Influence of Lateral Leaf Number on Vibration Characteristics and Energy Dissipation of the Walnut (Juglans regia) Branch–Leaf–Fruit Subsystem

Abstract

During the harvest period, the role of lateral leaves in the dynamic behavior of the walnut (Juglans regia) branch–leaf–fruit subsystem remains unclear, and vibration harvesting parameter selection still lacks targeted guidance. To address this issue, a local walnut branch–leaf–fruit subsystem was studied by combining a discrete dynamic model, free-vibration tests, forced-vibration tests, and MATLAB simulations to investigate the effects of lateral leaf number on system dynamics. A representative single-fruit subsystem with six lateral leaves was selected, and four leaf number conditions (zero, two, four, and six) were examined. High-speed imaging was used to identify leaf motion patterns, while natural frequencies and fruit tracking point displacement responses were measured. The results showed that lateral leaves mainly exhibited three motion modes during vibration: spin, swing, and spin–swing compound motion. Under the six-leaf condition, spin motion was dominant. As the number of lateral leaves increased from 0 to 6, the first-order natural frequency decreased from 13.92 ± 6.37 Hz to 8.79 ± 4.03 Hz, a reduction of 36.8%. Forced-vibration results showed that increasing lateral leaf number significantly reduced the displacement response of the fruit tracking point in the non-excitation directions. Under the six-leaf condition, the maximum displacements in the Y- and Z-directions were reduced by 56.0% and 55.8%, respectively, compared with the leafless condition, indicating that the forced response became more concentrated in the main excitation direction. In the original MATLAB model, lateral leaves were simplified as fixed lumped mass damping elements, and the predicted results differed from the experimental trends. After introducing dynamic damping parameters matched to leaf motion patterns, the simulated trends became closer to the experimental results. These findings indicate that lateral leaf number is an important structural factor affecting the natural characteristics and directional forced responses of the walnut branch–leaf–fruit subsystem. The results provide theoretical and experimental references for optimizing vibration parameters and supporting low-damage, high-efficiency walnut vibration harvesting.

1. Introduction

The forest fruit industry is an important component of China’s forestry economy, generating not only economic output but also considerable ecological benefits. According to the China Statistical Yearbook 2023, China’s orchard area reached 13.01 million hectares in 2022, and the total economic output value of forestry crops reached 682.08 billion yuan [1]. The continuous expansion of forest fruit cultivation has also created opportunities for the mechanization of fruit harvesting. At present, most forest fruit orchards in China are characterized by low-stature and dense planting patterns, with limited spacing between trees, which makes the operation of large-scale machinery inconvenient. As a result, harvesting is still mainly carried out manually. However, with the continuous rise in labor costs, this harvesting method is gradually becoming unable to meet the growing demand generated by increasing fruit production. In regions such as Xinjiang and Gansu, where large-scale and intensive cultivation has been implemented, mechanized harvesting has already been adopted, with vibration harvesting being the dominant method [2]. Nevertheless, the overall level of mechanization remains relatively low. Therefore, theoretical research on vibration-based harvesting machinery for forest fruits in China is of great significance for improving vibration harvesting technology and promoting the further development of the forest fruit industry.

Regarding vibration harvesting technology, Adrian and Fridley [3] first proposed the fundamental theory of inertia-type tree shakers and discussed the dynamic principles and design criteria of vibration-based harvesting systems. On this basis, recent studies have further combined field experiments, response surface analysis, and rigid–flexible coupling simulations to investigate the relationship between tree morphology, excitation parameters, and harvesting performance. For example, Dang et al. [4] analyzed the vibration harvesting process of olive trees using response surface methodology and rigid–flexible coupling simulation, showing that numerical simulation can provide an effective tool for optimizing vibration harvesting parameters. Liu et al. [5] investigated the fracture type, excitation force, and detachment force of Chinese walnuts (Juglans regia) under forced-vibration conditions, indicating that vibration parameters directly affect walnut detachment behavior and harvesting effectiveness. In addition, Liu et al. [6] analyzed the vibration response of walnuts during vibration harvesting, while Jia et al. [7] further investigated walnut dynamic responses under different excitation forms. These studies provide direct references for understanding the relationship between excitation parameters and walnut fruit responses.

For branch–fruit vibration response and fruit detachment mechanisms, He et al. [8] used high-speed imaging to record the dynamic response of goji berry branches and carried out vibration fruit removal experiments, providing experimental evidence for the relationship between excitation parameters and fruit removal. Zhao et al. [9] conducted modal analysis and experimental verification of Lycium barbarum shrubs, showing that natural frequencies and vibration modes are closely related to efficient vibration harvesting. Han et al. [10,11] recorded the vibration trajectories and shedding characteristics of chestnut fruits using high-speed photography and analyzed the relationship between fruit motion state and vibration-induced detachment. These studies indicate that the dynamic characteristics of fruit-bearing branches, including natural frequency, vibration mode, and response amplitude, are key factors affecting vibration harvesting performance.

Dynamic modeling has also been widely used to reveal the vibration response mechanism of fruit–branch systems. Villibor et al. [12] established a flexible body model of the coffee fruit–stem system and analyzed the dynamic behavior of the fruit–stem structure under vibration. Castro-García et al. [13] investigated the frequency response of late-season “Valencia” orange fruits and identified effective frequency ranges for selective vibration harvesting. These studies suggest that fruit detachment is not only related to the external excitation frequency and amplitude but is also strongly affected by the local dynamic properties of the fruit–pedicel–branch system. More recently, equivalent dynamic modeling has also been used to simplify complex tree structures into mass–spring–damper systems. Grande and Franceschini [14] calibrated mass–spring–damper equivalent systems for real-time assessment of tree dynamics, indicating that equivalent modeling can be useful for describing tree vibration behavior when the full structural model is difficult to construct.

In recent years, increasing attention has been paid to the influence of leaves on the dynamic response of fruit-bearing branches. Castro-García et al. [15] investigated the contribution of fruits and leaves to the dynamic response of secondary orange branches. Their results showed that leaves reduced the first natural frequency and markedly suppressed acceleration transmission along the branch. Sola-Guirado et al. [16] further studied the effect of leaves on the dynamic response of olive tree branches and their computational model, demonstrating that leaves can significantly change the damping behavior and vibration transmission characteristics of branches. Xu et al. [17] constructed a vibration model of fruited and leafy fruit trees for vibratory harvesting, emphasizing that fruits and leaves should be considered in the dynamic modeling of fruit trees. These studies indicate that leaves are not negligible appendages during vibration harvesting but may act as important components that affect mass distribution, damping, local constraints, and energy dissipation.

With the development of high-speed imaging and numerical simulation, the research object has gradually extended from simple branch–fruit systems to more complex branch–fruit–flower–leaf systems. Chen et al. [18] analyzed the fruit–flower–leaf dynamic response of Lycium barbarum for vibration harvesting, showing that attached organs such as leaves and flowers may participate in the vibration response and affect energy transfer. Zhou et al. [19] used finite element explicit dynamics to simulate the motion and shedding of jujube fruits under forced vibration, providing a numerical reference for analyzing fruit motion and detachment under dynamic excitation. Yu et al. [20] optimized vibration parameters for red jujube trees with different diameters, demonstrating that structural differences among trees should be considered when determining vibration harvesting parameters. Liu et al. [21] further investigated the effect of excitation trajectories on fruit–tree system response, indicating that excitation direction and trajectory form can significantly influence the vibration response distribution of fruit trees.

