#### Abstract

This paper presents the finite element algorithm and results of dynamical analysis of cracked plate subjected to moving oscillator with a constant velocity and any motion orbit. There are many surveys considering the dynamic response of the plate when there is a change in number of cracks and the stiffness of the spring k. The numerical survey results show that the effect of cracks on the plate's vibration is significant. The results of this article can be used as a reference for calculating and designing traffic structures such as road surface and bridge surface panels.

#### 1. Introduction

There are several types of plate structures affected by the vehicle load: pavement, railway system, and bridge floor, etc. Calculating these types of structures and the means of loading are modeled by different kinds of forces such as force, mass, and moving oscillators. Typically, the tracked vehicle is a moving mass, while the wheeled vehicle is described as a moving oscillator. Accordingly, structural dynamic analysis under the influence of mobile loads has been considered by many scientists. Nguyen Thai Chung and Le Pham Binh [1] analyzed the cracked beam on the elastic foundation under moving mass by using the finite element method (FEM). S.R. Mohebpour and P. Malekzadeh [2], P. Malekzadeh, A.R. Fiouz, H. Razi [3], Qinghua Song, Zhanqiang Liu, Jiahao Shi, Yi Wan [4], Qinghua Song, Jiahao Shi, and Zhanqiang Liu [5] presented a finite element model based on the first order shear deformation theory to investigate the dynamic behavior of laminated composite, FGM plates traversed by a moving oscillator, and a moving mass. Ahmad Mamandi, Ruhollah Mohsenzadeh, and Mohammad H. Kargarnovin [6] used finite element methods and Ansys software to simulate the nonlinear dynamic of rectangular plates subjected to accelerated or decelerated moving load. A.R. Vosoughi, P. Malekzadeh, and H. Razi [7] analyzed the moderately thick laminated composite plates on the elastic foundation subjected to moving load. G.L. Oian, S.N. Gu, J.S. Jiang [8], and Marek Krawczuk [9] analyzed the cracked plate subjected to dynamic loads by FEM. Yin T. and Lam H.F [10, 11] used a new solution method for investigation of the vibration characteristics of finite-length circular cylindrical shells with a circumferential part-through crack with four representative sets of boundary conditions being considered: simply supported, clamped-clamped, clamped-simply supported, and clamped-free. Li D. H., Yang X., Qian R. L., and Xu D. [12, 13] used the extended layer method (XLWM) to analyze the static reaction, free vibration, and transient response of cracked FGM plates.

#### 2. Finite Element Simulation and Dominant Equations

Figure 1 shows the cracked plate under the moving oscillator on the plate in the general coordinate system (X,Y,Z).

For finite element model formulation, the following assumptions are made:(i)The materials of the system are linear elastic.(ii)The load and pavement are not speared in the activity duration of system.

##### 2.1. Cracked Plate Element Subjected to Moving Oscillator

###### 2.1.1. Cracked Plate Element Subjected to Dynamic Loads

Plate is described by bending rectangular four-node elements (Figure 2). Arbitrary point in the element has positions (x, y) in global coordinate and positions (r, s) in local coordinate [14]. We assume that the thickness of plate element is a constant and the conditions of Mindlin–Reissner plate theory are satisfied.

The displacement fields are written as [15]where u_{0}, v_{0}, w_{0} are the displacements of the mid plane and , are rotations of normal about, respectively, the y and x axes.

The strain vector is presented in the formwhere

The constitutive equation can be written aswhere is stress vector without shear deformation: is stress vector of shear stress:with E being elastic modulus of longitudinal deformation and being Poisson ratio.

Using (7) and (8), the components of internal force vector are determined as

so that one obtainswhere - strain matrix, is the vector of curvatures and shear strains, and *α* is the shear strain correction factor, usually equal to *α* = 5/6.

According to the FEM procedure, the displacement of a point of the element is represented as [14, 16]where , , are displacements w, , at node, respectively, and are shape functions, which allows us to obtain where is a matrix for the internal force determination, and is a vector of the node displacement, with , (i = 1,2,3,4).

Substituting (12) into (10) leads to are matrices corresponding to bending moment and shear force, respectively, [10, 11].

The dynamic equation of plate element can be derived by using Hamilton’s principle [14, 17]:where , are kinetic energy and total potential energy of the element, respectively.

The kinetic energy of the element level is defined as [14]with being stiffness matrix and node loading vector of the element, respectively, [N] is mode shape function matrix of element, p is pressure of intensity, , and is the surface area of the plate elements.

Kinetic energy of element is determined by [14]where - mass matrix, *ρ* - mass density and -velocity vector.

Substituting (16) and (17) into (15), the dynamic matrix equation of the plate element without damping can be written as

In the case of cracked plate element, the stiffness matrix of the element can be written as [8, 18]where is the transformation matrix, given in Appendix A, in which is the flexibility matrix of the noncracked element, given in Appendix B, and is the flexibility matrix due to the presence of the crack, given in Appendix C [8].

Now, the dynamic matrix equations of the cracked plate element subjected to dynamic loads become

###### 2.1.2. Cracked Plate Element Subjected to Moving Oscillator

The force of the moving oscillator on the plate at the time t is determined as follows:where g is acceleration due to gravity, is acceleration of the mass m_{2}, and is acceleration of the plate at the force set point given by

Substituting into (22) yieldswhere and are the velocity and acceleration of the loads along x, y axes, respectively.

