# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 1 Introduction  
With the rapid development of modern science and technology, there has been an increasing number of large and complex engineering structures [1, 2]. When these structures become damaged, the consequences are catastrophic, leading to a significant loss of human lives and property [3, 4]. Therefore, it is necessary to adopt effective health-monitoring methods for such structures [5], and damage identification is a crucial aspect of structural health monitoring (SHM) [6, 7]. Reliable and efficient damage identification methods are especially required to achieve the safety and integrity of structures [8]. The most widely applied vibration theory in structural damage identification diagnoses damages by measuring the dynamic response and modal parameters of structures [9, 10]. As the basic characteristics of structures, modal parameters do not change with the excitation form [11], hence, the damage identification method based on modal parameters is reliable [12, 13]. [PERSON] et al. [14] analyzed the experimental and analytical modes of a cantilever beam using an artificial neural network based on the vibration theory to identify structural damages. [PERSON] et al. [15]. assessed structural damage by comparing the dynamic response parameters of the finite element model in damaged and undamaged states based on the experimental natural frequency and vibration mode of the structure and verified the model using the cantilever beam model. [PERSON] et al. [16]. identified the crack location and extension depth ofa cantilever beam by comparing the ratio variations in two adjacent natural frequencies. [PERSON] et al. [17] analyzed the variations in the structural frequency and mode shape of steel truss bridge structures under different damage distribution conditions and analyzed the reduction in precision after considering the damping ratio. [PERSON] et al. [18]. used the first-order feature perturbation method in updating the feature space to evaluate the potential structural damage and verified the stability and reliability of the recursive canonical correlation analysis. [PERSON] et al. [19]. developed an objective function of the natural frequency and mode shape of structures using the covariance matrix adaptive evolutionary optimization method. The method was applied to truss and frame structures, and its robustness was verified experimentally. [PERSON] et al. [20]. established the modal mode. In practical engineering applications, the damage identification method based on the dynamic response and modal parameters of the structure has some limitations. First, the number of structural modes is often numerous. Because of the effects of the structural scale, sensor distribution, and other factors, only a small amount of low-order modal information can be applied effectively, leading to incomplete modal information for damage identification. Second, owing to the influence of sensor measurement accuracy, environmental noise, and other factors, the damage identification method based on structural modal information is not sufficiently accurate and sensitive to the structural local damage. Hence, there is more serious damage identification precision with more structural modal data. It is feasible to add physical parameters, such as stiffness and mass, to obtain multiple structures with similar parameters and expand the data obtained experimentally. [PERSON] et al. [21] identified the shear damage of a four-story frame through numerical simulations and model experiments by adding a specific mass on the structure and determining its modal parameters. [PERSON] [22] analyzed the effects of an additional mass position and weight on the rotational mode and damage identification of glass fiber composite beams. [PERSON] et al. [23]. added controllable parameters, such as support, load, and temperature, to the original structure considering the mass, stiffness, and other physical parameters and observed an improvement in the damage identification accuracy.