Abstract
Combined with the coordinate transformation, the dynamic governing equations for elastic foundation under a moving load are established by using the reverberation ray matrix method, and the isolation effect of a graded wave impeding block is studied. The displacement and the stress are obtained in the time domain by using numerical inverse Fourier transformation. Via numerical examples, the effectiveness of vibration isolation of graded wave impeding block is compared to conventional single phase solid homogenous wave impeding block. The results show that compared with the single phase solid homogenous wave impeding block isolation system, the graded wave impeding block can effectively reduce the vibration amplitude and has more advantages and is effective under a moving load.
1. Introduction
With the rapid development of urbanization and modern industry, ground vibration caused by moving load is becoming more and more frequent and it has become public considerable attention and engineering concern [1]. Therefore, numerous studies have been conducted on the vibration isolation induced by moving loads. For a moving load, there are different ways to reduce the ground vibrations [2]. For example, the effectiveness of three different wave barriers, namely, the open trench, in-filled trench, and elastic foundation, in reducing the ground vibrations caused by the passage of trains is studied [3–7], which indicate that open trenches are more efficient than in-filled trenches or soil stiffening-even at low frequencies.
Besides barrier for isolating vibration, artificial setting of the wave impedance block (WIB) in the foundation to study the vibration isolation performance of vibration control has made great progress. Chouw and Schmid [8, 9] firstly suggested placing an artificial bed rock called “wave impeding block” (WIB) under the vibration foundation to screen the low frequency vibrations in the wave propagation regime. Their results showed that the WIB was more effective at low frequencies than a buried wall barrier of similar dimensions. Peplow [10] investigated the vibration isolation of WIB in the far-field of the surface of the ground due to a harmonic load acting over a strip by boundary element (BE) method. Takemiya [11] studied the response on the surface of the ground in the time domain with a Ricker wavelet type loading by using a boundary element model; their results showed that an effective width of the WIB should be chosen in relation to the wavelength of the given load. Based on a semianalytical BEM, the vibration effect of WIB in stratified half-space under the load of the travelling train is analyzed by Gao [12, 13].
It should be noted that WIB vibration isolation system as an effective measure for vibration pollution control is applied in practical engineering, but the previous studies mostly focused on the homogenous materials; few research works have been conducted on the vibration isolation performance of wave impeding block with materials properties that have a continuous variation along space relatively [14]. As we all know, the functionally graded materials (FGMs) have the advantages of relieving thermal stress, avoiding or reducing stress concentration, and being well designed [15, 16]. In particular, the gradient factor of FGMs has a significant influence on the dispersion curve and attenuation curve of the wave and causes the dissipation of energy in the process of vibration [17].
In this study, based on the FGMs, the isolation effect of graded WIB on reducing vibration induced by a moving load in elastic foundations is studied. This paper is organized as follows: combined with the coordinate transformation, the dynamic governing equations for elastic foundation under a moving load are established in Section 2. Using the reverberation ray matrix method (RRMM), the expressions of displacement and stress in the frequency domain are derived to study the vibration isolation of graded WIB in Section 3. Section 4 shows the effectiveness of vibration isolation of graded WIB compared to the homogenous WIB and without WIB via the numerical example. Finally, the conclusion is drawn in Section 5.
2. General Solution of Elastic Medium under Moving Load
Considering setting the graded WIB in elastic foundations subjected to a moving load as depicted in Figure 1, the thickness of foundations is , where the thickness of graded WIB is , the upper layer has a thickness , and the lower layer has a thickness in direction. A moving load, which moves in the -direction at constant speed , rests on a homogenous elastic soils stratum; the load amplitude is , the distribution length is is Cartesian coordinate, and is the moving coordinate which moves in the -direction with moving load at constant speed .
The basic equations for elastic medium are as follows:
equations of motion:
strain-displacement relationship:
constitutive law:In the above equations, are stress components, respectively. is the Dirac delta function, represents the components of displacements, , denotes the strain components, is the density, and and are Lamé constants.
In the absence of body forces, these can be written as
In order to solve governing equation (4), according to the Helmholtz decomposition method, the displacements vectors are represented by scalar and vector potentials as follows:
Substituting the above resolutions into (4), we obtain two sets of coupled equation in terms of the scalar potentials as follows:
The displacement vectors in the Cartesian coordinate and the scalar potentials and can be expressed as the harmonic functions of the angular frequency in the case of steady state by introducing the moving coordinate ; i.e., , .where .
Thus, the derivatives are changed as follows:where the subscript , denotes the partial differentiation with respect to .
The Fourier transform with respect to is defined as
Performing the Fourier transform with respect to the horizontal coordinates on (6a) and (6b), the governing (6a) and (6b) can be uncoupled aswhere , , , , .
Therefore, solution of the ordinary differential equations in (10) and (11) yields the following general solutions for the Fourier transform domain potentials:
In the moving coordinate , the displacement can be expressed by using the potentials and :
Combining (12), (13), (14a), (14b), and (1), the generation solution of the displacement and stress of the dynamic of soil in the Fourier transform domain can be obtained:where .
