# A Developed Jerk Sensor for Seismic Vibration Measurements: Modeling, Simulation and Experimental Verification

^{1}

^{2}

^{3}

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## Abstract

**:**

^{3}) is based on differentiating the time–acceleration signal. However, this technique is prone to errors especially in small amplitude and low frequency signals, and is deemed not suitable when online feedback is required. Here, we show that direct measurement of the jerk can be achieved using a metal cantilever and a gyroscope. In addition, we focus on the development of the jerk sensor for seismic vibrations. The adopted methodology optimized the dimensions of an austenitic stainless steel cantilever and enhanced the performance in terms of sensitivity and the jerk measurable range. We found, after several analytical and FE analyses, that an L-35 cantilever model with dimensions 35 × 20 × 0.5 (mm

^{3}) and a natural frequency of 139 (Hz) has a remarkable performance for seismic measurements. Our theoretical and experimental results show that the L-35 jerk sensor has a constant sensitivity value of 0.05 ((deg/s)/(G/s)) with ±2% error in the seismic frequency bandwidth of 0.1~40 (Hz) and for amplitudes in between 0.1 and 2 (G). Furthermore, the theoretical and experimental calibration curves show linear trends with a high correlation factor of 0.99 and 0.98, respectively. These findings demonstrate the enhanced sensitivity of the jerk sensor, which surpasses previously reported sensitivities in the literature.

## 1. Introduction

^{3})) for a bandwidth of 0~100 (Hz). Kubota et al. [22] developed a servo-type jerk sensor for an air-type anti-vibration system. This type of servo jerk sensor is mainly based on integral feedback control. They improved their jerk sensor’s phase and sensitivity characteristics by increasing the stiffness of the plate spring of the pendulum in the sensor. Manabe et al. [23,24] proposed a servo-type jerk sensor based on controlling the zero position of a pendulum through velocity feedback control (i.e., PI compensator), which auto corrects the position, enhances stability and reduces oscillation. Li et al. [25] proposed a direct method for measuring the jerk using a fiber optic jerk sensor (FOJS) based on a differentiating Mach–Zehnder interferometer. If the sensing probe detects a jerk signal, the fiber winding on the probe stretches and modulates the phase of the interferometer. Thus, the jerk is measured by measuring the absolute phase of the interference light. In 1999, Tamura et al. [26] proposed a direct method for sensing the jerk with a cantilever-based jerk sensor. They measured the output voltage of a vibratory gyroscope mounted on an aluminum cantilever to detect discontinuities in the response of buildings under earthquakes based on the continuous vibration system theory. They reported two sensor models named A and B. However, the model with the higher sensitivity has significantly large dimensions of 145 × 30 × 5 (mm

^{3}) and a low sensitivity of 0.01 ((deg/s)/(G/s)). However, in practical applications, lower dimensions and a higher sensitivity are needed. Therefore, more research is required to enhance the sensitivity of the cantilever-based jerk sensor proposed by Tamura [26].

## 2. Materials and Methods

#### 2.1. Sensor Structure

#### 2.2. Sensor Analytical Model

^{3}. The rate of change of force can be related to the jerk when mass is constant by differentiating Newton’s second law (${F}_{\left(t\right)}=m{a}_{\left(t\right)}$) with respect to time as described in Equation (1).

#### 2.3. Sensor Design

#### 2.4. Sensor FE Modeling and Simulation Work

#### 2.4.1. Material Properties

#### 2.4.2. Geometrical Model

^{3}) was constructed and attached to a fixed base of dimensions 5 × 20 × 10 (mm

^{3}), as shown in Figure 6. The printed circuit board (PCB) size for the suggested MPU9250 gyroscope mounted on the cantilever free tip was 25.5 × 15.4 × 1.5 (mm

^{3}). This size was considered when representing the affecting area of the PCB mass. An optimum mesh sized at 1800 three-dimensional cubic elements of length 1 (mm) was enough to have an absolute error of less than 1% compared to the analytical solution (Equation (17)).

