# Damage Identification Method Using Additional Virtual Mass Based on Damage Sparsity
## 1 Introduction
However, it is challenging to design, install, and disassemble actual physical parameters in engineering practice. [PERSON] et al. [24]. developed a damage identification method using an additional virtual mass based on the virtual deformation method. The frequency-domain response of the structure was determined by applying excitation on the actual damaged structure, and the frequency-domain equation relating to the additional virtual mass was derived. The frequency-domain equations of different virtual structures were established by adding various virtual masses at different points of the original structures to expand the modal information. On the other hand, damage in structures is generally local with a sparse distribution [25]. The damage identification objective function translates into over-determined form when structural modal information is expanded by the additional virtual mass method, which will lead to non-sparse result because of noise. Thus, the sparse damage identification method can highlight the sparsity of optimization results, to improve accuracy. The deterministic sparse damage identification method introduces a sparse constraint term, usually the \(l_{p}\) norm of damage-factor, into the objective function for determining the sparse solution. [PERSON] et al. [26]. proposed a new Tikhonov iterative method for solving ill-conditioned equations of damage identification and developed a singular-value dichotomy program to determine regularization parameters. [PERSON] et al. [27]. modified a structural model and identified its damage using the \(l_{1}\) regularization method of sparse recovery theory based on the structural frequency and mode shape. [PERSON] et al. [28]. updated a structural model using the regularization method and performed damage identification of three-dimensional truss towers based on the sensitivity. [PERSON] et al. [29]. established two methods to determine the parameters of the \(l_{1}\) regularization method. One method ensures that the remaining norm and solution norm of the optimization problem are both small, and the other method makes the variance between the theoretical and measured responses close to each other. In this study, the additional virtual mass and the sparse damage identification methods were combined for higher identification precision and consistency with actual damage distribution. At first, these traditional sparse methods, such as the greedy iteration-orthogonal matching pursuit (OMP) algorithm with the damage-factor \(l_{0}\) norm as the sparse constraint, the Lasso regression model with the \(l_{1}\) norm as the constraint item, and the ridge regression model with the \(l_{2}\) norm as the constraint term, were compared and analyzed based on the additional virtual mass method. Moreover, aiming at the shortage of above traditional sparse methods that the Lasso regression and the ridge regression need to obtain regularization coefficient with complex process, and lack of integrity in the recognition process of the OMP method, the improved OMP (IOMP) method was developed. Finally, through numerical simulations and experimental verification, the advantages and disadvantages of each method were proved.