Improving Dynamic Performance of a Small Rhizome Chinese Herbs Harvesting Machine via Analysis, Testing, and Experimentation

Abstract

The small rhizome Chinese herbal medicine harvesting machine is used for excavating the underground rhizomes of Chinese medicinal plants. The reliability of an existing machine of this type was found to be suboptimal due to high vibration levels, as confirmed by direct measurements. To remedy this issue, the differential equation of motion was derived, solved, and visualized using MATLAB software (R2017a). The impact of various parameters on the equation of motion was analyzed through both time and frequency domain plots, as well as experimental analysis. The parameters studied included the rotational speed of the tractor’s power take-off (PTO) shaft, the machine’s overall mass and stiffness, the transmission ratio, and the excitation force generated by the machine’s reciprocating parts. To reduce vibration in the non-resonant state and avoid resonance as the natural frequency changes, several modifications were necessary: the PTO speed needed to be controlled, the stiffness of the machine had to be increased, and the mass of the reciprocating parts had to be decreased. Additionally, expanding the transmission-ratio range of the operational machinery was essential. Sandbags were added to the machine’s frame to increase its overall mass. The above measures have reduced the vibration speed of the harvester during operation. The vibration speeds V_max and V_RMS of the harvester under both working and non-working conditions have decreased by half compared to their original values, reducing the occurrence of resonance in the harvester and effectively mitigating vibration damage, thereby enhancing operational reliability.

1. Introduction

Root-stem Chinese herbal medicine harvesters are specialized agricultural machinery equipment, primarily used for excavating the underground stems of medicinal plants, which generally grow at depths of 200 to 600 mm [1]. Currently, the widely used machines of this type are modified from potato diggers, with targeted improvements. The technical development trend is towards low-resistance excavation, efficient separation [2], and low-damage collection to enhance efficiency and reduce losses. Such as Chen et al. [3] reduce the digging resistance of the digging shovel through vibration, and Zhang et al. [4] adopted vibration methods for both the digging part and the soil-crop root separation part to reduce resistance and achieve effective separation of soil and crop roots. Users of the 4Y-100 Small Rhizome Chinese Herbs Harvesting Machine reported relatively low reliability of these machines. Operational surveys and replacement methods—designed to rule out product quality, material issues, personnel, land plot conditions, and other similar factors—have shown through measurements and theoretical analysis that the 4Y-100 harvesters experience excessive vibration during operation. This adversely impacts the working precision and dynamic stability of the mechanical structure, leading to component damage and fatigue failures.

The operational survey also revealed that some operators had retrofitted the gearboxes and transmission components of their tractors by altering the transmission ratio and adding counterweights to increase the harvester’s weight. These modifications aimed to reduce vibrations and improve the overall reliability of the machinery. However, due to a lack of understanding of the machine’s fundamental dynamics, the precise mechanisms by which these adjustments reduce vibration remain unclear, preventing more effective implementation of improvements.

Numerous studies have examined the vibration behavior of agricultural machinery to mitigate resonance and reduce excessive vibration during operation. Fukushima [5] and Wang et al. [6] performed finite element analyses and experimental validations to assess the dynamic characteristics of both individual components and entire machines. Fu et al. [7] investigated the vibration characteristics of potato harvesters using a combination of field measurements and software simulations, concluding that vibration intensity and frequency vary with operating conditions. Zhu et al. [8] experimentally determined that the natural frequency of agricultural machinery systems decreases as mass increases. Yao et al. [9] applied finite element modal analysis and vibration testing to study vibrational phenomena in grain combine harvesters, identifying engine vibrations, road excitations, and the movement of operational parts as primary sources of vibration. Xu et al. [10], through experimental work, found that the reciprocating motion of components such as engine pistons, vibrating screens, and sickle bars significantly contributes to vibration in rice combine harvesters. Chowdhury et al. [11] examined the vibration characteristics of a radish harvester in field conditions, highlighting that load conditions and variations in conveyor belt speed and position influence vibration levels.

In their research on the characteristics, influence, and control methods of agricultural machinery vibration, Wang et al. [12] conducted a vibration frequency analysis of mechanical corn precision seeders. They found that when operating speeds exceed 5 km/h, the seeder’s vibration energy is concentrated in the low-frequency range. Furthermore, while the intensity of vibration increases with higher speeds, the frequency distribution of the vibrational energy remains unchanged. Adam et al. [13] recorded vibration signals from both the tractor chassis and the driver’s thigh under driving and field operation conditions, identifying a seat resonance frequency of 2-3 Hz. Xin [14] and Qiu et al. [15] examined the dynamic properties of subsoilers, concluding that variations in soil resistance and other factors have minimal impact on subsoiler vibration. Zhang et al. [16] developed a dynamic equation for the vibrating screen of a potato digger, finding that when the excitation frequency matches the natural frequency, the screen’s vibration amplitude increases significantly, which could shorten its service life. Inoue et al. [17] proposed adding mass to the cutter driving mechanism of a combine’s blade as a strategy to reduce vibration.

Currently, in terms of the study of the basic dynamics behind the vibration problem of harvesters, Chen et al. [18] studied the vibration of rice harvesters under working conditions and found that the frame would vibrate significantly under the action of multiple sources of excitation force. Research on the dynamics modeling of agricultural machinery and equipment is limited, mainly relying on software motion simulation to establish the dynamics model, such as the study by Zhao et al. [19]. The effects of soil damping and other factors on the vibration of agricultural machinery in the actual operation process are not considered. In the field of vehicle vibration studies, linear differential equations based on Newton’s second law are usually used to study vehicle vibration issues.

However, there is a notable lack of research addressing vibration reduction in non-resonant states during the operation of agricultural machinery. Although practical applications exist, their effectiveness is hampered by a lack of theoretical underpinning.

This paper delves into the vibration challenges of the 4Y-100 Small Rhizome Chinese Herbs Harvesting Machine during field operations. By analyzing the factors influencing vibration and the machine’s vibration dynamics, the aim is to mitigate resonance and reduce vibrations under working conditions, thereby enhancing operational reliability. The paper explores engineering methods to modify these factors, reduce vibration levels, and prevent resonance, offering a theoretical framework for improving machine reliability. To accomplish this, the paper undertakes the following:

The paper first analyzes the vibration characteristics of the implement during tractor-driven field operations, deriving the governing differential equations. These equations are then solved using MATLAB to generate graphical representations of the solutions. The effects of various parameters are examined through time and frequency domain plots and experimental analysis. To validate the analytical findings, indoor experiments are conducted using an earth-bin test machine. Finally, the paper proposes engineering measures to modify influencing factors, thereby reducing vibration levels and preventing resonance.

2. Materials and Methods

2.1. Machine Composition and Working Principles of a Small Rhizome Chinese Herb Harvesting Machine

The 4Y-100 small rhizome Chinese herbal medicine harvester, illustrated in Figure 1, consists of a frame, digging shovel, swing screen, and eccentric-wheel swing mechanism. The swing screen is operated by an eccentric wheel connected to an oscillating rod mechanism. As the two-stage swing screen reciprocates, it separates the soil and rhizomes collected by the digging shovel. The soil falls through the screen gaps, while the rhizomes are transported backward and laid out on the ground. The relevant parameters of the harvester are shown in

Table 1. Parameters related to vibration test work equipment.

Device Name	Parameter	Parameter Value
4Y-100 small rhizome Chinese herbal medicine harvester	Overall length × width × height (mm)	3500 × 1300 × 1500
	Mating power (kW)	44–58.8
	Working width (mm)	1000
TCC-2.5 indoor soil bin	Working width (mm)	2500
	Motor drive power (kW)	90
	Hydraulic motor speed range (rpm)	2–2000

.

2.2. Experimental Setup and Conditions

2.2.1. Test Field

The field trial was conducted in September 2019, while the indoor soil bin experiment took place from May 2021 to January 2022. In this experiment, field operations and vibration data collection for the small root herbal harvester were carried out in an area with minimal impurities and a more uniform soil environment. The test site was located in Majiagou Village, Tong’anyi Township, Longxi County, Gansu Province, China (latitude 35.31° N, longitude 104.71° E). The performance test of the harvester, prior to its improvement, was conducted on flat, dry land with a soil moisture content of 13%, typical of loess soil. The harvested crop was Astragalus membranaceus, which had been cultivated for three years. The conditions of the experimental field and the growth of the harvested crop are shown in Figure 2a. To ensure the repeatability and accuracy of the experiment, environmental factors were minimized. An additional experiment to study factors affecting harvester vibration was conducted indoors in a soil bin, using loess soil in both cases. The soil properties are detailed in

Table 2. Parameters related to field characteristics of the soil.

