Finite Element Modelling Of Riveted Lap Joint Plates Samaneh Hosseingholizadeha, Aghil YousefiKomaa*, Seyed Saeid Mohatsebia a
Advanced Dynamic and Control System Laboratory, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran. * Corresponding author e-mail:
[email protected]
Abstract This paper presents a finite element modelling of a riveted lap joint using Abaqus 6.13-1. The model consists of two plates jointing each other by a riveted lap, a piezoelectric sensor and an actuator. Mode and frequency selection is conducted based on the wave structure and also wave length. In presented model, the effect of rivet’s diameter on excited frequency is investigated and the relationship between them is drawn and also transmission coefficients for different diameters also have been calculated. The main issue of modelling of this kind is the combining the induced voltage causing by rivets’ pressure and excitation wave which is not happen in real situation. To deal with this problem, two models are run. First one is the real model and the second one is the ideal model including tie plates which let the excited signal to completely transmit through the overlap part; subtracting the output signals of these two models can overlooks the pressure effect. Keywords: riveted lap joint; finite element modelling; transmission coefficient;
1. Introduction Ultrasonic waves practically have an important role in none-destructive evaluation. Their speed of inspection, sensitivity and low cost are much better than other ultrasonic techniques [1]. Lamb waves are guided waves emanating from the longitudinal and transverse waves in a thin plate. These waves were introduced by Horace Lamb [2] and more details can be found in ref [1] and [2]. There are several papers regarding to faults detection in plates, however, without considering any fasteners. It is not deniable that in real industries like aerospace, gas and oil industries, most plates are jointed to other parts of the structure by fasteners. So it is vital to consider the existence of these fasteners in the procedure of fault detection. Rivets, bolts, adhesives and welding are some examples of used fasteners. In this part, some researchers which investigated the effects of fasteners in fault detection are presented.
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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 An experimental study has been done on riveted lap joint species which uses the multi-feature integration method for fatigue crack detection and crack length estimation using Lamb waves [3]. Detection of the notches in rivet holes of thin plates using Lamb waves scanning method has been experimentally carried out in [4]. Simulations of the propagation of Lamb wave energy were illustrated. Experimental results on propagation of guided waves through a bolted joint under various bolt load have been reported in [5]. Three different torque loads were studied. The waves in sensors showed different arrival time which had linear dependency to the applied torques. Seunghee Park et al [6] used both impedance and Lamb wave methods to investigate the damage detection of lab-size steel bridge members. In [7] the propagation of ultrasonic guided waves in adhesively bonded joints has been discussed. Aluminium lap-shear joints were tested with the three bond states of fully cured epoxy, poorly cured epoxy and slip (water) interface. The Gabor Wavelet transform was employed as the joint time-frequency analysis to extract ultrasonic energy transmission coefficients over the specific frequency range. The results showed that mode a0 is a good candidate for such an inspection .In [8], a simplified model of a weld-strip between two identical semi-infinite elastic layers was investigated. The reflected and transmitted ultrasonic fields were expressed by modal series. Most studies in health monitoring systems which considered the effect of fasteners were done experimental, because it has been concluded that the finite element modelling of these structures is challenging due to some reasons. Generally, considering the effect of fasteners load makes the modelling to be difficult because this effect and travelling wave’s effect by actuator are combined in the sensors which should be distinguished. In addition, fasteners have numerous geometries and types whose effect should be considered in actuated waves, so this paper is organized as follows. Firstly, the effect of rivet holes diameter in lap joint plates on the excited frequency is studied. Secondly, the transmission coefficients are defined and calculated for different diameters.
2. Physics of the Problem In recent years, there has been an increasing interest in the generation of Lamb waves with piezoelectric (PZT) ceramic wafers from SHM community [3]. In this study,2 plates which make a lap joint structure are considered. It is assumed that riveted plates are made of 2 mm thickness aluminium sheets. They have 20 mm overlap with each other. Two piezoelectric disks are used as a sensor and actuator. There are three holes in the overlap area which are considered as the place of rivets. This structure is shown in Figure 1.
