This paper explores several methods for the passive localisation of Lamb waves in thin plate-like structures. Lamb waves are a form of non-linear, elastic, guided wave that propagate in thin plate-like structures [1]. The ability to localise Lamb waves is a powerful method for the structural health monitoring (SHM) of an aerospace vehicle such that the damage can be interrogated further or flagged for inspection at a regular maintenance interval. The nature of Lamb wave propagation is such that the behaviour is dispersive, i.e., each frequency travels at a different velocity. The presence of dispersion within Lamb waves makes their analysis difficult with traditional methods. Several techniques have been proposed for the removal of dispersion using post-processing analysis, however, few methods have been proposed in which the sensor configuration is optimised for the minimisation of dispersion [2]. As such, this paper proposes a technique that utilises the close spacing of a phased array style sensor array such that the dispersion of the Lamb wave signal may be considered negligible. The ability to make this assumption provides new opportunities for the localisation of AEs with this type of sensor array. The array could also be used for the analysis of other forms of acoustic emissions such as Rayleigh, Love and shear waves [3].
1.1. Structural Health Monitoring
SHM for aerospace vehicles is an invaluable tool in maintaining the integrity and health of a structure. With many similarities to non-destructive testing and evaluation (NDT&E), SHM operates in-situ, providing real-time information about the health of a structure to render damage assessments and provide prognostics [4–6]. The environment in which aerospace vehicles operate and the increasing demand for lighter structures to meet emissions targets and reduce costs motivate the development of an SHM network [7]. Such a network can provide real-time information regarding the health of the vehicle to allow for improved decision making and to aid in the development of condition-based maintenance schedules. Additional benefits from the sensor networks required for SHM include the ability to inform digital twin models and power an Industry 4.0 movement that could rapidly decrease vehicle down-time as well as improving safety [8]. The information collected can also contribute to powering future data-driven design innovations.
Bartelds [9] explored the applications of a piezoelectric transducer (PZT) network for inspection, reflecting upon how the inspection process remained the weakest link in the reliability chain. Bartelds [9] found that such a network could reduce inspection time by up to 50% for military aircraft and that total maintenance and inspection costs could be reduced by up to 20% for the civil sector. Dong and Kim [10] completed a cost-benefit analysis of a SHM network for fuselage structures in civil aviation aircraft and found that despite inspection and maintenance amounting for more than 27% of the overall life cycle costs of aircraft, such a network would not be cost effective with current technology limitations [11]. Dong concluded that more work should be performed to develop methods that would decrease the number of sensors required for monitoring. The proposed passive phased array (PPA) sensor configuration in this work presents as a useful method for the monitoring of large areas with a relatively small array, requiring few sensors.
With an estimated 55% of aircraft structural failures accredited to some form of fatigue damage, fatigue remains a persistent problem for the aviation industry [12]. Moreover, uncertainty in the loading and environmental conditions that the vehicle is exposed to can render design work inadequate for the application. Goranson [13] showed that differences between operating stresses and fatigue allowables resulted in up to 85% of service problems for long-life structures.
Moreover, the increasing use of composite materials in aerospace structures, whilst presenting new opportunities, also presents with new challenges that must be overcome [4, 14–16]. Fibre composites, whilst typically less affected from fatigue than metals, introduce new damage modes such as delamination, matrix cracking, fibre pull-out and fibre breakage, requiring new methods for monitoring [17].
Furthermore, recent work has revealed the growing issue of fatigue damage in propulsion systems [18], unveiling the importance of future work into not just the health of the structure but that of the entire vehicle with the implementation of an integrated vehicular health management (IVHM) system. Ultimately, the development of new methods for the NDT&E and SHM of aerospace structures and systems is important for improving the safety and performance of aerospace vehicles.
