Identification of impact force on a thick plate based on the elastodynamic and higher-order time-frequency analysis

Transcription

1 1249 Identification of impact force on a thick plate based on the elastodynamic and higher-order time-frequency analysis S-K Lee 1, S Banerjee 2, and AMal 3 1 Department of Mechanical Engineering, Inha University, Inchon, Republic of Korea 2 Department of Engineering Technology, Saint Louis University, Saint Louis, Missouri, USA 3 Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California, USA The manuscript was received on 23 March 2007 and was accepted after revision for publication on 1 August DOI: / JMES674 Abstract: The current paper presents a novel approach to precisely locate and characterize an impact load in thick plates. The approach is based on the analysis of the acoustic waveforms measured by a sensor array located on the plate surface in combination with the theoretical Green s function for the plate. The Green s functions are derived based on either the exact elastodynamic theory or an approximate shear deformation plate theory. For accurate estimation of the location of the impact source, the time differences in the arrival times of the waves at the sensors and their propagation velocities are determined first. This is accomplished through the use of a combined higher-order time-frequency (CHOTF) method, which is capable of detecting signals with lower signal to noise ratio compared with other available methods. Since most of the energy in the wave is carried by the flexural waves (A 0 mode), the group velocity of this mode is extracted using the CHOTF technique for estimating the impact source location. The estimates are shown to be in excellent agreement with the actual locations and time histories of the applied impact loads. Keywords: impact detection, thick plate, lamb wave, time-frequency analysis, plate theories 1 INTRODUCTION Elastic plates are used extensively in aircraft, aerospace, automotive, industrial, and power plant structures. One of the major concerns in these platelike structures is the damage due to impact and fatigue loads. Impact damage that may be superficial in appearance can frequently have strong detrimental effects on the performance of the structure. For instance, low-velocity impact can cause hidden delaminations inside composite structures. If undetected, the damage can grow, leading to catastrophic failure of the structure. In general, the analysis of structures subject to dynamic loads can be categorized into three basic problems. Corresponding author: Department of Mechanical Engineering, Inha University, 253 Yonghyun Dong, Nam Gu, Inchon , Republic of Korea. sangkwon@inha.ac.kr 1. Given the input and the system parameters, determine the output. 2. Given the input and the output, determine the system parameters. 3. Given the system parameters and the output, determine the input. While problems of the first category are described as direct problems, the second and third types are inverse problems. Inverse problems are of great current interest in a variety of applications and they have been investigated extensively by many authors [1 5]. As an example, the inversion of acoustic emission waves generated by foreign object impact in structural components can be used to monitor the location and severity of impact induced damage in defects-critical structures [1 5]. Therefore, a system that can detect the location of impact and estimate its energy can be very helpful in developing health monitoring systems for advanced structures. Michaels and Pao [1] describe a method to determine the dynamic force acting at a known location on

