#### Abstract

A new model is presented for studying the effects of crack parameters on the dynamics of a cracked beam structure. The model is established by the finite element displacement method. In particular, the stiffness matrix of the cracked beam element is firstly derived by the displacement method, which does not need the flexibility matrix inversion calculation compared with the previous local flexibility approaches based on the force method. Starting with a finite element model of cracked beam element, the equation of strain energy of a cracked beam element is formed by the displacement method combined with the linear fracture mechanics. Then, based on the finite element method, the dynamic model of the cracked beam structure is obtained. The results show that the dynamic model discovers the internal relation between the dynamic characteristics of cracked beam structure and structural parameters, material parameters, and crack parameters. Finally, an example is presented to validate the proposed dynamic model.

#### 1. Introduction

Beam structures are widely used in many applications, such as automotive, aerospace, and civil engineering [1–4]. Many beam structures are subjected to cyclic loading, which results in fatigue cracks [5, 6]. The presence of the crack not only changes the regional stress and strain fields of the crack tip but also affects structural dynamics. If these fatigue cracks cannot be timely detected and repaired, the subsequent fracture can bring catastrophic failure to the beam structures [7]. Recently, the vibration-based damage detection has become one of the commonly used tools for crack detection and diagnosis [8]. This approach is mainly based on changes in dynamic characteristics, such as natural frequency and mode shape [9].

In recent decades, the vibration-based methods for detecting cracks in beam structures have been proposed. These methods can be classified as analytical, numerical, and experimental [10, 11]. No matter what method is used, crack modelling is the most crucial step that is the base for subsequent vibration analysis. Based on the crack modelling theory, the crack models can be classified as “local flexibility model” and “consistent continuous cracked beam model” [12]. In particular, the local flexibility model has been widely used for vibration analysis of cracked beams [13].

Based on the local flexibility model, the theoretical modelling techniques of cracked beams can be grouped into the “continuous beam models” and “finite element models of cracked beams.” The first category models a crack as a massless rotational spring; therefore, the beam is divided into components from the crack sections represented by a rotational spring. Then, the classical Bernoulli–Euler beam theory or Timoshenko beam theory is used in the equation of motion for each beam segment. At last, the dynamic equation of cracked beam is determined from the boundary conditions and the compatibility conditions at the cracked section. For example, Rizos et al. [14] modelled the crack as a massless rotational spring and presented a method to detect the crack location and depth in structures by measuring amplitude at two points. Ostachowicz and Krawzuk [15] presented a continuous model of a cracked cantilever beam with two edge cracks. They also used massless rotational spring representation of crack in the beam. Chang and Chen [16] used a continuous model to represent the mode shapes of cracked beam with multicracks and presented a spatial wavelet-based method to detect the depths and positions of the cracks. In their study, the cracks were also modelled by massless rotational springs.

In the second category, the finite element models of cracked beam represent the presence of a crack by means of a reduction in the beam bending stiffness. First, a local flexibility matrix of cracked beam element is derived from the integration of stress intensity factors which is a function of basic unknown force at cracked beam element node [17–21]. Thus, the stiffness matrix of cracked beam element is transformed from the local flexibility matrix based on the equilibrium equation, and then a finite element model of cracked beam is established based on the finite element method. For instance, Qian et al. [17] presented a finite element model of a cracked beam. The 4 × 4 stiffness matrix is derived only considering the bending moment and shear force, which is calculated through the inverse of the compliance. Lee and Chung [18] proposed a procedure for detecting the crack location and size in a beam structure based on only the natural frequency. They also used inverse of the flexibility matrix and transformation matrix to calculate the 4 × 4 stiffness matrix of cracked beam element. Cacciola and Muscolino [19] used a finite element model of a cracked beam to study the deterministic and stochastic response of a beam with certain and uncertain no-propagating crack. The cracked part of the beam was also modelled by Euler type finite elements with two nodes and two basic unknown forces (bending and shear internal forces) at each node in their model.

