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Numerical and experimental investigation of wave propagation in rod-systems with cracks
Engineering Fracture Mechanics 77 (2010) 3532–3540
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Numerical and experimental investigation of wave propagation in rod-systems with cracks L. Gaul *, S. Bischoff, H. Sprenger, T. Haag Institute of Applied and Experimental Mechanics, Universitaet Stuttgart, Germany
a r t i c l e
i n f o
Article history: Available online 8 July 2010 The first author dedicates this article to Professor Dietmar Gross on the occasion of his retirement. Keywords: Structural health monitoring Reflection coefficient Friction modeling Longitudinal wave Piezoelectric transducer Multi-wire cable
a b s t r a c t In this study, the possibility of continuously monitoring load-carrying cables in bridges is considered. A sending/receiving transducer is used to generate an ultrasonic, longitudinal, elastic wave through the cable. The interaction between the L(0, 1)-wave and vertical cracks in a single rod is investigated using the Waveguide-FE-Method to predict the reflection and transmission coefficients. Moreover, this work analyzes how the elastic energy of a propagating wave is distributed between adjacent rods via friction. An energy-based model is developed to approximate the coupling behavior in a two-rod system. Finally, the numerical predictions are verified by experimental data. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The focus of this paper is crack detection in single and two-wire cables, which are widely used in numerous engineering applications, for example, as load-carrying structures of bridges, on elevators and as overhead power transmission lines. Cracks are usually caused by excessive mechanical loads from wind excitation or corrosive influences and can grow to large defects as cable age increases. Therefore, the structural health monitoring of cable stayed and suspension bridges will be particularly examined in this paper. An experimental setup for identifying wire cracks in bridges is illustrated in Fig. 1. A more detailed description can be found in [1]. As shown in Fig. 1, a transducer serves as sender and receiver of elastic waves. The sending transducer converts the electric excitation signal into mechanical energy via the piezo-electric effect. In this way, an ultrasonic wave propagating in the axial direction is generated in the multi-wire cable, which is reflected at the surface crack and returns to the receiving transducer. If the amplitude of the reflected wave, sensed by the receiving transducer, is above a threshold value, the presence of a defect can be assumed. In Fig. 2 the electrical output from the receiving transducer is depicted for an experimental investigation of a cable with several wires. The initial signal burst (A) corresponds to the left bound wave after reflection, while the second signal burst (B) results from the right bound wave which has been reflected at a surface crack. This paper covers the several aspects of analyzing defect cable structures. First, the interaction of guided waves in cylindrical structures with cracks is tackled, following a section about modelling energy propagation in friction coupled rods. Findings of both sections are incorporated in finite element and experimental analyses.
* Corresponding author. Address: Institute of Applied and Experimental Mechanics, Pfaffenwaldring 9, 70550 Stuttgart, Germany. E-mail address: [email protected] (L. Gaul). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.07.003
L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
3533
Fig. 1. Interrogation of load-carrying structures of bridges using ultrasonic signals.
0.5 0.4 0.3
Voltage (V)
0.2 0.1 0 -0.1
D
E
C
B
-0.2 -0.3 -0.4 A
-0.5 0
100 200 300 400 500 600 700 800 900 1000
Time (µs) Fig. 2. Time signal of excited wave (A), reflection at crack (B), reflection at free end (C) and further reflections (D and E).
2. Theory 2.1. Survey of relevant work One of the first experimental studies in this field of research, by Meitzler considered the propagation of elastic pulses in wires having a circular cross section [8]. He attributed pulse distortion to the propagation of several modes. His experimental and theoretical results suggest that coupling between the various modes of propagation were responsible for the observed pulse distortion. In a later study, Rizzo and Lanza di Scalea describe the generation and detection of ultrasonic waves in single wires and seven-wire cables using magnetostrictive sensors [9]. A formulation based on the Pochhammer–Chree theory is used to predict the change in velocity of longitudinal waves as a function of applied stress (acoustoelastic effect). Results from their acoustoelastic experiments are presented and compared to theoretical predictions. Ways to enhance the sensitivity of the acoustoelastic measurements were also investigated. Deviations between experiment and theory in the multi-wire case led the authors to conclude that the extension of their acoustoelastic theory was necessary. Washer et al. utilized the acoustoelastic effect for measuring the stress levels in post-tensioned rods and seven-wire cables [12]. In another study, Rizzo and Lanza di Scalea examined the wave propagation problem in seven-wire cables at the level of the individual wires [10]. They used a broadband ultrasonic setup and a time–frequency analysis based on the wavelet transform to characterize the dispersion and attenuation of longitudinal and flexural waves. They identified the vibration modes which propagate with minimal losses. Such modes are particularly useful for long-range inspection of the wires. Furthermore, they found that because dispersion curves are sensitive to the load level, elastic waves could be used for continuous load monitoring. In a recent paper, Rizzo and Lanza di Scalea employed a time–frequency analysis based on the discrete wavelet transform (DWT) for analyzing the ultrasonic signals. They found the de-noising property of the DWT to be particularly useful in their analyses [11]. Kuttig et al. incorporate the chirplet transform to cope with dispersive signals [6]. 2.2. Theoretical study of the reflection at a crack of an infinite rod The wave propagation in a cylindrical rod can be described analytically based on the Pochhammer–Chree-Theory. In contrast to the analytical approach, a numerical method according [7] is used in the following which yields circular wavenumbers, as well as displacement and force mode shapes.
