Vibration Characteristics Analysis of the Header Assembly of Combine Harvester Under Multi-Source Coupled Excitation

Abstract

The vibration of the combine harvester header assembly directly affects harvesting efficiency and operational quality. To address the insufficient dynamic characterization of the cantilever conveying trough under complex field excitations, this study systematically analyzes the vibration response characteristics of the header assembly under multi-source coupled excitation through field experiments and theoretical modeling. Acceleration sensors arranged at three measurement points on the header bottom collected vibration data, revealing that the dominant vibration frequency of the header has a deterministic harmonic relationship with the threshing drum’s operating frequency (3rd harmonic on the left side, 1.5th harmonic on the right side), demonstrating dynamic coupling effects within the integrated system. Through acceleration response analysis at four symmetric measurement points on the connection, the external excitation force was quantified as a sinusoidal function correlated to the feed quantity (F = 1094.4 sin(50πt/3)). A damped pendulum model of the cantilever conveying trough was established using the Lagrange method. Validation results show that the error between the predicted steady-state swing amplitude and measured values is only 1.11–4.3%, confirming the effectiveness of this simplified model in characterizing the system’s steady-state response. This research provides a theoretical foundation and methodological support for dynamic characterization, parameter optimization, and stability control of the cantilever header system in combine harvesters.

1. Introduction

As a core piece of equipment in modern large-scale agricultural production, the combine harvester’s operational efficiency and stability is directly related to the profitability and safety of grain production [1,2,3,4]. The header assembly, as the front-end key component of the combine harvester, performs critical functions such as crop cutting, conveying, and feeding [5,6,7,8]. Its performance significantly influences the machine’s harvesting quality and overall efficiency. However, the working environment of a combine harvester is complex and highly variable [9,10,11]. Factors such as uneven field terrain, heterogeneous crop density, and sudden fluctuations in feed rate and quantity subject the header assembly to complex and time-varying external excitations, which can induce severe vibrations [12,13,14]. These vibrations not only reduce operational comfort and accelerate structural fatigue but may also lead to inaccurate height control of the header and clogging during crop transport, thereby constraining improvements in the harvester’s working efficiency and reliability [15,16,17]. Hence, an in-depth investigation into the vibration response and stability mechanism of the header assembly under complex excitations is of considerable theoretical and practical value for enhancing the overall performance of combine harvesters [18,19,20,21].

In terms of dynamic research on the header assembly, scholars worldwide have carried out a series of exploratory studies [22,23,24,25]. Chen et al. [26] established a dynamic model for the vibration problem of the rice combine harvester frame under multi-source excitation, verified its validity through experiments, found that the constrained modal frequencies of the complete frame are directly related to the independent modal frequencies of each sub-frame and there is a risk of resonance. Taking the 4LZ-5.0E combine harvester as the research object, Gao et al. [27] identified the main vibration sources in each direction and the influence of feeding quantity on vibration through multi-speed and multi-measuring point vibration tests combined with spectrum analysis, put forward targeted vibration isolation and feeding quantity control suggestions. To address the drawbacks of traditional structural stress detection methods, Wang et al. [28] proposed a deep learning framework for total stress detection of steel components, providing a new approach for obtaining the total stress that is difficult to measure with traditional sensors. Moreover, the cantilever conveying trough of a combine harvester shares structural similarities with other cantilever systems in engineering machinery, allowing for methodological cross-reference. Zong et al. [29] established a dynamic model of the attitude angles of the boom-type road header based on the Lagrange equation, combined cutting load calculation with Simulink simulation to analyze the attitude response characteristics of the road header in a 30° steep coal seam, verified the model’s validity. Zhao et al. [30] investigated the static, nonlinear, and modal characteristics of an all-terrain crane boom under actual loading conditions. Popescu [31] incorporated a Rayleigh damping model to analyze vibrations in a bucket-wheel excavator boom. Wei et al. [32] performed a static analysis of a rocker arm using ANSYS2022R2 Workbench, determining stress and deformation to validate structural reliability. Liu et al. [33] studied the dynamic stability of a telescopic crane boom under periodic loading, deriving a critical frequency equation for dynamic instability based on Hamilton’s principle and further analyzing the influence of damping on stability. Although the studies provide a rich methodological reference for analyzing the dynamic characteristics of cantilever structures in engineering, there is a notable scarcity of in-depth research concerning the cantilever conveying trough itself, particularly studies involving dynamic modeling and stability analysis under complex field random excitations [34,35,36,37,38].

