# Dynamic Characteristics of a Simulated Sugarcane Field Exciter for Small Sugarcane Harvesters

Authors: Hanning Mo; Shangping Li; Guiqing He; Bang Zeng; Chen Qiu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

The sugarcane harvester vibration has a bad effect on the sugarcane cutting quality. The effect of sugarcane field roughness on the sugarcane harvester vibration is much more significant than those of cutting forces and the engine. In order to simulate sugarcane field roughness, a simulated sugarcane field exciter (SSFE) was developed to actuate a self-developed sugarcane harvester experiment platform (SHEP). The dynamics and the mathematical models of the SHEP were established. Simulations of the mathematical model show these two models are reasonable. The dynamic characteristic experiment of the SSFE shows it matches characteristics of sugarcane field roughness, but great lateral oscillations existed when it worked. Then the SSFE II was developed. The dynamic characteristic experiment of the SSFE II shows it matches characteristics of sugarcane field roughness and improves the SSFE. The modal test of the SHEP was done to further study dynamic characteristics of the SSFE II. With the SSFE II, simulated experiments of sugarcane harvesters under complete vibration causing conditions can be done in labs instead of sugarcane fields to avoid the low efficiency, poor security, and bad reliability during experiments in sugarcane fields.


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## Body

## 1. Introduction

The mechanized sugarcane harvesting area is only 5% in the whole planting area [1]. The most important reason for difficult mechanized sugarcane harvesting promotion is that mechanized sugarcane cutting brings a poor cutting quality.Aimed at the poor cutting quality of mechanized sugarcane harvesting, related research was done by scholars all over the world. Kroes [2, 3] and Mello and Harris [4] studied effects of cutting forces, cutting energies, and different cutting edge positions on the sugarcane cutting quality through experiments. Kroes and Harris [5] studied effects of sugarcane-pressing rollers, cutters, the cutter installing angle, and the space distance among saw teeth of cutting edges on the sugarcane cutting quality through simulations. Mello and Harris [6] studied the effect of penetrating cutting on the sugarcane cutting quality. Liu et al. [7, 8] studied broken forms of sugarcanes suffering from tensions, compressions, bending, and torsional moments and kinematics of sugarcanes cut by plain knives. Liu et al. [9] explored the cutting mechanism and mechanical properties of the sugarcane stem material. Yang Jian [10] studied effects of sugarcane fields and machine structural factors on the sugarcane ratoon breaking rate. Wang et al. [11] studied the sugarcane cutting mechanism through an emulator of the sugarcane-cutter system with changeable structural and kinematic parameters.Moreover, Thanomputra and Kiatiwat [12] used the high-pressure water cutting method with abrasive sands added to improve the sugarcane cutting efficiency. Mello and Harris [13] and Momin et al. [14] used different blades to carry on experimental research on the sugarcane cutting quality and found bending-angle-shape blades with or without saw teeth can improve the sugarcane cutting quality. Ripoli et al. [15] designed a cutter whose cutting depth into soils can be adjusted according to the gradient variation of sugarcane fields to reduce the sugarcane wastage and the impurity content. Johnson et al. [16] and Mathanker et al. [17] studied effects of the sugarcane cutting velocity and the cutting edge angle on the cutting energy. Kroes and Harris [18] designed a double-cutter model to study the kinematic trajectory of the cutters and calculated the maximum velocity ratio between the sugarcane harvester moving velocity and the cutter rotation velocity to improve the cutting quality. Silva [19] evaluated the sugarcane root damage degree caused by cutting height differences through experiments. Xiao et al. [20, 21] found the sugarcane field excitation has a bad effect on the cutting system of sugarcane harvesters, causes the vertical cutter frame vibration, and deteriorates the sugarcane cutting quality. Peng and Daogao [22] designed a cutter vibration model to study the effect of the bearing clearance on the cutting system vibration. Yang et al. [23] and Pelloso et al. [24] studied effects of the cutter rotation velocity and the sugarcane harvester moving velocity on the cutting quality.It is shown through research mentioned above that sugarcane field roughness has a bad effect on the sugarcane cutting quality. In the sugarcane cutting process, besides sugarcane field roughness, cutting forces and the engine also cause vibrations. It is shown through experiments done by our research group that the sugarcane harvester vibration has a bad effect on the sugarcane cutting quality and the effect of sugarcane field roughness on the sugarcane harvester vibration is much more significant than those of cutting forces and the engine. However, none of research mentioned above was focused on characteristics of sugarcane field roughness and how the sugarcane field excitation can be obtained in labs instead of sugarcane fields to avoid the low efficiency, poor security, and bad reliability during experiments in sugarcane fields and then to study how to improve the sugarcane cutting quality under simulated experimental conditions.For purposes above, a sugarcane harvester experiment platform (SHEP) was developed by our research group. On the SHEP, an actuating engine is used to simulate the engine excitation and a simulated sugarcane field exciter (SSFE) developed based on sugarcane field roughness signals collected in sugarcane fields is used to simulate the sugarcane field excitation. That is, the sugarcane field excitation can be obtained in labs, which has never been achieved in previous research. Thus, simulated experiments of sugarcane harvesters under complete vibration causing conditions of the engine, cutting forces existing in the sugarcane cutting process, and sugarcane field roughness can be done in labs instead of sugarcane fields.


