Damage Detection of a Plate Using Migration Technique X. LIN*
AND
F. G. YUAN
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695 ABSTRACT: This paper demonstrates a feasibility study of applying migration, which is widely used as an advanced data processing technique in geophysical exploration, to detect damage in a plate. This technique suggests a scheme that may build a real time, in situ, and quantitative health monitoring system. A linear actuator/sensor array is placed along a horizontal central axis on a square homogeneous isotropic plate. A technique is proposed to utilize piezoelectrics as both actuators and sensors to generate and collect the flexural waves in the plate. Migration technique is then adopted to interpret the recorded data and image the damage that formed in the plate. The wave field reflected from the damage is synthesized by using a two-dimensional explicit finite difference method to model the plate using Mindlin plate theory. A one-way version of flexural wave equation is derived. The one-way wave equation based poststack reverse-time migration is then used to back-propagate the synthetic wave field to their secondary sources, and the damage is imaged by applying appropriate imaging condition. The data pre-processing procedures before migration, such as muting direct arrival, deconvolution and stacking, are also discussed. The satisfactory determination of the locations and dimensions of the damages in several numerical simulation examples validates the feasibility of proposed monitoring system.
INTRODUCTION health monitoring for the aircraft has been attracting much attention in recent years [1–3]. For achieving the maximum flight safety and in-service life while reducing the maintenance cost, the current preventive maintenance will inevitably be replaced by the predictive maintenance, which is based on the monitoring information of the working state of the aircraft. Structural health monitoring systems (SHMS) may be developed from conventional Nondestructive Testing/Evaluation/Inspection technologies. Although the technologies have served well for years, they encounter three major drawbacks [4–5]. First, these techniques are built on an empirical basis, thus they could provide only qualitative, not quantitative information. Secondly, the exclusive testing equipment and method, which usually are available only in the lab, make the in situ monitoring unlikely. Thirdly, probably most unfavorable one, they cannot provide the real-time monitoring, leaving the SHMS a blank for the ability to detect the sudden damages which may be incurred in the battlefield. The advances in sensor material and smart structure technologies give a new prospective to structural health monitoring. Combining the smart structure concept and structural health monitoring together, which is called
S
TRUCTURAL
*Author to whom correspondence should be addressed. E-mail:
[email protected]
JOURNAL
OF INTELLIGENT
smart SHMS, may provide the potential to overcome some of the drawbacks in the current monitoring programs. Many new ideas have been studied [6–12]. These ideas may vary in selecting sensors, signals and interpretation methods, but they all share the common goal: to develop accurate, automatic and rapid monitoring techniques. In this paper, an innovative SHM method to detect damage in a plate structure is presented. A linear sensor/ actuator array is placed along a horizontal central axis on a homogeneous isotropic plate. The piezoelectrics serve as actuators to generate incident flexural waves in the plate and are also acted as sensors to receive the waves scattered from the damage. Migration technique, which is widely used in geophysics, is then adopted to interpret the recorded wave field data and then to produce an image of damage that occurred in the plate. Because the piezoelectrics acting as sensors and actuators can be either bonded on the surface or embedded in the structure, this system offers an opportunity for the situation where in situ and real time monitoring are required. Further, the migration is processed based on the wave equation, thus the extracted information from the received signals after migration is not qualitative, but quantitative, and gives an exact evaluation of the working status in every position of the structure. Especially, the output of migration provides the necessary data for imaging the damage in the plate, which can interpret outcome of the monitoring much easier and more direct than other methods.
MATERIAL SYSTEMS
AND
STRUCTURES, Vol. 12—July 2001
1045-389X/01/07 0469-14 $10.00/0 DOI: 10.1106/L388-OYY3-VDQC-4V5C ß 2002 Sage Publications
469
470
X. LIN
AND
The procedures of applying migration technique in SHM are outlined first. Then, based on Mindlin plate theory, a 2–6 order MacCormack finite difference algorithm is introduced to synthesize the waves reflected from the damage. Poststack reverse-time migration is used to back-propagate the reflection wave field and image the damage in the plate. A matrix form of the one-way wave equation for flexural waves is derived to proceed the migration. Some data processing procedures that are related to migration are also discussed. Examples of numerical simulation are given to show the promising features of this technique. It should be noted that this study is limited on numerical simulation, and experimental verification on the proposed system will be reported in the authors’ other papers.
