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The use of multisine excitations to characterise damage in structures
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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 43–57 www.elsevier.com/locate/jnlabr/ymssp
The use of multisine excitations to characterise damage in structures K. Vanhoenackera,*, J. Schoukensa, P. Guillaumeb, S. Vanlanduita a
Department of Electrical Engineering, ELEC, Vrije Universiteit Brussel, Pleinlaan 02, Brussels 1050, Belgium b Department of Mechanical Engineering, WERK, Vrije Universiteit Brussel, Belgium Received 4 August 2000; received in revised form 5 November 2002; accepted 11 March 2003
Abstract In order to detect the presence of damage and imperfections in materials, a new and promising method for non-destructive material testing has been developed. The technique focuses on the non-linear distortions that are present in the results of a frequency response function (FRF) or transfer function measurement of the sample. The kernel idea in the described method is to use well-chosen periodic excitations where only some of the considered frequency components are excited. The non-excited frequency lines are used to detect, qualify (even or odd non-linear distortions) and quantify (What is the level of the non-linear distortions?) the non-linear distortions. Undamaged materials are often essentially linear in their response. However, the non-linear behaviour of the same material increases significantly when damage appears. The method is applied in the field of damage detection and health monitoring. The method is illustrated by experiments on uncracked and cracked artificial slate beams used in civil constructions and during mechanical cyclic fatigue loading. The developed technique demonstrated to be a very fast and efficient tool to assess global damage in a material. r 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction A desire to monitor a structure and detect damage at the earliest possible stage is pervasive throughout the civil, mechanical and aerospace engineering communities. From the early 1970s on, different attempts were made to detect damage in structures from vibration measurements (see [1] for an overview). The vast majority of the work has considered only linear types of structural damage, such as a reduction in stiffness, a non-closing notch, or a change in geometry like removal of a member in a truss structure. The main idea is that a change in a structure should *Corresponding author. Fax: +32-2-629-28-50. E-mail addresses: [email protected] (K. Vanhoenacker), [email protected] (J. Schoukens), [email protected] (P. Guillaume), [email protected] (S. Vanlanduit). 0888-3270/04/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0888-3270(03)00044-X
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imply a change in the modal parameters of the structure (resonance frequency, mode shape, mode shape curvature, etc.). Because different damage scenarios result in a non-linear behaviour, the modal parameters are not always suitable as a damage detection feature. In the past, many studies have illustrated that a crack in a structure may cause the structure to exhibit non-linear behaviour if the crack is alternately open during a part of the response and closed during the remaining time intervals [1–3]. Recently, non-linear damage detection methods were introduced in the literature. In [4], it was demonstrated that the non-linear damage detection techniques are much more sensitive to the presence of damage than the classical linear methods. A disadvantage of using non-linear damage detection techniques is the assumption that a fault causes the structure to behave non-linearly. A beam with a crack that is open throughout the response cycles cannot be detected with this method. In this article a new and fast method, that combines the advantages of the existing linear and non-linear damage detection methods, will be proposed to detect damage in materials and to observe the impact of cyclic fatigue loading on the material. The method uses a broadband excitation signal that allows estimating, in the response, immediately the best linear approximation (also called related linear dynamic system—RLDS) of the overall system, the even and the odd non-linear contributions, and the noise contributions. The paper is organised as follows. In Section 2 we briefly discuss some linear and non-linear damage detection techniques. Section 3 presents a novel non-linear damage detection technique. The theory is illustrated by some real measurement examples in Section 4. Conclusions drawn from the research can be found in Section 5.
2. Overview of damage detection techniques 2.1. Linear damage detection techniques Damage or fault detection, as determined by changes in the dynamic properties or response of structures, is a subject that has received considerable attention in the literature. The basic idea is that modal parameters (notably frequencies, mode shapes, mode shape curvature and flexibility matrices) are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause changes in the modal properties. The observation that changes in structural properties cause changes in vibration frequencies was the impetus for using modal methods for damage identification and health monitoring [1,15]. The use of frequency shifts for the detection and succession of damage have significant practical limitations. First, the somewhat low sensitivity of frequency shifts to damage requires either very precise measurements or large levels of damage. Secondly, since the modal frequencies are a global property of the structure, it is not clear that shifts in this parameter can be used to identify more than the existence of damage. The underlying principle behind the use of mode shapes for fault detection is that changes in modes are sensitive indicators of changes in the physical integrity of any mechanical structure [1,14,15]. When vibrational changes take place in a structure, these changes can be quantified with commonly understood frequency domain measurements that are curve fit to extract the modal properties of the damaged structure. These modal properties are then compared with
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baseline (undamaged) properties, and the changes can be used to determine the severity of the damage. Mode shape derivates, such as the curvature can be used to obtain spatially distributed features sensitive to damage [1,15]. Mode shape curvature can be computed by numerically differentiating the identified mode shape vectors twice to obtain an estimate of the curvature. This method is motivated by the fact that the second derivate of the mode shape is more sensitive to small perturbations in the system than the mode shape itself. Changes in the flexibility matrix indices have been used as damage sensitive features [1,15]. The flexibility matrix is estimated from the mass normalised measured mode shapes and the measured eigenvalue matrix. The formulation of the flexibility matrix is approximate because in most cases not all the modes of the structure are measured. Using flexibility matrices, damage is detected by comparing the flexibility matrix indices computed using the modes of the damaged structure with the flexibility matrix indices computed using the modes of the undamaged structure. Because of the inverse relationship to the square of the modal frequencies, the measured flexibility matrix is most sensitive to changes in the lower frequency modes of the structure. 