## 2. Characteristics of Sugarcane Field Roughness

Characteristics of sugarcane field roughness were obtained through sugarcane field roughness signals collected in a flat and a hilly sugarcane fields. The flat and the hilly sugarcane field roughness signals are continuous random signals. It is shown by proportion analysis on frequency bands of the flat and the hilly sugarcane field roughness signals through MATLAB that their main excitation frequency band is 1∼6 Hz, in which the excitation frequency band with the greatest contribution is 0.5∼3.5 Hz. Therefore, sugarcane field roughness signals are also low-frequency vibration signals which generate great vibration displacements with low frequencies while small vibration displacements with high frequencies. This founds the theoretical basis for design on the SSFE.The piecewise fitting method was used to obtain fitting equations every 20 seconds to respectively match the flat and the hilly sugarcane field roughness signals accurately. In this paper, the piecewise fitting equation of the hilly sugarcane field roughness signal written as (1) was used.(1)ξt=∑i=18aisin2πfit1+φi,t1=0∼20s,ξt=∑i=18aisin2πfit2+φi,t2=20∼40s,ξt=∑i=18aisin2πfit3+φi,t3=40∼60s,ξt=∑i=18aisin2πfit4+φi,t4=60∼80s,ξt=∑i=18aisin2πfit5+φi,t5=80∼100s,ξt=∑i=18aisin2πfit6+φi,t6=100∼120s,ξt=∑i=18aisin2πfit7+φi,t7=120∼140s,ξt=∑i=18aisin2πfit8+φi,t8=140∼160s,where ai is amplitude with the unit of m; fi is the frequency with the unit of Hz; and φi is the initial phase with the unit of rad.Values of 24 fitting parameters in (1) every 20 seconds are shown in Tables 1 to 8.Table 1
Values of 24 fitting parameters in equation (1) in 0∼20 s.
i12345678ai0.013680.009650.010760.013780.008430.009570.009480.01076fi3.800962.775481.7006372.017522.226111.648092.179943.15605φi−1.567−2.8942.532−0.91753.175−2.476−0.932−2.754Table 2
Values of 24 fitting parameters in equation (1) in 20∼40 s.
i12345678ai0.011430.021670.013450.010430.009140.007890.009840.01653fi2.097132.157641.718152.335992.730892.576432.140132.49522φi−1.169−2.548−3.576−7.9532.686−3.496−1.495−2.458Table 3
Values of 24 fitting parameters in equation (1) in 40∼60 s.
i12345678ai0.012670.010580.013560.008530.008750.009580.007390.08598fi2.143312.065292.194272.337582.457012.257961.971342.53025φi4.5051.563−4.2580.2653−1.654−3.6810.8765−2.764Table 4
Values of 24 fitting parameters in equation (1) in 60∼80 s.
i12345678ai0.010450.007860.008040.011760.007650.012130.008430.01256fi2.606692.792991.985672.654462.778662.463382.41722.36624φi−1.2460.843−1.4760.9447−5.4590.7565−7.4563.476Table 5
Values of 24 fitting parameters in equation (1) in 80∼100 s.
i12345678ai0.011650.011540.009760.008970.009430.006640.011760.01147fi2.533442.156052.173572.616242.426753.14652.919112.44268φi−1.2240.8134−3.635−0.76190.9364−0.07442.8653.464Table 6
Values of 24 fitting parameters in equation (1) in 100∼120 s.
i12345678ai0.010430.011670.009660.013560.006320.009630.011760.01197fi2.566882.47932.192682.337582.622612.480892.350322.57484φi0.3469−1.542−6.532−3.2431.686−2.643−2.476−1.476Table 7
Values of 24 fitting parameters in equation (1) in 120∼140 s.
i12345678ai0.007340.011540.010980.013650.007890.009320.012870.01143fi2.098732.321662.095542.313692.589172.433122.350322.96815φi−5.163−1.4321.132−3.1541.795−1.563−7.432−1.743Table 8
Values of 24 fitting parameters in equation (1) in 140∼160 s.
i12345678ai0.011430.012410.011630.012850.013180.012760.009540.00857fi2.348732.91722.600322.587582.461782.616243.238852.36783φi−1.168−2.647−1.809−3.3620.95430.6143−2.754−4.432The fitting effect of (1) in 0∼60 s obtained through MATLAB is shown in Figures 1(a) to 1(c). Amplitude-frequency curves of the hilly sugarcane field roughness signal and (1) obtained through MATLAB are shown in Figures 1(d) and 1(e). The curve of (1) drawn through MATLAB is shown in Figure 1(f).Figure 1
The fitting effect and curves of equation (1) in the frequency and the time domains. (a) The fitting effect of equation (1) in 0∼20 s. (b) The fitting effect of equation (1) in 20∼40 s. (c) The fitting effect of equation (1) in 40∼60 s. (d) The amplitude-frequency curve of the hilly sugarcane field roughness signal. (e) The amplitude-frequency curve of equation (1). (f) The curve of equation (1).
(a)(b)(c)(d)(e)(f)According to Figures1(a) to 1(c), R-square values of (1) are all greater than 0.5 in 0∼60 s, showing (1) matches the hilly sugarcane field roughness signal every 20 seconds. According to Figures 1(d) and 1(e), the frequency band with great amplitudes of (1) matches that of the hilly sugarcane field roughness signal, showing (1) also matches the hilly sugarcane field roughness signal in the frequency domain. Therefore, (1) has a good fitting effect.The sugarcane field excitation is generated by sugarcane field roughness and calculated through the following equation:(2)Fwt=Kξt+Cξ˙t,where ξ˙t is the first derivative of ξt and K, C are the stiffness and damping coefficients between wheels of a sugarcane harvester and a sugarcane field.


