Parameter optimization for vibratory harvesting of wolfberry branches based on dual low-frequency vibration excitation and singular value spectrum

Abstract

To overcome empirical and discrete parameter selection and severe energy attenuation in wolfberry (Lycium barbarum L.) harvesting using single-source vibration, in this study, a dual-source low-frequency excitation method is proposed. Using ‘Ningqi No.7’ branches and a two-point synchronous excitation device, the effects of the amplitude (28 to 80 mm) and phase (-180° to 180°) of the upper and lower vibration sources (UVS and LVS) on the detachment percentages of the middle section, lower section, and the total detachment (TD) were investigated via response surface methodology. Singular value spectrum analysis of the acceleration signals extracted the maximum singular value (MSV) to quantify the overall branch vibration energy. Two main low-frequency modes near 4 Hz and 8 Hz with high damping were identified. The MSV was strongly correlated with TD (r = 0.751), confirming its reliability for effectiveness evaluation. The optimal parameters found were a UVS of 80 mm, LVS of 68 mm, and phase of 135°, yielding a TD of 85.9% in validation. This demonstrates that the synergistic control of amplitude and phase at a low frequency enhances the harvest efficiency, offering a new approach for intelligent parameter optimization based on vibration monitoring.

Full Article

Parameter Optimization for Vibratory Harvesting of Wolfberry Branches Based on Dual Low-Frequency Vibration Excitation and Singular Value Spectrum

Guangrui Hu  ,^a,† Shilong Feng,^b,† Qianwen Kou,^b Shuling Zhang  ,^b Yun Chen,^b

Lizhu Jin  ,^b Teng Zhang,^b Li Zhao,^b and Lingxin Bu  ^b,*

To overcome empirical and discrete parameter selection and severe energy attenuation in wolfberry (Lycium barbarum L.) harvesting using single-source vibration, in this study, a dual-source low-frequency excitation method is proposed. Using ‘Ningqi No.7’ branches and a two-point synchronous excitation device, the effects of the amplitude (28 to 80 mm) and phase (-180° to 180°) of the upper and lower vibration sources (UVS and LVS) on the detachment percentages of the middle section, lower section, and the total detachment (TD) were investigated via response surface methodology. Singular value spectrum analysis of the acceleration signals extracted the maximum singular value (MSV) to quantify the overall branch vibration energy. Two main low-frequency modes near 4 Hz and 8 Hz with high damping were identified. The MSV was strongly correlated with TD (r = 0.751), confirming its reliability for effectiveness evaluation. The optimal parameters found were a UVS of 80 mm, LVS of 68 mm, and phase of 135°, yielding a TD of 85.9% in validation. This demonstrates that the synergistic control of amplitude and phase at a low frequency enhances the harvest efficiency, offering a new approach for intelligent parameter optimization based on vibration monitoring.

DOI: 10.15376/biores.21.2.3931-3953

Keywords: Lycium barbarum L.; Vibrating harvest; Parameter experiment; Singular value decomposition; Response surface methodology

Contact information: a: School of Design, Xi’an Technological University, Xi’an; b: College of Mechatronic Engineering, North Minzu University, Yinchuan;

* Corresponding author: 2021023@nmu.edu.cn;

†These authors contributed equally to this work and should be considered co-first authors.

INTRODUCTION

Wolfberries (Lycium barbarum L.) are recognized worldwide as a functional food with medicinal and nutritional properties. According to the Report on the High-Quality Development of China’s Modern Wolfberry Industry (2024), China’s total fresh wolfberry production reached 1.40 million tons in 2023, and dried wolfberry production amounted to 240,000 tons. Export revenue reached RMB 750 million, representing a year-on-year increase of 23.1% compared to 2022. Gansu, Qinghai, Ningxia, and Xinjiang Provinces account for more than 98% of the wolfberry planting area in China. In major wolfberry-producing counties, approximately 50% of farmers’ average operating income is derived from wolfberries. Thus, this industry is both a characteristic and advantageous sector in northwestern China. It is of significant importance for regional industrial structure optimization and income growth among farmers and herders. Wolfberries are predominantly harvested manually, which is highly labor-intensive. Harvesting labor accounts for more than 50% of the total labor input in wolfberry production (Li et al. 2024) and harvesting costs exceed 40% of the total labor cost (Chen et al. 2025a). Consequently, the development of mechanized harvesting technologies has become an urgent need in the wolfberry industry.