2.2. Non-linear damage detection techniques Identification of the basic modal properties, mode shape curvature changes, and the flexibility matrices are based on the assumption that a linear model represents the structural response before and after damage. However, in many cases the damage will cause the structure to exhibit a nonlinear response. Therefore, the identification of features indicative of non-linear response can be a very effective means of identifying damage in a structure that originally exhibited a linear response. In the last years, a number of articles were published which use the measured non-linear contributions to predict if there is damage present or not. Assuming that the damage causes the structure to behave non-linearly (i.e. not as in the case of a beam with a crack which is open throughout the response cycles), these non-linear techniques provide several distinct advantages over the other methods for detecting faults in structures. In some cases these non-linear contributions are even used to localise and to quantify the damage. A first technique is based on measuring the higher order frequency response functions (FRFs) of the structure [5,6]. The analysis of these functions shows that they are highly dependent upon the size and the position of the crack and thus can be used as damage indicators. The correlation of the shape and the value of these functions with the crack size and position in the analytical model of the structure can be used for a sensitive structural damage-identification procedure (determination of the location, quantification of the severity of the damage and the prediction of the remaining service life of these structures). A disadvantage of this method is that the higher order FRFs are measured using stepped sine excitations, which implicate that the measurement time is rather high. The biggest disadvantage of this technique is that an analytical model of the structure or damage case must be available, which reduces seriously the applicability of this method. A second technique, Non-linear Elastic Wave Spectroscopy (NEWS), studies the amplitudedependent frequency response in dynamic wave experiments. Two different methods can be distinguished:
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One method is Non-linear Wave Modulation Spectroscopy (NWMS) [3,7]. It consists of exciting a sample with continuous waves of two separate frequencies simultaneously, and inspecting the harmonics of the two waves, and their sum and difference frequencies (side bands). Undamaged materials are essentially linear in their response to the two waves, while the same material, when damaged, becomes highly non-linear, manifested by harmonics and side band generation. Qualitatively we can say that the more damaged a material is, the larger is its nonlinear response. This method can be quickly applied and is ideally suited to applications where the question of damaged versus undamaged must be quickly addressed. This method can be applied to any type of geometry. The second method is Non-linear Resonant Ultrasound Spectroscopy (NRUS) [8], and depends on the study of the non-linear response of a single, or a group of, resonant modes within the material. Undamaged materials are essentially linear in their resonant response. The same material, however, becomes highly non-linear when damaged, manifested by amplitudedependent resonance frequency shifts, the generation of (principally third-order) harmonics and non-linear attenuation. Also in this method, the amount of non-linearity is highly correlated to the damaged state of the material. The method is extremely useful for basic research and specific applications that do not have strict requirements in terms of speed and application. This method can also be applied to any type of geometry and can be used during mechanical cyclic fatigue loading. The disadvantage of this method is the complex measurement set-up and the fact that a stepped sine is used as excitation signal.
3. The proposed technique: characterizing non-linear distortions during FRF measurements 3.1. Characterizing the non-linear distortions The presented damage detection technique is based on a method that was developed in [9] and that allows the detection and the qualification of non-linear distortions on the results of an FRF measurement without wasting measurement time. The kernel idea in this method to detect the non-linear distortions is to use well-chosen periodic excitations, called random multisines: ! ! N X 1 ð1Þ Xk ej2pðfmax =NÞkt xðtÞ ¼ pffiffiffiffiffi N k¼N with Xk ¼ Xk ¼ X ðkÞejjk ; fmax the maximum frequency of the excitation signal, NAN the number of frequency components, X ðkÞARþ ; and the phases jk are a realization of an independent uniformly distributed random process on ½0; 2p; so that E½ejjk ¼ 0: These multisines combine the advantage of a random behaviour (allows the measurement of the best linear approximation of the non-linear system [11]) while they still guarantee the generation of the user selected power spectrum in each realization. A nice property of the random multisines is their possibility to pick up linear and non-linear information in a separable way. This can be done within the same excitation signal. The main principle in the technique to visualise the non-linear distortions is to excite only a well-chosen set of frequency lines in the multisine. This implies that some frequency lines (detection lines) are
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consciously not excited. These will be used as detection lines to qualify (even or odd) and quantify (what is the level?) the non-linear distortions on the neighbouring excited lines (called measurement lines). It can be shown that random multisines, exciting only at a selected set of the odd frequency lines, can be used for the detection, qualification and the quantification of the non-linear distortions. The even non-linearities will under these conditions only contribute to the even frequency lines, and hence they do not disturb any more the FRF measurements [11]. Under perfect circumstances, the power of the even and the odd non-linear distortions in the output can be determined, respectively, at the even and the odd detection lines. Linear and odd non-linear contributions are present at the measurement lines. With the knowledge of the amplitude of the non-linear distortions at the odd detection lines it is, under the ideal circumstances, immediately possible to estimate the level of the non-linear contributions present at the excitation lines. In [11] it was shown that the random multisines can also be used to determine the RLDS of the device under test (DUT). It was also shown that the influence of non-linear distortions on the FRF-measurements with random multisines can be split in systematic and stochastic contributions. Since we do not use these data for the damage detection and health monitoring, we do not discuss these extended techniques here. 3.2. Example The most simple random multisine is a special-odd multisine [9,13], where only the frequencies 8k þ 1 and 8k þ 3; k ¼ 0; 1; 2; y; kmax in Eq. (1) have amplitudes different from zero. Next the output spectrum is calculated with a DFT (implemented as an FFT) using a rectangular window. In [11] it was shown that the even non-linearities excite only the even harmonics at the output ð2k; k ¼ 1; 2; yÞ; while the odd non-linearities appear only at the odd harmonics ð2k þ 1; k ¼ 1; 2; yÞ: Due to the choice of the excitation signal we get the following possibilities: * * * * * * *
at at at at at at at
lines lines lines lines lines lines lines
8k þ 1: 8k þ 2: 8k þ 3: 8k þ 4: 8k þ 5: 8k þ 6: 8k þ 7:
the output consist of linear contributions+odd non-linear distortions, only even non-linear distortions appear, the output consist of linear contributions+odd non-linear distortions, only even non-linear distortions appear, only odd non-linear distortions appear, only even non-linear distortions appear, only odd non-linear distortions appear.
This allows getting an idea of the non-linear behaviour of the system. If at least MX2 successive periods are measured in one block, it is still possible to make the same conclusions (respectively at lines Mð8k þ 1Þ; Mð8k þ 2Þ; Mð8k þ 3Þ; Mð8k þ 4Þ; Mð8k þ 5Þ; Mð8k þ 6Þ and Mð8k þ 7Þ). On top of that, also the noise level can be characterised by looking at the lines that are no multiple of M since these cannot be excited by a signal with M periods in the window. So only the noise (having a non-periodic behaviour) can contribute these. Other types of multisines (as described in [13]) can also be used for the detection, qualification and quantification of the non-linear distortions.