Field Type	Items	Parameter Value
Field harvesting	Average soil moisture content (%)	12.8
	Average soil firmness (kPa)	635.2
	Permissible soil weight (g/cm3)	1.21
Indoor soil bin	Average soil moisture content (%)	11.3
	Average soil firmness (kPa)	737.5
	Permissible soil weight (g/cm3)	1.31

. Indoor soil bin test machine-related parameters are shown in Table 1.

2.2.2. Test Equipment

The following equipment and software have been used:

• The 4Y-100 small rhizome Chinese herbal medicine harvester, produced by Gansu Weisheng Agricultural Equipment Manufacturing Co., Ltd. in Lanzhou, China.
• A 4Y-100 small rhizome Chinese herbal medicine harvester, powered by a Dong Fang Hong MF704 tractor manufactured by YTO Co., Ltd. in Luoyang, China.
• An indoor soil bin Tcc-2.5 manufactured by Heilongjiang Agricultural Machinery Research Institute, Harbin, China.
• A vibration velocity acquisition system, Donghua DH5922D, manufactured by Donghua Testing Technology Co., Ltd. in Jingjiang, China, with a sampling rate of up to 4 kHz and a frequency response range of DC to 100 kHz (±0.5 dB) at a 4 kHz sampling frequency.
• A magneto-electric speed sensor, Donghua 2D002, is manufactured by Donghua Testing Technology Co., Ltd., Jingjiang, China, with a measurement range of 0 to 0.5 m/s and a frequency range of −3 to +1 dB within 1 to 1000 Hz.
• Donghua Test DHDAS dynamic signal acquisition and analyzing system (DSAAS) and software supplied by Donghua Testing Technology Co., Ltd. in Jingjiang, China, running on a Dell laptop computer.

2.2.3. Experimental Crops

The 4Y-100 small rhizome Chinese herbal medicine harvester is primarily used for harvesting the underground tubers of Astragalus membranaceus, Codonopsis pilosula, and Glycyrrhiza uralensis. After harvest, these underground tubers are used as medicinal herbs in traditional Chinese medicine. The crops harvested during the field operation experiment are the underground tuber parts of Astragalus membranaceus that have grown for three years.

2.2.4. Test Method

To eliminate potential interference from vibration signals caused by damaged gears in the gearbox, defective bearings, loose connection bolts, cracking welds, and other similar defects, a thorough inspection of these components was conducted before and after each experimental trial to ensure their integrity.

Before each test, the vibration test system was calibrated using a vibration test bench with a known frequency and amplitude. The test system was warmed up before testing, and once the system stabilized, the signals during the no-vibration period were recorded. The channels were then zeroed based on this recording. During the test, the test conditions were maintained stable, data were collected without any abnormalities, and the data analysis process excluded any abnormal signal data.

In a test using an indoor soil bin tester, the power take-off shaft started at 40 rpm and then accelerated to 400 rpm at a constant speed. This took five minutes. The same tractor and driver were used in the field trials and machine-idling experimental tests. It was observed that there were differences in vibration speed amplitudes, but the differences in peak resonance frequencies were negligible. The experimental data, where the amplitude was at the median value, was selected.

For both field experiments and indoor soil bin tests, the small root and tuber Chinese herbal medicine harvester was initially operated at a reduced speed upon entering the soil, targeting an excavation depth of 350 mm while maintaining a steady walking pace of 0.25 m/s. The speed of the power output shaft was then gradually increased from a low speed to 400 rpm in a uniform manner. After zeroing the instrument channels, the dynamic signal acquisition system began collecting vibration velocity data from the lower speeds of the power output shaft, continuing until the shaft reached 400 rpm, at which point data collection was stopped to record the relevant values.

During the idle test, the small rhizome Chinese herbal medicine harvester was kept attached to the suspension device, raised off the ground. The power output shaft speed was uniformly increased from a low speed to 400 rpm, with the corresponding data being documented throughout the process.

2.3. Experimental Results

The variation in vibration velocity of the harvester was measured in the working condition with the soil damping effect and in the non-working idling condition without the soil damping effect. The following experimental results were obtained and are presented in the table below.

3. Vibration Analysis

3.1. Vibration Analysis of the Harvesting Machine

The machine is powered by the tractor’s power take-off (PTO). In the working state (Figure 3a), the operating frequency ranges between 2.79 and 2.89 Hz, while in the non-working state (Figure 3b), it is about 4.14 Hz. At these frequencies, the machine exhibits significant peaks in vibration speed amplitude. It was observed that resonance occurs at approximately 250 rpm of the PTO shaft input. During field operations, the tractor PTO’s output speed typically ranges from 200 to 300 rpm, placing the harvesting machine near or within a resonance state. As shown in Figure 4a, even when the operating speed is outside of resonance, the vibration level of the harvesting machine remains above the acceptable limits for mechanical vibrations according to industry standards [20]. This suggests that even when operating outside the resonance range, the machine’s structure and components can experience accelerated wear and damage due to high vibration levels. Taking these observations into account, when the tractor’s PTO speed is between 50 and 400 rpm, the vibration levels remain elevated, which is a primary cause of the poor reliability of the small rhizome Chinese herb harvesting machine.

3.2. Analysis of Vibration Characteristics and Establishment of Mechanical Equations

Comparing the amplitudes in Figure 4a,b, it is clear that the damping effect of the soil has led to a decrease in the vibration speed level of the harvester. This suggests that the vibration of the harvester is reduced under the influence of soil damping. By examining the resonance band peak frequencies and the sharpness of the resonance peaks in Figure 4a,b, the findings indicate that the resonance band peak frequency during the working state, under the influence of soil damping, is lower than that of the idling state without soil damping, and the sharpness of the resonance peak is diminished. This corresponds to the pattern of changes in resonance frequency and the sharpness of resonance peaks for systems with damping compared to undamped systems under the same excitation.

The test results presented in

Table 3. Experimental test results with a tractor as the power source.

Harvester Status		Resonance Band Peak Frequency Hz	Vibration Velocity Vmax (mm/s)	Vibration Speed Mean Square Value VRMS (mm/s)
1	Tractor machine traction drive work mode before improvement	2.89	66.931	104.58
2	The harvester is driven by the tractor, suspended off the ground by the tractor, and the harvester is in a non-working idling state.	4.14	118.85	154.277
3	Work mode of the traction drive of the indoor soil bin testing machine before improvement	3.086	97.606	152.187
4	The non-working, idling state of the indoor soil bin testing machine without spring	4.219	170.664	268.759

and Figure 3 indicate that the resonance frequency of the harvester has changed. These analyses suggest that the harvester’s vibration under operating conditions is influenced by damping effects. Consequently, the intrinsic frequency of the harvester varies and is not a fixed constant.

Figure 4a shows that the vibration velocity of the harvester changes with the rotational speed of the PTO, alternating between positive and negative values. This indicates that the vibration of the harvesting machine is generated by a reciprocating force. The vibration velocity levels in Figure 4a,b demonstrate that the excitation of the harvesting machine is caused by reciprocating forces.

When the machine is in operation, it mostly moves in a straight line at a constant speed, resulting in a net external force of zero. The excitation force and its reaction force are generated by the interaction between components, both of which are reciprocating forces associated with reciprocating motion. The reaction force of the excitation force acts on the movable parts, causing them to perform reciprocating motions, while the excitation force itself acts on the frame. Since the entire machine is based on the frame, the vibration of the frame drives the vibration of the whole machine. The reciprocating components of the harvesting machine include the eccentric wheel and swing rod mechanism, as shown in Figure 5.

