Where a small dimensionless parameter
was introduced, Equation (15) is transformed using multi-scale perturbation theory [
1] as follows:
where mass, gyroscope, and linear stiffness operators
are introduced. By substituting it into Equation (16), the dynamic equation of the composite structure is obtained as follows:
The general solution of Equation (23) is expressed in the complex form of separated variables as follows:
where
is the nth order modal function of the system;
is the
nth order vibration amplitude of the system;
is the nth natural frequency;
aa is the complex conjugate of the previous term.
3.1. Solution Based on Galerkin Discrete Analysis
The dynamic Equation (12) of the structure contains the variables of time and space coordinates, and it is a differential equation that is not convenient for analysis and calculation. Therefore, the equation was discretized and decoupled using the Galerkin method [
6], which obtained a general ordinary differential equation that was easy to solve and calculate. According to Equation (27), the solution
of the vibration equation is obtained by using the time function
and spatial functions
represents as follows:
The spatial function is the main mode of vibration
of the lateral motion, where
r = 1, 2, 3, …
n in Equation (28),
is expressed as follows:
Since the vibration of the vertical material-conveying system mainly affects the structural stability with low-frequency vibration, the first two modes of the system are mainly used. Equation (29) is substituted into Equation (14), the Galerkin of the first two orders is truncated, and the orthogonality of the modal function is used to multiply
on both sides of the equation. The partial differential Equation (14) is converted into a second-order nonlinear ordinary differential equation after integral operation on the interval [0, 1] as follows:
Equations (30) and (31) are the second-order nonlinear ordinary differential equation of the transverse vibration of the vertical material-conveying system. The first- and second-order vibration responses of the structure are obtained based on differential equations.
Based on the experimental method presented in [
23], the elastic modulus of common varieties of soybean, wheat, and corn was measured in this study, and the average elastic modulus values obtained are shown in
Table 1.
Obtain the elastic modulus and density of different grains as laminated structures from
Table 1 and Equation (1), and substitute them into Equation (32) to obtain different parameters as shown in
Table 2,
Table 3 and
Table 4.
Take the gravity of the laminated structure for
F0, then
where
is the Poisson’s ratio [
24].
3.2. Discussion of Results
The displacement responses of structure after normalization were calculated by using Equations (30) and (31), and they are shown in
Table 2,
Table 3 and
Table 4, where soybean, wheat, and corn are regarded as vertical lifting material. Then, the structural response
q1 and
q2 curves were obtained, which was the relationship curve between the displacement responses of structure after normalization and different filling degrees as shown in
Figure 6,
Figure 7 and
Figure 8.
Figure 6a,b show the first-order and second-order responses of different filling degrees k, respectively, where the conveyed material was soybean particles. And the fluctuation period changed significantly when the k value increased from 0.1 to 0.2. The frequency of
q1 and
q2 was calculated, which showed the frequency of
q1 increases from 0.2823 Hz to 0.3283 Hz and the frequency of
q2 increases from 0.5501 to 0.6462 Hz, following the increase in k value from 0.1 to 0.2. In addition, the wave patterns of
q1 and
q2 were consistent with different filling degrees, and the fluctuation period of
q1 was approximately twice that of
q2.
Figure 7a,b show the first-order and second-order responses of different filling degrees k, respectively, where the conveyed material was wheat particles. And the fluctuation period change significantly when the k value increased from 0.2 to 0.3. The frequency of
q1 and
q2 was calculated, which showed that the frequency of
q1 increases from 0.2815 Hz to 0.3205 Hz and the frequency of
q1 increases from 0.5567 to 0.6349 Hz, following the increase in the k value from 0.2 to 0.3. In addition, the wave patterns of
q1 and
q2 were consistent with different filling degrees, and the fluctuation period of
q1 was approximately twice that of
q2.
Figure 8a,b show the first-order and second-order responses of different filling degrees k, where the conveyed material was corn particles. And the fluctuation period changed insignificantly when the k value increased from 0.1 to 0.3. The frequency of
q1 and
q2 was calculated, which showed that the frequency of
q1 increases from 0.2804 to 0.2880 Hz and the frequency of
q2 increases from 0.5487 to 0.5653 Hz, following the increase in the k value from 0.1 to 0.3. The vibration frequencies of
q1 and
q2 varied less at different filling degrees. In addition, the wave patterns of
q1 and
q2 were consistent with different filling degrees, and the fluctuation period of
q1 was approximately twice that of
q2.
The correspondence curves of the first- and second-order response frequencies of the system obtained after normalization treatment and the filling degree of different materials are shown in
Figure 9.
Figure 9a shows the relationship between the frequency of
q1 and the filling degree of different materials.
Figure 9b shows the relationship between the frequency of
q2 and the filling degree of different materials. By comparing
Figure 9a,b, it was found that the first- and second-order response frequency curves under a different material filling degree k were completely consistent. The difference is that the second-order response frequency was twice that of the first-order response frequency.
Through the analysis of three different grain materials, we found the following: when the filling degree is between 0.1 and 0.3, the frequency fluctuates greatly; when the filling degree is greater than 0.3, the frequency tends to be stable. It can be seen that the lower the filling degree, the more unstable the frequency, and the frequency tends to be stable as the filling degree increases. In addition, when soybean and corn were used as lifting materials, the frequency of structural response first increases, then decreases with the change of material filling degree, and finally tends to be stable. When wheat particles are used as lifting materials, the structural response frequency continues to increase with the change of filling degree, and then becomes stable.
When soybean and corn are used as lifting materials, the stable value of their structural response frequency increases with the increase in filling degree, which is also lower than the stable value of structural response frequency when wheat is used as lifting material.
In this study, the first- and second-order response frequencies of the structure were obtained under different filling degrees of three kinds of lifting materials. It can be used as a reference to guide the frequency design of the moving speed of the synchronous belt and the pressure-supply device of the clamping force during the operation of the clamping and lifting structure, which avoids the working speed of the engineered synchronous belt and the pressure-supply frequency of the pressure-supply device being consistent with or close to the structural frequency, avoids the resonance and strong flutter of the structure during operation, and avoids the instability of the equipment’s clamping materials.
In addition, the difference between the three materials has a significant impact on the response frequency of each order of the laminated structure. Therefore, the design of the equipment’s future operating parameters needs to be combined with the material properties, which will avoid the instability caused by the consistent response frequency of each order of the laminated structure, so as to ensure the intelligence and efficiency of the design equipment.