#### 2.4.3. Boundary Conditions

#### 2.4.4. Simulation Details

#### 2.4.5. Post-Processing of Data

#### 2.5. Experimental Work

#### 2.5.1. Sensor Fabrication

^{3}), a base of dimensions 30 × 30 × 15 (mm

^{3}), a gyroscope and an accelerometer of dimensions 25.5 × 15.4 × 1.5 (mm

^{3}), as shown in Figure 7. All body parts were made of austenitic spring stainless steel 304. The cantilever strip was fixed to the base inside a built-in grove with a thickness of 0.6 (mm) and depth of 5 (mm) using 3 M5 bolts. The fixation method was designed to facilitate the trial testing of different cantilevers. The mass of the gyroscope was 3 (gm). The gyroscope was installed on the cantilever where the z-axis direction was out of the plane, as shown in the left photo of Figure 7.

#### 2.5.2. Sensor Setup

#### 2.5.3. Experimental Work Details

## 3. Results

#### 3.1. FE Results

#### 3.1.1. Modal Analysis

#### 3.1.2. Harmonic Analysis of the L-35 Cantilever

#### 3.1.3. Theoretical Sensitivity of the L-35 Jerk Sensor

#### 3.1.4. Theoretical Calibration Curve of the L-35 Jerk Sensor

^{3}) at a 19 (G) input acceleration (${a}_{i}$) and 40 (Hz) excitation frequency (${\omega}_{i}$). The theoretical jerk limit is based on the gyroscope’s reference voltage limit of ±1250 (mV) at the measuring range of ±250 (deg/s).

#### 3.2. Experimental Verification

#### 3.2.1. Experimental Harmonic Analysis

#### 3.2.2. Experimental Sensitivity Curves

#### 3.2.3. Experimental Calibration Curve

#### 3.3. Performance Comparison of the L-35 Jerk Sensor

## 4. Summary and Conclusions

^{3}) was selected and fabricated to work in the bandwidth of seismic measurements (0.1~40 (Hz)), and its natural frequency was experimentally found to be 111 (Hz). Analytical and FE analyses of the L-35 jerk sensor were performed to predict the theoretical sensitivity (${S}_{j}$) and calibration (${j}_{\left(t\right)}$) curves. The L-35 was tested on a high excitation computer-controlled shaker to obtain the experimental sensitivity and calibration curves. The jerk sensor’s theoretical and experimental calibration curves were found to be linear in the bandwidth of 0.1~40 (Hz) with a linearity r-squared value of 0.99 and 0.98, respectively. The experimental and theoretical jerk sensitivity (${S}_{j}$) was found to be in excellent agreement with a ±2% error and had a constant value of 0.05 ((deg/s)/(G/s)) for the excitation frequencies below 40 (Hz). The sensitivity of the enhanced jerk sensor has reached five times the sensitivity of the cantilever-based jerk sensor reported in [26], with a wider jerk measurable range of 47,000 (m/s

^{3}). The sensor can be utilized in other applications that require direct measurement of the force rate of change such as detection of discontinuities in signals during low cyclic fatigue testing.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic diagram showing the change in deflection angle along the length of a cantilever beam.

**Figure 3.**Schematic diagram showing dynamic oscillation of the cantilever beam due to input excitation.

**Figure 4.**Graphical representations of the input acceleration and the output deflection and angular velocity responses at the cantilever tip (

**a**) at 1 (G) and 1 (Hz); (

**b**) at 1 (G) and 10 (Hz); (

**c**) at 1 (G) and 40 (Hz); and (

**d**) at 1 (G) and 120 (Hz).

**Figure 8.**Photo of the high-excitation computer-controlled shaker for testing the L-35 jerk sensor and obtaining its calibration curve.