In structural analysis, the analytical methods can be divided into the force method, displacement method, and mixed method, depending on the type of the basic unknowns [22–24]. In this sense, the above works on the cracked beam element that are based on local flexibility matrix coefficient belong to the force method as the basic unknown is the nodal force. The equation of stiffness matrix of cracked beam elements by the force method can be summarized as follows. First, the unknown force is assumed at the node of the cracked beam element, and the equation of the additional stress energy due to the crack can be obtained by fracture mechanics. Second, the terms of the flexibility matrix of cracked beam are expressed by using Castigliano’s theorem. At last, the stiffness matrix of cracked beam elements is obtained based on the inverse of the compliance matrix and transformation matrix.

However, the force method has certain drawbacks which need to overcome. First of all, the stiffness matrix of cracked beam is at most the 6 × 6 matrix by transforming the 3 × 3 flexibility matrix. In that case, the nodal curvature condition cannot be used as one of its boundary conditions in the force method. In other words, this finite element method cannot accurately reflect the true dynamic deformation of beam structures because of the complexity of the vibration curves [25]. Finally, when using the force method for obtaining the stiffness matrix of cracked beam element, researchers are required to formulate not only the flexibility matrix of the cracked beam element but also further calculate the stiffness matrix of the cracked beam element based on the equilibrium condition and matrix inversion calculation, which is very inconvenient. Therefore, the force method is called an “indirect” method [22]. So, it is necessary to formulate an accurate and effective stiffness matrix equation of cracked beam element by a new method.

Unlike the force method, in the displacement method, the unknown displacement is firstly determined. When curvatures of axes at the nodes are used as one of the basic unknown displacements, the lateral displacement of cracked beam element can be assumed by quintic-Hermite’s interpolation function. Thus, the 8 × 8 stiffness matrix of cracked beam will be formed, which can accurately model the vibration of the cracked beam. Since it is also known that in structural analysis, the displacement method is preferred to the force method because the basic theory of the displacement method is more suitable by using computational procedures [22]. Namely, the displacement method has been widely used in finite element modelling as it is easy for computer automation. However, because the crack caused discontinuity of the body, this effective method has not yet been used to establish the dynamic model of the cracked beam element, which further limits the finite element displacement method (FEDM) on dynamic modelling of the cracked beam structures.

The purpose of this paper is to use the finite element displacement method to establish the dynamic equation of the cracked beam structures. The originality of this study is that the 8 × 8 stiffness matrix of cracked beam element is “directly” obtained by using the displacement method, which does not need the flexibility matrix inversion calculation. Based on this formulation, the new dynamic model of the cracked beam structure is obtained by the finite element displacement method, which builds the internal relation between the dynamic characters of cracked beam structure and the structural parameters, material parameters, and crack parameters. A numerical example is presented. It is shown that the influences of crack depth and crack location on natural frequencies of cantilever beam can be effectively estimated by the proposed dynamic model.

#### 2. The Finite Element Model of Cracked Beam Element

##### 2.1. Displacement Model of the Cracked Beam Element

Using the displacement method, the displacement model of the cracked beam element is shown in Figure 1. According to the finite element method, the generalized coordinate vector of the cracked beam element can be expressed aswhere and are longitudinal displacements of the nodes; and are the transverse displacement of the nodes; and are the elastic rotation angles of neutral axis at the nodes; and and are the curvatures of neutral axis at the nodes.

The elastic transverse displacement *W* (*x*, *t*) and elastic longitudinal displacements *V* (*x*, *t*) at the center point of any cross section of the element are expressed aswhere given in Appendix are the shape functions, respectively, and is the coordinate of the cracked beam element in the local coordinate system.

##### 2.2. Kinetic Energy of the Cracked Beam Element

Considering that the crack section does not affect the mass of the cracked beam element, the kinetic energy of the cracked beam element is [26]where is the length of the cracked beam element; is the mass distribution function of the cracked beam element; is the transverse absolute velocity at the central point of any cross section of the element; and is the longitudinal absolute velocity at the central point of any cross section of the element.

