# Vibration Fatigue Analysis of a Simply Supported Cracked Beam Subjected to a Typical Load
###### Abstract  
In order to realize the accurate prediction of the vibration fatigue life of the beam in service, a loose coupling analysis method is proposed to carry out vibration fatigue analysis of a beam with an initial crack. In modal analysis, the initial crack segment is replaced with a torsion spring, and the damping loss factor is introduced by the complex modulus of elasticity; for the simply supported beam, the inherent vibration characteristic equation of the cracked beam is derived. In vibration fatigue analysis, the interaction between the crack's growth and vibration analysis is considered, and a loose coupling analysis method is proposed to conduct modal dynamic response and vibration fatigue analysis simultaneously. Results indicate that the crack's relative location and depth determine the modal of the cracked beam, and crack parameters, damping loss factor and external excitation frequency are important factors for the vibration fatigue life of the beam. c 2022  
cracked beam for verification. [PERSON] and [PERSON] [8] designed a rectangular cross-section cracked beam, and applied the [PERSON]'s damage model to investigate a random vibration fatigue mechanism in the frequency domain numerically and experimentally. [PERSON] and [PERSON] [9] used the energy principle to derive the friction damping loss factor based on the Coulomb friction model, and investigated the coupling mechanism of vibration analysis and crack's growth. [PERSON] et al. [10] established the high frequency vibration-induced fatigue failure experimental platform of the cracked beam, and analyzed the influence of dynamic stress, acceleration and operational modal on the vibration fatigue life. References above provided a variety of solutions for vibration fatigue analysis of the cracked beam, but these methods had certain defects. In the theoretical method, the interaction between crack growth and the stress field is not considered; in the numerical method, the crack growth cannot be tracked; in the experimental method, the workload is huge and the cost is high. Structural vibration induces fatigue crack propagation, and fatigue crack propagation also changes the vibration characteristics and stress field of a structure, and then the crack growth behavior is affected. Scholars have carried out a lot of analysis and research to consider the complexity of the interaction between vibration and fatigue crack propagation, [PERSON] and [PERSON] [11] considered the interaction between stress intensity factor and crack axis, and established a model for fatigue life and crack axis analysis. [PERSON] and [PERSON] [12] used the generalized Forman equation [13] to simulate the crack growth, and analyzed the influence of the vibration on the crack growth of a rectangular plate with a unilateral crack. [PERSON] and [PERSON] [14] assumed the periodic load excitation was acting on the rectangular plate with a unilateral crack, and studied the non-linear vibration response characteristics. [PERSON] and [PERSON] [15] considered the coupling effect of vibration analysis and crack's growth, and proposed a new method to conduct the vibration fatigue analysis of cracked beams by introducing the complex elastic modulus. Assuming that the cyclic loading was acting on a V-notched cantilever beam, [PERSON] and [PERSON] [16] investigated the crack growth characteristics by experimental method, and proposed a new method to predict remaining life. [PERSON] and [PERSON] [17] designed the shaker vibration fatigue experimental plan of the cracked beam, and investigated non-linear behavior and remaining life. Considering the coupling effect between vibration analysis and the crack's growth, the references above provide several solutions to predict the remaining life of a beam. In the vibrating environment, there is interaction between the stress field and the crack field, and the modal dynamic response and vibration fatigue analysis need to be carried out at the same time.