#### Abstract

A prediction method for the remaining life of a V-notched beam using measured modal frequencies is proposed in this article. The main purpose is to provide a new monitoring method of crack growth for a cantilever beam. At first, the fatigue crack growth characteristic and the change law of modal frequencies of a V-notched cantilever beam under cyclic loading were studied by experiments. Subsequently, the relation of modal frequencies and crack growth were analyzed. Thereafter, the decrease ratio of the first modal frequency was employed to reflect the crack damage so as to set up the relation of the modal frequency and the crack damage, and the evolution model between the crack damage and the cyclic loading numbers were set up. A prediction method for crack growth life was proposed for a cantilever beam based on the decrease ratio of the first modal frequency in the end. The remaining life of a V-notched cantilever beam can be obtained using the proposed method with a given endured loading cycles and the corresponding measured modal frequency. Results indicate that the remaining fatigue life of the V-notched cantilever beam is related to the decrease ratio of the modal frequency, and the predicted remaining life is in good agreement with the measured remaining life as the crack depth extends to a certain value.

#### 1. Introduction

Beams are one of the basic components which are widely used in the field of aerospace, mechanical engineering, and civil engineering. Many components, such as the airplane wing, engine blades, and bridges are frequently simplified into beam models for fatigue and dynamic design. During operation, some beams have to work in a harsh environment. For example, the blade of a turbine engine has to endure the action of high temperature, high pressure, and high-speed environments. Due to the action of complicated forces, cracks can develop in a beam. The crack can make the loading capacity reduced and the beam’s life shorten greatly. It can also complicate the dynamic characteristics of the cracked beam. These changes result in difficulties in monitoring the crack growth life of a beam-like component.

A crack is one of the commonly observed flaws to beam-like components. Cracks not only affect the dynamics of the beam but also influence the fatigue crack growth. Thus, much related work has been done on the analytical methods of dynamic characteristics of cracked beam-like components. Nakhaei et al. presented an equation of dynamic motion for a beam-like component having a breathing crack, and crack parameters were considered in the equation [1]. Ma and Chen proposed an analytical way to study modal frequencies of a beam having a breathing oblique crack by using finite element software [2]. Zhang et al. used different contact elements to simulate the breathing behaviour of crack, and used spring elements to describe the elastic support for the sake of establishing a finite element model of an elastic support beam with cracks and offset boundary [3]. Liu and Barkey proposed a new stiffness equation for simulating the breathing behaviour of a crack, and the magnitude-based stiffness model was employed to analyze the nonlinear dynamic response of a cantilever beam with crack [4]. After that, Liu and Barkey investigated the coupling mechanism of the dynamic response and crack growth by using a breathing crack stiffness model and a friction damping model based on a single degree of freedom system of the beam [5]. Matsko et al. examined the forced vibration response of a cracked beam through statistical methods for the periodically correlated random process [6]. Jena and Parhi studied the dynamic response of a cracked cantilever beam having multiple transverse open cracks and a traversing mass [7]. Zeng et al. analyzed the complicated dynamic characteristics of the cantilever beams based on three types of cracks having different crack positions and depths [8]. Ma et al. examined the dynamic response of a slant-cracked cantilever beam using a new finite element model, where both plane and beam elements were used to reduce the computational costs [9]. Shankar and Pandey analyzed the complicated nonlinear dynamic response such as bifurcations and chaos by using the piecewise linear system, with the bilinear natural frequency, and the geometric nonlinearities were included in a cubic Duffing’s term [10]. Zeng et al. discussed the effects of aerodynamic force, angular acceleration, crack position, and crack depth on the nonlinear vibrational response of a cracked compressor blade during run-up process [11].