Substituting equation (2) into equation (3) yields the following equation:where is the absolute velocity vector of the cracked beam element, **m** is the mass matrix of the cracked beam element, and

By the above derivation, it is obtained that the mass matrix of intact beam element is also **m** as the assumption that the crack does not affect the mass.

##### 2.3. Potential Energy of the Cracked Beam Element

Considering a beam with a ratio of length to height larger than 10, the shearing deformation energy can be neglected. Therefore, the elastic potential energy of a beam element without crack can be expressed as [26]where is the elastic modulus of the cracked beam; is the area moment of inertia function of cracked beam element; and is the cross-sectional area function of cracked beam element.

Substituting equation (2) into equation (6) results inwherewhere is a 8 × 8 stiffness matrix of beam element by using the condition of curvature.

Based on the linear elastic fracture mechanics, the releasing energy due to crack propagation can be expressed as [27]where represents the area of the crack region and represents the strain energy release rate.

According to the deformation characteristics of the cracked beam element as shown in Figure 1, only the influence of the crack opening mode (also referred to mode I) is concerned. In this mode, the crack surfaces move apart in the direction perpendicular to the crack. Based on the general form of stain energy release rate function from reference [27], the equation for the strain energy release rate *G* is defined aswhere for plane strain, for plane stress, is the Poisson ratio, and and are tensile stress intensity factor of mode I and bending stress intensity factor of mode I, respectively. Then,where and are tensile stress and bending stresses on the cross-sectional section of the element, respectively, is the crack depth, and the correction function of the stress intensity factor and are defined as [28]where is the dimensionless crack depth, , and is the height of beam element.

In the elastic range, with the shearing action neglected, the strain at any point of the cross section of cracked beam element consists of two parts, namely, the tensile strain and bending strain. The tensile strain at any point on the cross section can be written as [25]

Based on the theory of materials mechanics, bending strain at any point on the cracked beam element can be expressed as follows [25]:where is the distance from any point on the cracked beam element to the neutral axis.

Substituting equation (2) into equations (14) and (15) yields the strain vectorwhere , is the strain transformation matrix, andwherewhere is the distance from the fractured cross section to the node.

Next, is ordered such that . Based on Hooke’s law, the relationship between the stress vector and strain vector can be written as

By substituting equations (10)–(12), (16), and (17) into equation (9), the releasing energy due to crack propagation *W*_{2} can then be transformed aswhere

It should be noted that the stiffness due to crack propagation is an 8 × 8 stiffness matrix by using the condition of curvature.

##### 2.4. Stiffness of Cracked Beam Element with Open Crack

The open crack model and breathing crack model are commonly used to describe the crack surface behavior during vibration. Which of the two models is suitable depends on the crack size and actual loaded state. The open crack model assumes that the cracks of beam structures are always open, and thus the closing of crack surfaces does not occur. Therefore, the total potential energy of the cracked beam element with open crack *W* consists of the difference between the elastic potential energy of beam element *W*_{1} and the releasing energy of crack propagation *W*_{2}; that is,

By substituting equations (7) and (20) into equation (22), the expression of the total strain energy of the cracked beam element with open crack is obtainedwhere is the stiffness matrix of the cracked beam element with an open crack and .

From the above derivation, is the element stiffness matrix component, which is related to the elastic potential energy of the beam element. And is another element stiffness matrix component, which is related to the beam element’s crack. Thus, for beam element with no crack, its stiffness matrix is .

##### 2.5. Stiffness of Cracked Beam with Breathing Crack

Breathing crack is a case where there is a repetitive opening and closing of the crack surface [28]. When the crack is at a repetitive opening and closing state, it shows different mechanical properties. Based on the fracture mechanics, the dynamic system of the cracked beam is a complex problem which is related to the stress and strain field of the crack tip, the shape of the crack interface, and the degree of crack closure. To simplify the dynamic analysis, the crack closure behavior can be simulated by a varying stiffness model as the change between the fully open and fully closed instantaneously gives rise to a bilinear-type stiffness [17].

