Vibration–Collision Coupling Modeling in Grape Clusters for Non-Damage Harvesting Operations

Abstract

In the table grape production process, issues such as berry detachment and damage caused by cluster vibrations and berry collisions are significant challenges. To investigate the underlying mechanisms and dynamics of these phenomena, a vibration–collision coupling method for table grape clusters was developed. Based on the vibration model of a grape cluster, the smallest vibration–collision coupling unit—referred to as the dual-twig–berry system—was proposed. This system was described using a “(viscoelastic hinge)–(rigid bar)–(flexible sphere)–(viscoelastic link)” model. The dynamic vibration–collision coupling equation of the dual-twig–berry system was derived by incorporating expressions for the viscoelastic vibration of the twigs, viscoelastic collision of the berries, and a generalized collision force (based on the Kelvin model) into the framework of the Lagrange equation. A computational-simulation method for solving this dynamic vibration–collision coupling equation was also developed. The simulation results revealed that the vibration–collision coupling pattern exhibited a shorter vibration period, smaller vibration amplitude, and higher vibration frequency compared to the vibration pattern without coupling. A reduction in vibration amplitude mitigates berry detachment caused by excessive instantaneous loads. However, the increase in vibration frequency exacerbates berry detachment due to fatigue and causes varying degrees of berry damage. This study provides a theoretical foundation for understanding the mechanisms of berry detachment and damage, offering valuable insights for mitigating these issues in table grape production.

1. Introduction

Global grape production has expanded significantly over the past 40 years. According to the Food and Agriculture Organization of the United Nations (FAO), total grape production in 2019 reached 77.137 million tons, with table grapes accounting for approximately 33% of this volume [1]. However, traditional mechanical methods for table grape production are unsuitable due to their propensity to cause damage, necessitating substantial labor inputs to ensure damage-free handling. The production process for table grapes includes harvesting, post-harvest operations such as sorting, packing, transportation, and other handling activities at both the farm and industrial levels [2,3]. Among these challenges, berry detachment and damage during handling significantly reduce the storage life and marketability of table grape clusters [4]. Unlike individual fruits, the integrity and non-destructiveness of fruit clusters are critical indicators of table grape quality. During production, losses due to berry detachment and damage caused by grape cluster vibrations and inter-berry collisions can reach 20% to 30% [5]. These issues severely undermine the commercial value of table grapes, and addressing the problems of berry detachment and damage has thus become a critical bottleneck in the production and supply chain of fresh table grapes.

There are various approaches to mitigating berry detachment and damage during grape production. Currently, chemical methods are widely employed in the grape industry [6], primarily focusing on antioxidant properties during storage [7,8,9] and the impact of fungi on grape preservation [10,11,12]. However, in recent years, increasing attention has been directed toward non-chemical methods. Researchers have sought to understand the relationship between mechanical harvesting and berry damage, aiming to optimize mechanical handling techniques to reduce berry detachment and damage. Studies on collision dynamics and fruit damage have been extensively conducted across different types of fruit berries [13,14]. Hertz’s contact theory and virtual simulation methods have been widely applied to investigate dynamic collisions and viscoelastic materials. Researchers have developed various pressure, deformation, and energy equations for different collision types, including berry–berry collisions, falling berry–board collisions, and berry–buffer material collisions. These models have been integrated into simulations based on the discrete element method (DEM) and finite element method (FEM) [15,16,17]. For instance, in vertical drop experiments involving berry–buffer material collisions, Vinokur et al. [16] demonstrated that the fruit detachment rate is proportional to the height. Traditional methods for measuring berry collisions are often impractical for certain scenarios, leading to the adoption of indirect techniques such as electronic fruit, piezoelectric films, and high-speed cameras [18,19,20]. Pezzi et al. [20] used measurement devices like electronic fruit to study berry collision mechanisms during mechanical harvesting and transportation without the challenges of direct measurement. Lu et al. [21] employed piezoelectric films to observe collisions in grape clusters during emergency stop and vertical drop tests. Edward et al. [22] used finite element modeling to analyze the dynamic collision process of apples, either with each other or against rigid surfaces. Similarly, Bao et al. [23] developed a finite element model of blueberries to assess injury levels via collision simulations. While these studies have primarily examined collisions between free-falling fruits or berries and surfaces, or between two free fruits or berries, they have largely overlooked the unique constraints present in table grape clusters. During harvesting and post-harvest handling, berries within a cluster remain connected to the stem, resulting in collisions under constrained conditions. Such constrained collisions differ significantly from free-fruit collisions in terms of dynamics and complexity. However, specific research on berries’ collisions under these constrained conditions is currently lacking, highlighting an important gap in the literature.