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
The ultrasonic displacement vector, u, and force vector, f, in a cylindrical rod can be expressed as a superposition of the eigenmodes un and fn at a given circular frequency x [2], as
uðx; y; z; tÞ ¼
X
r n un ðx; yÞejðkn zxtÞ ;
n
f ðx; y; z; tÞ ¼
X
ð2:1Þ
rn f n ðx; yÞejðkn zxtÞ ;
n
where kn and rn are the circular wavenumber and the dimensionless amplitude of the nth mode, respectively. rn denote reflection and transmission coefficients defined in (2.2). At any given frequency, a finite number of propagating and an infinite number of evanescent modes constitute the basis of the solution. The reflection at a crack of an incident wave propagating through an infinite rod in the positive z-direction is shown in Fig. 3. The simulation of scattering by a vertical crack is conducted by using the ultrasonic fields, such that the relevant boundary conditions are fulfilled. These boundary conditions require the crack surface (denoted by the superscript c) to be traction free, while the displacement and force field at the remainder of the interface (denoted by the superscript r) must be continuous. For the left and right surface of the crack formulations (2.2) are obtained
F crefl r refl þ F cinc r inc ¼ 0;
F ctrans r trans ¼ 0;
ð2:2Þ
respectively. For the points outside of the crack continuity conditions hold. For the displacement as well as for the force field, this leads to the expressions
U rinc r inc þ U rrefl rrefl þ U rtrans r trans ¼ 0;
ð2:3Þ
F rinc r inc þ F rrefl rrefl þ F rtrans r trans ¼ 0:
ð2:4Þ
The scattered wave fields are described by the unknown reflection coefficient, rrefl, and transmission coefficient, rtrans. To obtain rrefl and rtrans, the cross-section of the rod is discretized, which allows for formulation of (2.2)–(2.4) in one linear system of equations,
3 2 c F inc rinc 7 6 0 60 F ctrans 7 7 rrefl 7 6 6 ¼ 6 r 7 7: 6 r 4 F refl F rtrans 5 rtrans 4 F inc rinc 5 |fflfflfflffl ffl {zfflfflfflffl ffl } r U rrefl U rtrans U r rinc |fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflinc ffl{zfflfflfflfflfflfflfflffl} 2
F crefl
0
3
A
ð2:5Þ
b
This overdetermined system of equations can be solved in a least-square sense with
ri ¼ ðAH AÞ1 ðAH bÞ:
ð2:6Þ
Energy conservation is used to validate the results: for an arbitrary circular. Using modes normalized to unity, the computed reflection and transmission coefficients must fulfil rHr = 1. In this work, all results fulfilled energy conservation with a maximal deviation of 0.5%. Calculations are carried out for a cylindrical rod with radius a = 2 mm. The material of the rod is stainless steel (Young’s Modulus E = 210 GPa, mass density q = 7800 kg m3 and Poisson’s Ratio m = 0.3). Fig. 4 shows the reflection and transmission coefficients for the fundamental longitudinal incident mode L(0, 1), with circumferential order n = 0 and mode number m = 1, as a function of the crack depth. For the considered circular frequency, x = 0.45 106 s1, two types of waves can propagate through the medium, namely, the longitudinal mode L(0, 1) and the flexural mode F(1, 1).
Fig. 3. Infinite cylindrical rod with vertical crack.
L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
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Fig. 4. Reflection and transmission coefficients for surface cracks with various depth and incident L(0, 1) mode (fundamental longitudinal mode).
These coefficients can be identified from measurements (s.f. Figs. 1 and 2) and provide information about the onset and crack status prior to visual inspection. 3. Energy-based model Due to computational limitations, transient analysis of ultrasonic wave propagation in real multi-wire cables using finite elements is virtually impossible. An energy based method has thus been developed to model wave propagation in adjacent rods. An extensive list of literature on energy flow analysis techniques can be found in [5]. To gain a better understanding of the coupling which occurs between adjacent wires in a cable, a simplified model is considered which consists of two straight rods having a friction contact between them. This can be considered as a precursor to multi-wire modeling. For the case of longitudinal wave propagation in an elastic rod, the total mechanical energy, E, in a given volume, V, at some instant in time equals twice the kinetic energy [3],
E¼
Z
v i v i q dV;
i ¼ 1; 2;
ð3:1Þ
V
where q is the density, vi is the ith component of the velocity vector. The subscripts 1, 2 may be interpreted as the cylindrical coordinate directions, respectively (the fundamental longitudinal mode has no velocity component in the circumferential direction). This integral involves the entire velocity field, which is generally unknown. Assume now that it is possible to measure the radial velocity component, vs, on the surface of the rod in the axial coordinate range, (zo, z1). For the case of a steadystate modal distribution of energy in the rod, the integrand can be expressed as the three term product of: (i) the surface radial velocity component, (ii) a shape function of the radius, r, and (iii) a shape function of the axial coordinate, z. After carrying out the integration in the r- and h-coordinate directions, the integral (3.1) reduces to
