# Vibration Theoretical Analysis of Elastically Connected Multiple Beam System under the Moving Oscillator

Authors: Binbin He; Yulin Feng
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

To investigate the vibration analysis of elastically connected multiple beam system (ECMB) under the moving oscillator, finite sine-Fourier transform has been applied to the dynamic partial differential equations of ECMB with respect to space coordinates. Then, the numerical integration has been used to solve the equations. Finally, the expression for vibration analysis of ECMB under the moving oscillator has been derived based on finite sine-Fourier inverse transform. Using the method developed in this study and ANSYS numerical method, the vibration analysis of a four-layer beam system under moving oscillator with different speeds has been calculated. The results show that the calculated results from the method developed in this study are in good agreement with the ANSYS numerical calculation results, and the differences between the two calculation results are all less than 2%, which verified the correctness of the method developed in this study. The method developed in this study has been applied to the beam-rail system on a railway line in China. The effect of the train speed and interlayer stiffness on the vibration of beam-rail system has been investigated. The results show that the maximum dynamic deflection of the rail under the train load is always near the midspan, while the maximum dynamic deflections of the track plate, base plate, and bridge occur after the train travel through the midspan. The interlayer stiffness has a larger impact on the vibration of the rail and the track plate and little impact on the vibration of the base plate and the bridge.


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## Body

## 1. Introduction

Beam-type structures are widely used in mechanical, civil, rail transportation, material engineering, and other fields. In engineering applications, it is of great importance to predict the dynamic response of beam-type structures accurately [1, 2].Hitherto, there are many studies conducted by researchers all over the world on the dynamic response of beam-type structures under a moving oscillator. Also, there are many studies on the dynamic response of single-beam under a moving oscillator [3–5]. An important extension to the concept of single-beam is the double-beam system. The dynamic response of double-beam under a moving oscillator was investigated by different methods [6, 7]. Due to the complexity of the three-beam, the previous studies on the vibration problems of three-beam under a moving oscillator were mostly aimed at its vibration characteristics [8].In many engineering applications, the vibration theory of single-beam, double-beam, and three-beam cannot meet the requirement. Therefore, it is urgently required to develop a method which can determine the vibration response of ECMB. Hitherto, there are only a few studies on the vibration problem of ECMB. Rao [9] investigated the natural vibrations of elastically connected multi-Timoshenko beams. Kelly and Srinivas [10] developed a general theory for the free response of elastically connected axially loaded beams. A set of coupled partial differential equations that govern the free response of a set of n elastically connected axially loaded beams has been developed and nondimensionalized. Using Timoshenko and high-order shear deformation theory, Stojanović et al. [11] investigated a general procedure for the determination of the natural frequencies and buckling load for a set of beam system under compressive axial loading. Ariaei et al. [12] investigated the dynamic behavior of n parallel identical elastically connected Timoshenko beams subjected to a moving load. A relatively new computed approach called the Adomian modified decomposition method has been used to analyze the free vibration problem for an elastically connected multiple beam with arbitrary boundary conditions [13]. Through a differential transform method, a new procedure for determining natural frequencies and mode shapes of a system of elastically connected multiple rotating tapered beams has been presented [14]. Bakhshi Khaniki and Hosseini-Hashemi [15] investigated the dynamical behavior of ECMBs with respect to a moving mass. For double- and three-layered microbridge systems, using Laplace transform, an analytical solution has been developed. Moreover, for higher-layered microbridge systems, a state space method has been used. Hashemi and Bakhshi Khaniki [16, 17] studied the forced vibration of multilayered nanobeam systems (MLNBS) under a moving nanoparticle. Eringen’s nonlocal theory and Euler Bernoulli beam theory were used to model each layer of the MLNBS. It was shown that nonlocality has a significant effect in analyzing such systems. Kargarnovin et al. [18] proposed a closed-form solution to study the dynamics of a composite beam with a single delamination under the action of a moving constant force; the delaminated beam was divided into four interconnected beams using the delamination limits as their boundaries. In summary, most of the current research methods for the vibration response of double-beam and three-beam systems under a moving oscillator have complicated derivation, with low computational efficiency and many limiting conditions. There are only a few studies on the vibration response of ECMBs.In the present paper, based on the finite sine-Fourier transform, a method for calculating the vertical vibration response of ECMB under the moving oscillator has been developed. This method has the advantages of fewer constraints, higher computational efficiency, and a wider range of application. A comparison of the calculated results using this method, ANSYS numerical method, and existed literature showed that they are in good agreement. Therefore, the correctness of the method developed in this study has been verified. The problem of a dynamic response of beams subjected to moving oscillators is a reasonable idealization of how railway tracks and pavements behave under wheel load. This problem has been studied in some of above-mentioned studies [19–22]. Further, a beam-rail system on a railway line in China has been used as a case study. By simplifying the train into an oscillator system, the effect of the train speed on the dynamic deflection and amplification coefficient of beam-rail system and the effect of interlayer stiffness on the dynamic deflection of the midspan of beam-rail system have been analyzed.The original contribution of this article is to consider an elastically connected multiple beam system subjected to a moving oscillator, and the semianalytical expression form of this article is easy to solve, which results in high computational efficiency. The semianalytical method can provide a theoretical basis for deriving a practical formula for engineering calculation and make up the deficiency of numerical simulation analysis. This study could be useful in different subjects of science, e.g., mechanical, civil, traffic, and material engineering and many other fields. For example, the dynamic behavior of the bridge-track system of railway (maglev) under the train load or multilayered nanobeam systems as a part of nanostructures has achieved more usages with improvements done in nanotechnology and the usage of smaller devices in designing different machines.