Grape cluster vibrations play a crucial role in the mechanical harvesting and post-harvest handling of table grapes, as it contributes to berry detachment and damage. Existing research primarily focuses on four areas: the vibratory motion of the cluster, vibration frequency, the influence of vibration and excitation on fruit falling, and cluster modeling. Crooke et al. [24] and Ramli et al. [25] suggested that although the motion patterns of fruit clusters are highly complex, there are five basic motion patterns. Many scholars have examined the mechanical damage of different fruits under varying vibration frequencies. Kubilay et al. [26] and Opara et al. [27] conducted extensive research on the relationship between vibration frequency and the mechanical damage of packaged apples. Cao et al. [28] studied the mechanical damage of Huangguan pears under different vibration frequencies, analyzing the relationship between the vibration acceleration of the pears and the resonance frequency of the test bench. Zhou et al. [29] investigated the effect of vibration frequency on the motion of fruit, detachment time, and damage during mechanical sweet cherry harvesting with a shaker using high-speed cameras. Their findings indicated that shorter durations of high-level mechanical impacts induced greater fruit damage. Fernando et al. [30] explored the mechanical damage to bananas caused by vibration transmissibility and packaging materials through experimental studies. However, these studies largely regard the fruit cluster as a whole, neglecting the internal interactions among the individual fruits within the cluster. These interactions are fundamentally different from the excitation, transmission, and vibration responses in a complex fruit cluster. To explore the vibration response of the cluster system, Fischer et al. [31] conducted berry falling experiments with packaged table grapes at different vibration frequencies. They found that when the vibration frequency was between 5 and 10 Hz, the collision damage rate of the berries peaked. Furthermore, Liu et al. [32] discovered the influence of vibrations on the falling rate of grape clusters, depending on different holding positions during the start and stop phases of clamping and when transferring the clusters. They also determined the relationship between the angle deviation of the grape cluster and excitation transmission using high-speed cameras, in addition to the correlation between velocity and acceleration with respect to vibration. To explore the influence of excitation on vibration, simulation modeling methods have proven valuable in addition to experimental methods. Simulations facilitate the study of the excitation and vibration transmission routes within the cluster and can provide insights into the impact of mechanical handling on grape clusters. One common approach for modeling and simulating fruit clusters is three-dimensional reconstruction. Schöler et al. [33] obtained point cloud data of the stalk system through three-dimensional scanning from a phenotypic perspective and reconstructed the table grape cluster based on a model of the coffeeberry–stem–branch system. Huang et al. [34] created three-dimensional models of grape clusters based on the Open L system, which can be used to adjust the growth directions of the main rachis, stalk, and berry to obtain different bunch shapes. However, they focused only on the vibration characteristics of individual fruits and neglected the vibration behavior of entire fruit clusters. The primary aim of their three-dimensional reconstruction was to represent plant shapes, making it unsuitable for subsequent vibration analysis. Kondo et al. [35] modeled a tomato cluster as a pendulum system with two degrees of freedom. Yiannis et al. [36] built a discrete element method of a cluster comprising multiple berries for the stem removal of wine grape clusters. Hoshyarmanesh et al. [37] considered olive boughs, main branches, subshrubs, and twigs as a whole, treating the fruit–stem system as a hanging pendulum when building a 3D model and collecting vibration data. However, these studies treated the fruit cluster as a single entity and investigated excitation and vibration frequency without considering the complexities of individual fruit vibrations within the cluster, and this oversimplification could overlook critical aspects of berry detachment and damage dynamics. Li et al. [38] established a three-dimensional grape cluster model by discretizing the stem into 5 mm segments and connecting them with bushings to simulate the flexibility of the virtual model. They applied a spring-damping model to connect the berries and stems. However, the model used uniform mechanical properties for the fruit and stalk and assumed rigid connections at all rachises, which fails to reflect the actual deformation and variability of real grape clusters. Such models cannot therefore accurately describe the vibration characteristics or berry detachment behavior in practical production scenarios. In a previous study [39], we constructed a compound mechanical model of a table grape cluster, named the “(flexible rod)–(viscoelastic hinge)–(rigid bar)–(rigid ball)” system, which simulates the excitation and vibration transmission characteristics during the post-harvest transport stage. The model successfully captures the viscoelastic interactions between fruit stems and individual berries, reflecting the individual variability of cluster components, with the results showing that external picking forces could induce severe vibrations in the cluster model, leading to berry detachment. The accuracy of the model was validated through multibody dynamic software (Adams2018) and real-world experiments. However, this model did not consider berry–berry collisions or the effect of viscoelasticity between berries, which are key factors influencing berry detachment and damage. To better understand the mechanisms of berry detachment and damage in table grape clusters, it is crucial to further investigate the vibration–collision coupling behavior of grape clusters, which could provide essential insights for optimizing berry handling and reducing damage during mechanical harvesting and post-harvest processing.

In previous studies [40], the authors described the grape cluster vibration–collision coupling model from a finite element perspective; however, earlier modeling approaches did not allow for the direct observation of vibrations and collisions within the cluster, and significant challenges were also raised during programming in MATLAB2016a. To address these issues, this study approaches the grape cluster vibration–collision coupling model from an analytical mechanics perspective, ultimately deriving a dynamic vibration–collision coupling equation based on the Lagrange equation. A new vibration–collision coupling unit, named the “dual-twig–berry system”, was established by simplifying the characteristics of the table grape cluster while accounting for the viscoelastic properties of both the twigs and the berries. As a result, the dynamic vibration–collision coupling equation for the dual-twig–berry system was obtained by introducing the expressions for viscoelastic twig vibrations, viscoelastic collisions between berries, and the generalized collision force (based on the Kelvin model) into the framework of the Lagrange equation. Following the derivation of the dynamic equation, a simulation program was developed on the MATLAB2016a platform. Furthermore, a comparison between the simulation results and experimental data was conducted to verify the model’s feasibility. Finally, the dynamic characteristics of both the pure vibration pattern and the vibration–collision coupling pattern were compared, facilitating an in-depth analysis of the mechanisms behind berry detachment and damage within the dual-twig–berry system.

2. Materials and Methods

The vibration–collision coupling behavior of table grape clusters, along with the problem description, modeling, and computational approach, is presented in this section.