Figure 1.Geometry of aluminium sheets
The shank diameter is equal to the diameter of the hole and the head diameter is the diameter which limits the applied force. These two diameters are shown in Figure 2(a). In the finite element modelling, there is a partitioned circle as a surface of applied load. This is shown in Figure 2(b). 2
4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014
(a)
(b)
Figure 2.(a) Geometry of a rivet sample (b) Applied pressure by the rivets in finite element modelling
3. Mode and Frequency Selection As mentioned before, the goal of this paper is to receive a signal by the sensor due to discontinuities made by the geometry, so in order to select a proper mode we have to consider the wave structure and after that the group velocity and phase velocity. 3.1 Wave Structure Wave structures present in-plane and out-of-plane displacements. Because of the evident of lap joint, the high out-of-plane displacement can be very helpful in the transition of the excited wave through the lap joint part. In Figure 3, the wave structures for mode s0 and a0 for aluminium plates with 2mm thickness and 200 kHz excitation wave are shown; in which f is the frequency of excited wave and d is the thickness of the plate. S0 mode, Thickness=2 mm, Frequency= 200KHz Inplane displacement Out of plane displacement
2 1 0 -1 -2
Inplane displacement Out of plane displacement
3 Thickness (mm)
Thickness (mm)
3
-3
A0 mode, Thickness=2 mm, Frequency= 200KHz
2 1 0 -1 -2
-1
-0.5 0 0.5 Normalized Displacement
(a)
1
-3
-1
-0.5 0 0.5 Normalized Displacement
1
(b)
Figure 3. Wave structure of (a) S0 and (b) A0 modes for an aluminium plate with 2mm thickness and 200 KHz excitation
A suitable wave structure is one which has an acceptable in-plane and out-of-plane displacements, but there are some limitations in mode selection. These limitations are due to the available facilities in experiment; the first one is the thickness of available plates and the second one is frequency provided by actuators. The thicknesses of the plates in the laboratory are 2mm. For example, although the wave structure of the plate with f.d=1.5 has an appropriate in-plane and out-ofplane displacements, we cannot provide this situation owing to need a piezoelectric actuator with the frequency of 750 kHz. Maximum frequency of the available actuator in our laboratory was 200 kHz. This leads us to choose f.d=0.4 with the wave structure shown in Figure 3. It is acceptable to have this wave structure and it is obvious that mode a0 is a good candidate for such an inspection owing to the ease of transduction associated with the predominant out-of-plane displacements.
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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 3.2 Group and Phase Velocity dispersion curves Dispersion curves for aluminium plates are shown in Figure 4. The group velocity is used to determine the arrival time to the sensor and phase velocity is used to determine the wave length. Group Velocity Dispersion Curve for Aluminum Plate
Phase Velocity Dispersion Curve for Aluminum Plate 5.5
12 Symmetric Modes Anti-Symmetric Modes
Symmetric Modes Anti-Symmetric Modes
5
10
Group Velocity (km/sec)
Phase Velocity (km/sec)
4.5
8
6
4
4 3.5 3 2.5 2
2
0
1.5 1
0
1
2 3 4 Frequency x Thickness (MHz.mm)
5
6
0
1
2
(a)
3 fd(MHz*mm)
4
5
6
(b)
Figure 4.(a)Phase velocity and (b) group velocity dispersion curves for an aluminium plate
The group velocity for f.d=0.4 in s0 and a0 modes are 2500 and 5416 m/s respectively. The phase velocity for f.d=0.4 in s0 and a0 modes are 1733 and 5200 m/s respectively. For the calculation of wave length, we have: Phase velocity= (wave length × frequency) so wave length=Phase velocity/frequency The maximum wave length is for a0 mode which has 26 mm length. The distance between the center of actuator and sensor is 100 mm, so we can choose a proper excited wave for actuator. In a word, if we choose a 3.5 cycle tone burst for exciting, we are sure that the excited out coming signal finishes when it receives sensor. This exciting wave is shown in Figure 5. Tone burst- 3.5 cycle- 200k 1 0.8 0.6
Amplitude
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
0.2
0.4
0.6
0.8
1 Time
1.2
1.4
1.6
1.8 -5
x 10
Figure 5. A tone burst signal of 3.5 cycles and 200 kHz central frequency
As it was mentioned before, one of the goals of this paper is to see the effects of diameter in receiving the signal. For this purpose, we should model the structure in a way that all circumstances are equal and just the diameter would change. We choose five different diameters for the rivets which are chosen from the available standard sizes. These five sizes are given in Table 1.