E¼a
Z
z1
ðv s ðzÞÞ2 dz;
ð3:2Þ
z0
where a is a constant resulting from the integration. An equivalent integral results if the surface velocity is measured at a single point on the rod surface in the time range [t0, t1] that it takes the [z0, z1]-bounded energy package to propagate past the measurement point. This integral is
E¼a
Z
t1
cg ½v s ðtÞ2 dt;
ð3:3Þ
t0
where cg is the group velocity of the wave package. It is assumed in this step that dispersion effects are negligible. The time average power, P, associated with the energy package is obtained by dividing both sides of (3.3) by the pulse width, so that
P¼
E 1 ¼ cg a t1 t0 t1 t0
Z
t1
t0
½v s ðtÞ2 dt ¼ cg ahv 2s i:
For the case of time harmonic wave propagation, (3.4) simplifies to
ð3:4Þ
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
P ¼ cg a
jv s j2 : 2
ð3:5Þ
According to (3.5), if there are no energy losses in the rod, the time average mechanical power associated with harmonic, longitudinal, elastic wave propagation is directly proportional to the square of the radial surface velocity amplitude. Fig. 5 depicts the power balance for a differential section of the two-rod assembly. Here, elements 1 and 2 are assumed to be cross sections of the active and passive rods, respectively (see Fig. 8). As a finite pulse of elastic energy traverses the element pair, a loss of energy in each element due to material damping and an exchange of energy due to friction coupling occur. These energy flows are indicated by the vertical arrows in Fig. 5. Including the effect of material damping, (3.5) becomes
PðzÞ ¼ cg a
jv s j2 2k2 z e 2
ð3:6Þ
where k2 is the imaginary part of a complex circular wavenumber. The power loss from the ith rod element due to material damping is then given by
Pm i ¼ P i ðzÞ P i ðz þ dzÞ ¼
@P i ðzÞ dz ¼ cm Pi ðzÞdz; @z
i ¼ 1; 2;
ð3:7Þ
where the superscript m indicates a power loss caused by material damping, and cm = 2k2, the material damping coefficient. According to (3.7), the average power loss due to material damping in a rod element is proportional to the input power and distance the elastic wave propagates. The energy coupling mechanism is modelled using a distributed dashpot which connects the differential elements. The instantaneous mechanical power (product of force and velocity) transferred from/to the elements is
Fig. 5. Power balance for a differential section of the two-rod system.
k
k c
k c
k c
c
Fig. 6. Finite element model of two-rod system.
L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
Force input at z = 0 m
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Surface velocity at z = 0.9 m
Fig. 7. Force input and velocity output waveforms for the FE model.
Fig. 8. Experimental setup for measuring wave propagation in contacting rods.
e c ¼ cd v 1 ðv 1 v 2 Þdz; P 1 e c ¼ cd v 2 ðv 1 v 2 Þdz; P 2
ð3:8Þ
where the superscript c indicates a power loss/addition due to inter-element coupling, cd is the distributed dashpot coefficient (units Ns/m2), and v1, v2 are the instantaneous velocities of the differential rod elements. The time average power then becomes
cd jv 1 jðjv 1 j jv 2 jÞdz; 2 e c i ¼ cd jv 2 jðjv 1 j jv 2 jÞdz; Pc2 ¼ h P 2 2
ec i ¼ Pc1 ¼ h P 1
ð3:9Þ
where it has been assumed the motions of the differential elements are harmonic and in-phase. Recognizing that the square root of the time average power is related to the velocity amplitude, (3.9) becomes
pffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi P1 P1 P2 dz; pffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi c P1 P2 dz; P 2 ¼ cc P 2
Pc1 ¼ cc
ð3:10Þ
where cc is the overall coupling coefficient. Finally, a balance of energy on the individual rod elements (i.e., the net time average power flowing into an element is equal to the net time average power flowing out of the element) yields
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
pffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi @P1 ¼ cm P1 cc P1 P1 P2 ; @z p ffiffiffiffiffi p ffiffiffiffiffi pffiffiffiffiffi @P2 P1 P2 : ¼ cm P2 þ cc P2 @z
ð3:11Þ
This set of nonlinear differential equations is solved using the ode45 routine in Matlab. The material damping parameter, cm, the coupling parameter, cc, and the boundary conditions, P1 ðzÞjz¼zo ¼ Po1 ; P2 ðzÞjz¼zo ¼ P o2 , are determined using a least squares fit with experimental data [4]. 4. Numerical and experimental analyses In the following sections, the previously described energy-based model of wave propagation in a two-rod system is used in simulations and predictions. Results are compared with experimental data and finite element simulations.
Power (normalized)
Fig. 9. Surface velocity measurements at several locations on the two-rod system.
Distance from transducer (m) Fig. 10. Measured and simulated time average mechanical power distributions in the two-rod system.