2.1. Problem Description

During the production of table grapes, various stages such as in-field harvesting (including picking, transporting, and placing) and post-production activities (such as sorting, packing, and transporting) involve significant mechanical handling. Throughout the harvesting and subsequent handling processes, external excitations generate simultaneous vibrations and collisions within the grape cluster. These excitations may include impulse, picking, collision, and periodic excitations, which are applied to the main rachis and the bottom berries, as illustrated in Figure 1. Each type of excitation leads to distinct vibration–collision coupling behaviors within the grape cluster, which in turn results in different responses in terms of berry detachment and berry damage. To investigate the underlying principles of excitation transmission and vibration–collision coupling motion in grape clusters, space–time dynamic descriptions and simulation studies were conducted. The findings of these simulations help us to better understand the mechanisms governing a berry falling and berry damage, providing insights into how external forces during handling influence the overall integrity of the grape cluster.

2.2. Modeling

2.2.1. Vibration–Collision Coupling Model of a Grape Cluster System

Compared to the single-stem fruit system, the fruit cluster system has distinct characteristics (Figure 2). Each fruit cluster is attached to the main peduncle, which branches into several branch peduncles. Each branch peduncle further divides into multiple fruit stalks, which are connected to the fruit particles via the fruit pedicels [32]. During harvesting and transportation, excitation is transmitted to the fruit particles through the main peduncle, branch peduncles, and fruit stalks. Understanding these factors is crucial for optimizing the handling processes and minimizing losses during harvesting and post-harvest handling.

To describe the vibration behavior under different excitations, Liu et al. [32]. proposed a modeling method called the “(flexible rod)–(viscoelastic hinge)–(rigid bar)–(rigid ball)” model, as shown in Figure 3a. In this model, the main rachis and sub-rachises are represented as a flexible rod and rigid bar, respectively. Fruits are randomly added based on a normal distribution of diameters and the properties of the primary and secondary hinges are also randomly defined according to the normal distribution of their elasticity and damping coefficients. As shown in Figure 3b, this results in a 3D-simulation model of a grape cluster with various components. However, this model does not account for collisions between berries, leading to discrepancies between the simulation results and the actual dynamic behavior of the grape cluster. To improve the accuracy of the model, it is necessary to incorporate vibration–collision coupling within the grape cluster.

2.2.2. Vibration–Collision Coupling Model of a Dual-Twig–Berry System

To establish a vibration–collision coupled model based on the collision model (Figure 3b), the key is to integrate berry collisions into the vibration model. The vibration–collision coupling model of the dual-twig–berry system was thus first developed under the constraint conditions of the grape cluster. The dual-twig–berry system, the smallest vibration–collision coupling unit, is described using the “(viscoelastic hinge)–(rigid bar)–(flexible ball)–(viscoelastic link)” model, which consists of two rigid bars and two flexible balls. The viscoelastic hinge between the bars and the viscoelastic collision between the flexible balls are illustrated in Figure 4.

2.2.3. Assumptions of Dual-Twig–Berry System Modeling

The basic assumptions within the vibration–collision coupling model of the dual-twig–berry system are as follows:

(1) Under actual excitation, collisions within the grape cluster primarily occur between the berries, so collisions between the berries and twigs are neglected.

(2) The rotation of the twig around its center line is ignored due to the constraint of the viscoelastic hinge between the twigs.

(3) The mass of the twig and berry is assumed to be equivalent to the mass of the berry, neglecting the mass and size of the twig.

(4) For the study of collisions between smooth grape skins, friction is ignored.

Based on these assumptions, the vibration–collision coupling model of the dual-twig–berry system is derived, as shown in Figure 5. In this model (Figure 4), there is a viscoelastic constraint between the fixed bar and the twig, and a viscoelastic collision between the flexible balls. However, since the berry itself exhibits viscoelastic properties, the complex relationship between viscoelastic constraints and collision needs to be simplified. Using experimental methods, the equivalent elastic coefficient based on k₁ and k₂ can be obtained as k₃, and the equivalent viscoelastic coefficient based on c₁ and c₂ can be obtained as c₃. Similarly, the viscoelastic coefficient and damping coefficient for the two berries, as well as the viscoelastic collision, can be equivalent to k₄ and c₄. As a result, under the viscoelastic collision of two berries, the “(viscoelastic hinge)–(rigid bar)–(flexible ball)–(viscoelastic link)” model can be simplified and equivalent to the “(viscoelastic hinge)–(rigid bar)–(rigid ball)–(viscoelastic link)” model.

2.2.4. Motion Stages of the Vibration–Collision Coupling Model

To study the law of a berry falling and of berry damage, it is crucial to observe the motion patterns during cluster vibration and collision. There are two possible berry–berry relations and motion patterns in this context: one relation occurs when two adjacent berries remain in contact, and the other involves a changing interaction between non-contact and contact states. For the first relation, where the berries stay in contact, only the vibration–collision coupling pattern is observed. However, for the latter relation, the motion consists of a combination of both the pure vibration pattern and the vibration–collision coupling pattern.

In the dual-twig–berry system, the vibration pattern is defined as the vibratory motion of the twigs under the viscoelastic constraint between the twig and fixed rod, as shown in Figure 6a. The vibration–collision coupling pattern, on the other hand, is the result of the coupling between the vibratory motion (due to the viscoelastic hinge constraint) and the berry collisions (under viscoelastic contact constraint), as shown in Figure 6b.

2.3. Dynamic Equation of Vibration Patterns

The dynamic vibration–collision coupling model of the dual-twig–berry system, described in this manuscript, was proposed based on the assumptions mentioned in Section 2.2.3. The theoretical derivation and analysis of this model have been conducted, considering the actual conditions of Kyoho grapes.

The Lagrange method is an excellent tool for solving the motion of multi-body constrained systems, and the basic Lagrangian function is:

$$L=T-V$$

(1)

where T is the kinetic energy function of the dual-twig–berry system, and V is the potential energy function of the dual-twig–berry system.