The dynamic behavior of rotating machinery is often influenced by multiple external forces such as unbalanced forces and bearing forces [39,40,41]. For the conveying trough of a combine harvester, the excitations during operation are complex and stochastic [42,43,44]. Accurately quantifying these external excitations is a prerequisite for constructing a precise dynamic model. For a classical pendulum system with irrational and fractional nonlinear restoring force, Han et al. [45] proposed a simplified approximate system to describe the local dynamic characteristics of its small-angle vibration, studied the periodic solutions by combining analytical analysis with the averaging method and numerical simulation. Pavlaková et al. [46] provided sufficient conditions for the solvability of the impulsive Dirichlet boundary value problem of forced nonlinear differential equations involving the combination of viscous and dry frictions. Bot et al. [47] addressed the radial vibration of a rolling cylinder under time-varying point force, equating it to an orthotropic pre-stressed plate on a viscoelastic foundation, derived a closed-form analytical solution, and analyzed the effect of rolling speed on vibrational energy merging. Rubio et al. [48] and Liang et al. [49] studied the analytical motion of a pendulum on a surface of constant curvature without external forces and the existence of stable periodic solutions for a forced pendulum, respectively. While these studies have made notable progress in the theoretical analysis and numerical computation of pendulum systems, the majority focus on idealized periodic excitations or force-free conditions [50,51,52]. In contrast, the excitations acting on the conveying trough of a combine harvester during actual operation are non-ideal, multi-source, and stochastic. Presently, there is still a lack of comprehensive research concerning the dynamic response, parameter identification, and stability evaluation of the conveying trough system under these realistic working conditions. Furthermore, the integration of field-test data for the purpose of model validation and refinement remains insufficient [53,54,55].

This study aims to systematically conduct vibration characteristic testing and dynamic modeling of the combine harvester header assembly. Emphasis is placed on field experiments to quantitatively analyze the vibration response characteristics of the header, conveying trough, and hydraulic cylinder under different working conditions, accurately identify the dynamic parameters of key components, and establish a damped pendulum dynamic model of the cantilever conveying trough using the Lagrange method. By comparing theoretical predictions with field-measured data, the accuracy and effectiveness of the proposed model are validated. This research is expected to provide solid theoretical support and practical guidance for the dynamic design, parameter optimization, and vibration control of the cantilever header system in combine harvesters.

2. Materials and Methods

2.1. Experimental Setup and Instruments

This study takes the header assembly as the research object, which comprises a reel, header, conveying trough, hydraulic cylinder, and other components. The structure is shown in Figure 1. As a critical component of the combine harvester, the header assembly is primarily responsible for cutting crops and transporting them to the threshing drum. To investigate the external excitation sources acting on the header assembly, vibration signals from its components were analyzed.

The vibration signal acquisition system used in the experiment was developed by Jiangsu Donghua Testing Technology Co., Ltd. (Taizhou, China), including dedicated vibration testing software and the matching DH5902 dynamic signal acquisition and analyzer, as shown in Figure 2. The acquisition system consists of a computer for data analysis, one dynamic signal acquisition instrument, one impact hammer, and several connecting cables. The acquisition unit provides 38 signal channels in total, including two tachometer channels and 36 vibration measurement channels, with a sampling frequency of 1 kHz. The sensors employed were triaxial accelerometers (model: 1A312E). Detailed analysis of the vibration signals acquired from three orthogonal directions enabled the characterization of the vibration properties and acceleration response of the conveying trough under external excitation. The accelerometer specifications are as follows: a range of ±500 g, a resolution of 0.001 g, and a sensitivity of 1 mV/(m/s²). The three sensing axes are mutually perpendicular. Since the experimental objective was to capture the acceleration response of the conveying trough in the direction perpendicular to it due to external excitation, the sensor’s Z-axis was aligned perpendicular to the conveying trough, rather than adopting the conventional horizontal, vertical, and upward orientations.

2.2. Output Response Test of the Header at Different Positions

The field experiment for measuring the acceleration response of the combine harvester header was conducted on farmland in Xinfeng Town, Dantu District, Zhenjiang City, Jiangsu Province. Three sensors were uniformly distributed along the bottom of the header, with all measurement points positioned on the same horizontal line, which facilitates the comparison of vibration distribution characteristics along the header width and the analysis of potential lateral vibration modes.

Using the forward direction of the combine harvester as the positive reference, vibration measurements were primarily taken along three axes: the forward direction, the horizontal direction, and the vertical direction. The specific locations of the measurement points are shown in Figure 3, while the corresponding acquisition channels for each sensor direction and the measured vibration directions are summarized in

Table 1. Vibration measurement direction of sensor.

