Vibration Response of Metal Plate and Shell Structure under Multi-Source Excitation with Welding and Bolt Connection

Abstract

There are many excitation sources and complex vibration environments in combine harvesters. The coupling and superposition of different vibration signals on the plate and shell seriously affect the working parts of the body. This also reduces the reliability of the whole machine. At present, domestic and foreign research on existing harvesters mainly focuses on harvesting performance, with less research on vibration characteristics. Therefore, in this paper, the vibration response of the metal plate–shell under the two connection modes of bolt connection and welding is studied, in order to optimize the design and structure of the plate–shell structure of the combine harvester and improve the overall performance. First, the welded and bolted plates are numerically modeled using Hypermesh pre-processing functions. Then, the boundary conditions are simulated by continuous variable stiffness elastic constraint experiments. Finally, the intrinsic vibration dynamic model of the four-sided simply supported plate and four-sided solidly supported plate is established using the modal superposition method. By analyzing the modal frequencies and vibration patterns, the following results are obtained. The connection method between the plate and the frame has a significant impact on the inherent vibration characteristics of the plate. The bolt connection will make the plate’s intrinsic vibration frequency higher than that of the welding method, but the effect on the plate’s intrinsic vibration pattern is more minor. At the same time, in order to verify the accuracy of the model, the actual modal vibration patterns and frequencies of the same proportion of plates in the modal test are compared with the results of modal vibration patterns and frequencies obtained by Ansys. The errors of the two dynamic model analytical methods are within 1% and 3%, respectively. This result verifies the accuracy of the dynamic model of the metal plate and shell structure under different connection methods.

1. Introduction

With the rapid development of modern agricultural production, the harvester, as a critical piece of agricultural machinery and equipment, plays a vital role in the grain harvesting process [1]. The combine harvester is a harvesting machine with multiple excitation sources. The axial vibration, radial vibration and vibration generated by the unbalanced force of the excitation source are transmitted to the frame through the connection point, and further transmitted to the plate and shell structure through the frame [2]. The plate and shell structure is one of the three major components on the combine, accounting for more than 55% of the whole machine. Its main functions in the combine harvester are as follows: (a) bearing functions, such as the cutting table bottom plate, grain box bottom plate, etc.; (b) sealing effects, such as the thresher shell, driving outside wall, etc.; and (c) protective effects, such as the engine cover, transmission end cover, etc. The rest of the parts also contain part of the plate and shell structure. The performance of the plate and shell structure will largely affect the performance of the machine. However, different connection methods can have a significant effect on the vibration characteristics of the plate and shell structure, thus affecting the operational stability and lifetime of the combine harvester. In the vibration environment, the plate and shell structure will cause fatigue damage to the mechanical equipment after a long period of vibration. And then, the surface of the mechanical structure will form cracks or even fracture, which is also very harmful to the human body. Therefore, the study of the vibration characteristics has crucial scientific significance [3].

Several recent studies have examined combine harvester vibration problems, mainly focusing on the analysis of the excitation source of the combine harvester vibration. However, few have examined the plate and shell structure. In other fields of engineering machinery, such as the aerospace [4], marine [5] and automotive [6] industries, the study of the vibration mechanism of plate and shell structures on machinery has been extended to various aspects and has been quite in-depth. As the application of plate and shell structures is more and more extensive, several scholars have begun to research the vibration of plate and shell structure, such as the free vibration characteristics of composite shells [7], the vibration analysis of rectangular plate coupling [8], the vibration characteristics of cylindrical shells with internal structure [9] and so on. Compared with the complex plate and shell structures in other engineering fields, the vibration generated by the combine harvester during operation is relatively large. The main connection methods are usually welding and bolting [10]. Accordingly, it is essential to establish the dynamic equations and numerical modeling for both bolted and welded plate connections.

Various theories have been developed for modeling the dynamics of various effective plate–shell structures, such as the classical plate–shell theory, thin-plate bending theory, first-order shear deformation theory and higher-order shear deformation theory [11,12,13]. After considering the shear effect of the plate, many other theories have been developed successively, among which the more famous ones include Mindlin plate theory [14,15], modified Mindlin plate theory [16], etc. The theories have been applied to different machines. Domestic and foreign scholars have already conducted more studies on the free vibration of plate and shell structures on different machines, among which the more famous ones are Riley’s method [17] and Ritz’s method [18,19]. However, due to the fact that the initial studies on the vibration problems of plate and shell structures neglected the effect of shear deformation, the results of this type of solution lacked a certain degree of accuracy.

Theoretical studies on the vibration of basic plate and shell structures have been developed in various directions, including different shape classes, differences in boundary conditions and changes in load states [20,21,22,23]. Kim et al. proposed a frequency-domain spectral element method for the vibration analysis of thin plate structures under the action of moving point forces [24]. Wang et al. solved the problem of solving the vibration characteristics of a cracked rectangular plate by substituting the double-sine Fourier series into the vibration mode function of the cracked rectangular plate. The problem of solving the vibration characteristics of the cracked rectangular plate was finally solved. In Ref. [25], Cao et al. investigated the vibration response of a sandwich plate with a nanocore cladding [26]. Marc A. Eitner et al. investigated the interaction between a flexible plate under compressive oblique impulse excitation and boundary layer interaction (SBLI) in Mach flow. Zheng et al. solved the transverse displacement function and the vibration response matrix of a rectangular thin plate on a Winkler foundation by selecting the modified Fourier series and Rayleigh–Ritz method [27]. Ahmad et al. investigated the free and forced vibration response of laminated composite plates. They combined the first-order shear deformation theory (FSDT) [28] with the finite element method to obtain a steady-state response in the time domain using the Newmark method [29].

