# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 1 Introduction
To fully understand the vibration characteristics caused by the breathing crack, increasing research activities have been focused on the use of nonlinear models for cracked beams. The modelling of the breathing crack is the key step that provides a solid foundation for the following analysis. There are two common closing crack models, namely a \"square-type\" model and a \"simple periodic function\" model [26]. The first model assumes that the breathing crack can only be fully open or fully closed, and the states of partial closure or opening are excluded [27; 28; 29]. For example, [PERSON] and [PERSON] [27] presented the square-type model to simulate the stiffness change, and they formulated a closed-form solution of the cracked beam. [PERSON] [30] used the perturbation approach to predict the amplitude of spectral peaks of the weakly bilinear oscillator. [PERSON] et al. [31] developed a dynamic model of a cracked beam with a breathing crack by using consistent one-dimensional theory. Their results showed that the natural frequencies obtained from analysis are in accordance with the experimental results. However, this assumption is not accurate in most cases. For fatigue due to crack breathing, a partial crack closure occurs in reality. In particular, [PERSON] et al. [32] and [PERSON] et al. [33] have experimentally measured the state of partial closure. The second model simulates the stiffness changes as a simple trigonometric function [26] and has been adopted in [15; 34]. The crack in this simplified model can be not only in a fully open state or fully closed state, but also in a state of being partially open or closed. However, this model can be applied only to the case of the fundamental mode [35]. The most common models for the vibration analysis of a cracked beam include the finite element model (FEM), in which a special cracked beam finite element is used. On that basis, several approaches in conjunction with the FEM have been proposed. The commonly-used approaches include the direct numerical integration method [36], the frequency response function (HRF) [20], the harmonic balance method [37], and mode superposition [38]. For example, [PERSON] et al. [20] used the FEM combined with the higher-order HRF to analyze the vibrational response of a cracked cantilever beam with a closing crack, and they found that the appearance of super-harmonic resonances strongly depends upon the crack size and position. [PERSON] et al. [36] established the FEM of a cracked beam, and they compared the time response of the beam between the open crack and the breathing crack by numerical solution. Although the FEM of the cracked beam is by far the most commonly used method, this model has certain drawbacks which limit its application. For example, the FEM of a cracked beam with a breathing crack does not consider the states of partial crack closure. Besides this, the solving of the FEM of a beam with a breathing crack is relatively complex and time-consuming [25]. In this paper, a new stiffness model that considers the influence of partial crack closure is proposed to closely represent the stiffness variation of the cracked beam. Based on the proposed stiffness model, the equation of motion of a beam with a breathing crack subjected to a harmonic excitation is established. The multiple-scales method is applied to determine the resonance conditions. The relationships between the nonlinear behaviors and the system parameters are examined via both the numerical simulation and experimental study. The study shows that the frequent factors [39; 40; 41] may be used as a qualitative damage indicator for crack detection.