In Figure 5 and Figure 6, the x-axis of the Cartesian coordinate system represents the direction in which the harvester moves forward, while the y-axis represents the direction perpendicular to the ground. The link BD swings about a pin joint at C and performs a reciprocating motion. The harvester typically operates in a state of uniform linear motion. Taking the main frame as the reference system, since pivot point C does not move relative to the main frame, it can be concluded from the equations of static equilibrium that the forces and moments at pivot point C are in equilibrium. Based on the motion and force diagram and the vibration mechanism in Figure 5, we get:

$${\sum \overrightarrow{Mc}=0\to \sum \overrightarrow{Mcx}=0\to \mathrm{F}}_{\mathrm{x}\mathrm{D}}·{\mathrm{L}}_{\mathrm{C}\mathrm{D}}·\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\beta }={\mathrm{F}}_{\mathrm{x}\mathrm{B}}·{\mathrm{L}}_{\mathrm{B}\mathrm{C}}·\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\beta }$$

(1)

$${\mathrm{F}}_{\mathrm{x}\mathrm{D}}={\mathrm{F}}_{\mathrm{x}\mathrm{B}}·{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}$$

(2)

$$\sum \overrightarrow{F}=0\to \sum \overrightarrow{Fx}=0\to {\mathrm{F}}_{\mathrm{c}}={\mathrm{F}}_{\mathrm{x}\mathrm{B}}+{\mathrm{F}}_{\mathrm{x}\mathrm{D}}={\mathrm{F}}_{\mathrm{x}\mathrm{B}}\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)$$

(3)

where F_xB and F_xD are the horizontal forces exerted by the pendulum BD at points B and D, $\sum \overrightarrow{Mc}$ represents the total moment at point C, and F_C is the force exerted in the x-direction at point c of the frame. ${{\mathrm{L}}_{\mathrm{A}\mathrm{B}};\mathrm{L}}_{\mathrm{B}\mathrm{C}}{;\mathrm{L}}_{\mathrm{C}\mathrm{D}}$ are the lengths of the pendulum bar AB, BC, and CD segments, respectively, and ω is the angular velocity of the eccentric wheel.

From kinematic analysis, the pendulum BD displacement can be expressed as:

$$\begin{array}{c}{\mathrm{x}}_{\mathrm{B}}=\mathrm{r} \mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\omega }\mathrm{t}\right)+{\mathrm{L}}_{\mathrm{A}\mathrm{B}}\mathrm{c}\mathrm{o}\mathrm{s}\pm \mathsf{\alpha }\\ {\mathrm{y}}_{\mathrm{B}}=\mathrm{r} \mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\omega }\mathrm{t}\right)+{\mathrm{L}}_{\mathrm{A}\mathrm{B}}\mathrm{s}\mathrm{i}\mathrm{n}\pm \mathsf{\alpha }\end{array}$$

(4)

The Taylor series expansion of the square root yields

$$\begin{gathered} \mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }={\left(1-{\displaystyle \frac{{\mathrm{r}}^2}{{\mathrm{L}}_{\mathrm{A}\mathrm{B}}^2}}{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\mathsf{\omega }\mathrm{t})\right)}^{1/2}=1-{\displaystyle \frac{{\mathrm{r}}^2}{{2\mathrm{L}}_{\mathrm{A}\mathrm{B}}^2}}{\mathrm{s}\mathrm{i}\mathrm{n}}^2(\mathsf{\omega }\mathrm{t}) + \\ {\displaystyle \frac{{\mathrm{r}}^4}{{8\mathrm{L}}_{\mathrm{A}\mathrm{B}}^4}}{\mathrm{s}\mathrm{i}\mathrm{n}}^4(\mathsf{\omega }\mathrm{t})-{\displaystyle \frac{{\mathrm{r}}^6}{{16\mathrm{L}}_{\mathrm{A}\mathrm{B}}^6}}{\mathrm{s}\mathrm{i}\mathrm{n}}^6(\mathsf{\omega }\mathrm{t})+\dots \end{gathered}$$

(5)

Which for ${\displaystyle \frac{\mathrm{r}}{{\mathrm{L}}_{\mathrm{A}\mathrm{B}}}}={\displaystyle \frac{1}{10}}$ can be approximated with:

$$\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\alpha }=1-{\displaystyle \frac{{\mathrm{r}}^2}{{2\mathrm{L}}_{\mathrm{A}\mathrm{B}}^2}}{\mathrm{s}\mathrm{i}\mathrm{n}}^2\mathsf{\omega }\mathrm{t}=1-{\displaystyle \frac{{\mathrm{r}}^2}{{4\mathrm{L}}_{\mathrm{A}\mathrm{B}}^2}}\left(1-\mathrm{c}\mathrm{o}\mathrm{s}2\mathsf{\omega }\mathrm{t}\right)$$

(6)

By combining Equations (6) and (4), the equation of motion in the x direction at point B is obtained:

$${\mathrm{x}}_{\mathrm{B}}=\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\omega }\mathrm{t}+{\displaystyle \frac{{\mathrm{r}}^2}{4{\mathrm{L}}_{\mathrm{A}\mathrm{B}}}}\mathrm{c}\mathrm{o}\mathrm{s}2\mathsf{\omega }\mathrm{t}+{\mathrm{L}}_{\mathrm{A}\mathrm{B}}-{\displaystyle \frac{{\mathrm{r}}^2}{4{\mathrm{L}}_{\mathrm{A}\mathrm{B}}}}$$

(7)

While its acceleration is:

$${\displaystyle \frac{{\mathrm{d}}^2{\mathrm{x}}_{\mathrm{B}}}{{\mathrm{d}\mathrm{t}}^2}}=-\mathrm{r}{\mathsf{\omega }}^2\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\omega }\mathrm{t}+{\displaystyle \frac{\mathrm{r}}{{\mathrm{L}}_{\mathrm{A}\mathrm{B}}}}\mathrm{c}\mathrm{o}\mathrm{s}2\mathsf{\omega }\mathrm{t}\right)$$

(8)

Which for ${\displaystyle \frac{\mathrm{r}}{{\mathrm{L}}_{\mathrm{A}\mathrm{B}}}}={\displaystyle \frac{1}{10}}$; $\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\omega }\mathrm{t}\gg {\displaystyle \frac{\mathrm{r}}{{\mathrm{L}}_{\mathrm{A}\mathrm{B}}}}\mathrm{c}\mathrm{o}\mathrm{s}2\mathsf{\omega }\mathrm{t}$ becomes:

$${\displaystyle \frac{{\mathrm{d}}^2{\mathrm{x}}_{\mathrm{B}}}{{\mathrm{d}\mathrm{t}}^2}}\approx -\mathrm{r}{\mathsf{\omega }}^2\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\omega }\mathrm{t}$$

(9)

It can be inferred from force analysis that:

$${\mathrm{F}}_{\mathrm{x}\mathrm{B}}=\sum {\mathrm{F}}_{\mathrm{x}\mathrm{B}}\approx -\mathrm{m}\mathrm{r}{\mathsf{\omega }}^2\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\omega }\mathrm{t}\right)$$

(10)

where the eccentric wheel’s angular velocity is

$$\mathsf{\omega }={\displaystyle \frac{2\mathsf{\pi }\mathrm{n}}{60}}·{\displaystyle \frac{1}{i}}={\displaystyle \frac{\mathsf{\pi }\mathrm{n}}{30i}}$$

It can be further inferred from Equation (3) that

$$\begin{array}{c}{\mathrm{F}}_{\mathrm{c}}={\mathrm{F}}_{\mathrm{x}\mathrm{B}}\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)\approx -\mathrm{m}\mathrm{r}{\mathsf{\omega }}^2\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\omega }\mathrm{t}=\\ -\mathrm{m}\mathrm{r}{\left({\displaystyle \frac{\mathrm{n}}{30i}}\mathsf{\pi }\right)}^2\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\mathrm{n}}{30i}}\mathsf{\pi }\right)\mathrm{t}\end{array}$$

(11)

In Equation (11), m is the equivalent mass of rod BC, r is the radius of the eccentric wheel, ω is the rotational angular velocity of the eccentric wheel, L is the length of the corresponding rod, n is the rotational speed of the PTO of the tractor, and i is the transmission ratio of the gear box of the harvester. When the harvesting harvester is operating in steady state, all the above parameters have fixed values. From Equation (11), it can be seen that the size and value of F_C vary periodically, and its reaction force,${\mathrm{F}}_{\mathrm{C}}^{\mathrm{\prime }}$, is the main excitation force for the vibration of the harvesting machine in the forward direction of operation.