However, compared with mechanical plate and shell structures in other fields, the vibration of combine harvester plate and shell structures is more complicated, and the above studies cannot be fully applied. Meanwhile, there are few studies on the plate–shell structure and plate–shell coupling structure of combine harvesters with different connection modes.

Therefore, based on the theory of shell dynamics, the vibration response mechanism of combine harvesters is studied around bolted plates and welded plates. At the same time, the natural vibration function of the plate and shell structure of combine harvesters is solved by combining the two connection methods. The dynamic model is established under the corresponding constraint conditions and verified by experiment. The results are helpful in improving the study of the vibration characteristics of the combine shell structure, optimizing the design and structure of the combine shell structure and improving the overall performance of the combine.

2. Material and Methods

2.1. Selection of the Material Parameters

The material parameters used in this paper only consider their linear properties. The density of the electro-welded material is taken to be zero. This is due to the fact that spot welding does not produce additional weight in actual production. All other sheet metal parts are ordinary carbon steel materials. The material mechanical parameters used in the original model are shown in

Table 1. Mechanical parameters of materials.

Material	$\mathbf{Density} \mathbf{\rho }$$(\mathbf{g}·{\mathbf{cm}}^{-3})$	Modulus of Elasticity E(MPa)	$\mathbf{Poisson}’\mathbf{s} \mathbf{Ratio} \mathbf{\mu }$
Plain carbon steel (45 steel)	$7.85$	$2.1\times {10}^{-5}$	0.3
Electric welding material	0	$2.1\times {10}^{-5}$	0.3

.

According to the measurement of the actual parameters of the combine harvester frame and various types of plate and shell structures, the combine harvester frame was measured as a rectangular square tube with a width of 30mm, a height of 30 mm and a thickness of 2 mm. The plate and shell structure of the two connection modes is connected to a simple frame, which is composed of two kinds of rectangular square tubes (a total of four) with a length of 1000 mm and 800 mm. The plate is an 820 × 820 mm square plate with a thickness of 2 mm (in which the bolt connection plate is bolted to the square tube and plate in advance to make good bolt holes).

In order to test the intrinsic vibration characteristics of two types of plate structures, the experimental equipment used in this paper mainly includes a DH5902 vibration signal acquisition instrument, piezoelectric acceleration sensor, force sensor and excitation force hammer.

2.2. Types of Combine Shell Structures

According to the different ways of mechanical connection, the plate and shell structure of combine harvesters is divided into the following four major categories: welded plates, bolted plates, hooked plates and riveted plates. The welded plate is a type of plate–shell whose surface is bonded to the surface of the frame by welding joints. The hooked plate is a plate that is connected to the frame through auxiliary connectors such as clasps. The bolted connecting plate is a plate–shell that forms relatively fixed constraints between the plate surface and the frame surface through the bolted structure. The riveted plate is a plate–shell that fixes the plate by using axial force to nail the rod block in the rivet hole of the part and form a nail head. The plate and shell structure on combine harvesters is mainly bolted and welded, as shown in Figure 1. Riveted joints appear less frequently on existing combines. Hook-ups, on the other hand, are a unique way of connecting the cowling to the body and are also relatively rare. Taking the bolted and welded plate as the primary research objects, this paper carries out a study of its dynamic characteristics. It also analyzes the influence of different connection methods on the dynamic characteristics of the plate–shell.

2.3. Combine Harvester Plate and Shell Structure Modal Simulation

Finite element modal simulation of the plate and shell structure is carried out by using Hyperwork platform. Since the research object is a plate and shell structure, its thickness is much smaller than the length and width dimensions, so the simulation is carried out using shell cells. Shell units (PSHELLs) are used to divide the plate–shells. The body unit (PSOLID) is used to divide the rectangular square tubes of the racks. The cell size is 5 mm. The following is a simulation of the plate and shell structure with two different connection methods.

(1)

Simulation of spot-welding unit

The most used connection method on combine structures is welding. Spot-welded structures are more often used in the connection of sheet metal parts due to their high static strength, light weight, good reliability, stable performance and easy automation. The CWELD unit is used in the Hypermesh version 2020 software to simulate the connection relationship of spot welding, as shown in Figure 2.

(2)

Simulation of bolt connection

For the plates that need to be disassembled on the combine harvester, the plates are usually connected to the body by bolts. It is easy to simulate the bolted connection in Hypermesh by using RBE2 rigid unit. First, the central node of the bolt hole corresponding to the plate and frame is set as an Independent Node. Next, the node at the boundary of the circular hole is designated as a Dependent Node, creating a rigid connection between the two. Then, the Independent and Dependent Nodes are connected through the RBE2 cell. The RBE2 rigid unit is shown in Figure 3.

Optistruct version 2020 software is used to simulate the modal characteristics of the welded plate and bolted plate, respectively, in a free state. The simple model of the welded plate obtained by Hypermesh pre-processing is shown in Figure 4. The simple model of the bolted connecting plate is shown in Figure 5.

2.4. Equivalent Modeling of Plate and Shell Structures

This paper focuses on the transverse vibration of combine harvester plate and shell structures in a low frequency state, considering only the axial component of the unbalanced force but not the radial component of the unbalanced force. Further, the vibration problem of the combine harvester plate and shell structure can be transformed into the transverse forced vibration problem of a four-sided simply supported plate and four-sided solidly supported plate under the action of multi-point concentrated simple harmonic force.