In addition, finite element methods, energy transfer analysis, and intelligent sensing technologies have provided new tools for vibration harvesting research. Zhou et al. [22] analyzed shaking-induced cherry fruit motion and damage, showing that fruit motion patterns are closely related to vibration-induced damage. Chen et al. [23] used the finite element method to simulate the bending shape of Lycium barbarum fruit branches, while Zhao et al. [24] combined finite element simulations and experiments to analyze the detachment of Lycium barbarum fruits. Deng et al. [25] analyzed energy transfer characteristics in multi-level branches under vibration excitation, providing a reference for understanding how vibration energy propagates through complex branch systems. Shi et al. [26] proposed an AI binocular vision method for walnut tree posture determination and positioning in vibration harvesting, indicating that walnut harvesting research is moving toward intelligent perception, dynamic parameter matching, and precise excitation control. These studies provide useful methodological references for modeling flexible branches, fruit-bearing structures, vibration-induced detachment, energy transfer, and intelligent vibration harvesting.

In summary, although existing studies on vibration harvesting have made progress in excitation parameter optimization, fruit motion characterization, and branch–fruit system dynamics, they have mainly focused on leafless branches or research objects in which leaf effects were not explicitly considered. There is still a lack of systematic discussion on whether lateral leaves remain widely present on walnut branches during the harvest period and whether they may alter the natural characteristics, damping properties, and forced responses of the branch–leaf–fruit subsystem through added mass, petiole connection constraints, and aerodynamic energy dissipation. If lateral leaves have already largely fallen off at harvest, then studying their dynamic effects alone would have limited engineering significance. However, if large numbers of lateral leaves remain during harvesting, ignoring their role may lead to an incomplete understanding of system response rates and harvesting parameter matching. Unlike many previous vibration harvesting studies based on leafless branches or systems in which leaf effects were not explicitly considered, lateral leaves on walnut branches during the harvest period do not necessarily disappear completely. Their retained state may significantly alter the mass distribution, additional damping, and vibration energy transfer pathways of the branch–leaf–fruit subsystem.

Based on the above background, this study focuses on a local walnut branch–leaf–fruit subsystem and, by combining free-vibration tests, forced-vibration tests, and numerical simulations, addresses the following key questions:

(1)

How does variation in lateral leaf number affect the natural frequency and damping characteristics of the walnut branch–leaf–fruit subsystem?

(2)

Under external excitation, how do lateral leaf motion patterns regulate energy distribution in the excitation and non-excitation directions, as well as fruit response trajectories?

(3)

How can the regulatory effects of lateral leaves on system dynamic characteristics be used to provide a basis for selecting and optimizing excitation parameters in walnut vibration harvesting?

The purpose of this study is to clarify, from the perspective of local dynamic mechanisms, how lateral leaves during the harvest period influence the vibration response of the walnut subsystem and to provide experimental and theoretical references for low-damage and high-efficiency vibration harvesting.

2. Dynamic Model of the Branch–Leaf–Fruit Subsystem

To characterize the combined effects of changes in lateral leaf number on the mass distribution, local constraints, and damping energy dissipation of the walnut branch–leaf–fruit subsystem, a discrete dynamic model was established by coupling discretized branch beam elements with equivalent lumped parameters of lateral leaves. In this model, the main branch was represented by spatial beam elements, while each lateral leaf was equivalently modeled as an added mass–stiffness–damping unit at its attachment node, so as to reflect the local modulation of system dynamic parameters induced by lateral leaves. According to the beam vibration theory [12,17], the transverse vibration equation of a uniform branch beam is given in Equation (1):

$$\mathrm{\rho }A{\displaystyle \frac{{𝜕}^{2}w\left(x,t\right)}{𝜕t^2}}+EI{\displaystyle \frac{{𝜕}^{4}w\left(x,t\right)}{𝜕x^4}}+c_s{\displaystyle \frac{𝜕w\left(x,t\right)}{𝜕t}}=0$$

(1)

where $w\left(x,t\right)$ is the transverse displacement at position $x$ from the fixed end at time $t$; $\rho$ is the material density of the branch; $A=\pi R^2$ is the cross-sectional area; and $E$ is Young’s modulus. The second moment of area is defined in Equation (2):

$$I={\displaystyle \frac{\pi R^4}{4}}$$

(2)

where I is the second moment of area and $c_s$ is the structural damping coefficient characterizing the internal energy dissipation of the material.

To enable numerical solution and introduce discrete lateral leaf interaction points, the main branch was uniformly discretized into $N$ spatial beam elements. The coordinate system is shown in Figure 1, where the $X$-axis is aligned with the longitudinal axis of the main branch, the $Y$-axis lies in the horizontal plane and is perpendicular to the $X$-axis, and the $Z$-axis is perpendicular to the $XY$ plane. Each node has six degrees of freedom, namely, translational displacements along the $x$-, $y$-, and $z$-directions and rotational displacements about the corresponding axes. Thus, the displacement vector of the $i$-th node is expressed in Equation (3):

$$d_i={\left[u_{xi},u_{yi},u_{zi},{\mathrm{\theta }}_{xi},{\mathrm{\theta }}_{yi},{\mathrm{\theta }}_{zi}\right]}^T$$

(3)

Accordingly, each three-dimensional spatial beam element has a total of 12 degrees of freedom, with 6 degrees of freedom at each end node. Based on the finite element method, the global stiffness matrix $K_0$, mass matrix $M_0$, and gyroscopic matrix $G_0$ of the leafless system can be derived and assembled. For a typical spatial beam element, the element stiffness matrix $k_e$, consistent mass matrix $m_e$, and gyroscopic matrix $g_e$ take standard forms, generally written in Equations (4) and (5):

$$k_e={\left[{\displaystyle \genfrac{}{}{0pt}{}{k_{11}}{k_{21}}}{\displaystyle \genfrac{}{}{0pt}{}{k_{12}}{k_{22}}}\right]}_{12\times 12}$$

(4)

$$m_e={\left[{\displaystyle \genfrac{}{}{0pt}{}{m_{11}}{m_{21}}}{\displaystyle \genfrac{}{}{0pt}{}{m_{12}}{m_{22}}}\right]}_{12\times 12}$$

(5)

Using the direct stiffness method, the element matrices $k_e$ and $m_e$ are assembled into the global matrices according to node numbering:

$$K_0={\displaystyle \sum _{e=1}^N}{T}_e^T{k}_e{T}_e$$

(6)

$$M_0={\displaystyle \sum _{e=1}^N}{T}_e^T{m}_e{T}_e$$

(7)

where $T_e$ is the coordinate transformation matrix of element $e$. Since the subsequent analysis focuses on the updates of mass, stiffness, and damping parameters caused by lateral leaves, as well as the resulting changes in natural frequency and damping, Equations (3)–(7) are retained mainly as the basis of discrete modeling, and their detailed matrix expansions are omitted.

The basic structural damping of the system is represented by a Rayleigh damping model:

$$C_0=\mathrm{\alpha }{M}_0+\mathrm{\beta }{K}_0$$

(8)

where $\alpha$ and $\beta$ are the Rayleigh damping coefficients, which can be determined from the first two natural frequencies ${\omega }_1$ and ${\omega }_2$ and the corresponding damping ratios ${\zeta }_1$ and ${\zeta }_2$ of the leafless system:

$$\left[{\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\beta }}\right]={\displaystyle \frac{2}{{\omega }_2^2-{\omega }_1^2}}\left[{\displaystyle \genfrac{}{}{0pt}{}{{\omega }_1{\omega }_2}{-1}}{\displaystyle \genfrac{}{}{0pt}{}{-{\omega }_1{\omega }_2}{1}}\right]\left[\begin{array}{c}{\omega }_2{\zeta }_1\\ {\omega }_1{\zeta }_2\end{array}\right]$$

(9)

Assume that the system contains $S$ lateral leaves. The j-th lateral leaf is attached at node $p_j$ and is represented by equivalent mass $m_j$, equivalent damping $c_j$, and equivalent stiffness $k_j$. In this study, the equivalent stiffness of lateral leaves was introduced as a modeling parameter rather than an independently measured material parameter. This treatment was adopted because previous studies have shown that plant leaves and petioles exhibit measurable flexural rigidity and that leaves can significantly affect the natural frequency, damping, and vibration transmissibility of fruit-bearing branches. Therefore, each lateral leaf was simplified as an equivalent mass–stiffness–damping unit connected to the main branch through the petiole attachment node. The equivalent stiffness $k_j$ was used to represent the local constraint effect of the petiole–leaf connection and was introduced into the global stiffness matrix through node mapping.