The basic Lagrangian equation of an ideal and complete system is:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{q}}}\right)-{\displaystyle \frac{\partial L}{\partial q}}=Q$$

(2)

where q is the generalized coordinates including θ and φ, and $Q$ is the generalized external force.

The kinetic energy generated by the speed of the berries at the vibration pattern can be expressed as follows:

$$T={\displaystyle \frac{1}{2}}{m}_1{l}_1^2{\dot{\theta }}^2+{\displaystyle \frac{1}{2}}{m}_2{l}_2^2{\dot{\phi }}^2$$

(3)

where m₁ and l₁ are the mass and length of the left berry, m₂ and l₂ are the mass and length of the right berry, and θ and φ are the generalized coordinates (Figure 5).

In contrast to the traditional Lagrangian method, the vibration feature caused by the viscoelastic hinge between the two twigs is fully considered, whereby the elasticity of viscoelasticity is integrated into potential energy V. As a result, the potential energy generated by the viscoelastic action of the twigs and the gravity of the berries is as follows:

$$V=m_{1}g{l}_1\cos \theta +m_{1}g{l}_1\cos \phi +{\displaystyle \frac{1}{2}}{k}_3{\left(l_1\sin \theta +l_2\sin \phi -d_0\right)}^2$$

(4)

where g is the acceleration of gravity, d₀ is the distance between the two hinge points, and k₃ is the equivalent elastic coefficient of k₁ and k₂ coupling.

The traditional Lagrangian method does not consider the application of damping to the basic Lagrangian equation. This manuscript therefore introduces a method for integrating the viscoelastic action damping of the twigs into the theoretical framework of the Lagrangian equation by introducing dissipation function D. As a result, the system dissipation function is obtained as follows:

$$D={\displaystyle \frac{1}{2}}{c}_3\left(l_1^2{\dot{\theta }}^2-2l_1{l}_2\dot{\theta }\dot{\phi }+l_2^2{\dot{\phi }}^2\right)$$

(5)

where c₃ is the equivalent collision damping coefficient of c₁ and c₂ coupling.

By considering the elasticity and the damping of viscoelastic action of the twigs, the Lagrangian equation can finally be obtained as follows:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{q}}}\right)-{\displaystyle \frac{\partial L}{\partial q}}+{\displaystyle \frac{\partial D}{\partial \dot{q}}}=Q$$

(6)

In contrast again to the traditional method, the viscoelastic vibration is based on the Lagrangian equation. As a result, by substituting Equations (1)–(5) into Equation (6) and simplifying the calculation solution, the dynamic differential equations for the vibration pattern can be finally obtained as follows:

$$\begin{gathered} m_1{l}_1^2\ddot{\theta }-m_{1}g{l}_1\sin \theta +k_3{l}_1\left(l_1\sin \theta +l_2\sin \phi -d_0\right)\cos \theta +c_3{l}_1^2\dot{\theta }-c_3{l}_1{l}_2\dot{\phi }=Q \\ {m}_2{l}_2^2\ddot{\phi }-m_{2}g{l}_2\sin \phi +k_3{l}_2\left(l_1\sin \theta +l_2\sin \phi -d_0\right)\cos \phi +c_3{l}_2^2\dot{\phi }-c_3{l}_1{l}_2\dot{\theta }=Q \end{gathered}$$

(7)

2.4. Dynamic Equation of Vibration–Collision Coupling Pattern

The collision force is an external force in the dual-twig–berry system. When a collision occurs, the dual-twig–berry system will be affected by a sudden change force, which will change the vibration pattern. How to integrate the collision factors is therefore the key to establishing the dynamic vibration–collision coupling equation.

2.4.1. Collision Checking

In the vibration–collision coupling pattern, the collision of berries will significantly increase the chances of a berry falling and of berry damage. It is thus important to determine whether or not the berries collide. When berries A and B move relative to each other, if a collision occurs, there will be contact at points M and N; if not, there will be no contact point, as shown in Figure 7.

If the berries are spherical, the collision interference depth of the contact point can be obtained through geometric relations between two berries, which can be shown as follows:

$$\delta ={\left({\left(l_1\sin \theta -l_2\sin \phi \right)}^2+{\left(l_1\cos \theta -l_2\cos \phi \right)}^2\right)}^{{\displaystyle \frac{1}{2}}}-r_1-r_2$$

(8)

where δ is the amount of deformation along the radius of the berries that will squeeze each other and cause deformation after the contact, and $r_1$ and $r_2$ are the radius of the left berry and right berry, respectively.

To detect whether a collision happens at a collision point or not, the criterion for collision is

$$\delta <0$$

(9)

2.4.2. Collision Expression

After confirming that the berries collide, it is necessary to calculate the contact force. According to the Hertz contact theory, as the influence of the surface friction of the berries is ignored, the value of the contact force at the time is thus equal to the value of the collision force. Due to the viscoelasticity between the berries, the value of the collision force between the two berries A and B in the vibration–collision coupling model is calculated by using the Kelvin model, as shown in Figure 7.

The collision force can be calculated according to the basic Kelvin model. The basic Kelvin equation is

$$F_n=k_4\delta +c_4\dot{\delta }$$

(10)

where k₄ is the equivalent elastic coefficient between two berries, and c₄ is the equivalent collision damping coefficient between two berries.

Generalized collision force $Q_n$ is a kind of generalized force $Q$. The collision force can be introduced into the Lagrangian framework as a generalized external force, and according to the Lagrangian principle the collision force acting on the contact point of the berries can be transformed into the generalized collision force as follows:

$$Q_n=-{\displaystyle \frac{\partial F_n}{\partial \dot{q}}}$$

(11)

where $Q_n$ is the generalized collision force.