Location	Axis	Channel	Direction of Vibration
Left side	X	1	Horizontal direction
	Y	2	Working direction
	Z	3	Vertical direction
Middle	X	4	Horizontal direction
	Y	5	Working direction
	Z	6	Vertical direction
Right side	X	7	Horizontal direction
	Y	8	Working direction
	Z	9	Vertical direction

. During the harvesting process, the travel speed of the harvester was gradually increased. Vibration signals from the header were sampled continuously in real-time over a period of approximately 3 min. For analysis, signal segments that were relatively stable were selected.

2.3. Output Response Test of the Conveying Trough at Different Positions

During field operation, the combine harvester experiences complex working conditions, and the unreasonable length of the conveying trough causes the header assembly to not only undergo rotational motion but also induce vertical oscillations, thereby reducing harvesting efficiency. Therefore, it is necessary to analyze the structural stability of the cantilever conveying trough under external disturbances. Due to the structural characteristics of the connection between the header and the conveying trough, placing conventional pressure sensors at this interface to directly measure the excitation force acting on the conveying trough system is inaccurate. Moreover, the rigid connection between the header and the conveying trough, combined with the randomness of squeezing during operation, makes it unreliable to estimate the pressure value using strain gauges based on deformation at the connection. Thus, this study employs Newton’s second law to determine the acceleration at the header–conveying trough connection point and the mass of the header itself to derive the external excitation force acting on the conveying trough.

Acceleration responses at the junction between the header and the conveying trough were collected during the operation of a rice combine harvester under different field conditions. After filtering, the vibration characteristics of the conveying trough during operation were analyzed. The mass parameters of the header and the conveying trough were 330 kg and 120 kg, respectively. Four symmetric measurement points were arranged at the connection between the header and the conveying trough. This connection is a critical interface for force and motion transmission, and a potential structural weak point and vibration amplification area. As shown in Figure 4, these points are used to identify and quantify the external excitation input to the cantilever conveying trough system.

To account for the influence of feed quantity on the acceleration at the header–conveying trough connection, the combine harvester was operated at a constant speed under two conditions: half-cutting width and full-cutting width, with the engine speed maintained at the rated level. Each test covered 35 m, and a 5 m section of crops was reserved before the working area to ensure a stable feed quantity during the tests. Vibration signals were recorded over a 25 s period within the test section using the acquisition instrument, as illustrated in Figure 5.

2.4. Output Response Test of the Hydraulic Cylinder at Different Positions

The hydraulic cylinder within the combine harvester’s header assembly plays a critical role in supporting both the header and the conveying trough. It dynamically adjusts the cutting height of the cantilever header in response to crop height variations. Therefore, analyzing the influence of the hydraulic cylinder on the overall swing balance of the cantilever header is significant. Key parameters of the harvester’s hydraulic cylinder, including the cylinder barrel length, piston rod length, and working stroke, were obtained through measurement. The specific parameters are as follows: the piston rod length is 450 mm, the cylinder barrel length is 700 mm, the piston rod radius is 20 mm, the cylinder radius is 32.5 mm, the piston length is 50 mm, and the working stroke is 300 mm. Additionally, the inner cavity length is 450 mm with a radius of 25 mm, the Max. rod-side length is 450 mm, and the Max. cap-side length is 150 mm.

The DH5902 signal acquisition instrument was connected to the processing computer via cables. For this test, two accelerometers were utilized, with their respective channels connected to the acquisition instrument. The two accelerometers were mounted at the front and rear ends of the hydraulic cylinder barrel, respectively, as shown in Figure 6.

2.5. Damped Pendulum Model of the Conveying Trough Based on Lagrange Method

The header assembly of the combine harvester features a complex structure with unclear force origins. To analyze the factors causing significant swing of the cantilever conveying trough during operation, it is necessary to establish the dynamic differential equations of the cantilever header under external excitation during harvesting. During field operation, the working height of the header is frequently adjusted by varying the extension/retraction length of the hydraulic cylinder. The mechanical properties of the hydraulic cylinder, such as its stiffness and damping, significantly influence the swinging behavior of the cantilever header. Consequently, the simplified damped model of the conveying trough is illustrated in Figure 7.