Wolfberry harvesting device types include comb-brush, pneumatic, brush–vibration, and vibratory. Handheld comb-brush harvesters have low efficiency and impose a labor intensity comparable to that of manual harvesting (Jiang et al. 2024). Pneumatic harvesters require high energy consumption (Chen et al. 2021). Brush–vibration harvesters, in which rubber rollers simultaneously rotate and reciprocate vertically to strike wolfberry branches, cause direct impact damage to the fruit (Zhao et al. 2021a). Vibratory harvesting devices have attracted increasing attention because of their simple structure and high operational efficiency. Table 1 summarizes the key parameters and harvesting performance of recently developed vibratory wolfberry harvesters. Previous studies have focused on the picking and damage percentages of ripe fruit and the detachment percentage of unripe fruit. However, the selection of critical excitation parameters, such as the vibration frequency (2.5 to 48 Hz) and amplitude (15 to 70 mm), relies heavily on researchers’ experience, resulting in substantial discrepancies across studies. Although increasing the vibration frequency and amplitude facilitate the fruit detachment to some extent, it also leads to higher percentages of unripe fruit detachment, increased damage to ripe fruit, and a greater risk of plant structural damage (Deng et al. 2026).

Table 1. Research Status of Selected Vibratory Wolfberry Harvesting Devices

To better match the excitation parameters with plant characteristics, previous studies have primarily employed single-source excitation to determine the mechanical properties of wolfberry plants, focusing on their natural frequencies and energy transmission behavior. Zhao et al. (2021) developed a finite element model for wolfberries based on a transversely isotropic constitutive model. Modal analysis has been conducted to determine the natural frequencies of different modes, and experimental modal testing using accelerometers and an impact hammer have been performed to identify the resonance frequencies of wolfberries. However, variations in plant architecture and the simplifications introduced during model construction significantly affect the accuracy of modal analysis results (Macoretta et al. 2025). He et al. (2018) investigated the acceleration responses of fruit-bearing branches when excitation was applied at different locations on the plant. They showed that acceleration attenuation from primary third-order branches to fruit-bearing branches was approximately five-fold, and that attenuation from lateral third-order branches to fruit-bearing branches reached nearly six-fold. Wang et al. (2018) analyzed the relative motion between wolfberry fruit and branches during the detachment process using high-speed photography and revealed that fruit on the same fruit-bearing branch detached in different sequences, with fruit located closer to the vibration source detaching earlier. These studies collectively indicate that wolfberries undergo forced vibration under external excitation, and that energy transmission along branches is strongly influenced by internal damping and external factors, such as flowers, leaves, and fruit. As a result, excitation forces experience severe attenuation along the branch, making it difficult for vibrational energy to effectively propagate to distal regions, which leads to poor fruit detachment at locations far from the vibration source (Sola-Guirado et al. 2022).

Given the risk of plant damage associated with excessively high vibration frequencies and the inefficiency of fruit detachment under single-source excitation (Deng et al. 2025), prolonged operation of vibratory harvesting devices has been reported to induce numbness in the upper limbs of operators (Zhao et al. 2021b). Vibration exposure has been demonstrated to cause temporary alterations in vibrotactile sensitivity (Morioka and Griffin 2002). Specifically, exposure to vibration at 31.5 Hz and above for 32 min results in a temporary threshold shift (TTS) in the vibration perception threshold (VPT), accompanied by paresthesia and numbness (Malchaire et al. 1998). Therefore, it is essential to reduce the vibration frequency and limit the continuous working duration of operators. This study proposes the hypothesis: for wolfberry branches excited at a fixed low frequency (near the first-order natural frequency), the simultaneous application of two vibration sources with controlled amplitudes and phase relationships can create a superimposed vibration state that enhances the overall branch vibration energy and consequently improves the fruit detachment percentage, compared to single-source excitation at the same frequency. To test this hypothesis, this study explored the application of dual-source excitation to wolfberry branches. The specific objectives of this study were to: (1) characterize the low-frequency vibration modes and damping properties of wolfberry branches using singular value spectrum analysis; (2) investigate the effects of the amplitudes of the two vibration sources and their phase relationship on ripe fruit detachment percentages through response surface methodology; and (3) establish a quantitative correlation between the maximum singular value (MSV) extracted from the acceleration signals and the fruit detachment percentage, thereby evaluating the feasibility of using MSV as a vibration-state indicator for parameter optimization.