To couple the $j$-th lateral leaf into the main branch system, a node selection matrix $T_j$ is defined to extract the six degrees of freedom corresponding to node $p_j$. Starting from the leafless system matrices $M_0$, $K_0$, and $C_0$, the system matrices are updated after the addition of each lateral leaf as follows:

$$M_j=M_{j-1}+T_j^T{m}_j{I}_6{T}_j$$

(10)

$$K_j=K_{j-1}+T_j^T{k}_j{I}_6{T}_j$$

(11)

$$C_j=C_{j-1}+T_j^T{c}_j{I}_6{T}_j$$

(12)

where $I_6$ is the $6\times 6$ identity matrix. It can thus be seen that a change in lateral leaf number updates not only the mass matrix but also the stiffness and damping matrices. Therefore, the effect of lateral leaves cannot be simply interpreted as “only adding extra mass” but should instead be regarded as a synchronous reconstruction of the equivalent parameters of the discrete coupled system.

After integrating all $S$lateral leaves, the overall dynamic equation of the system can be written as:

$$M_S\ddot{q}+C_S\dot{q}+K_{Sq}=F\left(t\right)$$

(13)

where $M_S$, $K_S$, and $C_S$ are the overall mass, stiffness, and damping matrices of the system with lateral leaves, respectively; $q$ is the generalized displacement vector; and $F\left(t\right)$ is the external excitation vector.

The natural frequencies $f_n$ and damping ratios ${\zeta }_n$ of the system can be obtained by solving the following complex eigenvalue problem:

$$\left(K_S+i\mathrm{\omega }{C}_S-{\mathrm{\omega }}^2{M}_S\right)\mathrm{\varphi }=0$$

(14)

where $\omega =2\pi f$ and $\varphi$ is the modal shape vector. The damping ratio ${\zeta }_n$ is calculated using the modal damping formula:

$${\mathrm{\zeta }}_n={\displaystyle \frac{{\mathrm{\varphi }}_n^T{C}_S{\varphi }_n}{2{\mathrm{\omega }}_n{\varphi }_n^T{M}_S{\varphi }_n}}$$

(15)

Meanwhile, to correspond to the free-vibration test results, the logarithmic decrement $\delta$ is defined from the free-vibration decay curve as:

$$\mathrm{\delta }={\displaystyle \frac{1}{n}}\mathrm{l}n{\displaystyle \frac{A_i}{A_{i+n}}}$$

(16)

where $A_i$ and $A_{i+n}$ are the amplitudes of the $i$-th and $\left(i+n\right)$-th cycles, respectively. The damping ratio $\zeta$ can then be approximately calculated from $\delta$ as:

$$\mathrm{\zeta }={\displaystyle \frac{\mathrm{\delta }}{\sqrt{4{\mathrm{\pi }}^2+{\mathrm{\delta }}^2}}}$$

(17)

To further characterize the difference in damping energy dissipation under different lateral leaf number conditions during free decay, the energy dissipated per cycle and the relative energy dissipation rate were introduced as auxiliary evaluation indices. For a damped system, the energy dissipated by the damping term during one vibration cycle can be expressed as:

$$W_d={\int }_{t_i}^{t_{i+1}}{\dot{q}}^T{C}_S\dot{q} dt$$

(18)

where $W_d$ is the energy dissipated through the damping matrix $C_S$ during the $i$-th vibration cycle; $\dot{q}$ is the generalized velocity vector of the system; and $t_i$ and $t_{i+1}$ are two adjacent instants with the same phase.

Since the peak values of the free-decay envelope are more easily obtained experimentally, the relative energy dissipation rate is further characterized by the amplitude variation between adjacent cycles. If the peak amplitudes of the $i$-th and $\left(i+1\right)$-th cycles are $A_i$ and $A_{i+1}$, respectively, then the relative energy dissipation rate between adjacent cycles can be expressed as:

$${\eta }_i={\displaystyle \frac{E_i-E_{i+1}}{E_i}}$$

(19)

where $E_i$ and $E_{i+1}$ denote the equivalent vibration energies corresponding to the $i$-th and $\left(i+1\right)$-th cycles, respectively. If the vibration energy at the same measurement point is approximately assumed to be proportional to the square of the amplitude, then:

$${\eta }_i=1-{\left({\displaystyle \frac{A_{i+1}}{A_i}}\right)}^2$$

(20)

Combining Equation (16) with the definition of logarithmic decrement $\delta$, Equation (20) can be further written as:

$${\eta }_i=1-e^{-2\delta }$$

(21)

Equation (18) is used to characterize the damping energy dissipated per cycle from the model perspective, whereas Equations (19)–(21) are used to compare the relative energy dissipation levels under different lateral leaf number conditions based on the variation of adjacent peak amplitudes in the free-decay experiment.

3. Materials and Methods

3.1. Experimental Site and Experimental Objects

The experiments were conducted at Meitao Fruit Professional Cooperative, Houbai Town, Jurong City, Zhenjiang, Jiangsu Province, China (the American walnut experimental base of Nanjing Forestry University). The tests were carried out in mid-October 2025, during the walnut fruit maturity and harvest period, under clear weather conditions, with temperatures ranging from 17 to 28 °C. During the tests, the ambient wind speed was below 2.0 m s⁻¹, corresponding to a light-air to light-breeze condition, and its influence on the free-vibration and forced-vibration responses of the local branch–leaf–fruit subsystem was considered limited.

To reduce the interference of multi-fruit collision, mutual occlusion, and mass distribution differences on trajectory identification and parameter recognition, a single-fruit branch–leaf–fruit subsystem was selected as the basic experimental object in this study. All test samples were collected before fruit drop, so as to ensure good consistency in fruit maturity, structural integrity, and geometric morphology, thereby improving the comparability of data among different test conditions and providing a stable basis for subsequent free-vibration tests, forced-vibration tests, and model validation.

A total of 30 walnut branch–leaf–fruit subsystem samples were collected, with the number of lateral leaves ranging from 5 to 9. Statistical analysis showed that samples with 6 lateral leaves occurred more frequently in the measured population and exhibited relatively complete structures. Therefore, the single-fruit subsystem with 6 lateral leaves was selected as the basic sample state.

To investigate the robustness of the influence of lateral leaf number under different structural parameters, two additional structural variables, branch length and fruit mass, were introduced for parallel validation. Based on the measured distribution range of the 30 samples and considering the number of samples in each interval, branch length and fruit mass were each divided into three intervals to cover the typical structural differences among short, medium, and long branches as well as light, medium, and heavy fruits. Branch length was divided into 350–400 mm, 400–450 mm, and 450–500 mm; fruit mass was divided into 15–20 g, 20–25 g, and 25–30 g.

On this basis, one sample with 6 lateral leaves, complete leaf distribution, and typical structural morphology was selected from the sample library as the representative sample, which was used to fully demonstrate the variations in free-vibration and forced-vibration responses under the 0-, 2-, 4-, and 6-lateral-leaf conditions. Meanwhile, the other four samples with differences in branch length or fruit mass were selected as parallel samples to verify the consistency of these laws under different structural parameter conditions. The representative sample was used for mechanism demonstration, while the parallel samples were used for repeated validation. The structural parameters of all samples are listed in

Table 1. Dimensional parameters of the branch–fruit subsystem.

Subsystem No.	Length/mm	Radius/mm	Branch Mass/g	Fruit Mass/g
1	492.7	4.5	23.6	20.8
2	388.6	5.3	18.9	19.7
3	549.3	4.1	27.8	21.3
4	489.1	3.7	22.1	16.9
5	511.5	5.2	24.3	23.9

.