## 2. Mathematical Model and Governing Equations

### 2.1. Governing Equations of ECMB under the Moving Oscillator

The schematic diagram for ECMB under a moving oscillator is shown in Figure1. All of beams are homogeneous and prismatic and have the same length L, but they can have different mass or flexural rigidities, which makes the model more real in reflecting some actual engineering projects. According to the d’Alembert principle, the dynamic equilibrium equation of an oscillator system is(1)M2∂2Zt∂t2+k0Zt−y1=0.Figure 1
The schematic diagram for ECMB.According to the d’Alembert principle, the dynamic equilibrium equations of ECMB are(2)E1I1∂4y1∂x4+m1∂2y1∂t2+k1y1−y2=P0t,(3)E2I2∂4y2∂x4+m2∂2y2∂t2−k1y1−y2+k2y2−y3=0,(4)E3I3∂4y3∂x4+m3∂2y3∂t2−k2y2−y3+k3y3−y4=0,⋯(5)En−1In−1∂4yn−1∂x4+mn−1∂2yn−1∂t2−kn−2yn−2−yn−1+kn−1yn−1−yn=0,(6)EnIn∂4yn∂x4+mn∂2yn∂t2+kn−1yn−yn−1=0,(7)P0t=δx−vtM1+M2g−M1∂2y1∂t2+2v∂2y1∂x∂t+v2∂2y1∂x2+k0Zt−y1,where ynx,t,En,In,andmn with n=2,3,4,… are the dynamic deflection, elastic modulus, horizontal inertia moment, and beam mass per unit length for each layer of beam, respectively; M1andM2 are the unsprung mass and sprung mass, respectively; and Zt is the dynamic deflection of the sprung mass, and it is assumed that the unsprung mass moves along the beam without separating from the beam body. In this way, the deflection of unsprung mass is the same as that of the beam segment at the same location; k0 and c0 are the spring coefficient and damping coefficient of an oscillator system, respectively; kn−1 with n=2,3,4,… is the interlayer spring stiffness; δ is Dirac function; v is the moving speed of the oscillator; P0t is the external force; and g is the gravitational acceleration.


### 2.2. Finite Sine-Fourier Transform

In order to solve equations (1)–(7), the finite sine-Fourier transform has been used and the transformation is defined as follows [23]:(8)℘ynx,t=Un,kt=∫0Lynx,tsinξkxdx,(9)℘−1Un,kt=ynx,t=2L∑k=1∞Un,ktsinξkx,(10)ξk=kπL,k=1,2,3,….For a small deformation, the boundary conditions of ECMB are(11)ynx,tx=0,L=0,EIy′′nx,tx=0,L=0.Based on the boundary conditions [24], the finite sine-Fourier transform of the fourth derivative of the displacement function with respect to the coordinate x is(12)℘d4ynx,tdx2=ξk4Un,kt.Applying finite sine-Fourier transform with respect to the space coordinatex to both sides of equations (1)–(7) gives(13)L2M2Z¨t+L2k0Zt−k0∑i=1∞U1,itsinξivt=0,(14)E1I1ξk4U1,kt−M12Lv2∑i=1∞ξi2U1,itsinξivtsinξkvt+k02L∑i=1∞U1,itsinξivtsinξkvt+m1U¨1,kt+M1sinξkvt2L∑i=1∞U¨1,itsinξivt+k1U1,kt−k1U2,kt+M14Lvsinξkvt∑i=1∞ξiU˙1,itcosξivt−k0Ztsinξkvt=M1+M2gsinξkvt,(15)E2I2ξk4U2,kt+m2∂2∂t2U2,kt−k1U1,kt−U2,kt+k2U2,kt−U3,kt=0,(16)E3I3ξk4U3,kt+m3∂2∂t2U3,kt−k2U2,kt−U3,kt+k3U3,kt−U4,kt=0,⋯(17)En−1In−1ξk4Un−1,kt+mn−1∂2∂t2Un−1,kt−kn−2Un−2,kt−Un−1,kt+kn−1Un−1,kt−Un,kt=0,(18)EnInξk4Un,kt+mn∂2∂t2Un,kt+kn−2Un,kt−Un−2,kt=0.Althoughk and i in the above equations are between 1∼∞, taking N items can generally meet the accuracy requirement.Further, equations (13)–(18) can be expressed as(19)Mi,kU¨k+Ci,kU˙k+Ki,kUk=Pkt,where(20)Uk=U1,kU2,kU3,k⋯Un−1,kUn,kZT,Pkt=F1F2F3⋯Fn−1FnFzT,Mi,k=diagM11M22M33⋯Mn−1n−1MnnMzz,Ci,k=diagC110i×k0i×k⋯0i×k0i×k0,Ki,k=K11−K120i×k⋯0i×k0i×k−K1zK22−K23⋯0i×k0i×k0i×1K33⋯0i×k0i×k0i×1⋱⋮⋮⋮Kn−1n−1−Kn−1n0i×1Knn0i×1symsKzz,where M11, M22, M33, Mn−1n−1, Mnn, C11, K22, K33, Κ12, K23, K21, K32, Kn−1n−1, Knn, Kn−1n, and Knn−1 are all i×k matrices and K1z and F1,F2,F3,…,Fn−1,Fn are i×1 matrices.