The derivation of the system’s dynamic differential equations begins with the Lagrange method. For a system with n degrees of freedom, the Lagrange equation can be expressed as shown in Equation (1). Where T is the total kinetic energy of the system, V is the total potential energy of the system, q_j is the generalized coordinates, $\dot{q_j}$ is the generalized velocities and $Q_j^{\left(n\right)}$ is the non-conservative generalized force corresponding to the generalized coordinate, which may include dissipative (damping) forces or other external forces not derivable from a potential function.

This formulation provides the fundamental basis for analyzing the dynamic behavior of the cantilever conveying trough system under operational excitations, incorporating the effects of damping and hydraulic cylinder characteristics.

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\displaystyle \frac{\partial T}{\partial \dot{q_j}}}\right)-{\displaystyle \frac{\partial T}{\partial q_j}}+{\displaystyle \frac{\partial V}{\partial q_j}}=Q_j^{\left(n\right)}, j=1, 2, 3, \dots , n$$

(1)

3. Results and Discussion

3.1. Header Output Response Test Results and Analysis

The time-domain acceleration responses from all channels are shown in Figure 8. The waveforms from all three measurement points exhibit an increasing trend in excitation amplitude. This trend is most pronounced at the left-side measurement point, while the middle and right-side points show a more gradual increase. The stable difference in acceleration across the three directions at all three points is approximately 13 m/s². Noticeable interference fluctuations are observed at the middle and right-side measurement points, which may be attributed to contact between the sensors mounted on the bottom of the header and the rice stubble during operation. Furthermore, natural undulations of the field terrain may cause overall machine tilting or swaying, inducing additional vibrations at the middle and right-side measurement points; simultaneously, uneven crop density distribution leads to real-time load variations on the header, and such asymmetric feeding impacts can generate fluctuating interference during structural transmission.

To further analyze the factors disturbing the vibration response of the header during operation, the acceleration signals from the three sensors for the same direction were subjected to Fourier transform to obtain the frequency domain responses, as shown in Figure 9. The dominant frequencies vary with sensor location. The vibration response frequencies for the left and middle sections of the header are primarily within the 45–65 Hz. The operational speed of the threshing drum is typically between 600 and 800 r/min, corresponding to a fundamental frequency of approximately 15 Hz. The observed response frequencies in these areas coincide with the 3rd harmonic of the drum’s fundamental frequency. Vibration is also significant on the right side, with peak frequencies in all three directions appearing at around 22.5 Hz, which aligns with the 1.5th harmonic of the drum’s fundamental frequency. Vibration signals are present across the entire frequency spectrum at all three points. Therefore, the vibration characteristics of the header components indicate that the header is excited not only by the crop material but also by other components of the combine harvester. Similarly, it can be inferred that the cantilever conveying trough is also subject to mutual influences from various components and external excitations during operation.

When the combine harvester operates in the field, the conveying trough is relatively stationary to the main machine body. Therefore, its horizontal displacement and velocity are zero, and its vertical displacement and velocity are also effectively zero. Let F represent the external excitation force on the conveying trough, including the weight of the header. Let c and k represent the damping and stiffness of the connecting elements, respectively, l the length of the conveying trough, d the distance from the pivot axis to the hydraulic cylinder connection point, φ the pendulum angle, φ₁ the initial pendulum angle, m the conveyor chute mass and J the moment of inertia of the rigid body about the pivot axis. Introducing the x and y coordinates, where x_s and y_s are the horizontal and vertical displacements of the conveying trough, respectively, the displacement components can be expressed as Equation (2).

$$\left\{\begin{array}{c}{x}_s={\displaystyle \frac{1}{2}}l\left[\sin \left(\phi +{\phi }_1\right)-\sin {\phi }_1\right]\\ y_s={\displaystyle \frac{1}{2}}l\left[\cos {\phi }_1-\cos \left(\phi +{\phi }_1\right)\right]\end{array}\right.$$

(2)

Differentiating Equation (2) with respect to time yields the velocity components of the conveying trough, as shown in Equation (3), where is ẋ_s the horizontal velocity and ẏ_s is the vertical velocity.

$$\left\{\begin{array}{c}\dot{x_s}={\displaystyle \frac{1}{2}}\dot{\phi }\cos \left(\phi +{\phi }_1\right)\\ \dot{y_s}={\displaystyle \frac{1}{2}}\dot{\phi }\sin \left(\phi +{\phi }_1\right)\end{array}\right.$$

(3)

When the angular displacement variation is φ, force analysis of the header assembly shows that the kinetic energy T of the conveying trough is given by Equation (4), where the expression for J is given by Equation (5).