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3.3. Applying this multisine technique in the case of a non-ideal behaviour of the input signal In practice, it can happen that the real input differs from the desired input. Due to the presence of a low-quality generator, the presence of a non-linear actuator or due to the interaction of the generator with the non-linear structure, unwanted excitation power can be present at the detection lines in the excitation signal. At that moment, the output at the detection lines consist not only out of non-linear contributions, but also out of linear contributions originating from the unwanted excitation power at the detection lines at the input of the structure. So, it is necessary to compensate for these distortions (using for example a time consuming software feedback mechanism [10]), or to make a fast first-order correction of the measurements with a compensation method. Hereby, the linear contribution on the detection frequencies is eliminated, so that the compensated output depends only on the non-linear contributions. This first-order compensation method is extensively described and illustrated in [9].
4. Experiments and configuration The described multisine excitation method is illustrated on measurements of a vibrating rectangular beam. In a first experiment, the amplitude of the non-linear contributions in the response obtained with a damaged and an undamaged slate beam will be compared. In a second experiment, the evolution of the non-linear contributions in the response of the FRF measurements will be followed in the case that an originally undamaged slate beam is subject to cyclic fatigue loading. In both cases, we will investigate if they come into being and the presence of damage in the beam can be detected by an increasing of the non-linear contributions in the response of the FRF measurements. 4.1. Experimental set-up The experimental equipment used to obtain the results discussed in this paper is shown in Fig. 1. The experiments are applied on thin, rectangular beams of artificial slate used in civil constructions. The centre of the slate samples are placed on a B&K impedance head (type 8001), which is assembled on a B&K vibration exciter (type 4809). The measurements were driven by a Pentium II 400 MHz. A special-odd random multisine was used for the measurement of the non-linear contributions. Only the frequencies fk ¼ ð8k þ 1Þf0 and ð8k þ 3Þf0 ; k ¼ 0; 1; 2; y; 72 (harmonics 139111719?577) are excited with constant amplitude and random phase. The sampling frequency was set to fs ¼ 3200 Hz. The fundamental frequency, f0 ¼ ðfs =NÞE0:391 Hz, so that one period of the signal contains N ¼ 8196 points. The generator signals were designed in MATLAB and sent to a generator/acquisition rack, which consists of HPE1433A, HPE1434A and National Instruments VXI1394 cards. Communication between PC and VXI rack was established using home-developed software [12]. The generator signals were sent to a power amplifier (B&K type 2706) before being fed to the shaker. Similarly, the signals coming from the force and the acceleration sensors were first amplified with a B&K charge amplifier (type 2635)
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Sample
Acceleration
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Power Amplifier VXI generator/acquisition rack
Matlab Pentium II 400MHz
Fig. 1. Diagram of the experimental set-up.
350 mm 30 mm
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shaker position
8 mm
extra weight (100 g)
Fig. 2. Specimen geometry.
before being sent to the acquisition rack. One single measurement requires a measurement time of about 2.6 s. 4.2. Comparison of a damaged with an undamaged sample In a first experiment to qualify the value of multisine excitations in a damage detection process, a comparison was made between the results obtained with a damaged and an undamaged slate sample. The slate samples consist of two 4 mm thick slate layers. The second layer is separated in two parts (see Fig. 2) and glued under the first layer. This enables the simulation of a breathing crack. The undamaged state was obtained by glueing the separation in the second layer. Two extra weights of each 100 g were glued on the ends of the device in order to assure that a breathing crack was obtained during vibration. The nominal dimensions of the beam used in the present study are 350 mm 30 mm (2 4) mm. The crack was located 87 mm from one end. The beam was constructed from pieces of equal thickness, so the crack penetrates to half the beam thickness. The amplitude level of the special-odd random multisine excitation signal was chosen to be very small, so that no extra damage was generated in the structure. In both cases, the experiment consisted out of measuring 10 consecutive periods of the same multisine. The measurements were
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Fig. 3. Results of the proposed technique on an intact (left) and on a cracked slate beam (right). These figures depict an overview of the results obtained in one single experiment, which consisted out of 10 consecutive measurements. Top row: measured FRF; Second row: measured odd non-linear contributions (after compensation); Bottom row: measured even non-linear contributions (after compensation).
started once the transient disappeared. Fig. 3 illustrates the experimental data obtained with the cracked and the uncracked slate samples. It can be seen that the shape of the transfer functions of both samples are almost equal (Fig. 3, top). In both cases, the resonance frequency appears at
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68 Hz. No frequency shift can be observed. The variation of the FRF in the frequency range between 0 and 10 Hz can be explained due to the fact that the B&K vibration exciter (type 4809) gives only accurate results between 10 Hz and 20kHz. The only obvious difference in the FRFs of both samples is that the cracked beam shows a much larger value of the FRF at the resonance peak. The same remarks can be very clearly noticed when the even and the odd non-linear contributions obtained with the undamaged sample are compared with these of the cracked sample. The even and the odd non-linear contributions plotted in Fig. 3 are these that were obtained after compensation. At the resonance frequency, the even and the odd non-linear contributions of the cracked sample are almost 10 dB larger than the even and the odd non-linear contributions of the intact sample. Remark again that the difference in non-linear contributions appears only around the resonance frequency. In the other frequency bands, no clear difference in non-linear contributions can be observed. Comparing the amplitude of the linear contributions (not shown) with the even and the odd non-linear contributions reveals that the non-linear contributions are in the full frequency band smaller than the linear contributions, so it was permitted to use the compensation method. The same experiment, consisting out of 200 consecutive measurements of the FRF of both samples, provided the same results. 4.3. Evolution of the non-linear contributions during a cyclic fatigue loading experiment The second experiment aimed at damage assessment with multisine excitations during a controlled fatigue loading. The nominal dimensions of the beam used in this experiment are 400 mm 22 mm 4 mm. Again two extra weights of each 100 g were glued on the ends of the device to subject the beam to mechanical aging (see Fig. 4). To validate the proposed technique, the response of this beam is investigated in the frequency range between 5 and 225 Hz. In this range, the first and the second bending mode of the beam are present. The fatigue loading was generated by the excitation signal, in combination with the two extra weights. The amplitude level of the special-odd random multisine excitation signal was chosen high enough to induce permanent fatigue damage after several tens of cycles. After 6 min, a macro crack was visible at the surface. During these 6 min, 90 successive periods of the same multisine were measured in order to monitor the evolution of the state of the beam. Just like in case of the comparison of the damaged with an undamaged sample, we followed the evolution of the FRF and the compensated even and the odd non-linear contributions in the output. The measurements were started once the transient disappeared. The evolution of the FRF of the beam during the fatigue loading is plotted in Fig. 5. A significant increase of the first resonance frequency (at 14 Hz) can be observed. The increase of the 400 mm 4 mm
22 mm
extra weight (100g)
shaker position
Fig. 4. Specimen geometry.