3.3. Establishment of the Vibration Equation for the Harvester

During operation, the ground wheels and tractor suspension system, the wall of the trench formed by the work of the excavating shovel, and soil buildup on the side walls of the harvesting machine have a limiting and constraining effect, so that the left and right movement of the machine is negligible. From Figure 6, a load analysis of the implement in the vertical direction yields:

$$\mathrm{N}+{\mathrm{m}}_{\mathrm{O}\mathrm{A}}\mathrm{r}{\mathsf{\omega }}^2\sin \mathsf{\omega }\mathrm{t}-\mathrm{M}\mathrm{g}=\mathrm{F}\mathrm{y}$$

$$\mathrm{M}⨠{\mathrm{m}}_{\mathrm{O}\mathrm{A}}\to \mathrm{M}\mathrm{g}⨠{\mathrm{m}}_{\mathrm{O}\mathrm{A}}\mathrm{r}{\mathsf{\omega }}^2\sin \mathsf{\omega }\mathrm{t}$$

→Fy = 0→Vy = 0

(12)

In Equation (12), M is the overall mass of the harvesting machine, and N is the ground reaction force. m_oA is the mass of the eccentric wheel. The parameters r, ω have the same meaning as in Equation (11). Fy is the combined external force in the vertical direction of the machine. Vy is the speed of movement of the machine in the vertical direction. It can be seen that when the rotational speed n of the PTO is small, the gravitational force of the harvesting machine is much larger than the vertical forces generated by the eccentric wheel, and it cannot cause the movement of the harvesting machine in the vertical direction. Figure 4a,b show that in the working mode, there is only a first-order critical speed and no higher-order critical speeds, as one degree of freedom corresponds to only one critical speed [21]. It shows that the working vibration of the harvester meets the single input and single output requirements of the linear constant value system in the working speed range. Based on the above analysis, the vibration of the harvester is simplified as a single degree of freedom vibration along the forward direction of the machine.

The harvester digs deeply and has a lot of contact between the machine body and the soil. The influence of the soil can be expressed by the damping force Fz. There is an interaction force between the component and the body, which can be reacted to by the elastic force Ft. When the harvesting machine is working, the resistance received by the frame in the forward movement direction is represented by R. The force equation of the frame is:

$${\mathrm{F}}_{\mathrm{c}}^{\mathrm{\prime }}-\mathrm{c}\left.\left\{{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}\right.\right\}-\mathrm{k}\left.\left\{\mathrm{x}\right.\left(\mathrm{t}\right)\right\}+\left({\mathrm{F}}_{\mathrm{q}}-\mathrm{R}\right)=\mathrm{M}\left.\left\{{\displaystyle \frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{d}\mathrm{t}}^2}}\right.\right\}$$

(13)

where x(t) is the displacement, c is damping, k is the stiffness, and M is the total mass of the harvesting machine. The damping force is ${\mathrm{F}}_{\mathrm{z}}=\mathrm{c}\left.\left\{{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}\right.\right\}$, the elastic force is ${\mathrm{F}}_{\mathrm{t}}=\mathrm{k}\left.\left\{\mathrm{x}\right.\left(\mathrm{t}\right)\right\}$, the forward resistance is R, and the traction force is F_q. When the harvester is running in a steady operating mode (i.e., the speed and direction of the unit’s forward movement remain constant), the resistance and traction forces are equal, i.e., R = Fq.

$$\mathrm{M}\left.\left\{{\displaystyle \frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{d}\mathrm{t}}^2}}\right.\right\}+\mathrm{c}\left.\left\{{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}\right.\right\}+\mathrm{k}\left.\left\{\mathrm{x}\right.\left(\mathrm{t}\right)\right\}={\mathrm{F}}_{\mathrm{C}}^{\mathrm{\prime }}=-{\mathrm{F}}_{\mathrm{C}}$$

(14)

For ${\mathsf{\omega }}_{\mathrm{n}}$ the natural frequency of the harvesting machine and δ the damping ratio of the machine’s working system, we have ${\displaystyle \frac{\mathrm{k}}{\mathrm{M}}}={\mathsf{\omega }}_{\mathrm{n}}^2$ and ${\displaystyle \frac{\mathrm{C}}{\mathrm{M}}}=2\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}$ [22]. The elastic force is interactive and can be expressed by the elastic force of the harvester body. Using Equation (11), we can obtain the differential equation of motion in the walking direction of the harvesting machine in a steady operating state:

$${\displaystyle \frac{{\mathrm{d}}^2\mathrm{x}}{ {\mathrm{d}\mathrm{t}}^2}}+2\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}{\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}+{\mathsf{\omega }}_{\mathrm{n}}^2\mathrm{x}=\left({\displaystyle \frac{\mathrm{m}}{\mathrm{M}}}\right)\mathrm{r}{\mathsf{\omega }}^2\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\omega }\mathrm{t}$$

(15)

The complete solution of the differential equation is $x=x_1+x_2$ with $x_1$ the special solution and $x_2$ is the general solution of the homogeneous equation. The solution to Equation (15) is as follows:

$$\begin{array}{c}\mathrm{x}\left(\mathrm{t}\right)=\sqrt{{\mathrm{x}}_0+{{\displaystyle \frac{\left({\mathrm{v}}_0+\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}{\mathrm{x}}_0\right)}{{{\mathrm{w}}_{\mathrm{d}}}^2}}}^2}\mathsf{\delta }{\mathrm{e}}^{-\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}\mathrm{t}}{\mathrm{w}}_{\mathrm{d}}\cos \left({\mathsf{\omega }}_{\mathrm{d}}\mathrm{t}-{\mathsf{\phi }}_1\right)-\\ {\displaystyle \frac{\mathrm{r}\mathsf{\omega }}{\sqrt{{\left(1-{\left(\frac{\mathsf{\omega }}{{\mathsf{\omega }}_{\mathrm{n}}}\right)}^2\right)}^2+{\left(2\mathsf{\delta }\frac{\mathsf{\omega }}{{\mathsf{\omega }}_{\mathrm{n}}}\right)}^2}}}\left({\displaystyle \frac{\mathrm{m}}{\mathrm{M}}}\right)\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)\cos \left(\mathsf{\omega }\mathrm{t}-{\mathsf{\phi }}_2\right)\end{array}$$

(16)

$$\mathrm{V}={\displaystyle \frac{\mathrm{d}\mathrm{x}}{\mathrm{d}\mathrm{t}}}={\displaystyle \frac{\mathrm{r}\frac{\mathrm{n}}{i30}\mathsf{\pi }}{\sqrt{{\left(1-{\left(\frac{\frac{\mathsf{\pi }\mathrm{n}}{30i}}{\sqrt{\mathrm{K}/\mathrm{M}}}\right)}^2\right)}^2+{\left(2\mathsf{\delta }\frac{\frac{\mathsf{\pi }\mathrm{n}}{30i}}{\sqrt{\mathrm{K}/\mathrm{M}}}\right)}^2}}}\left({\displaystyle \frac{\mathrm{m}}{\mathrm{M}}}\right)$$

$$\left(1+{\displaystyle \frac{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}\right)\sin \left({\displaystyle \frac{\mathsf{\pi }\mathrm{n}}{30i}}\mathrm{t}-{\mathsf{\phi }}_2\right)-\mathrm{r}{\mathsf{\omega }}_0\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}{\mathrm{e}}^{-\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}\mathrm{t}}\sin \left[\left(\sqrt{1-{\mathsf{\delta }}^2}\right)\sqrt{{\displaystyle \frac{\mathrm{K}}{\mathrm{M}}}}\mathrm{t}\right]$$

(17)

From Equation (11), we get:

$$\mathrm{V}=-{\displaystyle \frac{\frac{{\mathrm{F}}_{\mathrm{c}}}{\mathrm{r}\frac{\mathrm{n}}{i30}\mathsf{\pi }·\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathrm{n}}{30i}\mathsf{\pi }\right)\mathrm{t}}}{\sqrt{{\left(1-{\left(\frac{\frac{\mathsf{\pi }\mathrm{n}}{30i}}{\sqrt{\mathrm{K}/\mathrm{M}}}\right)}^2\right)}^2+{\left(2\mathsf{\delta }\frac{\frac{\mathsf{\pi }\mathrm{n}}{30i}}{\sqrt{\mathrm{K}/\mathrm{M}}}\right)}^2}}}$$

$$\sin \left({\displaystyle \frac{\mathrm{n}}{i30}}\mathsf{\pi }\mathrm{t}-{\mathsf{\phi }}_2\right)-\mathrm{r}{\mathsf{\omega }}_0\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}{\mathrm{e}}^{-\mathsf{\delta }{\mathsf{\omega }}_{\mathrm{n}}\mathrm{t}}\sin \left[\left(\sqrt{1-{\mathsf{\delta }}^2}\right)\sqrt{{\displaystyle \frac{\mathrm{K}}{\mathrm{M}}}}\mathrm{t}\right]$$

(18)

where ω is the angular velocity of the eccentric wheel, ${\mathsf{\omega }}_0$ is the initial angular velocity of the eccentric wheel, n is the rotational speed of the PTO shaft of the tractor, i is the transmission ratio of the harvester’s gear box, r is the radius of the eccentric wheel, ${\mathsf{\omega }}_{\mathrm{n}}$ is the natural frequency of the harvester, K signifies the stiffness of the harvester, “δ” represents the soil damping coefficient, m is the mass of the pendulum BC, and M is the total mass of the harvesting machine.