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
4.1. Finite element model A section of the finite element (FE) model of a two-rod system is depicted in Fig. 6. The rods are 1 m long, 4 mm in diameter, and are assumed to have typical density and Young’s Modulus values for aluminum. The FE model is comprised of 4002 nodes with one displacement degree of freedom (DOF) in the axial direction. The model is discretized in space such that the element lengths are 1/20th of the elastic wavelength and in time such that the time step is 1/50th of the wave period. A 1DOF model allows that the wave modes, which lead to signal dispersion, are disregarded. However, the attenuation effect caused by material damping is accounted for in the model by multiplying the overall nodal displacement vector by enDt after each step in the integration scheme, where n is a material damping constant, and Dt is the time step size. Inter-rod coupling is achieved using parallel spring-dashpot combinations at adjacent nodes. The spring constant, k, dashpot constant, c, and material damping parameter, n, are determined using a least squares fit with experimental data of a two-rod system without normal load for a certain excitation frequency. The equations of motion for this system are solved with a linear Newmark integration scheme in Matlab [2]. Fig. 7 depicts the force input waveform along with the computed surface velocity at a location on the active rod. The force input is a four-period Hanning-windowed sinusoidal burst with frequency 450 kHz and is applied to one end of the active rod. The similarity of the waveforms is an indication of good convergence for the chosen spatial and temporal discretizations. In order to make comparisons with the energy-based model, the time average power is computed from the nodal velocity at a number of axial locations according to (3.4). 4.2. Experiment The two-rod experimental setup is illustrated in Fig. 8. The rods are made from aluminum and have a length of 3 m and a diameter of 4 mm. They are pressed together along the entire length using rubber bands. A piezoelectric transducer disc (10 mm diameter, 2 mm thick,) is glued to one end of the active rod. The piezoelectric transducer is driven with 2 cycles of a 450 kHz sinusoid. A longitudinal elastic wave is thereby generated in the active rod. The energy is coupled to the passive rod along the line of contact as the elastic wave propagates. The radial surface velocity of the rods is measured at several points along the axial direction using a laser Doppler vibrometer. The surface velocity measurements are depicted in Fig. 9. The coupling of energy between the rods and the dispersive nature of the incident longitudinal wave is evident. The dispersive behavior of elastic waves in the cylindrical wave guide is predicted by the Pochhammer–Chree theory. An experimental group velocity of 4310 m/s has been determined at 450 kHz, which agrees well with predictions using the Pochhammer–Chree theory. At each measurement point, the time average power is computed from the measured velocity according to (3.4). The dispersive nature of wave propagation com-
-3
x 10
Sensor response without notch
Amplitude [V]
5
0
-5
0
1
2
3
4
5
6
7
8 -4
x 10
Time [s] -3
x 10
Sensor response with notch
Amplitude [V]
5
0
-5
0
1
2
3
4
5
6
7
8 -4
Time [s]
x 10
Fig. 11. Measurement at two-rod system before and after introducing notch in passive rod.
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
plicated this computation. Therefore, the raw velocity signal is multiplied by a roving four-period Hanning window with a four-period width (based on a center frequency of 450 kHz) and is assumed to propagate with the experimentally determined group velocity at 450 kHz. Because the frequency content of the windowed signal is sufficiently narrowband, it propagates nondispersively. Thus, the theoretical development in Section 3 may be applied to compute the time average power. In Fig. 10, the experimental power distribution in the two-rod system is compared to that computed by the energy-based model and the FE model. There is consistent agreement between the experiment and the simulations. The FE simulation required 2 h, whereas the energy-based model simulation required 2 min. It is clear from these investigations that the energy-based model can be used to accurately and efficiently predict wave propagation in a cable structure with several wires. Fig. 11 shows results from measurements with a two-rod system with friction contact. Wave motion was excited and measured at one (active) rod, a notch was introduced in the second (passive) rod. Comparison of signals in the undamaged and damaged case show, that the notch can be detected, even if it is present in the rod which is not directly monitored. 5. Conclusions In this study, a better understanding of the interwire coupling in cable structures with several wires was gained. The power flow in a two-rod system was modeled with finite elements, and the results were confirmed experimentally. It was shown that a wave travelling in one wire of a two-wire cable transmits energy to neighboring wires. This allows for the interrogation of subsurface cracks in a multi-wire cable, which cannot be detected by visual inspection. The two-rod system treated in this study is somewhat simple, but nevertheless, it served as a substantion for energy based descriptions of wave propagation. In future efforts, the energy-based model will be validated for other frequencies and multi-wire configurations, including real cable structures, such as overhead transmission lines and bridge cables. References [1] Branham SL, Wilson MS, Hurlebaus S, Beadle BM, Gaul L. Nondestructive testing of overhead transmission lines. In: Conference on damage in composite materials, Stuttgart; 2006. [2] Castaings M, Le Clezio E, Hosten B. Modal decomposition method for modeling the interaction of Lamb waves with cracks. J Acoust Soc Am 2002;112(6):2567–82. [3] Craig RR. Structural dynamics. Englewood Cliffs, New Jersey: Prentice Hall; 1981. [4] Czichos H, Hennecke M. Hütte – das ingenieurwissen. Berlin: Springer-Verlag; 2007. [5] Graff KF. Wave motion in elastic solids. New York: Dover Publications, Inc.; 1991. [6] Kerber F, Sprenger H, Niethammer M, Luangvilai K, Jacobs LJ. Attenuation analysis of lamb waves using the chirplet transform. EURASIP J Adv Signal Process 2010;2010:6 [Article ID 375171]. [7] Mace BR, Duhamel D, Brennan MJ, Hinke L. Wavenumber prediction of wave motion in structural waveguides. J Acoust Soc Am 2005;117(5):2835–43. [8] Meitzler AH. Mode coupling occurring in the propagation of elastic pulses in wires. J Acoust Soc Am 1961;33(4):435–45. [9] Rizzo P, Lanza di Scalea F. Monitoring steel strands via ultrasonic measurements. In: Proceedings of SPIE – the international society for optical engineering, vol. 4696; 2002. p. 62–73. [10] Rizzo P, Lanza di Scalea F. Wave propagation in multi-wire strands by wavelet-based laser ultrasound. Exp Mech 2004;44(4):407–15. [11] Rizzo P, Lanza di Scalea F. Ultrasonic inspection of multi-wire steel strands with the aid of the wavelet transform. Smart Mater Struct 2005;14:685–95. [12] Washer GA, Green RE, Pond RB. Velocity constants for ultrasonic stress measurement in pre-stressing tendons. Res Nondestruct Eval 2002;14:81–94.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Numerical and experimental investigation of wave propagation in rod-systems with cracks L. Gaul *, S. Bischoff, H. Sprenger, T. Haag Institute of Applied and Experimental Mechanics, Universitaet Stuttgart, Germany