2 1250 S-K Lee, S Banerjee, and A Mal a plate from its wave motion measurements. In many problems of greater concern, however, is that the location of the load is generally not known. In the acoustic emission technique, the standard method to identify the source location in a plate is to place three or more transducers on the surface of the plate and triangulate the source location by using the differences in the arrival times of acoustic waves at the sensors [6 8]. Thus the arrival time or the propagation velocity of the waves is needed in the estimation of the source location. However, the flexural waves generated by an impact load are highly dispersive and if this is not taken into account in the measurement of their travel time, then large errors can occur in the source location [6]. In the impact location identification problem, an accurate estimation of the propagation velocity of the energy carried by the A 0 mode, i.e. its group velocity is difficult [9, 10]. A number of techniques for the measurement of the phase and group velocity of the flexural waves have been developed [6, 11 15]. The major drawbacks of some of these techniques have been reviewed by Prosser et al. [11]. Many of these drawbacks are absent in recently developed methods based on the time-scale time-frequency approaches and they have been used to measure the group velocity of flexural modes more accurately in comparison with conventional techniques. The higher order-time frequency method [16, 17] is more effective in detecting signals with lower signal to noise ratio. Therefore, the combined higher-order time-frequency method (CHOTF) is used in the current paper. After location of the impact load is determined, the theoretical solution to the impact load is used for the estimation of the impact energy. Several approaches have been used to determine the impact energy [3, 4]. These methods are based on a structural model that characterizes the relationship between the input and the sensor output to predict the response to the impact load. In some of these methods, strain gauges [3] or piezoelectric films [4] are used as sensors. Gaul and Hurlebaus [2] apply the classic plate theory (CPT) as a theoretical solution to an impact load after identifying the location of the impact load using the wavelet transform (WT). If the plate is very thin, then a sufficiently accurate response can be obtained using the CPT. However, for thicker plates, i.e. when the wavelengths are comparable to the plate thickness, the exact elastodynamic theory needs to be employed in the solution of the problem in order to obtain meaningful results [18]. Since the exact solution is computationally very intensive the approximate shear deformation plate theory (SDPT), retaining the transverse shear and rotary inertia, can also be used as an alternative in modelling the dynamic deformations across the thickness of the plate [19]. In an earlier attempt [17], WT in combination with the CPT were used to determine the location and the time history of the impact load on a thick (10 mm) aluminium plate. It was found that the WT has disadvantage of having bad time resolution at low frequency and bad frequency resolution at high frequency. The CPT can be used for the analysis of thin plate (3 mm) with relatively low error [20], but is not accurate for thick plate. In order to avoid the above shortcomings, a novel and effective approach is developed in this paper to determine the location and the time history of the impact load on a thick aluminium plate. The dispersion relations based on the exact and the SDPT and the CHOTF are employed to determine the group velocity of the dispersive flexural mode and the location of load. The advantages of the proposed approach over the other available methods are clearly discussed and the improvement in the impact location estimation is pointed out. The time history of the impact load is determined using the theoretical Green s function (based on SDPT and exact plate theory) and the response measured at the sensor array located on the plate surface. The estimated impact location for the plate is shown to have an excellent agreement with the actual location of the load. 2 IMPACT RESPONSE OF A PLATE 2.1 Thick plate theory The theoretical solution to the response of a plate subject to surface or subsurface sources is a topic of considerable current research interest. The solution of three-dimensional problems involving homogeneous or multi-layered isotropic and anisotropic plates of finite thickness and large lateral dimensions subjected to various types of surface loads has been given earlier in references [19] to[25] and will not be repeated here. For a homogeneous, isotropic plate of infinite dimensions in the x 1 and x 2 directions (Fig. 1), the elastodynamic field can be conveniently expressed in terms of a six-dimensional, stress-displacement vector Fig. 1 Geometry of the concentrated load problem