4. Results

When the harvester operates in steady-state mode, the parameters in Equation (17) remain constant, with n and t as variables. Utilizing MATLAB software, we have solved Equation (17) and plotted graphs to analyze the relationship between the factors represented by the variables in the equation and the vibration speed of the harvester.

4.1. Effect of Tractor PTO Speed on Vibration Velocity

The test results presented in Figure 4b reveal that when the harvester is idling, the machine’s natural frequency is ω_n = 4.14 Hz. For analytical purposes, given the known natural frequency of the harvester (ω_n) and its mass (M), the stiffness (K) of the harvester can be approximated to K = 7712, as per equation (${\displaystyle \frac{\mathrm{k}}{\mathrm{M}}}={\mathsf{\omega }}_{\mathrm{n}}^2$). The damping in the machine’s operating system is predominantly due to soil damping. For the soil type in the harvester’s operational area, the average soil damping ratio ranges from 0.2 to 0.4 [23], with a chosen value of δ = 0.3 for this analysis. In Equation (17), all parameters except V, n, and t are constant structural or operational parameters of the harvester. To simulate the harvester’s operational states, we considered M values of 450 kg, 550 kg, and 650 kg, which correspond to the initial soil entry, the intermediate state, and the steady operating state, respectively. Equation (17) was solved using MATLAB, and the results were plotted in a three-dimensional graph (Figure 7a) of the effect of the tractor PTO speed (n) on the vibration velocity. To facilitate observation, taking the output rotational speed of the tractor PTO n as the abscissa and the vibration speed V of the harvesting machine in the forward direction as the ordinate, Figure 7a is transformed into a two-dimensional plot in Figure 7b.

As shown in Figure 7a,b, the change rule of vibration speed with the tractor PTO output rotational speed is as follows: the initial speed of 50 rpm results in a larger vibration speed value, and as the tractor PTO output speed increases, the vibration velocity decreases under soil damping. When the PTO speed approaches the harvesting machine’s resonance zone, the vibration speed of the harvesting machine increases sharply until resonance is reached. Once the output rotational speed of the tractor PTO crosses the resonance zone speed, the vibration speed value drops rapidly from the peak and remains greater than the vibration speed before entering the resonance zone. The above change rules are consistent with the trends of the actual measurement results shown in Figure 3a and Figure 4a. The time taken to reach this speed and the quality changes accompanying the digging process do not affect the trend of the vibration speed, which varies with the output rotational speed of the tractor PTO.

The analysis in Equation (12) is the same as described in the literature [24] and is similar to the working principle of plate compactors known as frog-tamping machines studied in reference [25]. It has been confirmed through experiments that if the output rotational speed of the tractor PTO is too high, the vertical force generated by the eccentric mechanism will be greater than the gravity of the harvesting machine. The harvesting machine is caused to reciprocate in the vertical direction, and the digging shovel collides with the unexcavated soil in the vertical direction, causing the digging shovel to be damaged. In summary, in order to prevent the occurrence of resonance and vibration speeds exceeding the amount of vibration damage to the harvesting machine and to avoid the vertical direction of the digging shovel and soil impact damage, we need to strictly control the output rotational speed of the tractor PTO. By controlling the tractor PTO output speed below the resonance speed, it can effectively reduce the vibration damage to the harvester caused by resonance and extend the machine’s service life. Avoiding vertical collisions between the digging shovel and the soil reduces the risk of damage to the shovel and ensures the machine’s continuous operation capability. It also reduces the risk of injury to the operator due to shovel failure during operation. This improves the overall safety of the operation and provides a safer working environment for the operator.

4.2. Effect of Mass M on Vibration Velocity

From Equation (17), it is evident that mass M affects the vibration speed of the harvesting machine. When all other parameters within Equation (17) are held constant, as illustrated in Figure 8a,b, it is observed that, outside the resonance section, at a consistent rotation speed of the tractor PTO, the machine with a higher mass, M exhibits a lower vibration speed compared to one with a lower mass M. However, an increase in the harvesting machine’s mass results in a decrease in its natural frequency and maximum vibration speed, shifting the tractor PTO’s output rotational speed corresponding to the resonance zone towards the lower end of the speed spectrum. This indicates that, within the range of output rotational speeds of the tractor PTO, excluding the resonance zone, the vibration speed of the harvesting machine is inversely proportional to the total mass M. While increasing the mass of the harvesting machine can effectively reduce its vibration speed, it is crucial to prevent the mass from increasing to the extent that it causes the resonance zone’s corresponding PTO output rotational speed to migrate to the lower speed range, potentially triggering resonance at an earlier stage. Similar to the use of mass dampers (TMD) to mitigate vibration in washing machines [26], the overall mass M of the harvester can be augmented by adding counterweights to reduce the vibration rate. The working mass of the harvester M is the total mass of the machine in working condition. To facilitate adjustments, the total mass M can be increased by introducing a temporary counterweight, allowing operators to use that increase in the temporary counterweight to reduce the harvester’s vibration speed. To verify the above analysis, the harvester was operated by an indoor soil box test machine. The digging depth, forward speed of the whole machine, and soil state remained constant. The speed of the power output shaft of the indoor soil box test machine was varied evenly from low to high. The harvester was loaded with 50 kg, 100 kg, and 150 kg counterweights for testing.

The data collected from the test was processed using the Songhua Test DHDAS dynamic signal acquisition and analysis system software to obtain the vibration velocity time domain diagram and frequency domain diagram. The results of the resonance band peak frequency (Hz), vibration speed maximum value (V_max), and vibration speed mean square value (V_RMS) are shown in

Table 4. Load mass test.

Harvester Status		Resonance Band Peak Frequency Hz	Vibration Velocity Vmax (mm/s)	Vibration Speed Mean Square Value VRMS (mm/s)
1	Work mode of the traction drive of the indoor soil bin testing machine before improvement	3.086	97.606	152.187
2	When the indoor soil bin testing machine is traction-driven, the harvester is loaded with a 50 kg counterweight for testing.	2.656	21.878	37.465
3	When the indoor soil bin testing machine is traction driven, the harvester is loaded with a 100 kg counterweight for testing.	2.559	12.340	27.818
4	When the indoor soil bin testing machine is traction driven, the harvester is loaded with a 150 kg counterweight for testing.	2.461	9.431	22.128

. Except for the range of counterweight masses, the other conditions of the soil bin test remained unchanged. From Table 4, it can be seen that the resonance zone and vibration speed of the harvester vary with the range of the loaded counterweight mass, and the resonance zone, resonance frequency, and the corresponding rotational speed of the power output shaft change to low frequency and low rotational speed as the counterweight mass increases. The V_RMS and V_max data show that the trend of decreasing vibration velocity remains constant with the increase in extra mass. This explains why it was found during the field investigation of the harvesting machine that the operator could reduce the vibration velocity of the harvesting machine by installing a counterweight.

4.3. Effect of Stiffness k on Vibration Velocity

According to [27], the stiffness of the car body has a significant influence on the vibration characteristics of the vehicle. Reducing the stiffness of the car body causes the natural frequency to decrease, and the car body is more likely to resonate under external excitation [28]. The stiffness of the car body structure plays a very important role in ensuring structural fatigue strength [29]. Other parameters in Equation (17) remain unchanged, as shown in Figure 9a. As the stiffness k of the harvesting machine increases, the natural frequency of the harvesting machine increases, and the output rotational speed of the tractor PTO corresponding to the resonance area moves to the high-speed range. The operating speed range of the tractor PTO is extended to the high-speed end. At the same time, increasing the stiffness can improve the vibration resistance of the structure and reduce weld fatigue damage [30]. Based on the above two points, increasing the stiffness can improve the vibration resistance of the harvesting machine.