a r t i c l e
i n f o
Article history: Available online 8 July 2010 The first author dedicates this article to Professor Dietmar Gross on the occasion of his retirement. Keywords: Structural health monitoring Reflection coefficient Friction modeling Longitudinal wave Piezoelectric transducer Multi-wire cable
a b s t r a c t In this study, the possibility of continuously monitoring load-carrying cables in bridges is considered. A sending/receiving transducer is used to generate an ultrasonic, longitudinal, elastic wave through the cable. The interaction between the L(0, 1)-wave and vertical cracks in a single rod is investigated using the Waveguide-FE-Method to predict the reflection and transmission coefficients. Moreover, this work analyzes how the elastic energy of a propagating wave is distributed between adjacent rods via friction. An energy-based model is developed to approximate the coupling behavior in a two-rod system. Finally, the numerical predictions are verified by experimental data. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The focus of this paper is crack detection in single and two-wire cables, which are widely used in numerous engineering applications, for example, as load-carrying structures of bridges, on elevators and as overhead power transmission lines. Cracks are usually caused by excessive mechanical loads from wind excitation or corrosive influences and can grow to large defects as cable age increases. Therefore, the structural health monitoring of cable stayed and suspension bridges will be particularly examined in this paper. An experimental setup for identifying wire cracks in bridges is illustrated in Fig. 1. A more detailed description can be found in [1]. As shown in Fig. 1, a transducer serves as sender and receiver of elastic waves. The sending transducer converts the electric excitation signal into mechanical energy via the piezo-electric effect. In this way, an ultrasonic wave propagating in the axial direction is generated in the multi-wire cable, which is reflected at the surface crack and returns to the receiving transducer. If the amplitude of the reflected wave, sensed by the receiving transducer, is above a threshold value, the presence of a defect can be assumed. In Fig. 2 the electrical output from the receiving transducer is depicted for an experimental investigation of a cable with several wires. The initial signal burst (A) corresponds to the left bound wave after reflection, while the second signal burst (B) results from the right bound wave which has been reflected at a surface crack. This paper covers the several aspects of analyzing defect cable structures. First, the interaction of guided waves in cylindrical structures with cracks is tackled, following a section about modelling energy propagation in friction coupled rods. Findings of both sections are incorporated in finite element and experimental analyses.
* Corresponding author. Address: Institute of Applied and Experimental Mechanics, Pfaffenwaldring 9, 70550 Stuttgart, Germany. E-mail address: [email protected] (L. Gaul). 0013-7944/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2010.07.003
L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
3533
Fig. 1. Interrogation of load-carrying structures of bridges using ultrasonic signals.
0.5 0.4 0.3
Voltage (V)
0.2 0.1 0 -0.1
D
E
C
B
-0.2 -0.3 -0.4 A
-0.5 0
100 200 300 400 500 600 700 800 900 1000
Time (µs) Fig. 2. Time signal of excited wave (A), reflection at crack (B), reflection at free end (C) and further reflections (D and E).
2. Theory 2.1. Survey of relevant work One of the first experimental studies in this field of research, by Meitzler considered the propagation of elastic pulses in wires having a circular cross section [8]. He attributed pulse distortion to the propagation of several modes. His experimental and theoretical results suggest that coupling between the various modes of propagation were responsible for the observed pulse distortion. In a later study, Rizzo and Lanza di Scalea describe the generation and detection of ultrasonic waves in single wires and seven-wire cables using magnetostrictive sensors [9]. A formulation based on the Pochhammer–Chree theory is used to predict the change in velocity of longitudinal waves as a function of applied stress (acoustoelastic effect). Results from their acoustoelastic experiments are presented and compared to theoretical predictions. Ways to enhance the sensitivity of the acoustoelastic measurements were also investigated. Deviations between experiment and theory in the multi-wire case led the authors to conclude that the extension of their acoustoelastic theory was necessary. Washer et al. utilized the acoustoelastic effect for measuring the stress levels in post-tensioned rods and seven-wire cables [12]. In another study, Rizzo and Lanza di Scalea examined the wave propagation problem in seven-wire cables at the level of the individual wires [10]. They used a broadband ultrasonic setup and a time–frequency analysis based on the wavelet transform to characterize the dispersion and attenuation of longitudinal and flexural waves. They identified the vibration modes which propagate with minimal losses. Such modes are particularly useful for long-range inspection of the wires. Furthermore, they found that because dispersion curves are sensitive to the load level, elastic waves could be used for continuous load monitoring. In a recent paper, Rizzo and Lanza di Scalea employed a time–frequency analysis based on the discrete wavelet transform (DWT) for analyzing the ultrasonic signals. They found the de-noising property of the DWT to be particularly useful in their analyses [11]. Kuttig et al. incorporate the chirplet transform to cope with dispersive signals [6]. 2.2. Theoretical study of the reflection at a crack of an infinite rod The wave propagation in a cylindrical rod can be described analytically based on the Pochhammer–Chree-Theory. In contrast to the analytical approach, a numerical method according [7] is used in the following which yields circular wavenumbers, as well as displacement and force mode shapes.