3 Identification of impact force on a thick plate 1251 {Ŝ} in the frequency domain {Ŝ(k 1, k 2, x 3, ω)} ={û i ˆσ i3 }, i = 1, 2, 3 (1) where û i, ˆσ ij are the displacement and stress components, k 1 and k 2 are the wavenumbers in 1 and 2 directions, respectively, and ω is the circular frequency. The stress-displacement vector can be obtained in the frequency domain through the solution of a system of first-order equations supplemented by appropriate boundary conditions, resulting in a system of linear equations of order six. The boundary conditions on the faces of the plate are ˆσ 1 i3 (x 1, x 2,0,ω) = ˆF i (x 1, x 2, ω) ˆσ 1 i3 (x 1, x 2, H, ω) = 0 (2a) (2b) where, ˆF i is the Fourier-time transform of the spatially distributed impact load. In general the surface displacement can be expressed in the wavenumber integral form I = 1 4π 2 f (k 1, k 2, ω) g(k 1, k 2, ω) ei(k 1x 1 +k 2 x 2 ) dk 1 dk 2 (3) where the functions f and g are obtained by solving the system of linear equations as indicated earlier. This integral must be evaluated numerically for a large number of frequency points and the resulting spectra can then be inverted by fast Fourier transform (FFT) to determine the time-dependent displacement and stress components. 2.2 Recovery of the impact load For the plate shown in Fig. 1, the Green s function appropriate for the normal impact load problem represents the x 3 component of the displacement at (x 1, x 2, 0) to a concentrated normal force of δ(t) time dependence acting at the origin along the x 3 direction. It can be obtained in the frequency domain using the method described above and the time dependent Green s function, g3 3(x 1, x 2,0,t), can then be obtained by taking the inverse Fourier transform (FT) of the of the frequency domain Green s function, ĝ3 3(x 1, x 2,0,ω). Theoretically, if the time history of impact load is known, then the displacement can be calculated by convolving the load time history with the theoretical Green s function. Conversely, if the displacement u 3 (x 1, x 2,0,t) is measured on the plate, then the vertical impact load can be calculated by deconvolving the measured displacement with the theoretical Green s function. However, in the foreign object impact problem, the location of impact load is unknown and this must be determined first in order to recover its time history. This is done here by using the acoustic waveforms measured by three or more sensors distributed on the surface of the plate. In the calculation of the Green s function, the origin of the coordinates is placed at the impact point. 2.3 Source location In order to locate the impact source at an arbitrary point on the surface of an isotropic plate, three sensors are placed on the same surface on its surface and three intersecting circles are drawn, each centred at a sensor, with their radii determined by the travel times of the wave energy from the impact source to the respective sensor. By solving the three resulting simultaneous equations, a closed form expression for the source location can be obtained [6, 26]. If the sensors are located at S 0 (0, 0, 0), S 1 (x 1, y 1, 0), and S 2 (x 2, y 2, 0), then the polar coordinates of the source point, (r, θ) are given by A 1 r = 2(x 1 cos θ + y 1 sin θ + δ 1 ) A 12 = 2(x 2 cos θ + y 2 sin θ + δ 2 ) (4) where A 1 = x y2 1 δ2 1 and A 2 = x y2 2 δ2 2, δ 1 = t 1 c and δ 21 = t 2 c, c is the group velocity of the waves, and t 1 and t 2 are the time differences between sensor S 0 S 1 and S 0 S 2 respectively. The angle θ is obtained form the equation cos(θ ϕ) = K (5) where [ ] (A2 δ 1 A 1 δ 2 ) K = B (6) B =[(A 1 x 2 A 2 x 1 ) 2 + (A 1 y 2 A 2 y 1 ) 2 ] 1/2 (7) and tan ϕ = [ ] (A1 y 2 A 2 y 1 ) (A 1 x 2 A 2 x 1 ) (8) Thus the quantities to be extracted from the signals are the differences between the wave arrival times at the sensors. For the estimation of the radii of the circles, the group velocity of the waves is also required in addition to their arrival times. It is difficult to measure the group velocity of dispersive waves using traditional methods. The time-frequency method provides an effective method for determining the group velocity of the dispersive Lamb waves propagating across the plate in the impact test [6 8].