HOW MIGRATION WORKS IN SHMS Figure 1a shows a plate with a piezoelectric array either bonded on the surface or embedded between layers of the structure. These piezoelectrics act as actuators and sensors alternately. This configuration is an analog to the reflection seismic prospecting as shown in Figure 1b. In Figure 1a, a source wavelet (voltage) is applied on one of the piezoelectric actuators and induces change in strain. By properly choosing piezoelectric built-in method and voltage polarity, waves with specific mode such as flexural waves will generate and propagate in the plate. If the waves encounter the interface of an inhomogeneity or damage, they will be reflected, refracted or diffracted. When the reflection waves hit the piezoelectrics acting as sensors, the strain change sensed on the piezoelectrics will generate a voltage output, which will be amplified and recorded for further processing. In seismic exploration (Figure 1b), however a wave source is exploded on the surface of the Earth and the response of the waves from the Earth interior is received by an array of geophones or hydrophones (receivers). The recorded wave field is processed by a series of data processing steps, and then finally
F. G. YUAN
produces an image of the geophysical structure by the use of migration technique. Migration in geophysics includes two basic steps: extrapolation and imaging. Extrapolation means treating the recorded wave field as an excitation and reconstructing the spatial wave field on the vertical section of the Earth. Imaging means building and displaying the strength and location of the reflector by extracting a variable from the extrapolated wave field, which is governed by an imaging condition. SHMS can use this migration concept to develop the image of the damage in a plate from the wave field recorded by the piezoelectrics through similar data collection and processing methods. The resulting image of the plate will graphically show the size and location of the damage, if it does exist, as one of the goals of SHMS required. In SHMS, migration can be applied more robustly than in geophysical prospecting. First of all, in geophysics field, migration is based on a priori estimation of the velocity distribution of the Earth’s underneath layers, which is critical prerequisite to obtain correct image through migration. For SHMS, it is no longer a concern. Once the structure is to be targeted, the material properties and structure layout are essentially known, and thus the wave propagation velocity can be accurately predicted. Another advantage is that the actuators and sensors can distribute at an arbitrary location, which provide more flexibility using migration methods. Further, it is relatively easy in SHMS to generate various types of source wavelets and to control the excitation period and different energy transfer direction of the waves, while it is extremely difficult in geophysics field. These characteristics of input wavelet can be properly selected to detect different possible shapes and forms of the damages. Therefore, intelligently choosing these characteristics will make the detection more sensitive to different types of damages. The study objects in SHMS and in geophysics or seismology are different, hence some of the data processing procedures in geophysics or seismology
Figure 1. An analog between plate HMS and geophysical exploration.
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Damage Detection of a Plate Using Migration Technique
field, which are directly relative to the Earth, may either unsuitable or unnecessary in SHMS, such as velocity analysis, residual statics correction, ground roll muffling, etc. However, other procedures, such as deconvolution, CMP sorting, NMO correction and stacking, which are essential to migration methods, must be adopted when migration is applied to SHMS field. Especially, some procedures, such as stacking, whose purpose is to enhance the signal to noise ratio, are always desirable in SHMS.