## 3. Design and Improvement on the SSFE

The SSFE is shown in Figure2.Figure 2
The SSFE. 1: the shorter axis; 2: the axis coupler; 3: upper springs; 4: an eccentric mass block; 5: the middle support platform; 6: lower springs; 7: the longer axis.Lower springs are used to simulate stiffness and damping parts between wheels and a sugarcane field to simulate their moving. Upper springs are used to simulate stiffness and damping parts between wheels and the body frame of a sugarcane harvester. The SSFE vibration is produced through unbalanced forces generated by the high-speed rotation of eccentric mass blocks. Unbalanced forces can be changed through the number of eccentric mass blocks.Lateral oscillations of the SSFE can simulate lateral swings of a sugarcane harvester working in a sugarcane field. It is found through experiments done by our research group that the vertical cutter vibration is bad for the sugarcane cutting quality while lateral cutter oscillations are good for cutting off sugarcanes. Therefore, lateral oscillations of the SSFE are unavoidable and needed, but they should not be great, or the SSFE may turn over laterally, so the SSFE vibration displacements along lateral directions should be smaller than that along the vertical direction. The SSFE with lateral-oscillation-limiting devices (SSFE II) was developed based on the SSFE to improve it so that much more accurate simulated sugarcane field excitation can be achieved in the lab. Lateral-oscillation-limiting devices are external and internal sleeves of the upper and lower springs. The SSFE II is shown in Figure3. There are one upper spring and four lower springs in it.Figure 3
The SSFE II. 1 and 2: the external and the internal sleeves of the upper spring; 3: eccentric mass blocks; 4: the axis coupler; 5, 6, and 9: the upper, the middle, and the lower support platforms; 7 and 8: the external and the internal sleeves of a lower spring.The SHEP with two SSFE IIs is shown in Figure4.Figure 4
The SHEP. 1: the sugarcane-feeding device; 2: the sugarcane-clamping device; 3: the sugarcane-feeding pathway; 4: the actuating engine; 5: the cutter frame; 6: a vibration absorber; 7: the body frame; 8: the SSFE II; 9: a cutter.Two SSFE IIs are in the front of the SHEP while two vibration absorbers are at the back. The actuating engine is at the top. The SHEP is equivalent to a sugarcane harvester working in a sugarcane field.