4 1252 S-K Lee, S Banerjee, and A Mal 3 HIGHER-ORDER TIME-FREQUENCY METHOD FT is one of the most useful tools for the spectral analysis of stationary signals associated with the dynamic response of structures. However, wave propagation in structural solids usually results in non-stationary signals since the wave speed in most structural components depends on frequency. Although the short time FT can be used for the analysis of nonstationary signals [27], it does not provide good time and frequency resolutions simultaneously because of the principle of uncertainty [28]. An approach based on the Wigner Ville distribution (WVD) [29] provides better time and frequency resolutions. The WVD defined through two equivalent forms given below W (t, f ) = s(t τ /2)s (t + τ /2)e 2πif τ dτ (9a) or = S( f ν/2)s ( f + ν/2)e 2πif ν dν (9b) where s(t) is the signal being analysed, S( f ) is its FT and denotes complex conjugation. The WVD has many desirable properties including good timefrequency resolution. However, its direct application is limited by the fact that it generates cross-terms. Specifically, the WVD of a signal s 1 (t) + s 2 (t) contains components due solely to s 1 (t) and s 2 (t), referred to as the auto-terms, along with an additional crossterm arising from the interaction between s 1 (t) and s 2 (t). They are distinguished from the auto-terms by virtue of the fact that the cross-terms are oscillatory, whereas the auto-terms are not. Hence applying a two-dimensional low pass filter to the WVD reduces the size of the cross-terms relative to the autoterms. The general form of a bilinear time-frequency representation is C(t, f ) = W (t, f ) ( f f, t t) dt df (10) where C(t, f ) is a general bilinear distribution and (t, f ) is a two-dimensional low pass filter. Different choices for the function (t, f ) result in a plethora of bilinear distributions. As an example, the Psuedo Wigner Ville distribution (PWVD) method mentioned in the introduction is one of the family of general bilinear distributions C(t, f ). The bilinear methods decompose a signal s energy, i.e. s(t) 2 dt, into a function of time and frequency. An extension of such analysis techniques is to consider signal decompositions that relate to higher moments, s(t) k dt, where k is an integer greater than two. The auto-terms appear in unexpected locations in the time-frequency plane. For this reason it is common to restrict one s attention to even order distributions. The current paper will concentrate on fourth-order distributions, being next even order greater than two. There are two forms of the fourth-order distribution [30, 31]. One is the sliced multi-frequency distribution SW f (t, f ) = S( f ν/4) 2 S ( f + ν/4) 2 e 2πiνt dν (11) and the other is the sliced multi-time distribution SW t (t, f ) = s(t τ /4) 2 s (t + τ /4) 2 e 2πif τ dτ (12) These forms of the representations have been chosen since they most clearly illustrate their connections with the WVD. The above signal representations can be thought of as decompositions of a signal s fourth moment, since SW t (t, f ) dt df = s(t) 4 dt (13a) and SW f (t, f ) dt df = S( f ) 4 df (13b) It should be noted that for the fourth moment there is no equivalent Parseval s theorem, that is S( f ) 4 df = s(t) 4 dt (14) This highlights the differences between the representations (11) and (12) they are decompositions of different quantities. From equations (11) and (12), one can obtain an intuitive notion about the suitability of the methods for different classes of signals. Consider the SW f (t, f ) of a narrowband signal, so that its contribution is centred on some peak in the FT domain.then the operation of squaring the FT, seen in equation (11), will serve to accentuate this peak. Similarly the operations involved in SW t (t, f ) serve to accentuate short duration transient signals. The former distribution is referred to as the sliced Wigner fourth-order moment spectrum (SWFOMS) [27]. The latter distribution was initially proposed as one of a more general class of distributions referred to as the L-Wigner distribution (LWD) [32]. An alternative approach is also needed to reduce the cross-term for these higher-order distributions. One method proposed for achieving this is based on the observation that SW f (t, f ) = 1 W ( f, t τ)w ( f, t + τ)dτ (15a) 2 SW t (t, f ) = 1 W (t, f ν)w (t, f + ν)dν (15b) 2 The above equations imply that the SWFOMS can be obtained by convolving the second-order WVD in