THE REFLECTION WAVE FIELD Governing Equations Mindlin plate theory [13] is used to model the flexural wave propagation in a thin elastic isotropic plate. By using the stress resultants (Qx, Qy, Mx, My and Mxy) and the plate displacement components (w, x and y), the equations of motion are written as @2 w @Qx @Qy þ þq ¼ @t2 @x @y
ð1aÞ
h3 @2 x @Mx @Mxy þ Qx ¼ 12 @t2 @x @y
ð1bÞ
h3 @2 y @Mxy @My þ Qy ¼ 12 @t2 @x @y
ð1cÞ
h
2 8 9 1 v < Mx = 6v 1 M y ¼ D6 4 : ; Mxy 0 0
Qx ¼ 2 Gh
@w þ @x
@w þ Qy ¼ 2 Gh @y
ð2Þ
where G0 ¼ 2G. Equation (2) provides an approach to describe flexural and thickness-shear waves in the plate. The Mindlin plate theory gives an adequate approximation to three dimensional elasticity theory under the frequency of lowest antisymmetric thickness-shear mode. If both the transverse shear deformation and rotatory inertia terms are omitted, Equation (2) reduces to classic thin plate theory, which is only valid to predict the flexural waves for the case of small wavenumber and characteristic dimension of plate much smaller than the wavelength: Dr4 w þ h
@2 w ¼ qðx, y, tÞ @t2
ð3Þ
qT ¼ fq, 0, 0, 0, 0, 0, 0, 0g and differentiating Equations (1d)–(1f ), Equation (1) can be re-written as a first-order system equations in a matrix form: E0
ð1dÞ
ð1eÞ
y
@2 h3 @2 @2 w r2 0 2 Dr2 w þ h 2 2 G @t @t 12 @t 2 2 2 Dr h @ ¼ 1 0 þ q G h 12G0 @t2
uT ¼ fw_ , _ x , _ y , Qy , Qx , M x , M y , M xy g
x
in which r4 ¼ r2r2. Defining
and the constitutive equations: 9 8 @ x > > > > > > 3> > @x > > 0 > > > > = < @ y 0 7 7 @y > 1 v 5> > > > > > > > > > 2 @ @ y> x > > ; : þ @y @x
obtained. Then, by eliminating x and y from the three motion equations, a single differential equation in terms of transverse displacement w is written as:
ð1fÞ
In Equation (1), E and G are Young’s modulus and shear modulus respectively, h is the plate thickness, v is the Poisson’s ratio, is the mass density, is a shear correction factor, 2 ¼ 2/12, and D is the plate bending stiffness, i.e., D ¼ Eh3/[12(1 v2)]. Substituting the plate stress components Equations (1d–f ) into Equations (1a–c), the equations of motions in terms of plate displacement components can be
@u @u @u ¼ A0 þ B 0 þ C 0 u þ q @t @x @y
ð4Þ
where E0 ¼ 2 h 6 6 60 6 6 6 60 6 6 6 6 60 6 6 6 60 6 6 6 6 60 6 6 6 60 6 6 6 4 0
0
0
0
0
0
0
0
h3 12
0
0
0
0
0
0
0
h3 12
0
0
0
0
0
0
0
1 2 Gh
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 2 Gh
1 v ð1 v2 ÞD ð1 v2 ÞD v 1 ð1 v2 ÞD ð1 v2 ÞD 0
0
0 0 2 ð1 vÞD
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
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X. LIN
2
0
6 60 6 6 6 60 6 6 60 6 A0 ¼ 6 61 6 6 6 60 6 6 60 4 0 2
0 0
0
1
0
0
0
1
0
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
1 0
0
0
0
0
0 0
0
0
0
0
0 1
0
0
0
0 0
0 0
1
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 0
0
0
0
0 1
0
0
0
0
1 0
0
0
0
0
0 0
6 60 6 6 6 60 6 6 60 6 C0 ¼ 6 60 6 6 6 60 6 6 60 4 0
0
0
3
0 0
@U @U @U ¼ At þ Bt @t @x @y
U t ¼ f x þ gy
3
0 0
0
1
0
0
0 0
1
0
0
0
0 1
0
0
0
0
1 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0 0
0
0
0
0
0
3
predictor
7 07 7 7 7 07 7 7 07 7 7 07 7 7 7 07 7 7 07 5
ðnÞ U ð1Þ i,j ¼ U i,j þ
t h ðnÞ ðnÞ ðnÞ 37 f i,j f ðnÞ f 8 f i1,j i1,j i2,j 30x i ðnÞ þ f ðnÞ i2,j f i3,j
ð8aÞ
corrector U ðnþ1=2Þ ¼ i,j
0
1 Further defining U ¼ E 0 u, At ¼ A0 E 1 0 , Bt ¼ B0 E 0 , 1 C t ¼ C 0 E 0 , Equation (4) can be rewritten as
@U @U @U ¼ At þ Bt þ C tU þ q @t @x @y
ð7Þ
where f and g are AtU and BtU respectively. Subscripts x, y and t represent the derivatives with respect to these variables. A MacCormack explicit finite difference algorithm is used to solve this two-dimensional wave equation. In each time step t, this algorithm updates Equation (7) by first solving the equation in the x direction (Ut ¼ fx) and then in the y-direction (Ut ¼ gy), thus the algorithm is a dimensional splitting method and has a smaller phase error than unsplit schemes. Defining Fx as backward–forward operator for onedimensional problem Ut ¼ AtUx, which includes a ðnþ1=2Þ : predictor U ð1Þ i,j and a corrector U i,j