A simply supported plate refers to the flat plate support that can only transfer the horizontal and vertical direction of the force, such as that of the combine harvester frame on the plate through the bolt connected to the support. A solidly supported fingerboard transmits not only horizontal and vertical forces but also bending moments at the support of the fingerboard. For example, the combine thresher frame is welded to the chassis beam simultaneously, or the plate is connected to the support by welding.

Through model simplification, the bolted plate can be simplified to a four-sided simply supported plate, and the welded plate can be simplified to a four-sided solidly supported plate. The simplified model is shown in Figure 6 and Figure 7.

According to the boundary conditions of the combine harvester in practice, combined with the classical thin plate theory, it can be seen that if the boundary of the thin plate is a simply supported edge, i.e., regardless of whether it can slide in the horizontal direction or not, the deflection as well as the bending moment at each point on the edge of the plate is zero. In the rectangular coordinate system, the boundary conditions are shown in Equation (1).

$${(w)}_{y=y_0}=0 {(\frac{\partial w}{\partial y})}_{y=y_0}=0$$

(1)

Also, if the sheet boundary is a fixed edge, i.e., the sheet boundary is a wholly fixed edge or a flat clamped edge (which is free to slide along the plane direction), the deflection at each point on the edge and the slope of the deflection along the perpendicular direction of that edge are zero. In the right-angle coordinate system, the boundary conditions are shown in Equation (2).

$${(w)}_{y=y_0}=0, {(\frac{{\partial }^{2}w}{{\partial }^{2}y})}_{y=y_0}=0$$

(2)

In this paper, the intrinsic vibration and forced vibration of a four-sided simply supported plate and a four-sided solidly supported plate are investigated through dynamic modeling. At the same time, multiple simple harmonic forces are used as excitation sources to analyze the dynamic response of such plates and shells under multiple excitation sources from the perspective of the analytical method. Based on the above simplification of the vibration sources and plate and shell structures, this paper transforms the vibration problem of plate and shell structures on combine harvesters into a low-frequency transverse vibration problem of a rectangular simply supported (solidly supported) plate under the action of multi-point centralized simple harmonic forces. Only the axial component of the vibration sources is considered.

2.4.1. Dynamic Modeling of a Four-Sided Simply Supported Plate

Based on the classical plate and shell theory, the basic assumptions of Kirchhoff’s thin plate theory are used, and the following assumptions are made in this paper: (a) in plate bending, the standard line perpendicular to the midplane before and after deformation still maintains a straight vertical line; (b) when the plate is bending and deforming, σ_x, σ_y and τ_xy are the principal stresses, τ_xz and τ_yz are the secondary stresses and σ_z is even smaller; (c) when the plate is bending, the transverse displacements $w$ are identical at all points in the straight line perpendicular to the midplane; (d) no deformation occurs at the midplane during force bending or bending vibration, and there is no displacement of all points in the surface relative to the midplane; (e) when the plate is bent, the transverse displacement w is the same as that of the points on the line perpendicular to the center plane; and (f) the center plane does not deform when the plate is subjected to bending or bending vibration, and there is no displacement relative to the center plane at each point in the plane. According to Kirchhoff’s thin plate theory, the primary differential equation for the transverse vibration of a rectangular thin plate can be expressed as shown in Equation (3).

$$\frac{{\partial }^{4}w}{\partial x^4}+2\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w}{\partial y^4}+\frac{\rho h}{D}\frac{{\partial }^{2}w}{\partial t^2}={\nabla }^2{\nabla }^{2}w+\frac{\rho h}{D}\frac{{\partial }^{2}w}{\partial t^2}=\frac{q}{D}(x,y,z)$$

(3)

In Equation (3), ρ is the mass density per unit area of the rectangular thin plate, h is the thickness of the rectangular thin plate and q is the dynamic load applied on the rectangular thin plate per unit area in the transverse direction.

$${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$$

(4)

where ${\nabla }^2$ is the Laplace operator.

$$D=\frac{E{h}^3}{12(1-{\mu }^2)}$$

(5)

In Equation (5), D is the bending stiffness or flexural rigidity of the plate.

For the inherent vibration of a four-sided simply supported plate, the solution of the basic equation is shown in Equation (6).

$$w=W\sin (\omega t+\phi )$$

(6)

Therefore, the vibration equation for the intrinsic vibration of the rectangular plate is shown in Equation (7).

$$\frac{{\partial }^{4}w}{\partial x^4}+2\frac{{\partial }^{4}w}{\partial x^2\partial y^2}+\frac{{\partial }^{4}w}{\partial y^4}-{\alpha }^{4}W=0$$

(7)

Included among these are

$${\alpha }^4={\omega }^2\frac{\rho h}{D}$$

(8)

For a four-sided simply supported plate, the vibration boundary conditions can be expressed as shown in Equation (9).

$$\left\{\begin{array}{l}x=0,x=a:W=\frac{{\partial }^{2}W}{\partial x^2}=0\\ y=0,y=b:W=\frac{{\partial }^{2}W}{\partial y^2}=0\end{array}\right.$$

(9)

The vibrational solution can be expressed as a double trigonometric function, as shown in Equation (10).

$$W(x,y)=A\sin \frac{m\pi x}{a}\sin \frac{n\pi y}{b}$$

(10)

where A is a constant. Substitution into Equation (7) gives the result shown in Equation (11).

$$A({\pi }^4{(\frac{m^2}{a^2}+\frac{n^2}{b^2})}^2-{\alpha }^4)\sin \frac{m\pi x}{a}\sin \frac{n\pi y}{b}=0$$

(11)

For the above equation to hold for any point on the plate with a non-zero solution for the vibration pattern ($A$ ≠ 0), the condition is shown in Equation (12).