As a result, move the generalized collision force from the generalized force to the left of Equation (7). In the dual-twig–berry system, the generalized force mainly includes the generalized collision force and the generalized force except the generalized collision force because of the viscoelastic collision of berries:

$$Q_n=Q-Q_j$$

(12)

where $Q_j$ is the generalized force without the generalized collision force.

2.4.3. Dynamic Equation of Vibration–Collision Coupling Patterns

In this study, by introducing the collision force into the vibration dynamic equation as the external force of the dual-twig–berry system, the dynamic modeling of vibration–collision coupling based on the Lagrange method is realized. The vibration–collision coupling equation can be obtained by substituting the general collision force into the generalized external force $Q$ through the Lagrangian principle. From Equations (7), (10) and (11), the dynamic differential equations can be obtained by simplifying the calculation solution of the dynamic equation as follows:

$$\begin{gathered} m_1{l}_1^2\ddot{\theta }-m_{1}g{l}_1\sin \theta +k{l}_1\left(l_1\sin \theta +l_2\sin \phi -d_0\right)\cos \theta +c_3{l}_1^2\dot{\theta }-c_3{l}_1{l}_2\dot{\phi }+{\displaystyle \frac{\partial F_n}{\partial \dot{\theta }}}=Q_j \\ {m}_2{l}_2^2\ddot{\phi }-m_{2}g{l}_2\sin \phi +k{l}_2\left(l_1\sin \theta +l_2\sin \phi -d_0\right)\cos \phi +c_3{l}_1^2\dot{\phi }-c_3{l}_1{l}_2\dot{\theta }+{\displaystyle \frac{\partial F_n}{\partial \dot{\phi }}}=Q_j \end{gathered}$$

(13)

By substituting Equations (10)–(12) into Equation (13) and simplifying the calculation solution, the viscoelastic vibration expressions of two twigs, the viscoelastic collision of two berries, and the generalized collision force are introduced in the framework of Lagrangian equation. Finally, the dynamic equation representing the vibration–collision coupling pattern of the dual-twig–berry system is obtained as follows:

$$\begin{gathered} m_1{l}_1^2\ddot{\theta }-m_{1}g{l}_1\sin \theta +k{l}_1\left(l_1\sin \theta +l_2\sin \phi -d_0\right)\cos \theta +c_3{l}_1^2\dot{\theta }-c_3{l}_1{l}_2\dot{\phi }-c_4{l}_1{l}_2\left(\cos \theta \sin \phi +\sin \theta \cos \phi \right){\left(l_1^2+l_2^2-2l_1{l}_2\sin \theta \sin \phi -2l_1{l}_2\cos \theta \cos \phi \right)}^{-{\displaystyle \frac{1}{2}}}=Q_j \\ {m}_2{l}_2^2\ddot{\phi }-m_{2}g{l}_2\sin \phi +k{l}_2\left(l_1\sin \theta +l_2\sin \phi -d_0\right)\cos \phi +c_3{l}_1^2\dot{\phi }-c_3{l}_1{l}_2\dot{\theta }-c_4{l}_1{l}_2\left(\cos \theta \sin \phi +\sin \theta \cos \phi \right){\left(l_1^2+l_2^2-2l_1{l}_2\sin \theta \sin \phi -2l_1{l}_2\cos \theta \cos \phi \right)}^{-{\displaystyle \frac{1}{2}}}=Q_j \end{gathered}$$

(14)

2.5. Model Computing

2.5.1. Computation Platform

Based on the dynamic equation of the vibration–collision coupling model, a simulation program was established. For the simulation, the experimental equipment used was a Dell-T7920 workstation, and the operating system was Windows 10. The workstation hardware configuration included two Intel Xeon Gold 6248R CPUs, 64 GB of RAM, and two NVIDIA Quadro RTX 5000 graphics cards. The Matlab version used for the simulation was 2016a.

2.5.2. Computation

Equations (7) and (14) both indicate that the dynamic equations of the vibration–collision coupling model of the dual-twig–berry system are composed of typical nonlinear differential-algebraic equations with constant coefficients. In this study, the Runge–Kutta method was applied to solve the dynamic equation as follows:

$$\begin{gathered} y_{n+1}=y_n+h\sum _{j=1}^s{b}_j{k}_j \\ {k}_j=f\left(x_n+c_{j}h, y_n+h\sum _{i=1}^s{a}_{ij}{k}_i\right), j=1,2,\cdots ,s; \\ {c}_j=\sum _{i=1}^s{a}_{ij} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\hspace{1em}j=1,2,\cdots ,s; \end{gathered}$$

(15)

From Equation (15), the more widely used standard fourth-order Runge–Kutta formula can be simplified as follows [41]:

$$\begin{gathered} y_j=y_j+{\displaystyle \frac{h}{6}}\left(K_1+{2K}_2+2K_3+K_4\right) \\ {z}_{j+1}=z_j+{\displaystyle \frac{h}{6}}\left(L_1+{2L}_2+{2L}_3+L_4\right) \\ {K}_1=z_j \\ {L}_1=f\left(x_j,y_j,z_j\right) \\ {K}_2=z_j+{\displaystyle \frac{h}{2}}{L}_1 \\ {L}_2=f\left(x_{j+\frac{1}{2}},y_j+{\displaystyle \frac{h}{2}}{z}_j,z_j+{\displaystyle \frac{h}{2}}{L}_1\right) \\ {K}_3=z_j+{\displaystyle \frac{h}{2}}{L}_2 \\ {L}_3=f\left(x_{j+\frac{1}{2}},y_j+{\displaystyle \frac{h}{2}}{z}_j+{\displaystyle \frac{h^2}{4}}{L}_1,z_j+{\displaystyle \frac{h}{2}}{L}_2\right) \\ {K}_4=z_j+h{L}_3 \\ {L}_4=f\left(x_{j+\frac{1}{2}},y_j+h{z}_j+{\displaystyle \frac{h^2}{2}}{L}_2,z_j+h{L}_3\right) \end{gathered}$$

