# Monitoring of the Static and Dynamic Displacements of Railway Bridges with the Use of Inertial Sensors

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## Abstract

**:**

## 1. Introduction

- validate design conceptions,
- evaluate condition state,
- assess behavior under high-speed train loadings,
- carry out damage detection based on vibration analysis.

## 2. Description of Monitoring System

#### 2.1. Hardware

- uniaxial gravity inclinometer with measuring range of ±1°;
- triaxial piezoelectric accelerometer with measuring range of ±5g;
- MEMS-type triaxial accelerometer with measuring range of ±3g;
- resistance strain gauge;
- temperature sensor with a measuring range from −55 °C to +125 °C.

#### 2.2. Algorithm and Software

- (1)
- 5 s time registration of all signals before the train enters the bridge;
- (2)
- force vibration registration when the train is on the bridge;
- (3)
- free vibration registration after train leaves the bridge.

#### 2.2.1. Static Displacement Determination

_{i}of the inclinometer arrangement and the coordinates of the supports (x

_{s1}, z

_{s1}) and (x

_{s2}, z

_{s2}). For the sake of simplification, we assume zero coordinates of the first support: x

_{s1}= 0, z

_{s1}= 0, and the zero coordinate of the second support z

_{s2}= 0. To determine the deflection line, we use indications of α

_{i}(t) with i of that inclinometer in the time function.

_{j}(x) spline curves, which represent the analyzed cross-section along the structure [34,42]:

_{3,j}, d

_{2,j}, d

_{1,j}, d

_{0,j}are the expansion coefficients of the displacement curve function.

_{j}(x), we assume support points, inclinometer installation points, and curve connection points. Due to unfavorable properties of spline curves to create an excessive number of inflection points, in the case of the measurement data of angles distorted by measurement errors, we try to minimize the number of spline curves.

_{1}(x), D

_{2}(x), and D

_{k}(x) (k = 3) were used. The points of the location of inclinometers i = 2 and 3 were taken as the points of the curve connection.

- points of inclinometers’ location (coordinates (x
_{i},z_{i})) with D_{j}(x) curve:

- 2.
- points of connection of curves D
_{j}(x) and D_{j+}_{1}(x):

- 3.
- support points (coordinates (x
_{s}_{1}, z_{s}_{1}) or (x_{s}_{2}, z_{s}_{2})) belonging to curve D_{1}(x) or D_{k}(x):

- tangent to the angle indicated by the inclinometer, according to Equation (2);
- smooth curves (first derivative), according to Equation (3);
- curves with continuous curvature (second derivation), according to Equation (4);
- curve continuity, according to Equation (5);
- a known (zero) coordinate of support z
_{s}, according to Equation (6); - angle on the outer support equal to the indications of the nearest inclinometer, according to Equation (7).

_{s}is calculated as a point of the D

_{2}(x

_{d}) spline curve:

#### 2.2.2. Dynamic Displacement Determination

_{sc}(t) of dynamic displacement d

_{d}(t). This part of the signal processing is shown at the central path of the flow chart (Figure 1).

_{i}(t) is the inclinometer readings, α

_{i LP}(t) is the inclinometer readings after low-pass filtration, and F

_{I CLP}and n

_{IL}are the cut frequency and filter order, respectively.

_{i ZLP}(t)—inclinometer readings after low-pass filtration and adjustment to zero signal.

_{sc}(t) is calculated as

_{s}are scale coefficients of the quasi-static component.

_{dc}(t) of dynamic displacement d

_{d}(t), the accelerometer is located at the point of the displacement’s examination (Figure 2—point (x

_{d},z

_{d})). The forced and free vibration parts of the signal from the accelerometer are used for calculations (phase of the train detection at Figure 1). This part of the signal processing is shown in the right path of the flow chart (Figure 1).

_{i}(t) is the accelerometer readings, a

_{i HLP}(t) is the accelerometer readings after high- and low-pass filtration, and F

_{A CHP}, F

_{A CLP}, and n

_{AH}, n

_{AL}are high- and low-pass cut frequencies, and filter orders, respectively.

_{dI}(t)—result of double integration, t

_{1}—time of the beginning of the forced vibration, t

_{2}—time of the end of the free vibration.

_{dt}(t)—result of removing the linear trend.

