# Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurement–Model Link

#### 2.1.1. The Case Where the Displacement, Velocity, and Acceleration are Measured

#### 2.1.2. The Case Where the Strain is Measured

#### 2.2. Model Updating

#### 2.2.1. Eigenvalue Residual

#### 2.2.2. Mode Shape Residual

## 3. Numerical Example

^{3}. The structural damping ratio for the 1st and 2nd modes was 5%. The strain gauges were installed at positions $S1\left(2.5,-0.05\right)$, $S2\left(7.5,-0.3\right)$, and $S3\left(12.5,-0.05\right)$. Assuming that the strain gauges were installed on the tendon for the introduction of the pre-stress force in a concrete structure, this layout allows not only the section strain but also the pre-stress force to be measured [16,17]. Accelerometers were installed vertically at the top of the center of each element.

#### 3.1. Modal Identification

#### 3.2. Construction of Transform Matrices

#### 3.3. Model Updating

## 4. Numerical Application to Bridge

^{3}. The damping ratio for the 1st and 2nd modes was 5%. A total of 50 strain sensors were installed longitudinally along the section bottom centerline at a spacing of 1 m, as shown in Figure 5b,c. This sensor layout can be achieved by only one quasi-distributed fiber optic sensor [18] based on resonance frequency mapping.

#### 4.1. Without Uncertainties

#### 4.2. With Uncertainties

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Two-Dimensional Euler–Bernoulli Beam Element

## References

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**Figure 4.**Model updating process (

**a**) when the strain was measured and (

**b**) when the acceleration was measured.

**Figure 5.**Numerical application: (

**a**) shape of the bridge structure, (

**b**) layout of the strain sensors, (

**c**) segments of the structure, and (

**d**) first mode shape of the structure.

**Figure 6.**Modal identification: (

**a**) measured strains; (

**b**) stabilization diagram; and (

**c**) mode shape obtained by modal identification.

**Figure 7.**Model updating results: (

**a**) model updating process; (

**b**) comparison of elastic moduli of the structure and updated model; and (

**c**) error ratio of the structure-updated model elastic moduli.

**Figure 8.**Error ratio in model updating as a result of considering uncertainties in sensor position and direction and measurement noise: (

**a**) without measurement error; (

**b**) with standard deviation of $1\times {10}^{-6}\mathrm{m}/\mathrm{m}$ in measurement error; (

**c**) with standard deviation of $2\times {10}^{-6}\mathrm{m}/\mathrm{m}$ in measurement error.

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**MDPI and ACS Style**

Cho, K.; Park, Y.-H.; Cho, J.-R.
Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions. *Appl. Sci.* **2019**, *9*, 4309.
https://doi.org/10.3390/app9204309

**AMA Style**

Cho K, Park Y-H, Cho J-R.
Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions. *Applied Sciences*. 2019; 9(20):4309.
https://doi.org/10.3390/app9204309

**Chicago/Turabian Style**

Cho, Keunhee, Young-Hwan Park, and Jeong-Rae Cho.
2019. "Model Updating Using Measurements from Sensors Installed in Arbitrary Positions and Directions" *Applied Sciences* 9, no. 20: 4309.
https://doi.org/10.3390/app9204309