# Response of Composite Beam Structure under Unbalanced Excitation of Rice Threshing System

Authors: Xujun Zhai; Xingqin You; Hao Zhang; Zhong Tang; Xinzhong Wang; Guoqiang Wang
Journal: Shock and Vibration
 (2022)

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## Abstract

This paper studies the dynamic response of the support beam under the unbalanced excitation of the rice threshing system. The differential equation of motion of the supporting beam under the action of harmonic force is established by the modal superposition method, and its dynamic response is analyzed. It is found that the initial vibration of the supporting beam will fluctuate greatly when it is excited by the unbalanced force. In response to this problem, the application of composite beams as the support structure of the threshing system is proposed. The composite beam is mainly composed of uniform and isotropic Euler–Bernoulli beams that are parallel to each other, and the two beams are connected by elastic spring elements. Then, the vibration response of the composite beam structure under unbalanced forces is studied. The results show that the vibration of the composite beam is more stable when it is excited by the unbalanced drum. The overall vibration response amplitude is smaller than that with single support, and the vibration reduction effect is obvious. When the stiffness changes, the change trend of the response amplitude of beam 1 and beam 2 is opposite, that is, the stiffness increases, the vibration amplitude of beam 1 decreases, and the amplitude of beam 2 increases. This shows that the composite beam is suitable for the support structure of the harvester.


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## Body

## 1. Introduction

The threshing device is the core component of the combine harvester. Its main function is to remove the grains from the ears and at the same time separate the grains in the output mixture (grain grains, short stalks, husks, and other debris). During the working process of the threshing drum, the fluctuation of the grain feeding amount and the entanglement of the stalks will cause the drum to produce unbalanced force. Under the excitation of the unbalanced force, forced vibration occurs at the place separated from the support of the roller, which seriously affects the stability of the whole machine. Therefore, the research on the vibration suppression and damping method caused by the unbalanced force of the threshing drum is of great significance.In practical engineering applications, many structures can be simplified to elastic beam structures [1]. In the work of the threshing drum of rice combine harvester, due to the dynamic imbalance generated under its rotating state, the steel structure supporting at its two ends can also be regarded as an elastic beam system excited by simple harmonics. For the study of elastic beams, researchers have proposed many beam models to predict the mechanical response of beam structures. Among them, the Euler–Bernoulli beam [2–5] and Timoshenko beam [6, 7] considering shear deformation are the most classic ones. With the deepening of research, high-order shear deformation theories have been proposed, such as Lee beam theory (LBT) [8] and functionally graded beam [9, 10]. According to various mechanical models of the beam, the natural frequency, mode shape [11, 12], and dynamic response under different loads [13] can be analyzed. In order to obtain the relevant parameters of the beam structure, there are three commonly used analysis methods, including the Rayleigh–Ritz method, the Gorman superposition method, and the accurate dynamic stiffness method [14]. In addition, in order to improve the accuracy of solving the beam model, researchers have proposed more accurate numerical and analytical methods. For example, in order to solve the precise dynamic response of the beam, the improved element method of the frequency-domain spectral element method is used [15]. The dynamic stiffness method is improved, and a new calculation method of dynamic beam element expansion is proposed, which can be used to accurately solve the dynamic stiffness matrix [16].In addition to the study of classical beam theory, composite beams have also been a research hotspot in recent years. The composite beam can be regarded as an important extension of the single beam structure. The system can simplify a complex continuous system formed by connecting two independent systems through a layer of elastic elements. It has been found in many engineering applications, including composite beams [17–21], double-walled carbon nanotubes [22–24], cranes [25], and floating slab railway tracks [26, 27]. A beam and elastic layer of the composite system can be regarded as a continuous dynamic absorber to suppress the forced vibration of the other beam [28]. In different engineering practices, the upper and lower beams have the same or different stiffness, boundary conditions, and linear mass density [29]. Nedri et al. studied the free vibration of composite laminates on elastic foundations using an improved hyperbolic shear deformation theory and presented the numerical results of the free vibrations of the composite slabs [30]. For the establishment of the dynamic model of the composite beam system, Han et al. proposed an accurate analysis method for the double beam system based on the dynamic stiffness method to establish the beam model [31], and through the improved Wittrick–Williams algorithm, its dynamic characteristics were accurately determined by analysis in [32]. Based on this, the dynamic characteristics of the inclined tension double beam system as well as the influence of factors such as sag, bending stiffness, boundary conditions, and actual inclined beam inclination angle on the characteristics are studied [33, 34]. Gohari et al. synthesized higher-order Fourier integrals and higher-order unit step function differential equations to solve the electromechanical bending response of smart piezoelectric composite laminated rectangular plates [35]. In addition, in view of the limitations in modeling and simulation of large-scale structure dynamics and vibration control, a new state-space method [36] was proposed to improve calculation accuracy and efficiency. In actual engineering practice, the composite beam system is subjected to different loads, such as random force [37], moving constant load [28], continuous moving load [38], and simple harmonic oscillator [39]. The dynamic response of the system will also have the difference. External conditions such as hot and humid environment [40] also have a certain impact on its connection layer and dynamic response.In this paper, the support system under the action of the unbalanced force of the threshing system is studied. Based on Euler–Bernoulli beam theory, the dynamic response models of single beam support and composite beam support are established. The dynamic response of the two beams under the action of simple harmonic force is solved by the modal superposition method. The response states of two different support structures under the same excitation force are compared. It is found that the composite beam structure can be better applied to the support structure of the threshing device of the harvester.