$${\pi }^4{(\frac{m^2}{a^2}+\frac{n^2}{b^2})}^2-{\alpha }^4=0$$

(12)

By calculation, the (m,n)th-order vibration mode solution and the corresponding intrinsic frequency of the four-sided simply supported plate satisfying Equations (7) and (9) are shown in Equations (13) and (14).

$$W_{mn}(x,y)=\sin \frac{m\pi x}{a}\sin \frac{n\pi y}{b}$$

(13)

$${\omega }_{mn}^2=\frac{{\pi }^{4}D}{\rho h}{(\frac{m^2}{a^2}+\frac{n^2}{b^2})}^2$$

(14)

2.4.2. Dynamic Modeling of a Four-Sided Fixed-Support Plate

Due to the limitation of the boundary conditions, there is no definite analytical solution for the four-sided solidly supported rectangular plate, which can be solved by the above general solution method for solving the four-sided simply supported plate. Existing methods are usually based on the displacement variational principle of energy, and the exact solution can be approximated by utilizing the infinite term series sum. In this paper, the intrinsic vibration modes of a four-sided solidly supported plate are solved based on the multinomial combination method of beam functions. The beam function means that when one direction of a rectangular plate is very long, its vibration mode in the other direction will be very close to the vibration mode function of the corresponding boundary condition unidirectional plate or beam. The vibration mode functions of solidly supported beams at both ends in x and y directions are shown in Equations (15) and (16).

$$X_m(x)=\sin \frac{k_{m}x}{a}+A_m\cos \frac{k_{m}x}{a}+B_m\sin h\frac{k_{m}x}{a}+C_m\cos h\frac{k_{m}x}{a}$$

(15)

$$Y_n(y)=\sin \frac{k_{n}y}{b}+A_n\cos \frac{k_{n}y}{b}+B_n\sin h\frac{k_{n}y}{b}+C_n\cos h\frac{k_{n}y}{b}$$

(16)

Let the deflection vibration pattern of the rectangular plate be as shown in Equation (17).

$$W_{mn}(x,y)={\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y)$$

(17)

where X_m(x) and Y_n(y) are the mth- and nth-order shape functions of the solidly supported beams in the x and y directions and A_mn is the coefficient to be determined.

From Kirchhoff’s assumption combined with the geometric and physical equations of thin plate vibration, the specific energy per unit volume expression of thin plate vibration can be obtained as shown in Equation (18).

$$v=\frac{E{z}^2}{2(1-{\mu }^2)}[{(\frac{{\partial }^{2}w}{\partial x^2})}^2+{(\frac{{\partial }^{2}w}{\partial y^2})}^2+2\mu (\frac{{\partial }^{2}w}{\partial x^2})(\frac{{\partial }^{2}w}{\partial y^2})+2(1-\mu ){(\frac{{\partial }^{2}w}{\partial x\partial y})}^2]$$

(18)

Integrating Equation (18) over the thin plate surface, the result is shown in Equation (19).

$$\begin{gathered} V={\displaystyle \underset{V}{\iiint }vdV={\displaystyle \underset{A}{\iint }[{\displaystyle \underset{-h/2}{\overset{h/2}{\int }}Wdz}]}}dxdy \\ =\frac{D}{2}{\displaystyle \underset{A}{\iint }[{(\frac{{\partial }^{2}w}{\partial x^2})}^2+{(\frac{{\partial }^{2}w}{\partial y^2})}^2+2\mu (\frac{{\partial }^{2}w}{\partial x^2})(\frac{{\partial }^{2}w}{\partial y^2})+2(1-\mu ){(\frac{{\partial }^{2}w}{\partial x\partial y})}^2]}dxdy \end{gathered}$$

(19)

In the rectangular coordinate system, the vibration mode boundary conditions of the four solid supported edges are shown in Equation (20).

$$\left\{\begin{array}{l}x=0,x=a:W=\frac{\partial W}{\partial x}=0\\ y=0,y=b:W=\frac{\partial W}{\partial y}=0\end{array}\right.$$

(20)

Substituting the boundary conditions into Equation (19) yields the result shown in Equation (21).

$$V=\frac{D}{2}{\displaystyle \underset{A}{\iint }[{(\frac{{\partial }^{2}w}{\partial x^2})}^2+{(\frac{{\partial }^{2}w}{\partial y^2})}^2]}dxdy=\frac{D}{2}{{\displaystyle \underset{A}{\iint }(w)}}^{2}dxdy$$

(21)

Among them are

$$\begin{gathered} {\displaystyle {\int }_0^a{\displaystyle {\int }_0^b(\frac{{\partial }^{2}w}{\partial x^2})(\frac{{\partial }^{2}w}{\partial y^2})dxdy={\displaystyle {\int }_0^a[}}}(\frac{{\partial }^{2}w}{\partial x^2})(\frac{\partial w}{\partial y}){|}_0^b-{\displaystyle {\int }_0^b(\frac{{\partial }^{3}w}{\partial x^2\partial y})}(\frac{\partial w}{\partial y})dy)]dx \\ ={\displaystyle {\int }_0^a[(\frac{{\partial }^{2}w}{\partial x^2})(\frac{\partial w}{\partial y}){|}_0^b-(\frac{{\partial }^{3}w}{\partial x^2\partial y})(w){|}_0^b+{\displaystyle {\int }_0^b(\frac{{\partial }^{4}w}{\partial x^2\partial y^2})(w)}dy)]dx} \\ ={\displaystyle {\int }_0^a{\displaystyle {\int }_0^b(\frac{{\partial }^{4}w}{\partial x^2\partial y^2})(w)}}dxdy \end{gathered}$$