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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 Table 1.The using standards of rivets’ sizes rivet1 rivet2 rivet3 rivet4 rivet5
rivet dia (inch)
rivet dia(mm)
head dia(inch)
head dia(mm)
0.09375 0.15625 0.25 0.28125
2.34375 3.90625 6.25 7.03125
0.122 0.203 0.325 0.365
3.05 5.075
0.34375
8.59375
0.45
11.25
8.125 9.125
4. Signal processing The model has been run in two parts. Having applied the rivet force, the actuator releases the three and half cycle tone burst wave. This means that before the sensor see the effect of excitation wave, it sees the applied pressure effect. This is so grave an issue that needs to be dealt because in reality the sensor only sees the excitation wave. In order to eliminate the displacement causing by applied pressure, we have run 2 types of model. First one is the real model including applied pressure and excitation wave and the second one is the model consisting of only applied pressure with no exciting wave. The signal which is obtained by subtracting the result of sensor in second model from the first one can eliminate the pressure effect. These signals of the sensor for different rivets are shown in Figure 6. Induced voltage in sensors for 5 different rivets holes 0.3
0.2
Voltage(v)
0.1
0
-0.1 rivet1 rivet2 rivet3 rivet4 rivet5
-0.2
-0.3
-0.4
0
0.2
0.4
0.6 Time (s)
0.8
1
1.2 -4
x 10
Figure 6.Output signals for five different rivets’ diameters
For the data decomposition of signals in sensor, we use wavelet analysis which is one of the powerful tools in signal processing. This method is our priority over the others signal processing like Fast Fourier Transform owing to its ability to describe the signals in both time and frequency domain simultaneously. For this purpose, we used Morl wavelet to carry out the continuous wavelet transform on the output signal. This wavelet is chosen because of its similarity to the excited wave. The output signal in rivet3 with its continuous wavelet transform is shown in Figure 7 as a sample.
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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 Analyzed Signal 0.2 0.1 0 -0.1 10
20
30
40
50
60
70
80
90
100
Frequency (KHz)
Scalogram Percentage of energy for each wavelet coefficient 165 176 188 203 219 238 262 290 325 369 427 507 625 812 1160 2031 8125
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50 60 Time(sec)
70
80
90
100
Figure 7.Continuous wavelet transform for output signal in model of rivet2
The maximum coefficient in CWT of signals demonstrates the time and frequency of received excited signal. These results are given in Table 2 for all rivets. Table 2.The time and frequency of received signals for all rivets Rivet1 Rivet2 Rivet3 Rivet4 Rivet5
Frequency(KHz)
Time(µs)
253 253 238 225 128
64.1972 64.0035 65.1062 64.8975 65.8959
5. Results and Discussion This part discusses the results of finite element modelling which revolves around the effect of varied rivet diameter. 5.1 Frequency of Excitation Waves in Rivets Although the frequency of impulsive wave was 200 kHz, practically different excited frequencies are seen for five different structures. This leads us to consider the bandwidth of the excited signal. Bandwidth of signal is a range of frequency which is excited in some special frequency regarded to the circumstance and is determined by FFT 1. The bandwidth of 3.5 cycle tone burst signal is shown in Figure 8. This figure demonstrates that for the impulsive wave with 200k excitation has 107 kHz bandwidth.
1
6
Fast Fourier Transform
4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 FFT of 200 KHz - 3.5 cycle tone burst signal 170
165
Y(f)-db
160
155 B.W= 107KHz 150
145
140 0
0.5
1
1.5 2 2.5 Frequency (Hz)
3
3.5
4 5
x 10
Figure 8. Fast Fourier Transform of tone burst signal of 3.5 cycles and 200 KHz central frequency
The relationship between the diameter and frequency is shown in the Figure 9. Excited frequencies are obtained from the CWT of output signal explained in previous part.
Variation of Excited Frequcies versus Rivets Diameters 260
Excited Frequency (KHz)
240
220
200
180
160
140
120
2
3
4
5 6 Rivets Diameter (mm)
7
8
9
Figure 9.Variation of excited frequency versus rivets diameter
It is obvious that exponential interpolation is a good candidate to represent the relationship between excited frequency and rivet’s diameter, so exponential of second order is used to fit a curve among the data points. This relationship can be expressed by the Equation 1 in which x represent the diameter of rivets in mm and y is the excited frequency. The curves shows a descending approach of frequency when the rivet’s diameter increases. This curve demonstrates that there are different frequencies for different diameters. Therefore during frequency selection, size of rivet plays a key role which should be considered. (
)
(
)
(1)
The coefficients a, b, c and d are shown in Table 3, which are obtained by least square method.