5 Identification of impact force on a thick plate 1253 the time domain while the LWD can be obtained by convolving in the frequency domain. The generalized versions of the SWFOMS and the LWD are given by SSW f (t, f ) = 1 h(τ) 2 C( f, t τ)c( f, t + τ)dτ 2 (16a) SSW t (t, f ) = 1 H(ν) 2 C(t, f ν)c(t, f + ν)dν 2 (16b) where C(t, f ) is any bilinear distribution described by equation (10) and h(t) is a windowing function. If the bilinear distribution is suitably smoothed, so that its cross-terms are removed, then a prudent choice of the weighting function would ensure that no additional cross-terms are generated, and the SWFOMS and LWD are essentially cross-term free [33, 34]. In references [33] and [34] it is recommended that h(t) be selected as a rectangular window, with its width (in time and frequency) no larger than the minimum separation between two components. Evidently the cross-term reduction is highly effective even for the non-oscillating components. It has been shown [35] that the SWFOMS is affected by non-oscillatory crossterms along the time axis, but not along the frequency axis. On the other hand, the LWD is not affected by non-oscillatory cross-terms along the time axis. Therefore, to represent a non-stationary signal, SWFOMS is a good choice when the frequency separation is important, and the LWD is a good choice when the time separation is important. Consequently, for dispersive Lamb mode waves, SWFOMS is a good choice at very low frequencies and LWD is better at higher frequencies. For the representation of the A 0 mode, SWFOMS is employed as the first step, and LWD is used at the next step. The higher-order time frequency representation of these two-steps is defined as the CHOTF [16]. In order to demonstrate the CHOTF, the contour maps are explained in time-frequency space for three simulated signals and one measured signal as shown in Fig. 2. Three simulated signals and the measured signal used for time-frequency analysis are as follows. 1. Simulated chirp signals, x = exp[ j(a 1 t 2 + b 1 + a 2 t 2 + b 2 )] with a 1 = , b 1 = 1000, a 2 = , b 2 = Fig. 2 Comparison of the time-frequency analysis for three simulated waveform signals and one measured waveform signal using: (a) simulation signal no. 1 (b) simulation signal no. 2 (c) simulation signal no. 3 (d) dispersive signal

6 1254 S-K Lee, S Banerjee, and A Mal Fig. 3 Comparison of WT analysis for three simulated waveform signals and one measured waveform signal using: (a) simulation signal no. 1 (b) simulation signal no. 2 (c) simulation signal no. 3 (d) dispersive signal 2. The same simulated chirp signals as in 1, but with background noise. The background noise is a normal distribution with zero mean, variance one, and standard deviation one. 3. Simulated frequency-modulation signal, x = exp [ j2π(b cos(at) + ct)] with a = 0.05, b = 0.4, c = Dispersive wave signal measured on a simple beam [16]. Figures 2(a), (b), (c), and (d) show the CHOTF, respectively, for each signal. In the figures, the horizontal axis represents time and the vertical axis represents frequency. The Fig. 2(a) are the results of the timefrequency analysis for the simulated chirp signals. The Fig. 2(b) shows the same results for the same simulated chirp signals but with background noise. Figure 2(c) shows the same results for the simulated frequencymodulation signal. Figure 2(d) shows the same results for a dispersive wave measured on the beam. From demonstration, the CHOFT is effectively used for the identification of flexural waves (A 0 mode). Although it has cross-terms problem for the multiple signals because it is based on Wigner distribution, it has a good time and frequency resolution after smoothing the cross-terms. Therefore, the time-frequency characteristic is clear. On the other hand, Figs 3(a), (b), (c), and (d) show the WT for each signal, respectively. It is clearly seen that WT failed to resolve the time-frequency characteristics due its poor time and frequency resolution, although it has advantage not having cross-terms for multiple signals. 4 IMPACT TEST ON AN ALUMINIUM PLATE Impact tests were carried out on a 6061-T6 grade aluminium plate with lateral dimensions 1200 mm 1200 mm and 10 mm in thickness. The material properties of plate are given in Table 1. A schematic Table 1 Aluminium plate Material properties Young s modulus (E) 70 Gpa Poisson s ratio (ν) 0.3 Density (ρ) 2760 kg/m 3 Thickness (H ) 10mm