## 2. Response of Supporting Beam under Unbalanced Excitation

### 2.1. Unbalanced Excitation Force of Threshing System

First of all, in order to obtain the dynamic load of the threshing drum when it is working, it is necessary to conduct a vibration test on it. The test instrument and test system are shown in Figures1 and 2. The test instrument is mainly DH5902 vibration signal acquisition instrument. The sensor used in the measurement is an acceleration sensor, and its installation position is shown in Figure 3. It mainly measures the vibration response of the drum in three directions: lateral, vertical, and axial. In addition, the drum was run at three speeds of low, medium, and high to explore the dynamic response of the drum under changing operating conditions.Figure 1
Signal collector.Figure 2
Test system.Figure 3
Sensor installation location.The sensor data in the middle of the drum is selected for analysis, and the time-domain response in the vertical direction is shown in Figure4. It can be seen from the figure that when the drum runs stably at a certain speed, its response changes relatively smoothly, which can be approximated for periodic response. When the operating conditions changed, the load changes, and it affects the stability of its response. Further analysis of its spectral response is obtained, and its spectrogram is shown in Figure 5. It can be seen that its response frequency is mainly around 45 Hz.Figure 4
Time-domain response.Figure 5
Frequency-domain response.In order to further explore the response of the support structure under the drum excitation force, the drum excitation force needs to be simplified. Therefore, the dynamic model of the threshing drum is established according to the above test results. Rice grains and stalks enter the threshing drum from the conveying trough to complete the preliminary separation. Due to the influence of the gravity of the drum and its grains and the tension of the sprocket, the shaft end of the drum is not simply subjected to torsion moment. Therefore, the force on the shaft end can be equivalent to tensionT and torque Mq. In addition, during the working process of the threshing drum, the entanglement of the stalks and the change in the quality of the material in the drum will also form an eccentric mass, resulting in unbalanced centrifugal force. Under the excitation of the unbalanced force, the supporting beam produces a vibration response. The eccentric distance of the center of mass is e, the weight of the drum is m, and the centripetal acceleration is a. In order to explore the vibration response of the supporting beam under the unbalanced vibration of the drum, the mechanical model of the vibration system is simplified. Assuming that the centrifugal force changes periodically according to the simple harmonic law, under the action of the periodic load, the supporting beam produces forced vibration. The dynamic model of the threshing drum is shown in Figure 6. The eccentric mass of the rotor is Me, and the unbalanced centrifugal force generated during the rotation is(1)P=Meθ2e.Figure 6
Unbalanced force of the threshing system.According to the measured response spectrum of the drum under normal working conditions, the dynamic load generated by the threshing drum can be approximated as(2)ft≈2sin3×30π⋅t+2cos3×30π⋅t.


### 2.2. Response Model of a Single Beam under the Unbalanced Excitation

The support beam bears the unbalanced force generated during the working process of the drum. As shown in Figure7, the support beam produces plane bending deformation due to the unbalanced excitation of the threshing system. In this process, the shear deformation of the beam can be ignored, so the vibration response of the threshing drum support part can be studied through the Euler–Bernoulli beam model.Figure 7
Deformation of support beam under force.Assuming that the length of the supporting beam isL, its axis is taken as the x-axis, and the left end point is the origin of the x-axis. Suppose the cross-sectional area of the beam at x is A, the elastic modulus of the material is E, the density is ρ, and the moment of inertia of the section about the neutral axis is I. y(x, t) represents the lateral motion function of the beam segment, and f(x, t) represents the external force of the beam. According to Euler–Bernoulli beam theory, the vibration equation of the supporting beam is(3)EI∂4yx,t∂x2+m∂2yx,t∂t2+C∂yx,t∂t=fx,t.In engineering practice, the mode superposition method (modal analysis method) is often used to solve the differential equation of motion of the continuous system. This paper also adopts this method. Firstly, the differential equation of motion of the free vibration of the supporting system can be expressed as(4)∂2∂x2EI∂2y∂x2+m∂2yx,t∂t2=0.According to the actual installation position, the supporting beam is simply supported. The imbalance exciter is applied at the midpoint of the beam. Then, the boundary conditions of the system can be expressed as(5)y0,t=0,yL,t=0,∂2y0,t∂x2=0,∂2yL,t∂x2=0.Under the given boundary conditions, the natural frequency and mode shape function of the supporting beam can be obtained:(6)ωn=nπL4EIm,Ynx=sinnπxL.According to the orthogonality of the mode shape,(7)∫0LmYrxYsxdx=δrs,δrs=1，r=s0，r≠s.The steady-state response solution of the supporting beam under external forces can be expressed as(8)yx,t=∑r=1∞Yrxqrt,where qr(t) and Yr(x) are the generalized mode and the regular mode of the rth mode of the supporting beam under undamped motion, respectively. Substitute the above equation into equation (3) to obtain(9)∑r=1∞EId4Yrx∂x4qrt+cYrxq˙rt+mYrxq¨rt=fx,t.Substituting equation (4) into equation (8), we can obtain(10)∑r=1∞mq¨rt+cq˙rt+mω2qrtYrx=fx,t.Multiplying both sides of the above formula byYs(x), after integrating x from 0 to L, we obtain(11)∑r=1∞q¨rt+cmq˙rt+ω2qrt∫0LYrxYsxdx=∫0LYsxfx,tdx.According to equation (6), equation (10) can be changed to(12)q¨rt+2ξrωrq˙rt+ωr2qrt=frtmr.According to Duhamel’s integral [41], the solution of equation (12) is(13)qrt=∫0thrt−τfrtmrdτ,where hr(t) is the unit impulse response function of the rth model, which can be expressed as(14)hrt=1ωr1−ξr2e−ξrωrtsinωr1−ξr2t,t>0.By substituting equations (5) and (12) into equation (7), the vibration response of the support beam under the unbalanced excitation of the drum can be obtained:(15)yx,t=2ml∑r=1∞sinrπxL∫0t∫0Lhrt−τsinrπζLsinπζLfrdζdτ=2mlsinrπxL∫0thrt−τfrdτ.


