# Dynamic Parameter Identification of a Long-Span Arch Bridge Based on GNSS-RTK Combined with CEEMDAN-WP Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Stability Test of GNSS-RTK Receivers

^{−12}m

^{2}s to 10

^{−7}m

^{2}s. The natural frequency value of the Rainbow Bridge is far greater than 0.06 Hz, and thus the low frequency components less than 0.06 Hz in the measurement signal can be cognized as noise directly. As long as a reasonable threshold value is set, the above cognized noise can be dealt with using traditional denoising algorithms. However, high-frequency random noise, which presents a wide distribution in the frequency domain, is hard to remove. To address this problem, a combined denoising method, which addresses both low-frequency and high-frequency noise, is proposed in the next section.

## 3. The Principle of CEEMDAN-WP and RDT

#### 3.1. EEMD, CEEMD, and CEEMDAN Algorithms

- Add Gauss white noise $\omega \left(t\right)$ into the original signal $x\left(t\right)$ and thus produce a new signal $X\left(t\right)$.$$X\left(t\right)=x\left(t\right)+\omega \left(t\right)$$
- Using the EMD algorithm, $X\left(t\right)$ is decomposed into a number of IMF components and a residual component $r\left(t\right)$.$$X\left(t\right)={\displaystyle \sum _{i=1}^{N}im{f}_{i}\left(t\right)+r\left(t\right)}$$
- By repeating the above two steps, the IMFs $im{f}_{ij}\left(t\right)$ are obtained, where $i$ is the iteration number and $j$ is the mode.
- Compute the average of the IMF components to eliminate the effects of additional white noise.$$im{f}_{j}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}im{f}_{ij}\left(t\right)}$$

- Add positive and negative white noise ${\omega}^{\pm}\left(t\right)$ into the original signal $x\left(t\right)$, and generate two new signals ${X}^{\pm}\left(t\right)$.$$\{\begin{array}{l}{X}^{+}\left(t\right)=x\left(t\right)+{\omega}^{+}\left(t\right)\\ {X}^{-}\left(t\right)=x\left(t\right)+{\omega}^{-}\left(t\right)\end{array}$$
- Repeat the above step, and decompose the new signals using EMD.
- Derive two sets of IMF components for the new signals.
- Calculate decomposition results by averaging multiple components.$$im{f}_{j}\left(t\right)=\frac{1}{2N}{\displaystyle \sum _{i=1}^{2N}im{f}_{ij}\left(t\right)}$$

- Define ${E}_{j}(\cdot )$ as the operator which produces the $j$-th IMF which has been decomposed based on the EMD algorithm. The first mode $im{{f}^{\prime}}_{1}$ is derived by EMD from the signal $x\left(t\right)+{\epsilon}_{0}{\omega}_{i}\left(t\right)$.$$im{f}_{1}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}im{f}_{i1}\left(t\right)}$$
- Calculate the first residual signal.$${r}_{1}\left(t\right)=x\left(t\right)-im{{f}^{\prime}}_{1}\left(t\right)$$
- Decompose the signal ${r}_{1}\left(t\right)+{\epsilon}_{1}{E}_{1}\left({\omega}_{i}\left(t\right)\right)$ $\left(i=1,\cdots ,N\right)$ to derive the first mode, after which the second mode is defined.$$im{{f}^{\prime}}_{2}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{E}_{1}\left({r}_{1}\left(t\right)+{\epsilon}_{1}{E}_{1}\left({\omega}_{i}\left(t\right)\right)\right)}$$
- For $j=2,\cdots ,J$, calculate the $j$-th residual signal.$${r}_{j}\left(t\right)={r}_{j-1}\left(t\right)-im{{f}^{\prime}}_{j}\left(t\right)$$
- For $j=2,\cdots ,J$, decompose the signal ${r}_{j}\left(t\right)+{\epsilon}_{j}{E}_{j}\left({\omega}_{i}\left(t\right)\right)$ $\left(i=1,\cdots ,N\right)$ to derive the first mode, after which the $\left(j+1\right)$-th mode is defined.$$im{{f}^{\prime}}_{j+1}\left(t\right)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{E}_{1}\left({r}_{j}\left(t\right)+{\epsilon}_{j}{E}_{j}\left({\omega}_{i}\left(t\right)\right)\right)}$$
- Turn to step 4 for the next $j$.