### 2.3. Solution of Equations

As the position of the moving oscillator on ECMB changes with time, the mass matrixMi,k, the damping matrix Ci,k and the stiffness matrix Ki,k in equation (19) also all change with time t. Hence, the dynamic equilibrium equations for the coupling system of ECMB under moving oscillator are differential equations with time-varying coefficients. Further, the coefficient matrices Mi,k, Ci,k, and Ki,k also vary with the change in the speed v. The reaction of the entire structural system is calculated step by step at every time increment Δt, and the imbalance of the system within Δt is usually ignored in the calculation. On the other hand, the time-varying characteristic of the system is approximated by a series of constantly changing time-invariant systems. Therefore, the method of Newmark−β is used in this study to solve equation (19).U
k
t
+
Δ
t, U˙kt+Δt, and U¨kt+Δt are the displacement, velocity, and acceleration at time t+Δt, respectively, and they have been set to satisfy the dynamical equation (19) as follows:(21)Mi,kU¨kt+Δt+Ci,kU˙kt+Δt+Ki,kUkt+Δt=Pkt.The displacement, velocity, and acceleration at the initial time are as follows:(22)Uk0=0,U˙k0=0,U¨k0=0.Using theNewmark−β method, the velocity and displacement at time t+Δt are(23)U˙kt+Δt=U˙kt+1−γU¨kt+γU¨kt+ΔtΔt,Ukt+Δt=Ukt+U˙ktΔt+12−βU¨kt+βU¨kt+ΔtΔt2,where γandβ are the parameters of the Newmark−β method.Further, the acceleration, velocity, and displacement at time (t+Δt) are(24)U¨kt+Δt=Ukt+Δt−UktβΔt2−U˙ktβΔt−U¨kt2β−1,U˙kt+Δt=U˙kt+Δt1−γU¨kt+γΔtU¨kt+Δt,Ukt+Δt=Ki,k+Mi,kβΔt2+Ci,kγβΔt−1Pkt+Δt+Mi,kUktβΔt2+U˙ktβΔt+U¨kt2β−1+Ci,kUktγβΔt+U˙ktβ−1+U¨ktΔt2γβ−2.By substitutingUk into equation (9), we can obtain(25)y=2L∑k=1NUksinξkx,where y=y1,y2,y3,…,yn−1,ynT.