3. Reverberation Ray Matrix Method
It is usually very difficult to obtain analytical solutions to the governing equations for waves in a general graded WIB. When the material properties vary with the thickness coordinate only, we can adopt an accurate laminate plate to model the original graded WIB as shown in Figure 2 [18]; the graded WIB is divided into many layers of small thickness so that the material properties of each layer can be regarded as constant. The number of layers, N, is controlled by the accuracy of the model [18]. In recent years, RRMM is a new solution method which was originally developed by Pao et al. [19], who further [20] applied it to study the transient wave propagation in layered liquids as well as in laminated solids, and it has been demonstrated that RRMM has a great advantage in tracing the wave rays. In this paper, RRMM will be employed to analyze the graded WIB. One key step in RRMM is that two local coordinates should be employed for each layer. One locates at the upper surface of the layer and the other locates at the lower surface. Figure 3 shows the two local coordinates at the ith interface. The superscript of the coordinate indicates that it locates at the ith interface and directs to the (i-1)th interface.
In the local coordinates , according to (15a), (15b), (15c), and (15d), we have the solution as follows:
At the ith interface, the continuity conditions require
By virtue of (16a), (16b), (16c), (16d), (17a), and (17b), we get the so-called local scattering relations at the ith interface written in a matrix form aswhere is the amplitude vector of departing waves; is the amplitude vector of arriving waves; the matrix is called the scattering matrix at ith interface; and
Considering that the soil foundation surface subjected to a moving load and the bottom surface is fixed, the appropriate boundary conditions to be satisfied in the frequency domain and moving coordinate frame are given by
By substituting (20a) and (20b) into (16a), (16b), (16c), and (16d), we can obtainwhere , , , , , .
Combination of (18) , (21a), and (21b) leads towhere , , .
Equation (22) presents a global scattering relation between various waves at all interfaces and two boundary surfaces. It is shown that (22) contains a total of algebraic equations with totally unknowns. Thus, additional relations should be supplemented. Notice that a departing wave in the local coordinates becomes an arriving wave in the local coordinates . Hence, we can deduce the following phase relations:where is ith thickness.
By substituting (16b) into (23), we can obtain
Introducing a new departing wave vector,
Combining (22), (24), and (25), we can obtainwhere is the phase matrix defined by
It can be seen that the elements in are completely the same as those in , but the orders of elements change. Therefore, we can introduce the permutation matrix to link them together and define them by
It is then obtained by substituting (27) into (23) thatwhere is the reverberation matrix defined by and is the unit matrix.
Finally, solving (29), we can obtain and . Combining (16a), (16b), (16c), and (16d), the displacement and stress of the dynamic of soil in the frequency domain can be obtained, and then the corresponding physical quantities in the Fourier transform domain are obtained by applying the inverse Fourier transform.
4. The Numerical Example and Discussion
Assuming that the material properties of graded wave impeding block have an exponential law distribution along the thickness coordinate, there arewhere represents the physical-mechanical parameters of graded WIB; is a nonhomogeneous parameter, named gradient index, and different values of dictate that the material properties have different nonhomogeneous distributions through the thickness of WIB. From (30) we can know that graded materials become homogenous materials when .
The physical and mechanical parameters of elastic soil foundation are as follows [21]: , , . The parameters , , , , , are used in the following analysis. The physical-mechanical parameters of surface and bottom of graded WIB are shown in Table 1. For simplicity, assume that all physical-mechanical parameters of graded WIB have the same gradient index.
Figures 4-5 show the variations of horizontal displacement and vertical displacement along the horizontal direction at the ground surface with the homogenous WIB (where the gradient index ) and the graded WIB (where the gradient index ). Comparing between the homogenous WIB and graded WIB, from Figures 4 and 5, we can see that the graded WIB has changed the vibration phase of horizontal displacement and vertical displacement . Meanwhile, it is shown in Figures 4 and 5 that the and values at the ground surface in a graded WIB are much less than those in a homogeneous WIB. The maximum value is 0.368 when , while the maximum value is 0.099 when ; the maximum value is 4.032 when , while the maximum value is 1.168 when . It indicates that the amplitude of and is significantly reduced with the graded WIB, indicating a better vibration isolation of the graded WIB.
5. Conclusions
In this study, based on the FGMs, the isolation effect of graded WIB on reducing vibration induced by a moving load in elastic foundations is studied. Combined with the coordinate transformation, using the reverberation ray matrix method, the expressions of displacement and stress in the frequency domain are derived to study the vibration isolation of graded WIB. Based on the numerical example in this paper, the result shows that the and values when are smaller than those when ; it is interesting to note that the vibration isolation of the graded WIB is better than that of the homogeneous WIB.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Chinese Natural Science Foundation (Grant no. 51368038).