All samples were tested under four lateral leaf number conditions, namely, 0, 2, 4, and 6 leaves, following the same procedure, and each condition was repeated three times. The representative sample was used to demonstrate the influence of lateral leaf number on system response, while the parallel samples were used to verify the repeatability and stability of the results under different structural parameter conditions.

3.2. Experimental Methods

After sample selection and parameter measurement, free-vibration tests were first conducted on each sample, followed by forced-vibration tests, in order to compare the response variations in the system under different lateral leaf number conditions.

In the free-vibration test, the end of the subsystem was manually displaced from its equilibrium position to a controllable initial displacement and then released rapidly, allowing the system to undergo free-decay vibration without continuous external excitation until the vibration ceased, as shown in Figure 2. In the forced-vibration test, a periodic external excitation was applied to the branch–leaf–fruit subsystem in the horizontal direction to simulate the external vibration conditions during vibration harvesting, and the motion characteristics of the system in the steady response stage were recorded.

In the forced-vibration test, the basal end of the subsystem was fixed using the same clamping method as that used in the free-vibration test. The fixed end was treated as a fully constrained boundary, whereas the fruit-bearing end was treated as a free end. A sinusoidal displacement excitation with a frequency of 20 Hz and an amplitude of 10 mm was applied along the X-direction. The X-direction was defined as the main excitation direction in the image coordinate system, while the Y- and Z-directions were defined as non-excitation directions.

The excitation parameters were determined by considering both the engineering excitation range and the stability of the measured response. The frequency of 20 Hz falls within the engineering excitation range in which the low-order modes of the tested samples may participate in the response, and it can effectively excite the forced motion characteristics of the branch–leaf–fruit subsystem. Preliminary tests also showed that the system response under this condition was relatively stable, which facilitated steady-state trajectory extraction using high-speed imaging and enabled comparison among different lateral leaf number conditions.

During the forced-vibration test, the tracking point P near the pedicel–fruit connection, as shown in Figure 1, was selected as the response analysis point. This point was closest to the fruit end and relatively far from the fixed end, resulting in a more pronounced displacement response. Therefore, it was suitable for trajectory extraction and for comparing the forced responses under different lateral leaf number conditions. Before data extraction, the subsystem was excited for several cycles to reduce the influence of the initial transient response. The steady-state response was then recorded by high-speed cameras, and the fifth vibration cycle was selected for displacement analysis. Under each lateral leaf number condition, the free-vibration test was conducted first, followed by the forced-vibration test, and each condition was repeated three times. The main experimental equipment included M310 and VEO410 high-speed cameras (Vision Research Inc., Wayne, NJ, USA), a computer, TEMA high-speed video analysis software (Image Systems AB, Linköping, Sweden), and related data processing software. The image acquisition parameters were set as follows: sampling frequency of 500 fps, resolution of 1024 × 768, and post-trigger acquisition mode. Considering that the amplification effect of linear displacement on angular variation increases with distance from the fixed end, the farther the point is from the fixed end, the larger the amplitude becomes. Therefore, the terminal tracking point near the fruit–branch connection was selected as the fruit response analysis point. At the same time, markers were placed on the lateral leaves to extract their motion postures and trajectory characteristics under different test conditions.

For the free-vibration test, the damping-related parameters were extracted from the free-decay displacement curves of tracking point P. Five consecutive peak amplitudes after release were selected from each free-decay curve. The logarithmic decrement was calculated from adjacent peak amplitudes according to Equation (16), and the equivalent damping ratio was then obtained using Equation (17). To avoid the uncertainty caused by absolute stiffness estimation, the energy dissipation was characterized by the relative energy dissipation rate between adjacent cycles, which was calculated according to Equations (19)–(21). For each lateral leaf number condition, the damping-related parameters were calculated from repeated tests.

After each test, the number of lateral leaves was gradually reduced by symmetric removal on both sides, and the above free-vibration and forced-vibration test procedures were repeated until all lateral leaves had been removed. Taking the initial condition with 6 lateral leaves as an example, as shown in Figure 3, two lateral leaves were symmetrically removed each time, resulting in four experimental conditions with 6, 4, 2, and 0 lateral leaves. This symmetric leaf removal strategy made it possible to change the number of lateral leaves while preserving, as much as possible, the symmetry of the system’s mass and stiffness distribution, thereby reducing the additional dynamic coupling effects caused by asymmetric leaf distribution and allowing lateral leaf number to serve as the primary variable under investigation.

It should be noted that the actual experimental operation followed the sequence of 6, 4, 2, and 0 leaves during progressive leaf removal. However, in the subsequent results analysis, for the convenience of describing the influence of increasing lateral leaf number on the system dynamic characteristics, comparisons and discussions were uniformly conducted in the order of 0, 2, 4, and 6 leaves.

All tests followed the same procedure: under each lateral leaf number condition, free-vibration tests were conducted first, followed by forced-vibration tests, and each condition was repeated three times to reduce random error and ensure the repeatability and comparability of the results.

The experimental data were statistically analyzed to evaluate the repeatability and significance of the measured response parameters. The results are presented as mean ± standard deviation (SD). Since the same five branch–leaf–fruit subsystem samples were tested under the four lateral leaf number conditions of 0, 2, 4, and 6 leaves, one-way repeated-measures analysis of variance (ANOVA) was used to evaluate the effect of lateral leaf number on the first-order natural frequency and on the maximum displacement responses in the X-, Y-, and Z-directions. Statistical significance was determined at p < 0.05. The statistical analysis was performed using MATLAB (R2024b, The MathWorks Inc., Natick, MA, USA).

To further compare the effects of lateral leaf number variation on the forced response trajectories of the branch–leaf–fruit subsystem and to examine the applicability of the above discrete dynamic model under forced-vibration conditions, a three-dimensional dynamic simulation model of the branch–leaf–fruit subsystem was established on the MATLAB platform based on the experimentally measured branch structural parameters and fruit physical parameters. Using the system dynamic equation given in Equation (13) as the basis, harmonic external excitation was applied in the $X$-direction, and the steady-state response trajectories of the fruit tracking point under different lateral leaf number conditions were calculated for comparison with the forced-vibration experimental results.

Based on the relevant walnut-related calculations reported by Cui et al. [27], combined with the test data of the present study, the main parameters adopted in the simulation model are listed in

Table 2. MATLAB simulation parameters.

Branch Length/m	Radius/m	Density kg/m3	Poisson’s Ratio	Damping Ratio	Mass of a Single Lateral Leaf/g
0.5	0.01	1019	0.35	0.056	10

.

4. Results and Discussion

4.1. Analysis of Lateral Leaf Motion Patterns

To analyze the dynamic response characteristics of lateral leaves during free vibration, high-speed imaging was used to record the walnut branch–leaf–fruit local subsystem in real time, and the motion process of the lateral leaves was identified frame by frame with the aid of TEMA high-speed video analysis software. In this study, the term subsystem refers to a local dynamic unit composed of the terminal fruiting branch segment, its attached lateral leaves, and a single fruit. For the convenience of comparing how increasing lateral leaf number affects motion patterns, the results are discussed in the order of zero, two, four, and six lateral leaves.

Experimental observations showed that lateral leaves mainly exhibited three typical motion patterns during free vibration, namely, spin motion, swing motion, and spin–swing compound motion, as shown in Figure 4. Here, spin motion refers to the rotation of a leaf around a local axis near the connection region between the petiole and the branch; swing motion refers to the periodic reciprocating deflection of the leaf relative to the main branch axis; and spin–swing compound motion refers to the case in which an obvious rotational component accompanies the swing motion. Unlike the previous description based only on trajectory lines, the classification of these three motion patterns should be based on changes in the actual leaf posture and its angular displacement relative to the branch.