EXPERIMENTAL METHODS

Fresh Samples and Experimental Device

Field experiments were conducted from October 1 to 4, 2025, on wolfberry cultivar ‘Ningqi No. 7’ in an orchard located in Baiqiao Village, Zhongwei City, Ningxia Hui Autonomous Region, China (105.20° E, 37.56° N). Figure 1 shows the structure of the experimental apparatus. The linear motion module was equipped with two movable platforms, each mounting a crank–slider mechanism driven by a DC motor (DC 24 V, frequency range 0 to 4 Hz), which served as the upper and lower vibration sources (UVS and LVS). Branches bearing a relatively large amount of fruit were selected from orchard plants, cut, and fixed at the cut end to the experimental device so that the branch could hang freely under natural gravity. The heights of the two crank–slider mechanisms were adjusted to positions corresponding to the upper and lower one-third of the total branch length. The branch was secured to the ends of the crank–slider mechanisms using cable ties. Thus, the branch was divided into upper, middle, and lower sections, and the number of fruits in each section was recorded before and after the experiment. As the upper section contained few fruits, dynamic accelerometers were installed at the midpoints of the middle and lower sections of the branch. Acceleration signals were acquired and processed using a COCO-90 dynamic signal analyzer (Crystal Instruments Co., Ltd., Santa Clara, CA, USA). Branch morphological parameters, including the branch diameters at the vibration source fixation points (P_d1 and P_d2), branch diameters at the upper and lower endpoints (P_s1 and P_s2, denoted as D_d1, D_d2, D_s1, and D_s2), and the total branch length (L) were recorded as characteristic descriptors.

Fig. 1. Overview of the test device: (1) linear motion module, (2) upper vibration source (UVS); (3) lower vibration source (LVS), (4) base frame, (5) mobile platform, (6) upper acceleration sensor (UAS), and (7) lower acceleration sensor (LAS)

Experimental Design

In preliminary experiments, fruit detachment was evaluated under conditions in which both vibration sources operated at the maximum amplitude of 80 mm and frequencies of 1, 2, 3, and 3.5 Hz. The fruit detachment percentage remained below 20% at all of these frequencies. In contrast, fruit detachment increased when the excitation frequency increased to 4 Hz. This improvement is likely attributable to the proximity of 4 Hz to the natural frequency of wolfberry branches (Su et al. 2025). Thus, the excitation frequencies of both vibration sources were fixed at 4 Hz in subsequent experiments. It should be noted that 4 Hz was the upper operational frequency limit of the DC motor (DC 24 V) used in this study, which physically constrained the frequency range that could be explored. The UVS and LVS employed identical crank–slider mechanisms, as illustrated in Fig. 2(a). The lengths of connecting rod 2 and reciprocating rod 3 (L₂ and L₃) were 85 mm and 235 mm, respectively. The length of crank 1 (L₁) could be adjusted within 14 to 40 mm by changing the position of joint 1, resulting in a displacement amplitude of 28 to 80 mm for the reciprocating rod. In addition, the excitation phase relationship between the UVS and LVS was considered, as it could alter the vibration state of the branch and thereby influence fruit detachment. Three phase conditions, namely 180°, 0°, and −180°, were established, as shown in Figs. 2(b), (c), and (d), respectively.

Fig. 2. Structure of the vibratory excitation device and phase configurations of the upper and lower vibration sources during the experiment

Using Design-Expert software V13, a three-factor, three-level Box-Behnken design was developed with the amplitudes of the UVS and LVS and the excitation phase as the independent variables. A quadratic regression response surface methodology (RSM) was applied to determine the responses of the fruit detachment percentage in the middle (DPM, Y₁) and lower parts of the branches (DPL, Y₂), and the total fruit detachment percentage (TD, Y₃) to the experimental factors.