## 4. The Dynamics and the Mathematical Models of the SHEP

The dynamics model of the SHEP is simplified as a spring-mass system shown in Figure5. The body frame of the SHEP, two cutters, and the actuating engine are simplified as mass blocks. Two SSFE IIs and two vibration absorbers are simplified as spring dampers, equivalent to four wheels of a sugarcane harvester. The positive direction of the z axis is the upward vertical direction. The x axis is along radiuses of two cutters with the positive direction pointing to the left of the SHEP. The y axis is along the sugarcane-feeding pathway with the positive direction pointing to the back of the SHEP. The x and the y axes are along two lateral directions vertical to the z axis.m1, m2, and m3 are masses of the body frame, a cutter, and the actuating engine, m1=3600kg, m2=70kg, m3=400kg; Ix, Iy, Ix3, Iy3, Iz3are rotational inertias of the body frame and a cutter around the x, the y, and the z axes, Ix=Iy=3.4×105kg⋅m2, Ix3=65.39153867kg⋅m2, Iy3=23.05961249kg⋅m2, Iz3=54.79386107kg⋅m2; K1, K2, C1, C2 are stiffness and damping coefficients between two front wheels and the body frame, K1=K2=3.52×105N/m, C1=C2=2.686×103N/m/s; K3, K4, C3, C4 are stiffness and damping coefficients between two rear wheels and the body frame, K3=K4=5.28×105N/m, C3=C4=2.218×103N/m/s; K5, C5 are the stiffness and the damping coefficients between two cutters and the body frame, K5=9.093×106N/m, C5=6.3N/m/s; K6, C6 are the stiffness and the damping coefficients between the actuating engine and the body frame, K6=5.98×106N/m, C6=1.57×103N/m/s; Fw1, Fw2, Fw3, Fw4 are sugarcane field excitations acting on the four wheels, as calculated through (3) according to (2):(3)Fwit=Kiξt+Ciξ˙t,i=1,2,3,4.Figure 5
The dynamics model of the SHEP.Fe and Fe′ are periodical forces acting on the body frame by the engine and the engine by its internal structures, as calculated through the following equation:(4)Fet=55.17sin2π×40.0207006t,Fe′t=2493cos2×2π×29.1799363t.Fx, Fy, and Fz are cutting forces along the x, the y, and the z axes, as calculated through the following equation:(5)Fxt=517.2sin2π×19.0666667t,Fyt=414.2sin2π×19.0666667t,Fzt=373.2sin2π×19.0666667t.a, a′, L, and R are halves of the front wheel distance, the rear wheel distance, the length of the body frame, and the center distance of two cutters, a=0.68m, a′=0.605m, L=2.5m, R=0.27m; L1 is the distance between front wheels or rear wheels and the mass center of the body frame, L1=1.5m; L2 is the distance between the mass center of the cutter frame and that of the body frame, L2=0.9m; L3, ye are distances between the mass center of the body frame with the action line of Fe observed along the x axis and the y axis, L3=0.1m, ye=0.3025m; r is the length of the cutter axis, r=0.5m; z1, z2, z3, z4 are vertical displacements of connection points of four wheels and the body frame; z5 is vertical displacements of connection points of two cutters and the body frame. They are calculated through the following equation:(6)z1=z7−a2θ2+rθ2−L2θ1,z2=z7+a2θ2+rθ2−L2θ1,z3=z7−a′2θ2−rθ2+L2θ1,z4=z7+a′2θ2−rθ2+L2θ1,z5=z7+rθ2−L2θ1.z7, zq, and z6 are vertical displacements of the body frame, a cutter, and the actuating engine; θ1, θ2 are rotation angles of the body frame around the x and the y axes; θ1′, θ2′, and θ3 are rotation angles of a cutter around the x, the y, and the z axes. They are calculated through given parameters above.According to the D’Alembert principle, the mathematical model of Figure5 is written as (7) in a matrix form.(7)MZ¨+CZ˙+KZ=F,where(8)M=m1Ix1Iy1m2Iz3Ix3Iy3m3,Z=z7θ1θ2zqθ3θ1′θ2′z6,F=Fe+Fw1+Fw2+Fw3+Fw4FeL3+Fw1L1+Fw2L1+Fw3L1+Fw4L1Feye−Fw1a+Fw2a−Fw3a+Fw4aFzFxL2−FzL2+Fyr−FxrFe′,C=C1+C2+C3+C4+2C5+C6−L2C1−L2C2+L2C3+L2C4−2L2C5−a2C1+rC1+a2C2+rC2−a′2C3−rC3+a′2C4−rC4+2rC5−2C5000−C6−C1L1−C2L1+C3L1+C4L1+C6L3−2L2C5L2L1C1+L2L1C2+L2L1C3+L2L1C4+2L2C5a2L1C1−rL1C1−a2L1C2−rL1C2−a′2L1C3−rL1C3+a′2L1C4−rL1C4−2L2rC52L2C5000−C6L3−C1a+C2a−C3a′+C4a′+C6yeL2aC1−L2aC2−L2a′C3+L2a′C4a22C1−raC1+a22C2+raC2+a′22C3+ra′C3+a′22C4−ra′C40000−C6ye−C5L2C5−rC5C5000000000000C5L2−L22C5rL2C5−C5L20000C5R−L2RC5rRC5−C5R0000−C6000000C6,K=K1+K2+K3+K4+2K5+K6−L2K1−L2K2+L2K3+L2K4−2L2K5−a2K1+rK1+a2K2+rK2−a′2K3−rK3+a′2K4−rK4+2rK5−2K5000−K6−K1L1−K2L1+K3L1+K4L1+K6L3−2L2K5L2L1K1+L2L1K2+L2L1K3+L2L1K4+2L2K5a2L1K1−rL1K1−a2L1K2−rL1K2−a′2L1K3−rL1K3+a′2L1K4−rL1K4−2L2rK52L2K5000−K6L3−K1a+K2a−K3a′+K4a+K6yeL2aK1−L2aK2−L2a′K3+L2a′K4a22K1−raK1+a22K2+raK2+a′22K3+ra′K3+a′22K4−ra′K40000−K6ye−K5L2K5−rK5K5000000000000K5L2−L22K5rL2K5−K5L20000K5R−L2RK5rRK5−K5R0000−K6000000K6.Curves ofzq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ changing with time drawn through MATLAB are shown in Figure 6.Figure 6
Curves ofzq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ changing with time. (a) The curve of zq changing with time. (b) The curve of z7 changing with time. (c) The curve of z6 changing with time. (d) The curve of θ1 changing with time. (e) The curve of θ2 changing with time. (f) The curve of θ3 changing with time. (g) The curve of θ1′ changing with time. (h) The curve of θ2′ changing with time.
(a)(b)(c)(d)(e)(f)(g)(h)According to Figure6, zq, z7, z6, θ1, and θ2 become smaller and smaller, that is, convergent along with time, showing the dynamics and the mathematical models of the SHEP are reasonable. Finally, they change with time in approximately periodical variation laws in that Fw, Fe, and Fe’ are periodical excitations, showing displacements of two cutters; the body frame and the engine of a sugarcane harvester along the z axis finally change approximately in periodical variation laws along with time. θ1′ and θ2′ become greater and greater along with time in that two cutters keep rotating around their axes in the sugarcane cutting process.Surface diagrams ofzq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ changing with ai during continuous time drawn through MATLAB are shown in Figure 7.Figure 7
Surface diagrams ofzq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ changing with ai during continuous time. (a) The surface diagram of zq changing with ai. (b) The surface diagram of z7 changing with ai. (c) The surface diagram of z6 changing with ai. (d) The surface diagram of θ1 changing with ai. (e) The surface diagram of θ2 changing with ai. (f) The surface diagram of θ3 changing with ai. (g) The surface diagram of θ1′ changing with ai. (h) The surface diagram of θ2′ changing with ai.
(a)(b)(c)(d)(e)(f)(g)(h)According to Figure7, the greater ai is, the greater zq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ will be, showing the sugarcane field excitation is a kind of displacement excitations; that is, the hillier a sugarcane field is, the more severe the sugarcane harvester vibration will be.Surface diagrams ofzq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ changing with fi during continuous time drawn through MATLAB are shown in Figure 8.Figure 8
Surface diagrams ofzq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ changing with fi during continuous time. (a) The surface diagram of zq changing with fi. (b) The surface diagram of z7 changing with fi. (c) The surface diagram of z6 changing with fi. (d) The surface diagram of θ1 changing with fi. (e) The surface diagram of θ2 changing with fi. (f) The surface diagram of θ3 changing with fi. (g) The surface diagram of θ1′ changing with fi. (h) The surface diagram of θ2′ changing with fi.
(a)(b)(c)(d)(e)(f)(g)(h)According to Figure8, when fi is about 5 Hz, in the main excitation frequency band of sugarcane field roughness signals, 0∼6 Hz, zq, z7, z6, θ1, θ2, θ3, θ1′, and θ2′ are the greatest, further showing the dynamics and the mathematical models of the SHEP are reasonable and the SHEP can simulate a sugarcane harvester working in a sugarcane field.