## 2.1. Governing Equations of ECMB under the Moving Oscillator

The schematic diagram for ECMB under a moving oscillator is shown in Figure1. All of beams are homogeneous and prismatic and have the same length L, but they can have different mass or flexural rigidities, which makes the model more real in reflecting some actual engineering projects. According to the d’Alembert principle, the dynamic equilibrium equation of an oscillator system is(1)M2∂2Zt∂t2+k0Zt−y1=0.Figure 1
The schematic diagram for ECMB.According to the d’Alembert principle, the dynamic equilibrium equations of ECMB are(2)E1I1∂4y1∂x4+m1∂2y1∂t2+k1y1−y2=P0t,(3)E2I2∂4y2∂x4+m2∂2y2∂t2−k1y1−y2+k2y2−y3=0,(4)E3I3∂4y3∂x4+m3∂2y3∂t2−k2y2−y3+k3y3−y4=0,⋯(5)En−1In−1∂4yn−1∂x4+mn−1∂2yn−1∂t2−kn−2yn−2−yn−1+kn−1yn−1−yn=0,(6)EnIn∂4yn∂x4+mn∂2yn∂t2+kn−1yn−yn−1=0,(7)P0t=δx−vtM1+M2g−M1∂2y1∂t2+2v∂2y1∂x∂t+v2∂2y1∂x2+k0Zt−y1,where ynx,t,En,In,andmn with n=2,3,4,… are the dynamic deflection, elastic modulus, horizontal inertia moment, and beam mass per unit length for each layer of beam, respectively; M1andM2 are the unsprung mass and sprung mass, respectively; and Zt is the dynamic deflection of the sprung mass, and it is assumed that the unsprung mass moves along the beam without separating from the beam body. In this way, the deflection of unsprung mass is the same as that of the beam segment at the same location; k0 and c0 are the spring coefficient and damping coefficient of an oscillator system, respectively; kn−1 with n=2,3,4,… is the interlayer spring stiffness; δ is Dirac function; v is the moving speed of the oscillator; P0t is the external force; and g is the gravitational acceleration.


## 2.2. Finite Sine-Fourier Transform

In order to solve equations (1)–(7), the finite sine-Fourier transform has been used and the transformation is defined as follows [23]:(8)℘ynx,t=Un,kt=∫0Lynx,tsinξkxdx,(9)℘−1Un,kt=ynx,t=2L∑k=1∞Un,ktsinξkx,(10)ξk=kπL,k=1,2,3,….For a small deformation, the boundary conditions of ECMB are(11)ynx,tx=0,L=0,EIy′′nx,tx=0,L=0.Based on the boundary conditions [24], the finite sine-Fourier transform of the fourth derivative of the displacement function with respect to the coordinate x is(12)℘d4ynx,tdx2=ξk4Un,kt.Applying finite sine-Fourier transform with respect to the space coordinatex to both sides of equations (1)–(7) gives(13)L2M2Z¨t+L2k0Zt−k0∑i=1∞U1,itsinξivt=0,(14)E1I1ξk4U1,kt−M12Lv2∑i=1∞ξi2U1,itsinξivtsinξkvt+k02L∑i=1∞U1,itsinξivtsinξkvt+m1U¨1,kt+M1sinξkvt2L∑i=1∞U¨1,itsinξivt+k1U1,kt−k1U2,kt+M14Lvsinξkvt∑i=1∞ξiU˙1,itcosξivt−k0Ztsinξkvt=M1+M2gsinξkvt,(15)E2I2ξk4U2,kt+m2∂2∂t2U2,kt−k1U1,kt−U2,kt+k2U2,kt−U3,kt=0,(16)E3I3ξk4U3,kt+m3∂2∂t2U3,kt−k2U2,kt−U3,kt+k3U3,kt−U4,kt=0,⋯(17)En−1In−1ξk4Un−1,kt+mn−1∂2∂t2Un−1,kt−kn−2Un−2,kt−Un−1,kt+kn−1Un−1,kt−Un,kt=0,(18)EnInξk4Un,kt+mn∂2∂t2Un,kt+kn−2Un,kt−Un−2,kt=0.Althoughk and i in the above equations are between 1∼∞, taking N items can generally meet the accuracy requirement.Further, equations (13)–(18) can be expressed as(19)Mi,kU¨k+Ci,kU˙k+Ki,kUk=Pkt,where(20)Uk=U1,kU2,kU3,k⋯Un−1,kUn,kZT,Pkt=F1F2F3⋯Fn−1FnFzT,Mi,k=diagM11M22M33⋯Mn−1n−1MnnMzz,Ci,k=diagC110i×k0i×k⋯0i×k0i×k0,Ki,k=K11−K120i×k⋯0i×k0i×k−K1zK22−K23⋯0i×k0i×k0i×1K33⋯0i×k0i×k0i×1⋱⋮⋮⋮Kn−1n−1−Kn−1n0i×1Knn0i×1symsKzz,where M11, M22, M33, Mn−1n−1, Mnn, C11, K22, K33, Κ12, K23, K21, K32, Kn−1n−1, Knn, Kn−1n, and Knn−1 are all i×k matrices and K1z and F1,F2,F3,…,Fn−1,Fn are i×1 matrices.


