# Optimal Sensor Placement in Reduced-Order Models Using Modal Constraint Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Formulations

**M**and

**K**denote the $N\times N$ mass and stiffness matrices, respectively, and

**u**and $\ddot{u}$ denote the $N\times 1$ generalized displacement and acceleration vectors, respectively. The mathematical model is formulated using the modal characteristics of the natural frequency ${\omega}_{i}\left(i=1,2,\dots ,N\right),$ and the corresponding normalized mode shape vector ${\Phi}_{i}\left(i=1,2,\dots ,N\right)$. ${\Phi}_{i}$ denotes the i-th column of the mode shape matrix.

**u**in terms of $r\times 1$ modal coordinate vector $y$ and $N\times r$ mode shape matrix $\Phi $ is written as

**y**is the $r\times 1$ modal displacement vector and $r$ indicates the number of target modes. The subscripts s and m denote the slave and master modes, respectively. ${\Phi}_{s}$ and ${\Phi}_{m}$ denote the s $\times r$ slave mode shape matrix and $\left(N-s\right)\times r$ master mode shape matrix, respectively, and $\left(N-s\right)$ represents the number of candidate sensor locations. Here,$\left(N-s\right)\ge r$.

#### 2.1. Modal Reduction–Effective Independence (MR–EI) Approach

**R**, the displacements at the master DOFs can be determined, where

**R**denotes the Boolean matrix to define the master DOFs.

**u**, of Equation (4a)

**,**and $\widehat{u}$ of Equation (5), respectively, is expressed as

#### 2.2. CDE–EI Approach

**I**denotes the $\mathrm{s}\times \mathrm{s}$ identity matrix.

**=**$\left[\begin{array}{cc}I& -{\Phi}_{s}{\Phi}_{m}^{+}\end{array}\right]$, and $b$ is the right-hand-side term in Equation (9), $b=\mathbf{0}$.

**V**and

**U**are eigenvectors of ${Q}^{T}Q$ or $Q{Q}^{T}$. By inserting the eigenvectors $\chi $ of ${Q}^{T}Q$ or $Q{Q}^{T}$ and $\xi =Q=I-{M}^{-1/2}{\left(A{M}^{-1/2}\right)}^{+}A$ into Equation (7), the EI method based on the CDE approach can be iteratively applied, and the objective function of Equation (8) can be obtained. The DOF with low contribution to the EID is relocated to the slave DOFs. The final sensor locations are obtained by iterating the same process as the EI method until they match the initial number of sensors.

#### 2.3. Modified MKE Method

#### 2.4. Modified MSE Method

**P**is moved to the slave DOFs. The remainder of the procedure is similar to that of the MMKE method by iteration.

#### 2.5. Modified MAC Method

## 3. Numerical Example

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A plane truss. The number without parentheses indicates the node and the number with parentheses the element.

**Figure 2.**OSPs obtained from the proposed methods: (

**a**) MR–EI, (

**b**) CDE–EI, (

**c**) MMKE27, (

**d**) MMKE11, (

**e**) MMSE, (

**f**) MMAC, (

**g**) Hao. The number without parentheses indicates the node and the number with parentheses the element.

Horizontal Node | Vertical Node | OSP Criterion | Objective Function | |
---|---|---|---|---|

MR-EI | 2, 4 | 3, 4, 9, 10, 12, 13 | Difference between unbiased and biased displacement data | Equation (8) |

CDE-EI | 3, 5, 13, 14 | 10, 12, 13, 14 | ||

* MMKE 27 | 2, 6, 13, 14 | 4, 6, 9, 12 | Modal kinetic energy | Equation (14) |

* MMKE 11 | 2, 6, 14, 15 | 4, 6, 9, 12 | ||

MMSE | 2, 4, 13 | 3, 6, 7, 9, 13 | Modal strain energy | Equation (14) |

MMAC | 4, 5, 8, 14 | 2, 5, 6, 7 | Matching degree of mode shape vector | Equation (19) |

Sun and Buyukozturk | 2, 7, 12, 14 | 4, 7, 10, 13 |

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Lee, E.-T.; Eun, H.-C.
Optimal Sensor Placement in Reduced-Order Models Using Modal Constraint Conditions. *Sensors* **2022**, *22*, 589.
https://doi.org/10.3390/s22020589

**AMA Style**

Lee E-T, Eun H-C.
Optimal Sensor Placement in Reduced-Order Models Using Modal Constraint Conditions. *Sensors*. 2022; 22(2):589.
https://doi.org/10.3390/s22020589

**Chicago/Turabian Style**

Lee, Eun-Taik, and Hee-Chang Eun.
2022. "Optimal Sensor Placement in Reduced-Order Models Using Modal Constraint Conditions" *Sensors* 22, no. 2: 589.
https://doi.org/10.3390/s22020589