Each experimental condition was repeated five times. Each vibration test lasted 10 s. The total number of ripe fruits in the middle (N₁) and lower parts of the branch (N₂) were recorded before each test, and the remaining number of ripe fruits in the middle (n₁) and lower parts (n₂) were recorded after the test. The fruit detachment percentages were calculated using Eqs. 1 to 3:

where i denotes the replicate number within each experimental condition, and i = 1 to 5.

The response variables were fitted using a general quadratic polynomial model, expressed in Eq. 4 (Bu et al. 2020). Table 2 presents coded levels of the experimental factors.

where Y is the response variable measured for each combination of factors; and β₀, β_i, β_ii, and β_ij are terms of regression coefficients for intercept, linearity, square, and interaction, respectively.

Table 2. Factor Codes of the Independent Variable Levels

Acceleration Signal Processing and Singular Value Spectrum Calculation

Acceleration signals recorded by the upper and lower accelerometers (UAS and LAS, respectively) were used to compute singular value spectra under different experimental conditions, providing a quantitative representation of branch vibration states. Singular value decomposition (SVD) is based on energy with the characteristic that useful signal singular values are significantly larger than noise singular values, which makes it suitable for vibration signal denoising (Liu et al. 2017). SVD is commonly employed in engineering applications for identifying natural frequencies, ranking modal energy contributions, and estimating system damping (Miao et al. 2015). In the COCO-90 vibration signal analyzer, a Hanning window was applied to the time-domain acceleration responses of the wolfberry branches to suppress spectral leakage. The signals from different channels were superimposed using Post Analyzer software (Version: 2016) to enhance the strength of the effective signals. The processed signals were truncated to remove invalid data, such as noise at the beginning and end of the signals and abnormal interference segments, retaining only valid vibration response data. Non-stationary signals were denoised using wavelet processing in MATLAB 2023. This approach removed noise components (e.g., hardware noise, environmental interference) and preserved key features of the vibration signal, including natural frequencies, damping, and time/frequency-domain characteristics related to modal shapes. After multiple practical validations and a comparison of reconstructed signals at different decomposition levels, a decomposition level of five was selected for analysis, as shown in Fig. 3.

Fig. 3. Signal decomposition process

In the wavelet-based de-noising procedure, the bior3.5 biorthogonal wavelet was selected to maximize the fidelity of the denoised signal to the original time-domain waveform and to preserve the physical characteristics of branch vibration. This selection enabled accurate decomposition and reconstruction of the vibration waveform and retained signal details, particularly for analyses of time-domain decay features, such as damping ratio estimation and modal identification of wolfberry branches. The branch vibration signal was decomposed into low-frequency coefficients, which represent the main vibrational components (e.g., waveforms corresponding to natural frequencies), and high-frequency coefficients, which contain noise (e.g., wind, electronic interference) and local branch features. Soft-threshold function S_Th(Z_K) was defined. Threshold T_h > 0 was applied to the wavelet coefficients, such that coefficients with absolute values smaller than T_h were set to zero and coefficients with absolute values larger than Th (|Z_K| > T_h) were processed according to Eq. 5:

where N is the signal length and σ is the standard deviation of the noise, estimated as σ = MAD(b₁) / 0.6745 with MAD representing the median absolute deviation. Specifically, the median of b₁ was computed, and the absolute deviations of each element in b1 were then calculated from this median to form a new list. The median of this new list was the MAD value.