$$T={\displaystyle \frac{1}{2}}m\left({\dot{x_s}}^2+{\dot{y_s}}^2\right)+{\displaystyle \frac{1}{2}}J{\dot{\phi }}^2$$

(4)

$$J={\displaystyle \frac{1}{3}}m{l}^2$$

(5)

Substituting Equations (3) and (5) into Equation (4) yields the kinetic energy expression for the conveying trough, Equation (6).

$$T={\displaystyle \frac{7}{24}}m{l}^2{\dot{\phi }}^2$$

(6)

The potential energy of the combine harvester’s conveying trough includes its own gravitational potential energy and the elastic potential energy of the hydraulic cylinder. The swinging motion during operation causes continuous changes in these potential energies. Selecting the lowest point of the cutting height during operation as the zero potential energy reference, the gravitational potential energy of the conveying trough system is given by Equation (7). The elastic potential energy of the system is given by Equation (8), where x_y is the horizontal displacement of the hydraulic cylinder, expressed by Equation (9).

$$V_1={\displaystyle \frac{1}{2}}mgl\left[\cos {\phi }_1-\cos \left(\phi +{\phi }_1\right)\right]$$

(7)

$$V_2={\displaystyle \frac{1}{2}}k{x_y}^2$$

(8)

$$x_y=d\left[\sin \left(\phi +{\phi }_1\right)-\sin {\phi }_1\right]$$

(9)

Summing Equations (7) and (8) gives the total potential energy of the conveying trough system, Equation (10).

$$V=V_1+V_2={\displaystyle \frac{1}{2}}k{d}^2{\left[\sin \left(\phi +{\phi }_1\right)-\sin {\phi }_1\right]}^2+{\displaystyle \frac{1}{2}}mgl\left[\cos {\phi }_1-\cos \left(\phi +{\phi }_1\right)\right]$$

(10)

According to the Lagrange formulation, it is necessary to compute the partial derivatives of the kinetic energy with respect to the generalized angular velocity and angular displacement, and the partial derivative of the potential energy with respect to the generalized angular displacement. The resulting system of dynamic differential equations for the cantilever conveying trough system derived via Lagrange method are presented as Equation (11) in the φ coordinate. The parameters F, c, and k in this system of differential equations need to be determined.

$$\begin{array}{l}{\displaystyle \frac{7}{12}}m{l}^2\ddot{\phi }+{\displaystyle \frac{1}{2}}k{d}^2\left[2\sin \left(\phi +{\phi }_1\right)\cos \left(\phi +{\phi }_1\right)-2\cos \left(\phi +{\phi }_1\right)\sin {\phi }_1\right]+{\displaystyle \frac{1}{2}}mgl\sin \left(\phi +{\phi }_1\right)\\ =Fl-c{d}^2\dot{\phi }{\left[\cos \left(\phi +{\phi }_1\right)\right]}^2\end{array}$$

(11)

3.2. Conveying Trough Output Response Test Results and Analysis

Preliminary analysis of the acquired signals revealed sudden transient acceleration peaks and troughs occurring simultaneously during the test, likely caused by external impacts on the combine harvester during operation. To better reflect the actual force conditions on the cantilever conveying trough, a relatively stable signal segment (15 s to 50 s) was selected for analysis. The signal processing was performed in MATLAB2022b, with the following steps: The raw acceleration signals were first detrended to remove DC offsets or baseline drift. A 4th-order Butterworth low-pass filter with a cutoff frequency of 30 Hz was applied, using the filtfilt function for zero-phase filtering to eliminate transient spikes while preserving key vibration characteristics. The valid signal in the figures, representing a processed segment rather than individual raw data points, better reflects the overall trend. The original signal and the acceleration response signals of the four channels are shown in Figure 10.

The stable peak values of the valid signals from sensors 1 and 2, located above the connection, were approximately 3.34 m/s² and 3.91 m/s², respectively, with stable trough values of approximately −4.43 m/s² and −4.17 m/s². For sensors 3 and 4, located below the connection, the stable peak values were approximately 3.58 m/s² and 5.56 m/s², with stable trough values of approximately −5.23 m/s² and −5.71 m/s², respectively. The absolute values of the stable peak and trough values from the lower sensors 3 and 4 were consistently higher than those from the upper sensors 1 and 2.

Gradually increasing the feed quantity operated at a constant speed in the field under the full-cutting width working condition, while the sensor arrangement and signal acquisition channel settings were identical to those in the previous test. Additionally, after processing the collected abrupt acceleration signals, the extracted valid signals of acceleration responses in the Z-axis direction from the four sensors are shown in Figure 11.