## 5. Design on Experiments

### 5.1. Design on the SSFE and the SSFE II Output Frequency Calibration Experiments

The SSFE and the SSFE II output frequency calibration experiments were done to obtain their output frequencies. The SSFE and the SSFE II input frequencies were controlled through a digital frequency convertor (Model: F1000-G0055T3B). Corresponding to every input frequency of the SSFE and the SSFE II, a laser tachometer (Model: DT-2234B) was used to measure eccentric axis rotation velocities of the SSFE and the SSFE II which were used to calculate output frequencies corresponding to their input frequencies at this moment.


### 5.2. Design on Dynamic Characteristic Experiments of the SSFE and the SSFE II

Dynamic characteristic experiments of the SSFE and the SSFE II were designed as vibration displacement measuring experiments to study their dynamic characteristics and verify whether they match characteristics of sugarcane field roughness and whether lateral-oscillation-limiting devices can limit lateral oscillations of the SSFE II. The block diagram of dynamic characteristic experiments of the SSFE and the SSFE II is shown in Figure9.Figure 9
The block diagram of dynamic characteristic experiments of the SSFE and the SSFE II.In Figure9, the laser displacement sensor was used to measure the SSFE and the SSFE II vibration displacements along the x, the y, and the z axes. The SSFE and the SSFE II vibration displacement measuring experiments were designed as complete cross grouping experiments. The SSFE and the SSFE II input frequencies and eccentric masses were two experimental factors. Levels of the SSFE and the SSFE II input frequencies were 5∼15 Hz and 8∼25 Hz (the step size is 1). Levels of the eccentric mass were 1∼4 kg (the step size is 1).


### 5.3. Design on the LMS Modal Test of the SHEP

The LMS modal test of the SHEP was done to further study dynamic characteristics of the SSFE II. The LMS modal test system of the SHEP and its block diagram are shown in Figures10 to 11. The force hammer was used to knock on the SHEP to obtain its modal information, as is shown in Figure 12. Three-axis acceleration sensors were pasted on the SHEP according to the paste point arrangement diagram drawn through Geometry of LMS Test.Lab, as is shown in Figure 13.Figure 10
The LMS modal test system of the SHEP.Figure 11
The block diagram of the LMS modal test system of the SHEP.Figure 12
The SHEP with three-axis acceleration sensors pasted on it knocked with the force hammer.Figure 13
The paste point arrangement diagram.


## 5.1. Design on the SSFE and the SSFE II Output Frequency Calibration Experiments

The SSFE and the SSFE II output frequency calibration experiments were done to obtain their output frequencies. The SSFE and the SSFE II input frequencies were controlled through a digital frequency convertor (Model: F1000-G0055T3B). Corresponding to every input frequency of the SSFE and the SSFE II, a laser tachometer (Model: DT-2234B) was used to measure eccentric axis rotation velocities of the SSFE and the SSFE II which were used to calculate output frequencies corresponding to their input frequencies at this moment.


## 5.2. Design on Dynamic Characteristic Experiments of the SSFE and the SSFE II

Dynamic characteristic experiments of the SSFE and the SSFE II were designed as vibration displacement measuring experiments to study their dynamic characteristics and verify whether they match characteristics of sugarcane field roughness and whether lateral-oscillation-limiting devices can limit lateral oscillations of the SSFE II. The block diagram of dynamic characteristic experiments of the SSFE and the SSFE II is shown in Figure9.Figure 9
The block diagram of dynamic characteristic experiments of the SSFE and the SSFE II.In Figure9, the laser displacement sensor was used to measure the SSFE and the SSFE II vibration displacements along the x, the y, and the z axes. The SSFE and the SSFE II vibration displacement measuring experiments were designed as complete cross grouping experiments. The SSFE and the SSFE II input frequencies and eccentric masses were two experimental factors. Levels of the SSFE and the SSFE II input frequencies were 5∼15 Hz and 8∼25 Hz (the step size is 1). Levels of the eccentric mass were 1∼4 kg (the step size is 1).