extra weight (100 g)
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Fig. 5. 3-dimensional representation of the evolution of the measured FRF during cyclic fatigue loading experiment on a slate beam.
first resonance frequency starts already from the beginning of the experiment and proceeds faster when the experiment makes progress. A significant increase of the amplitude of the first mode appears 15 measurements before the end of the experiment. At the end, the amplitude of the first mode is at least 15 dB larger compared with the beginning. No frequency shift of the first resonance frequency can be observed. The second mode (around 176 Hz) however, shows a completely different behaviour. The amplitude of the second mode does not change significantly during the experiment and even decreases at the end. On the other side, small frequency shifts can occasionally be observed (see Fig. 6). These are probably due to the development of micro-cracks in the beam. Around measurement 75, a significant frequency shift appears. This corresponds with the remarkable increase of the amplitude of the first resonance frequency. Fig. 7 illustrates the evolution of the odd non-linear contributions as function of the time (number of measurements) and the frequency. The highest values of the odd non-linear contributions are present at the resonance frequencies. The odd non-linear contributions at the first resonance frequency exhibit the same behaviour as the FRF at this resonance frequency. The odd non-linear contributions increase during the experiment, and adopt during the last 30 measurements extreme large values, which are almost 17 dB larger compared to their starting values. Remark that these large values of the odd non-linear contributions appear in a very narrow frequency band around the first resonance frequency. The expansion of the amplitude of the odd non-linear contributions around the second resonance frequency follows the same evolution as the odd non-linear contributions at the first resonance frequency. However, there are some serious differences. The maximal values of amplitude of the odd non-linear at the second resonance frequency are smaller compared to the values obtained at the first resonance frequency. The values of the odd non-linear contributions at the end of the experiment adopt values who are merely 10 dB larger compared to the beginning of the experiment. On the other side, it is clearly
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Fig. 6. Partial enlargement (of the second mode) of the evolution of the measured FRF during cyclic fatigue loading experiment on a slate beam.
Fig. 7. Three-dimensional representation of the evolution of the measured odd non-linear contributions during cyclic fatigue loading experiment on a slate beam.
visible (see Figs. 7 and 8) that the frequency band in which larger values of the odd non-linear contributions are attained becomes much wider during the fatigue loading experiment. Significant higher values for the odd non-linear contributions can be observed in a frequency range up to 40 Hz at both sides of the second resonance frequency. A small shift of the second resonance frequency can also be observed. In the remaining part of the measured frequency band no changes of the odd non-linear contributions worth mentioning are observed. The evolution of the even non-linear contributions as function of the time (number of measurements) and the frequency is plotted in Fig. 9. The highest values of the even non-linear contributions are present at the resonance frequencies. The even non-linear contributions at the
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Fig. 8. Evolution of the odd non-linear contributions at certain frequencies during the cyclic fatigue loading experiment on a slate beam.
Fig. 9. Three-dimensional representation of the evolution of the measured even non-linear contributions during cyclic fatigue loading experiment on a slate beam.
first resonance frequency exhibit almost the same behaviour as the odd non-linear contributions at this first resonance frequency. The only difference that can be observed is that the extreme large values, which are obtained in the last 20 measurements are merely 13 dB larger compared to their starting values. Remark again that these large values of the even non-linear contributions appear in a very narrow frequency band around the first resonance frequency. The expansion of the amplitude of the even non-linear contributions around the second resonance frequency is smaller than the expansion around the first resonance frequency. The values of the even non-linear contributions at the end of the experiment adopt values, which are
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Fig. 10. Evolution of the even non-linear contributions at certain frequencies during the cyclic fatigue loading experiment on a slate beam.
5 dB larger compared to the beginning of the experiment. From Figs. 9 and 10, it can also be seen that the frequency band in which larger values of the even non-linear contributions are attained does not become wider during the fatigue loading experiment. However, a clear frequency shift of about 5 Hz can be observed. In the remaining part of the measured frequency band no changes of the odd non-linear contributions worth mentioning are observed.
5. Conclusions In this article a new and promising method for non-destructive material testing has been developed and applied in the field of damage detection and succession. The method uses a periodic broadband multisine excitation that allows estimating, in the response, immediately the FRF of the overall system and the even and the odd non-linear contributions. We have shown by means of two case studies that the use of multisine excitations can be used to give additional information about the presence of damage in a structure. Experimental data from a beam with an opening and a closing crack, and a mechanical cyclic fatigue loading experiment on a beam have demonstrated that undamaged materials are essentially linear in their response. However, the non-linear behaviour of the same material increases significantly when damage appears. The progressive damage/fatigue experiment discussed in this paper clearly illustrate that the sensitivity of the odd non-linear contributions to the detection of damage features is, for this case superior than the even non-linear contributions and the FRF of the overall system. The described multisine excitation technique is a very fast and efficient technique to assess and to follow the evolution of global damage in a material. This multisine excitation technique allows the user to examine the non-linear behaviour of the structure, while it is on the same time, with the
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same results possible to investigate the FRFs with the linear damage detection techniques. This means also that the results of this method can be used in the case that the damage does not introduce non-linear behaviour. This method is also superior compared to the other nonlinear methods due to the fact that it is possible to assess and to follow the evolution of the damage in a material in situ and in working conditions. The ability to use the compensation method minimises the influence of the disturbing contributions originated from the measurement equipment.
Acknowledgements This research was supported by a grant of the Flemish Institute for the improvement of the scientific-technological research in industry (IWT), the Belgian National Fund for Scientific Research, Flanders, the Flemish government (GOA-IMMI), and the Belgian government as a part of the Belgian program on Interuniversity Poles of attraction (IUAP4/2) initiated by the Belgian State, Prime Ministers Office, Science Policy Programming.