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
The ultrasonic displacement vector, u, and force vector, f, in a cylindrical rod can be expressed as a superposition of the eigenmodes un and fn at a given circular frequency x [2], as
uðx; y; z; tÞ ¼
X
r n un ðx; yÞejðkn zxtÞ ;
n
f ðx; y; z; tÞ ¼
X
ð2:1Þ
rn f n ðx; yÞejðkn zxtÞ ;
n
where kn and rn are the circular wavenumber and the dimensionless amplitude of the nth mode, respectively. rn denote reflection and transmission coefficients defined in (2.2). At any given frequency, a finite number of propagating and an infinite number of evanescent modes constitute the basis of the solution. The reflection at a crack of an incident wave propagating through an infinite rod in the positive z-direction is shown in Fig. 3. The simulation of scattering by a vertical crack is conducted by using the ultrasonic fields, such that the relevant boundary conditions are fulfilled. These boundary conditions require the crack surface (denoted by the superscript c) to be traction free, while the displacement and force field at the remainder of the interface (denoted by the superscript r) must be continuous. For the left and right surface of the crack formulations (2.2) are obtained
F crefl r refl þ F cinc r inc ¼ 0;
F ctrans r trans ¼ 0;
ð2:2Þ
respectively. For the points outside of the crack continuity conditions hold. For the displacement as well as for the force field, this leads to the expressions
U rinc r inc þ U rrefl rrefl þ U rtrans r trans ¼ 0;
ð2:3Þ
F rinc r inc þ F rrefl rrefl þ F rtrans r trans ¼ 0:
ð2:4Þ
The scattered wave fields are described by the unknown reflection coefficient, rrefl, and transmission coefficient, rtrans. To obtain rrefl and rtrans, the cross-section of the rod is discretized, which allows for formulation of (2.2)–(2.4) in one linear system of equations,
3 2 c F inc rinc 7 6 0 60 F ctrans 7 7 rrefl 7 6 6 ¼ 6 r 7 7: 6 r 4 F refl F rtrans 5 rtrans 4 F inc rinc 5 |fflfflfflffl ffl {zfflfflfflffl ffl } r U rrefl U rtrans U r rinc |fflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflinc ffl{zfflfflfflfflfflfflfflffl} 2
F crefl
0
3
A
ð2:5Þ
b
This overdetermined system of equations can be solved in a least-square sense with
ri ¼ ðAH AÞ1 ðAH bÞ:
ð2:6Þ
Energy conservation is used to validate the results: for an arbitrary circular. Using modes normalized to unity, the computed reflection and transmission coefficients must fulfil rHr = 1. In this work, all results fulfilled energy conservation with a maximal deviation of 0.5%. Calculations are carried out for a cylindrical rod with radius a = 2 mm. The material of the rod is stainless steel (Young’s Modulus E = 210 GPa, mass density q = 7800 kg m3 and Poisson’s Ratio m = 0.3). Fig. 4 shows the reflection and transmission coefficients for the fundamental longitudinal incident mode L(0, 1), with circumferential order n = 0 and mode number m = 1, as a function of the crack depth. For the considered circular frequency, x = 0.45 106 s1, two types of waves can propagate through the medium, namely, the longitudinal mode L(0, 1) and the flexural mode F(1, 1).
Fig. 3. Infinite cylindrical rod with vertical crack.
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Fig. 4. Reflection and transmission coefficients for surface cracks with various depth and incident L(0, 1) mode (fundamental longitudinal mode).
These coefficients can be identified from measurements (s.f. Figs. 1 and 2) and provide information about the onset and crack status prior to visual inspection. 3. Energy-based model Due to computational limitations, transient analysis of ultrasonic wave propagation in real multi-wire cables using finite elements is virtually impossible. An energy based method has thus been developed to model wave propagation in adjacent rods. An extensive list of literature on energy flow analysis techniques can be found in [5]. To gain a better understanding of the coupling which occurs between adjacent wires in a cable, a simplified model is considered which consists of two straight rods having a friction contact between them. This can be considered as a precursor to multi-wire modeling. For the case of longitudinal wave propagation in an elastic rod, the total mechanical energy, E, in a given volume, V, at some instant in time equals twice the kinetic energy [3],
E¼
Z
v i v i q dV;
i ¼ 1; 2;
ð3:1Þ
V
where q is the density, vi is the ith component of the velocity vector. The subscripts 1, 2 may be interpreted as the cylindrical coordinate directions, respectively (the fundamental longitudinal mode has no velocity component in the circumferential direction). This integral involves the entire velocity field, which is generally unknown. Assume now that it is possible to measure the radial velocity component, vs, on the surface of the rod in the axial coordinate range, (zo, z1). For the case of a steadystate modal distribution of energy in the rod, the integrand can be expressed as the three term product of: (i) the surface radial velocity component, (ii) a shape function of the radius, r, and (iii) a shape function of the axial coordinate, z. After carrying out the integration in the r- and h-coordinate directions, the integral (3.1) reduces to
