# Application of Rotation Rate Sensors in Modal and Vibration Analyses of Reinforced Concrete Beams

^{*}

## Abstract

**:**

## 1. Introduction

^{T}, where superscript T denotes the transversal character of the first natural mode of this beam. Calculating the first spatial derivative of the transversal mode shape $r={w}^{\prime}$ with respect to x (slope), one obtains so called first rotational mode shape (1

^{R}), representing the rotations of the beam axis along the first mode. Analogously, one can obtain the so-called “curvature” mode (1

^{κ}), which describes the changes of the curvature along the first mode. The curvature of the beam can be obtained from its transversal displacements, w = w(x,t), by the familiar formula of the structural mechanics, as follows:

^{κ}* mode). The smaller the slope, r, the closer the approximate curvature, ${\kappa}^{*}$, to the actual curvature, $\kappa $. In fact, for typical deflections of civil engineering structures, the curvature can be successfully obtained using the ${\kappa}^{*}$ approximation.

- direct extraction of the rotational modes (spatial derivatives of translational modes),
- direct slope measurements (rotational deflection shape) during vibration measurements with an inertial exciter,
- average strain control during vibrations of cracked reinforced concrete rods.

## 2. Materials and Methods

#### 2.1. General Description of the Analysed Beams

#### 2.2. Modal Parameter Extraction Using Impact Hammer

#### 2.3. Inertial Vibration Exciter Tests

#### 2.4. Strain Measurements Using Rotation Rate Sensors

#### 2.5. Comparing the Modes

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Simply supported Euler–Bernoulli beam and its (1

^{T}) first transversal natural mode, (1

^{R}) first rotational natural mode (slope of the first transversal mode), (1

^{κ}) first curvature natural mode, and (1

^{κ}*) approximation of the first curvature natural mode as a second derivative of the first transversal mode.

**Figure 3.**A sketch showing the infliction of damage to the beams using an Instron actuator (photograph on the right). Dimensions in mm.

**Figure 5.**Sensors configuration for (

**a**) beam UHPC1_L and (

**b**) beam UHPC2_L. Dashed lines show the theoretical mode shapes computed for the Euler–Bernoulli beam.

**Figure 6.**The exemplary frequency response functions (FRFs) of the analyzed beam UHPC2_L in configuration 1.

**Figure 8.**Comparison of the first (

**a**) translational and (

**b**) rotational mode of (1) beam UHPC1_L and (2) beam UHPC2_L. Results for the intact beams.

**Figure 9.**Comparison of the second (

**a**) translational and (

**b**) rotational modes of (1) beam UHPC1_L and (2) beam UHPC2_L. Results for the intact beams.

**Figure 10.**Sketch of cracks in (

**a**) UHPC1_L after the test procedure and (

**b**) UHPC2_L after the test procedure.

**Figure 11.**Comparison of the first (

**a**) translational and (

**b**) rotational modes of the intact and damaged (1) beam UHPC1_L and (2) beam UHPC2_L.

**Figure 12.**Amplitudes of (

**a**) translational acceleration and (

**b**) rotation rate of (1) beam UHPC1_L and (2) beam UHPC2_L during the inertial vibration exciter tests. Results for the intact beams.

**Figure 13.**Result for the intact UHPC2_L beam induced by the inertial vibration exciter with 27.26 Hz. (

**a**) Rotation rate at axis three, (

**b**) rotation rate at the axis, (

**c**) difference in rotation rate between axis three and axis four, (

**d**) average strain rate between axis three and axis four, (

**e**) average strains between axis three and axis four, and (

**f**) short time window of average strains between axis three and axis four.

Beam Symbol | Fibre Type | Density (kg/m^{3}) | Compressive Strength (MPa) | Tensile Strength in Flexure (MPa) |
---|---|---|---|---|

UHPC1_L | Glass | 2 301 | 122 | 8.60 |

UHPC2_L | Steel | 2 625 | 238 | 10.13 |

Sensor Name | Type | Measurement Range | Bandwidth | Resolution |
---|---|---|---|---|

Horizon HZ1-100-100 | MEMS | ±100 deg/s | 60 Hz | 0.0005°/s |

Gladiator G150z | MEMS | ±100 deg/s | 200 Hz | 0.004°/s |

Axis | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|

Sensor Type | UHPC1_L | G | G | G | H | H | H | G | G | G |

UHPC2_L | H | G | G | G | H | G | G | G | H |

**Table 4.**Modal parameters for the first three modes (least squares complex exponential (LSCE) method) in the intact state of the beams.