(16)

The fourth-order Runge–Kutta method, obtained from Equation (15), can be used to solve the differential equations of the vibration and vibration–collision coupling patterns, but Equation (16) is widely used in the core algorithm of this study. The main principle of the Runge–Kutta method is to apply the Taylor series of the finite items to approximate the solution function, and the source of the error is the Taylor’s truncation term, where the error refers to the truncation error. The Taylor series is used to calculate the approximate value of the variable stage at the terminal of each integration step, and the value of the Taylor series is used for subtraction. The error obtained through the steps above is then applied as the criterion for calculating the error. If the error estimate is greater than the beginning set value of the system, then the integration step length is decreased, and the error estimate is recalculated. If the error is much smaller than the beginning set value of the system, then the integration step length is increased. As a result, the nonlinear equations can be solved.

2.5.3. Simulation Method

Traditional vibration-pattern-calculation methods generally focus only on the vibration pattern, neglecting the vibration–collision coupling pattern, as the combination of vibrations and collisions complicates the calculation process. Consequently, few methods exist for calculating the vibration–collision coupling pattern. This study thus proposes a novel calculation method for both the pure vibration pattern and the vibration–collision coupling pattern in a plane coordinate system using Matlab2016a.

The key principle of this method is that the vibration pattern and the vibration–collision coupling pattern are treated as two distinct but interconnected motion patterns. These patterns are seamlessly combined by monitoring the distance between the berries and determining the simulation time T. At each stage, given the system coordinates $\left(\theta ,\phi ,\dot{\theta },\dot{\phi }\right)$, set T = 0. According to Equation (7), the vibration pattern can be solved, and the distance between the berries can be calculated. If the distance is less than zero, it indicates a collision, and the vibration–collision coupling pattern has been triggered. In such cases, the magnitude of the collision force is computed using Equation (10), and the post-collision stage $\left({\theta }_0,{\phi }_0,{\dot{\theta }}_0,{\dot{\phi }}_0\right)$ is determined using Equation (13). This new stage is reset as the initial condition T = 0, and the process repeats to obtain all the stages of the vibration–collision coupling model based on Matlab2016a.

2.5.4. Simulation Parameters and Process

In this study, the relevant parameters of each component were obtained from [32]. The solving process of the dynamic vibration–collision coupling equation is shown in Figure 8, and the specific solution steps are as follows:

(1) Set generalized coordinates ($\theta ,\phi$). Analyze the dynamic model and apply the Lagrangian method to obtain the viscoelastic-vibration dynamic equation and the vibration–collision coupling dynamic equation on the basis of the generalized coordinates ($\theta ,\phi$).

(2) Enter the system parameters and the initial values. The system parameters mainly include the radius and mass of the berries, the viscoelastic constraint coefficients k3 and c3 between the two twigs, and the viscoelastic collision coefficients k4 and c4 between the two berries. The initial value was $\left(\theta ,\phi ,\dot{\theta },\dot{\phi }\right)$, corresponding to the time T = 0 in the first step of the simulation.

(3) Solve the generalized coordinate position and velocity at the next moment. The ode23 function based on the Runge–Kutta method was applied to solve the dynamic vibration equation.

(4) Collision judgment. According to Equation (7), determine the generalized coordinates $\left(\theta ,\phi ,\dot{\theta },\dot{\phi }\right)$ at each time step in the simulation, and calculate the distance between the center of the two berries at every step. Using Equation (8), if the distance is less than the sum of the radii of the berries, then calculate the generalized coordinate of every step of the vibration–collision coupling pattern of the dual-twig–berry system. Next, return to step (3) again to solve the dynamic vibration–collision coupling equation based on the ode23 function of Runge–Kutta method. If it is greater than the sum of the radii, then proceed to the step (5).

(5) Simulation time judgment. In this step, determine whether the time T_{(n = I)} at one time step has reached the beginning set final time T0 or not. If the final time is not reached, return to step (3). If the beginning set time is reached, the output is achieved. Then save the generalized coordinates $\left({\theta }_0,{\phi }_0,{\dot{\theta }}_0,{\dot{\phi }}_0\right)$ at each moment. This completes the whole process of solving the dynamic equation [42].

2.6. Simulation Design

To study the dynamic characteristics of the vibration–collision coupling model for the dual-twig–berry system, a dynamic vibration–collision coupling model was established using Matlab2016a and numerically solved. Four types of simulation experiments were designed, as follows:

(1) Simulation of motion trajectory. To analyze the motion trajectory, the vibration test of one berry and one twig under the viscoelastic constraint with no collision and the vibration–collision coupling test of the dual-twig–berry system was designed. The number of steps for the whole solution was set to 350 and the initial angle of the right twigs was set at 45° in the vibration test (test 1), while in the vibration–collision coupling test (test 2), the number of steps was set to 350, and the initial angle of two twigs was set at −45° and 45°, respectively. The total of 350 steps was determined to ensure convergence within a reasonable time based on multiple simulations. Comparing experimental and simulation results (with initial angles of 30°, 45°, 60°, and 75° between the fruit and the vertical direction), the smallest error occurred when the initial angle was 45°, and this angle was thus selected as the initial angle for subsequent simulations. The gravitational potential energy was set as the external excitation in the dual-twig–berry system.