In engineering practice, increasing the stiffness of a harvester may increase the amount of steel used and improve the quality of the harvester. As shown in Figure 9, increasing the mass while keeping the stiffness constant decreases the natural frequency of the harvester, and the range of tractor PTO output speeds corresponding to the resonance zone shifts towards the lower end of the speed range. The output rotational speed of the tractor PTO that was previously outside the resonance zone now falls within that range, leading to an increase in the equipment’s vibration speed at those speeds. Therefore, when enhancing the stiffness of the harvester, it is essential to strive for a balance that avoids an excessive increase in the machine’s mass. To verify the impact of changes in the harvester’s stiffness on vibration speed, structural reinforcement was applied to strengthen the harvester’s structure (as shown in Figure 10). All other working conditions and structural parameters of the harvester remained unchanged. Experiments were conducted in two scenarios: the indoor soil bin machine’s traction-driven operating state and the tractor-driven harvester’s non-operational idling state. The changes in vibration speed and the resonance zone, as well as the corresponding power take-off (PTO) shaft rotational speeds, were observed.

Comparing the data in row 9 (harvester structural reinforcement stiffness changes, the state of traction-driven operation of indoor soil bin tester) with row 3 (work mode of traction drive of indoor soil bin testing machine before improvement) of Table 3. Row 10 (the tractor drive is in a non-working idling state after the structural stiffness of the harvesting machine is changed) is compared with the data in row 2 (the tractor drive is in a non-operational idling condition before improvement), where the peak frequency of the resonance frequency band has increased. The structural parameters of the harvester remain unchanged after the structural reinforcement, and the increase in mass is only 10 kg, which is only 2% of the total mass of the machine and can be approximated as negligible. The stiffness k is directly proportional to the resonant frequency. The results show that under the two working conditions before and after structural reinforcement, the peak frequency of the resonance frequency band increases, and the values of V_max and V_RMS decrease after the overall stiffness of the harvester is increased, indicating that the increase in stiffness can reduce the vibration speed and improve the vibration resistance without excessive increase in mass.

To verify the changes in the vibration velocity of the harvester after structural enhancement that resulted in increased rigidity, the vibration velocity was measured using an indoor soil bin test machine and a tractor, respectively, while keeping all other experimental conditions the same. As shown in

Table 5. Stiffness increase test.

Harvester Status		Resonance Band Peak Frequency Hz	Vibration Velocity Vmax (mm/s)	Vibration Speed Mean Square Value VRMS (mm/s)
1	The harvester is driven by the tractor, suspended off the ground by the tractor, and the harvester is in a non-working idling state.	4.14	118.85	154.277
2	After the stiffness of the harvester is altered, the harvester is driven by the tractor, suspended off the ground by the tractor, and the harvester is in a non-working idling state.	5.352	91.997	94.607
3	Work mode of the traction drive of the indoor soil bin testing machine before improvement	3.086	97.606	152.187
4	Harvester structural reinforcement stiffness changes; the state of traction-driven operation of indoor soil bin tester	4.945	32.576	50.836

, after the structural reinforcement and the enhancement of rigidity, the peak frequency of the resonance frequency band shifted towards the higher frequency range, while both V_RMS and V_max decreased, which is in accordance with the analysis results mentioned earlier.

4.4. Effect of Transmission Ratio i on the Distribution of the Resonance Zone

In a stable operating state of machinery, the frequency of mechanical vibration is inherently linked to the movement patterns of the components that comprise the machinery [31]. Concurrently, the transmission ratio serves as an indicator of the distinct motion laws governing these components. With other parameters in Equation (17) held constant and the natural frequency of the harvesting machine remaining unchanged, an increase in the transmission ratio I is observed. As shown in Figure 11, the speed value n of the tractor’s power take-off (PTO) that triggers resonance also rises, indicating a direct proportionality between the PTO’s resonant speed value n and the transmission ratio i. When the transmission ratio is increased, the output rotational speed of the tractor PTO that previously resonated at the original ratio no longer does so at the new ratio. As the transmission ratio i increases, the rotational speed range of the power output shaft corresponding to the resonance zone shifts towards the higher rotational speed end. This suggests that, given the inherent frequency of the harvester remains constant by increasing the transmission ratio, the resonance speed n of the tractor PTO can be shifted to a higher rotational speed, thereby avoiding resonance in the harvester. Given that the working condition is stable and the inherent frequency of the harvester remains unchanged, the engine can be operated at a higher speed range by increasing the transmission ratio without triggering resonance, thus giving full play to the power performance of the tractor.

This is particularly beneficial for harvesting machinery operating under conditions of deep digging and high resistance. As can be seen from the comparison of the images in Figure 11 (i3, i4, and i5), when the intrinsic frequency is changed because of the change in the quality of the harvester, the tendency for the intrinsic frequency to decrease can be counteracted by increasing the transmission ratio, thus realizing that, when the intrinsic frequency of the machine is changed, the occurrence of the resonance can be avoided by changing the transmission ratio. Expanding the range of counterweights without causing resonance and reducing the vibration speed of the harvester by using larger counterweights.

To control the vibration of the harvester, it is essential to manage the primary factor. Figure 4 illustrates the changes in the vibration speed of the harvester. Observations reveal that the resonance in the harvester is related to the working frequency of its operating parts. As shown in Figure 11a and expressed in Equation (11), changes in the rotational speed of the harvester’s working parts simultaneously alter the working frequency, the excitation force generated by the movement of these parts, and the frequency of this force. For a harvester operating stably with other status values constant, the rotational speed of the working parts is the primary factor affecting vibration and is the independent variable. The exciting force generated by the movement of the components, the frequency of the exciting force, and the working frequency of the working components are all dependent on the rotational speed.

The data from Table 3, which includes field tests, soil bin tests, and no-load test, demonstrates that the natural frequency of the harvesting machine is not a constant value under different operating conditions. When the harvesting machine is in operation, variations in constraint conditions and overall machine quality can lead to changes in the natural frequency. This will change the specific output rotational speed of the tractor’s PTO, which corresponds to the resonance zone of the harvester and triggers resonance in the harvester. To meet the power demands of the harvesting machine, the engine speed must remain within a specific range. In order to avoid rotational speeds that trigger harvester resonance, the transmission ratio must be adjusted. If the existing transmission ratio does not meet the requirements, the transmission mechanism must be modified to broaden the range of available transmission ratios. This ensures that there is a match between the power source’s rotational speed required for the harvester’s operation and the rotational speed of the working parts that avoids inducing resonance in the harvester, thereby achieving resonance avoidance and vibration reduction. During the course of investigating harvester operations, it was observed that operators retrofitted the tractor’s gearbox and the harvester’s transmission mechanism to modify the transmission ratio, successfully preventing resonance and reducing vibration speeds. The aforementioned analysis elucidates the underlying mechanism of action, providing a method for avoiding resonance when the inherent frequency of the power machine changes.

4.5. The Relationship Between Excitation Force F_c Variation and Vibration Velocity

Reducing the excitation force and reducing the fluctuation amplitude of the excitation force is one of the measures for reducing vibration [32]. When the harvesting machine is in a steady operating state, n, i, M, r, L_BC, and L_CD remain unchanged. From Equation (18), it can be seen that the change in the excitation force F_c causes the vibration speed to change. Matlab was used to solve Equation (17) to make a graphical representation, and the results are shown in Figure 12a,b. The magnitude of the excitation force produced by the reciprocating components of the harvester is directly proportional to the magnitude of the vibration speed induced by this force, while the output rotation speed of the tractor PTO that induces resonance remains constant. When the exciting force generated by the moving parts decreases, the vibration speed value of the harvester decreases across the entire rotational speed range, whether in resonance or non-resonance states. The resonance phenomenon in the harvester is associated with the rotational speed of these reciprocating parts and is independent of the magnitude of the excitation force they generate. As shown in Figure 12b, a smaller excitation force resulting from the movement of the reciprocating parts corresponds to a lower vibration speed. The vibration speed is depicted in Figure 12c. When numerous reciprocating parts are present and multiple excitation forces impact the harvesting machine, even if the excitation force from the movement of a particular reciprocating part is minor, it is crucial to be vigilant about the resonance. This can lead to an increase in vibration throughout the machine.

