatively correlated with the moisture content and positively correlated with the size. The outer cells of sorghum seeds are keratinized and harder, with a tighter structure along the Y-axis, requiring more destructive force. At the same moisture content, the Z-axis can withstand the largest compression force, followed by the Y-axis, and the X-axis can withstand the smallest compression force. When compressing in the X-axis direction, the compression probe was in contact with the pointed relics at the top of the sorghum grain. Since the contact area between the two contact points is small, stress concentration is likely to occur. The top of the sorghum grain is weakly compressed, so it is easily damaged and cracked, and the contact area is smallest when compressed in the X-axis direction, so the smallest damage force is required when compressed in the X-axis direction. Therefore, in preparing the research sorghum combine harvester, the harvester fan and drum speed need to be determined based on the maximum compression force that the X-axis can withstand to minimize the sorghum breakage rate.

Where X-axis, Y-axis, and Z-axis are the direction of action of the compression force.

According to the maximum compression force of the XYZ three axes, the damage energy and elastic modulus of the XYZ axis were calculated by combining Equations (6) and (7), as shown in

Table 2. Triaxial compression damage the energy and elastic modulus of sorghum seeds.

Moisture Content/%	Destructive Energy/J			Apparent Contact Modulus of Elasticity/M pa
	X-Axis	Y-Axis	Z-Axis	X-Axis	Y-Axis	Z-Axis
12.41%	19.229 ± 0.910	21.409 ± 1.072	24.741 ± 0.899	32.262 ± 0.925	25.216 ± 1.121	72.426 ± 1.245
14.35%	17.985 ± 1.556	20.397 ± 0.109	22.953 ± 0.798	26.955 ± 1.301	26.585 ± 1.201	60.147 ± 1.031
17.19%	16.551 ± 0.874	18.186 ± 1.343	21.717 ± 0.357	20.155 ± 0.977	19.576 ± 0.915	55.812 ± 1.168
19.62%	14.369 ± 1.914	17.452 ± 1.330	18.802 ± 0.661	15.039 ± 1.302	13.761 ± 1.0.13	42.997 ± 0.931
22.53%	13.409 ± 1.371	16.313 ± 0.425	17.609 ± 0.510	10.355 ± 1.012	8.648 ± 1.203	25.677 ± 1.141

. According to Table 2, the fitting curves of moisture content destruction energy and moisture content elastic modulus are established, as shown in Figure 9 and Figure 10.

Figure 9 shows that at the same moisture content, the energy required to break the sorghum kernels along the Z-axis direction is the largest, followed by the Y-axis direction, and the X-axis direction is the smallest. The modulus of elasticity of the Z-axis is much greater than that of the X and Y-axis at the same moisture content and decreases with increasing moisture content.

In the case of the same moisture content, the damage energy required for compression in the Z-axis direction is the largest, followed by the Y-axis. The damage energy required for compression in the X-axis direction is the smallest. The law of variation is similar to the law of variation in damage force when compressing in different directions. In the Y- and Z-axis directions of compression, the endosperm portion of the sorghum grain is compressed, but the area of the sorghum grain compressed in the Y-axis direction is smaller than the area compressed in the X-axis direction, resulting in the same external load. At this point, the Y-axis direction is more susceptible to stress concentration than the Z-axis, and the kernels are more susceptible to damage from external forces. Hence, the destructive force is more significant in the Z-axis direction of compression than in the Y-axis. Similarly, the modulus of elasticity in the Z-axis is also the largest.

The equations of the fitted curves shown in Figure 10 are in

Table 3. Triaxial modulus of elasticity fitting equation.

Direction	Fitting Equation	R2
X-axis	y = −425.75x + 124.65	0.96
Y-axis	y = −219.10x + 58.69	0.99
Z-axis	y = −180.63x + 49.81	0.94

. The equations R² are all greater than 0.93, which is a good fit and can be used to estimate the apparent contact elastic modulus of sorghum seeds at arbitrary moisture content for discrete element modeling.

3.2.2. Compression Deformation and Yield Load

The deformation variables and yield loads of sorghum seeds of different varieties in compression tests at different moisture contents are shown in

Table 4. Results of sorghum seed compression test.