### 2.3. Vibration Response Model of Composite Supporting Beam System

The single support beam will produce greater vibration under the action of the unbalanced force of the threshing drum. Therefore, a composite beam structure can be considered. The composite beam system consists of two upper and lower beams with elastic support in the middle. The simplified model is shown in Figure8.Figure 8
Equivalent model of the composite beam system.According to Euler–Bernoulli beam theory, differential equations of motion can be established for the upper and lower beams in the system. At the same time, considering the coupling effect of the middle elastic connecting layer, the differential equation of motion of the combined system is established as follows:(16)EI∂4y1∂x4+m∂2y1∂t2+c∂y1∂t−∂y2∂t+ky1−y2=fx,t,EI∂4y2∂x4+m∂2y2∂t2−c∂y1∂t−∂y2∂t−ky1−y2=0,where y′x,t=y1x,t−y2x,t, which represents the relative displacement of the main beam with respect to the auxiliary beam, and we can obtain(17)EI∂4yx,t∂x4+m∂2yx,t∂t2+2c∂y1−y2∂t+2ky′x,t=fx,t,(18)EI∂4y2x,t∂x4+m∂2y2x,t∂t2=ky′+c∂y1−y2∂t.According to the actual installation position, the composite beam is simply supported, and the application position of the unbalanced excitation is at the midpoint of the main beam. Then, the boundary conditions of the system can be expressed as(19)y10,t=y20,t=0,y1L,t=y2l,t=0,∂2y10,t∂x2=∂2y20,t∂x2=0,∂2y1L,t∂x2=∂2y2L,t∂x2.The external excitation force can be expressed as(20)fx,t=Fxcosωx=Pδx−L2cosωt,where P is a constant, δ is the Dirac delta function, and ω is the excitation frequency.The forced vibration of the relative motion system is solved by the modal superposition method. Firstly, the inherent characteristics of the relative motion system under undamped free vibration are solved. At this time, the differential equation of motion can be expressed as(21)EI∂4yx,t∂x4+m∂2yx,t∂t2+2ky′x,t=0,where y′x,t is the natural vibration mode of the above motion differential equation system, and its solution form can be expressed as(22)y′x,t=Y′xAcosωt+Bsinωt.Substituting equation (21) into equation (20), we can obtain(23)EI∂4Yx∂x4−mω2Y′x+2kY′x=0.Equation (22) can be changed to(24)∂4Y′x∂x4−λ4Y′x=0,where λ4=mω2−2k/EI, and the solution of equation (24) is(25)Y′x=Ceγx.The four complex roots ofγ can be expressed as γ1,2,3,4=α,jα,−α,−jα, α=λ=mω2−2k/EI1/4.Then, the solution of the equation can be expressed as(26)Y′x=C1eαx+C2e−αx+C3ejαx+C4e−jαx.It can be expressed as(27)Y′x=D1coshαx+D2sinhαx+D3cosαx+D4sinαx.Bring in the boundary conditions to obtainD1 =D2 = D3 = 0.Therefore, the available frequency equation is(28)sinαL=0.The characteristic root is(29)αn=nπLn=1,2,3,…,∞.The following can be obtained by calculation:(30)α=λ=mω2−2kEI1/4=nπL.Therefore, the natural frequency can be obtained as(31)ωn=nπL4EIm+2km.The corresponding mode shape is(32)Yn′x=2mLsinnπxL.According to the orthogonality of the mode shape, the following can be obtained:(33)∫0LmYr′xYs′xdx=δrs,where δrs=1，r=s0，r≠s.According to the modal superposition method, the solution of equation (17) can be expressed as(34)y′x,t=∑r=1∞Yr′xqrt,where Yr′x is the mode shape function of the rth mode and qrt is the undetermined time function. Substituting equation (34) into equation (17), we can obtain(35)∑r=1∞EId4Yrx∂x4+2kYrxqrt+2cYrxq˙rt+mYrxq¨rt=fx,t.Combining equations (35) and (23), we can obtain(36)∑r=1∞mq¨rt+2cq˙rt+mω2qrtYrx=fx,t.Multiply both sides of the above equation byYs′x and integrate x from 0 to L to obtain(37)∑r=1∞q¨rt+2cmq˙rt+ω2qrt∫0LYrxYsxdx=∫0LYsxfx,tdx.According to equation (33), the above equation can be simplified to(38)q¨rt+2ξsωsq˙rt+ωs2qst=∫0LYsxfx,tdx.From (20) and (32), we can obtain(39)∫0LYsxfx,tdx=2mLPcosωt∫0Lδx−L2sinsπxLdx.Equation (39) can be further transformed into(40)q¨st+2ξsωsq˙st+ωs2qst=2mLPsinsπLcosωt.The stable solution of the above equation is(41)qst=2mLPωs2sinsπL11−ω/ωs22+2ξsω/ωs2cosωt+ϕs,where(42)ϕs=−tan−12ξsω/ωs1−ω/ωs2.From this, the relative displacement response of the main beam and the auxiliary beam can be obtained as(43)y′x,t=∑r=1∞ArsinrπxLcosωt+ϕr,where(44)Ar=2Psinrπ/2/mLωr21−ω/ωr22+2ξrω/ωr2.After obtaining the relative displacement curve between the two beams, according to equations (18) and (43), the forcing function acting on the auxiliary beam is(45)f2x,t=∑r=1∞Ark2+c2ω2sinrπxLcosωt+ϕr+θ,where θ=tan−1cω/k, and equation (18) can be transformed into(46)EI∂4y2x,t∂x4+m∂2y2x,t∂t2=f2x,t.According to the mode superposition method, the solution of this equation can be expressed as(47)y2x,t=∑s=1∞Y2sxq2st.Similar to solving the relative displacement, equation (48) can be transformed into(48)q¨2st+ω2s2q2st=∫0LY2sxf2x,tdx.The stable solution of equation (48) is(49)q2st=L/2mAsk2+c2ω2/ω2s21−ω/ωs2cosωt+ϕs+θ.Substituting equation (49) and mode function into equation (47), the steady-state response of the auxiliary beam under unbalanced excitation can be obtained:(50)y2x,t=∑r=1∞1mω2r2Ark2+c2ω21−ω/ω2r2sinrπxLcosωt+ϕr+θ.Finally, the steady-state vibration response of the main beam can be obtained by equations (50) and (43) and the relationship between the two beams:(51)y1x,t=∑r=1∞ArsinrπxL1mω2r2k2+c2ω21−ω/ω2r2cosωt+ϕr+θ+cosωt+ϕr.