(22)

$$\begin{gathered} {\displaystyle {\int }_0^a{\displaystyle {\int }_0^b{(\frac{{\partial }^{2}w}{\partial x\partial y})}^{2}dxdy={\displaystyle {\int }_0^a(\frac{{\partial }^{2}w}{\partial x^2})(\frac{\partial w}{\partial y}){|}_0^{b}dx}}}-{\displaystyle {\int }_0^a{\displaystyle {\int }_0^b(\frac{{\partial }^{3}w}{\partial x\partial y^2})(\frac{\partial w}{\partial x})dxdy}}) \\ ={\displaystyle {\int }_0^a(\frac{{\partial }^{2}w}{\partial x\partial y})(\frac{\partial w}{\partial y}){|}_0^{b}dx-{\displaystyle {\int }_0^b(\frac{{\partial }^{3}w}{\partial x\partial y^2})(w)}{|}_0^{a}dy+{\displaystyle {\int }_0^a{\displaystyle {\int }_0^b(\frac{{\partial }^{4}w}{\partial x^2\partial y^2})(w)}dxdy}} \\ ={\displaystyle {\int }_0^a{\displaystyle {\int }_0^b(\frac{{\partial }^{4}w}{\partial x^2\partial y^2})(w)}}dxdy \end{gathered}$$

(23)

From Hamilton’s principle, we can get as shown in Equation (24).

$$\begin{gathered} \delta {\displaystyle {\int }_{t_0}^{t_f}(T-U)}dt+{\displaystyle {\int }_{t_0}^{t_f}{\displaystyle {\iint }_S{P}_Z\delta wdsdt=0}} \\ T=\frac{1}{2}{{\displaystyle \underset{\Omega }{\iint }\rho h(\frac{\partial w}{\partial t})}}^{2}dxdy \end{gathered}$$

(24)

where T is the kinetic energy of the sheet, U is the elastic potential energy of the sheet, P_z is the transverse load, for free vibration P_z = 0.

Let the vibration deflection of the thin plate be as shown in Equation (25).

$$w(x,y,t)=W(x,y)\sin (\omega t+\phi )$$

(25)

Substituting Equation (25) into Equation (24) yields Equation (26).

$$U_m=\frac{D}{2}{\displaystyle \underset{S}{\iint }(\frac{{\partial }^{2}w}{\partial x^2}+\frac{{\partial }^{2}w}{\partial y^2}}{)}^{2}dxdy,T_m=\frac{1}{2}{{\displaystyle \underset{S}{\iint }\rho hW}}^{2}dxdy$$

(26)

Bringing Equation (17) into Equation (26), the result obtained is shown in Equation (27).

$$\begin{gathered} \delta (U_m-{\omega }^2{T}_m)=\delta [U_m({\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y))-{\omega }^2{T}_m{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y)] \\ =\frac{\partial }{\partial A_{mn}}[U_m({\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y)-{\omega }^2{T}_m{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y)]\delta A_{mn} \\ =0 \end{gathered}$$

(27)

Due to the arbitrariness of δA_mn, all the coefficients in parentheses in Equation (27) are zero, so Equation (28) can be obtained.

$$[U_m({\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y)-{\omega }^2{T}_m{\displaystyle \sum _{m=1}^{\infty }{\displaystyle \sum _{n=1}^{\infty }{A}_{mn}}}{X}_m(x)Y_n(y)]=0$$

(28)

Thus, the equation yields a system of nth-order chi-square linear equations about A_mn, whose eigenvalue is the square of the modal vibration frequency of the four-sided solidly supported rectangular plate. Substituting the eigenvectors into Equation (17) yields the mode intrinsic vibration pattern.

2.4.3. Modal Analysis of Plate and Shell with Different Connections Designed

In order to verify the influence of different connection methods on the inherent vibration characteristics of the plate and shell structure, this section will verify the test modal analysis of the welded plate and bolted plate. In this paper, the modal test of the combine harvester plate and shell structure adopts the vibration signal acquisition method of single-point excitation and multi-point pickup. The hammering method is used as the excitation method. Before the test, a reasonable layout is needed to accurately outline the frame structure and modal vibration pattern of the experimental object. The excitation point should be arranged in the place where the whole structure can be vibrated, and at the same time avoiding the essential vibration mode of the structure.

The relevant information of the central test apparatus is shown in

Table 2. Equipment and its parameters required for modal test.

No.	Equipment Name	Model	Manufacturer	Main Parameters
1	Exciting force hammer (nylon hammer head)	LC02	PCB	Sensitivity: 10 mV/1 bf; Range: ±500 1bf pk
2	Force transducer	3A102	PCB	Sensitivity: 4.512 pC/N Range: ±500 N
3	Piezoelectric acceleration sensor	1A312E	PCB	Sensitivity: 100 mV/g Range: ±500 g
4	Signal acquisition instrument	DH5902	DongHua Test	Number of channels: 36 Sampling bandwidth: 16,100 kHz

.

An image of the main test apparatus in the modal test is shown in Figure 8.

The actual plate structure was fabricated on a 1:1 scale based on the numerical model of the plate and shell structure used in the computational modal analysis, as shown in Figure 9 and Figure 10. Figure 9 shows the plate structure connected by welding. Figure 10 shows the plate structure connected by bolting. The connection points correspond to the numerically modeled connection points.

By hammering the selected excitation point in advance, the plate structure is forced to vibrate by providing an exciting force. The three-way acceleration sensor on the board will collect the vibration response signal of each measuring point under the exciting force. After the measured frequency response function is imported into the modal analysis software, the inherent properties of the plate structure can be obtained by subsequent analysis.