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[13] K. Vanhoenacker, T. Dobrowiecki, J. Schoukens, Design of multisine excitations to characterize the nonlinear distortions during FRF-measurements, IEEE Transactions on instrumentation and measurement 50 (5) (2001) 1097–1102. [14] M.A. Mannan, P. McHargue, M.H. Richardson, Continuous monitoring of modal parameters to quantify structural damage, Proceedings of the 12th International Modal Analysis Conference, Honolulu, Hawaii, 1993, pp. 59–65. [15] C.R. Farrar, S.W. Doebling, T.A. Duffey, Vibration-based damage detection, presented at the SD2000 Structural Dynamics Forum, April 11–17, 1999.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 18 (2004) 43–57 www.elsevier.com/locate/jnlabr/ymssp
The use of multisine excitations to characterise damage in structures K. Vanhoenackera,*, J. Schoukensa, P. Guillaumeb, S. Vanlanduita a
Department of Electrical Engineering, ELEC, Vrije Universiteit Brussel, Pleinlaan 02, Brussels 1050, Belgium b Department of Mechanical Engineering, WERK, Vrije Universiteit Brussel, Belgium Received 4 August 2000; received in revised form 5 November 2002; accepted 11 March 2003
Abstract In order to detect the presence of damage and imperfections in materials, a new and promising method for non-destructive material testing has been developed. The technique focuses on the non-linear distortions that are present in the results of a frequency response function (FRF) or transfer function measurement of the sample. The kernel idea in the described method is to use well-chosen periodic excitations where only some of the considered frequency components are excited. The non-excited frequency lines are used to detect, qualify (even or odd non-linear distortions) and quantify (What is the level of the non-linear distortions?) the non-linear distortions. Undamaged materials are often essentially linear in their response. However, the non-linear behaviour of the same material increases significantly when damage appears. The method is applied in the field of damage detection and health monitoring. The method is illustrated by experiments on uncracked and cracked artificial slate beams used in civil constructions and during mechanical cyclic fatigue loading. The developed technique demonstrated to be a very fast and efficient tool to assess global damage in a material. r 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction A desire to monitor a structure and detect damage at the earliest possible stage is pervasive throughout the civil, mechanical and aerospace engineering communities. From the early 1970s on, different attempts were made to detect damage in structures from vibration measurements (see [1] for an overview). The vast majority of the work has considered only linear types of structural damage, such as a reduction in stiffness, a non-closing notch, or a change in geometry like removal of a member in a truss structure. The main idea is that a change in a structure should *Corresponding author. Fax: +32-2-629-28-50. E-mail addresses: [email protected] (K. Vanhoenacker), [email protected] (J. Schoukens), [email protected] (P. Guillaume), [email protected] (S. Vanlanduit). 0888-3270/04/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0888-3270(03)00044-X
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imply a change in the modal parameters of the structure (resonance frequency, mode shape, mode shape curvature, etc.). Because different damage scenarios result in a non-linear behaviour, the modal parameters are not always suitable as a damage detection feature. In the past, many studies have illustrated that a crack in a structure may cause the structure to exhibit non-linear behaviour if the crack is alternately open during a part of the response and closed during the remaining time intervals [1–3]. Recently, non-linear damage detection methods were introduced in the literature. In [4], it was demonstrated that the non-linear damage detection techniques are much more sensitive to the presence of damage than the classical linear methods. A disadvantage of using non-linear damage detection techniques is the assumption that a fault causes the structure to behave non-linearly. A beam with a crack that is open throughout the response cycles cannot be detected with this method. In this article a new and fast method, that combines the advantages of the existing linear and non-linear damage detection methods, will be proposed to detect damage in materials and to observe the impact of cyclic fatigue loading on the material. The method uses a broadband excitation signal that allows estimating, in the response, immediately the best linear approximation (also called related linear dynamic system—RLDS) of the overall system, the even and the odd non-linear contributions, and the noise contributions. The paper is organised as follows. In Section 2 we briefly discuss some linear and non-linear damage detection techniques. Section 3 presents a novel non-linear damage detection technique. The theory is illustrated by some real measurement examples in Section 4. Conclusions drawn from the research can be found in Section 5.
2. Overview of damage detection techniques 2.1. Linear damage detection techniques Damage or fault detection, as determined by changes in the dynamic properties or response of structures, is a subject that has received considerable attention in the literature. The basic idea is that modal parameters (notably frequencies, mode shapes, mode shape curvature and flexibility matrices) are functions of the physical properties of the structure (mass, damping, and stiffness). Therefore, changes in the physical properties will cause changes in the modal properties. The observation that changes in structural properties cause changes in vibration frequencies was the impetus for using modal methods for damage identification and health monitoring [1,15]. The use of frequency shifts for the detection and succession of damage have significant practical limitations. First, the somewhat low sensitivity of frequency shifts to damage requires either very precise measurements or large levels of damage. Secondly, since the modal frequencies are a global property of the structure, it is not clear that shifts in this parameter can be used to identify more than the existence of damage. The underlying principle behind the use of mode shapes for fault detection is that changes in modes are sensitive indicators of changes in the physical integrity of any mechanical structure [1,14,15]. When vibrational changes take place in a structure, these changes can be quantified with commonly understood frequency domain measurements that are curve fit to extract the modal properties of the damaged structure. These modal properties are then compared with
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baseline (undamaged) properties, and the changes can be used to determine the severity of the damage. Mode shape derivates, such as the curvature can be used to obtain spatially distributed features sensitive to damage [1,15]. Mode shape curvature can be computed by numerically differentiating the identified mode shape vectors twice to obtain an estimate of the curvature. This method is motivated by the fact that the second derivate of the mode shape is more sensitive to small perturbations in the system than the mode shape itself. Changes in the flexibility matrix indices have been used as damage sensitive features [1,15]. The flexibility matrix is estimated from the mass normalised measured mode shapes and the measured eigenvalue matrix. The formulation of the flexibility matrix is approximate because in most cases not all the modes of the structure are measured. Using flexibility matrices, damage is detected by comparing the flexibility matrix indices computed using the modes of the damaged structure with the flexibility matrix indices computed using the modes of the undamaged structure. Because of the inverse relationship to the square of the modal frequencies, the measured flexibility matrix is most sensitive to changes in the lower frequency modes of the structure. 