Using soft-threshold function S_Th(Z_K), the high-frequency coefficients—where noise was primarily concentrated—were either set to zero or shrunk, and the coefficients corresponding to valid signal components were retained. In the high-frequency coefficients of wolfberry branch vibrations, noise appeared as small, irregular-amplitude coefficients, and local branch vibrations exhibited larger, regular-amplitude coefficients. The soft-threshold function effectively removed small-amplitude noise coefficients and preserved significant feature coefficients, achieving smooth shrinkage of coefficients and producing a denoised signal that more accurately reflected the actual branch vibration waveform. Low-frequency coefficients were retained unchanged, and an inverse wavelet transform was applied to reconstruct a clean signal with the noise removed. Covariance-driven frequency-domain modal analysis was then conducted. The acceleration responses from the upper and lower accelerometers were combined into an information matrix, and their cross-power spectral density matrix was computed, capturing amplitude and phase relationships (covariance information) between the two sensors. SVD was applied to this cross-power spectral density matrix at each frequency point to obtain the singular value spectrum (Klema and Laub 1980).

Table 3. Core Parameters for Singular Value Spectrum Calculation

The time-domain acceleration signals of wolfberry branches measured by the UAS and LAS were denoted as x₁(t) and x₂(t), respectively, with sampling frequency F_s and total number of samples L_s. Table 3 summarizes the core parameters used in the analysis are (Wall et al. 2003). The preprocessed signals were segmented with a 50% window overlap to ensure smoother spectral continuity between adjacent frames. Each segment was windowed individually, and a discrete Fourier transform was used to obtain the frequency-domain complex values:

The frequency-domain values of the K segments were averaged and normalized to obtain the cross-power spectral density at frequency f_m. The positive semi-definite (PSD) matrix was computed according to Eq. 8,

where X^* denotes the complex conjugate.

For m-th frequency point f_m, a 2 × 2 complex covariance matrix corresponding to this single frequency point was extracted from the PSD matrix, denoted as follows:

Due to numerical computation errors, H_m may not satisfy the Hermitian symmetry requirement,  ; that is, H^H is the conjugate transpose. Therefore, a correction must be applied:

After decomposition, elements σ_m1 and σ_m2 on the diagonal of the diagonal matrix correspond to the first and second singular values, respectively. The singular values across the entire frequency range were obtained by repeating this procedure.

RESULTS AND DISCUSSION

Data Statistics

Based on the experimental design in Design-Expert 13, 17 experimental settings were evaluated, each with five replicates. Y₁, Y₂, and Y₃ were calculated based on the total number of fruits in the middle and lower sections before the test and the number of fruits remaining after the test. The MSV of each test was calculated based on the acceleration data recorded by the sensors.

Table 4. Statistical Summary of the Experimental Results

The mean ± standard deviation of N₁, N₂, L, D_S1, D_d1, D_d2, and D_s2 were 9.27 ± 4.75, 7.35 ± 4.49, 480.71 ± 71.01, 2.60 ± 0.49, 2.18 ± 0.44, 1.61 ± 0.33, and 0.73 ± 0.20 mm, respectively. Table 4 summarizes the experimental and statistical results.

Correlation Analysis between the Maximum Singular Value and Other Parameters

Using SPSS software (IBM SPSS 27.0, Chicago, IL, USA), multiple linear regression analysis was conducted with the MSV as the dependent variable. The excitation parameters (X₁, X₂, and X₃), response outcomes (Y₁, Y₂, and Y₃), and branch morphological parameters (L, D_S1, D_d1, D_d2, and D_s2) were treated as independent variables. Figure 4 shows the Pearson correlation heatmap. The correlation coefficients between the MSV and branch morphological parameters (L, D_S1, D_d1, D_d2, and D_s2) were all less than or equal to the absolute value of 0.125, indicating extremely weak or negligible linear correlations.

Fig. 4. Pearson correlation coefficients among maximum singular value, excitation parameters, response outcomes, and branch morphological parameters

In contrast, the fruit detachment percentages exhibited stronger correlations with the MSV. The Pearson correlation coefficients of Y₁, Y₂, and Y₃ with the MSV were 0.761, 0.565, and 0.751, respectively, indicating the strong positive linear relationship of the MSV with Y₁ and Y₃. This suggests that increases in the MSV correspond to a synchronous increase in these detachment percentages. The Y₂ had a moderate positive correlation with the MSV but a weaker association compared with Y₁ and Y₃. These results demonstrate a strong positive relationship between the branch vibration state and harvesting performance, suggesting that the MSV can serve as an indicator of wolfberry harvesting efficiency and can be used to optimize excitation parameters during non-harvest periods or in simulation environments.