7 Identification of impact force on a thick plate 1255 Fig. 4 Schematics of the experimental setup of the experimental setup is shown in Fig. 4. The impact load was excited by an instrumented impact hammer ENDEVECO (Model 2302) with a hemispherical aluminium tip. Figure 5 shows the sensor arrangement and the location of the impact. The sensors used to detect the wave signals in the plate were B&K accelerometer type The outputs from the accelerometers were amplified using the charge and the signal conditioner of ENDEVECO model number 133. The signals were then digitized using a TDS channels digital oscilloscope at a sampling rate of 50 khz. The software for the data transfer from oscilloscope to the computer is supplied by Texas Instrument Company. The test is repeated for 11 times by impacting the plate at the same point. The recorded data show similar results in each case, and omitted for brevity. 5 RESULTS AND DISCUSSION 5.1 Impulse response: theoretical and experimental validation The locations of the impact and three accelerometers, S 0, S 1, and S 2, are shown in Fig. 5. The theoretical displacements at the sensor locations are calculated using the approximate SDPT method as described by Mal [18] and the accelerations are obtained through careful double differentiations of displacements. Since the cut-off frequency of the measured accelerations is 25 khz, the theoretical accelerations are also calculated up to 25 khz only. The frequency resolutions and the sampling rates in both the theoretical and experimental accelerations are 20 Hz and 50 khz, respectively. Figure 6 shows the comparison between the calculated waveforms using SDPT and the measured waveforms. The waveforms in the top and bottom rows are the non-filtered and filtered signals, respectively. The measured waveforms show reflected waves from the edges of the plate while the calculated waveforms have only the direct waves from the source. The time history and spectrum of the impact load ˆF j=3 (x 1, x 2, ω) used for the simulation and measurement are shown in Fig. 7. Fig. 5 Locations of impact load, O, and the accelerometers (S 0, S 1, S 2 ) on an aluminium plate. OS 0 = 447 mm, OS 1 = 447 mm, OS 2 = 200 mm 5.2 Estimation of impact location The location of the impact source can be obtained through triangulation using the travel times of the waves from the source to the three sensors. For dispersive waves, the travel times are those of Lamb wave packets generated by the impact load and it is necessary to determine their group velocity. In order to estimate the arrival time and group velocity, the CHOTF is used on the filtered wave signals shown in

8 1256 S-K Lee, S Banerjee, and A Mal Fig. 6 Comparison between the theoretical and measured waveforms at the three sensors: (a) S 2, (b) S 0, and (c) S 1 Fig. 7 The time history and spectrum of the impact force measured by the force transducer Fig. 6. The contour map and image analysis for the signals in the time-frequency space are shown in Fig. 8. In the figures, the horizontal axis presents time and the vertical axis presents frequency. The dashed lines in the centre part in each map correspond to the peak amplitude, and these are plotted together in Fig. 9(a). The time difference as shown in Fig. 9 becomes the arrival time difference for the waveform measured by the sensors. The group velocity corresponding to these arrival time differences is calculated and the results are shown in Fig. 9(b). The group velocity of the Lamb waves propagating in a plate can be obtained from the solution of their dispersion equations, which can be derived either using the exact elasticity theory or various approximate formulations. The approximate formulations are generally more efficient in producing results, but may lack the desired accuracy. In the frequency range of interest here the measured signals are dominated by the first flexural (A 0 ) mode of the Lamb waves. The dispersion curves for these waves in the aluminium plate calculated by using the exact theory and the approximate SDPT are plotted in Fig. 10. It can be seen that the two results are very close in the frequency and velocity ranges considered here. Thus, the approximate theory is adequate for the present application. The group velocity (dashed line in Fig. 10) is obtained by calculating slope of the angular frequency as a function of the wave number for the A 0 mode [24]. The location of impact load is estimated using the group velocity and the arrival time difference calculated by the CHOTF. Figure 11 shows the location of the impact load estimated by using the group velocity at 2900 Hz. Figure 12 shows the location of impact load estimated by using the arrival times and the group velocity calculated by the CHOTF. Figure 12(a) shows the estimated location of the source and Fig. 12(b) shows the distance error between true location and estimated location. According to these results, the estimation of the location of impact load

