·Table of Contents ·Methods and Instrumentation | Acoustoelastic Measurement of Bolt Axial Load with Velocity Ratio MethodHajime YasuiInstrumentation Engineering Division, Toyota Motor Corporation, Toyota 471-8571 Koichiro Kawashima Department of Mechanical Engineering Nagoya Institute of Technology, Nagoya 466-8555 Contact |
Fig 1: Example of short bolts used for automobiles |
The spread of high-speed digital signal acquisition and signal processing systems enables us to measure easily precise time-of-flight of ultrasonic pulses which propagates between the bolt head and end. In the present paper, the acoustoelastic velocity ratio method is used for estimating quantitatively the axial load in short bolts used for automobiles.
Fig 2: Model of axially loaded bolts. |
Longitudinal wave : Transverse wave: (1)
where V_{L0} and V _{T0} are the longitudinal and transverse wave velocities in the unstressed portion, and a _{L} and a_{ T} are the acoustoelastic constants of the longitudinal and transverse waves. (Hereafter, subscripts L and T will be omitted if the equations for the longitudinal wave and the transverse wave are identical.) Thus, the pulse-echo time-of-flight required for the total bolt length is expressed by the following equation, including the time-of-flight of the unstressed portion ( L-Le):
(2) |
3-1 Differential of time-of-flight method[1][2]
The relationship between F and DT (= the change of time-of-flight due to axial load) is measured before the determination the axial load in the bolt. Then, the load value is determined from DT to be based on the calibration curve.
(3) |
This method is advantageous in that variations in bolt length do not affect the value of axial load. However, its application is limited due to the necessity of time-of-flight measurement in stressed and unstressed states. This method is called the " DT-method" hereafter.
3-2 Velocity ratio method[3][4]
The velocity ratio method which uses the difference in the acoustoelastic coefficients of longitudinal wave and transverse waves. The ratio of the time-of-flight of transverse wave to that of longitudinal wave is approximated by the following equation:
(4) |
4-1 Bolts and measurement system
The geometry of the forged bolts is shown in Fig.3. The end faces of the bolts were
finished by a milling machine to be flat and parallel. The tensile strength, yield stress and the
Young's modulus of the bolt's material SCM440 are 1.1GPa, 1.0GPa and 210GPa, respectively.
Fig 3: Geometry of bolts. | Fig 4: Schematic measurement system. |
Atorque wrench was used to apply an axial load to the bolt fitted into a load cell, as shown in Fig. 4. We confirmed that the error in the axial load indicated by the load cell is within ±1% using the master cell as a standard. We applied axial load greater than 5 kN for BoltAand 10 kN for Bolt B, because of poor reproducibility at low load due to unsteady contact conditions at the bolt /jig interfaces.
4-2 Measurement of time of flight
The broad-band normal incidence transducer (6.3 mm in diameter; Panametrics-X1045) was used
in the measurement, which excites and receives simultaneously 10MHz longitudinal wave and 5MHz
transverse wave. To ensure a stable contact condition between the bolt head and the transducer, the transducer was attached with Nd-Fe-B magnet and a grip. Viscous couplant was used at the
transducer/bolt head interface.
The time-of-flight measurement was carried out with an ultrasonic pulser-receiver (Tungaloy TSB97001-1) and an A/D converter (Tungaloy TSB97001-2) of a 8-bit resolution and a minimum sampling interval of 10ns (1.0ns for 1GS/s in equivalent sampling rate) as shown in Fig.4. During the measurement, the B1 and B2 echoes were averaged from 64 to 1024 times to improve S/N ratio.
Fig 5: Received waveforms of bolt B |
5-1 DT-method
The calibration curves obtained by the DT-method for 10 bolts of A and B types are shown in Fig. 7. Calibration curve of Bolt A shows nice linearity in the
load range less than 40kN, and the slopes agree within 0.2% . The error value, i.e., the predicted range of axial load values at 95% reliability, with linear regression in
the 5-40 kN load range (10% to 85% of the yield stress) is 1.4%.