Under the leafless condition, no leaf-related motion existed in the system, and vibration energy was mainly dissipated through branch structural damping and air resistance. When the number of lateral leaves increased to two, leaf motion was dominated by large-amplitude swinging, with a weak rotational component observed locally, indicating that the constraining effect of the leaves on the system was still relatively weak. Under this condition, energy dissipation was mainly associated with the additional aerodynamic resistance generated during swinging. When the number of lateral leaves further increased to four, leaf motion gradually changed from pure swinging to a compound pattern involving both swinging and rotation, and differences in posture response among individual leaves began to appear. This suggests that, with an increase in lateral leaf number, the coupling between leaves and the branch became stronger, and the system energy dissipation pathways became more complex than those under the two-leaf condition.

When the number of lateral leaves increased to six, lateral leaf motion was dominated by spin, while the swing amplitude was relatively reduced, indicating that leaves were more likely to form a relatively stable rotational response when a more complete leaf retention condition was maintained. This phenomenon suggests that, under this condition, lateral leaves not only acted as added mass but also participated in system energy dissipation through local constraints at the petiole connection, aerodynamic resistance during leaf rotation, and coupling interaction with the main branch vibration. Combined with the subsequent free-vibration and forced-vibration results, it can be seen that the non-excitation direction responses were more strongly suppressed under the six-leaf condition, suggesting that a spin-dominated leaf motion pattern was more favorable for reducing vibration dispersion in non-dominant directions, whereas the 2–4-leaf conditions represented an intermediate stage in the transition from swing-dominated motion to compound motion.

Therefore, lateral leaf number changes not only whether leaves participate in the vibration response but also the specific way in which they participate in system energy dissipation: under the leafless condition, structural damping dominates; under the two-leaf condition, swing-related dissipation dominates; under the four-leaf condition, swinging and spinning act together; and under the six-leaf condition, relatively stable spin response dominates. These results indicate that lateral leaf number is an important structural factor determining the distribution of leaf motion patterns and their associated energy-dissipation modes. This also provides a morphological basis for the subsequent analysis of natural frequency changes and forced response differences.

4.2. Analysis of Free-Vibration Characteristics and Parameter Variations in the Branch–Leaf–Fruit Subsystem

To analyze the influence of lateral leaf number variation on the free-vibration response of the walnut branch–leaf–fruit local subsystem, the tracking point $P$ near the fruit–pedicel junction shown in Figure 1 was selected as the response analysis point. For the same subsystem, the free end was displaced to the same initial position and then released, allowing the system to undergo free-decay vibration without continuous external excitation. High-speed imaging and trajectory extraction methods were used to record the spatial motion of tracking point $P$.

The experimental results showed that after free release, the motion trajectory of tracking point $P$ generally exhibited a nearly closed path that gradually converged toward the equilibrium position over time. It should be noted that, due to the coupled responses in two transverse directions, the projected trajectory of tracking point $P$ in the observation plane showed a nearly closed pattern that gradually contracted inward as the vibration decayed, rather than a strict geometric ellipse. In the initial stage of free vibration, the system possessed obvious response components in both transverse directions, together with the effects of gravity and geometric nonlinearity. Therefore, the trajectory exhibited skewness, contraction, and partial non-closure. As vibration energy was continuously dissipated by structural damping, air resistance, and leaf-related energy dissipation, the trajectory envelope gradually collapsed inward and finally converged to the static equilibrium position.

The convergence process of tracking point $P$ differed under different lateral leaf number conditions. Under the leafless condition, the free vibration of the system was mainly governed by the structural damping of the branch itself and air resistance; the trajectory converged more rapidly, but the initial rebound amplitude was relatively large, indicating that the system was more sensitive to the initial disturbance in this case. When the number of lateral leaves increased to two, the trajectory was still dominated by a relatively large-amplitude swinging response, but compared with the leafless condition, the contraction speed of the trajectory became slower, indicating that the leaves had begun to participate in energy dissipation and local coupling. When the number of lateral leaves further increased to four, the trajectory pattern was generally similar to that under the two-leaf condition, still exhibiting a relatively evident ellipse-like converging feature; however, the local trajectory became smoother and the rebound process more stable, indicating that the system had gradually evolved from a low-leaf, weak-coupling state to an intermediate state in which part of the leaves participated in energy dissipation. When the number of lateral leaves increased to six, the overall trajectory became more concentrated and the convergence process more uniform, suggesting that under a more complete leaf retention condition, the local constraints at the petiole connection, the energy dissipation associated with leaf spin–swing, and the branch–leaf coupling jointly affected the free-vibration decay process.

Overall, variation in lateral leaf number not only altered the added mass of the system but also changed the energy dissipation path and the trajectory evolution mode during free vibration. Under the leafless condition, the system mainly relied on the structural damping of the branch for energy dissipation. As the number of lateral leaves increased, the additional aerodynamic resistance caused by leaf swinging and rotation, the local constraints at the petiole connection, and the dynamic coupling between the branch and leaves jointly participated in vibration attenuation. As a result, the trajectory envelope, rebound height, and convergence rate of tracking point $P$ all changed. Quantitatively, these differences can be further characterized by the logarithmic decrement $\delta$, the equivalent damping ratio $\zeta$, and the number of cycles required for the response to decay to a given amplitude. Therefore, lateral leaf number is an important structural parameter that affects the free-vibration response pattern and energy dissipation characteristics of the system.

To further quantify the effects of lateral leaf number on the natural characteristics and damping-related energy dissipation of the system, modal frequencies under different conditions were obtained from Equation (13) based on the aforementioned discrete dynamic model, while the experimental equivalent damping ratio was identified from the free-vibration decay curves according to Equations (16) and (17). In addition, differences in damping energy dissipation can be further characterized by Equations (18)–(20) combined with variations in successive peak amplitudes during free decay. Since the main focus of this section is to compare the variation in natural frequencies under different lateral leaf number conditions,

Table 3. Ten natural frequencies (Hz) under different lateral leaf number conditions.

Frequency Order	No Lateral Leaves/Hz	Two Lateral Leaves/Hz	Four Lateral Leaves/Hz	Six Lateral Leaves/Hz
1	13.92 ± 6.37	10.71 ± 4.91	10.61 ± 4.86	8.79 ± 4.03
2	17.39 ± 7.95	13.39 ± 6.13	13.26 ± 6.07	10.98 ± 4.93
3	21.54 ± 9.85	16.58 ± 7.59	16.41 ± 7.51	13.60 ± 6.01
4	31.89 ± 14.15	24.54 ± 11.22	24.30 ± 11.08	20.14 ± 9.02
5	54.73 ± 25.01	42.13 ± 19.26	41.71 ± 19.02	34.57 ± 15.71
6	82.93 ± 37.87	63.83 ± 28.72	63.21 ± 28.35	52.38 ± 23.92
7	83.92 ± 38.32	65.15 ± 29.33	64.98 ± 28.92	53.67 ± 24.16
8	85.77 ± 39.16	66.02 ± 30.15	65.37 ± 29.77	54.17 ± 24.83
9	106.09 ± 43.72	81.66 ± 33.61	80.59 ± 33.16	67.01 ± 27.24
10	167.35 ± 30.24	128.81 ± 23.17	127.54 ± 22.83	105.70 ± 18.63

first summarizes the first ten natural frequencies of the system under four typical conditions, namely, zero, two, four, and six lateral leaves.

As shown in Table 3, when the number of lateral leaves increased from zero to two, four, and six, the mean natural frequencies of all modes showed an overall decreasing trend. Taking the first-order natural frequency as an example, the value decreased from 13.92 ± 6.37 Hz under the leafless condition to 8.79 ± 4.03 Hz under the six-leaf condition, corresponding to a reduction of 36.76%. The one-way repeated measures ANOVA indicated that lateral leaf number had a significant effect on the first-order natural frequency; F_3,12 = 29.50 and p < 0.001. Although the standard deviations were relatively large because of differences in branch length, radius, and fruit mass among samples, the decreasing trend with increasing lateral leaf number was consistent among the five samples.