Mode | Beam UHPC1_L | Beam UHPC2_L | ||
---|---|---|---|---|

f_{n} [Hz] | ξ [%] | f_{n} [Hz] | ξ [%] | |

1 | 23.51 | 0.555 | 27.06 | 0.345 |

2 | 64.52 | 0.511 | 74.18 | 0.504 |

3 | 127.33 | 0.577 | 140.66 | 0.818 |

Direct Hammer | Direct FEM | Derivative Hammer | Derivative FEM | |
---|---|---|---|---|

Direct Hammer | 1.000 | 0.999 | 0.998 | 0.997 |

Direct FEM | 0.999 | 1.000 | 0.999 | 0.999 |

Derivative Hammer | 0.998 | 0.999 | 1.000 | 1.000 |

Derivative FEM | 0.997 | 0.999 | 1.000 | 1.000 |

Direct Hammer | Direct FEM | Derivative Hammer | Derivative FEM | |
---|---|---|---|---|

Direct Hammer | 1.000 | 0.996 | 0.995 | 0.994 |

Direct FEM | 0.996 | 1.000 | 0.999 | 0.999 |

Derivative Hammer | 0.995 | 0.999 | 1.000 | 1.000 |

Derivative FEM | 0.994 | 0.999 | 1.000 | 1.000 |

Direct Hammer | Direct FEM | Derivative Hammer | Derivative FEM | |
---|---|---|---|---|

Direct Hammer | 1.000 | 0.991 | 0.972 | 0.986 |

Direct FEM | 0.991 | 1.000 | 0.975 | 0.986 |

Derivative Hammer | 0.972 | 0.975 | 1.000 | 0.995 |

Derivative FEM | 0.986 | 0.986 | 0.995 | 1.000 |

Direct Hammer | Direct FEM | Derivative Hammer | Derivative FEM | |
---|---|---|---|---|

Direct Hammer | 1.000 | 0.944 | 0.833 | 0.898 |

Direct FEM | 0.944 | 1.000 | 0.947 | 0.986 |

Derivative Hammer | 0.833 | 0.947 | 1.000 | 0.986 |

Derivative FEM | 0.898 | 0.986 | 0.986 | 1.000 |

**Table 9.**Modal parameters for the first three modes (LSCE method) in the damaged state of the beams.

Mode | Beam UHPC1_L | Beam UHPC2_L | ||
---|---|---|---|---|

f_{n} [Hz] | ξ [%] | f_{n} [Hz] | ξ [%] | |

1 | 19.40 | 1.78 | 23.56 | 0.883 |

2 | 55.44 | 1.41 | 65.11 | 1.318 |

3 | 111.08 | 1.71 | 131.45 | 0.793 |

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**MDPI and ACS Style**

Bońkowski, P.A.; Bobra, P.; Zembaty, Z.; Jędraszak, B. Application of Rotation Rate Sensors in Modal and Vibration Analyses of Reinforced Concrete Beams. *Sensors* **2020**, *20*, 4711.
https://doi.org/10.3390/s20174711

**AMA Style**

Bońkowski PA, Bobra P, Zembaty Z, Jędraszak B. Application of Rotation Rate Sensors in Modal and Vibration Analyses of Reinforced Concrete Beams. *Sensors*. 2020; 20(17):4711.
https://doi.org/10.3390/s20174711

**Chicago/Turabian Style**

Bońkowski, Piotr Adam, Piotr Bobra, Zbigniew Zembaty, and Bronisław Jędraszak. 2020. "Application of Rotation Rate Sensors in Modal and Vibration Analyses of Reinforced Concrete Beams" *Sensors* 20, no. 17: 4711.
https://doi.org/10.3390/s20174711