The test adopts the method of single-point excitation and multi-point pickup, and the measurement is carried out in batches according to the number of points and the number of sensors. According to the actual number of sensors, 4 acceleration sensors are installed in each batch. During the test, each batch was tapped three times, respectively, and the data collected from these three times were linearly averaged. The sampling mode is set to continuous sampling, and the sampling frequency is 2.56 kHz. The trigger mode is a signal trigger, and the number of analysis points is 4096. The frequency response type is selected as H1 (unbiased estimation).

After the test sampling begins, the corresponding test point is hammered (trying to keep the same force each time). The fluctuations in frequency response function, impulse force signal and coherence function of each measuring point were observed in real time by the signal acquisition instrument. The validity of the signal acquisition is judged according to signal fluctuation. During the experiment, the square tube supports of the two types of plate structures were fixed, and the mode test of the middle plate structure was carried out by the hammer method. This experiment adopts the technique of single-point excitation and multi-point pickup. During hammering, we avoided a large amplitude of vibration. Figure 11 shows the field experiment diagram during the modal test of the two types of plate structures. Point A is the signal analysis system. Point B is the signal acquisition instrument. Point C is the acceleration sensor.

3. Results and Discussions

3.1. Analysis of Simulation Results of Plate and Shell Structure

(1)

Analysis of modal frequencies and vibration patterns

After simulation and analysis by Hyperwork version 2020 software, the first six orders of elastic modes of the simple model of welded plate and bolted plate in the free state were obtained. The description of the simulation calculation is shown in

Table 3. The first 12 modal frequencies and modal shapes of the welded plate and bolted plate.

Object of Analysis	Order	Frequency/Hz	Mode Shape
Bolt connection plate	1	4.24 × 10−3	Overall lower left corner of the front and rear vibration
	2	4.53 × 10−3	Overall front–back vibration
	3	4.56 × 10−3	Overall up and down translation
	4	4.65 × 10−3	Overall left–right translation
	5	5.05 × 10−3	Overall left–right–front–right torsion
	6	5.37 × 10−3	Overall up and down/back and forth twisting
Weld plate	1	4.28 × 10−3	Overall left–right translation
	2	4.52 × 10−3	Integral front–back translation
	3	4.63 × 10−3	Overall first-order torsional mode
	4	4.90 × 10−3	Overall up and down translation
	5	5.63 × 10−3	Overall left–right–front–right torsion
	6	5.74 × 10−3	Overall up and down/back and forth torsion

.

Corresponding to the above-mentioned 1st–6th-order intrinsic frequencies, the modal vibration patterns of each order of this simple model of welded plate are shown in Figure 12.

Corresponding to the above-mentioned 1st–6th-order intrinsic frequencies, the modal vibration shapes of each order of the simple model of this bolted plate are shown in Figure 13.

According to the natural frequency, except for the rigid body, the first 1-6 modes of the simple model of the welded plate and bolted plate are free modes. The first- and third-order natural modes are the first- and second-order bending modes, respectively. The second-, fourth- and sixth-order natural modes are the first-, second-, third- and fourth-order torsional modes, respectively. The modal results of the bolted plates are similar to those of the welded plates.

(2)

Comparative analysis of welded and bolted plates

The intrinsic vibration frequencies (except rigid body modes) obtained from the simulation of the two connection types of the plate and shell structures are compared as described above.

As shown in Figure 14, the intrinsic vibration frequencies of the plate and shell structures with different connection methods have significant differences. The first six orders of the intrinsic frequency of the bolted plate are larger than the intrinsic frequency of the welded plate. From the comparison of Figure 12 and Figure 13, it can be seen that the two types of plate and shell structures with different connection methods are basically similar in terms of the modal shapes of each order (except for the rigid body modes). Therefore, the different connection methods only affect the intrinsic vibration frequency of the plate and shell structure, and the intrinsic vibration modes are less affected.

Through the free mode analysis of the welded plate and bolted plate (including the square tube frame), the constrained mode of the plate itself is further simulated. The obtained natural modes of the first six orders of the welded plate and bolted plate are shown in Figure 15.

As can be shown from the constrained mode vibration patterns of the welded and bolted plates in Figure 15, the constrained mode intrinsic vibration patterns of the welded and bolted plates are the same. At the same time, comparing Figure 12, Figure 13 and Figure 15, it can be seen that the vibration modes of the welded plate and the bolted plate are the same when the square tube frame is included and not included. It can be seen that the square tube frame has no effect on the vibration modes of the plates.

Table 4. Natural mode shapes in confinement modes of welded and bolted plates.

Object of Analysis Order	Order	Constrained Mode Frequency/Hz	Free Mode Frequency/Hz
Welded plate	1	6.3257	9.10
	2	7.9732	10.13
	3	17.06	18.32
	4	19.028	19.53
	5	20.33	22.69
	6	33.898	34.80
Bolt-on plate	1	9.4655	13.21
	2	9.9113	13.55
	3	19.602	20.87
	4	22.499	21.75
	5	22.825	24.91
	6	35.393	37.31

shows the intrinsic vibration modes of the welded and bolted plates in the restrained mode. Comparing the intrinsic modes of the welded plate and the bolted plate in the restrained mode, it can be seen that the intrinsic frequency of the bolted plate is higher than that of the welded plate in the first six orders. This is the same as the result obtained by comparing the free-mode intrinsic frequencies of the plates containing the frame. However, comparing the restrained mode intrinsic frequency of the plate only and the free mode intrinsic frequency of the plate with the frame, it can be seen that the first 6^th-order intrinsic frequency of the welded plate is lower than that of the bolted plate, which indicates that the presence of the frame has an effect on the mode intrinsic frequency of the plate.