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4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 Table 3.Coefficients of exponential function of frequency and rivet’s diameter Coefficient
Value
a b c d
-0.03703 0.9446 254.5 -0.00103
5.2 Transmission Coefficients One question that needs to be asked, however, is that how much energy is transmitted through lap joint. To determine the transmission coefficient, it is necessary to have a definition. In doing this, it would be better to compare a quantity which can represent the energy of the wave. In this paper, the maximum amplitude of output signal is considered. This maximum is obtained regarding to the receiving time of a0 mode as it was calculated in previous part. To compare this quantity with another one, an ideal modelling is carried out including tie plates. In this modelling the wave uses the whole overlap region to transmit from lap joint part, so the maximum amplitude in receiving time in second model can complete the relationship. The definition of transmission coefficient is given in Equation 2. (2)
2
Transmission coefficients for different rivets’ diameters are given in Table 4. Table 4.Transmission coefficients for different rivets’ diameters real-pick
tie-pick
TC (%)
0.2015 0.207 0.2191 0.1792
1.1075 1.2494 1.5076 1.4084
18.194131 16.567953 14.533033 12.723658
0.1495
1.7542
8.5224034
rivet1 rivet2 rivet3 rivet4 rivet5
The table shows that by increasing the diameter of holes, the transmission coefficient is decreasing. This result stems from the fact that fewer overlap area causes less wave transmission. This trend is demonstrated in Figure 10. Variation of transmission coefficient versus rivets Diameters 20
18
TC (%)
16
14
12
10
8
2
3
4
5 6 Rivets Diameter (mm)
7
8
9
Figure 10.Variation of transmission coefficient versus rivets Diameters
2
8
Transmission Coefficient
4th International Conference on Acoustics & Vibration (ISAV2014), Tehran, Iran, 10-11 Dec. 2014 The variation of TC shows an exponential descending approach versus rivet’s diameter. Equation 1 shows this relationship and its coefficients are given in Table 5. Table 5.Transmission coefficients for different rivets’ diameters
Coefficient
Value
a b c d
-0.01658 0.6716 20.17 -0.04395
6. Conclusion This paper presents a finite element modelling of a riveted lap joint using Abaqus 6.13-1. This model consists of two plates jointing each other and a piezoelectric sensor and actuator. In mode selection out of plane displacements in lap joint play a key role, because the impulsive wave should transmit through the lap joint. Taking into account this factor and owing to the existing limitation, the 200 KHz with 3.5 cycle tone burst has been chosen as the excited signal. Five models with five different diameters have been run. The continuous wavelet transform with Morl wavelet has been used for drawing the feature of output signal. The relationship between diameter and the frequency is determined by second order exponential function. The transmission coefficients for five models have calculated which are fitted by an exponential function.
REFERENCES 1.
2. 3.
4.
5.
6.
7.
8.
J. L. Rose, “Ultrasonic Waves in Solid Media”, Cambridge University Press, Cambridge, UK, 1999. H. Lamb, “Waves in Elastic Plate”, Proceeding of the Royal Society A93,114-128 (1917) J. He, X. Guan, “A multi-feature integration method for fatigue crack detection and crack length estimation in riveted lap joints using Lamb waves”, Smart Material and Structures, 22(2013) R.Osegueda, V.Kreinovich, “Detection of cracks at rivet holes in thin plates using Lambwave scanning”, Journal of Smart Non-destructive Evaluation and Health Monitoring of Structural and Biological Systems II,55(2003) J. Bao, V. Giurgiutiu, “Effects of fastener load on wave propagation through lap joint”, Journal of Health Monitoring of Structural and Biological Systems, 8695(2013) S. Park, C. Yun, “PZT-based active damage detection techniques for steel bridge components”, 957–966(2006) F. Lanza di Scalea, P. Rizzo, “Propagation of Ultrasonic Guided Waves in Lap-Shear Adhesive Joints”, Journal of Acoustical Society of America,146-56(2004) M. Valentin Predoi, M. Rousseau, “Lamb waves propagation in elastic plane layers with a joint strip”, Journal of Ultrasonic, 551–559(2005)
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