2.2. Non-linear damage detection techniques Identification of the basic modal properties, mode shape curvature changes, and the flexibility matrices are based on the assumption that a linear model represents the structural response before and after damage. However, in many cases the damage will cause the structure to exhibit a nonlinear response. Therefore, the identification of features indicative of non-linear response can be a very effective means of identifying damage in a structure that originally exhibited a linear response. In the last years, a number of articles were published which use the measured non-linear contributions to predict if there is damage present or not. Assuming that the damage causes the structure to behave non-linearly (i.e. not as in the case of a beam with a crack which is open throughout the response cycles), these non-linear techniques provide several distinct advantages over the other methods for detecting faults in structures. In some cases these non-linear contributions are even used to localise and to quantify the damage. A first technique is based on measuring the higher order frequency response functions (FRFs) of the structure [5,6]. The analysis of these functions shows that they are highly dependent upon the size and the position of the crack and thus can be used as damage indicators. The correlation of the shape and the value of these functions with the crack size and position in the analytical model of the structure can be used for a sensitive structural damage-identification procedure (determination of the location, quantification of the severity of the damage and the prediction of the remaining service life of these structures). A disadvantage of this method is that the higher order FRFs are measured using stepped sine excitations, which implicate that the measurement time is rather high. The biggest disadvantage of this technique is that an analytical model of the structure or damage case must be available, which reduces seriously the applicability of this method. A second technique, Non-linear Elastic Wave Spectroscopy (NEWS), studies the amplitudedependent frequency response in dynamic wave experiments. Two different methods can be distinguished:
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One method is Non-linear Wave Modulation Spectroscopy (NWMS) [3,7]. It consists of exciting a sample with continuous waves of two separate frequencies simultaneously, and inspecting the harmonics of the two waves, and their sum and difference frequencies (side bands). Undamaged materials are essentially linear in their response to the two waves, while the same material, when damaged, becomes highly non-linear, manifested by harmonics and side band generation. Qualitatively we can say that the more damaged a material is, the larger is its nonlinear response. This method can be quickly applied and is ideally suited to applications where the question of damaged versus undamaged must be quickly addressed. This method can be applied to any type of geometry. The second method is Non-linear Resonant Ultrasound Spectroscopy (NRUS) [8], and depends on the study of the non-linear response of a single, or a group of, resonant modes within the material. Undamaged materials are essentially linear in their resonant response. The same material, however, becomes highly non-linear when damaged, manifested by amplitudedependent resonance frequency shifts, the generation of (principally third-order) harmonics and non-linear attenuation. Also in this method, the amount of non-linearity is highly correlated to the damaged state of the material. The method is extremely useful for basic research and specific applications that do not have strict requirements in terms of speed and application. This method can also be applied to any type of geometry and can be used during mechanical cyclic fatigue loading. The disadvantage of this method is the complex measurement set-up and the fact that a stepped sine is used as excitation signal.
3. The proposed technique: characterizing non-linear distortions during FRF measurements 3.1. Characterizing the non-linear distortions The presented damage detection technique is based on a method that was developed in [9] and that allows the detection and the qualification of non-linear distortions on the results of an FRF measurement without wasting measurement time. The kernel idea in this method to detect the non-linear distortions is to use well-chosen periodic excitations, called random multisines: ! ! N X 1 ð1Þ Xk ej2pðfmax =NÞkt xðtÞ ¼ pffiffiffiffiffi N k¼N with Xk ¼ Xk ¼ X ðkÞejjk ; fmax the maximum frequency of the excitation signal, NAN the number of frequency components, X ðkÞARþ ; and the phases jk are a realization of an independent uniformly distributed random process on ½0; 2p; so that E½ejjk ¼ 0: These multisines combine the advantage of a random behaviour (allows the measurement of the best linear approximation of the non-linear system [11]) while they still guarantee the generation of the user selected power spectrum in each realization. A nice property of the random multisines is their possibility to pick up linear and non-linear information in a separable way. This can be done within the same excitation signal. The main principle in the technique to visualise the non-linear distortions is to excite only a well-chosen set of frequency lines in the multisine. This implies that some frequency lines (detection lines) are
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consciously not excited. These will be used as detection lines to qualify (even or odd) and quantify (what is the level?) the non-linear distortions on the neighbouring excited lines (called measurement lines). It can be shown that random multisines, exciting only at a selected set of the odd frequency lines, can be used for the detection, qualification and the quantification of the non-linear distortions. The even non-linearities will under these conditions only contribute to the even frequency lines, and hence they do not disturb any more the FRF measurements [11]. Under perfect circumstances, the power of the even and the odd non-linear distortions in the output can be determined, respectively, at the even and the odd detection lines. Linear and odd non-linear contributions are present at the measurement lines. With the knowledge of the amplitude of the non-linear distortions at the odd detection lines it is, under the ideal circumstances, immediately possible to estimate the level of the non-linear contributions present at the excitation lines. In [11] it was shown that the random multisines can also be used to determine the RLDS of the device under test (DUT). It was also shown that the influence of non-linear distortions on the FRF-measurements with random multisines can be split in systematic and stochastic contributions. Since we do not use these data for the damage detection and health monitoring, we do not discuss these extended techniques here. 3.2. Example The most simple random multisine is a special-odd multisine [9,13], where only the frequencies 8k þ 1 and 8k þ 3; k ¼ 0; 1; 2; y; kmax in Eq. (1) have amplitudes different from zero. Next the output spectrum is calculated with a DFT (implemented as an FFT) using a rectangular window. In [11] it was shown that the even non-linearities excite only the even harmonics at the output ð2k; k ¼ 1; 2; yÞ; while the odd non-linearities appear only at the odd harmonics ð2k þ 1; k ¼ 1; 2; yÞ: Due to the choice of the excitation signal we get the following possibilities: * * * * * * *
at at at at at at at
lines lines lines lines lines lines lines
8k þ 1: 8k þ 2: 8k þ 3: 8k þ 4: 8k þ 5: 8k þ 6: 8k þ 7:
the output consist of linear contributions+odd non-linear distortions, only even non-linear distortions appear, the output consist of linear contributions+odd non-linear distortions, only even non-linear distortions appear, only odd non-linear distortions appear, only even non-linear distortions appear, only odd non-linear distortions appear.
This allows getting an idea of the non-linear behaviour of the system. If at least MX2 successive periods are measured in one block, it is still possible to make the same conclusions (respectively at lines Mð8k þ 1Þ; Mð8k þ 2Þ; Mð8k þ 3Þ; Mð8k þ 4Þ; Mð8k þ 5Þ; Mð8k þ 6Þ and Mð8k þ 7Þ). On top of that, also the noise level can be characterised by looking at the lines that are no multiple of M since these cannot be excited by a signal with M periods in the window. So only the noise (having a non-periodic behaviour) can contribute these. Other types of multisines (as described in [13]) can also be used for the detection, qualification and quantification of the non-linear distortions.