Therefore, through the above modal simulation, it can be concluded that the connection method of the plate and the frame has a large impact on the inherent vibration characteristics of the plate. The bolt connection will make the inherent vibration frequency of the plate higher than that of the welding method, but the impact on the inherent vibration pattern of the plate is small.

3.2. Validation of Test Results

(1)

Parametric identification of frequency response curves of received vibration signals

The modal assurance criterion (MAC) is a statistical index that is more sensitive to significant differences in modal modes and less sensitive to minor differences. Therefore, it can be used as a statistical index to show the degree of consistency between modes. After the vibration signal of the shell structure is obtained by the acceleration sensor, it is necessary to identify the parameters of the frequency response curve of the received vibration signal by the relevant software to obtain the natural vibration properties of the structure. At the same time, the accuracy of the parameter identification was judged by the modal assurance criterion (MAC) graph.

Figure 16 shows the MAC diagrams obtained from the modal tests of the welded and bolted plates. As shown in Figure 16, the correlation between the modal frequencies of each order is close to 0, and the modal shapes of each frequency are independent of each other, which indicates that the modal parameters calculated in this modal test are valid.

(2)

Comparison between dynamic model and simulation model.

The above expressions for the intrinsic vibration modes of the four-sided simply supported plate and the four-sided solidly supported plate are expressed as image data output through MATLAB 2022, which is then compared with the simulation results to analyze the accuracy of the dynamic model.

The material parameters of the rectangular thin plate used for data comparison in this paper are shown in

Table 5. Material parameters of rectangular thin plates.

No.	Parameter	Value
1	Size (length a, width b)	a $\times$ b = 0.8 m $\times$ 0.5 m
2	Thickness (h)	H = 0.001 m
3	Mass density ($\rho$)	$\rho$= 7850 kg/${\mathrm{m}}^3$
4	Modulus of elasticity (E)	E = 2 ×${10}^{11}$ Pa
5	Poisson’s ratio ($\upsilon$)	$\upsilon$= 0.3

.

Firstly, the intrinsic shapes and intrinsic frequencies of the four-sided simply supported rectangular thin plate are calculated, and the expressions of the intrinsic shapes are input into MATLAB. Then, the analytical solution of the first six natural modes of the four-sided simply supported plate is obtained by MATLAB through the input of each condition parameter. At the same time, Ansys Workbench is used to establish the corresponding numerical model, set up the boundary conditions and solve the first six orders of intrinsic shapes of the four-sided simply supported plate. The results are shown in Figure 17.

As can be seen from Figure 16, the analytical solution of the natural vibration mode of the four-sided simply supported plate calculated by MATLAB is basically consistent with the numerical solution obtained by Ansys. The first natural vibration mode of the plate is one and a half waves on the X-axis and one half on the Y-axis. The second natural mode is two half waves on the X-axis and one half on the Y-axis. The third-order natural mode is one and a half waves on the X-axis and two and a half on the Y-axis. The fourth natural mode is three and a half waves on the X-axis and one and a half on the Y-axis. The fifth-order natural mode is two half waves on the X-axis and two half waves on the Y-axis. The sixth natural mode is four and a half waves on the X-axis and one and a half on the Y-axis.

The intrinsic frequencies obtained from the simulation and analytical intrinsic frequencies are listed (in which the intrinsic frequencies obtained from the analytical method need to be converted into engineering frequencies), as shown in

Table 6. Frequency comparison.

Modal Order	Simulated Intrinsic Modal Order	Analytical Intrinsic Frequency	Error
1	13.346	13.346	0%
2	24.593	24.593	0%
3	42.153	42.138	0.36%
4	43.347	43.338	0.21%
5	53.396	53.385	0.21%
6	69.617	69.580	0.53%

. Comparison shows that the results obtained by the analytical method are similar to those obtained by simulation, and the error is within 1%. This shows the accuracy of the dynamic model of the four-sided simply supported plate established in this paper.

Similarly, MATLAB is used to calculate the intrinsic vibration mode and frequency of the four-sided solidly supported plate. At the same time, the intrinsic vibration mode and intrinsic frequency of the four-sided solidly supported plate are solved by Ansys Workbench. The obtained results are shown in Figure 18.

The comparison of the vibration modes is shown in Figure 18. The natural vibration modes of the four-sided fixed support plate obtained by the dynamic equation established by the analytical method in this paper are the same as those obtained by simulation. The first-order natural vibration mode of the four-sided fixed support plate is one and a half waves on the X-axis and one half on the Y-axis. The second natural mode is two half waves on the X-axis and one half on the Y-axis. The third-order natural mode is one and a half waves on the X-axis and two and a half on the Y-axis. The fourth natural mode is three and a half waves on the X-axis and one and a half on the Y-axis. The fifth-order natural mode is two half waves on the X-axis and two half waves on the Y-axis. The sixth natural mode is four and a half waves on the X-axis and one and a half on the Y-axis. By comparing Figure 17 and Figure 18, it can be seen that the first six natural vibration modes of the four-sided fixed support plate and the four-sided simple support plate in this calculation example are entirely consistent.

The simulated and analyzed intrinsic frequencies of this example are solved and the results are shown in

Table 7. Frequency comparison.