0 0 0
ð6Þ
since E0 is independent of time and spatial coordinates and A0, B0 do not depend on the spatial coordinates, then At and Bt are not dependent on spatial coordinates. Equation (6) can be rewritten in a divergence-free form:
7 0 17 7 7 7 1 07 7 7 0 07 7 7 0 07 7 7 7 0 07 7 7 0 07 5 0
F. G. YUAN
simulate the reflection waves and implement the migration. Considering the major portion of Equation (5)
7 07 7 7 7 17 7 7 07 7 7 07 7 7 7 07 7 7 07 5
0 0
0 6 60 6 6 6 60 6 6 61 6 B0 ¼ 6 60 6 6 6 60 6 6 60 4 2
0 0
AND
ð5Þ
In modeling the damage as a region of inhomogeneity in the plate, only the property matrix E0 needs to be altered in the above governing equations, which will make the modeling of wave reflection much easier in numerical calculation. Finite Difference Algorithm A 2–6 order finite difference algorithm developed from 2–4 MacCormack algorithm [14] is used in this study to
1 ðnÞ U i,j þ U ð1Þ i,j 2 t h ð1Þ ð1Þ 37 f iþ1,j f ð1Þ þ 8 f ð1Þ i,j iþ2,j f iþ1,j 60x i ð1Þ þ f ð1Þ iþ3,j f iþ2,j
ð8bÞ
and defining F þ x as forward–backward operator for Ut ¼ AtUx: predictor ðnÞ U ð1Þ i,j ¼ U i,j þ
t h ðnÞ ðnÞ ðnÞ 37 f iþ1,j f ðnÞ f 8 f i,j iþ2,j iþ1,j 30x i ðnÞ þ f ðnÞ iþ3,j f iþ2,j
ð8cÞ
corrector U ðnþ1=2Þ ¼ i,j
1 ðnÞ t h ð1Þ U i,j þ U ð1Þ 37 f i,j f ð1Þ þ i,j i1,j 2 60x i
ð1Þ ð1Þ ð1Þ 8 f ð1Þ i1,j f i2,j þ f i2,j f i3,j
ð8dÞ
473
Damage Detection of a Plate Using Migration Technique
where the two indices (i, j) in the subscript represent the x and y direction grid (i, j 2 [0, M], M is the maximum grid index) and the index (n) in the superscript represents the time step (n 2 [0, N], N is the maximum time step index). Further by replacing f with g and replacing the changes in index i with index j in Equation (8), it is straightforward to define the backward–forward operator Fy and forward–backward operator F þ y for one-dimensional problem Ut ¼ BtUy. Then the MacCormack splitting method can be expressed as U
nþ2
¼
þ n F xF yF þ y Fx U
ð9Þ nþ2
is Un is the output value of the nth time step and U for the (n þ 2)th time step. Each operator proceeds the calculation by a half time step; thus a complete update includes four operators in two time steps. In each sequential time step, the order of the x and y direction updates is reversed, so is the order of forward–backward and backward–forward operators. In this way, the MacCormack algorithm acquires second-order accuracy in time and sixth-order accuracy in space (referred to 2–6 scheme). For @U/@t ¼ CtU term in Equation (5), the increment Ct t of U due to Ct is calculated in advanced by U ðnÞ i,j ¼ e and then at each time step this term is added into each grid point. This method avoids the difference computation at each time step thus speeds up the calculation. If the plate is subjected to a distributed transverse loading, the q term in Equation (5) can be counted into the numerical result by inserting the amplitude into the iteration at each time step as a boundary condition. For a point force with amplitude P, which is desired in migration technique, the force term q with q ¼ P/a is applied as a distributed force on a very small area a to simulate the loading effect. Because the last two terms in the right hand side of Equation (5) are independent of spatial grid, they are updated after the splitting computation of Equation (6) and summed into the result for each time step. Although the explicit finite-difference scheme is computationally much more efficient than the implicit scheme, it is restricted by CFL (Courant–Friedrichs– Lewy) stability condition. CFL stability condition assumes that the waves are not allowed to propagate over two grids in just a single time step, so that the properties of the waves could be reserved in the numerical approximation and the stability is guaranteed. It requires the numerical propagation speed x/t to be less than fastest wave propagation wave. For the MacCormack scheme, the time step is limited by
t