When crack is in a closing state, the strain at *r* (see Figure 1) is tensile. Conversely, when crack is in an opening state, the strain at *r* is compressive. According to the mechanics of materials, the axial strain at any point of beam element is the sum of the tensile strain and the bending strain ; the strain at *r* can be expressed as

When element meshing is proceeded, the crack is generally set on the middle of the element, and then the strain on the position *r* can be expressed as

In this paper, the strain state on the *r* coordinate of *h*/2 is used as the criterion for judging whether the crack is opening or closing, and then the crack closing condition of the cracked beam element iswhere . In contrast, the crack opening condition is . Therefore, the potential energy of closing crack in the beam element is expressed as

Substituting equations (7) and (27) into equation (22) and rearranging results in the total potential energy of the cracked beam element with breathing crack aswhere is the stiffness matrix of beam element with breathing crack, is defined as state function, i.e., , and .

As mentioned above, is the element stiffness matrix of the intact beam element. And is the element stiffness matrix component, which is related to the beam element's crack. Thus, for the cracked beam element with breathing crack in a closing state, its stiffness matrix **k**_{1b} is , which is the same as noncrack beam element. And the cracked beam element stiffness matrix **k**_{1b} is if the breathing crack is in an opening state, which is the same as the always open crack.

##### 2.6. The Kinetic Equation of the Cracked Beam Element

Lagrange equation for the cracked beam element can be written as [26]where is the generalized force vector of applied load and is the element node force vector by the other elements connecting to the cracked beam element.

Substituting equations (4), (23), and (28) into equation (29) yields the following equation:where is the rigid body acceleration of the cracked beam element, is the stiffness matrix of the cracked beam element, i.e., if crack is open one, and if crack is breathing one.

#### 3. Equations of Motion for Cracked Beam Structure

In the finite element analysis of the beam structure with cracked beam, the beam structure is divided into *n* beam elements. Without loss of generality, a straight beam with transverse cracks is used for analysis, as shown in Figure 2. Assuming that **B**_{i} is the coordinate matrix of the *i*-th unit between local numbering and system number and **R**_{i} is the transformation matrix between the *i*-th unit coordinate with the global coordinate, the differential equations of motion of the *i*-th element in the global coordinate system can be expressed as [26]

In the formula,where , , and are the generalized coordinate vector in the global coordinate system, the vibration acceleration vector, and the rigid body acceleration vector, respectively, and is the *i*-th beam element stiffness. That is, when the *i*-th unit is cracked beam element, then is . When the *i*-th unit is noncrack of the beam element, then is ; and are the external force and element node force vector of the *i*-th beam element in the local coordinate system, respectively.

Stacking up all elements’ differential equations of motion, the differential equation of motion of the beam structure containing cracks members can be written aswhere , , and .

It is important to note that the element force in equation (31) offsets each other.

Assuming that the damping force is proportional to the speed, the dynamic equation of the beam structure with cracked member included in damping iswhere and are damping matrix and generalized velocity vector of cracked beam structure, respectively.

It should be noted that the stiffness matrix of cracked beam structure is composed of the 8 × 8 stiffness matrix of cracked beam element and several 8 × 8 stiffness matrix of noncrack beam elements. As described above, the stiffness matrix of cracked beam element is a function of crack size, so the dynamic performance of the cracked beam structure is not only related to its structure parameters and material but also crack parameters.

#### 4. Numerical Validation and Discussion

Cantilever beam is the one of the simple structures in beam structures. Without loss of generality, a rectangular cross section cantilever beam with transverse cracks is used for numerical analysis. The length of the beam is 300 mm, height and width of the beam are 20 mm, the elasticity modulus of the material is *E* *=* 206 GPa, and the mass density is *ρ* *=* 7750 kg/m^{3}, which was validated by experimental data by Rizos et al. [14] and by data in [18, 29, 30].