Figure 12d illustrates that, apart from the area where the resonance is triggered, the value of the vibration speed of this harvester is mainly determined by the magnitude of the excitation force generated by the reciprocating moving parts that trigger the larger excitation force. In order to reduce the vibration speed of this harvester in the non-resonant state, the focus should be on the reciprocating moving parts that generate large excitation forces. Reducing the vibration speed of the harvester can be achieved by reducing the excitation force that causes the harvester to vibrate. As expressed in Equation (11), reducing the excitation force $F_c$ can be achieved by reducing r, L_BC, and m and increasing L_CD. R, L_BC, and L_CD are structural parameters of the harvesting machine, which are limited by the structural relationship of the components of the harvesting machine and the conditions for the movement of soil and medicine plants on the oscillating screen and are not suitable for change. m is the mass of the rod BC, and reducing m can decrease the excitation force and thus reduce the vibration.

Mass is a universal parameter. Reducing mass can reduce vibration speed. In other mechanical structures, a similar phenomenon is observed: the lighter the mass of the moving body, the smaller the vibration. For example, the smaller the mass of the automobile engine piston and the oscillating mass of the crankshaft connecting rod mechanism, the weaker the vibration [33]. The lightweight of the unsprung mass of the automobile shock absorption system and the lightweight of the gears in the gear transmission will affect the dynamic characteristics and operational stability. From the above analysis, it can be observed that when the harvesting machine is in a stable working state, the change in the speed of the reciprocating parts with respect to time, that is, the acceleration, is within a certain range of constant change, while other states and conditions remain unchanged. The mass decreases, the combined force decreases, one of the components of the combined force, which is the reaction force to the excitation force, decreases, the excitation force decreases, and the vibration speed generated by the excitation force decreases. To reduce vibration, while other conditions remain constant and the rigidity, strength, and dynamic balance of the components are maintained, the mass of the moving parts of the harvester should be minimized.

To reduce the vibration speed of the equipment in non-resonance states, the excitation force generated by the moving parts can be decreased. The reduction in the excitation force can also be achieved by applying an external force to the harvesting machine. Springs, as one of the commonly used components for reducing vibration, can be installed on the machine based on the aforementioned principles to determine the specific location and the structural parameters of the springs. As shown in Figure 5, a spring is added at the end of the rod BD, D. Because the movement of the rod BD is a forced movement and is limited by the structure of the linkage, the spring’s rebound movement is limited rather than free to shake. The spring force is F_K, obtained by force analysis.

$${F_{xB}{\mathrm{L}}_{\mathrm{B}\mathrm{C}}+F}_c{\mathrm{L}}_{\mathrm{B}\mathrm{C}}=F_c{\mathrm{L}}_{\mathrm{C}\mathrm{D}}+\left(F_{xD}-F_K\right){\mathrm{L}}_{\mathrm{C}\mathrm{D}}$$

(19)

$$F_K=\mathrm{K}{X}_D$$

(20)

$$X_D={\displaystyle \frac{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}}{X}_B={\displaystyle \frac{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}}rcos\omega t$$

(21)

$${ F}_K=\mathrm{K}{X}_D=\mathrm{K}{\displaystyle \frac{{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}}rcos\omega t$$

(22)

$$F_c={\displaystyle \frac{\left(F_{xD}-F_K\right){\mathrm{L}}_{\mathrm{C}\mathrm{D}}-F_{xB}{\mathrm{L}}_{\mathrm{B}\mathrm{C}}}{{\mathrm{L}}_{\mathrm{B}\mathrm{C}}-{\mathrm{L}}_{\mathrm{C}\mathrm{D}}}}$$

(23)

The reaction force F_C is the excitation force after loading the spring. Comparing Equation (23) with Equation (11) shows that the value of the excitation force decreases. The spring force F_K acts on the rod BD, reducing the excitation force. In addition to selecting the spring parameters according to the manual, the spring natural frequency ft should be avoided from overlapping with the harvesting machine working frequency, resulting in spring resonance. The frequency should be avoided from being 10 times the harvesting machine working vibration frequency, fg [34].

In order to verify the effect of loading the spring, the vibration speed of the harvester in its working state was measured while driving the indoor soil bin testing machine while the soil conditions and working conditions remained constant. The results are shown in

Table 6. Vibration Velocity Measurement Values with Spring Installation and Comprehensive Vibration Reduction Measures.

Harvester Status		Resonance Band Peak Frequency Hz	Vibration Velocity Vmax (mm/s)	Vibration Speed Mean Square Value VRMS (mm/s)
1	Work mode of the traction drive of the indoor soil bin testing machine before improvement	3.086	97.606	152.187
2	The working status of the traction drive of the indoor soil bin testing machine after adding a spring to the harvesting machine	3.086	48.805	107.554
3	The non-working, idling state of the indoor soil bin testing machine without spring	4.219	170.664	268.759
4	Under the traction drive of the indoor soil bin test machine, the improved harvester, equipped with springs and a 100-kg counterweight, undergoes a non-working, idling state test.	3.203	10.814	15.414
5	The improved harvester, equipped with springs and a 100-kg counterweight, undergoes a working state test under the traction drive of the indoor soil bin test machine.	4.375	50.507	75.550

. Compared with the time when no spring was added, after the spring was installed, both V_RMS and V_max decreased, indicating that the components and position of the spring mounted on the harvester are appropriate, and it has the effect of reducing vibration.

From the above analysis, adding counterweight to the harvester, strengthening the structure to increase rigidity, and installing springs can all contribute to reducing the vibration speed of the harvester. To verify the combined effect of the above measures, 100 kg of counterweight was added, the structure was strengthened, and springs were installed. Under the condition that other conditions remained unchanged, the harvester was driven by an indoor soil trough test machine. The vibration speed of the machine was measured in the working state and idling in non-working state. The results are shown in Table 6.

As shown in Figure 13a,b (Table 6, rows 1 and 5, data),The above measures reduce the vibration speed of the harvesting machine’s V_max and V_RMS in work mode to half of the value in the non-improvement state. The natural frequency of the harvesting machine has increased. To prevent resonance, the upper limit of the permissible working speed range of the PTO is extended from 250 rpm to the section of 300 rpm to 400 rpm. The test results show that the above measures play a role in reducing the vibration speed of the harvesting machine and reducing its vibration damage to the machine.

5. Discussion

To prevent mechanical resonance, through finite element analysis and simulation, by modifying and optimizing the structure during shutdown [35] or by adjusting the system’s mass and stiffness in a non-operational state [36], the natural frequency of the system can be altered to prevent it from being equal to the frequency or multiples of the excitation force frequency, thus avoiding resonance. Once the adjustments are made, the system’s natural frequency does not change when it is operational. Alternatively, if the natural frequency of the system remains unchanged, the frequency of the external excitation force can be modified to prevent resonance [37]. However, this paper’s experimental results indicate that the natural frequency of the small rhizome Chinese herbs harvesting machine is not stable during operation and is variable. However, unlike the results of previous studies, the experimental results in this paper show that the inherent frequency of the small rhizome Chinese herbs harvesting machine is not stable during operation and is variable. The results of this paper show that resonance can be avoided by changing the transmission ratio under the prerequisite of ensuring the operational dynamics requirements when the inherent frequency of the implement is changed during the operation process. The natural frequency is influenced by the mass of the machine [38]. Under working conditions, the total mass of the harvester is not constant, consisting of the empty vehicle mass, the mass of harvested soil, and the mass of harvested crops. During operation, the harvester’s vibrations are affected by the damping properties of the soil, causing its natural frequency to fluctuate. Field test results and soil bin experiments confirm that the natural frequency is not fixed during operation. To avoid resonance, the physical parameters of the structure are often adjusted to alter its dynamic characteristics, thus shifting the natural frequency away from the range of external forces [39]. However, in practical working conditions, it is not feasible to rapidly modify the structural parameters to avoid resonance-inducing frequency ranges. When the natural frequency of the harvester changes, the corresponding rotational speed values also shift. To prevent resonance, these speed values should be avoided.

As outlined in the analysis in Section 4.4, to maintain the tractor’s power performance, the engine speed must stay within a specific range. By adjusting the transmission ratios, the rotational speed of the harvester’s moving parts can be regulated to avoid resonance. Chinese herbs have a well-developed underground root system and high digging resistance, requiring the tractor engine speed to remain high in order to provide the necessary power for the task. If the tractor’s engine speed is constant, the transmission ratios of both the harvester and tractor should be adjusted based on operating conditions. The greater the adjustments and the broader the range, the more effective the performance. This allows the harvester’s working parts to operate over a wide range of rotational speeds, preventing resonance and reducing vibration. It ensures the tractor’s power output, mechanical operating speed, and unit performance are aligned with the task requirements.