## 2.1. Unbalanced Excitation Force of Threshing System

First of all, in order to obtain the dynamic load of the threshing drum when it is working, it is necessary to conduct a vibration test on it. The test instrument and test system are shown in Figures1 and 2. The test instrument is mainly DH5902 vibration signal acquisition instrument. The sensor used in the measurement is an acceleration sensor, and its installation position is shown in Figure 3. It mainly measures the vibration response of the drum in three directions: lateral, vertical, and axial. In addition, the drum was run at three speeds of low, medium, and high to explore the dynamic response of the drum under changing operating conditions.Figure 1
Signal collector.Figure 2
Test system.Figure 3
Sensor installation location.The sensor data in the middle of the drum is selected for analysis, and the time-domain response in the vertical direction is shown in Figure4. It can be seen from the figure that when the drum runs stably at a certain speed, its response changes relatively smoothly, which can be approximated for periodic response. When the operating conditions changed, the load changes, and it affects the stability of its response. Further analysis of its spectral response is obtained, and its spectrogram is shown in Figure 5. It can be seen that its response frequency is mainly around 45 Hz.Figure 4
Time-domain response.Figure 5
Frequency-domain response.In order to further explore the response of the support structure under the drum excitation force, the drum excitation force needs to be simplified. Therefore, the dynamic model of the threshing drum is established according to the above test results. Rice grains and stalks enter the threshing drum from the conveying trough to complete the preliminary separation. Due to the influence of the gravity of the drum and its grains and the tension of the sprocket, the shaft end of the drum is not simply subjected to torsion moment. Therefore, the force on the shaft end can be equivalent to tensionT and torque Mq. In addition, during the working process of the threshing drum, the entanglement of the stalks and the change in the quality of the material in the drum will also form an eccentric mass, resulting in unbalanced centrifugal force. Under the excitation of the unbalanced force, the supporting beam produces a vibration response. The eccentric distance of the center of mass is e, the weight of the drum is m, and the centripetal acceleration is a. In order to explore the vibration response of the supporting beam under the unbalanced vibration of the drum, the mechanical model of the vibration system is simplified. Assuming that the centrifugal force changes periodically according to the simple harmonic law, under the action of the periodic load, the supporting beam produces forced vibration. The dynamic model of the threshing drum is shown in Figure 6. The eccentric mass of the rotor is Me, and the unbalanced centrifugal force generated during the rotation is(1)P=Meθ2e.Figure 6
Unbalanced force of the threshing system.According to the measured response spectrum of the drum under normal working conditions, the dynamic load generated by the threshing drum can be approximated as(2)ft≈2sin3×30π⋅t+2cos3×30π⋅t.


## 2.2. Response Model of a Single Beam under the Unbalanced Excitation

The support beam bears the unbalanced force generated during the working process of the drum. As shown in Figure7, the support beam produces plane bending deformation due to the unbalanced excitation of the threshing system. In this process, the shear deformation of the beam can be ignored, so the vibration response of the threshing drum support part can be studied through the Euler–Bernoulli beam model.Figure 7
Deformation of support beam under force.Assuming that the length of the supporting beam isL, its axis is taken as the x-axis, and the left end point is the origin of the x-axis. Suppose the cross-sectional area of the beam at x is A, the elastic modulus of the material is E, the density is ρ, and the moment of inertia of the section about the neutral axis is I. y(x, t) represents the lateral motion function of the beam segment, and f(x, t) represents the external force of the beam. According to Euler–Bernoulli beam theory, the vibration equation of the supporting beam is(3)EI∂4yx,t∂x2+m∂2yx,t∂t2+C∂yx,t∂t=fx,t.In engineering practice, the mode superposition method (modal analysis method) is often used to solve the differential equation of motion of the continuous system. This paper also adopts this method. Firstly, the differential equation of motion of the free vibration of the supporting system can be expressed as(4)∂2∂x2EI∂2y∂x2+m∂2yx,t∂t2=0.According to the actual installation position, the supporting beam is simply supported. The imbalance exciter is applied at the midpoint of the beam. Then, the boundary conditions of the system can be expressed as(5)y0,t=0,yL,t=0,∂2y0,t∂x2=0,∂2yL,t∂x2=0.Under the given boundary conditions, the natural frequency and mode shape function of the supporting beam can be obtained:(6)ωn=nπL4EIm,Ynx=sinnπxL.According to the orthogonality of the mode shape,(7)∫0LmYrxYsxdx=δrs,δrs=1，r=s0，r≠s.The steady-state response solution of the supporting beam under external forces can be expressed as(8)yx,t=∑r=1∞Yrxqrt,where qr(t) and Yr(x) are the generalized mode and the regular mode of the rth mode of the supporting beam under undamped motion, respectively. Substitute the above equation into equation (3) to obtain(9)∑r=1∞EId4Yrx∂x4qrt+cYrxq˙rt+mYrxq¨rt=fx,t.Substituting equation (4) into equation (8), we can obtain(10)∑r=1∞mq¨rt+cq˙rt+mω2qrtYrx=fx,t.Multiplying both sides of the above formula byYs(x), after integrating x from 0 to L, we obtain(11)∑r=1∞q¨rt+cmq˙rt+ω2qrt∫0LYrxYsxdx=∫0LYsxfx,tdx.According to equation (6), equation (10) can be changed to(12)q¨rt+2ξrωrq˙rt+ωr2qrt=frtmr.According to Duhamel’s integral [41], the solution of equation (12) is(13)qrt=∫0thrt−τfrtmrdτ,where hr(t) is the unit impulse response function of the rth model, which can be expressed as(14)hrt=1ωr1−ξr2e−ξrωrtsinωr1−ξr2t,t>0.By substituting equations (5) and (12) into equation (7), the vibration response of the support beam under the unbalanced excitation of the drum can be obtained:(15)yx,t=2ml∑r=1∞sinrπxL∫0t∫0Lhrt−τsinrπζLsinπζLfrdζdτ=2mlsinrπxL∫0thrt−τfrdτ.