##### 4.1. Open Crack

The cantilever beam is divided into seven beam elements; the crack is located at the middle of element, as shown in Figure 3. Based on the mass matrix and stiffness matrix of equation (34), the first two natural frequencies of the cracked beam with open crack are obtained by the proposed finite element model with displacement method (FEDM) through MATLAB programming. To validate the proposed model, the first two natural frequencies obtained from the proposed model are compared to the experimental results in Kam and Lee [29] and the numerical results by finite element model with force method (FEFM) in Lee and Chung [18]; the FEFM is composed of 4 × 4 stiffness matrix of cracked beam element.

The results are shown in Table 1, and the dimensionless natural frequencies (cracked beam frequency/undamaged beam frequencies) are shown in Figures 4–6.

**(a)**

**(b)**

**(a)**

**(b)**

**(a)**

**(b)**

As shown in Table 1, the differences between the first two natural frequencies from the FEFM and experimental investigation are from 0 to 1.37%, and those between the proposed model and experimental investigation are from 0 to 0.34%; in most cases, it is less than 0.1%. It is noted that the results from the proposed model are in satisfactory agreement for all cases. In some cases, the results from the proposed model are more accurate than the results from the FEFM, as shown in Figures 4(a) and 5(b). Results from the experimental investigation and the FEFM validate to some extent the correctness of the proposed model.

Next, the influences of crack depth and crack location on the natural frequencies of cantilever beam are analyzed. For simplicity, calculations are only carried out using the proposed model as the natural frequency from proposed model accord with the experimental results. Considering the cantilever beam with a crack, the dimensionless crack depth *a/h* is chosen to be from 0 to 0.5 and the dimensionless crack location *x/L* is chosen to be from 0 to 1. Influences of crack depths and the crack location on the first two natural frequencies of the cracked beam are shown in Figure 7. It can be seen from Figures 7(a) and 7(b) that the first two dimensionless natural frequencies decrease with crack depth*.* Figure 7(a) depicts that the first natural frequency of the crack cantilever beam decreases parabolically with the crack location from the fix end to free end of the cantilever beam when the crack depth is kept constant. This indicates that the first natural frequency influenced by crack location decreases gradually with increase of the crack location (away from the fix end of the cracked cantilever beam), which becomes a minimum value when the crack is near the fixed end. Figure 7(b) depicts that the second natural frequency exhibits wavelike variations with crack location; the maximum natural frequency is about at the crack location *x*/*L* of 0.2, while the natural frequency value at the middle of the cantilever beam is a low value.

**(a)**

**(b)**

To further verify the accuracy of the proposed method, the natural frequencies of the cracked cantilever beam with a smaller ratio of width to height (*b*/*h*) are evaluated. The beam parameters used are *L* = 200 mm, *b* = 1 mm, and *h* = 7.8 mm, with various crack ratios of *x*/*L* and *a*/*h*. The natural frequencies of the cantilever beam have been calculated using the FEFM by Qian et al. [17], and they showed that the numerical results from the FEFM agree quite well with the experimental data. Table 2 shows the comparison of the natural frequencies results from the FEFM and proposed method. It can be seen that the differences between the results from the FEFM and those from the proposed method are less than 1.35%, which demonstrates that the proposed method is valid for a beam with a small ratio of width to height.

##### 4.2. Closing Crack

In this section, the fundamental frequency of the cracked beam with a breathing crack is determined by the proposed model and is compared with that found by the finite element analysis (FEA) used by Andreaus [31]. The physical model parameters are the same as those described in the first paragraph of Section 4. The finite element model of the cracked cantilever beam with a breathing crack is also the same as the cracked cantilever beam with an open crack, as shown in Figure 3. As described in Section 2.5, the closing crack is considered as the “bilinear-type” model which is fully open or fully close when the strain near the crack tip is tensile or compressive. Thus, the fundamental frequency of a beam with a breathing crack *f*_{b} can be written as [32]where is the natural frequency of the cracked cantilever beam when the crack closes and *f*_{o} is the natural frequency of the cracked cantilever beam when the crack opens.