## 2.3. Solution of Equations

As the position of the moving oscillator on ECMB changes with time, the mass matrixMi,k, the damping matrix Ci,k and the stiffness matrix Ki,k in equation (19) also all change with time t. Hence, the dynamic equilibrium equations for the coupling system of ECMB under moving oscillator are differential equations with time-varying coefficients. Further, the coefficient matrices Mi,k, Ci,k, and Ki,k also vary with the change in the speed v. The reaction of the entire structural system is calculated step by step at every time increment Δt, and the imbalance of the system within Δt is usually ignored in the calculation. On the other hand, the time-varying characteristic of the system is approximated by a series of constantly changing time-invariant systems. Therefore, the method of Newmark−β is used in this study to solve equation (19).U
k
t
+
Δ
t, U˙kt+Δt, and U¨kt+Δt are the displacement, velocity, and acceleration at time t+Δt, respectively, and they have been set to satisfy the dynamical equation (19) as follows:(21)Mi,kU¨kt+Δt+Ci,kU˙kt+Δt+Ki,kUkt+Δt=Pkt.The displacement, velocity, and acceleration at the initial time are as follows:(22)Uk0=0,U˙k0=0,U¨k0=0.Using theNewmark−β method, the velocity and displacement at time t+Δt are(23)U˙kt+Δt=U˙kt+1−γU¨kt+γU¨kt+ΔtΔt,Ukt+Δt=Ukt+U˙ktΔt+12−βU¨kt+βU¨kt+ΔtΔt2,where γandβ are the parameters of the Newmark−β method.Further, the acceleration, velocity, and displacement at time (t+Δt) are(24)U¨kt+Δt=Ukt+Δt−UktβΔt2−U˙ktβΔt−U¨kt2β−1,U˙kt+Δt=U˙kt+Δt1−γU¨kt+γΔtU¨kt+Δt,Ukt+Δt=Ki,k+Mi,kβΔt2+Ci,kγβΔt−1Pkt+Δt+Mi,kUktβΔt2+U˙ktβΔt+U¨kt2β−1+Ci,kUktγβΔt+U˙ktβ−1+U¨ktΔt2γβ−2.By substitutingUk into equation (9), we can obtain(25)y=2L∑k=1NUksinξkx,where y=y1,y2,y3,…,yn−1,ynT.


## 3. Method Validation

To validate the accuracy of the method developed in this study for calculating the dynamic response of ECMB under a moving oscillator, a four-layer beam system is taken as an example. Using the MATLAB program and ANSYS numerical method, the vibration analysis of a four-layer beam system has been calculated for four moving speeds. Taking a four-story beam system as an example, using the MATLAB program and ANSYS numerical method, the vibration analysis of a four-layer beam system has been calculated for four moving speeds (i.e., 16 m/s, 32 m/s, 128 m/s, and 160 m/s). Furthermore, the results of the dynamic response time-history curves and the peak of vibration analysis of the midspan deflection have been compared [25]. A comparison of results from the Matlab program and ANSYS numerical calculations is shown in Table 1 and Figure 2. In addition, the multilayer beam system was degraded to a double-layer beam to compare with the methods of the existed literature conveniently, and then the dynamic response of double-layer beam under the moving oscillator was calculated and compared with the results of Wu and Gao’s method [6]. The dynamic response peaks of each layer at the positions of x = 2.5 m, 5.0 m, and 7.5 m were, respectively, extracted as shown in Table 2.Table 1
Comparison of the dynamic response peaks of the midspan deflections of the four-layer beam system.
Computed result
Speed (m/s)
16
32
128
160
L
Ia mm
−0.55500
−0.55500
−0.93800
−0.98700
L
Im mm
−0.55183
−0.54953
−0.92658
−1.00572
L
IIa mm
−0.12100
−0.11500
−0.21300
−0.20500
L
IIm mm
−0.12082
−0.11464
−0.21318
−0.20751
L
IIIa mm
−0.10200
−0.09300
−0.17100
−0.15800
L
IIIm mm
−0.10225
−0.09290
−0.17102
−0.15944
L
IIIIa mm
−0.09060
−0.09230
−0.15000
−0.15200
L
IIIIm mm
−0.09070
−0.09210
−0.14960
−0.14975
E
I %
−0.57
−0.98
−1.22
1.89
E
II %
−0.15
−0.32
0.09
1.23
E
III %
0.25
−0.11
0.01
0.91
E
IIII %
0.11
−0.22
−0.27
−1.48Figure 2
Comparison of dynamic response time-history curves of four-layer beam system under four moving speeds.
(a)
(b)
(c)
(d)Table 2
Comparison of dynamic response peaks at the positions ofx = 2.5 m, 5.0 m, and 7.5 m.
Computed result
Calculation method
2.5
5.0
7.5
First layer
The proposed method/m
0.00753
0.01075
0.00743
Wu and Gao’s method [6]/m
0.00766
0.01083
0.00730
Deviation
−1.69%
−0.74%
1.78%
Second layer
The proposed method/m
0.00481
0.006825
0.00484
Wu and Gao’s method [6]/m
0.00488
0.006820
0.00488
Deviation
−1.43%
0.07%
−0.82%In Table1, LIa, LIIa, LIIIa, and LIIIIa are ANSYS calculation results of the peak of vibration analysis of the midspan deflections of the first layer, second layer, third layer, and fourth layer, respectively. LIm, LIIm, LIIIm, and LIIIIm are the Matlab calculation results of the peak of vibration analysis of the midspan deflections of the first layer, second layer, third layer, and fourth layer, respectively. EI, EII, EIII, and EIIII are the calculation errors between the Matlab program and ANSYS calculation results of the peaks of vibration analysis of the midspan deflections of the first layer, second layer, third layer, and fourth layer, respectively. In Figure 2, uI, uII, uIII, and uIIII are the midspan dynamic deflections of the first layer, second layer, third layer, and fourth layer, respectively.From Table1, it can be seen that EI, EII, EIII, and EIIII of four moving speeds are less than 1.89%, 1.23%, 0.91%, and −0.48%, respectively. Since the differences are small, the correctness of the method developed in this study has been verified. From Figure 2, it can be seen for the dynamic time-history curves of the midspan deflections at the first layer, second layer, third layer, and fourth layer, the Matlab program results are in good agreement with the ANSYS numerical calculation results. From Table 2, it could be seen that the deviation of the calculation results of two methods at different positions (x = 2.5 m, 5.0 m, and 7.5 m) is less than 1.8%. Since the differences are small, the correctness of the method developed in this study has been verified again.The advantages of the method developed in this study in terms of higher number of layers in this paper for specific performance are as follows: the system studied in this paper consists ofn identical beams, in general, when much more layers are considered, the coupled partial equations are difficult to solve. Compared with the literature [12], this paper does not need to build a special relation of the stiffness of intermediate springs for decoupling the equation of motion; with the finite sine-Fourier transform, the equations can be uncoupled and solved. Compared with the literature [6], when studying the vibration of double-beam system, this paper does not need to divide into the vibration of two single-beam systems subjected to the same force, interlayer stiffness, damping, material properties of each layer, and moving oscillator [19]. When the number of layers increases, only the increased layer needs to be input into Formula (19) after the finite sine-Fourier transform; as long as the appropriate value of k is chosen, the vibration of elastically connected multiple beam system under the moving oscillator can be carried out, where, although k is between 1∼∞, taking N≥30 items can generally meet the accuracy requirement of engineering calculation [26].