## 2.3. Vibration Response Model of Composite Supporting Beam System

The single support beam will produce greater vibration under the action of the unbalanced force of the threshing drum. Therefore, a composite beam structure can be considered. The composite beam system consists of two upper and lower beams with elastic support in the middle. The simplified model is shown in Figure8.Figure 8
Equivalent model of the composite beam system.According to Euler–Bernoulli beam theory, differential equations of motion can be established for the upper and lower beams in the system. At the same time, considering the coupling effect of the middle elastic connecting layer, the differential equation of motion of the combined system is established as follows:(16)EI∂4y1∂x4+m∂2y1∂t2+c∂y1∂t−∂y2∂t+ky1−y2=fx,t,EI∂4y2∂x4+m∂2y2∂t2−c∂y1∂t−∂y2∂t−ky1−y2=0,where y′x,t=y1x,t−y2x,t, which represents the relative displacement of the main beam with respect to the auxiliary beam, and we can obtain(17)EI∂4yx,t∂x4+m∂2yx,t∂t2+2c∂y1−y2∂t+2ky′x,t=fx,t,(18)EI∂4y2x,t∂x4+m∂2y2x,t∂t2=ky′+c∂y1−y2∂t.According to the actual installation position, the composite beam is simply supported, and the application position of the unbalanced excitation is at the midpoint of the main beam. Then, the boundary conditions of the system can be expressed as(19)y10,t=y20,t=0,y1L,t=y2l,t=0,∂2y10,t∂x2=∂2y20,t∂x2=0,∂2y1L,t∂x2=∂2y2L,t∂x2.The external excitation force can be expressed as(20)fx,t=Fxcosωx=Pδx−L2cosωt,where P is a constant, δ is the Dirac delta function, and ω is the excitation frequency.The forced vibration of the relative motion system is solved by the modal superposition method. Firstly, the inherent characteristics of the relative motion system under undamped free vibration are solved. At this time, the differential equation of motion can be expressed as(21)EI∂4yx,t∂x4+m∂2yx,t∂t2+2ky′x,t=0,where y′x,t is the natural vibration mode of the above motion differential equation system, and its solution form can be expressed as(22)y′x,t=Y′xAcosωt+Bsinωt.Substituting equation (21) into equation (20), we can obtain(23)EI∂4Yx∂x4−mω2Y′x+2kY′x=0.Equation (22) can be changed to(24)∂4Y′x∂x4−λ4Y′x=0,where λ4=mω2−2k/EI, and the solution of equation (24) is(25)Y′x=Ceγx.The four complex roots ofγ can be expressed as γ1,2,3,4=α,jα,−α,−jα, α=λ=mω2−2k/EI1/4.Then, the solution of the equation can be expressed as(26)Y′x=C1eαx+C2e−αx+C3ejαx+C4e−jαx.It can be expressed as(27)Y′x=D1coshαx+D2sinhαx+D3cosαx+D4sinαx.Bring in the boundary conditions to obtainD1 =D2 = D3 = 0.Therefore, the available frequency equation is(28)sinαL=0.The characteristic root is(29)αn=nπLn=1,2,3,…,∞.The following can be obtained by calculation:(30)α=λ=mω2−2kEI1/4=nπL.Therefore, the natural frequency can be obtained as(31)ωn=nπL4EIm+2km.The corresponding mode shape is(32)Yn′x=2mLsinnπxL.According to the orthogonality of the mode shape, the following can be obtained:(33)∫0LmYr′xYs′xdx=δrs,where δrs=1，r=s0，r≠s.According to the modal superposition method, the solution of equation (17) can be expressed as(34)y′x,t=∑r=1∞Yr′xqrt,where Yr′x is the mode shape function of the rth mode and qrt is the undetermined time function. Substituting equation (34) into equation (17), we can obtain(35)∑r=1∞EId4Yrx∂x4+2kYrxqrt+2cYrxq˙rt+mYrxq¨rt=fx,t.Combining equations (35) and (23), we can obtain(36)∑r=1∞mq¨rt+2cq˙rt+mω2qrtYrx=fx,t.Multiply both sides of the above equation byYs′x and integrate x from 0 to L to obtain(37)∑r=1∞q¨rt+2cmq˙rt+ω2qrt∫0LYrxYsxdx=∫0LYsxfx,tdx.According to equation (33), the above equation can be simplified to(38)q¨rt+2ξsωsq˙rt+ωs2qst=∫0LYsxfx,tdx.From (20) and (32), we can obtain(39)∫0LYsxfx,tdx=2mLPcosωt∫0Lδx−L2sinsπxLdx.Equation (39) can be further transformed into(40)q¨st+2ξsωsq˙st+ωs2qst=2mLPsinsπLcosωt.The stable solution of the above equation is(41)qst=2mLPωs2sinsπL11−ω/ωs22+2ξsω/ωs2cosωt+ϕs,where(42)ϕs=−tan−12ξsω/ωs1−ω/ωs2.From this, the relative displacement response of the main beam and the auxiliary beam can be obtained as(43)y′x,t=∑r=1∞ArsinrπxLcosωt+ϕr,where(44)Ar=2Psinrπ/2/mLωr21−ω/ωr22+2ξrω/ωr2.After obtaining the relative displacement curve between the two beams, according to equations (18) and (43), the forcing function acting on the auxiliary beam is(45)f2x,t=∑r=1∞Ark2+c2ω2sinrπxLcosωt+ϕr+θ,where θ=tan−1cω/k, and equation (18) can be transformed into(46)EI∂4y2x,t∂x4+m∂2y2x,t∂t2=f2x,t.According to the mode superposition method, the solution of this equation can be expressed as(47)y2x,t=∑s=1∞Y2sxq2st.Similar to solving the relative displacement, equation (48) can be transformed into(48)q¨2st+ω2s2q2st=∫0LY2sxf2x,tdx.The stable solution of equation (48) is(49)q2st=L/2mAsk2+c2ω2/ω2s21−ω/ωs2cosωt+ϕs+θ.Substituting equation (49) and mode function into equation (47), the steady-state response of the auxiliary beam under unbalanced excitation can be obtained:(50)y2x,t=∑r=1∞1mω2r2Ark2+c2ω21−ω/ω2r2sinrπxLcosωt+ϕr+θ.Finally, the steady-state vibration response of the main beam can be obtained by equations (50) and (43) and the relationship between the two beams:(51)y1x,t=∑r=1∞ArsinrπxL1mω2r2k2+c2ω21−ω/ω2r2cosωt+ϕr+θ+cosωt+ϕr.


## 3. Vibration Response Analysis of Support System Subjected to Unbalanced Excitation

### 3.1. Vibration Response of Single Beam

Support beam material is Q235 steel, elastic modulusE = 200 GPa, and density ρ = 7.858 × 103 kg/m3. The length of the beam is L = 0.5 m, and the section shape is rectangular. For the moment of inertia of the rectangular section I = (bh3)/12, b is the width and h is the height. Take b = 0.06 m and h = 0.003 m; then, I = I = 1.35 × 10−10 m4. A simple harmonic excitation with a frequency of ω = 30 π rad/s and an amplitude force of 500N is applied to the center of the beam, and the displacement of each point on the supporting beam varies with time as shown in Figure 9.Figure 9
Time-domain response of the supporting beam.From the time-domain response of the displacement of each point of the supporting beam, it can be seen that the amplitude oscillation of the supporting beam is attenuated due to the instability of the load within the load of 0 to 0.4 s. Then, it stabilized. Further analysis of the midpoint of the support beam, as shown in Figure10, shows a more obvious attenuation trend. Furthermore, the effect of damping on its vibration response is further explored, as shown in Figure 11, and the vibration amplitude of the support beam decreases as the material damping increases.Figure 10
Midpoint response process.Figure 11
Time-domain response of midpoint displacement under different damping.During the working process of the threshing system, its speed will be adjusted according to the field conditions, and the excitation frequency of the unbalanced force produced by it will change accordingly. Therefore, the excitation frequency of the external force is changed, and the vibration response of the single beam support at different frequencies is analyzed. The time-domain response is shown in Figure12. It can be seen from the figure that as the excitation frequency increases, the instability of the vibration response becomes more obvious. The vibration amplitude also increases accordingly.Figure 12
Response of beams under different excitation frequencies.


