# Structural Damage Detection through Dual-Channel Pseudo-Supervised Learning
## 2 Methodology  
### The Equation of Motion  
The equation that describes the motion of a linear system with \(n\) degrees of freedom (DOFs) [38] is as follows:  
\[\mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\mathbf{u}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{F}(t) \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) denote \(n\times n\) matrices representing mass, damping, and stiffness. Meanwhile, \(\mathbf{u}(t)\), \(\dot{\mathbf{u}}(t)\), and \(\ddot{\mathbf{u}}(t)\) are \(n\times 1\) vectors representing displacement, velocity, and acceleration. Additionally, \(\mathbf{F}(t)\) is \(n\times 1\) a vector representing excitation.  
It is generally assumed that damage to a structure only results in a decrease in stiffness and does not cause a change in mass. Prior to damage, the stiffness matrix and flexibility matrix of the structure are denoted as \(\mathbf{K}_{u}\) and \(\mathbf{F}_{u}\), respectively. After damage, the stiffness matrix and flexibility matrix of the structure are denoted as \(\mathbf{K}_{d}\) and \(\mathbf{F}_{d}\), respectively. The subscripts \(d\) and \(u\) represent the states after damage and before damage, respectively. The change in the overall stiffness matrix of the structure, before and after damage, is denoted as \(\Delta\mathbf{K}\)[39].  
\[\Delta\mathbf{K}=\mathbf{K}_{u}-\mathbf{K}_{d} \tag{2}\]  
This can be further expressed as the linear superposition of elemental stiffness matrices.  
\[\Delta\mathbf{K}=\sum_{i=1}^{n}a_{i}K_{u}^{i} \tag{3}\]  
where \(K_{u}^{i}\) denotes the stiffness matrix of the \(i\)-th element in the structure, and \(a_{i}(0\leq a_{i}\leq 1)\) represents the damage coefficient.  
Structural damage detection and the acceleration response are closely related in the field of SHM. The acceleration response refers to the measurement of the acceleration generated by a structure under various loads or external forces. In the context of structural damage detection, response data are collected using acceleration sensors placed at critical points on the structure. The collected data are then analyzed to identify any abnormal patterns or deviations from the expected behavior of a healthy structure. This analysis can reveal valuable insights into the presence and location of structural damage. Different methods and algorithms, such as the modal analysis, frequency-based analysis, and time-frequency analysis, can be applied to the response data for damage detection and localization. By comparing the response data with baseline or reference data, engineers can identify changes in the structural dynamics that may indicate damages. Additionally, the amplitude and frequency characteristics of the acceleration response can provide information about the severity and type of damage, such as cracks, fractures, or deformations. Overall, the acceleration response data play a crucial role in the process of structural damage detection, enabling engineers to assess the structural integrity and make informed decisions regarding the maintenance, repair, or further investigation of the damaged structure.  
This study proposes a method for detecting structural damage by combining a dual-channel pseudo-supervised learning strategy (DCPS) with a Conformer model, using structural acceleration response data. The DCPS is applied to structural damage detection, enabling the model to acquire noise immunity instead of manual denoising.