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3.3. Applying this multisine technique in the case of a non-ideal behaviour of the input signal In practice, it can happen that the real input differs from the desired input. Due to the presence of a low-quality generator, the presence of a non-linear actuator or due to the interaction of the generator with the non-linear structure, unwanted excitation power can be present at the detection lines in the excitation signal. At that moment, the output at the detection lines consist not only out of non-linear contributions, but also out of linear contributions originating from the unwanted excitation power at the detection lines at the input of the structure. So, it is necessary to compensate for these distortions (using for example a time consuming software feedback mechanism [10]), or to make a fast first-order correction of the measurements with a compensation method. Hereby, the linear contribution on the detection frequencies is eliminated, so that the compensated output depends only on the non-linear contributions. This first-order compensation method is extensively described and illustrated in [9].
4. Experiments and configuration The described multisine excitation method is illustrated on measurements of a vibrating rectangular beam. In a first experiment, the amplitude of the non-linear contributions in the response obtained with a damaged and an undamaged slate beam will be compared. In a second experiment, the evolution of the non-linear contributions in the response of the FRF measurements will be followed in the case that an originally undamaged slate beam is subject to cyclic fatigue loading. In both cases, we will investigate if they come into being and the presence of damage in the beam can be detected by an increasing of the non-linear contributions in the response of the FRF measurements. 4.1. Experimental set-up The experimental equipment used to obtain the results discussed in this paper is shown in Fig. 1. The experiments are applied on thin, rectangular beams of artificial slate used in civil constructions. The centre of the slate samples are placed on a B&K impedance head (type 8001), which is assembled on a B&K vibration exciter (type 4809). The measurements were driven by a Pentium II 400 MHz. A special-odd random multisine was used for the measurement of the non-linear contributions. Only the frequencies fk ¼ ð8k þ 1Þf0 and ð8k þ 3Þf0 ; k ¼ 0; 1; 2; y; 72 (harmonics 139111719?577) are excited with constant amplitude and random phase. The sampling frequency was set to fs ¼ 3200 Hz. The fundamental frequency, f0 ¼ ðfs =NÞE0:391 Hz, so that one period of the signal contains N ¼ 8196 points. The generator signals were designed in MATLAB and sent to a generator/acquisition rack, which consists of HPE1433A, HPE1434A and National Instruments VXI1394 cards. Communication between PC and VXI rack was established using home-developed software [12]. The generator signals were sent to a power amplifier (B&K type 2706) before being fed to the shaker. Similarly, the signals coming from the force and the acceleration sensors were first amplified with a B&K charge amplifier (type 2635)
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Sample
Acceleration
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Matlab Pentium II 400MHz
Fig. 1. Diagram of the experimental set-up.
350 mm 30 mm
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8 mm
extra weight (100 g)
Fig. 2. Specimen geometry.
before being sent to the acquisition rack. One single measurement requires a measurement time of about 2.6 s. 4.2. Comparison of a damaged with an undamaged sample In a first experiment to qualify the value of multisine excitations in a damage detection process, a comparison was made between the results obtained with a damaged and an undamaged slate sample. The slate samples consist of two 4 mm thick slate layers. The second layer is separated in two parts (see Fig. 2) and glued under the first layer. This enables the simulation of a breathing crack. The undamaged state was obtained by glueing the separation in the second layer. Two extra weights of each 100 g were glued on the ends of the device in order to assure that a breathing crack was obtained during vibration. The nominal dimensions of the beam used in the present study are 350 mm 30 mm (2 4) mm. The crack was located 87 mm from one end. The beam was constructed from pieces of equal thickness, so the crack penetrates to half the beam thickness. The amplitude level of the special-odd random multisine excitation signal was chosen to be very small, so that no extra damage was generated in the structure. In both cases, the experiment consisted out of measuring 10 consecutive periods of the same multisine. The measurements were
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Fig. 3. Results of the proposed technique on an intact (left) and on a cracked slate beam (right). These figures depict an overview of the results obtained in one single experiment, which consisted out of 10 consecutive measurements. Top row: measured FRF; Second row: measured odd non-linear contributions (after compensation); Bottom row: measured even non-linear contributions (after compensation).
started once the transient disappeared. Fig. 3 illustrates the experimental data obtained with the cracked and the uncracked slate samples. It can be seen that the shape of the transfer functions of both samples are almost equal (Fig. 3, top). In both cases, the resonance frequency appears at
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68 Hz. No frequency shift can be observed. The variation of the FRF in the frequency range between 0 and 10 Hz can be explained due to the fact that the B&K vibration exciter (type 4809) gives only accurate results between 10 Hz and 20kHz. The only obvious difference in the FRFs of both samples is that the cracked beam shows a much larger value of the FRF at the resonance peak. The same remarks can be very clearly noticed when the even and the odd non-linear contributions obtained with the undamaged sample are compared with these of the cracked sample. The even and the odd non-linear contributions plotted in Fig. 3 are these that were obtained after compensation. At the resonance frequency, the even and the odd non-linear contributions of the cracked sample are almost 10 dB larger than the even and the odd non-linear contributions of the intact sample. Remark again that the difference in non-linear contributions appears only around the resonance frequency. In the other frequency bands, no clear difference in non-linear contributions can be observed. Comparing the amplitude of the linear contributions (not shown) with the even and the odd non-linear contributions reveals that the non-linear contributions are in the full frequency band smaller than the linear contributions, so it was permitted to use the compensation method. The same experiment, consisting out of 200 consecutive measurements of the FRF of both samples, provided the same results. 4.3. Evolution of the non-linear contributions during a cyclic fatigue loading experiment The second experiment aimed at damage assessment with multisine excitations during a controlled fatigue loading. The nominal dimensions of the beam used in this experiment are 400 mm 22 mm 4 mm. Again two extra weights of each 100 g were glued on the ends of the device to subject the beam to mechanical aging (see Fig. 4). To validate the proposed technique, the response of this beam is investigated in the frequency range between 5 and 225 Hz. In this range, the first and the second bending mode of the beam are present. The fatigue loading was generated by the excitation signal, in combination with the two extra weights. The amplitude level of the special-odd random multisine excitation signal was chosen high enough to induce permanent fatigue damage after several tens of cycles. After 6 min, a macro crack was visible at the surface. During these 6 min, 90 successive periods of the same multisine were measured in order to monitor the evolution of the state of the beam. Just like in case of the comparison of the damaged with an undamaged sample, we followed the evolution of the FRF and the compensated even and the odd non-linear contributions in the output. The measurements were started once the transient disappeared. The evolution of the FRF of the beam during the fatigue loading is plotted in Fig. 5. A significant increase of the first resonance frequency (at 14 Hz) can be observed. The increase of the 400 mm 4 mm
22 mm
extra weight (100g)
shaker position
Fig. 4. Specimen geometry.