By substituting (23) into (22), the force of the moving oscillator on the plate at the time t can be written as

Concentrated force (24) is described by the uniformly distributed load as follows [12–14]:where *δ*(.) is the Dirac’s delta function, and

Substituting (24) into (25) leads to

The element nodal load vector is [14]

Substituting (28) into (29) leads to the nodal load vector equationwhere

Substituting (30) into (20) leads to the dynamic equation of the cracked plate element subjected to moving oscillator, which is

The dynamic equation of mass m_{2} can be written as

By combining (1) and (2), the dynamic system of equations of the cracked plate and mass m_{2} are presented as follows:

Or

##### 2.2. Governing Differential Equations for Total System

By assembling all elements matrices and nodal force vectors, the governing equations of motions of the total system can be derived aswhere is the overall structural mass matrix, is the overall structural stiffness matrix with total uncracked elements, is the overall structural stiffness matrix with total cracked elements, is the overall mass matrix due to moving mass m_{1}, is the overall mass matrix due to moving mass m_{2}, is the overall damping matrix due to moving mass m_{1}, and is the overall structural damping matrix [14, 17].

This is a linear differential equation system with time dependence coefficient, which can be solved by using direct integration Newmark’s method. A MATLAB program named Cracked_Plates_Moving_2019 was conducted to solve (40).

#### 3. Numerical Analysis

We consider a rectangular cracked plate with size L = 3.0 m, W = 1.6 m, thickness h = 0.025 m, crack length = 0.5 m, and it appears in the middle of the plate (Figure 1). Material parameters of the plate: Young Modulus E = 2.0x10^{11} N/m^{2}, Poisson coefficient *ν* = 0.28, density *ρ* = 7800 kg/m^{3}. The boundary condition of the plate is SFSF (simply support, free/ simply support, free). The moving oscillator has mass m_{1} = 300 kg connected with m_{2} = 200 kg via spring with stiffness k = 1.5x10^{5} N/m and damping element with resistance coefficient, c = 4.5x10^{3} Ns/m, and k and c are parallel. Moving oscillator moves at velocity v = 10 m/s along the centerline y = W/2 of the plate.

The results of the vibration of the plate subjected to moving oscillator (MO) and the effect of moving mass (MM) (M = m1 + m2 = 500 kg) are shown in Table 1 and Figures 3, 4, 5, and 6, in which the “int” symbol represents the intensity value (stress intensity and train intensity).

*Comment*: Compared to the case of cracked plate subjected to moving oscillator, the case of cracked plate subjected to moving mass indicates greater response of the plate. Therefore, the destructive capacity of the structure is greater.

##### 3.1. The Effect of the Number of Cracks

To evaluate the effect of the number of cracks on vibration of cracked plate under moving oscillator, the three cases were investigated: Case 1: the plate has one crack in the middle (X = L/2, the basic problem); Case 2: the plate has one crack in the middle and one same size crack at X = L/4; Case 3: the plate has 3 cracks at X = L/4, L/2, 3/4. The results of the vibration of the plate are shown in Table 2 and Figures 7, 8, 9, and 10.

*Comment:* Strain, displacement, stress, and acceleration at point A increase as the number of cracks increases, but these values fluctuate at the crack edge, sometimes increase and sometimes decrease.

##### 3.2. The Effect of the Stiffness of the Spring k

To evaluate the effect of k hardness in the oscillation system on the response of the system, the authors examine the problem when k varies from 1x10^{5} N/m to 9.0x10^{5} N/m. Response of the system at points A and B is shown in Table 3 and Figures 11, 12, 13, 14, 15, and 16.

*Comment:* When the k hardness changes, the oscillation of the system varies considerably. With the parameters of the given plate, the displacement response, acceleration, stress, and strain at the computed points are the greatest when k = 2.5x10^{5} N/m.

##### 3.3. The Effect of Loading Velocity

The authors analyzed the dynamics of the plate with the speed of the oscillator system varying from 6 m/s to 14 m/s; the results are shown in Table 4 and Figures 17, 18, and 19.

*Comment:* When the speed of the oscillation system increases, the displacement and stress of the plate decrease, but there is no clear rule. According to the authors, the main reason may be due to the influence of the plate's free vibration frequency and the moving oscillator; this is the difference with the case of the plate subjected to moving mass.

#### 4. Conclusions

In the end, with the set of survey parameters, the case of the cracked plate under the moving mass is more dangerous than the case of moving oscillators operates. However, the problem of texture affected by the oscillation system is complex. In each case, a reevaluation is needed. The response of the system depends on the interrelation between the frequency of the stimulus and the natural frequency of the system.

The results show that stress and strain at the crack head are much larger than they are at other sites. These values vary considerably when the number of cracks and k hardness are changed.

#### Appendix

#### A.

The transformation matrix [T] is

#### B.

The flexibility matrix of the noncracked element is

#### C.

The flexibility matrix due to the presence of the crack is

(i) Crack parallel to the x-axis of the element:

(ii) Crack parallel to the y-axis of the element:wherewith when the crack is parallel to the x-axis, and when the crack is parallel to the y-axis of the element, and (i=1,2) are correction functions given in [8].

#### 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 there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was supported by Le Quy Don University.