As indicated by Equations (10)–(12), variation in lateral leaf number updates not only the mass matrix but also the stiffness and damping matrices. Therefore, the change in frequency cannot be simply interpreted by the intuition of a single-degree-of-freedom system; instead, variation in lateral leaf number systematically changes the frequency distribution of the low-order dominant modes.

It should be noted that if the system were simply approximated as a lumped single-degree-of-freedom system, reducing the mass $m$ would generally increase the frequency when the stiffness $k$ remains essentially unchanged. However, the object investigated in this study is not a purely lumped single-degree-of-freedom system, but rather a discrete coupled system composed of discretized main-branch beam elements, lumped fruit mass, and equivalent mass–stiffness–damping units representing lateral leaves. According to the matrix update relations of the model, variation in lateral leaf number changes not only the mass matrix but also the stiffness and damping matrices. Consequently, although removing lateral leaves reduces the added mass, it also weakens the local constraints at the petiole connection, the additional restoring effect of lateral leaves on the main branch, and the coupling effects related to leaf motion patterns. Thus, the decrease in equivalent stiffness exceeds the frequency-increasing effect caused by reduced mass, and the final result is a decrease in natural frequency. In other words, the frequency change observed here cannot be simply explained by the single-degree-of-freedom intuition of “lower mass leads to higher frequency”; it should instead be understood as the result of equivalent parameter reconstruction in a discrete coupled system.

In addition, the natural frequencies under the four-leaf and two-leaf conditions were very close to each other, and this phenomenon was consistent across different samples. Taking the first-order frequency as an example, the mean values under the two-leaf and four-leaf conditions were 10.71 ± 4.91 Hz and 10.61 ± 4.86 Hz, respectively, with a relative difference of only about 1.01%; the differences in the remaining modal frequencies were also generally small. This indicates that when the number of lateral leaves increased from the leafless condition to a small number of leaves, the natural characteristics of the system changed more noticeably; however, when the number of leaves increased further from two to four, the frequency reduction caused by additional leaves entered a stage of diminishing marginal effect. In other words, the 2–4-leaf conditions correspond to an intermediate transition stage from a “few-leaf, swing-dominated” state to a state in which partial leaves participate in coupled energy dissipation. At this stage, the system reaches a new temporary balance among added mass, local constraints, and energy dissipation mode, so that the frequency responses under the two conditions are close but not identical. The natural frequencies of parallel samples are summarized in

Table 4. Four groups of experimental data.

Test No.	Leaf Count	Order/Hz
		1	2	3	4	5	6	7	8	9	10
2	0	26.43	32.98	40.85	59.42	103.67	156.98	158.86	162.35	189.76	199.99
	2	20.35	25.42	31.46	46.50	79.82	119.85	122.34	124.99	145.97	153.59
	4	20.15	25.17	31.14	45.97	78.92	118.45	120.91	123.54	144.21	151.75
	6	16.71	20.62	25.25	37.63	65.24	99.17	100.63	102.78	118.99	124.79
3	0	10.37	12.94	16.01	23.68	40.63	61.52	62.25	63.62	78.69	124.08
	2	7.99	9.97	12.33	18.22	31.28	47.35	48.32	48.97	60.58	95.51
	4	7.91	9.87	12.20	18.03	30.96	46.87	47.83	48.48	59.97	94.55
	6	6.56	8.18	10.01	14.76	25.66	38.86	39.61	40.18	49.72	78.37
4	0	11.78	14.69	18.18	26.88	46.13	69.82	70.66	72.21	89.31	140.83
	2	9.06	11.32	13.99	20.68	35.50	53.75	54.86	55.59	68.76	108.42
	4	8.97	11.20	13.85	20.47	35.14	53.21	54.31	55.03	68.07	107.15
	6	7.47	9.27	11.47	16.97	29.02	44.11	44.98	45.60	56.43	88.97
5	0	14.98	18.69	23.13	34.21	58.71	88.92	89.98	91.95	113.73	179.32
	2	11.53	14.40	17.81	26.33	45.19	68.45	69.86	70.79	87.56	138.05
	4	11.41	14.25	17.63	26.06	44.73	67.76	69.16	70.08	86.68	136.38
	6	9.47	11.81	14.61	21.62	37.07	56.17	57.54	58.28	71.85	113.28

.

As shown in Table 4, after the same procedure was applied to the other four parallel samples, the absolute values of natural frequencies differed among samples due to differences in branch length, radius, and fruit mass. Among them, short and thick branches exhibited higher overall frequencies because of their greater structural stiffness, whereas slender and long branches showed relatively lower frequencies due to increased flexibility. However, under different structural parameter conditions, the overall trend remained consistent: increasing the number of lateral leaves corresponded to a decrease in frequency, while reducing the number of lateral leaves corresponded to an increase in frequency. This indicates that the influence of lateral leaf number on the natural characteristics of the system has good stability and consistency.

At the same time, as the modal order increased, the frequency difference among different leaf number conditions gradually enlarged. Taking the representative sample as an example, the difference in the tenth-order frequency between the six-leaf and leafless conditions reached 61.65 Hz, whereas the difference in the first-order frequency was only 5.13 Hz; the former was approximately 12.0 times the latter. This indicates that variation in lateral leaf number has a greater influence on higher-order modes, suggesting that the role of leaves in system dynamic regulation is not limited to low-order dominant modes but is also significant in higher-order local responses.

In summary, the free-vibration tests showed that variation in lateral leaf number simultaneously affected the trajectory pattern of tracking point P, the distribution of natural frequencies, and the damping-related energy dissipation parameters. The mean first-order natural frequency decreased with increasing lateral leaf number, whereas the logarithmic decrement, equivalent damping ratio, and relative energy dissipation rate showed an increasing trend. These results indicate that lateral leaves not only changed the equivalent dynamic parameters of the system but also enhanced vibration attenuation during free decay. Among these conditions, the zero-leaf condition corresponded to higher natural frequencies and weaker leaf-related damping, the six-leaf condition corresponded to lower natural frequencies and stronger branch–leaf coupled dissipation, and the 2–4-leaf conditions exhibited obvious intermediate transition characteristics.

This decreasing trend of natural frequency with increasing lateral leaf number is generally consistent with the findings of Castro-García et al. [15], who reported that the presence of leaves on secondary orange branches reduced the first natural frequency and markedly suppressed vibration transmission. Although the present study focused on a walnut branch–leaf–fruit subsystem rather than citrus secondary branches, both studies indicate that leaves should not be regarded as negligible appendages in vibration harvesting analysis. The difference is that the present study further quantified the influence of lateral leaf number by comparing the zero-, two-, four-, and six-leaf conditions, showing that the frequency reduction was gradually strengthened as more lateral leaves were retained. This suggests that, for walnut vibration harvesting, the leaf retention state during the harvest period may directly affect the selection of excitation parameters.

4.3. Analysis of Forced-Motion Characteristics and Parameter Variations in the Branch–Leaf–Fruit Subsystem

As described in Section 3.2, a sinusoidal displacement excitation with a frequency of 20 Hz and an amplitude of 10 mm was applied along the X-direction to obtain the forced vibration response of the branch–leaf–fruit subsystem. Unlike the free-vibration test, which mainly reflected the inherent characteristics and decay behavior of the system, the forced-vibration test was used to analyze the directional response, non-excitation direction deviation, and trajectory morphology of the fruit tracking point under continuous external excitation. Therefore, this section focuses on the displacement response of tracking point P in the X-, Y-, and Z-directions under different lateral leaf number conditions. For the four typical lateral leaf number conditions of zero, two, four, and six leaves, the obtained steady-state trajectories are shown in Figure 5.