extra weight (100 g)
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Fig. 5. 3-dimensional representation of the evolution of the measured FRF during cyclic fatigue loading experiment on a slate beam.
first resonance frequency starts already from the beginning of the experiment and proceeds faster when the experiment makes progress. A significant increase of the amplitude of the first mode appears 15 measurements before the end of the experiment. At the end, the amplitude of the first mode is at least 15 dB larger compared with the beginning. No frequency shift of the first resonance frequency can be observed. The second mode (around 176 Hz) however, shows a completely different behaviour. The amplitude of the second mode does not change significantly during the experiment and even decreases at the end. On the other side, small frequency shifts can occasionally be observed (see Fig. 6). These are probably due to the development of micro-cracks in the beam. Around measurement 75, a significant frequency shift appears. This corresponds with the remarkable increase of the amplitude of the first resonance frequency. Fig. 7 illustrates the evolution of the odd non-linear contributions as function of the time (number of measurements) and the frequency. The highest values of the odd non-linear contributions are present at the resonance frequencies. The odd non-linear contributions at the first resonance frequency exhibit the same behaviour as the FRF at this resonance frequency. The odd non-linear contributions increase during the experiment, and adopt during the last 30 measurements extreme large values, which are almost 17 dB larger compared to their starting values. Remark that these large values of the odd non-linear contributions appear in a very narrow frequency band around the first resonance frequency. The expansion of the amplitude of the odd non-linear contributions around the second resonance frequency follows the same evolution as the odd non-linear contributions at the first resonance frequency. However, there are some serious differences. The maximal values of amplitude of the odd non-linear at the second resonance frequency are smaller compared to the values obtained at the first resonance frequency. The values of the odd non-linear contributions at the end of the experiment adopt values who are merely 10 dB larger compared to the beginning of the experiment. On the other side, it is clearly
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Fig. 6. Partial enlargement (of the second mode) of the evolution of the measured FRF during cyclic fatigue loading experiment on a slate beam.
Fig. 7. Three-dimensional representation of the evolution of the measured odd non-linear contributions during cyclic fatigue loading experiment on a slate beam.
visible (see Figs. 7 and 8) that the frequency band in which larger values of the odd non-linear contributions are attained becomes much wider during the fatigue loading experiment. Significant higher values for the odd non-linear contributions can be observed in a frequency range up to 40 Hz at both sides of the second resonance frequency. A small shift of the second resonance frequency can also be observed. In the remaining part of the measured frequency band no changes of the odd non-linear contributions worth mentioning are observed. The evolution of the even non-linear contributions as function of the time (number of measurements) and the frequency is plotted in Fig. 9. The highest values of the even non-linear contributions are present at the resonance frequencies. The even non-linear contributions at the
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Fig. 8. Evolution of the odd non-linear contributions at certain frequencies during the cyclic fatigue loading experiment on a slate beam.
Fig. 9. Three-dimensional representation of the evolution of the measured even non-linear contributions during cyclic fatigue loading experiment on a slate beam.
first resonance frequency exhibit almost the same behaviour as the odd non-linear contributions at this first resonance frequency. The only difference that can be observed is that the extreme large values, which are obtained in the last 20 measurements are merely 13 dB larger compared to their starting values. Remark again that these large values of the even non-linear contributions appear in a very narrow frequency band around the first resonance frequency. The expansion of the amplitude of the even non-linear contributions around the second resonance frequency is smaller than the expansion around the first resonance frequency. The values of the even non-linear contributions at the end of the experiment adopt values, which are
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Fig. 10. Evolution of the even non-linear contributions at certain frequencies during the cyclic fatigue loading experiment on a slate beam.
5 dB larger compared to the beginning of the experiment. From Figs. 9 and 10, it can also be seen that the frequency band in which larger values of the even non-linear contributions are attained does not become wider during the fatigue loading experiment. However, a clear frequency shift of about 5 Hz can be observed. In the remaining part of the measured frequency band no changes of the odd non-linear contributions worth mentioning are observed.
5. Conclusions In this article a new and promising method for non-destructive material testing has been developed and applied in the field of damage detection and succession. The method uses a periodic broadband multisine excitation that allows estimating, in the response, immediately the FRF of the overall system and the even and the odd non-linear contributions. We have shown by means of two case studies that the use of multisine excitations can be used to give additional information about the presence of damage in a structure. Experimental data from a beam with an opening and a closing crack, and a mechanical cyclic fatigue loading experiment on a beam have demonstrated that undamaged materials are essentially linear in their response. However, the non-linear behaviour of the same material increases significantly when damage appears. The progressive damage/fatigue experiment discussed in this paper clearly illustrate that the sensitivity of the odd non-linear contributions to the detection of damage features is, for this case superior than the even non-linear contributions and the FRF of the overall system. The described multisine excitation technique is a very fast and efficient technique to assess and to follow the evolution of global damage in a material. This multisine excitation technique allows the user to examine the non-linear behaviour of the structure, while it is on the same time, with the
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same results possible to investigate the FRFs with the linear damage detection techniques. This means also that the results of this method can be used in the case that the damage does not introduce non-linear behaviour. This method is also superior compared to the other nonlinear methods due to the fact that it is possible to assess and to follow the evolution of the damage in a material in situ and in working conditions. The ability to use the compensation method minimises the influence of the disturbing contributions originated from the measurement equipment.
Acknowledgements This research was supported by a grant of the Flemish Institute for the improvement of the scientific-technological research in industry (IWT), the Belgian National Fund for Scientific Research, Flanders, the Flemish government (GOA-IMMI), and the Belgian government as a part of the Belgian program on Interuniversity Poles of attraction (IUAP4/2) initiated by the Belgian State, Prime Ministers Office, Science Policy Programming.
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