**Figure 9.**An illustration showing the extraction of the data samples from a recorded gyroscope signal.

**Figure 11.**The combined effect of different magnitudes and frequencies of input acceleration on (

**a**) the dynamic deflection with (

**b**) a zoomed view into 0~40 (Hz), and (

**c**) the angular velocity with (

**d**) a zoomed view into 0~40 (Hz) at the tip of the L-35 stainless steel cantilever with damping ratio of 0.02.

**Figure 12.**Bode diagrams as obtained from FEA for (

**a**) sensitivity of jerk sensor for acceleration $\left|{S}_{a}\right|$; (

**b**) $\left|{S}_{a}\right|$ phase angle; (

**c**) sensitivity of jerk sensor for jerk $\left|{S}_{j}\right|$; and (

**d**) $\left|{S}_{j}\right|$ phase angle.

**Figure 13.**Jerk sensor sensitivity over the frequency range of 0~300 (Hz) for (

**a**) input acceleration $\left|{S}_{a}\right|$ with (

**b**) a zoomed view into 0~120 (Hz) and (

**c**) input jerk $\left|{S}_{j}\right|$ with (

**d**) a zoomed view into 0~120 (Hz).

**Figure 14.**Theoretical calibration curve of the L-35 jerk sensor for input jerk at frequency range of 0.1~40 (Hz) and acceleration range of 0.1~2 (G) as predicted from the FE model.

**Figure 15.**FFT analysis of a sample of the (

**a**) accelerometer signal with its (

**b**) amplitude spectrum, and (

**c**) gyroscope signal with its (

**d**) amplitude spectrum; signals were recorded at 60 (Hz) and 100% torque.

**Figure 16.**Maximum FFT peaks of (

**a**) accelerometer and (

**b**) gyroscope; signals were recorded in the frequency range of 5~60 (Hz).

**Figure 17.**L-35 jerk sensor sensitivities for (

**a**) input acceleration with (

**b**) a zoomed view into 1~40 (Hz), and (

**c**) input jerk with (

**d**) a zoomed view into 1~40 (Hz).

**Figure 18.**Experimental calibration curve of the L-35 jerk sensor for input jerk at frequency range of 0.1~40 (Hz) and acceleration range of 0.1~1.2 (G).

**Table 1.**First mode natural frequency of 18 cantilever dimension sets as estimated from Equation (26) and modal FEA.

Model | Length $\left(\mathit{L}\right)$ (mm) | Width $\left(\mathit{w}\right)$ (mm) | Thickness $\left(\mathit{t}\right)$ (mm) | First Mode Natural Frequency $\left({\mathit{f}}_{\mathit{n}}\right)$ (Hz) | |
---|---|---|---|---|---|

Equation (26) | FEA | ||||

1 | 30 | 15 | 0.5 | 166.55 | 162.21 |

2 | 1 | 419.01 | 425.81 | ||

3 | 1.5 | 700.13 | 724.35 | ||

4 | 20 | 0.5 | 184.37 | 183.61 | |

5 | 1 | 453.30 | 474.7 | ||

6 | 1.5 | 746.52 | 798.0 | ||

7 | 35 | 15 | 0.5 | 127.27 | 122.82 |

8 | 1 | 316.34 | 321.95 | ||

9 | 1.5 | 524.49 | 548.05 | ||

10 * (L-35) | 20 | 0.5 | 140.23 | 138.95 | |

11 | 1 | 340.35 | 358.44 | ||

12 | 1.5 | 556.22 | 602.68 | ||

13 | 40 | 15 | 0.5 | 100.90 | 96.932 |

14 | 1 | 248.06 | 253.52 | ||

15 | 1.5 | 408.52 | 431.38 | ||

16 | 20 | 0.5 | 110.69 | 109.56 | |

17 | 1 | 265.61 | 281.76 | ||

18 | 1.5 | 431.27 | 473.34 |

Property | Value | Unit |
---|---|---|

Density | 7969 | (kg/m^{3}) |

Young’s Modulus | 195 | (GPa) |

Poisson’s Ratio | 0.27 | |

Bulk Modulus | 141.3 | (GPa) |

Shear Modulus | 76.772 | (GPa) |

Tensile Yield Strength | 252.1 | (MPa) |

Tensile Ultimate Strength | 565.1 | (MPa) |

Damping Ratio [35] | 0.02 | |

Constant Structural Damping Coefficient [35] | 0.04 | (kg/m^{3}) |

**Table 3.**Deflection $\left({U}_{tip}\right)$ and angular velocity $\left({\Omega}_{tip}\right)$ amplitudes at the tip of the L-35 cantilever as obtained from FE harmonic analysis at 1, 40 and 139 (Hz) under different input accelerations.