The stable peak values of the valid signals from sensors 1 and 2, located above the header-conveying trough connection, were approximately 5.25 m/s² and 5.95 m/s², respectively, with stable trough values of approximately −5.53 m/s² and −5.07 m/s². For sensors 3 and 4, located below the connection, the stable peak values were approximately 6.01 m/s² and 9.15 m/s², with stable trough values of approximately −7.78 m/s² and −4.29 m/s², respectively. The absolute acceleration values measured by the lower sensors were consistently higher than those from the upper sensors. Furthermore, under the full-cutting width condition, the absolute values of the stable measurements increased.

Acceleration response signals at the connection under two working conditions were collected to account for the influence of feed quantity. After filtering the transient signals, the excitation force range at the connection was determined through data fitting, with a summary analysis provided in

Table 2. Acceleration response under different cutting widths.

Working Condition		Sensor 1	Sensor 2	Sensor 3	Sensor 4
Half-cut width	Peak (m/s2)	3.34	3.91	3.58	5.56
	Valley (m/s2)	−4.43	−4.17	−5.23	−5.71
Full-cut width	Peak (m/s2)	5.25	5.95	6.01	9.15
	Valley (m/s2)	−5.53	−5.07	−7.78	−4.29

.

The acceleration response signals acquired by each sensor under the two working conditions were extracted and presented in the form of histograms, as shown in Figure 12.

Analysis of the acceleration response signals and their distribution at the header-conveying trough connection under different working conditions showed minimal variation between the positive and negative peak values recorded by sensors at the same horizontal level, indicating consistent vibration response horizontally. The absolute peak accelerations measured by sensors 3 and 4 (below the connection) were generally greater than those from sensors 1 and 2 (above), suggesting more significant vibration at the lower position. Therefore, data from the lower region should be prioritized when determining the acceleration value range. Additionally, the absolute peak values of the acceleration response generally increased with higher feed quantities, indicating strongest external excitation on the conveying trough during full-cutting width operation. Consequently, the acceleration range should primarily be based on the full-cutting width condition.

To simplify the analysis, the external excitation on the conveying trough was approximated as a sinusoidal function. Its amplitude was determined jointly from the positive and negative peaks of the acceleration signals measured by all four sensors under the full-cutting width condition. Stable segments from the signals of all four sensors under full-cutting width were selected, and two different time intervals were extracted, as shown in Figure 13.

Within the 24–26 s interval, the time between adjacent peaks was approximately 0.121 s, while in the 38–40 s interval, it was about 0.119 s. Thus, the acceleration signal generated at the header-conveying trough connection under external force has a period of approximately 0.12 s. Combining the component mass and the acceleration amplitude range (the signal from sensor 4 under full-cutting width), the expression for the external excitation force on the combine harvester’s conveying trough was derived as Equation (12).

$$F=1094.4\sin {\displaystyle \frac{50}{3}}\pi t$$

(12)

The acceleration signals from the four sensors under both full and half-cutting width conditions were double integrated to obtain the amplitude response. Further analysis using Fast Fourier Transform (FFT) characterized the frequency composition and energy distribution, as shown in Figure 14 and Figure 15.

The amplitude response results demonstrated that amplitudes at all measurement points were larger under the full-cutting width condition than under the half-cutting width condition. Specifically, the amplitude at Sensor 1 (upper left) was 0.079 mm under the full-cutting width condition versus 0.073 mm under the half-cutting width condition; the amplitude at Sensor 2 (upper right) was 0.076 mm versus 0.070 mm; the amplitude at Sensor 3 (lower left) was 0.100 mm versus 0.080 mm; and the amplitude at Sensor 4 (lower right) was 0.099 mm versus 0.075 mm. This validates that an increased feed quantity intensifies vibration at the connection.

Frequency analysis revealed that vibration energy was primarily concentrated within the low-frequency range of 50–100 Hz. The dominant vibration frequencies for sensors 1 and 2 were approximately 72.226 Hz and 73.242 Hz, respectively, indicating proximity. In contrast, sensors 3 and 4 both showed a dominant frequency of 56.641 Hz, demonstrating frequency consistency for sensors at the same height. The energy peaks in the frequency spectrum further validated the dominance of low-frequency vibrations.