## 5.3. Design on the LMS Modal Test of the SHEP

The LMS modal test of the SHEP was done to further study dynamic characteristics of the SSFE II. The LMS modal test system of the SHEP and its block diagram are shown in Figures10 to 11. The force hammer was used to knock on the SHEP to obtain its modal information, as is shown in Figure 12. Three-axis acceleration sensors were pasted on the SHEP according to the paste point arrangement diagram drawn through Geometry of LMS Test.Lab, as is shown in Figure 13.Figure 10
The LMS modal test system of the SHEP.Figure 11
The block diagram of the LMS modal test system of the SHEP.Figure 12
The SHEP with three-axis acceleration sensors pasted on it knocked with the force hammer.Figure 13
The paste point arrangement diagram.


## 6. Analysis on Experiment Results

### 6.1. Analysis on the SSFE and the SSFE II Output Frequency Calibration Experiment Results

The SSFE and the SSFE II output frequency calibration curves drawn through Excel are shown in Figure14.Figure 14
The SSFE and the SSFE II output frequency calibration curves. (a) The SSFE output frequency calibration curve. (b) The SSFE II output frequency calibration curve.
(a)(b)According to Figure14, the SSFE output frequencies range from 1.3 Hz to 4.81 Hz and the SSFE II output frequencies range from 2.24 Hz to 7.35 Hz, approximately in the main excitation frequency band of sugarcane field roughness signals, 1∼6 Hz, showing the SSFE and the SSFE II input frequencies were chosen reasonably.


### 6.2. Analysis on the Dynamic Characteristic Experiment Result of the SSFE

Under every eccentric mass, curves of the SSFE vibration displacements along the three directions changing with the SSFE input frequency drawn through Excel are shown in Figure15.According to Figure15, under the same eccentric mass, the SSFE vibration displacements along two lateral directions are greater than that along the vertical direction, so the SSFE may turn over laterally. Therefore, the SSFE needs improvement.Figure 15
Under every eccentric mass, curves of the SSFE vibration displacements along the three directions changing with the SSFE input frequency. (a) Under a 1 kg eccentric mass. (b) Under a 2 kg eccentric mass. (c) Under a 3 kg eccentric mass. (d) Under a 4 kg eccentric mass.
(a)(b)(c)(d)Along every direction, curves of the SSFE vibration displacements changing with the SSFE input frequency under different eccentric masses drawn through Excel are shown in Figure16.Figure 16
Along every direction, curves of the SSFE vibration displacements changing with the SSFE input frequency under different eccentric masses. (a) Along thex axis. (b) Along the y axis. (c) Along the z axis.
(a)(b)(c)According to Figure16, along the same direction, the greater the eccentric mass is, the greater the SSFE vibration displacement will be. According to Figures 15 and 16, under the same eccentric mass or along the same direction, the greater the SSFE input frequency is, the greater the SSFE vibration displacement will be.Moreover, according to Figures15 and 16, the SSFE vibration displacement has an obvious increasing trend when the SSFE input frequency is greater than 8 Hz. According to Figure 14(a), the SSFE output frequencies corresponding to 8∼10 Hz are 2.58∼3.16 Hz, in the excitation frequency band of sugarcane field roughness signals with the greatest contribution, 0.5∼3.5 Hz, and the SSFE output frequencies corresponding to 11∼15 Hz are 3.53∼4.81 Hz, in the main excitation frequency band of sugarcane field roughness signals, 1∼6 Hz, making the SSFE vibration displacement have an obvious increasing trend. Besides, the SSFE output frequencies are smaller than 10 Hz, so with low output frequencies, the SSFE generated vibrations along the three directions. Therefore, the SSFE can generate low-frequency vibration signals, matching characteristics of sugarcane field roughness.


### 6.3. Analysis on the Dynamic Characteristic Experiment Result of the SSFE II

Under every eccentric mass, curves of the SSFE II vibration displacements along the three directions changing with the SSFE II input frequency drawn through Excel are shown in Figure17.According to Figure17, under the same eccentric, the SSFE II vibration displacements along two lateral directions are smaller than that along the vertical direction, so lateral-oscillation-limiting devices can limit lateral oscillations of the SSFE II; that is, the SSFE was improved. Therefore, the SSFE II simulates the sugarcane field excitation much more accurately and makes the SHEP much more similar to a sugarcane harvester working in a sugarcane field.Figure 17
Under every eccentric mass, curves of the SSFE II vibration displacements along the three directions changing with the SSFE II input frequency. (a) Under a 1 kg eccentric mass. (b) Under a 2 kg eccentric mass. (c) Under a 3 kg eccentric mass. (d) Under a 4 kg eccentric mass.
(a)(b)(c)(d)Along every direction, curves of the SSFE II vibration displacements changing with the SSFE II input frequency under different eccentric masses drawn through Excel are shown in Figure18.Figure 18
Along every direction, curves of the SSFE II vibration displacements changing with the SSFE II input frequency under different eccentric masses. (a) Along thex axis. (b) Along the y axis. (c) Along the z axis.
(a)(b)(c)According to Figure18, along the same direction, the greater the eccentric mass is, the greater the SSFE II vibration displacement will be. According to Figures 17 and 18, under the same eccentric mass or along the same direction, the greater the SSFE II input frequency is, the greater the SSFE II vibration displacement will be.Moreover, according to Figures17 and 18, wave crests of the SSFE II vibration displacement appear when the SSFE II input frequencies are 16 Hz, 17 Hz, 20 Hz, 21 Hz, 23 Hz, and 24 Hz and the SSFE II vibration displacement has an obvious increasing trend when the SSFE II input frequency is greater than 16 Hz. According to Figure 14(b), the SSFE II output frequencies corresponding to these six input frequencies are 4.69∼7.04 Hz, approximately in the main excitation frequency band of sugarcane field roughness signals, 1∼6 Hz, making peak values of the SSFE II vibration displacement appear and the SSFE II vibration displacement have an obvious increasing trend. Besides, the SSFE II output frequencies are smaller than 10 Hz, so with low output frequencies, the SSFE II generated vibrations along the three directions. Therefore, the SSFE II can generate low-frequency vibration signals, matching characteristics of sugarcane field roughness.