#### 3.2. WP Method

#### 3.3. The CEEMDAN-WP Model

- CEEMDAN is employed for decomposition to obtain a series of IMFs.
- Because of the existence of background noise, some IMF components are noise dominated. Reconstruct signals after removing the components which are noise dominated.
- A three-level WP is used to decompose the signal obtained in Step 2.
- Determine the classic wavelet basis.
- Select proper thresholds and quantify the decomposed coefficients.
- Reconstruct the signal and export.

#### 3.4. The RDT Method

## 4. Performance Evaluation of the CEEMDAN-WP

## 5. Structural Dynamic Deformation Monitoring of Rainbow Bridge

#### 5.1. Bridge Description and Test Plan

#### 5.2. FEM of the Bridge

#### 5.3. Vibration Signal Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The amplitudes of $r\left(t\right)$ and ${r}_{1}\left(t\right)$ along with their PSD functions.

**Figure 5.**The intrinsic mode function (IMF) components of signal $r\left(t\right)$ and the corresponding PSD functions.

**Figure 8.**(

**a**) The arrangement of the GNSS-RTK reference station; (

**b**) the arrangement of GNSS-RTK rover stations; and (

**c**) the schematic locations of the GNSS-RTK rover stations.

**Figure 10.**(

**a**) The first order of the mode shape, (

**b**) the second order of the mode shape, (

**c**) the third order of the mode shape, (

**d**) the fourth order of the mode shape, (

**e**) the fifth order of the mode shape, and (

**f**) the sixth order of the mode shape.

CEEMDAN | WP | CEEMDAN-WP | |
---|---|---|---|

SNR | 9.7908 | 9.3499 | 10.2086 |

RMSE (cm) | 0.3187 | 0.3258 | 0.3121 |

Steel Tube and Wind Brace | Concrete Inside the Steel Tube | Crossbeam and Stringer | Tie Bar | Bridge Deck | Pier Column | |
---|---|---|---|---|---|---|

Elasticity modulus (Pa) | 2.1 × 10^{11} | 3.5 × 10^{10} | 3.0 × 10^{10} | 2.1 × 10^{11} | 2.85 × 10^{10} | 3.3 × 10^{10} |

Density (kg/m^{3}) | 7800 | 2600 | 2600 | 7800 | 2500 | 2600 |

Poisson coefficient | 0.3 | 0.1667 | 0.1667 | 0.3 | 0.1667 | 0.1667 |

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**MDPI and ACS Style**

Xiong, C.; Yu, L.; Niu, Y.
Dynamic Parameter Identification of a Long-Span Arch Bridge Based on GNSS-RTK Combined with CEEMDAN-WP Analysis. *Appl. Sci.* **2019**, *9*, 1301.
https://doi.org/10.3390/app9071301

**AMA Style**

Xiong C, Yu L, Niu Y.
Dynamic Parameter Identification of a Long-Span Arch Bridge Based on GNSS-RTK Combined with CEEMDAN-WP Analysis. *Applied Sciences*. 2019; 9(7):1301.
https://doi.org/10.3390/app9071301

**Chicago/Turabian Style**

Xiong, Chunbao, Lina Yu, and Yanbo Niu.
2019. "Dynamic Parameter Identification of a Long-Span Arch Bridge Based on GNSS-RTK Combined with CEEMDAN-WP Analysis" *Applied Sciences* 9, no. 7: 1301.
https://doi.org/10.3390/app9071301