# Quantification of Bolt Tension by Surface Acoustic Waves: An Experimentally Verified Simulation Study

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of Ultrasonic Waves

_{L}) and shear velocity (C

_{s}) are related to the material properties which can be expressed in Equations (1) and (2). The approximation of the Rayleigh wave velocity (C

_{R}), which depends on the shear wave speed and the material properties, can be expressed in Equation (3) [7]:

## 3. Simulation Work

#### 3.1. Initial Simulation Studies

^{3}[10]. For more consistent meshing control, the top surface was sliced to 3 mm in depth, and then the two slices were consolidated with a bonded contact to be as one part [11]. The incorporation of slices not only allows the localization of a dense mesh for the top surface and less dense mesh along with the remaining thickness of the plate, but it also allows for reducing the computational time to solve the finite element (FE) model since the number of nodes is significantly decreased. Initially, a single nodal point on the upper left surface was selected to represent a single transducer to excite the wave. However, the authors observed that the generated wave is circular which leads to undesirable and non-realistic reflections from the plate sides. These reflections interacted with the incident wave, making the results more complicated to analyze. For this reason, twelve evenly distributed point sources near the upper left end of the plate were selected to excite SAW where each point source represents a single transducer. The generated wave is a straight line when it leaves the source and becomes a curved line as it travels far from the source, similar to our experimental observations. However, large plates necessitate additional point sources to excite a particular wave mode and reduce the noise level [12]. In addition to twelve transmitting point sources, evenly spaced three receivers were introduced to the model. These are nodal points that are located at the top surface of the plate. Both the left and right end of the plate had a fixed boundary condition. Figure 1 illustrates the geometrical details and boundary conditions of the initial verification studies.

#### 3.2. Preload Analysis

_{i}is the maximum allowable preload (proof load), A

_{t}is tensile stress area which is a function of thread pitch, S

_{y}is yield strength of the bolt, and d is the nominal bolt diameter. In this study, a standard medium carbon steel bolt and nut with 10 mm nominal diameter were imported from the commercially available CAD software library, Inventor Autodesk 2018. Based on the bolt material and nominal diameter, the value of S

_{y}and A

_{t}are 640 N/mm

^{2}[15] and 57.1 mm

^{2}respectively. After substituting all values into Equation (5), the maximum allowable preload that bolt can withstand is obtained as 23.3 kN, and it is rounded up to 24 kN. By substituting this value into Equation (4), the maximum allowable torque on bolt is 43 N.m. For comparison purposes, six values of preload were utilized in this study. The preload values varied between 4 and 24 kN in the steps of 4 kN.

_{f }, is assumed to be 0.2 for all frictional contact type [16]. Table 1 shows the details of the contacts used in this study.

#### 3.3. Transient Simulations

_{R}, that mainly depends on the mechanical properties of the material, and it can be estimated using Equation (3). The second parameter is the wavelength (λ) which is equal to the ratio of the ultrasonic wave speed (C

_{R}) to the frequency (ƒ) in a perfectly elastic material $\mathsf{\lambda}={C}_{R}/f$. By utilizing the material properties for steel and Equation (3), the Rayleigh wave speed (C

_{R}) was found to be 2917.58 m/s. The corresponding wavelength for this wave speed and the central frequency of interest (100 kHz) was obtained as 29.1 mm, which was rounded to 29 mm.

_{max}is the maximum frequency, l

_{e}is the element size, and λ

_{min}is the shortest wavelength. In the case of this simulation, the wavelength was found to be 29 mm. Thus, the estimated values for time step and element size were obtained from Equations (6) and (7) as 0.5 µs and 2.9 mm, respectively.

_{H}is the number of cycles divided by the frequency. Substituting the time step of 0.5 µs, the frequency of 100 kHz, and several cycles of 2 into the Equation (8), the time function of the input signal is obtained, as shown in Figure 2. Figure 2a illustrates that the input signal completes two cycles in 20 µs, and after this time, the displacement should be set to be zero. The total duration time of the simulation, 60 µs, was selected based on the initial simulation results that showed the incident wave reaching the far end of the plate at the time of 60 µs. A 42 ns (0.042 µs) time step was taken between 20 and 60 µs to record additional data of the wave amplitude; hence, an accurate comparison can be achieved.

## 4. Results and Discussion

#### 4.1. Verification Studies

_{1}, R

_{2}, and R

_{3}as illustrated in Figure 1) and the SAW source are 20.22, 40.44, and 60.667 mm, respectively. Figure 4a shows the response of each receiver (displacement in the Z direction) and the time of zero-crossing for the three signals to verify the phase velocity based on a zero-crossing algorithm using Equation (9) [25,26].

_{3}. It is worth noting that the central frequency is almost 100 kHz, which is in a good agreement with the central frequency of the incident wave.