_{d}is applied:

_{d}(t) caused by the train passage on the bridge is calculated as the sum of the quasi-static d

_{sc}(t) and the dynamic component d

_{dc}(t):

## 3. The First Tests

#### 3.1. The Tested Bridge

#### 3.2. Algorithm Adaptation to the Tested Bridge

- a single 3° curve (designation 40),
- a single 3° curve with an angle of rotation at the support equal to that of inclinometer I1 (designation 41),
- two spline curves connected together at the point of location of inclinometer I2 with an angle of rotation at a support equal to that of inclinometer I1 (designation 42),
- three spline curves combined at the points of the locations of inclinometers I1 and I2 with an angle of rotation at a support equal to that of inclinometer I1 (designation 43).

_{d min}(negative displacement and extreme value were marked “min”) and extreme upward (positive displacement and extreme value were marked “max”) from the respective deflection values measured with the reference method (inductive sensor) d

_{r min}and d

_{r max}:

_{I CLP}cut-off frequency of signals from inclinometers and the high-pass filter F

_{A CHP}cut-off frequency of signals from the accelerometer, with other parameters according to Table 1.

## 4. Results of the Bridge Monitoring

#### 4.1. Static Displacement Measurement Results

- tachymetric measurement before arrival of the train,
- 5 s inclinometer readings before the train reaches the bridge,
- calculation of displacement for both methods and comparison of results.

- uniform change in the temperature of the bridge (steel structure and concrete plate) generates mainly longitudinal deformation;
- temperature change of steel structure (excluding concrete plate) causes bending mode with 6.4 mm for 10 °C at the middle of the span length;
- temperature change of lateral wall of steel girder creates torsional deformation of the bridge.

#### 4.2. Dynamic Displacement Measurement Results—Single Train Passages

- Multiple-unit train: ∆
_{min}= 0.13 mm, ∆_{min}/d_{r min}= 1.3%, ∆_{max}= 0.36 mm, ∆_{max}/d_{r max}= 4.8%; - Separate locomotive and cars: ∆
_{min}= 0.36 mm, ∆_{min}/d_{r min}= 2.2%, ∆_{max}= 0.22 mm, ∆_{max}/d_{r max}= 3.3%.

_{max}, s

_{min}to the extreme minimum and maximum values—was determined from comparative studies:

_{min}or s

_{max}).

#### 4.3. Dynamic Displacement Measurement Results—Continuous Monitoring

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A flow chart of the static and dynamic displacement monitoring: The left path of the static displacement determination, the central path of the quasi static component, and the right path of the dynamic component of dynamic displacement determination.

**Figure 2.**Diagram of the structure deflection line outlining: Upper—the arrangement of inclinometers (

**1**) along the monitored span; (

**2**) lower—the spline curve outlined, where: (

**3**) Support point, (

**4**) point of the inclinometer’s location, (

**5**) point of the displacement examination; (x

_{s1},z

_{s1}) and (x

_{i},z

_{i}) respectively the coordinates of the span support points and inclinometers; a

_{i}, inclinometer readings; D

_{k}(x), spline curve.

**Figure 4.**Diagram of inclinometer locations (I1, I2, and I3) and accelerometer location (A1) for permanent installation (one year monitoring), inclinometers (I4, I5, and I6) and accelerometer (A2) for short-term installation (one day test); prisms (the measuring: P0- P6, and the reference system: P10, P11, P12 are located outside the drawing); TS—Total Station location.

**Figure 5.**View of inclinometers and accelerometer location: (

**a**) Inclinometer I1 location; (

**b**) accelerometer A1, and inclinometer I2 location.

**Figure 7.**Analysis of the accuracy of the structure quasi-static deflection mapping using spline curves in relation to the deflection curve determined with the use of numerical model load data (calc); 40—deflection curve determined by a single 3° curve, 41—line determined by a single 3° curve with an angle of rotation at support equal to that of inclinometer I1, 42—line determined by 2 spline curves with an angle of rotation at support equal to that of inclinometer I1, 43—line determined by 3 spline curves with an angle of rotation at support equal to that of inclinometer I1.

**Figure 8.**Analysis of the sensitivity to measurement errors of the structure quasi-static deflection mapping using spline curves in relation to the deflection curve determined with the use of numerical model load data (calc); 40—deflection curve determined by a single 3° curve, 41—line determined by a single 3° curve with an angle of rotation at support equal to that of inclinometer I1, 42—line determined by 2 spline curves with an angle of rotation at support equal to that of inclinometer I1, 43—line determined by 3 spline curves with an angle of rotation at support equal to that of inclinometer I1.