E¼a
Z
z1
ðv s ðzÞÞ2 dz;
ð3:2Þ
z0
where a is a constant resulting from the integration. An equivalent integral results if the surface velocity is measured at a single point on the rod surface in the time range [t0, t1] that it takes the [z0, z1]-bounded energy package to propagate past the measurement point. This integral is
E¼a
Z
t1
cg ½v s ðtÞ2 dt;
ð3:3Þ
t0
where cg is the group velocity of the wave package. It is assumed in this step that dispersion effects are negligible. The time average power, P, associated with the energy package is obtained by dividing both sides of (3.3) by the pulse width, so that
P¼
E 1 ¼ cg a t1 t0 t1 t0
Z
t1
t0
½v s ðtÞ2 dt ¼ cg ahv 2s i:
For the case of time harmonic wave propagation, (3.4) simplifies to
ð3:4Þ
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P ¼ cg a
jv s j2 : 2
ð3:5Þ
According to (3.5), if there are no energy losses in the rod, the time average mechanical power associated with harmonic, longitudinal, elastic wave propagation is directly proportional to the square of the radial surface velocity amplitude. Fig. 5 depicts the power balance for a differential section of the two-rod assembly. Here, elements 1 and 2 are assumed to be cross sections of the active and passive rods, respectively (see Fig. 8). As a finite pulse of elastic energy traverses the element pair, a loss of energy in each element due to material damping and an exchange of energy due to friction coupling occur. These energy flows are indicated by the vertical arrows in Fig. 5. Including the effect of material damping, (3.5) becomes
PðzÞ ¼ cg a
jv s j2 2k2 z e 2
ð3:6Þ
where k2 is the imaginary part of a complex circular wavenumber. The power loss from the ith rod element due to material damping is then given by
Pm i ¼ P i ðzÞ P i ðz þ dzÞ ¼
@P i ðzÞ dz ¼ cm Pi ðzÞdz; @z
i ¼ 1; 2;
ð3:7Þ
where the superscript m indicates a power loss caused by material damping, and cm = 2k2, the material damping coefficient. According to (3.7), the average power loss due to material damping in a rod element is proportional to the input power and distance the elastic wave propagates. The energy coupling mechanism is modelled using a distributed dashpot which connects the differential elements. The instantaneous mechanical power (product of force and velocity) transferred from/to the elements is
Fig. 5. Power balance for a differential section of the two-rod system.
k
k c
k c
k c
c
Fig. 6. Finite element model of two-rod system.
L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
Force input at z = 0 m
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Surface velocity at z = 0.9 m
Fig. 7. Force input and velocity output waveforms for the FE model.
Fig. 8. Experimental setup for measuring wave propagation in contacting rods.
e c ¼ cd v 1 ðv 1 v 2 Þdz; P 1 e c ¼ cd v 2 ðv 1 v 2 Þdz; P 2
ð3:8Þ
where the superscript c indicates a power loss/addition due to inter-element coupling, cd is the distributed dashpot coefficient (units Ns/m2), and v1, v2 are the instantaneous velocities of the differential rod elements. The time average power then becomes
cd jv 1 jðjv 1 j jv 2 jÞdz; 2 e c i ¼ cd jv 2 jðjv 1 j jv 2 jÞdz; Pc2 ¼ h P 2 2
ec i ¼ Pc1 ¼ h P 1
ð3:9Þ
where it has been assumed the motions of the differential elements are harmonic and in-phase. Recognizing that the square root of the time average power is related to the velocity amplitude, (3.9) becomes
pffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi P1 P1 P2 dz; pffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi c P1 P2 dz; P 2 ¼ cc P 2
Pc1 ¼ cc
ð3:10Þ
where cc is the overall coupling coefficient. Finally, a balance of energy on the individual rod elements (i.e., the net time average power flowing into an element is equal to the net time average power flowing out of the element) yields
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pffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi @P1 ¼ cm P1 cc P1 P1 P2 ; @z p ffiffiffiffiffi p ffiffiffiffiffi pffiffiffiffiffi @P2 P1 P2 : ¼ cm P2 þ cc P2 @z
ð3:11Þ
This set of nonlinear differential equations is solved using the ode45 routine in Matlab. The material damping parameter, cm, the coupling parameter, cc, and the boundary conditions, P1 ðzÞjz¼zo ¼ Po1 ; P2 ðzÞjz¼zo ¼ P o2 , are determined using a least squares fit with experimental data [4]. 4. Numerical and experimental analyses In the following sections, the previously described energy-based model of wave propagation in a two-rod system is used in simulations and predictions. Results are compared with experimental data and finite element simulations.
Power (normalized)
Fig. 9. Surface velocity measurements at several locations on the two-rod system.
Distance from transducer (m) Fig. 10. Measured and simulated time average mechanical power distributions in the two-rod system.