(2) Simulation of the symmetric vibration–collision coupling pattern. To research the law of the kinematic and the dynamic characteristics of the dual-twig–berry system, two tests were designed, and the number of steps for the whole solution was set to 350. In test 3, the angle of the right twigs was set at 45° in the dual-twig–berry system. The parameters of the two berries and two twigs were the same. As a result, test 3 was completely symmetrical at the initial stage. In contrast, test 4 was a control experiment, and the initial conditions were the same as those in test 3, but there was no interaction between the two berries. The gravitational potential energy was set at the external excitation for test 3 and test 4.

(3) Simulation for the effect of viscoelasticity on the vibration–collision coupling pattern. To study the effect of viscoelasticity of the hinges on the vibration–collision coupling pattern, test 5 and test 6 were designed. In test 5, the number of steps for the whole solution was set to 350 and the initial angle of the right twigs was set at 45° in the dual-twig–berry system. The parameters of the two berries and two twigs were the same. In contrast, test 6 was a control experiment, and the initial conditions were the same as in test 5 but there was no viscoelasticity at the hinges. To analyze the effect of the viscoelasticity of the berries on the vibration–collision coupling pattern, tests 7 and 8 were designed. In test 7, the number of steps for the whole solution was set to 350 and the initial angle of the right twigs was set to 45° in the dual-twig–berry system. The parameters of the two berries and two twigs were the same. In contrast, test 8 was a control experiment, and the initial conditions were the same as those in test 7, but there was no viscoelasticity of the berries. The gravitational potential energy was set as the external excitation for tests 5, 6, 7, and 8.

(4) Simulation of unsymmetrical vibration–collision coupling pattern. To analyze the effect of the initial angle of the berries on the vibration–collision coupling pattern, test 9 was designed. In test 9, the number of steps for the whole solution was set to 350 and the initial angle of the left twigs was set to −45° in the dual-twig–berry system. The parameters of the two berries and two twigs were the same, but the initial angle of the right twig was 15°, 30°, 45°, 60°, and 75°, respectively, in the five experiments of test 9. The gravitational potential energy was set as the external excitation for test 9.

3. Results and Discussion

3.1. Simulation of Motion Trajectory

The simulation results of test 1 and test 2 are shown in Figure 9a,b. In Figure 10, the blue, yellow, and green circles all represent the berries in motion. As shown in Figure 9a, the initial stage of the right berry is a1, and it performs the vibration pattern under external excitation. The motion trajectory of the vibration pattern of the right berry can be expressed as a1→a2→a3→a4→a5. Based on a previous study [32], Figure 10 shows that the motion trajectory of the berry circled in blue is a1→a2→a3→a4→a5→a6→a7→a8, which corresponds to the motion trajectory of the berry in Figure 9a. As a result, it can be concluded that the fruit circled in blue exhibits the vibration pattern.

In Figure 9b, the initial stage of the berry is c1, and it performs the vibration pattern under external excitations when the motion trajectory is c1→c2→c3, and when the right berry reaches the critical position at c3, the berries collide and exhibit the vibration–collision coupling pattern. In a short time, the motion trajectory is c3→c4→c5. In contrast, based on our previous research [32], Figure 10 shows that the motion trajectory of the berry circled in green is c1→c2→c3→c4→c5→c6→c7, which corresponds to the motion trajectory of the right berry due to the collision with the berry circled in yellow. The same behavior is observed in the simulation and experiment: the berry vibrates because of the external excitation at first, and when the berry collides with other berries, the vibration–collision coupling pattern occurs. As a result, it can be concluded that the fruit circled in green exhibits the vibration–collision coupling pattern. These results indicate that the motion can be effectively described via the vibration–collision coupling model of the dual-twig–berry system.

3.2. Simulation of the Symmetric Vibration–Collision Coupling Pattern

3.2.1. Result

(1)

Angle

One of the simulation results from tests 3 and 4 is shown in Figure 11. In test 4, the maximal vibration angle exhibits periodic decay. Figure 12 presents experimental results from Faheem et al. [2], who observed a linear relationship between the hanging force and swing angle, with the hanging force displaying periodic decay in the pure vibration pattern. The result of test 4 is consistent with the findings of Faheem et al. [2]. In contrast, in test 3 the maximal vibration angle also shows periodic decay, but the vibration period is shorter compared to test 4. In both tests, the angle eventually stabilizes at a fixed value as the model moves. A common characteristic in both cases is that the angle change curves for the left and right berries are identical, at least when considering the angle values shown in Figure 11.

(2)

Kinetic energy

One of the simulation results from tests 3 and 4 is shown in Figure 13. As observed in Figure 13, the kinetic energy of the berry in test 4 fluctuates periodically, with the maximum kinetic energy in each cycle gradually decreasing linearly until all energy is eventually dissipated. In contrast, in test 3 the kinetic energy also follows a periodic decay pattern. However, after a collision, the kinetic energy experiences a sharp decrease. The vibration period in test 3 is shorter than that in test 4, and in both cases, the kinetic energy eventually stabilizes to a fixed value as the model progresses. A common feature of both tests is that the kinetic energy change curves for the left and right berries are identical.

(3)

Potential energy

One of the simulation results from tests 3 and 4 is shown in Figure 14, showing that the potential energy of the berry in test 4 changes periodically, with the minimum value of the potential energy in each cycle following a linear trend. In contrast, the potential energy in the vibration–collision coupling model shows periodic decay. The vibration period in test 3 is shorter than that in test 4, but in both cases, the potential energy eventually stabilizes at a fixed value as the model progresses. A common feature of both tests is that the potential energy change curves for the left and right berries are identical.