## 4. Analysis of Examples

Since the method developed in this study has been verified (Section3), it has been applied to a 32 m span beam-rail system (rail, track plate, base plate, and bridge) on a line in China [6, 27], as shown in Figure 3. The train model has been simplified into a system consisting of wheel, spring, damper, and sprung mass. The material properties and geometric parameters of the beam-rail system are shown in Table 3.Figure 3
Simplified model of a railway line.Table 3
Material properties and geometric parameters of the beam-rail system.
Parameter
Value
Rail
Elastic modulusE1
2.06
×
10
11
N
/
m
2
Line massm1
60
kg
/
m
Moment of inertiaI1
3.217
×
10
−
5
m
4
Densityρ1
7.85
×
10
3
kg
/
m
3
Track slab
Line massm2
1275
kg
/
m
Densityρ2
2.5
×
10
3
kg
/
m
3
Bed plate
Elastic modulusE3
3.00
×
10
10
N
/
m
2
Line massm3
1401.25
kg
/
m
Moment of inertiaI3
1.6862
×
10
−
3
m
4
Densityρ3
2.5
×
10
3
kg
/
m
3
Bridge
Elastic modulusE4
3.45
×
10
10
N
/
m
2
Line massm4
36
,
000
kg
/
m
Moment of inertiaI4
10.42
m
4
Densityρ4
2.5
×
10
3
kg
/
m
3
Interlayer stiffness
Stiffness of rail pad and fastenerk1
6
×
10
7
N
/
m
2
Stiffness of the sliding layerk3
1.375
×
10
12
N
/
m
2
Stiffness of the rubber pad under the supporting blockk2
9
×
10
8
N
/
m
2
Train
Wheel massM1
2000
kg
Spring stiffnessk0
2.352
×
10
6
N
/
m
Car body massM2
11
,
500
kg