9 Identification of impact force on a thick plate 1257 Fig. 9 Arrival time differences and group velocity for the waves measured by the sensors S 0, S 1, and S 2. (a) arrival time differences obtained by the CHOTF and (b) group velocity corresponding to the arrival time differences Fig. 8 CHOTF for the filtered signals of the measured waveforms at (a) sensor S 2, (b) sensor S 0, and (c) sensor S 1 space for each sensor location for these filtered waveforms using the CHOTF are shown in Fig. 13. According to these results, the dispersion curves for the A 0 Lamb modes are clearly shown in Fig. 13. As in Fig. 8, the dashed lines in the centre part of each map correspond to the peak amplitude, and these lines are re-plotted in Fig. 14(a) in order to calculate the arrival times. The group velocity corresponding to these arrival time by the CHOTF is effective since the distance errors are within 5 per cent. In the above example, the frequency of the measured signals is around 20 khz, which can be considered to be in the lower range for Lamb waves in a plate of 10 mm thickness. In order to apply the technique at higher frequencies simulated signals with higherfrequency content can be used. The simulated waveforms at the sensor locations due to a force of δ(t) time dependence acting at the origin are calculated using the approximate SDPT [18]. A band pass filter from 10 to 200 khz is applied to these waveforms. The contour map and image analysis in time-frequency Fig. 10 Theoretical phase and group velocities of Lamb waves in the A 0 mode in a 10 mm thick aluminium plate

10 1258 S-K Lee, S Banerjee, and A Mal Fig. 11 Estimation of the location of impact load using the measured arrival times at 2.9 khz and the corresponding group velocity differences is calculated and compared with theoretically calculated group velocity in Fig. 14(b). The location of the impact load is estimated using the group velocity and the arrival time differences calculated by the CHOTF. Figure 15 shows the location of the impact load estimated by using the group velocity at 48 khz. Figure 16 shows the location of impact load estimated by using the arrival times and the group velocity calculated by the CHOTF. Figure 16(a) shows the estimated location of the source and Fig. 16(b) shows the distance error between true location and estimated location. According to these results, the estimation of the location of impact load by the CHOTF is effective even at high frequencies since the distance errors are very small. The simulated waveform due to the same force is also calculated using the exact theory [18] under the same conditions as in the previous case. The results of the time-frequency analysis for these filtered waveforms using the CHOTF are shown in Figs 17 to 20. In this case the dispersion curves for the A 0 as well as other modes are present at higher frequencies (70 khz). According to these results, the Fig. 13 CHOTF for the waveforms simulated by the approximated SDPT (a) sensor S 2, (b) sensor S 0, and (c) sensor S 1 Fig. 12 Estimation of source location and error estimates using the group velocity for the waveforms measured at the sensors: (a) distance from true source (b) error in the estimated distance group velocity for A 0 mode calculated by CHOTF corresponds exactly with the theoretical group velocity up to 70 khz. Figure 19 shows the location of the impact load estimated by using the group velocity at 48 khz. Figure 20(a) shows the location of impact load estimated by using the arrival times and the group velocity calculated by the CHOTF. Figure 20(b) shows the estimated location of the source and distance error between true location and estimated location. These results indicate that the distance error estimated by using exact theory is smaller than that estimated by SDPT from 10 khz to 70 khz, due to the fact that the group velocity calculated from SDPT is overestimated at higher frequencies [21]. Figure 21 shows this overestimation through comparison between the waveforms obtained from SDPT and form exact solution at higher frequencies. At lower frequencies this overestimation