Fig 6: Frequency spectra. | Fig 7: Calibration curves of DT method. |
On the other hand, nonlinearity is obvious for the calibration curve of Bolt B , in particular for high load range (50kN). The variation of slopes is 1.6% in the 10-50 kN load range (10% to 50% of the yield stress) and the variation in the 50-80 kN range is greater than 10%.
To apply these calibration curves for high load, we should take account of the nonlinearily. Figure 8 shows the influence of the approximate orders on the relative error of the calibration curve and shows the relationship between the approximate orders and the dispersion degree expressed by 1-r ( r is the correlation coefficient ) in parallel referring. The relative error was expressed by the predicted range at 95% reliability of the difference between the estimated calibration curve and the data the 10-85% yield stress range in the condition of the transducer fixed for each 1 bolt. The result shows that the nonlinear effect of Bolt B results in significant error when the conventional linear approximation, i.e. by Eq.(3) is used. Namely, when the Bolt-B is calibrated by linear approximation based on the data of 10-50kN, the errors of the axial load in the level of 70 kN appears greater than 12% or more even if perfect measurement would be carried out. Therefore, we performed second-order regression approximation in the 10-70kN range (10% to 70% of the yield stress), and confirmed that the variation at 95% reliability of the measured data to the value calculated based on the calibration curve is 11.6% for 10 bolts.
The nonlinearity results from nonlinear extension of the stressed part due to the geometry of bolts. With regard to bolt B, high stress appears locally at the threaded portion and at the corner of the seating as shown in Fig.9. Moreover, variations in stress distribution depends on slight differences in the geometry of bolts. This effect appears as the scattered slopes shown in Fig. 7.
Fig 8: Comparison of approximate order | Fig 9: Axial stress distribution obtained by FEM analysis. |
5-2 Velocity ratio method
The calibration curves obtained by the velocity ratio method are shown in Fig. 10, in which the
measurement was done for only one bolt of the type A and B with the transducer being fixed . The
predicted relative ranges of the axial load at 95% reliability are 0.4% for Bolt A and 2.6% for Bolt B. Both calibration curves show better linearity than those of the DT-method. Even in higher load over
60kN of Bolt B, the deviation from the linearity is quite small . Moreover, the calibration curve of bolt A has high value of the correlation (r 0.9994) and the slopes of 5 bolts agree within 0.2%. On the other hand, the calibration curves of bolt B are low correlation (r 0.992) and the scattering of the slope is about 4.9% as shown in Fig. 11. These results show that the slight variation of the geometry and material properties give non-negligible error of the axial load for short and highly stressed bolts.
Fig 10: Calibration curves of VR method | Fig 11: Calibration curves of VR method |
The calibration curves shown in Fig.12 were obtained for 10 bolts of type A and B under the condition where the transducer was repeatedly attached and removed for 20 times. The waveforms of each point were averaged 64 to 1024 times, depending on S/N ratio and the vertical resolution. The predicted relative range of the axial load (at 95% reliability) is 2.9% for Bolt A and 28.4% for Bolt B. The error increases markedly by frequent detaching the transducer due to the variation in couplant thickness between the transducer and the bolt head, therefore, the techniques of reducing this error and excluding the extraordinary value is required for practical application.
Fig 12: Calibration curves of VR method |
As an empirical procedure, the extraordinary results of the time-of-flight was excluded by the following process. The transducer was repeatedly attached to and removed from the bolt head for 20 times. The waveform of B1 and B2 echoes at each measurement was accumulated 64 times.
The data at particular load levels in this process is shown in Fig.13. The results is shown in Fig.14. The predicted relative range of axial load values (at 95% reliability) is 2.1% for Bolt A and 14.6% for Bolt B, which are much smaller than those in Fig.12.
Fig 13: Processing for reducing error value. | Fig 14: Calibration curve with processing. |
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