Species	Moisture Content (%)	Compression Deformation (mm)	Yield Load (N)
Jinza 34	10.3 ± 0.148	0.427 ± 0.012	77.693 ± 0.123
	14.0 ± 0.022	0.417 ± 0.013	72.368 ± 0.031
	17 ± 0.023	0.469 ± 0.013	58.649 ± 0.032
	20 ± 0.012	0.509 ± 0.023	50.188 ± 0.032
	23 ± 0.032	0.650 ± 0.024	42.033 ± 0.032
Liaonuo 11	12.65 ± 0.023	0.588 ± 0.025	100.116 ± 0.036
	14 ± 0.045	0.486 ± 0.032	93.593 ± 0.035
	17 ± 0.012	0.558 ± 0.036	85.435 ± 0.036
	20 ± 0.025	0.638 ± 0.032	73.706 ± 0.036
	23 ± 0.029	0.855 ± 0.036	63.016 ± 0.036
Longza 20	11.33 ± 0.023	0.342 ± 0.036	90.706 ± 0.035
	14 ± 0.025	0.340 ± 0.035	85.165 ± 0.035
	17 ± 0.036	0.324 ± 0.034	77.875 ± 0.036
	20 ± 0.025	0.355 ± 0.034	62.897 ± 0.037
	23 ± 0.031	0.499 ± 0.034	51.980 ± 0.026

.

As we can be seen from Table 4, as the moisture content increases, the compression deformation first decreases and then increases, and the yield load continues to decrease. To reduce the rate of sorghum grain breakage, the mechanized harvest of sorghum should be carried out during the period of low moisture content as much as possible, and the storage machine processing of sorghum seeds should keep the sorghum in a dry state.

Based on the data in Table 4, the fitted curves for the compressive deformation of the three sorghum seeds, Jinza 34, Liaonuo 11, and Longza 20, were established, as shown in Figure 11. As can be seen from the figure: with increasing moisture content, the compressive deformation of the three sorghum seed grains showed an increasing trend, which is due to the high moisture content of sorghum seed grains, soft texture, small apparent contact modulus of elasticity, which makes them more susceptible to deformation under yield load.

3.3. Impact Mechanics Test

3.3.1. Impact Damage Critical Angle and Impact Force

The impact test was conducted on 50 randomly selected intact, mold-free sorghum grains from the same variety with different moisture contents. The pendulum swing angle β and impact force F were recorded when the sorghum seeds were cracked, and the impact load F was calculated when the pendulum fell in contact with the sorghum seeds. The results are shown in

Table 5. Results of impact tests with different varieties of sorghum seeds at different moisture contents.

Species	Moisture Content (%)	Critical Angle of Impact Breakage β (°)	Impact Breakage Critical Force F (N)
Jinza 34	10.3 ± 0.148	3.57 ± 0.03	25.421 ± 0.011
	14.0 ± 0.022	3.69 ± 0.02	26.012 ± 0.012
	17 ± 0.023	3.97 ± 0.08	27.163 ± 0.023
	20 ± 0.012	4.43 ± 0.06	28.616 ± 0.024
	23 ± 0.032	5.18 ± 0.04	31.371 ± 0.021
Liaonuo 11	12.65 ± 0.023	5.03 ± 0.05	32.068 ± 0.026
	14 ± 0.045	5.12 ± 0.02	33.433 ± 0.025
	17 ± 0.012	5.68 ± 0.03	34.442 ± 0.023
	20 ± 0.025	6.29 ± 0.04	35.582 ± 0.021
	23 ± 0.029	7.41 ± 0.02	36.377 ± 0.027
Longza 20	11.33 ± 0.023	3.24 ± 0.07	26.426 ± 0.032
	14 ± 0.025	3.42 ± 0.08	27.427 ± 0.025
	17 ± 0.036	3.55 ± 0.05	28.480 ± 0.029
	20 ± 0.025	4.05 ± 0.01	30.045 ± 0.021
	23 ± 0.031	4.95 ± 0.04	32.921 ± 0.035

.

Based on the data in the table, equations for the moisture content impact load breakage critical angle of the three varieties of sorghum seeds were developed, as shown in Figure 12.

The critical angle fitting curves for the three sorghum seeds broken under impact loading are shown in Figure 12 which roughly show a linear relationship. To the fitting effect of the curve, this paper uses a cubic polynomial function for fitting, and the fitting correlation coefficients are above 0.995. The relevant results are incredibly close to the actual situation.