To further analyze the vibration energy distribution, power spectrum is performed on the data from sensor 4 under the full-cutting width condition, as shown in Figure 16. The results showed energy concentrated within the 65–70 Hz frequency band, consistent with the prior frequency analysis. The 3D energy waterfall diagram of the amplitude signal clearly showed energy concentration below 100 Hz, further confirming low-frequency vibration as the primary excitation source.

3.3. Hydraulic Cylinder Output Response Test Results and Analysis

The damping parameter of the hydraulic cylinder is a critical condition for solving the dynamic equations. In oscillatory or vibrational systems, the dimensionless parameter ζ, known as the damping ratio, is introduced to facilitate analysis. Its expression is given by Equation (13), where c is the hydraulic cylinder damping parameter, m₁ is the mass of the hydraulic cylinder (17.09 kg), ω_n is the system’s natural frequency, and c_c is the system’s critical damping coefficient.

$$\zeta ={\displaystyle \frac{c∕\left(2m_1\right)}{\sqrt{k/m_1}}}={\displaystyle \frac{c}{2\sqrt{m_{1}k}}}={\displaystyle \frac{c}{2m_1{\omega }_n}}={\displaystyle \frac{c}{c_c}}$$

(13)

The free vibration amplitude of a damped system decays exponentially. Assuming a natural period T_d, the amplitudes of two consecutive peaks in the same direction are A_i and A_i+1, as expressed in Equation (14), where ω is the undamped natural frequency, and ω_d is the damped natural frequency for 0 ≤ ζ < 1.

$$\left\{\begin{array}{c}{A}_i=A{e}^{-\zeta {\omega }_n{t}_i}\sin \left({\omega }_d{t}_i+\phi \right)\\ A_{i+1}=A{e}^{-\zeta {\omega }_n\left(t_i+T_d\right)}\sin \left[{\omega }_d\left(t_i+T_d\right)+\phi \right]\end{array}\right.$$

(14)

The ratio of two consecutive amplitudes is given by Equation (15), where η is the amplitude decay rate. The natural logarithm of the amplitude decay rate is defined as the logarithmic decrement δ, expressed in Equation (16). For small damping ratios, the logarithmic decrement can be approximated by Equation (17).

$$\eta ={\displaystyle \frac{A_i}{A_{i+1}}}=e^{\zeta {\omega }_n{T}_d}=e^{{\displaystyle \frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}}}$$

(15)

$$\delta =ln\eta =\zeta {\omega }_n{T}_d={\displaystyle \frac{2\pi \zeta }{\sqrt{1-{\zeta }^2}}}$$

(16)

$$\delta =2\pi \zeta$$

(17)

By combining Equations (13) to (17), the expression for the hydraulic cylinder damping parameter is derived as Equation (18).

$$c=2\zeta m_1{\omega }_n$$

(18)

As shown in Figure 17, the first six constrained modal frequencies of the hydraulic cylinder, obtained using ANSYS Workbench, are 60.604 Hz, 60.672 Hz, 291.92 Hz, 292.01 Hz, 702.77 Hz, and 702.83 Hz. The fundamental natural frequency corresponding to the primary mode shape of the hydraulic cylinder is 60.604 Hz.

After preprocessing the acquired signals, the acceleration response from a suitable time segment was selected, as shown in

Table 3. Hydraulic cylinder acceleration response.

Peak	X Value	Y Value
1	1.970	2.289
2	1.974	2.319
3	1.979	1.740

. The displacement response, obtained by double-integrating the acceleration signal, is also presented in

Table 4. Hydraulic cylinder amplitude response.

Peak	X Value	Y Value
1	1.970	0.001
2	1.974	0.002
3	1.979	0.001

.

The acquired acceleration response signal exhibits an overall decaying trend, consistent with the actual operational behavior of the hydraulic cylinder. The peak in the acceleration response corresponds temporally with the second peak in the displacement response; this discrepancy may be attributed to signal transmission delay or external interference during the impact hammer excitation. The first three peak values of the hydraulic cylinder acceleration response are 2.286 m/s², 2.319 m/s², and 1.740 m/s², while the corresponding displacement response peaks are 0.001 mm, 0.002 mm, and 0.001 mm. Consequently, the logarithmic decrement δ should be calculated using the second and third amplitude values. The calculated logarithmic decrement δ is ln2, yielding a hydraulic cylinder damping parameter c of 228.58 N·s/m.