3.2.2. Analysis

(1) Observing Figure 11, Figure 13, and Figure 14, due to berry collisions in the dual-twig–berry system, the vibration amplitude of the berry is reduced, the frequency of collisions increases, and mechanical energy loss also rises. Consequently, the kinetic energy decreases to zero, while the deflection angle and potential energy decrease periodically until they reach a fixed value. The sudden drop in kinetic energy during the vibration–collision cycle is primarily due to the exchange between kinetic energy and internal energy during collisions, where a portion of the kinetic energy is converted into internal energy. Ultimately, the deflection angle and potential energy stabilize due to the continuous loss of mechanical energy.

(2) During the vibration–collision coupling process, part of the mechanical energy is converted into internal energy, which leads to a decrease in the berry’s vibration amplitude. In the context of table grape harvesting and post-harvest handling, berry detachment is mainly caused by the overload or fatigue of the pedicel. A decrease in vibration amplitude will reduce the berry detachment caused by an excessive instantaneous load, whereas the increased collision frequency may increase fruit loss due to fatigue. To address the issue of a berry falling in the vibration–collision coupling process, further investigation into the comprehensive dynamics of a berry falling caused by this pattern is necessary.

(3) The transformation of mechanical energy into internal energy during vibration–collision coupling may cause varying degrees of berry damage, which can negatively impact the commodity quality and storage life of the fruit. The primary causes of berry damage include an excessive collision force and accumulated fatigue damage. In table grape production, collision forces rarely reach the threshold that would cause immediate berry damage, but fatigue damage due to repeated collisions is more common. Future research should therefore focus on minimizing collision-induced fatigue damage to improve berry handling and storage outcomes.

3.3. Simulation of Different Effect on Vibration–Collision Coupling Pattern

3.3.1. The Effort of Viscoelasticity

The simulation results of test 5 and test 6 are shown in Figure 15. It can be observed that the angle of the berry gradually decreases over time due to the exchange between mechanical energy and internal energy. Additionally, in test 5, a noticeable delay occurs before the collision happens, and the vibration range of the twig narrows. These results suggest that part of the mechanical energy is converted into the potential energy of the berries, while another portion is transformed into the internal energy of the twigs. Consequently, the viscoelastic vibration of the twigs influences the overall vibration behavior of the model.

The simulation results from tests 7 and 8 are shown in Figure 16. It can be observed that the angle of the berry gradually decreases over time due to the exchange between mechanical energy and internal energy. Additionally, in test 8 a delay occurs before the collision, as the vibration amplitude is larger compared to that in test 7. Furthermore, in test 7, the vibration range of the twig is reduced because some of the kinetic energy is converted into internal energy. This indicates that the viscoelastic collision of the berries influences the overall vibration behavior of the model.

3.3.2. Effect of the Unsymmetrical Angle

The simulation results from test 9 are shown in Figure 17. It can be observed that although the initial deflections differ, the angle change curves from steps 0 to 200 are similar. In the range from steps 200 to 350, there is no clear pattern in the angle change curves due to the interaction between kinetic and potential energy. However, the loss of mechanical energy ultimately causes the deflection of the berry to stabilize at a fixed value. Additionally, as observed for the right berry, the larger the initial deflection, the smaller the maximum deflection in the first vibration–collision cycle. In contrast, a larger initial deflection leads to a greater maximum angle in the second vibration–collision cycle.

4. Conclusions

In this study, cluster vibrations and collision dynamics in the proposed dual-twig–berry system were explored through the modeling, dynamic expression, and numerical computation of vibration–collision coupling behavior. The main findings and conclusions are summarized as follows:

(1) A vibration–collision coupling model for the dual-twig–berry system was proposed based on the “(flexible rod)–(viscoelastic hinge)–(rigid bar)–(rigid ball)” vibration model. The new model, termed “(viscoelastic hinge)–(rigid bar)–(rigid ball)–(viscoelastic link)”, successfully describes the coupling of vibrations and collisions in the dual-twig–berry system.

(2) The dynamic vibration–collision coupling equation was derived by incorporating the expressions for viscoelastic twig vibrations, viscoelastic collisions between berries, and generalized collision forces into the Lagrangian equation framework. A novel computational method was developed to determine the vibration–collision coupling pattern in the dual-twig–berry system.

(3) Simulations demonstrated that the vibration–collision coupling pattern results in shorter vibration periods and smaller amplitudes compared to pure vibration patterns. Based on these results, the effects of vibration–collision coupling on berry detachment and berry damage were analyzed. These findings provide a crucial theoretical foundation for studying berry detachment and damage mechanisms.

Under the vibration–collision coupling mode, the fruit’s vibration period is short, and the amplitude is small. This phenomenon indicates that the system rapidly dissipates energy after collisions, causing the fruit to reach a stable state in a short period of time. When designing grape harvesting machinery, special attention should therefore be paid to the collision force between fruits to reduce excessive collision loads and prevent the fatigue damage and fruit drop caused by frequent collisions.

The viscoelasticity of the twigs slows down the system’s response, making the vibration attenuation process of the fruit more gradual. This phenomenon is important for grape harvesting operations, as it suggests that the viscoelastic properties of the twigs should not be overlooked when designing a vibration–collision coupling system. Properly controlling the rigidity and viscoelastic characteristics of the twigs can enhance the quality of grape harvesting operations.

Future work will focus on extending this dual-twig–berry model to simulate the vibration–collision coupling behavior of entire grape clusters, which will aid in understanding the vibration and collision dynamics during table grape harvesting and transportation.