Input Acceleration(G) | 1 (Hz) | 40 (Hz) | 139 (Hz) | |||
---|---|---|---|---|---|---|

${U}_{tip}$ (mm) | ${\Omega}_{tip}$ (deg/s) | ${U}_{tip}$ (mm) | ${\Omega}_{tip}$ (deg/s) | ${U}_{tip}$ (mm) | ${\Omega}_{tip}$ (deg/s) | |

0.10 | 0.002 | 0.03 | 0.002 | 1.38 | 0.039 | 102.61 |

0.25 | 0.004 | 0.08 | 0.005 | 3.44 | 0.098 | 256.53 |

0.50 | 0.008 | 0.16 | 0.009 | 6.88 | 0.196 | 513.05 |

0.75 | 0.013 | 0.24 | 0.014 | 10.32 | 0.295 | 769.58 |

1.00 | 0.017 | 0.32 | 0.018 | 13.76 | 0.393 | 1026.11 |

1.25 | 0.021 | 0.39 | 0.023 | 17.21 | 0.491 | 1282.63 |

1.50 | 0.025 | 0.47 | 0.027 | 20.65 | 0.589 | 1539.6 |

1.75 | 0.029 | 0.55 | 0.032 | 24.09 | 0.687 | 1795.69 |

2.00 | 0.034 | 0.63 | 0.037 | 27.53 | 0.789 | 2052.21 |

Property | L-35 | A * | B * | JW-1 ** |
---|---|---|---|---|

Dimensions (mm) | 35 × 5 × 0.5 | 145 × 30 × 5 | 20 × 30 × 1.2 | N/A |

Material | Stainless steel 304 | Aluminum | Aluminum | N/A |

Natural frequency (Hz) | 111 | 90 | 160 | N/A |

Sensitivity ((deg/s)/(G/s)) | 0.053 | 0.01 | 0.004 | N/A |

Gyroscope sensitivity (mV/(deg/s)) | 5 | 25 | 25 | N/A |

Sensitivity (mV/(m/s^{3})) | 0.03 | 0.02 | 0.01 | 0.08 |

Bandwidth | 0.1–40 | 0.1–60 | 0.1–60 | 0.3–100 |

Measurable Range (m/s^{3}) | 47,000 | 14,000 | 35,000 | 10,000 |

Linearity (%) | ±1 | ±1 | ±1 | ±1 |

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## Share and Cite

**MDPI and ACS Style**

Geriesh, M.M.; Fath El-Bab, A.M.R.; Khair-Eldeen, W.; Mohamadien, H.A.; Hassan, M.A.
A Developed Jerk Sensor for Seismic Vibration Measurements: Modeling, Simulation and Experimental Verification. *Sensors* **2023**, *23*, 5730.
https://doi.org/10.3390/s23125730

**AMA Style**

Geriesh MM, Fath El-Bab AMR, Khair-Eldeen W, Mohamadien HA, Hassan MA.
A Developed Jerk Sensor for Seismic Vibration Measurements: Modeling, Simulation and Experimental Verification. *Sensors*. 2023; 23(12):5730.
https://doi.org/10.3390/s23125730

**Chicago/Turabian Style**

Geriesh, Mostafa M., Ahmed M. R. Fath El-Bab, Wael Khair-Eldeen, Hassan A. Mohamadien, and Mohsen A. Hassan.
2023. "A Developed Jerk Sensor for Seismic Vibration Measurements: Modeling, Simulation and Experimental Verification" *Sensors* 23, no. 12: 5730.
https://doi.org/10.3390/s23125730