As shown in Figure 17, for determining the hydraulic cylinder stiffness k, a model was created in Creo based on the cylinder’s parameters and subsequently imported into ANSYS Workbench for analysis. The maximum strain values under static and working conditions were found to be 4.3963 × 10⁻⁵ and 4.8848 × 10⁻⁵, respectively. The minimum stiffness values calculated for these two states are 1.02358802 × 10⁸ N/m and 1.02358336 × 10⁸ N/m. Therefore, the hydraulic cylinder stiffness k is taken as 1.0236 × 10⁸ N/m.

3.4. Output Response Analysis and Experimental Verification of the Conveying Trough

Through systematic analysis of field harvest test data, this study successfully obtained the acceleration response characteristics at the connection between the combine harvester header and the conveying trough. This revealed the excitation effects on the conveying trough from the header, rice crop, and other components, as well as the energy variation patterns of the cantilever header system. To validate the accuracy of the previously established damped pendulum model for the cantilever conveying trough, the parameters of the combine harvester’s cantilever conveying trough system were assigned to the corresponding variables in the function to analyze the influence of external excitation on the trough’s swing during operation. The initial parameters for the field test system were as follows: length of the cantilever conveying trough l is 1.504 m, hydraulic cylinder stiffness k is 1.0236 × 10⁸ N/m, distance from the pivot axis to the hydraulic cylinder connection point b is 1 m, hydraulic cylinder damping parameter c is 228.58 N·s/m, initial swing angle φ₁ is 46.40°, and excitation force from the header and rice crop on the conveying trough F is 1094.4 sin(50πt/3) N.

The damped pendulum model of the cantilever conveying trough was processed in MATLAB using the ODE45 solver to obtain the time-dependent functions for the swing angle and angular velocity of the conveying trough, as plotted in Figure 18.

Analysis of the dynamic response within the first 6 s shows that both the swing angle variation and the angular velocity exhibit a decaying trend in the initial 3 s, before gradually stabilizing. The maximum swing angle variation is 9.05 × 10⁻⁵ rad, with a steady-state value of approximately 6.75 × 10⁻⁵ rad. This corresponds to an initial swing amplitude of 0.13126 mm and a steady-state amplitude of 0.0979 mm. Compared to the experimentally measured swing amplitude of 0.099 mm, the theoretical error in the steady state is only 1.11%, validating the model’s accuracy.

To mitigate experimental contingency errors, data from sensor 3 was further used for verification. The variations in swing angle and angular acceleration are shown in Figure 19.

The results indicate consistent dynamic characteristics with those from Measurement Point 4. The swing angle variation and angular velocity decrease within the first 3 s before stabilizing. The maximum swing angle variation is 8.81 × 10⁻⁵ rad, with a steady-state value of 6.60 × 10⁻⁵ rad, corresponding to a steady-state swing amplitude of 0.0957 mm. The error compared to the experimental result is 4.3%.

4. Conclusions

This study systematically investigated the vibration characteristics and dynamic responses of the combine harvester header assembly under different working conditions through field experiments and dynamic modeling, with a focus on the dynamic behaviors of the header, conveying trough, and hydraulic cylinder. The main conclusions are as follows:

(1)

The header vibration response exhibits distinct multi-source excitation characteristics. Time-domain and frequency-domain analyses indicate that the header is subjected not only to direct excitation from the rice crop but also to significant coupled excitation from other components of the combine harvester. Frequency analysis revealed that the dominant vibration frequencies on the left and right sides of the header coincide with the 3rd and 1.5th harmonics of the threshing drum’s operating frequency, respectively, confirming dynamic coupling effects within the integrated system.

(2)

The excitation force at the connection between the header and the conveying trough is positively correlated with the feed quantity. Acceleration response tests under different cutting-width conditions showed that the peak acceleration at the connection under full-cutting width was significantly higher than that under half-cutting width. The vibration was more pronounced at measurement points located below the connection. The excitation force can be approximately characterized as a sinusoidal function with an amplitude dependent on the feed quantity, providing an effective boundary condition for system modeling.

(3)

The damped pendulum model established based on the Lagrange equation demonstrates good predictive capability under the tested conditions. The theoretical steady-state swing amplitude calculated by the model differed from the field-measured values by only 1.11% to 4.3%, verifying that the model accurately reflects the swinging behavior of the conveying trough under external excitation. This model provides a reliable theoretical tool for the dynamic performance evaluation and optimization of the header assembly.

It should be noted that the established model is primarily applicable to steady-state response prediction under the described test conditions, and its predictive capability for transient processes requires further validation. Future work will involve testing under different machine models, wider feed rate ranges, and conditions containing transient events to further evaluate the model’s applicability boundaries and robustness.