Because of the effects of crack parameters on the vibrational response of beam, the workers prefer to use the modal parameters of the beam to identify the crack and its propagation in beam-like structures. Many works have been done in the past several decades. Zare investigated a method for the sake of identifying the crack parameters of a single-edge crack in a curved beam via both numerical and measured modal frequencies [12]. Hou and Lu proposed a way to identify the crack damage in beams by using a cracked beam element model, which is derived from the local flexibility model and fracture mechanic principle [13]. Altunışık et al. carried out a detailed analysis on the modal parameter identification and the vibration response-based crack identification of a cantilever beam with hollow circular cross section having multiple cracks [14]. According to the crack identification problem of a hollow circular cross-sectional cantilever beam, they solved the problem by the transfer matrix method analytically and the finite element method numerically [15]. And subsequently, Altunışık et al. presented an automated model updating method for crack identification of a beam with the box cross section [16]. Khiem and Huyen proposed a method to detect a single crack in a functionally graded Timoshenko beam by measurements of modal frequencies [17]. Liu and Barkey studied an identification method of a cantilever beam structure having a single-edge crack by the use of a frequency error function [18]. He and Ng proposed a guided wave damage identification method by using a model-based approach to detect cracks in beam-like structures [19]. Amiri et al. proposed an effective way for detection of multiple cracks in a cracked beam based on the discrete wavelet transform [20]. Andreaus and Casini investigated a novel method for crack identification of beam-like structures having cracks by examining the static deflection [21]. Mao studied the free vibration response and crack detection of the Euler–Bernoulli beam by using the Adomian decomposition method and high-pass filters [22]. He et al. developed a two-stage method with the ability to quantify crack parameters by using the quasistatic moving load-induced displacement response [23].

Many crack detection methods have been proposed by researchers. However, due to the diversity in beam-like components in engineering, no one method can be employed to identify all kinds of crack damages. Additionally, most of vibration-based crack identification methods are employed to find the parameters of static cracks. Few of them are suitable for monitoring crack growth. According to crack growth of a beam, Fu et al. investigated pure mode I fracture and mixed mode fracture of concrete bending beams under the impact load through experiment and numerical method [24]. Ru et al. presented an extended finite element method to discuss the crack growth problems for a beam subjected to three-point bending [25]. But these methods cannot be used to track the crack depth on real time. Therefore, it is still a relevant problem to study the crack growth prediction method by using the relation of crack parameters and vibrational response.

Since modal frequency is one of the easiest obtained vibration characteristics of a beam among various vibration parameters, it is the most commonly used parameter to predict the remaining life of the cracked beam. In this article, a V-shaped notch beam was designed, and the testing system of the modal frequency and the crack growth were set up at first. Thereafter, modal frequency and crack growth testing were carried out for several cantilever beams. Based on the relation of the measured modal frequencies and the crack growth, a prediction model for the remaining life of a V-notched beam was proposed in the end.

#### 2. Experiment of V-Notched Beams

##### 2.1. Test Specimen

In this article, a V-shaped notched beam with square cross section is considered. The material of the beam is AISI 1018 steel. The material properties of the steel are shown in Table 1.

As shown in Figure 1, the dimension of the beam is designed as *L* = 713.7 mm, *b* = 38.1 mm, and *h* = 38.1 mm. A V-shaped notch was cut on the surface of the beam so that the crack will be initiated and extends to the tip of the notch in the course of fatigue test. The depth of the notch *a* is 5 mm, and the width of the notch is 3.3 mm. The distance from the tip of notch to the left end of the beam *l* is 102 mm. Two through holes were drilled in the left end of the beam for bolting installation. Distances from the two holes to the end of the beam are *e*1 = 19.05 mm, *e*2 = 44.45 mm, and = 19.05 mm. The diameters of the two holes are *d* = 12.7 mm.

##### 2.2. Test Method

During testing, the left end of the beam is clamped completely by two bolts as shown in Figure 2. The beam is assembled to make the V-notched surface downward so as to generate tensile stress at the tip of notch when a concentrated force is applied to the free end of the notched beam. The beam is forced to vibrate up and down by a sine concentrated force produced by a hydraulic actuator in the right end of the beam. The hydraulic actuator is controlled by an MTS 407 controller. In this article, the maximum force applied to the beam is 2200 N, and the minimum force is 220 N. The loading frequency is 2 Hz. During the fatigue test, the 407 controller is employed to record the loading cyclic numbers.