Regarding the excitation parameters, the linear correlation coefficients of X₁, X₂, and X₃ with the MSV were 0.772, 0.437, and 0.075, respectively, indicating that the amplitude of the excitation had a moderate positive effect on the MSV, that the secondary amplitude had a weak positive effect, and that the phase had a negligible negative effect. This shows that excitation amplitude has a more significant impact on the MSV than the excitation phase. Significant correlations were observed between excitation parameters (X₁, X₂, and X₃) and response outcomes (Y₁, Y₂, and Y₃), which were further quantitatively analyzed using RSM (Section of Multiple Regression Analysis of Excitation Parameters).

Singular Value Spectrum Analysis

The upper curve represents the first singular value, corresponding to the component with the strongest energy at each frequency and reflecting the true physical mode of the system, i.e., the natural frequency. The lower curve represents the second singular value, corresponding to the next-strongest energy component, which is typically associated with noise, computational modes, or closely spaced modes. Using test 14-2 in Table 4 as an example, Fig. 5(a) shows that the first singular value exhibited a peak near 4 Hz (MSV approximately 2.15) and a smaller secondary peak near 8 Hz, indicating the presence of two separable modes. For wolfberry branches measured with a two-channel setup, the primary modal energy in low-order modes was concentrated in the first singular value. The peak frequency of the first singular value in the singular value spectrum is a key indicator for identifying the branch’s natural frequency. If the second singular value exhibits a peak coinciding with the first singular value peak or an independent secondary peak with significant energy, then it can be used to validate the natural frequency location. Figures 5(b) and 5(c), showing amplitude–frequency plots for the upper and lower acceleration sensors (UAS and LAS), display clear peaks near 4 Hz and 8 Hz, thereby improving the reliability of natural frequency identification. These two low-order modal frequencies are similar to the results reported by Su et al. (2025). An excitation frequency of 4 Hz almost coincides with the first natural frequency, producing a “quasi-resonance” effect, indicating that a relatively large vibration response can be achieved with minimal input energy under appropriate vibration frequency, phase, and location conditions. As shown in Fig. 5(a), the first singular value decreases sharply after the 4 Hz peak and drops to only 23.95% of the maximum value by 8 Hz, indicating substantial damping in the branch. Figures 5(b) and 5(c) show that the 4 Hz peak has a notable width, rather than a sharp “needle-like” spike, confirming the presence of damping, consistent with the findings of So (2003).

A notable difference was observed between the amplitude–frequency spectra of the upper and lower acceleration sensors. The spectrum from the LAS (Fig. 5(c)) was visibly smoother and simpler than that from the UAS (Fig. 5(b)), with the minor spectral features largely absent and only the dominant peaks near 4 Hz and 8 Hz remaining prominent. This phenomenon can be attributed to the frequency-dependent attenuation characteristics of the wolfberry branch as a viscoelastic biomaterial. As vibration energy propagates along the branch toward the distal (lower) region, higher-frequency components and minor vibrational modes are preferentially dissipated because the damping coefficient of plant tissues generally increases with frequency. As a result, by the time the vibration signal reaches the lower sensor position, the minor modes have largely dissipated, leaving a smoother spectrum dominated by the fundamental mode near 4 Hz. This phenomenon is consistent with the high damping characteristics revealed by the singular value spectrum analysis and the energy attenuation behavior documented in previous studies (He et al. 2018; Sola-Guirado et al. 2022).

(a)

(b)

(c)

Fig. 5. Acceleration signal processing results for test 14-2: (a) singular value spectrum, (b) amplitude–frequency plot of the upper acceleration sensor, and (c) amplitude–frequency plot of the lower acceleration sensor

Multiple Regression Analysis of Excitation Parameters

Response surface analysis of DPM (Y₁)

Table 5 reports the analysis of variance (ANOVA) results used to generate the quadratic model. X₁, X₂, X₃, X₁X₂, and X₂X₃ had significant effects (P < 0.05) on the DPM. The coefficients of the DPM model responses using the factor codes as variables were expressed as follows:

Table 5. ANOVA Results of the Quadratic Model Response to DPM

RSM was used to analyze the interactive effects of the amplitudes of the UVS and LVS and the excitation phase on DPM. As shown in Fig. 6(a), the interaction between the amplitudes of the UVS and LVS had a significant effect on DPM. When the excitation phase was zero and the LVS amplitude was high, DPM initially increased and then decreased with the increasing UVS amplitude. Figure 6(c) shows that when the UVS amplitude was zero, there was an interactive effect of the excitation phase and LVS amplitude on DPM. Specifically, when the excitation phase was low, DPM first increased and then decreased as the LVS amplitude increased. The middle section of the branch was subjected to the combined excitation of the UVS and LVS, achieving a superimposed vibration state. Changes in the amplitudes of the UVS and LVS and excitation phase altered the branch vibration mode. When the branch vibration pattern resembled a specific modal shape, it resulted in intense vibration, which was reflected by higher MSVs and increased fruit detachment percentages. Figure 6(b) shows that when the LVS amplitude was zero, DPM decreased as the UVS amplitude and excitation phase decreased. The influence of the UVS amplitude on DPM was slightly greater than that of the excitation phase.

Fig. 6. Response surfaces of the effects of interactive factors on DPM

Response surface analysis of DPL

Table 6 shows the ANOVA results used to determine the quadratic model. X₁, X₂, and X₁X₂ exhibited significant effects (P < 0.05) on the multiples index.

Table 6. ANOVA Results of the Quadratic Model Response to DPL

The coefficients of DPL model responses (Y₂) using the factor codes as variables were expressed as follows:

As shown in Fig. 7(a), the interaction between the UVS and LVS amplitudes had a significant effect on DPL. The lowest DPL was observed when both the UVS and LVS amplitudes were at low levels. According to Fig. 7(b), for a given excitation phase, DPL increased rapidly with an increasing UVS amplitude. Figure 7(c) shows that the variation in DPL considering the LVS amplitude was similar to the trend observed in Fig. 6(b) for DPL considering the UVS amplitude.

Fig. 7. Response surfaces of the effects of interactive factors on DPL

Response Surface Analysis of TD

Table 7 reports the ANOVA results used to determine the quadratic model. X₁, X₂, X₃, X₁X₂, X₂², and X₃² had significant effects (P < 0.05) on TD. The coefficients of the TDR model responses (Y₃) using the factor codes as variables were expressed as follows:

Table 7. ANOVA Results of the Quadratic Model Response to TD

For TD, the UVS and LVS amplitudes exhibited an interactive effect. As shown in Fig. 8(a), when the excitation phase was zero, TD increased with the LVS amplitude for a given UVS amplitude, and this increasing trend was more pronounced when the UVS amplitude was lower. When the LVS amplitude was high, changes in TDR with varying UVS amplitudes were relatively small, remaining at high values. Furthermore, as shown in Figs. 8(b) and 8(c), TD reached its maximum when the excitation phase was high and both the LVS and UVS amplitudes were high.

Fig. 8. Response surfaces of the effects of interactive factors on TD

Optimization analysis and experimental validation

To achieve optimal harvesting performance, parameter optimization was conducted using Design-Expert 13.0 software with factors X₁, X₂, and X₃ and the objectives of maximizing DPM (Y₁), DPL (Y₂), and TD (Y₃). The predicted optimal results showed a DPM of 0.920, DPL of 0.804, and TD of 0.868 when the UVS amplitude, LVS amplitude, and excitation phase were set at 79.43 mm, 67.39 mm, and 137.16°, respectively. Based on these optimized parameters, experimental trials were conducted with five replicates, the results of which are summarized in Table 8. The experimental outcomes were closely matched with the predicted values with relative deviations of 6.20% for DPM, 5.97% for DPL, and 1.04% for TD. A TD of 85.9% demonstrated acceptable harvesting performance under low-frequency vibration conditions, which is comparable to the harvesting percentages reported in recent studies (Chen et al. 2024a; Mei et al. 2024; Wei et al. 2025). The vibration frequency remains a critical factor influencing wolfberry fruit detachment.

Table 8. Comparison of the Simulation Test and Experimental Results