**Figure 9.**The distribution of average deviation at different cut-off frequencies of the F

_{I CLP}low-pass filter signal from inclinometers and the F

_{A CHP}high-pass filter signal from the accelerometer.

**Figure 10.**An example of an analysis of a one-day static displacement at ¼ span length point: (

**a**) Average temperature measurements versus time; (

**b**) indirect measurements of displacement with the use of inclinometer (green points) and total station measurements of displacement (red line with sharps) versus time.

**Figure 11.**An example of an analysis of a one-year static displacement at ¼ span length point versus temperature; indirect measurements of displacement with the use of inclinometer (green points) with linear fitting (green dashed line) and total station measurements of displacement (red line with sharps).

**Figure 12.**An example of an analysis of a multiple-unit train passage with a speed of about 190 km/h: (

**a**) Signals from inclinometers; (

**b**) signal from an accelerometer; (

**c**) determined quasi-static displacement component (from inclinometers) d

_{sc}, dynamic displacement component (from an accelerometer) d

_{dc}, and total dynamic displacement d

_{d}; (

**d**) comparison of total dynamic displacement d

_{d}with reference measurement d

_{r}(zoom of max and min values).

**Figure 13.**An example of an analysis of a separate locomotive and passenger cars passage with a speed of about 138 km/h: (

**a**) Signals from inclinometers; (

**b**) signal from an accelerometer; (

**c**) determined quasi-static displacement component (from inclinometers) d

_{sc}, dynamic displacement component (from an accelerometer) d

_{dc,}and total dynamic displacement d

_{d}; (

**d**) comparison of total dynamic displacement d

_{d}with reference measurement d

_{r}(zoom of max and min values).

**Figure 14.**Analysis of the distribution of the measurement result deviation of extreme values in relation to reference measurements for multiple-unit trains (ED250) passages: (

**a**) Depending on train speed and (

**b**) air temperature.

**Figure 15.**Analysis of the distribution of the measurement result deviation of the extreme values from the reference measurements for separate locomotive (EP09) and cars passages: (

**a**) Depending on train speed and (

**b**) air temperature.

**Figure 16.**An example of displacement monitoring under the load of multiple-unit trains ED250 for one month (December 2017): Extreme displacement versus speed; red dot—minimum, blue dot—maximum; three red and blue lines correspond to the numerically determined extreme deflections from an empty train (dotted line), the state of normal use (dashed line), and the overloaded state foreseen by the manufacturer (solid line).

Type of the Signal Processing | Inclinometers | Accelerometer |
---|---|---|

High-pass filtration | --- | F_{A CHP} = 1.67 Hz n_{AH} = 6 |

Low-pass filtration | F_{I CLP} = 0.73 Hz n_{IL} = 4 | F_{A CLP} = 30 Hz n_{AL} = 6 |

Integration | c_{s} = 1.01 | c_{d} = 1.30 |

**Table 2.**Root-mean-square deviation for all trains, multiple-unit trains, and separate locomotive and cars passages.

Train | Root-Mean-Square Deviation | |||
---|---|---|---|---|

s_{min} | s_{max} | s_{min}/d_{r min} | s_{max}/d_{r max} | |

[mm] | [mm] | [%] | [%] | |

All trains | 0.64 | 0.53 | 4.8% | 8.2% |

Multiple-unit trains ED250 | 0.41 | 0.54 | 3.8% | 7.5% |

Separate locomotive EP09 and cars | 0.84 | 0.36 | 4.9% | 6.0% |

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**MDPI and ACS Style**

Olaszek, P.; Wyczałek, I.; Sala, D.; Kokot, M.; Świercz, A.
Monitoring of the Static and Dynamic Displacements of Railway Bridges with the Use of Inertial Sensors. *Sensors* **2020**, *20*, 2767.
https://doi.org/10.3390/s20102767

**AMA Style**

Olaszek P, Wyczałek I, Sala D, Kokot M, Świercz A.
Monitoring of the Static and Dynamic Displacements of Railway Bridges with the Use of Inertial Sensors. *Sensors*. 2020; 20(10):2767.
https://doi.org/10.3390/s20102767

**Chicago/Turabian Style**

Olaszek, Piotr, Ireneusz Wyczałek, Damian Sala, Marek Kokot, and Andrzej Świercz.
2020. "Monitoring of the Static and Dynamic Displacements of Railway Bridges with the Use of Inertial Sensors" *Sensors* 20, no. 10: 2767.
https://doi.org/10.3390/s20102767