### 3.2. Vibration Response Analysis of Composite Beam Structure

The vibration response of the single beam support structure after being excited by the unbalanced threshing drum is unstable. During the initial response, the amplitude will fluctuate, showing nonperiodic vibration. The response of the composite beam structure with elastic support connection under unbalanced excitation is further analyzed. The response of each point of beam 1 under simple harmonic excitation in the composite system is shown in Figure13, and the response of each point on the bottom supporting beam (beam 2) is shown in Figure 14. It can be seen from the figure that under the action of simple harmonic excitation, the time-domain response of the composite beam structure is a stable sinusoidal wave, which is more stable than a single beam. Also, the overall vibration amplitude is smaller.Figure 13
Time-domain response of beam 1.Figure 14
Time-domain response of beam 2.To analyze the impact of the excitation frequency on the entire system, set the excitation frequencies asw1 = 30π, w2 = 60π, w3 = 90π, and w4 = 120π (rad/s). Set the response time to 0.5 s, and the vibration responses of beam 1 and beam 2 in the system are shown in Figures 15 and 16, respectively. When w1  = 30π, the vibration amplitude of beam 1 is larger. As the frequency increases, the vibration becomes more stable and the amplitude decreases more.Figure 15
Time-domain response of the main beam under different excitation frequencies.Figure 16
Time-domain response of the auxiliary beam under different excitation frequencies.Figure17 shows the time-domain response at the midpoint of the composite beam system. It can be seen that because the excitation force directly acts on beam 1, its vibration response is more obvious. However, beam 2 is less responsive due to the indirect effect of beam 1 and elastic support. At t = 0.5 s, the response mode of the composite system is shown in Figure 18. It can be seen from the figure that the excitation frequency excites the first-order mode of the system, the peak point of vibration is at the center of the system, and the vibration gradually decreases from the center to both sides. Set w1 = 100π, w2=200π, and w3=400π (rad/s), respectively. It can be seen from Figure 14 that when the excitation frequency is w1, the system mode response is a first-order mode. When the excitation frequency reaches w2 and w3, the second-order mode of the system is excited. The working speed of the harvester threshing system is generally 600 r/min∼1000 r/min, which is in the low-frequency vibration area. Therefore, when the composite beam structure is subjected to the unbalanced excitation caused by the threshing system, most of the first-order vibration modes are excited (Figure 19).Figure 17
Time-domain response of the composite beam system.Figure 18
The mode response of the system att = 0.5 s.Figure 19
Vibration shape response excited by different frequencies.In order to explore the influence of the support stiffness of the elastic layer on the vibration response of the system, the response time is selected as 0.5 s, and the support stiffnessK is 2 × 103 N/m, 2 × 104N/m, 2 × 105N/m, and 2 × 106N/m, respectively. The vibration shapes of beam 1 and beam 2 are shown in Figures 20 and 21. It can be seen from the figures that as the stiffness increases linearly, there is no obvious change in the response of the system. When the stiffness changes, the change trend of the response amplitude of beam 1 and beam 2 is opposite. When the stiffness is 2 × 105N/m, the vibration response of beam 1 is the smallest. At this time, the vibration amplitude of beam 2 is significantly larger than that of other stiffnesses. This shows that the elastic support layer and beam 2 work together to absorb part of the vibration energy of beam 1.Figure 20
Vibration shape of beam 1 under different support stiffnesses.Figure 21
Vibration shape of beam 2 under different support stiffnesses.Set the fixed support stiffness ask = 2 × 105 N/m and the excitation frequency as w = 60π. Under this condition, the influence of the damping change on the vibration response of the two beams in the system is analyzed. The response mode of the system at t = 0.5 s is shown in Figure 22. It can be seen from the figure that under this support stiffness, the increase in damping will cause the vibration response of beam 1 to decrease and at the same time cause the response of beam 2 to increase. This shows that while the material damping absorbs the overall vibration energy, the elastic support and beam 2 in the composite beam structure absorb the vibration energy on the beam 1. However, the overall change amplitude is small and basically negligible.Figure 22
Vibration shape of the composite system under different damping.Further analyze the effect of composite beam on vibration suppression and set the supporting stiffness of composite beam ask = 2 × 105N/m. Simultaneously, an unbalanced excitation with an excitation frequency of w  = 60 π·rad/s is applied to the single beam and the composite beam. The time-domain response signal obtained within 1 s is shown in Figure 23. It can be seen from the figure that a single beam exhibits obvious instability under unbalanced excitation, and the vibration response amplitude is relatively large. The vibration response of the composite beam is periodic simple harmonic vibration. Compared with single beams, composite beams can restrain vibration more obviously. A comparative analysis of the response amplitude of the two support systems (Figure 24) shows that in the early stage of the response, the amplitude of a single beam is too large due to the instability of vibration. Then, it stabilized and the amplitude declined. The composite beam is more stable and has a smaller vibration amplitude.Figure 23
Comparison of the time-domain response of two beams.Figure 24
Comparison of vibration response amplitude.


## 3.1. Vibration Response of Single Beam

Support beam material is Q235 steel, elastic modulusE = 200 GPa, and density ρ = 7.858 × 103 kg/m3. The length of the beam is L = 0.5 m, and the section shape is rectangular. For the moment of inertia of the rectangular section I = (bh3)/12, b is the width and h is the height. Take b = 0.06 m and h = 0.003 m; then, I = I = 1.35 × 10−10 m4. A simple harmonic excitation with a frequency of ω = 30 π rad/s and an amplitude force of 500N is applied to the center of the beam, and the displacement of each point on the supporting beam varies with time as shown in Figure 9.Figure 9
Time-domain response of the supporting beam.From the time-domain response of the displacement of each point of the supporting beam, it can be seen that the amplitude oscillation of the supporting beam is attenuated due to the instability of the load within the load of 0 to 0.4 s. Then, it stabilized. Further analysis of the midpoint of the support beam, as shown in Figure10, shows a more obvious attenuation trend. Furthermore, the effect of damping on its vibration response is further explored, as shown in Figure 11, and the vibration amplitude of the support beam decreases as the material damping increases.Figure 10
Midpoint response process.Figure 11
Time-domain response of midpoint displacement under different damping.During the working process of the threshing system, its speed will be adjusted according to the field conditions, and the excitation frequency of the unbalanced force produced by it will change accordingly. Therefore, the excitation frequency of the external force is changed, and the vibration response of the single beam support at different frequencies is analyzed. The time-domain response is shown in Figure12. It can be seen from the figure that as the excitation frequency increases, the instability of the vibration response becomes more obvious. The vibration amplitude also increases accordingly.Figure 12
Response of beams under different excitation frequencies.