11 Identification of impact force on a thick plate 1259 Fig. 16 Estimation of source location and distance error using the group velocity for the waveform simulated by SDPT (a) distance from true source and (b) error in distance estimate Fig. 14 Arrival time differences for the waveform simulated by SDPT. (a) Arrival time difference obtained by the CHOTF (b) group velocity corresponding to arrival time difference is not significant as shown in Fig. 22. At frequencies over 70 khz the presence of higher Lamb wave modes in addition to the A 0 mode in the exact simulation introduces errors in the estimation of the source location. The distance errors are within 1 per cent, which proves the effectiveness of the proposed approach in comparison to the errors of 10 per cent reported earlier by using WT and CPT [8, 17]. Fig. 15 Estimation of the location of impact load using the arrival time at 49 khz and its corresponding group velocity for the waveforms simulated by SDPT Fig. 17 CHOTF for the waveforms simulated by the exact solution (a) sensor S 2, (b) sensor S 0, and (c) sensor S 1

12 1260 S-K Lee, S Banerjee, and A Mal Fig. 20 Estimation of source location and the error in distance using the group velocity for the waveforms simulated by the exact theory and SDPT. (+) exact solution ( ) SDPT (a) distance from true source and (b) error in distance estimate Fig. 18 Arrival time difference for the waveforms simulated by the exact solution (a) arrival time difference obtained by the CHOTF (b) group velocity corresponding to arrival time difference 5.3 Recovered time history of the impact load As indicated earlier, the time dependence of the impact load can be recovered by deconvolving the waveforms measured at each sensor with the Green s functions. The theoretical Green s functions are the Fig. 21 The simulated waveforms for a force f (t) = δ(t) using SDPT in the high frequency range 10 to 200 khz at (a) S 2, (b) S 0, and (c) S 1 Fig. 19 Estimation of the location of impact load using the arrival time at 49 khz and the corresponding group velocity for the waveforms simulated by the exact solution Fig. 22 Simulated waveforms using SDPT in the low frequency range between 2 and 4 khz. (a) measured force, (b) waveform at sensor S 2, (c) waveform at sensor S 0, and (d) waveform at sensor S 1

13 Identification of impact force on a thick plate 1261 normal components of the displacement at the sensor locations due to a normal concentrated force of time dependence δ(t) at the impact location. They can be obtained from the exact formulation or the approximate SDPT using the procedure described in reference [18]. Since the accelerations are measured in the experiments, they are obtained from the displacements through careful double differentiations of the displacements. The time histories of these Green s functions using accelerations are plotted in Fig. 23. Since the measured accelerations include the waves reflected from the edges of the plate, while the calculated Green s functions are for a plate of infinite lateral dimensions, the time duration of the measured response used for the estimation the time history of impact load is selected to eliminate these reflections. In the current study, 0.6 ms is selected as the time duration because the propagation energy of the direct waveform is concentrated in that range as shown in Fig. 23. The reflected part in the measured is replaced by zero padding. The time history of the estimated impact load is compared with that of true impact in Fig. 24. The figures in the top row of Fig. 24 show the comparison between the measured impact load and the estimated impact load, while Fig. 23 Theoretical green functions g3 3 (t) between the impact point and sensors in the plate are obtained by using the approximated SDPT (a) sensor S 2, (b) sensor S 0, and (c) sensor S 1 Fig. 24 The comparison between the measured true impact load and the estimated impact load and the comparison between the spectrum of measured impact load ˆF j=3 (ω) and that of the estimated impact load. ( ) time history of true impact, ( ) time history of estimated impact load (a) sensor S 2, (b) sensor S 0, and (c) sensor S 1

14 1262 S-K Lee, S Banerjee, and A Mal those in the bottom row show their FTs. According to theses results, the impact load is very well estimated using the impulse response at S 2, although there is some error due to the error in the estimated location. The estimations of the impact load using the impulse response at S 0 and S 1 also have some errors. 6 CONCLUSIONS The location and time history of an impact load on a thick aluminium plate is identified through the use of the time-frequency method and deconvolution with the theoretical Green s function. An instrumented impact hammer is used as the application of the impact load on the plate surface and the waves generated by the load are recorded by three accelerometers located on the same surface. The location of the impact load is determined using the arrival time and group velocity of the waves using the time-frequency analysis. The CHOTF is examined for simulated as well as measured waveform signals. 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