1.2. Lamb Waves
The theory of Lamb waves originates from the work of Horace Lamb in 1917 [1]. By applying a Helmholtz decomposition to the elastodynamic constitutive equation for isotropic and homogenous materials with a traction free boundary condition, Lamb derived the governing equation for the behaviour of such waves [1, 3]. These governing equations, known as the Rayleigh-Lamb equations, revealed that Lamb waves are composed of two modes: symmetric and antisymmetric. An infinite number of modes can exist in a plate simultaneously[19]. The Rayleigh-Lamb equations are shown below for both modes and indicates that the behaviour of the waves is non-linear and dispersive within the medium.
$$\begin{array}{c}\frac{\text{tan}\left(qh\right)}{\text{tan}\left(ph\right)}=-\frac{4{k}^{2}qh}{{{(k}^{2}-{q}^{2}) }^{2}}\#\left(1.1\right)\end{array}$$
$$\begin{array}{c}\frac{\text{tan}\left(qh\right)}{\text{tan}\left(ph\right)}=-\frac{{{(k}^{2}-{q}^{2}) }^{2}}{4{k}^{2}qh}\#\left(1.2\right)\end{array}$$
These equations can be solved by applying a variety of numerical schemes. The depiction of the resulting solution is often given in the form of dispersion curves for a particular material and frequency-thickness product. An example of such a curve is shown in Fig. 1.
|
Figure 1: Lamb wave dispersion curves for an aluminium plate[19]. |
As is evident from the dispersion curve, the presence of higher order modes only exists above a certain frequency-thickness product, often known as the cut-off frequency. The restriction of analysis to only the fundamental, or zeroth order, wave modes simplifies their analysis greatly and is often applied in literature [20–22]. Whilst the fundamental order modes are the most studied, the analysis of higher order modes, although difficult, can provide useful information regarding the health of the structure [23, 24].
1.3. Phased Arrays
A phased array is a form of steered sensor array in which a phase delay between adjacent sensors can steer the direction of a wavefront [25, 26]. Phased arrays have been extensively used in radar technology because of their simplicity and absence of moving parts. While they are commonly used for steering electromagnetic wavefronts, phased arrays have also found applications in acoustics, such as sonography (biomedical imaging), seismology (oil and gas prospecting), and sonar. Phased arrays have also been applied in the fields of NDT&E and SHM for the steering of ultrasonic waves to scan a material for damage or to interrogate a specific location [4, 27–29]. The use of such an array allows for monitoring a large area with a relatively small sensor array. In all of these applications, the phased array is actively employed, in which the array is being used to generate the wavefront. The use of active sensing is generally employed with time of arrival (ToA) algorithms that can localise and interrogate a defect based upon the time required for the wave to return to the array[30].
As an alternative, this work has explored the use of phased arrays in a passive sense, in which the array passively collects AE information from the structure for interrogation. In this manner, ToA algorithms cannot be applied as the time that the signal was generated is unknown. Passive phased arrays rely on the measurement of time difference of arrival (TDoA), which determines the time difference between the reception of the signal by adjacent sensors. The use of TDoA can be applied to a variety of methods for the localisation of an emission. A minimum of three sensors is required for any such method to localise an acoustic source on a two-dimensional plane. An additional benefit of using a PPA is the availability of sensors for detecting emissions. The most common sensor used in active phased array monitoring is that of PZT, that can function as both actuators, for generating the wavefront, and sensors, for acquiring the returned signal. By eliminating the need to generate a signal, alternative sensors such as fibre Bragg gratings (FBGs), fibre lasers, or laser Doppler vibrometers (LDVs) can be employed, each with its own unique merits. Table 1 provides a summary of some passive sensors that can be used in a phased array configuration.
Table 1
A summary of sensors that could be employed for passive phased array localisation.
Sensor | Measurement Type |
PZT | Electrical – Piezoelectric effect |
Strain Gauge | Electrical - Resistivity |
CMUTs | Electrical - Capacitance |
FBG | Optical |
LDVR | Optical |