The finite element analysis is established by using a commercial FE software ADINA. As shown in reference [31], the finite element mesh consists of 2D solid eight-node isoparametric elements, resulting 380 elements and 1627 nodes.

Table 3 lists the fundamental natural frequencies calculated by the proposed model and FEA and experimental study, respectively. The dimensionless fundamental frequencies *f*_{o}*/f*_{c} and *f*_{b}*/f*_{c} are evaluated for both crack scenarios with four different values of depth and position. In addition to Table 3, Figures 8(a) and 8(b) show the effect of the crack depth *a*/*h* on the fundamental frequencies of the cracked beams having two different crack locations. Each plot contains five curves, three of which are showing this study, finite element analysis, and the experimental results with open crack; the rest two curves are showing this study and finite element analysis results with breathing crack. Curve for the experimental results of cracked beam with breathing crack is not depicted because of the lack of experimental data.

**(a)**

**(b)**

For the cantilever beam with open crack, differences between the dimensionless fundamental natural frequencies *f*_{o}/*f*_{c} from the FEA and experimental investigation are from 0.002 to 0.023, and those between the proposed model and experimental investigation are from 0.001 to 0.004. It is obvious from Table 3 and Figures 8(a) and 8(b) that for open crack, the results of this study are in good agreement with the experimental results. Moreover, comparing with the fundamental natural frequencies *f*_{o}/*f*_{c} from the experimental results of Table 3, the results obtained by the proposed model are closer to the experimental data than those by the commercial finite element software. Comparison results show that the proposed model is accurate and effective.

For the cantilever beam with breathing crack, the difference between the dimensionless fundamental natural frequencies *f*_{b}/*f*_{c} from the FEA and the proposed model are from 0.08% to 0.32%. It is also derived from Table 3 and Figures 8(a) and 8(b) that for breathing crack, the results of this study are generally close to the results of Andreaus and Baragatti [31].

To study the crack closure on the fundamental frequencies of the cracked beams with breathing crack further, the dimensionless fundamental natural frequencies of the cracked beam with open crack and breathing crack are plotted together in Figures 8(a) and 8(b). It is obvious that the difference between the fundamental frequencies of the cracked beam with open crack and breathing crack increases with increase of crack depth.

#### 5. Experimental Validation

In order to validate the simulation results, an experimental study is conducted. For this purpose, a vibration experiment platform is setup as shown in Figure 9. The cantilever beam vibration testing apparatus was produced by Donghua Testing Company with DHVTC in China.

As shown in Figure 9, the experimental setup consists of the solid base, the beam specimen, the impact hammer (type LC02), and the measuring system. The beam specimen is clamped on the solid base. An accelerometer (type IEPE) is attached at the one-sixth of the beam length from the fixed end. Free vibrations were induced by striking the free end of the specimen by using the impact hammer that is equipped with an internal force sensor (type 3A102). The measuring system consists of the signal conditioner, 6-channel signal collector, signal analyzer (DH5922), and the PC. The sampling frequency used in testing was 5000 Hz.

The three specimens of C45 E steel having a cross section of 15.6 × 20 mm^{2} were used in the test. The elasticity modulus of the material is *E* = 206 GPa, and the mass density is *ρ* = 7850 kg/m^{3}. The dimensions of the specimens are shown in Figure 10. The cantilevered specimens have a free length of 240 mm. To simulate an open crack, a notch with a width of 0.2 mm was cut by a wire electrical discharge machine (WEDM). The crack depths of the three specimens are 2 mm, 5 mm, and 7.8 mm, respectively, as shown in Figure 11. The experimental results show that the first natural frequencies of the cracked cantilever beam with crack depths 2 mm, 5 mm, and 7.8 mm are 207.52 Hz, 192.87 Hz, and 156.25 Hz, respectively, and the second natural frequencies of cracked cantilever beam with crack depths 2 mm, 5 mm, and 7.8 mm are 1237.0 Hz, 1232.9 Hz, and 1184.1 Hz, respectively.

