## 2 Methods  
### Theory of the SER Approach  
Modal reduction and expansion approaches have been conventionally used in modal analysis for validation and correlation. The SEREP is an approach in which the displacement mode shapes of a structure are employed and a transfer matrix is found using least-square minimization. The method can be used to expand or reduce (the inverse of the matrix) data. In this study, we expanded this approach and used strain mode shapes to develop a transformation matrix for reducing and expanding strain.  
To extract the equations for strain mode shapes, we first focused on displacement mode shapes. Equation (1) presents the basic expression of modal analysis. This theory states that the response of a structure to excitations is a linear combination of mode shapes.  
\[\{X_{n}\}{=}\ [\phi_{n}]\{\text{P}\}, \tag{1}\]where \([\phi_{n}]\) and \(\{X_{n}\}\) represent the displacement mode shapes and displacement response, respectively, of the structure at the full set degrees of freedom (DOFs), and \(\{\mathrm{P}\}\) denotes the contribution of each mode to the system's response. A similar equation for strain response is expressed as follows:  
\[\{\varepsilon_{n}\}{=[V_{n}]\{\mathrm{P}\}}, \tag{2}\]  
where \([V_{n}]\) denotes the full-space strain mode shapes of the system and \(\{\varepsilon_{n}\}\) is the full-field strain response of the structure. The full set of DOFs can be expressed as a combination of active a and deleted d DOFs. Active degrees of freedom are those included in the limited set of measurements as follows:  
\[\{\varepsilon_{n}\}{=\left\{\begin{array}{c}\varepsilon_{a}\\ \varepsilon_{d}\end{array}\right\}}. \tag{3}\]  
Combining (2) and (3), we obtain the following expression:  
\[\left\{\begin{array}{c}\varepsilon_{a}\\ \varepsilon_{d}\end{array}\right\}{=}[V_{n}]\{\mathrm{P}\}=\left[\begin{array} []{c}V_{a}\\ V_{d}\end{array}\right]\{\mathrm{P}\}, \tag{4}\]  
where \(V_{a}\) and \(V_{d}\) are the strain mode shapes of the structure for the active and deleted DOFs. For the active DOFs, (4) can be expressed as follows:  
\[\{\varepsilon_{a}\}{=[V_{a}]\{\mathrm{P}\}}. \tag{5}\]  
There are more unknowns in (5) than there are equations. Therefore, (5) should be solved using the generalized inverse method. In the generalized inverse method, these equations are solved using the least squares approach. Premultiplying both sides of (5) by \({[V_{a}]}^{T}\) results in the following:  
\[{[V_{a}]}^{T}\{\varepsilon_{a}\}{=[V_{a}]}^{T}{[V_{a}]\{\mathrm{P}\}}. \tag{6}\]  
Premultiplying (6) by \({([V_{a}]}^{T}{[V_{a}]})^{-1}\), we obtain  
\[{([V_{a}]}^{T}{[V_{a}])}^{-1}{[V_{a}]}^{T}\{\varepsilon_{a}\}{=([V_{a}]}^{T}{[ V_{a}])}^{-1}{[V_{a}]}^{T}{[V_{a}]\{\mathrm{P}\}}. \tag{7}\]  
Equation (7) can be expressed as follows:  
\[\{\mathrm{P}\}={([V_{a}]}^{T}{[V_{a}])}^{-1}{[V_{a}]}^{T}\{\varepsilon_{a}\}, \tag{8}\]  
\[\{\mathrm{P}\}=[V_{a}]^{S}\{\varepsilon_{a}\}, \tag{9}\]  
where \(g\) refers to the generalized inverse of the matrix. To determine the full-space response, we substitute \(\{\mathrm{P}\}\) from (9) into (2) to obtain  
\[\{\varepsilon_{n}\}{=[V_{n}]}{[V_{a}]}^{S}\{\varepsilon_{a}\}. \tag{10}\]  
Using (10), the following transformation matrix for the SER approach is formed:  
\[[\mathrm{T}]=[V_{n}][V_{a}]^{S}. \tag{11}\]  
Therefore, (11) can be used to expand real-time operating strain data measured using strain gauges \(\{\mathrm{RTOS}_{a}\}\) in the expanded (full-space) real-time operating data \(\{\mathrm{RTOS}_{n}\}\).  
\[\{\mathrm{RTOS}_{n}\}{=[\mathrm{T}]\{\mathrm{RTOS}_{a}\}} \tag{12}\]In the proposed technique, the strain mode shapes of the structure are obtained using the structure's finite element model.