During the experiment, an accelerometer was used to pick up the acceleration signal of the notched beam during vibration and fatigue tests. Therefore, the accelerometer is mounted on the top side of the beam as shown in Figure 2. For observing the stress near the cross section of the notch, two strain gauges are mounted on the surface of beam as shown in Figure 3. Strain gauge 1 is mounted on the top surface of the beam, and strain gauge 2 is mounted on the side surface of the beam close to half height of the beam. The purpose of strain gauge 1 is to detect the strain history at the cross section of notch. The purpose of strain gauge 2 is to provide a reference strain for stopping the fatigue test. Pushed by the sine force at the free end of the notched beam, the measured magnitude of initial tensile stress around the tip of V notch is about 140 MPa. This stress is sufficient to produce crack at the notch tip and to make it propagate.

##### 2.3. Test Procedure

Figure 4 shows the whole test system of the notched beam. The test works include fatigue testing and modal testing. In this experiment, the hammer impact method is used to measure the modal frequencies of the beam when the fatigue test is stopped. Prior to the fatigue testing, a modal frequency test has been done for the notched beam without crack. After modal frequency testing, the bending fatigue test of the notched beam has been carried out until a crack initiated and extended to a visible crack depth. Then, the modal frequency test and the fatigue test are repeated until the failure of the V-notched beams. Figure 5 shows the test procedure of the beam.

##### 2.4. Test Result

Three notched beams are used for vibration and fatigue tests. They are numbered as T1, T2, and T3. During the modal test, the sample period is set as 20 seconds and the sample frequency is set as 1024 Hz. The first two modal frequencies of the notched beams are obtained from the time-domain acceleration signals using the fast Fourier transform program. The results are listed in Table 2. Because the stress at the tip end of the V notch is tension, the crack remains open during the fatigue test. Figure 6 shows the photos about the crack growth path of the three V-shaped notched beams when the fatigue testing is over.

#### 3. Discussion

After analyzing the results of fatigue and modal tests, the relation between crack growth and cyclic numbers and the relations between modal frequencies and crack growth can be obtained.

Figure 7 studies the characteristics of crack growth of the V-notched beams. It can be seen that the crack growth rate increases with the increasing of cycle numbers. The crack growth rate of beam T1 is the fastest one, but that of beam T2 is the slowest one.

Figures 8 and 9 examine the relations of the cyclic numbers and the modal frequencies. When the beam reaches to failure, the decrease of the first modal frequency is about 8 Hz for T1, 8.53 Hz for T2, and 3 Hz for T3. The decrease of the second modal frequency is about 13 Hz for T1, 12.5 Hz for T2, and 2.3 Hz for T3. The first two modal frequencies decrease with the increasing of cyclic numbers, and the decrease value of frequency of T3 is the minimum one, and those of T2 and T3 are almost equal.

According to the fatigue test and the modal test, the first two modal frequencies of the V-notched beams decrease with the crack growth. Figures 10 and 11 present the relation between the first two modal frequencies and crack growth. It is shown that the decrease ratio of the first two modal frequencies of the beam T1 is the fastest one but that of T3 is the slowest one. This behaviour is not in agreement with that of the crack growth curves.

Through detailed analysis, it is found that the crack growth ratio of the V-notched beam is very slow in the beginning of fatigue, but the crack growth rate is becoming faster and faster with crack growth. Therefore, the remaining life can be predicted by using the decrease ratio of the modal frequencies in the crack growth phase when an initial crack is produced.

#### 4. Crack Damage Model

Based on the damage mechanical theory, if the ratio of the current crack depth and the fail crack depth is defined as the current damage, the damage value is zero when no crack happens, and the damage value is 1 when the crack is extended to the failure length. This assumption is conformed to the MINER’s criteria [26]. Therefore, the crack damage is defined aswhere *D* denotes the crack damage value; Δ*a*_{i} denotes the current crack depth in the crack growth of the *i*^{th} phase; and *a*_{k} denotes the crack depth when the beam failed. In this article, as the crack extends to the half height of the beam and the beam is assumed to be failed.