### 6.4. Analysis on the LMS Modal Test Result of the SHEP

The POLYMAX steady-state diagram of the SHEP and its MAC matrix obtained through modal analysis of LMS Test.Lab are shown in Figures19 and 20.Figure 19
The POLYMAX steady-state diagram of the SHEP.Figure 20
The MAC matrix of the SHEP.According to Figure19, seven stages of natural frequencies, that is, seven stages of vibration modes, were chosen in the POLYMAX steady-state diagram of the SHEP. According to Figure 20, these seven stages of vibration modes have low correlations between each other; that is, these seven stages of vibration modes have high accuracy, showing these seven stages of natural frequencies SHEP have high reliability. Seven stages of natural frequencies and correlation coefficients are shown in Table 9.Table 9
Seven stages of natural frequencies and correlation coefficients (%) of the SHEP.
StageNatural frequency (Hz)123456716.81910016.0862.8072.4076.6791.1052.334211.51416.08610012.6635.29516.2330.8264.441323.1982.80712.6631000.0383.6450.6434.271443.0482.4075.2950.0381006.9133.3054.455544.3146.67916.2333.6456.9131000.4564.776653.4811.1050.8260.6433.3050.4561005.353762.1852.3344.4414.2714.4554.7765.353100According to Table9, low correlation coefficients exist between every two natural frequencies of the SHEP, showing these seven stages of natural frequencies have high accuracy and reliability. Besides, the first natural frequency of the SHEP is 6.819 Hz, making the sympathetic vibration of the SHEP appear when the SSFE II output frequencies were 5.86 Hz, 6.14 Hz, 6.74 Hz, and 7.04 Hz close to 6∼7 Hz and then making peak values of the SSFE vibration displacement appear according to Figures 17 and 18.


## 6.1. Analysis on the SSFE and the SSFE II Output Frequency Calibration Experiment Results

The SSFE and the SSFE II output frequency calibration curves drawn through Excel are shown in Figure14.Figure 14
The SSFE and the SSFE II output frequency calibration curves. (a) The SSFE output frequency calibration curve. (b) The SSFE II output frequency calibration curve.
(a)(b)According to Figure14, the SSFE output frequencies range from 1.3 Hz to 4.81 Hz and the SSFE II output frequencies range from 2.24 Hz to 7.35 Hz, approximately in the main excitation frequency band of sugarcane field roughness signals, 1∼6 Hz, showing the SSFE and the SSFE II input frequencies were chosen reasonably.


## 6.2. Analysis on the Dynamic Characteristic Experiment Result of the SSFE

Under every eccentric mass, curves of the SSFE vibration displacements along the three directions changing with the SSFE input frequency drawn through Excel are shown in Figure15.According to Figure15, under the same eccentric mass, the SSFE vibration displacements along two lateral directions are greater than that along the vertical direction, so the SSFE may turn over laterally. Therefore, the SSFE needs improvement.Figure 15
Under every eccentric mass, curves of the SSFE vibration displacements along the three directions changing with the SSFE input frequency. (a) Under a 1 kg eccentric mass. (b) Under a 2 kg eccentric mass. (c) Under a 3 kg eccentric mass. (d) Under a 4 kg eccentric mass.
(a)(b)(c)(d)Along every direction, curves of the SSFE vibration displacements changing with the SSFE input frequency under different eccentric masses drawn through Excel are shown in Figure16.Figure 16
Along every direction, curves of the SSFE vibration displacements changing with the SSFE input frequency under different eccentric masses. (a) Along thex axis. (b) Along the y axis. (c) Along the z axis.
(a)(b)(c)According to Figure16, along the same direction, the greater the eccentric mass is, the greater the SSFE vibration displacement will be. According to Figures 15 and 16, under the same eccentric mass or along the same direction, the greater the SSFE input frequency is, the greater the SSFE vibration displacement will be.Moreover, according to Figures15 and 16, the SSFE vibration displacement has an obvious increasing trend when the SSFE input frequency is greater than 8 Hz. According to Figure 14(a), the SSFE output frequencies corresponding to 8∼10 Hz are 2.58∼3.16 Hz, in the excitation frequency band of sugarcane field roughness signals with the greatest contribution, 0.5∼3.5 Hz, and the SSFE output frequencies corresponding to 11∼15 Hz are 3.53∼4.81 Hz, in the main excitation frequency band of sugarcane field roughness signals, 1∼6 Hz, making the SSFE vibration displacement have an obvious increasing trend. Besides, the SSFE output frequencies are smaller than 10 Hz, so with low output frequencies, the SSFE generated vibrations along the three directions. Therefore, the SSFE can generate low-frequency vibration signals, matching characteristics of sugarcane field roughness.


