# Optimization of Sensor Placement for Modal Testing Using Machine Learning
## 2 A Review of Traditional Sensor Placement Techniques for Modal Testing
### Residual Kinetic Energy Method (RKE)  
The RKE method is a technique that provides information on the sensor location that exhibits the maximum response for each mode shape and may offer improved performance over EIM. It is commonly used by NASA to determine sensor placement for modal testing based on detailed FEA models [25]. The method ensures that the residual kinetic energy is minimized in all degrees of freedom and modes under consideration. When this is computed, DOFs with high residual kinetic energy indicate that additional refinement is needed in order to measure the corresponding degree of freedom in a given mode. After another sensor is added to cover that degree of freedom, the residual kinetic energy is recomputed. This process is repeated until the solution is suitably orthogonal [26]. The RKE method ensures all recorded modes are fully orthogonal and, therefore, independent from each other. The selection of the optimal sensor locations for modal testing using the RKE method is based on the partition of the modal vectors \(\mathbf{\Phi}\)  
\[\mathbf{\Phi}=\left\{ \begin{aligned} &\mathbf{\Phi}_{s}\\ &\mathbf{\Phi}_{o}\end{aligned} \right\} \tag{12}\]  
into \(\mathbf{\Phi}_{s}\), which corresponds to \"sensed\" modal vectors (contains the DOF where sensors will be placed), and \(\mathbf{\Phi}_{o}\), which corresponds to the \"omitted\" partition of the eigenvector (no sensor will be placed at these DOFs). The stiffness and mass matrix can also be partitioned accordingly, and Equation (1) becomes \[K\mathbf{\Phi}-\mathbf{M}\mathbf{\Phi}\lambda= \begin{bmatrix}\mathbf{K}_{ss}&\mathbf{K}_{so}\\ \mathbf{K}_{os}&\mathbf{K}_{oo}\end{bmatrix}\begin{Bmatrix}\mathbf{\Phi}_{s}\\ \mathbf{\Phi}_{o}\end{Bmatrix} - \begin{bmatrix}\mathbf{M}_{ss}&\mathbf{M}_{so}\\ \mathbf{M}_{os}&\mathbf{M}_{oo}\end{bmatrix}\begin{Bmatrix}\mathbf{\Phi}_{s}\\ \mathbf{\Phi}_{o}\end{Bmatrix} \lambda \tag{13}\]  
A static Guyan reduction transformation is used to obtain an approximation of the omitted modes \(\mathbf{\Psi}_{o}\):  
\[\mathbf{\Psi}_{o}=-\mathbf{K}_{oo}^{-1}\mathbf{K}_{os}\mathbf{\Phi}_{s} \tag{14}\]  
The mass matrix corresponding to the selected sensor location is \(\widetilde{\mathbf{M}}_{ss}\):  
\[\widetilde{\mathbf{M}}_{ss}= \begin{Bmatrix}\mathbf{I}\\ -\mathbf{K}_{oo}^{-1}\mathbf{K}_{os}\end{Bmatrix} ^{\mathbf{T}} \begin{bmatrix}\mathbf{M}_{ss}& \mathbf{M}_{so}\\ \mathbf{M}_{os}&\mathbf{M}_{oo}\end{bmatrix}\begin{Bmatrix}\mathbf{I}\\ -\mathbf{K}_{oo}^{-1}\mathbf{K}_{os}\end{Bmatrix}  \tag{15}\]  
The orthogonality of the reduced modes identified using the selected points \(\mathbf{\Phi}_{s}\) can be tested by parameter \(\mathbf{OR}_{s}\), which differ from the identity matrix because a reduced set of DOF is used to represent the modes:  
\[\mathbf{OR}_{s}=\mathbf{\Phi}_{s}^{T}\widetilde{\mathbf{M}}_{ss}\mathbf{\Phi}_{s}\ \
eq\mathbf{I} \tag{16}\]  
Industry and government standards require \(\left|OR_{s,\ ij}\right|\leq 10\%\) for \(i\
eq j\). The residual error can be defined as follows:  
\[\mathbf{R}= \begin{Bmatrix}\mathbf{\Phi}_{s}\\ \mathbf{\Phi}_{o}\end{Bmatrix} - \begin{Bmatrix}\mathbf{\Phi}_{s}\\ \mathbf{\Psi}_{o}\end{Bmatrix} = \begin{Bmatrix}0\\ \mathbf{\Phi}_{o}-\mathbf{\Psi}_{o}\end{Bmatrix} = \begin{Bmatrix}0\\ \mathbf{\Phi}_{o}+\mathbf{K}_{oo}^{-1}\mathbf{K}_{os}\mathbf{\Phi}_{s}\end{Bmatrix}  \tag{17}\]  
The error matrix is, therefore, a subtraction of the \"omitted\" DOF modes determined analytically (through finite elements or another analytical method) and the estimated \"omitted\" modes. As the modal kinetic energy for the complete system is defined by Equation (3), a residual kinetic energy \(\mathbf{R}\mathbf{K}\mathbf{E}\) matrix can be defined using the residual error \(\mathbf{R}\):  
\[\mathbf{R}\mathbf{K}\mathbf{E}=(\mathbf{M}\mathbf{R})\otimes\mathbf{R} \tag{18}\]  
Each column of the \(\mathbf{R}\mathbf{K}\mathbf{E}\) matrix represents the contribution of each degree of freedom to the residual kinetic energy of a specific mode.