According to the analysis of fatigue and modal tests of the V-notched beams, the decrease ratio of modal frequencies is closely related to the crack growth of the notched beam. In the present work, *f*_{m} is assumed to be the *m*^{th} modal frequency of beams without crack and *f*_{mi} is assumed to be the *m*^{th} modal frequency of beams with crack in the *i*^{th} phase.

Assume that *τ* is the decrease ratio of the modal frequencies of beams. It is defined as

Figures 12 and 13 present the relation of the frequency ratio of the *m*^{th} modal frequency and the crack depth. The results indicate that the relation of the decrease ratio of the first two modal frequencies and the crack depth of the beam is nonlinear. This behaviour is related to the parameters of the V-shaped notch and the propagation of the crack. Comparing Figure 12 with Figure 13, the ratio of the first modal frequency decreases about 15% when the crack extends to failure. It is much bigger than the decrease ratio of the second modal frequency. Thus, the first modal frequency is more preferable to be employed to model the crack damage. Plus, the first modal frequency is much easily obtained from the vibration mode test during the operation of machines, and the decrease ratio of first modal frequency is used to define the crack damage model.

In order to describe the impacts of crack growth on the decrease ratio of the first modal frequency, a polynomial function is used to set up the relation of *τ* and *D*. It is expressed aswhere *α* and *n* are the unknown coefficients, which are determined by the materials of the beam and its boundary conditions. *β* is the order number of the power function.

Using the measured data of three testing beams, the crack damage value of *D* varying with *τ* can be analyzed through equation (3) as shown in Figure 14. It can be seen from Figure 14 that the crack damage *D* increases with the decrease ratio of the first modal frequency during crack growth. The increasing of the crack damage value as *τ* is smaller than 0.05 is much bigger than that of the crack damage value as *τ* is between 0.05 and 0.1. When *τ* reaches 10%, the damage value *D* increases faster and faster until the beam fails. This trend is similar to the trends of the crack growth rate of the V-shaped notched beams. It can be divided into three phases. In the first stage, the increasing of crack damage decreases with *τ*. The increasing of the crack damage is stable in the second stage. But in the third stage, the increasing of the damage value increases very fast.

For using the decreasing ratio of the first modal frequency to predict the remaining life of beams, the relation of the crack damage model and the decrease ratio of the modal frequency is fitted by using three-order polynomial function equation. It is expressed as

Based on the definition of the crack damage model, the crack damage value *D* = 0 as the beam has no crack, that is, *τ* = 0. However, the damage value must be equal to 1 as the crack is extended to failure. Thus, as *τ* reaches *τ*_{k} when the beam fails, the decrease ratio of the first modal frequency must satisfy with

When *α*_{β} and *n* are determined in formula (3), *τ*_{k} can be solved for predicting the life of the beam. According to equation (5), the predicted value of *τ*_{k} is about 0.15. This value can be used as the criteria to judge the failure of the beam in engineering.

#### 5. Prediction on the Remaining Life

In this article, *N*_{i} is assumed to be the current fatigue life, and *N*_{k} is the final fatigue life. If *q* *=* *N*_{i}/*N*_{k}, *q* represents the fatigue life ratio of the beam. Based on the relation of the crack growth and the decrease ratio of the first modal frequency and the relation of loading cyclic numbers and the crack growth, Figure 15 shows the relation of the fatigue life ratio *q* and the decrease ratio of first modal frequency *τ*. The results show that the crack initiation life happens at about *q* = 0.6. During this stage, the decrease ratio of the first modal frequency is very slow. With the increasing of the fatigue life, the crack damage value increases. As the corresponding fatigue life ratio *q* is larger than 0.6, the decrease ratio of the modal frequency is accelerated. During the fatigue crack growth of each V-notched beam, the decrease ratio of modal frequency of T3 is the minimum one, and that of T2 is the maximum one.

From the above analysis, the decrease ratio of the first modal frequency is not only related to the fatigue life ratio of the beam but also is related to the crack damage of the notched beam. Thus, the crack damage value of the notched beam can be defined as the power function of the fatigue life ratio of the beam. It is defined aswhere *p* is the unknown coefficient which is related to the material and structure.