Modal Order	Simulated Intrinsic Frequency	Analytical Intrinsic Frequency	Error
1	25.564	26.1222	2.18%
2	37.79	38.6782	2.35%
3	58.736	60.0771	2.28%
4	63.744	65.1227	2.16%
5	75.21	77.0536	2.45%
6	87.905	89.7852	2.14%

. Comparing the first six orders of intrinsic frequency, it can be seen that the error between the simulation results and the analytical results is within 3%. It can be seen that the dynamic model of the four-sided simply supported plate and four-sided solidly supported plate established in this paper is accurate and effective.

(1)

Analysis of modal test data of plate and shell structure

Through the modal test, the analysis software supporting the modal test instrument obtained the intrinsic vibration frequency and vibration pattern of the welded plate and the simple model of the bolted plate under the constrained mode.

Table 8. Natural frequencies of welded plates and bolted plates.

Object of Analysis	Order	Intrinsic Frequency/Hz	Intrinsic Vibration Pattern
Welded plate	1	61.608	1st-order bending
	2	81.677	First-order torsion
	3	106.436	Second-order bending
	4	125.665	Second-order torsion
	5	138.327	Third-order torsion
	6	154.437	Fourth-order torsion
Bolted plate	1	73.294	1st-order bending
	2	79.551	First-order twist
	3	103.795	Second-order bending
	4	127.225	Second-order twist
	5	155.841	Third-order torsion
	6	176.777	Fourth-order torsion

shows the intrinsic vibration frequency of the welded plate and the bolted plate as well as the description of the modal vibration pattern.

By comparing the intrinsic vibration properties of the welded plate and the bolted plate mentioned above, the 1st–6th-order intrinsic vibration frequency of the bolted plate is significantly higher than that of the welded plate. This conclusion is also consistent with the previous simulation results. Comparing the simulation results with the experimental results, it is found that the experimental modal intrinsic frequency of the welded plate and the bolted plate is higher than the simulated modal intrinsic frequency. Combined with the experimental modal results, it can be seen that the intrinsic vibration characteristics of the plate and shell structure will be affected to a certain extent by different connection methods. The bolted connection has better intrinsic vibration characteristics than the welded connection, but it is limited to the intrinsic frequency and has less effect on the intrinsic mode.

3.3. Sections to Be Improved

(1)

In this paper, the forced response vibration modeling of welded and bolted plates under multi-source excitation is not well studied. In particular, the dynamic equations of the four-sided solidly supported plate under multi-point concentrated simple harmonic force need to be further investigated. In addition, the intrinsic vibration dynamic model of the simply supported plate and the solidly supported plate should be explored more deeply in order to obtain a more accurate expression of the vibration characteristics of the structure.

(2)

For the vibration response of the four-sided simply supported plate and the four-sided solidly supported plate under the action of multi-point concentrated simple harmonic force, the influencing factors need to be further investigated, and the influence of each structural parameter on the forced vibration of the plate and shell should be explored in depth. On this basis, the optimal design scheme of the plate and shell structure is further optimized, and a better scheme is proposed to design the plate and shell structure of the combine harvester. In order to optimize the structure of the plate on the combine harvester, the plate before and after optimization should be installed to the corresponding position on the combine harvester, and the vibration signals of the plate should be measured and compared in order to obtain more realistic data.

4. Conclusions

Combine harvesters, as an essential mechanized equipment for grain harvesting, greatly improve the efficiency of grain harvesting food, greatly reduce the input of labor and have become an indispensable production tool in the daily production life of farmers. In existing combine harvesters, there are still development bottlenecks, such as the whole harvesting process of vibration, low reliability, frequent failures and so on. As there are more rotating parts on the combine harvester, these excitation sources will cause severe vibration of the whole machine, which will have a profound impact on the machine’s work. Hence, it is urgent to solve the vibration problem of the combine harvester. In this regard, this paper conducts a study as follows.

(1)

Through the comparison and analysis of the first six orders of the intrinsic vibration pattern and intrinsic frequency, it is concluded that the connection of the plate and the frame has a significant impact on the intrinsic vibration characteristics of the plate. The bolt connection makes the intrinsic vibration frequency of the plate higher than that of the welding method, but the intrinsic vibration pattern of the plate has a small impact.

(2)

The expressions of natural modes of the four-sided simply supported plate and four-sided fixed supported plate are obtained from the angle of analytical method. Through the kinetic equations constructed in this paper, the intrinsic shapes of the four-sided (simply supported) solidly supported plate are the same as those obtained from the simulation model. At the same time, by comparing the natural frequency obtained by simulation and the natural frequency obtained by analysis, the error of the four-sided simply supported plate model is between 0 and 0.53%, which can be considered as less than 1%. The error of the four-sided fixed support plate is between 2.14 and 2.45%, which can be judged as less than 3%.

(3)

The vibration signal acquisition method of single-point excitation and multi-point pickup is adopted, and the hammer method is used as the excitation method to verify the influence of different connection methods on the natural vibration characteristics of the plate and shell structure. After parameter identification of the frequency response curve of the received vibration signal, the accuracy of parameter identification is judged by a MAC graph.

It can be seen from the results that the correlation between the modal frequencies of each order is close to 0, and the modal modes of each frequency are independent of each other, which indicates that the modal parameters calculated in this modal test are effective.

In summary, this paper has investigated the vibration response mechanism and structural optimization of the plate and shell structure of combine harvesters under the vibration environment of multi-source excitation. In practical applications, the findings of this paper on the vibration characteristics of welded (e.g., combine chassis) and bolted connections (e.g., the connection between the cutting table and the conveyor box) on combines can provide a basis for the design and optimization of the plate and shell structure, which in turn can improve the comprehensive performance of the plate and shell structure.