To avoid interference from the initial transient stage in the analysis, the fifth cycle was selected as the characteristic cycle for steady-state analysis. In this cycle, the amplitude variation between adjacent cycles had become relatively small, and the system response had entered a relatively stable state, thus reflecting the true forced response under continuous excitation more objectively. Based on the displacement data within the fifth cycle, the maximum displacements of tracking point $P$ in the $X$-, $Y$-, and $Z$-directions under different conditions were extracted, and the results are listed in

Table 5. Comparison of the maximum displacements of the fruit tracking point under different lateral leaf number conditions.

Number of Lateral Leaves	Maximum X Displacement/cm	Maximum Y Displacement/cm	Maximum Z Displacement/cm
0	9.10 ± 0.36	2.82 ± 0.62	4.30 ± 1.00
2	8.30 ± 0.19	1.72 ± 0.59	3.60 ± 0.80
4	7.60 ± 0.11	1.43 ± 0.41	2.50 ± 0.63
6	7.00 ± 0.07	1.24 ± 0.15	1.90 ± 0.23

.

As shown in Table 5, as the number of lateral leaves increased from zero to six, the maximum displacement of the tracking point in all three directions decreased, but the reductions differed markedly among directions. Along the excitation direction X, the mean maximum displacement decreased from 9.1 ± 0.36 cm under the leafless condition to 7.00 ± 0.07 cm under the six-leaf condition, corresponding to a reduction of 28.13%. Given that the input excitation amplitude was 10 mm, the response of the fruit end in the $X$-direction under all conditions did not completely coincide with the imposed displacement, and the following amplitude of the fruit end along the excitation direction further decreased as the number of lateral leaves increased. This indicates that the more complete the leaf retention, the less likely the system is to exhibit large synchronous following motion under continuous excitation.

In contrast, the reductions in the non-excitation directions were more pronounced. As the number of lateral leaves increased from zero to six, the maximum displacement in the Y-direction decreased from 2.82 ± 0.62 cm to 1.24 ± 0.15 cm, corresponding to a reduction of 65.43%. The maximum displacement in the Z-direction decreased from 4.3 ± 1.00 cm to 1.90 ± 0.23 cm, corresponding to a reduction of 67.45%. These results indicate that, with an increase in lateral leaf number, the suppression of vibration deviation in the non-excitation directions was much stronger than the weakening of the primary response in the excitation direction. In other words, the presence of lateral leaves did not simply reduce vibration in all directions; rather, it more effectively suppressed the additional deflections in the $Y$- and $Z$-directions, causing the forced response to become more concentrated in the main excitation direction in space.

The stronger suppression of the Y- and Z-direction responses observed in this study is also consistent with previous reports on vibration transmission in fruit-bearing branches. Castro-García et al. [15] found that leaves greatly suppressed acceleration transmission in orange branches, indicating that leaves can act as important energy-dissipating components during forced vibration. Similar to their conclusion, the present results show that the presence of lateral leaves reduced the spatial dispersion of the fruit-end response. However, unlike previous studies that mainly evaluated acceleration transmission along branches, the present study analyzed the three-dimensional displacement response of a local walnut branch–leaf–fruit subsystem and found that the suppression effect was more obvious in the non-excitation directions than in the main excitation direction. This provides a more direct explanation for why retained lateral leaves may improve the directionality of forced response during walnut vibration harvesting.

Further insight can be obtained from the motion trajectories shown in Figure 6. Under different lateral leaf number conditions, the trajectory pattern of tracking point $P$ differed significantly. Under the leafless condition, the trajectory exhibited a larger spread in the $Y$- and $Z$-directions, indicating a stronger spatially coupled deviation under continuous excitation. When the number of lateral leaves increased to two, the response in the non-excitation directions began to weaken, but the trajectory still showed obvious lateral expansion. Under the four-leaf condition, the trajectory further contracted toward the $X$-direction, indicating that the system had entered an intermediate stage from multidimensional dispersed response toward response concentration in the main direction. When the number of lateral leaves increased to six, the trajectory became the most concentrated, with the smallest spread in the non-excitation directions, indicating that under this condition, the lateral leaves exerted stronger constraint and energy-dissipation effects on non-dominant directional vibrations.

This trend is consistent with the evolution of lateral leaf motion patterns observed in Section 4.1. Under low-leaf-number conditions, leaf motion was dominated by swinging, and the system was more prone to deviations in the non-excitation directions. When a more complete leaf retention condition was maintained, the rotation and swinging of leaves and their coupling interaction with the branch became more sufficient, which was more favorable for suppressing additional vibration in non-dominant directions. It should be noted that this phenomenon should not be simply attributed to “larger aerodynamic damping leading to higher energy utilization efficiency” because the forced response was simultaneously affected by local constraints at the petiole connection, added mass distribution, and changes in leaf motion mode.

The same forced-vibration procedure was then applied to the other four parallel samples, with each condition repeated three times, in order to examine the consistency of the above laws among samples with different structural parameters. The results are shown in

Table 6. Repeated experimental results.

No.	Number of Lateral Leaves	Maximum X Displacement	Maximum Y Displacement	Maximum Z Displacement	No.	Number of Lateral Leaves	Maximum X Displacement	Maximum Y Displacement	Maximum Z Displacement
2	0	9.9	4.5	6.9	3	0	9.8	3.8	5.9
	2	8.7	3.3	5.5		2	8.6	2.6	4.0
	4	7.8	2.5	4.1		4	7.7	1.8	3.2
	6	7.0	1.1	1.6		6	6.9	1.5	2.2
4	0	10	4.1	6.5	5	0	9.9	3.7	6.2
	2	8.8	3.0	5.2		2	8.7	2.7	4.5
	4	7.9	2.2	3.9		4	7.8	1.9	3.4
	6	7.1	1.3	1.9		6	7.0	1.4	2.1

.

As shown in Table 6, although the parallel samples differed in branch length, radius, and fruit mass, the raw repeated test data showed the same trend as the statistical results in Table 4. For each parallel sample, the maximum displacement responses in the X-, Y-, and Z-directions generally decreased as the number of lateral leaves increased from zero to six. This consistency among samples supports the reliability of the statistical results and indicates that the effect of lateral leaf number on forced-vibration response was not limited to a single representative sample.

Overall, the forced-vibration tests showed that lateral leaf number had a significant effect on the displacement response of the branch–leaf–fruit subsystem. As the number of lateral leaves increased, the displacement response in the excitation direction weakened to some extent, whereas the suppression of deviation in the non-excitation directions was more pronounced. This was manifested as progressively more concentrated trajectories in the main excitation direction and weaker spatially dispersed responses. The zero-leaf condition corresponded to stronger multidimensional coupled deviation, the six-leaf condition corresponded to stronger directional constraint, and the 2–4-leaf conditions represented an intermediate transition stage between the two.

4.4. MATLAB-Based Analysis of Motion Trajectories of the Subsystem

In the original simulation model, lateral leaves were simplified as fixed equivalent mass–damping units connected to the main branch nodes, without explicitly considering the spin, swing, and time-varying coupling effects of leaves during vibration. The trajectory results obtained from this model are shown in Figure 6, and their projections in the $X$–$Z$ plane are shown in Figure 7. Overall, the original model could reflect the influence of the added mass of lateral leaves on the overall displacement distribution of the system, but the predicted motion trajectory patterns were obviously different from the experimental results.

Specifically, in the original simulation results, the leafless system exhibited a relatively regular ellipse-like trajectory. When the number of lateral leaves increased to six, the trajectory shifted downward by about 10.4 cm as a whole, showing an obvious vertical settlement feature. Corresponding displacement amplitude analysis showed that, after introducing lateral leaves, the amplitude in the $X$-direction decreased from 10.0 cm to 9.5 cm, a reduction of 5.0%, whereas the maximum response in the $Y$-direction increased from 0.050 cm to 0.287 cm, and the amplitude in the $Z$-direction increased from 3.0 cm to 6.5 cm, an increase of 116.7%. This indicates that, in the original model, the introduction of lateral leaves was mainly reflected in the effects of added mass and fixed damping on the system equilibrium position and the responses in non-excitation directions, but it failed to correctly reproduce the experimentally observed trend that “the more lateral leaves, the more strongly the $Y$- and $Z$-direction displacements are suppressed”.