### 4.1. Effect of Train Speed on the Vibration of the Beam-Rail System

Figure4 shows the variations of the dynamic deflections of rail, track plate, base plate, and bridge when the train moves across the bridge at the low speeds (i.e. 8 m/s and 32 m/s), medium-high speeds (64 m/s and 80 m/s), and high speeds (128 m/s and 160 m/s). In Figure 4, w is the dynamic deflection and Lp is the position of the train on the bridge. Figure 5 shows the comparison between the curves of the dynamic deflections of the rail, track plate, base plate, and bridge and the train position on the bridge with v=32m/s and v=160m/s.Figure 4
Dynamic deflections of rail, track plate, base plate, and bridge under different train speeds. Deflection of (a) rail, (b) track slab, (c) bed plate, and (d) bridge.
(a)
(b)
(c)
(d)Figure 5
Comparison of dynamic deflection responses of rail, track plate, base plate, and bridge under different train speeds.
(a)
(b)From Figures4 and 5, it can be seen that when the train speed increases from 8 m/s to 200 m/s, the peaks of the vibration analysis of the rail, track plate, base plate, and bridge deflection in the beam-rail system all increase initially and then subsequently decrease, which shows that for the beam-rail system, there is a critical train speed. The maximum dynamic deflection always occurs near the midspan of the rail, and the excitation effect of the train has a significant effect only on the first order frequency of the rail. Hence, the investigation on the maximum dynamic deflection of the rail can be done by analyzing the maximum dynamic deflection at midspan only. However, the shapes of dynamic deflection curves of the track plate, base plate, and bridge vary with the train speed, the dynamic response frequency increase with the train speed increase, and the position of the maximum dynamic deflection shifts from the midspan to the traveling direction of the train with the train speed increase. Further, it can be seen that the dynamic deflection response curves of the base plate and the bridge are basically the same. This is due to the large vertical compressive stiffness of the sliding layer, which causes the base plate and the bridge to act as one unit.


### 4.2. Effect of Stiffness of Each Layer under the Rail on the Vibration of Beam-Rail System

For the train speed of 100 m/s, the effects of the fastener under the rail, the rubber pad under the supporting block, and the sliding layer stiffnesses on the vibration of the beam-rail system have been investigated. In each calculation, only one of three above stiffnesses has been changed, and the other two remain unchanged. Then, three different calculation schemes have been designed. For the stiffness of the fastener under the railk1 and the stiffness of the rubber pad under the supporting block k2, the range is 40 MN/m2∼100 MN/m2, with an increment of 10 MN/m2. For the stiffness of the sliding layer k3, the range is 500 MN/m2∼2e5 MN/m2, with an increment of 100 MN/m2. Figures 6–8 show the maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the fastener under the rail and the rubber pad under the supporting block and the sliding layer, respectively, where Dpeak is the peak dynamic response of the midspan deflection.Figure 6
The maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the fastener under the rail.Figure 7
The maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the rubber pad under the supporting block.Figure 8
The maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the sliding layer.From Figure6, it can be seen that as the stiffness of the fastener under the rail increases, the maximum rail dynamic deflection decreases with a larger decrease in amplitude; the maximum dynamic deflection of the rail plate fluctuates at first and then gradually flattens, and the variations of the maximum dynamic deflections of the base plate and the bridge are very small and they are basically the same. Therefore, the use of fasteners with high elasticity can effectively reduce the vibration of the rail. As shown in Figure 7, the stiffness of the rubber pad under the supporting block mainly affects the vibrations of the rail and the track plate. When the stiffness of the rubber pad increases from 40 MN/m2 to 100 MN/m2, the maximum dynamic deflection of the rail first fluctuates sharply and then flattens after reaching to a certain amplitude, the maximum dynamic deflection of the track plate also fluctuates drastically at first and then flattens after dropping to a certain value, and the maximum dynamic deflections of the base plate and the bridge remain practically unchanged. From Figure 8, it can be seen that as the stiffness of the sliding layer increases, the maximum rail dynamic deflection increases at first and then tends towards a constant value, the maximum dynamic deflection of the track plate and the base plate first decreases and then tends towards a constant value, and the maximum dynamic deflection of the bridge remains practically unchanged.


## 4.1. Effect of Train Speed on the Vibration of the Beam-Rail System

Figure4 shows the variations of the dynamic deflections of rail, track plate, base plate, and bridge when the train moves across the bridge at the low speeds (i.e. 8 m/s and 32 m/s), medium-high speeds (64 m/s and 80 m/s), and high speeds (128 m/s and 160 m/s). In Figure 4, w is the dynamic deflection and Lp is the position of the train on the bridge. Figure 5 shows the comparison between the curves of the dynamic deflections of the rail, track plate, base plate, and bridge and the train position on the bridge with v=32m/s and v=160m/s.Figure 4
Dynamic deflections of rail, track plate, base plate, and bridge under different train speeds. Deflection of (a) rail, (b) track slab, (c) bed plate, and (d) bridge.
(a)
(b)
(c)
(d)Figure 5
Comparison of dynamic deflection responses of rail, track plate, base plate, and bridge under different train speeds.
(a)
(b)From Figures4 and 5, it can be seen that when the train speed increases from 8 m/s to 200 m/s, the peaks of the vibration analysis of the rail, track plate, base plate, and bridge deflection in the beam-rail system all increase initially and then subsequently decrease, which shows that for the beam-rail system, there is a critical train speed. The maximum dynamic deflection always occurs near the midspan of the rail, and the excitation effect of the train has a significant effect only on the first order frequency of the rail. Hence, the investigation on the maximum dynamic deflection of the rail can be done by analyzing the maximum dynamic deflection at midspan only. However, the shapes of dynamic deflection curves of the track plate, base plate, and bridge vary with the train speed, the dynamic response frequency increase with the train speed increase, and the position of the maximum dynamic deflection shifts from the midspan to the traveling direction of the train with the train speed increase. Further, it can be seen that the dynamic deflection response curves of the base plate and the bridge are basically the same. This is due to the large vertical compressive stiffness of the sliding layer, which causes the base plate and the bridge to act as one unit.