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4.1. Finite element model A section of the finite element (FE) model of a two-rod system is depicted in Fig. 6. The rods are 1 m long, 4 mm in diameter, and are assumed to have typical density and Young’s Modulus values for aluminum. The FE model is comprised of 4002 nodes with one displacement degree of freedom (DOF) in the axial direction. The model is discretized in space such that the element lengths are 1/20th of the elastic wavelength and in time such that the time step is 1/50th of the wave period. A 1DOF model allows that the wave modes, which lead to signal dispersion, are disregarded. However, the attenuation effect caused by material damping is accounted for in the model by multiplying the overall nodal displacement vector by enDt after each step in the integration scheme, where n is a material damping constant, and Dt is the time step size. Inter-rod coupling is achieved using parallel spring-dashpot combinations at adjacent nodes. The spring constant, k, dashpot constant, c, and material damping parameter, n, are determined using a least squares fit with experimental data of a two-rod system without normal load for a certain excitation frequency. The equations of motion for this system are solved with a linear Newmark integration scheme in Matlab [2]. Fig. 7 depicts the force input waveform along with the computed surface velocity at a location on the active rod. The force input is a four-period Hanning-windowed sinusoidal burst with frequency 450 kHz and is applied to one end of the active rod. The similarity of the waveforms is an indication of good convergence for the chosen spatial and temporal discretizations. In order to make comparisons with the energy-based model, the time average power is computed from the nodal velocity at a number of axial locations according to (3.4). 4.2. Experiment The two-rod experimental setup is illustrated in Fig. 8. The rods are made from aluminum and have a length of 3 m and a diameter of 4 mm. They are pressed together along the entire length using rubber bands. A piezoelectric transducer disc (10 mm diameter, 2 mm thick,) is glued to one end of the active rod. The piezoelectric transducer is driven with 2 cycles of a 450 kHz sinusoid. A longitudinal elastic wave is thereby generated in the active rod. The energy is coupled to the passive rod along the line of contact as the elastic wave propagates. The radial surface velocity of the rods is measured at several points along the axial direction using a laser Doppler vibrometer. The surface velocity measurements are depicted in Fig. 9. The coupling of energy between the rods and the dispersive nature of the incident longitudinal wave is evident. The dispersive behavior of elastic waves in the cylindrical wave guide is predicted by the Pochhammer–Chree theory. An experimental group velocity of 4310 m/s has been determined at 450 kHz, which agrees well with predictions using the Pochhammer–Chree theory. At each measurement point, the time average power is computed from the measured velocity according to (3.4). The dispersive nature of wave propagation com-
-3
x 10
Sensor response without notch
Amplitude [V]
5
0
-5
0
1
2
3
4
5
6
7
8 -4
x 10
Time [s] -3
x 10
Sensor response with notch
Amplitude [V]
5
0
-5
0
1
2
3
4
5
6
7
8 -4
Time [s]
x 10
Fig. 11. Measurement at two-rod system before and after introducing notch in passive rod.
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L. Gaul et al. / Engineering Fracture Mechanics 77 (2010) 3532–3540
plicated this computation. Therefore, the raw velocity signal is multiplied by a roving four-period Hanning window with a four-period width (based on a center frequency of 450 kHz) and is assumed to propagate with the experimentally determined group velocity at 450 kHz. Because the frequency content of the windowed signal is sufficiently narrowband, it propagates nondispersively. Thus, the theoretical development in Section 3 may be applied to compute the time average power. In Fig. 10, the experimental power distribution in the two-rod system is compared to that computed by the energy-based model and the FE model. There is consistent agreement between the experiment and the simulations. The FE simulation required 2 h, whereas the energy-based model simulation required 2 min. It is clear from these investigations that the energy-based model can be used to accurately and efficiently predict wave propagation in a cable structure with several wires. Fig. 11 shows results from measurements with a two-rod system with friction contact. Wave motion was excited and measured at one (active) rod, a notch was introduced in the second (passive) rod. Comparison of signals in the undamaged and damaged case show, that the notch can be detected, even if it is present in the rod which is not directly monitored. 5. Conclusions In this study, a better understanding of the interwire coupling in cable structures with several wires was gained. The power flow in a two-rod system was modeled with finite elements, and the results were confirmed experimentally. It was shown that a wave travelling in one wire of a two-wire cable transmits energy to neighboring wires. This allows for the interrogation of subsurface cracks in a multi-wire cable, which cannot be detected by visual inspection. The two-rod system treated in this study is somewhat simple, but nevertheless, it served as a substantion for energy based descriptions of wave propagation. In future efforts, the energy-based model will be validated for other frequencies and multi-wire configurations, including real cable structures, such as overhead transmission lines and bridge cables. References [1] Branham SL, Wilson MS, Hurlebaus S, Beadle BM, Gaul L. Nondestructive testing of overhead transmission lines. In: Conference on damage in composite materials, Stuttgart; 2006. [2] Castaings M, Le Clezio E, Hosten B. Modal decomposition method for modeling the interaction of Lamb waves with cracks. J Acoust Soc Am 2002;112(6):2567–82. [3] Craig RR. Structural dynamics. Englewood Cliffs, New Jersey: Prentice Hall; 1981. [4] Czichos H, Hennecke M. Hütte – das ingenieurwissen. Berlin: Springer-Verlag; 2007. [5] Graff KF. Wave motion in elastic solids. New York: Dover Publications, Inc.; 1991. [6] Kerber F, Sprenger H, Niethammer M, Luangvilai K, Jacobs LJ. Attenuation analysis of lamb waves using the chirplet transform. EURASIP J Adv Signal Process 2010;2010:6 [Article ID 375171]. [7] Mace BR, Duhamel D, Brennan MJ, Hinke L. Wavenumber prediction of wave motion in structural waveguides. J Acoust Soc Am 2005;117(5):2835–43. [8] Meitzler AH. Mode coupling occurring in the propagation of elastic pulses in wires. J Acoust Soc Am 1961;33(4):435–45. [9] Rizzo P, Lanza di Scalea F. Monitoring steel strands via ultrasonic measurements. In: Proceedings of SPIE – the international society for optical engineering, vol. 4696; 2002. p. 62–73. [10] Rizzo P, Lanza di Scalea F. Wave propagation in multi-wire strands by wavelet-based laser ultrasound. Exp Mech 2004;44(4):407–15. [11] Rizzo P, Lanza di Scalea F. Ultrasonic inspection of multi-wire steel strands with the aid of the wavelet transform. Smart Mater Struct 2005;14:685–95. [12] Washer GA, Green RE, Pond RB. Velocity constants for ultrasonic stress measurement in pre-stressing tendons. Res Nondestruct Eval 2002;14:81–94.