## 3.2. Vibration Response Analysis of Composite Beam Structure

The vibration response of the single beam support structure after being excited by the unbalanced threshing drum is unstable. During the initial response, the amplitude will fluctuate, showing nonperiodic vibration. The response of the composite beam structure with elastic support connection under unbalanced excitation is further analyzed. The response of each point of beam 1 under simple harmonic excitation in the composite system is shown in Figure13, and the response of each point on the bottom supporting beam (beam 2) is shown in Figure 14. It can be seen from the figure that under the action of simple harmonic excitation, the time-domain response of the composite beam structure is a stable sinusoidal wave, which is more stable than a single beam. Also, the overall vibration amplitude is smaller.Figure 13
Time-domain response of beam 1.Figure 14
Time-domain response of beam 2.To analyze the impact of the excitation frequency on the entire system, set the excitation frequencies asw1 = 30π, w2 = 60π, w3 = 90π, and w4 = 120π (rad/s). Set the response time to 0.5 s, and the vibration responses of beam 1 and beam 2 in the system are shown in Figures 15 and 16, respectively. When w1  = 30π, the vibration amplitude of beam 1 is larger. As the frequency increases, the vibration becomes more stable and the amplitude decreases more.Figure 15
Time-domain response of the main beam under different excitation frequencies.Figure 16
Time-domain response of the auxiliary beam under different excitation frequencies.Figure17 shows the time-domain response at the midpoint of the composite beam system. It can be seen that because the excitation force directly acts on beam 1, its vibration response is more obvious. However, beam 2 is less responsive due to the indirect effect of beam 1 and elastic support. At t = 0.5 s, the response mode of the composite system is shown in Figure 18. It can be seen from the figure that the excitation frequency excites the first-order mode of the system, the peak point of vibration is at the center of the system, and the vibration gradually decreases from the center to both sides. Set w1 = 100π, w2=200π, and w3=400π (rad/s), respectively. It can be seen from Figure 14 that when the excitation frequency is w1, the system mode response is a first-order mode. When the excitation frequency reaches w2 and w3, the second-order mode of the system is excited. The working speed of the harvester threshing system is generally 600 r/min∼1000 r/min, which is in the low-frequency vibration area. Therefore, when the composite beam structure is subjected to the unbalanced excitation caused by the threshing system, most of the first-order vibration modes are excited (Figure 19).Figure 17
Time-domain response of the composite beam system.Figure 18
The mode response of the system att = 0.5 s.Figure 19
Vibration shape response excited by different frequencies.In order to explore the influence of the support stiffness of the elastic layer on the vibration response of the system, the response time is selected as 0.5 s, and the support stiffnessK is 2 × 103 N/m, 2 × 104N/m, 2 × 105N/m, and 2 × 106N/m, respectively. The vibration shapes of beam 1 and beam 2 are shown in Figures 20 and 21. It can be seen from the figures that as the stiffness increases linearly, there is no obvious change in the response of the system. When the stiffness changes, the change trend of the response amplitude of beam 1 and beam 2 is opposite. When the stiffness is 2 × 105N/m, the vibration response of beam 1 is the smallest. At this time, the vibration amplitude of beam 2 is significantly larger than that of other stiffnesses. This shows that the elastic support layer and beam 2 work together to absorb part of the vibration energy of beam 1.Figure 20
Vibration shape of beam 1 under different support stiffnesses.Figure 21
Vibration shape of beam 2 under different support stiffnesses.Set the fixed support stiffness ask = 2 × 105 N/m and the excitation frequency as w = 60π. Under this condition, the influence of the damping change on the vibration response of the two beams in the system is analyzed. The response mode of the system at t = 0.5 s is shown in Figure 22. It can be seen from the figure that under this support stiffness, the increase in damping will cause the vibration response of beam 1 to decrease and at the same time cause the response of beam 2 to increase. This shows that while the material damping absorbs the overall vibration energy, the elastic support and beam 2 in the composite beam structure absorb the vibration energy on the beam 1. However, the overall change amplitude is small and basically negligible.Figure 22
Vibration shape of the composite system under different damping.Further analyze the effect of composite beam on vibration suppression and set the supporting stiffness of composite beam ask = 2 × 105N/m. Simultaneously, an unbalanced excitation with an excitation frequency of w  = 60 π·rad/s is applied to the single beam and the composite beam. The time-domain response signal obtained within 1 s is shown in Figure 23. It can be seen from the figure that a single beam exhibits obvious instability under unbalanced excitation, and the vibration response amplitude is relatively large. The vibration response of the composite beam is periodic simple harmonic vibration. Compared with single beams, composite beams can restrain vibration more obviously. A comparative analysis of the response amplitude of the two support systems (Figure 24) shows that in the early stage of the response, the amplitude of a single beam is too large due to the instability of vibration. Then, it stabilized and the amplitude declined. The composite beam is more stable and has a smaller vibration amplitude.Figure 23
Comparison of the time-domain response of two beams.Figure 24
Comparison of vibration response amplitude.