As shown in Figure 12d, under stable working conditions, even if the natural frequency of the harvester remains constant, various moving parts generate their own excitation forces, and the rotational speeds at which resonance occurs for each part differ. To prevent triggering resonance, the range of transmission ratios should be broad, allowing for a larger selection of transmission ratios to avoid the resonant speeds associated with each moving part. With a fixed range of transmission ratios, more available options lead to greater mechanical stability. However, an increase in transmission ratio options also raises structural complexity, which can reduce reliability. Therefore, it is essential to strike a balance between these factors based on specific operational needs and design requirements.

When selecting a tractor for a small rhizome Chinese herbal harvester, it is important that the tractor’s power meets the necessary requirements. To expand the tractor’s transmission ratio, you may choose a high-horsepower continuously variable transmission tractor or one with more gears. If the transmission ratio of the existing tractor and implement does not meet these needs, the issue can be resolved by modifying the transmission or selecting a tractor with additional gear options. Replacing the tractor with one that has more gears is more expensive, while modifying the existing tractor is more cost-effective. However, modified tractors may experience issues during operations other than herb harvesting.

To expand the range of transmission ratios, using gears or sprockets in the harvester’s transmission system often involves large gears or multi-stage transmissions. Adjusting the transmission ratio by replacing components can be inconvenient, and such mechanisms typically have poor vibration resistance and high costs.

For a broader transmission ratio range, easier adjustment, and overload protection during high-resistance operations, belt transmission or hydraulic transmission can be utilized. Hydraulic transmission, employing a hydraulic motor, throttle valve, and relief valve, provides multiple transmission ratios and overload protection. This system overcomes the limitations of traditional transmission ratios, allowing implements to be compatible with various tractor models to meet power performance and operational needs.

However, due to cost and reliability considerations, belt drives can also be employed. By replacing pulleys of different diameters, belt drives offer adjustable ratios and additionally provide buffering and vibration absorption [40].

To enhance resistance to structural vibrations and reduce welding fatigue, increasing the rigidity of the harvester is an effective strategy. By boosting rigidity, the harvester’s resistance to vibrations can be improved, weld fatigue damage can be minimized, and its natural frequency can be increased. However, it is crucial to balance changes in stiffness with the impact on mass. Reinforcing the structure to increase rigidity may require additional steel, which could add to the harvester’s mass and potentially lower its natural frequency. This reduction in natural frequency could lead to resonance when the harvester operates, causing a decrease in the rotational speed of the tractor’s power take-off (PTO) that induces resonance. As a result, the harvester’s vibration speed could increase, potentially leading to more severe vibration damage and diminishing the benefits gained from increased rigidity.

When evaluating the materials used in the harvester, it is important to consider the vehicle’s lightweight practices. Utilizing high-strength steel can enhance the structural stiffness of the harvester while maintaining or even reducing its overall mass. This approach increases the harvester’s natural frequency, which in turn raises the corresponding output rotational speed of the tractor’s PTO. As a result, the operational range of the tractor’s PTO output speed is expanded, improving operational efficiency and helping to prevent weld cracking.

To reduce the vibration velocity in the non-resonant state of a harvester, it is important not only to avoid resonance but also to manage the vibration speed outside the resonant zone. A harvester cannot operate in the resonance zone for extended periods, and minimizing the vibration speed in the non-resonant state is crucial. By reducing the vibration speed, we ensure that bolts and other fasteners remain securely connected and extend the lifespan of the bearings, thereby enhancing the overall reliability of the harvesting machine.

Reducing the vibration speed outside the resonance section can be achieved by increasing the harvester’s mass. Adding a counterweight acts as a non-movable passive mass damper, raising the harvester’s overall mass and lowering its intrinsic frequency. This adjustment helps reduce vibration speed across the full speed range of the tractor’s power output shaft, including the resonance region. However, this increase in mass should be carefully managed, as it does not change the harvester’s stiffness, but it does shift the resonance band to a lower frequency range (lower speed range).

To avoid premature resonance due to increased mass, the mass addition should be limited. Additionally, a counterweight that is too large can lead to higher energy consumption. Therefore, when aiming to reduce the vibration speed of the harvesting machine, it is important to balance the added mass carefully; more mass is not always better.

When adding ballast weight, attention should be paid to the method and location of loading. The added weight should not create new sources of vibration or introduce new excitation forces. For convenience, adjustability, and cost effectiveness, it is best to use bags filled with soil as counterweights. These should be placed on a stable frame of the harvester that can support the load without shifting relative to the harvester.

Based on the analysis above, it can be concluded that to reduce the vibration speed of the harvester across its entire speed range—including both resonance and non-resonance states—the key is to minimize the excitation force generated by the moving parts. This can be accomplished by adjusting the relevant structural parameters of the machine. However, modifying any structural parameters other than the mass of the reciprocating parts can influence their movement, thereby affecting the machine’s overall performance. Therefore, any changes to the structural parameters should be approached with caution.

Unlike previous studies, the above research shows that changes in the structural parameters of moving parts lead to changes in excitation forces, which in turn affect the overall vibration of the machinery. It also provides an explanation based on changes in excitation forces for the phenomenon that smaller inertial quantities of mechanical structures result in lighter vibrations. By studying the relationship between changes in the structural parameters of moving parts and the magnitude of excitation forces, it is possible to determine reasonable values for these parameters to reduce vibration.

When the harvester is in operation, both increasing the overall mass within a certain range and decreasing the mass of the reciprocating components can help reduce vibration. However, the optimal balance between these two adjustments is not necessarily at the extremes of highest or lowest values. The reduction in reciprocating component mass is limited by the strength requirements and cost considerations of the working parts, imposing a practical lower limit. Moreover, increasing the overall mass while the machine is operating is constrained by the risk of earlier onset of resonance and higher energy consumption. Similar to how the ratio of unsprung to sprung mass is optimized in a vehicle, the mass ratio in the harvester should be carefully balanced.

Apply an external force to counteract the excitation force. This external force should vary with the same period as the excitation force but act in the opposite direction. Focus on the moving parts of the harvesting machine that generate the highest excitation force. Currently, the use of springs to reduce vibration is more about altering the path of vibration transmission and the dynamic response characteristics of the system rather than merely reducing the magnitude of the exciting force. If using springs to decrease the exciting force, it is essential to ensure the correct configuration of the target component, installation position, and related parameters.

6. Conclusions

(1)

Before the improvement, the vibration speed of the harvester was high in both the resonant and non-resonant states. The machine was sometimes in resonance in the working state, which was the main reason for damage and poor working reliability. In the operating mode, the harvester intrinsic period is a variable value.

(2)

The PTO rotational speed affecting resonance in the harvester is proportional to its transmission ratio. In the case of changes in the inherent frequency of the harvester, resonance can be avoided by a reasonable choice of transmission ratios. To reduce vibration and prevent resonance, the PTO output speed should be controlled. Reducing the mass of reciprocating moving parts reduces the speed of vibration. Reducing the vibration speed of the harvester in the non-resonant state can be achieved by increasing the mass of the machine. Except for the resonance area, the vibration speed of the harvesting machine is inversely proportional to the total mass of the harvesting machine. Increasing stiffness enhances vibration resistance. The excitation force from reciprocating parts is proportional to the vibration velocity. Modification of harvester structural parameters can change this force.

(3)

Increase the weight of the machine under operating conditions by loading sandbags onto a frame that has no relative motion to the harvester and can bear weight stably. The moving parts on the harvester reduce the mass of the moving parts while ensuring the stiffness and strength of the parts and their dynamic equilibrium; expand the range of choices of ratios for the harvester under operating conditions; and all of the above engineering measures reduce the vibration speed of the harvester and help to prevent the occurrence of resonance.

The findings of this research offer an initial insight into how to mitigate mechanical vibrations and improve the operational reliability of herbal harvesting equipment. These results provide valuable guidance for future design enhancements and engineering approaches aimed at reducing vibrations in similar agricultural machinery. However, it is important to acknowledge that this study was confined to a single machine type and did not account for the full spectrum of environmental variables, which may limit the generalizability of the findings. Furthermore, the precision of the measuring instruments and variations in operational conditions could have impacted the results. To achieve a comprehensive understanding and practical application, future research should examine the interaction of various factors and their collective influence on the vibration of agricultural machinery.