## 6.3. Analysis on the Dynamic Characteristic Experiment Result of the SSFE II

Under every eccentric mass, curves of the SSFE II vibration displacements along the three directions changing with the SSFE II input frequency drawn through Excel are shown in Figure17.According to Figure17, under the same eccentric, the SSFE II vibration displacements along two lateral directions are smaller than that along the vertical direction, so lateral-oscillation-limiting devices can limit lateral oscillations of the SSFE II; that is, the SSFE was improved. Therefore, the SSFE II simulates the sugarcane field excitation much more accurately and makes the SHEP much more similar to a sugarcane harvester working in a sugarcane field.Figure 17
Under every eccentric mass, curves of the SSFE II vibration displacements along the three directions changing with the SSFE II input frequency. (a) Under a 1 kg eccentric mass. (b) Under a 2 kg eccentric mass. (c) Under a 3 kg eccentric mass. (d) Under a 4 kg eccentric mass.
(a)(b)(c)(d)Along every direction, curves of the SSFE II vibration displacements changing with the SSFE II input frequency under different eccentric masses drawn through Excel are shown in Figure18.Figure 18
Along every direction, curves of the SSFE II vibration displacements changing with the SSFE II input frequency under different eccentric masses. (a) Along thex axis. (b) Along the y axis. (c) Along the z axis.
(a)(b)(c)According to Figure18, along the same direction, the greater the eccentric mass is, the greater the SSFE II vibration displacement will be. According to Figures 17 and 18, under the same eccentric mass or along the same direction, the greater the SSFE II input frequency is, the greater the SSFE II vibration displacement will be.Moreover, according to Figures17 and 18, wave crests of the SSFE II vibration displacement appear when the SSFE II input frequencies are 16 Hz, 17 Hz, 20 Hz, 21 Hz, 23 Hz, and 24 Hz and the SSFE II vibration displacement has an obvious increasing trend when the SSFE II input frequency is greater than 16 Hz. According to Figure 14(b), the SSFE II output frequencies corresponding to these six input frequencies are 4.69∼7.04 Hz, approximately in the main excitation frequency band of sugarcane field roughness signals, 1∼6 Hz, making peak values of the SSFE II vibration displacement appear and the SSFE II vibration displacement have an obvious increasing trend. Besides, the SSFE II output frequencies are smaller than 10 Hz, so with low output frequencies, the SSFE II generated vibrations along the three directions. Therefore, the SSFE II can generate low-frequency vibration signals, matching characteristics of sugarcane field roughness.


## 6.4. Analysis on the LMS Modal Test Result of the SHEP

The POLYMAX steady-state diagram of the SHEP and its MAC matrix obtained through modal analysis of LMS Test.Lab are shown in Figures19 and 20.Figure 19
The POLYMAX steady-state diagram of the SHEP.Figure 20
The MAC matrix of the SHEP.According to Figure19, seven stages of natural frequencies, that is, seven stages of vibration modes, were chosen in the POLYMAX steady-state diagram of the SHEP. According to Figure 20, these seven stages of vibration modes have low correlations between each other; that is, these seven stages of vibration modes have high accuracy, showing these seven stages of natural frequencies SHEP have high reliability. Seven stages of natural frequencies and correlation coefficients are shown in Table 9.Table 9
Seven stages of natural frequencies and correlation coefficients (%) of the SHEP.
StageNatural frequency (Hz)123456716.81910016.0862.8072.4076.6791.1052.334211.51416.08610012.6635.29516.2330.8264.441323.1982.80712.6631000.0383.6450.6434.271443.0482.4075.2950.0381006.9133.3054.455544.3146.67916.2333.6456.9131000.4564.776653.4811.1050.8260.6433.3050.4561005.353762.1852.3344.4414.2714.4554.7765.353100According to Table9, low correlation coefficients exist between every two natural frequencies of the SHEP, showing these seven stages of natural frequencies have high accuracy and reliability. Besides, the first natural frequency of the SHEP is 6.819 Hz, making the sympathetic vibration of the SHEP appear when the SSFE II output frequencies were 5.86 Hz, 6.14 Hz, 6.74 Hz, and 7.04 Hz close to 6∼7 Hz and then making peak values of the SSFE vibration displacement appear according to Figures 17 and 18.


## 7. Conclusions

(1)
Simulations of the mathematical model show the dynamics and the mathematical models of the SHEP are reasonable and the SHEP can simulate a sugarcane harvester working in a sugarcane field.(2)
The dynamic characteristic experiment of the SSFE shows it matches characteristics of sugarcane field roughness, but great lateral oscillations existed when it worked, so it needed improvement. Then the SSFE II were developed.(3)
The dynamic characteristic experiment of the SSFE II shows it matches characteristics of sugarcane field roughness and lateral-oscillation-limiting devices can limit lateral oscillations of the SSFE II, so the SSFE was improved.(4)
The LMS modal test of the SHEP shows its first natural frequency is 6.819 Hz, making the sympathetic vibration of the SHEP appear when the SSFE II output frequencies were 5.86 Hz, 6.14 Hz, 6.74 Hz, and 7.04 Hz close to 6∼7 Hz and then making peak values of the SSFE vibration displacement appear.With the SSFE II, simulated experiments of sugarcane harvesters under complete vibration causing conditions of the engine, cutting forces, and sugarcane field roughness can be done in labs instead of sugarcane fields to avoid the low efficiency, poor security, and bad reliability during experiments in sugarcane fields.



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*Source: 3209449-2022-02-12.xml*