#### 4.2. A Comparison Between No Hole and Fully-Loosened Bolt

#### 4.3. Detection of Wave Reflection

#### 4.4. Particle Motion Along the Reflected Wave Propagation Path

#### 4.5. Time History of the Reflected Wave Near the Bolt

_{i}is the amplitude of each preload, A

_{4kN}the reference amplitude (4 kN). It can be noted in Figure 8c that the maximum difference is found at time 46 µs, where the phase shifting of reflected wave occurs.

#### 4.6. Simulation and Experimental Comparison

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of the top surface of the plate, including the locations of the three receivers and twelve transmitting point sources. All dimensions are in mm.

**Figure 2.**(

**a**) The time domain of the excitation signal; (

**b**) the frequency spectrum of the excitation signal obtained by FFT.

**Figure 3.**(

**a**) Schematic representation of the bolted joint geometry; (

**b**) the top view of the geometry, including the receiver locations and the reflected wave propagation path. All dimensions are in mm.

**Figure 4.**(

**a**) Displacement responses in Z-axis for the three receivers R

_{1}, R

_{2}, and R

_{3}as illustrated in Figure 1; (

**b**) Frequency spectrum for R3 in Z-axis obtained by FFT.

**Figure 5.**(

**a**) Displacement responses in Z-axis for no hole and fully-loosened bolt cases; (

**b**) Zoomed-in view between the time of 30 and 60 µs for no hole and fully-loosened bolt cases.

**Figure 6.**Time history of the 4 kN signal that passes two neighboring receivers which are located before the bolt along the bolt center.

**Figure 7.**(

**a**) The reflected waves from the bolt at time 46 µs; (

**b**) The displacement in Z-direction along the reflected wave path at time 46 µs before the bolt; (

**c**) The average displacement in Z-direction along the path between time 42 and 50 µs.

**Figure 8.**(

**a**) Time history of the propagating wave for all preload values; (

**b**) Zoomed-in view of the reflection zone, located between 40 and 60 µs; (

**c**) The differential representation of all preload cases with 4 kN case as the reference.

**Figure 9.**The experimentally obtained distance of the first reflected wave from the transducer as a function of the applied torque.

Target Surface | Source Surface | Contact Type |
---|---|---|

Clearance Hole | The Bolt Body | Frictional with C_{f} = 0.2 |

Top Surface of the Plate | Bottom Surface of the Bolt Head | Frictional with C_{f} = 0.2 |

The Bolt Body | The Nut Inner Surface | Bonded |

Bottom Surface of the Plate | Top Surface of the Nut | Frictional with C_{f} = 0.2 |

${\mathit{C}}_{\mathit{p}\mathit{h}\mathit{a}\mathit{s}\mathit{e}}=\frac{\Delta {\mathit{x}}_{\mathit{i}}}{\Delta {\mathit{t}}_{\mathit{i}\mathit{m}}}$ | Theoretical Value (m/s) | Simulated Value (m/s) | Error (%) |
---|---|---|---|

R1&R2 | 2917.58 | 2905.6 | −0.4 |

R2&R3 | 2917.58 | 2926.48 | 0.3 |

R1&R3 | 2917.58 | 2916.01 | −0.05 |

**Table 3.**The distance of zero-crossing of each preload from the source and the corresponding wavelength. All dimensions in mm.

Preload (kN) Torque (N.m) | 4 7.2 | 8 14.4 | 12 21.6 | 16 28.8 | 20 36 | 24 43.3 |
---|---|---|---|---|---|---|

Distance from the source | 17.4 | 16.7 | 16.4 | 16.2 | 15.9 | 15.5 |

Wavelength (λ) | 22.31 | 22.25 | 25.23 | 25.12 | 25.23 | 25.5 |

34% | 67% | 100% | |
---|---|---|---|

Simulated shift from the max. allowable (mm) | 1.2 | 0.7 | 0 |

Experimental shift from the max. allowable (mm) | 0.9 | 0.8 | 0 |

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**MDPI and ACS Style**

Alhazmi, H.; Guldiken, R. Quantification of Bolt Tension by Surface Acoustic Waves: An Experimentally Verified Simulation Study. *Acoustics* **2019**, *1*, 794-807.
https://doi.org/10.3390/acoustics1040046

**AMA Style**

Alhazmi H, Guldiken R. Quantification of Bolt Tension by Surface Acoustic Waves: An Experimentally Verified Simulation Study. *Acoustics*. 2019; 1(4):794-807.
https://doi.org/10.3390/acoustics1040046

**Chicago/Turabian Style**

Alhazmi, Hani, and Rasim Guldiken. 2019. "Quantification of Bolt Tension by Surface Acoustic Waves: An Experimentally Verified Simulation Study" *Acoustics* 1, no. 4: 794-807.
https://doi.org/10.3390/acoustics1040046