## 4. Experimental Verification

In order to verify the validity of the solution results of the composite beam structure model, it is verified by bench experiments. The content of the experiment is mainly the vibration response of single-supported beam and composite beam under simple harmonic force. The content of experimental comparison is the vibration response of the two beams with different excitation frequencies and different support stiffnesses. First, analyze the influence of different excitation forces and excitation frequencies on the vibration response of the support system under ideal boundary conditions. The excitation force and speed are adjusted by the asynchronous vibration motor, so as to realize the measurement of the vibration response of the support structure under different excitation forces. The installation of the test bench and test equipment is shown in Figures25 and 26.Figure 25
Experimental signal acquisition system site.Figure 26
Composite beam structure and model.The speed of the eccentric vibration motor is increased from 800 r/min to 1200 r/min, and the signal acquisition time at each speed is 10 s. The maximum vibration acceleration values of the single beam/composite beam structure are shown in Table1.Table 1
Acceleration amplitude of each measuring point of the single beam/composite beam structure.
Rotating speedPoint 1Point 2Point 3Point 4Point 5Single beam8001.0611.5852.5431.6070.69910001.9034.3997.2924.7472.23712004.71511.47818.84911.7385.214Composite beam8000.6931.4031.9351.4130.61710003.0497.4158.3517.8163.61612004.57711.28513.54612.8566.703It can be seen from the data in the table that the vibration amplitude of each measuring point of a single beam increases significantly with the increase of the rotation speed. Measuring point 3 is the excitation position, so the amplitude is larger, and the amplitude on both sides gradually decreases. At the same speed, the vibration amplitude of the composite beam is smaller than the maximum amplitude of a single beam. It can be verified that the beam 2 and the supporting layer have a suppressive effect on the vibration response of the main beam.The stiffness of the elastic support layer in the middle can be regarded as the stiffness of multiple elastic elements connected in parallel. Therefore, in order to further verify the influence of the support stiffness on this restraint, the parallel stiffness of the upper and lower beams was changed by changing the number of springs in the middle connecting layer of the upper and lower beams. The measured vibration response amplitudes of composite beams under different support stiffnesses are shown in Table2.Table 2
Vibration response amplitude of composite beams with different support stiffnesses.
Rotating speedKPoint 1Point 2Point 3Point 4Point 5800k10.8970.8401.6831.4851.149k20.6931.4031.9351.4130.617k30.6511.2431.8361.2350.5821000k12.9326.9658.5747.3113.439k23.0497.4158.3517.8163.616k32.8526.9008.2317.2603.3751200k15.01611.85614.34213.4407.159k24.57711.28513.54612.8566.703k34.48310.66412.54811.4905.480The single-turn stiffness of the spring used in the test is 51.2 N/mm, and the calculation formula for the stiffness of its single root isk = (Pn − P1)/h, where pn is the maximum working load; p1is the minimum working load; and h is the working stroke; its values are standardized values. From this calculation, the stiffness of a single spring is 2 × 104 N/m. When three springs are connected in parallel, k1 = 6 × 104 N/m. From the data in the table, it can be seen that when the support stiffness is k2, the vibration response amplitude is the largest, and the amplitude at k3 is the smallest. The comparison results of the vibration response of the support beam under different stiffness are shown in Figure 27. It can be seen that when the stiffness reaches 105 N/m, the vibration suppression effect of the composite beam system is the most obvious.Figure 27
Vibration response of composite beams under differentk values. (a) n = 1000 r/min. (b) n = 1200 r/min.The vibration stability of the composite beam and the original support structure was further compared. The eigenvalues of the acceleration signal waveforms of the two beams at the same rotational speed measured experimentally are shown in Table3. As can be seen from the table, the maximum waveform RMS value of the combined beam is 1.948 m/s2, while the RMS value of the single beam is 4.097 m/s2. In addition, the comparison of the kurtosis value and peak-to-peak value of each measuring point shows that the vibration response of the composite beam is more stable.Table 3
Statistics of waveform characteristics of acceleration signals of two structures.
FeaturesPoint 1Point 2Point 3Point 4Point 5Single beamPeak-to-peak6.0128.9119.0035.1889.003RMS value1.1462.4111.7940.9124.097Average value−0.048−0.100−0.0240.0923.630Standard deviation1.1452.4111.7930.9081.901Kurtosis−0.751−0.789−0.832−0.693−0.802Composite beamPeak-to-peak5.4932.9919.6135.2499.217RMS value1.0260.5191.9420.9631.948Average value−0.0170.0710.1180.2360.028Standard deviation1.0250.5141.9380.9341.948Kurtosis−0.453−0.476−0.790−0.572−0.794The comparison of the vibration displacement signals at the midpoint of the two support structures is shown in Figure28. In the sampling time of 20 s, it can be seen that the vibration response of the composite beam support is obviously more stable when subjected to unbalanced force. Figure 29 shows the response curves of the two within 1 s. The vibration of the composite beam presents a periodic damped oscillation. The vibration peak-to-peak value of single beam support is 0.012 mm, and the vibration peak-to-peak value of composite beam is 0.007 mm.Figure 28
Comparison of time-domain response.Figure 29
Comparison of response stability.Spectrum analysis was performed on the test results of single beam and composite beam, and the results are shown in Figure30. It can be seen that the response frequency of the composite beam is stable at 50 Hz, while the single beam support mainly responds at low frequency.Figure 30
Spectral response analysis for two supports. (a) Composite beam. (b) Single beam support.


## 5. Conclusion

In this paper, the harvester threshing drum and its supporting structure are studied, and the vibration response of the threshing drum in the working process is tested. Based on this, the vibration response of the support beam under drum excitation is studied. The dynamic response model of the threshing drum support structure is established by Euler–Bernoulli beam theory. The dynamic responses of the two beams subjected to unbalanced forces are solved by the modal superposition method. In this way, the response characteristics of single beam support and composite beam support under unbalanced excitation are studied. The dynamic responses of composite beams with different support stiffness, different excitation frequencies, and different damping values are analyzed. The response states of two different support structures under the same excitation force are compared. The following conclusions can be drawn:(1)
A composite beam structure with elastic support connection is proposed, which replaces the original support structure at the shaft end of the harvester threshing cylinder. The numerical and experimental results show that the vibration response of the composite beam structure under the unbalanced excitation force of the threshing cylinder is more stable, and the vibration transmission of the threshing cylinder can be suppressed. When subjected to the unbalanced excitation of the threshing drum, a single beam will have fluctuations in response, and the vibration will be unstable. Also, with the increase of excitation frequency, the amplitude of its vibration response increases gradually. Instability is more pronounced. On the contrary, when the composite beam structure is subjected to the unbalanced excitation brought by the threshing system, the response is relatively stable and there is no fluctuation. As the excitation frequency increases, the dynamic response amplitude decreases gradually.(2)
The stiffness value of the elastic connecting layer of the composite beam structure has a certain influence on the dynamic response. When the stiffness changes, the change trend of the response amplitude of beam 1 and beam 2 is opposite, that is, the stiffness increases, the vibration amplitude of beam 1 decreases, and the amplitude of beam 2 increases. When the stiffness is 2 × 105 N/m, the vibration response of beam 1 is the smallest. This shows that the elastic support layer and beam 2 work together to absorb part of the vibration energy of beam 1, and the bench experiment also verifies this result.



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*Source: 5792102-2022-04-06.xml*