Vibration simulation and experiment of three open-center shape olive trees

2.1. Force analysis and Fruit-stalk separation conditions

The general eccentric block harvester is of single eccentric block type. The single eccentric block generates a constant centrifugal force under rotation to form circumferential excitation, It can be decomposed into component forces with equal size, phase difference of 90 degrees and located in $X$ and $Y$ directions respectively, as shown in Fig. 1, and the mechanical model is shown in Eq. (1):

where $m$ is the mass of eccentric block, $e$ is the eccentricity, $\omega$ is the rotational angular velocity, $t$ is the time, $F_x$ is the component force in the $x$-axis direction, and $F_y$ is the component force in the $y$-axis direction.

The plane stress diagram of olive fruit is shown in Fig. 2, and the fruit abscission condition is shown in Eq. (2). Ignoring the difference between $\alpha$ and $\beta$ and the influence of gravity [7], it can be reduced to Eq. (3):

where $F_c$ is the binding force between the fruit and the fruit stalk, $a$ is the fruit acceleration, $a_n$ and $a_t$ are components of acceleration, $F_a$ is the inertial force generated by the vibration of the fruit, $F_n$ and $F_t$ are components of inertial force.

Through experimental measurement, the average weight of olive fruit at maturity was 4.1 g, and the average binding force between fruit and fruit stalk was 3.73 N. Therefore, the average abscission acceleration of olive fruit is 909.76 m/s².

Fig. 1Schematic diagram of single eccentric block excitation:

Fig. 2Stress analysis diagram of olive fruit

2.2. Simulation experiment design

The pruned olive tree is a three open-center shape tree. Assuming that the main branches of olive tree are located in the plane perpendicular to the ground, the three main branches are the same, and the branch section is circular, the three-dimensional model of olive tree is established through the lofting function in SolidWorks, as shown in Fig. 3. According to the actual measurement and observation, there are differences in trunk thickness, height and included angle between main branches of different olive trees. Therefore, the trunk diameter D, trunk height H and main branch angle A and B are selected as the test factors, in which the ratio between the cross sections of each branch of olive tree and the trunk diameter is constant, the optimal vibration frequency and exciting force are selected as the evaluation indexes, and the four factor three-level experiment is designed by using Design-Expert 12.0.3 software, and a total of 27 groups of experiments were carried out. The horizontal code table is shown in Table 1.

Fig. 3Three dimensional model of olive tree

2.3. Vibration test and simulation

In order to verify the correlation between the simulation and the actual process, a olive tree in an olive garden was randomly selected for vibration test, The trunk diameter D is 153 mm; The height h of the trunk is 510 mm; The included angles of main branches a and B are 125° and 130° respectively. The vibration motor with adjustable speed is used to provide vibration force, and the vibration motor is fixed with U-bolts, the selected frequency and corresponding excitation force are shown in Table 2. The selected excitation point is point B in Fig. 4(b), and the position of acceleration measurement point is shown in Fig. 4(b). At the same time, the harmonic response analysis consistent with the test conditions is carried out. The established 3D model is imported into ANSYS 19.0 for simulation analysis. The density of olive tree is1024 kg/m³, and the stiffness matrix is shown in Eq. (4):

The damping ratio is 0.04. Apply fixed constraints to the bottom of the tree.

2.4. Simulation

Modal analysis can be used to determine the inherent properties of olive trees, so modal analysis is used to determine the optimal vibration frequency. The first 20 modes of olive tree are selected for natural frequency analysis.

The steps to determine the optimal vibration force are as follows: firstly, apply the excitation force with the size of 1000 N and the phase difference of 90 ° to the $X$ and $Y$ directions of point B in Fig. 3 respectively, and the acceleration of the endpoint (Fig. 3) of the branch is obtained by simulation. According to the linear nature of harmonic response analysis [8], amplify the acceleration to the acceleration required for fruit abscission, obtain the corresponding vibration force, set it as the simulated vibration force, and the simulated vibration force is applied to point B for verification.

Fig. 4The test of measuring acceleration

a) Testal setup

b) Distribution of measuring points

3.1. Analysis of vibration test and simulation results

The synthetic accelerations of different points on the branch are obtained through simulation and test. The correlation coefficient between simulated acceleration and simulated acceleration is calculated by SPSS v26.0, which is 0.724, which proves that there is a great correlation between test and simulation. The simulated acceleration is linearly fitted with the simulated acceleration, and the results are shown in Fig. 5. The average abscission acceleration (909.76 m/s²) of olive fruit was substituted into the fitting curve, and the abscission acceleration in the simulation was 803.02 m/s².

Fig. 5Comparison between test and simulation

3.2. Modal analysis

According to the results of modal analysis, the modes of olive are divided into three categories, as shown in Fig. 6. At the intermediate frequency, the vibration displacement is the largest and the whole olive tree is vibrating, which is conducive to vibration harvesting [9]. Therefore, this frequency is taken as the optimal vibration frequency.

Fig. 6Modal analysis results

a) Lower frequency

b) Intermediate frequency

c) Higher frequency

3.3. Harmonic response analysis

According to the steps of determining the optimal vibration force in Section 2.4, the excitation force with the size of 1000 N and the phase difference of 90° is applied to the $X$ and $Y$ directions at point B respectively. The acceleration frequency curve of the olive branch endpoint in Fig. 3 can be obtained, as shown in Fig. 7(a), the peak value corresponding to 17 Hz is the largest, and the acceleration of point 1 at 17 Hz is the smallest, which is 246.8 m/s². According to the linear nature of harmonic response analysis, the acceleration is proportionally amplified to 803.02 m/s². At this time, the applied simulated excitation force should be 3255 N. When a circumferential excitation force of 3255 N is applied at point B, the acceleration frequency curve of the corresponding point is shown in Fig. 7(b). Compared with Fig. 7(a), the linear nature of harmonic response analysis is verified.

Fig. 7Harmonic response analysis results

a)

b)

3.4. Analysis of simulation experiment results

27 groups of simulation experiments were carried out, and the analysis of variance was carried out. The results show that the p value of the two models is less than 0.05 and the values of lack of fit were more than 0.1, so the regression model is significant. The quadratic polynomial response regression models of the optimal vibration frequency ($f$) and the optimal vibration force$\left(F\right)$ are established respectively, the coding equation is shown in Eqs. (5-6):