3.3.2. Maximum Impact Load Prediction

The system has a system of masses with only gravitational potential. From the law of conservation of mechanical energy of the conservative system, Equation (11) is presented. From Equation (12), find the velocity of the impact pendulum when it comes into contact with the sorghum seeds.

$$\frac{1}{2}{m}_{1}gh\left(1-cos\beta \right)+m_{2}g\left(h+\frac{b}{2}\right)\left(1-cos\beta \right)=\frac{1}{2}\mathrm{J}{\omega }^2$$

(11)

where m₁ is the pendulum mass, kg. m₂ is the pendulum mass, kg. h is the pendulum length, m. b is the pendulum height, m. β is the angle between the pendulum’s axis and the vertical direction when the pendulum is falling, °. J is the rotational inertia when rotating on a fixed axis, kg-m². g is the acceleration of gravity, m/s². ω is the angular velocity of the pendulum at contact with the sorghum grain, rad/s.

$$\mathrm{J}=\frac{1}{3}{m}_1{h}^2+m_2{\left(h+\frac{b}{2}\right)}^2+\frac{1}{12}{m}_2\left(a^2+c^2\right)$$

(12)

where a is the pendulum length, m. c is the pendulum width, m. J is the rotational inertia when rotating on a fixed axis, kg-m². ω is the angular velocity of the pendulum at contact with the sorghum grain, rad/s.

$$\mathrm{v}=\left(h+\frac{b}{2}\right)\omega$$

(13)

where v is the impact speed of the pendulum, m/s. h is the pendulum length, m. b is the pendulum height, m. ω is the angular velocity of the pendulum at contact with the sorghum grain, rad/s.

The sorghum seed compression test obtained the static shape variables Δ_st for different varieties and moisture contents to reach the yield load. The dynamic load factor (14) was calculated from the impact velocity v (m/s) and the static shape variables of the seeds Δ_st (m).

$$K_d=\sqrt{\frac{{\mathrm{v}}^2}{{\mathrm{g}\mathsf{\Delta }}_{st}}}$$

(14)

The maximum impact load F_d(N) on the sorghum seeds is calculated from the dynamic load factor, impact pendulum, and pendulum mass m(kg), as shown in Equation (15).

$$F_d=K_d·m·\mathrm{g}$$

(15)

After measurement, the parameter values of the impact mechanics test bench are shown in

Table 6. Parameter values of impact mechanics test bench.

Swinging Rod Weight	Pendulum Weight	Swinging Rod Length	Length of Pendulum	Height of Pendulum	Width of the Pendulum
m1/kg	m2/kg	h/m	a/m	b/m	c/m
0.504	0.804	0.34	0.074	0.114	0.014

below.

Combined with Equations (11)–(15), the maximum impact load that grain can bear can be estimated through the compression test of sorghum grain. The comparison between the calculated prediction results and the experimental values is shown in Figure 13.

It can be seen from Figure 13 that the impact force of the damage to the three kinds of sorghum grains under impact load increased with the increase in moisture content because the endosperm structure of the grains was soft when the moisture content was high. The buffering effect against the impact load was strong. Therefore, harvesting was carried out during high moisture content, which could reduce the damage caused by impact load. In Figure 13, the average error of the prediction results of Jinza 34 is 2.28%, that of Liaonuo 11 is 1.88%, and that of Longza 20 is 1.34%. The prediction results are close to the actual values, and the prediction effect is good.

4. Conclusions

In this paper, compression and impact mechanics tests were carried out on sorghum grains with different moisture content, and the characteristics of grain damage were measured. The main contents can be summarized as follows:

(1)

The length, width and height of the three kinds of sorghum grains increased with moisture content. The geometric mean diameter of the three kinds of sorghum grains was 3.405–3.550 mm, and the sphericity increased with moisture content, all above 75.5%. When the discrete element method was used to simulate sorghum’s harvest and processing process, a sphere model with a diameter of 3.5 mm could be established for convenience.

(2)

By compression test, the maximum compression force of the XYZ axis of the grain was measured and fitted to obtain the binary surface graph; the failure energy of the X-axis is 13.409–19.229 J, the Y-axis is 16.313–21.409 J, and the Z-axis is 17.609–24.741 J. The apparent contact elastic modulus of the three axes is calculated, and the maximum of the Z-axis can reach 72 MPa, and the X-axis and Y-axis are roughly equal, with a range of 10–27 MPa.

(3)

The impact mechanics test measured and fitted the critical angle of sorghum grain damage under impact load. The fitting equation R² of the three sorghum grain varieties was more significant than 0.996. The mathematical model for predicting the critical load was established, and the error of the calculated value was compared with the experimental value. The error was less than 3%, and the prediction effect was good. Compression test results could predict the critical impact load of sorghum grain damage.

(4)

In compression tests, the magnitude of damage force showed a negative correlation with moisture content, and the amount of compression deformation increased with increasing moisture content, which is due to the high moisture content of sorghum seed grains, soft texture, small apparent contact modulus of elasticity. In impact mechanics tests, the critical angle of sorghum seed breakage and critical value of breakage impact force increased with increasing moisture content.