From the perspective of response distribution, the original simulation results showed that under the leafless condition, most of the vibration response was concentrated in the $X$-direction excitation. After introducing lateral leaves, the response proportion in the $X$-direction decreased significantly, whereas the response in the $Z$-direction increased markedly, showing a tendency of response transfer from the excitation direction to the vertical direction. This result is inconsistent with the laws obtained from the forced-vibration experiments in Section 4.3. The experiments showed that as the number of lateral leaves increased from zero to six, the displacement responses of the fruit tracking point in the $Y$- and $Z$-direction non-excitation directions decreased significantly, whereas the original simulation predicted a strong vertical deviation. This indicates that although simplifying the lateral leaves as fixed lumped mass–damping elements can describe their influence on the additional load of the system, it is insufficient to characterize the regulatory role of real leaf motion patterns on the directional forced response.

The main reason for the discrepancy between the simulations and experiments lies in the simplification of the model assumptions. First, in the experiments, leaf motion under the six-leaf condition was dominated by spin, whereas under the 2–4-leaf conditions it showed a compound pattern of swing and spin, indicating that the leaf energy dissipation mode changed with the number of leaves. However, the original simulation model used the same type of fixed damping parameter for all leaf-bearing conditions and did not reflect the time-varying damping and coupling effects corresponding to differences in leaf motion patterns. Second, the original model emphasized the static settlement and vertical deviation caused by added mass while insufficiently considering the local constraints at the petiole connection, the aerodynamic energy dissipation induced by leaf rotation, and the reconstruction of system parameters corresponding to different leaf numbers. As a result, the $Z$-direction response was overestimated and the ability of the model to explain the experimental trends was weakened.

To reduce the discrepancy between the original MATLAB simulation and the forced-vibration experimental results, an equivalent damping correction was introduced into the lateral leaf model. In the original model, lateral leaves were simplified as fixed lumped mass–damping units, which could not fully represent the additional energy dissipation caused by leaf swing, spin, and branch–leaf coupling. Therefore, the corrected equivalent damping coefficient was expressed as $c_{N,\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}={\lambda }_N{c}_N$, where $c_N$ is the original equivalent damping coefficient under the $N$-leaf condition and ${\lambda }_N$ is a leaf-number-dependent damping correction coefficient related to the observed leaf motion pattern. The correction coefficient was calibrated by reducing the relative error between the simulated and experimentally measured maximum displacements of tracking point $P$, especially in the Y- and Z-direction non-excitation responses. The relative error was calculated as $RE=\mid A_{\mathrm{s}\mathrm{i}\mathrm{m}}-A_{\mathrm{e}\mathrm{x}\mathrm{p}}\mid /A_{\mathrm{e}\mathrm{x}\mathrm{p}}\times 100\%$, where $A_{\mathrm{s}\mathrm{i}\mathrm{m}}$ and $A_{\mathrm{e}\mathrm{x}\mathrm{p}}$ are the simulated and experimental maximum displacements, respectively.

For the six-leaf condition, the original model predicted maximum displacements of 0.287 cm and 6.50 cm in the Y- and Z-directions, whereas the corresponding experimental values were 1.31 ± 0.15 cm and 1.94 ± 0.23 cm, respectively. The relative errors were therefore 78.1% and 235.1%, indicating that the original model could not accurately reproduce the non-excitation direction responses. After introducing the equivalent damping correction, the predicted maximum displacements in the Y- and Z-directions were 1.20 cm and 1.80 cm, and the corresponding errors decreased to 8.4% and 7.2%, respectively. These results indicate that the corrected model improved the prediction accuracy of the non-excitation direction responses. However, this correction should be regarded as an equivalent empirical treatment rather than a complete description of the fluid–structure coupling and nonlinear petiole motion of lateral leaves.

Taken together, the overall comparison among the zero-, two-, four-, and six-lateral-leaf conditions shows that the trajectory differences corresponding to different leaf numbers were relatively small in the original simulation. The fundamental reason is that the model did not distinguish the changes in leaf motion patterns and energy dissipation modes under different leaf number conditions, nor did it adequately represent the synchronous reconstruction of system mass, stiffness, and damping caused by changes in lateral leaf number. In contrast, after introducing the equivalent damping correction coefficient, the corrected model more reasonably reflected the experimentally observed trend that the displacement responses in the non-excitation directions decreased with increasing lateral leaf number. This indicates that in the forced-vibration analysis of the branch–leaf–fruit subsystem, if the evolution of lateral leaf posture and its corresponding time-varying energy dissipation effects are ignored, the model will be unable to accurately explain experimentally observed phenomena such as trajectory convergence, enhanced directionality, and suppression in the non-excitation directions.

In summary, the MATLAB simulation analysis indicates that the influence of lateral leaves on the system motion trajectory cannot be simply attributed to static settlement caused by added mass but should instead be understood as the combined result of added mass, local constraints, and motion-pattern-related energy dissipation. The original model can reflect the basic influence of the presence of lateral leaves on the system response, but its explanatory ability for trajectory morphology and directional differences under different leaf number conditions is limited. By introducing the leaf-number-dependent equivalent damping correction coefficient and using the relative error of maximum displacement as the validation criterion, the corrected model showed improved consistency with the experimental trends, especially in the Y- and Z-direction non-excitation responses under the six-leaf condition. This suggests that to further enhance model accuracy, future work still needs to improve the representation of leaf aerodynamic energy dissipation, petiole-connection nonlinearity, and leaf-group coupling effects.

5. Conclusions

This study investigated the influence of lateral leaf number on the dynamic characteristics of a local walnut branch–leaf–fruit subsystem by combining discrete dynamic modeling, free-vibration tests, forced-vibration tests, and MATLAB simulation. The results showed that lateral leaves changed both the natural characteristics and forced response directionality of the subsystem. As the number of lateral leaves increased, the natural frequency decreased, while the damping-related energy dissipation and suppression of non-excitation direction responses became more pronounced. Lateral leaves mainly exhibited three motion patterns during vibration, namely, spin, swing, and spin–swing compound motion. The six-leaf condition was dominated by spin motion, whereas the 2–4-leaf conditions showed a transitional state involving swing and spin–swing compound motion.

The forced-vibration results further showed that increasing lateral leaf number reduced the spatial dispersion of the fruit-end response. Compared with the leafless condition, the six-leaf condition produced stronger suppression of the Y- and Z-direction displacement responses, indicating that lateral leaves helped concentrate the forced response toward the main excitation direction. This finding has practical relevance for walnut vibration harvesting: the leaf retention state during the harvest period should be considered when selecting excitation parameters because retained leaves may alter the effective frequency response, damping level, and response directionality of the branch–leaf–fruit subsystem.

The MATLAB simulation showed that simplifying lateral leaves only as fixed lumped mass–damping units was insufficient to reproduce the experimentally observed suppression of non-excitation direction responses. After introducing an equivalent damping correction related to leaf motion patterns, the simulation results became more consistent with the experimental trends, suggesting that leaf-induced energy dissipation should be included in future vibration response models. These results provide a reference for selecting excitation frequency, avoiding unfavorable low-order response conditions, and optimizing low-damage walnut vibration harvesting devices.

Future work will introduce more advanced numerical methods, such as the finite element method (FEM), discrete element method (DEM), FEM–DEM coupling, and fluid–structure interaction simulation, to better describe branch flexibility, petiole connection nonlinearity, leaf–air interaction, and leaf-group coupling effects during vibration harvesting.