According to equation (6), *D* is zero before the force was applied to the notched beam. And *D* is 1 as the beam was vibrated up and down to fail. Figure 16 shows the relation of the fatigue life ratio *q* and the crack damage value *D*. The life prediction model is fitted by using the power function equation and the measured data. The life prediction model for three beams are fitted as

Thus, the fatigue life of the notched beam can be predicted by the decrease ratio of the first modal frequency

Through the prediction model for the fatigue life, the relation of crack damage and the crack growth life can be obtained. If Δ*N*^{∗}_{i} is assumed to be the current remaining life, the predicted remaining life is

Figure 17 compares the predicted remaining life and the measured remaining life. The results indicate that the proposed method can be used to predict the remaining fatigue life of the V-notched beams. The errors between them for three beams are different. In the beginning of the crack growth, the predicted error is larger than that of crack growth in the second stage. For three beams, the prediction errors of T1 and T3 are bigger than that of T2.

#### 6. Error Analysis

For better understanding the applicable of the proposed method, Δ*N*^{∗} represents the predicted remaining life and Δ*N* represents the measured remaining life. A relative error model of the predicted model of the remaining life is defined as

Figure 18 examines the relative error of the remaining fatigue life of the notched beams. The error is very large in the beginning of crack growth, but it decreases at the phase of stable crack growth. The error is below 10% in the stable crack growth phase. This implied that the proposed method is not suitable for predicting the remaining life of the V-notched beam at the beginning of crack growth, but it is helpful to analyze the remaining life of the crack growth in the stable phase. For three beams, the prediction precision of T2 is the best. For T1, the prediction error decreases with crack growth.

#### 7. Conclusions

Starting from the test of three notched beams, the relations of the modal frequencies and the crack growth and the cyclic loading numbers were analyzed. The crack damage model was proposed using the decrease ratio of the first modal frequency. A remaining life prediction model for the fatigue crack growth life was proposed in the end. The main conclusions are as follows:(1)The modal frequencies decrease with the crack growth, and the decrease ratio is very slow in the beginning of the crack growth. The sensitivity of the first modal frequency to the crack damage is more obvious than that of the second modal frequency. Therefore, it is preferable to use the first modal frequency to monitor the remaining life of the cracked beam in engineering.(2)The proposed crack damage model and the prediction model for the remaining fatigue life is not suitable for predicting the life in the beginning of crack growth. It is helpful for predicting the remaining life of the fatigue crack growth in the stable stage and the errors are below 10%.(3)The proposed model can be used to predict the remaining life of the beam. But it is necessary to obtain the measured current life and the current modal frequency first. Additional work should be done for this proposed prediction model to be more general in nature.

#### Nomenclature

L: | Length of the beam |

b: | Width of the beam |

h: | Height of the beam |

a: | Depth of the notch |

: | Width of the notch |

l: | Distance from the notch to the beam’s left end |

e1, e2: | Distances from the holes to the beam’s end |

: | Distances from the holes to the beam’s end |

d: | Diameter of the hole |

E: | Young’s modulus |

ν: | Poisson’s ratio |

ρ: | Material density |

σ_{s}: | Yield stress |

σ_{b}: | Ultimate stress |

D: | Crack damage value |

Δa_{i}: | Current crack depth |

a_{k}: | Crack depth when the beam failed |

f_{m}: | m^{th} modal frequency of the intact beam |

f_{mi}: | m^{th} modal frequency of the beam with crack |

Δf_{mi}: | Decrease of m^{th} modal frequency |

α: | Unknown coefficient |

n: | Order number of the polynomial |

p, β: | Power of the polynomial |

τ: | Decrease ratio of the modal frequency |

ΔN^{∗}: | Predicted remaining fatigue life |

ΔN: | Measured remaining fatigue life |

N_{i}: | Current fatigue life |

N_{k}: | Final fatigue life |

q: | Fatigue life ratio |

δ: | Relative error. |

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research work was supported by the National Science Foundation of China (51565039) and the Science and Technology Project of Jiangxi Provincial Education Department (GJJ170584).