Table 4 compares the results of the first two natural frequencies obtained by the experiment, the proposed method with 8 × 8 stiffness matrix, and the FEFM with 4 × 4 stiffness matrix, respectively. As shown in Table 4, the proposed method is better to estimate the first natural frequency as the differences between the values from the proposed method and those from the experiment are smaller. With respect to the second natural frequency, the proposed method is able to result in the accuracy comparable to that obtained from the FEFM.

Furthermore, the accuracy of the proposed method in predicting the first natural frequency becomes more evident with an increase of the crack depth. When a crack depth of *a* = 5 mm is present at *X* = 60 mm, the difference between the experimental value and that obtained from the proposed method is 1.188% while the difference between the experimental value and that obtained from the FEFM is 4.244%. When a crack depth of *a* = 7.8 mm is present at *X* = 60 mm, the difference between the experimental value and that obtained from the proposed method is 4.804% while the difference between the experimental value and that obtained from the FEFM is 11.578%. For the second natural frequency, the proposed method is able to provide an accuracy that is comparable with that by the FEFM. The experiment validates the effectiveness of the proposed method.

#### 6. Conclusions

In this paper, a new stiffness matrix of cracked beam element has been derived by using the displacement method, which does not require deriving the flexibility matrix inversion calculation that is needed with the usual force method. On that basis, a finite element model for dynamic analysis of a cracked beam structure has been proposed. This model allows to effectively determine the internal relation between the dynamic characters of cracked beam structure and the structural parameters, material parameters, and crack parameters.

The natural frequencies calculated by the proposed model agree quite well with the experimental data. And in some cases, the results from the proposed model are more accurate than the results from the finite element model with the force method. This indicates that the proposed model in this paper is an improved one compared with the existing models. Therefore, this proposed method may be extended to complex beam structures with various cracks.

The numeric results show that the first two natural frequencies of the cracked cantilever beam decrease with crack depth, but the amount of decreasing value is different for the different mode. When the crack depth is kept constant, the first natural frequency is decreased parabolically with the distance between the fixed end and the free end of the cantilever beam, whereas the second mode exhibits wavelike variations with crack location.

In comparison with a beam with open crack, the crack closure of breathing crack influences the dynamic characteristics of the cracked beam structure when the cracks breathe. Difference between the fundamental frequencies of the cracked beam with open crack and breathing crack increases with crack depth*.*

The numerical results show that the proposed method can also achieve a high accuracy with the cracked beam of a small width to height ratio. When the ratio of width to height becomes as small as plate size, a finite-length strip with edge crack will be used for modelling. The dynamic model of a cracked plate can also be obtained by using the pattern similar to modelling of the cracked beam. Therefore, the proposed method offers an alternative approach to further study the dynamic characteristics of the cracked plate.

The method proposed in this paper can also be extended to the Timoshenko beam with a crack, if the kinetic energy, potential energy, and strain energy release rate of crack propagation in the Timoshenko beam are known. This extension can be done by taking the advantage of the considerable research conducted for the Timoshenko beam.

The open crack model and closing crack model proposed in this paper can be easily extended to further study breathing crack effects and the interaction between open crack and closing crack.

#### Appendix

According to the finite element method, *W* (*x, t*) and *V* (*x, t*) can be expressed as [25]where are the shape functions, andwhere and *l* is the length of the cracked beam element.

#### 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 research was supported by the National Natural Science Foundation of China under grant numbers 51375500 and 51465001.