## 4.2. Effect of Stiffness of Each Layer under the Rail on the Vibration of Beam-Rail System

For the train speed of 100 m/s, the effects of the fastener under the rail, the rubber pad under the supporting block, and the sliding layer stiffnesses on the vibration of the beam-rail system have been investigated. In each calculation, only one of three above stiffnesses has been changed, and the other two remain unchanged. Then, three different calculation schemes have been designed. For the stiffness of the fastener under the railk1 and the stiffness of the rubber pad under the supporting block k2, the range is 40 MN/m2∼100 MN/m2, with an increment of 10 MN/m2. For the stiffness of the sliding layer k3, the range is 500 MN/m2∼2e5 MN/m2, with an increment of 100 MN/m2. Figures 6–8 show the maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the fastener under the rail and the rubber pad under the supporting block and the sliding layer, respectively, where Dpeak is the peak dynamic response of the midspan deflection.Figure 6
The maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the fastener under the rail.Figure 7
The maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the rubber pad under the supporting block.Figure 8
The maximum dynamic deflections of the rail, track plate, base plate, and bridge corresponding to different stiffnesses of the sliding layer.From Figure6, it can be seen that as the stiffness of the fastener under the rail increases, the maximum rail dynamic deflection decreases with a larger decrease in amplitude; the maximum dynamic deflection of the rail plate fluctuates at first and then gradually flattens, and the variations of the maximum dynamic deflections of the base plate and the bridge are very small and they are basically the same. Therefore, the use of fasteners with high elasticity can effectively reduce the vibration of the rail. As shown in Figure 7, the stiffness of the rubber pad under the supporting block mainly affects the vibrations of the rail and the track plate. When the stiffness of the rubber pad increases from 40 MN/m2 to 100 MN/m2, the maximum dynamic deflection of the rail first fluctuates sharply and then flattens after reaching to a certain amplitude, the maximum dynamic deflection of the track plate also fluctuates drastically at first and then flattens after dropping to a certain value, and the maximum dynamic deflections of the base plate and the bridge remain practically unchanged. From Figure 8, it can be seen that as the stiffness of the sliding layer increases, the maximum rail dynamic deflection increases at first and then tends towards a constant value, the maximum dynamic deflection of the track plate and the base plate first decreases and then tends towards a constant value, and the maximum dynamic deflection of the bridge remains practically unchanged.


## 5. Conclusions

Based on finite sine-Fourier inverse transform, the expression for dynamic response of a ECMB under a moving oscillator has been derived. Further, the vibration analysis of a four-layer simply supported beam system under moving oscillator with different speeds and the vibration analysis of a double-layer simply supported beam system under moving oscillator with different positions have been calculated. Comparing with the numerical methods developed in earlier studies, the method developed in this study has the advantages of clearer concept, simpler programming, and less computational effort. A beam-rail system on a line in China has been used as a case study, and the effect of the train speed and interlayer stiffness on the dynamic deflection and amplification coefficient of the midspan of the beam-rail system has been analyzed. Moreover, from the case study, the following conclusions have been obtained, which are meaningful to engineering design.(1)
The results show that the calculated results are in good agreement with those of ANSYS numerical and the existing literature. Hence, the correctness of the method developed in this study has been verified.(2)
This study could be useful in different subjects of science, e.g., mechanical, civil, traffic, and material engineering and many other fields. For example, the dynamic behavior of the bridge-track system of railway (maglev) under the train load or multi layered nanobeam systems as a part of nanostructures has achieved more usages with improvements done in nanotechnology and the usage of smaller devices in designing different machines.(3)
The vertical dynamic deflection of the rail in the beam-rail system under the train load always reaches a maximum near the midspan. The maximum dynamic deflections of the track plate, base plate, and bridge may not occur when the train is located at the midspan but occurs after the train travels through the midspan.(4)
In the beam-rail system, there are several critical speeds which should be paid attention to and avoided in the practical engineering. The interlayer stiffness has a larger impact on the vibration of the rail and the track plate while it has a smaller impact on the vibration of the base plate and the bridge.



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*Source: 4950841-2019-10-16.xml*