Chasing the high-resolution mapping of rotational and strain FRFs as receptance processing from different full-field optical measuring technologies
Introduction
The accurate sensing of vibration related features in broad frequency band structural dynamics is a demanding task that can affect the reliability of the most advanced physical quantities of interest. Upon the reliability of the latter depend many predictive speculations by means of advanced inferences: simulations about the behaviour of complex parts in a broad frequency range; life cycle estimations; quality assurance of the manufacturing processes, especially when tight dynamic targets need to be reached in working conditions. Beside more traditional quantities like accelerations, velocities and displacements, the rotational dofs are gaining interest (see [1], [2], [3], [4], [5], [6], [7]) for their contribution in refining reliable models of non-conventional structures, despite the difficulties faced in their sensing with traditional transducers. In more recent times, many authors [8], [9], [10], [11], [12] have proved the advantage of using also the rotational dofs in frequency based substructuring, to increase the precision of the hybrid coupling in automotive applications; in [13], [14] only the displacements were available for the hybrid coupling, indeed with lower agreement with the real measurements.
Other authors (see [15], [16], [17], [18]) addressed the rotational dof measurements by adding a T-block-like arrangement of accelerometers, or by putting precisely known distances in between, to use a finite differences approach to evaluate the quantities, but loading the specimen with relevant inertia and damping. In this way, though, the procedure to obtain a many rotational dofs map becomes certainly cumbersome and not practical. The interest for rotational dofs is increased when they are coupled with model updating procedures (see [19], [20], [21], [22]), with the aim to improve the finite element formulation in anisotropic compounds and complex shaped parts. As in [18], the right estimate of rotational Impedances has specific interest for torsional vibrations of shafts and for extended modal analysis of coupled torsional and bending dynamics, although these enquiries were only made with the aid of lumped sensor measurements.
Before the optical full-field techniques appeared in the first viable industrial prototypes, up to recent times, most authors in engineering literature (e.g. [23], [24], [25]) have used lumped sensors for spatially coarse data, like strain gauges or rosettes, coupled with tuned FE models to simulate the missing data on locations where no real data were acquired, also in the spatially dense sensing of strains. This has been the procedure also in structural dynamics for fatigue life estimations, with the appropriate material constitutive parameters. This procedure certainly brings the known drawbacks of finding an effective sensor placement, of spatial averaging, of mass burden, of local distortions, of cross sensitivities to the surface curvature, of the accuracy of the FE model and of its effective updating to have reliable predictions. To answer part of these questions, the author started to work towards enhancing the full-field measurements in broad frequency bands (see [26], [27], [28], [29], [30], [31]) and the predictions on the structural reliability of the components (see [7], [32], [33], [34], [35], [36], [37]). This specific work, instead, explains in details the latest comparative assessment among full-field technologies when the numerical derivatives of raw datasets are taken into account.
Shearography [38] was used in [39] by means of integration along the shear direction to obtain vibration displacement patterns in stroboscopic lighting, while it could have been proficiently used as a direct optical derivative technique to implement the rate of deflection of the tangent plane at each point in the surface map, thus measuring directly the rotations for dynamically deformed flat surfaces. As can be checked instead in [40], the industrial instrumentation manufacturers specialised only in NDT applications of shearography, and no system (to the knowledge of the author) is produced with the mated electronics of dynamic ESPI for complex-valued dynamic rotation distribution. Whereas, also for typical shearography’s applications, the dynamic excitation can be relevant, where faults might be even more easily revealed, because linked to anomalies in higher frequency operative deflection shapes (see [41], [42]). Further derivations might also have led to complex-valued dynamic strain mapping, for the Impedance FRF models, with less expansion of derivative errors, as highlighted below.
The TEFFMA1 fundamental research project aimed at making a comparison between the state-of-the-art of industrial implementations in optical native full-field technologies and the SLDV as reference, to understand at which point of their development these experimental procedures can provide enhanced peculiarities in NVH applications. It was out of the project’s scope, though, to investigate all the specific techniques’ implementations. On the contrary, the TEFFMA project aimed to assess the real potentialities of the data obtainable from commercial instruments of more general use, even if at an immature, but advanced, prototype evolutionary stage, as highlighted in Table 1. The reader interested instead in the basic techniques behind the instrumentations can find the needed survey in scientific literature (e.g. [43], [44] for SLDV, [45] for photogrammetry, [46], [47] for DIC, [48], [49] for ESPI).
In the well equipped laboratory of the TU-Wien, as summarised in Section 2 below, it was possible to build a complex set-up for the comparison of the peculiarities of the different optical technologies there available (SLDV, DIC and ESPI) in acquiring full-field FRFs. No standard sensors were tested to make a cross comparison with the optical technologies, as was done instead for accelerometers in [50] or for strains, with Shearography and fibre Bragg gratings (not SLDV, DIC & ESPI), in [51]. As it will be clearer in Section 2, the set-up would have been otherwise compromised by the adhesion and by the inertial distortions of traditional lumped sensors, like accelerometers or strain gauges and their cabling. It must be remarked that a lot of work was done [52], [53], or is still under refinement, to find the best agreement with optical techniques and strain measurements. For more details the interested reader is invited to look at Refs. [7], [28], [29], [30], [31], [54], [55], [56], [57]. Indeed, the quality achieved in full-field Receptances (FRFs of displacement over force), in terms of field continuity, proved to be a feasible starting point for robust numerical derivative operators to obtain new experiment-based quantities, here in the FRFs’ map shape of complex-valued dynamic rotations and strains over force. All the three tested optical technologies offered an increasing spatial definition with respect to traditional lumped sensors, from a very good to an extreme sampling of the surface, coupled with a detailed frequency domain description, clearly addressing the posed question about the gauge location and the capabilities to improve the reliability of the predictions, beyond the forecast developments highlighted in [58]. Furthermore, most important for lightweight structures, those techniques offer non-contacting and -distorting measurements. The high spatial density solved the sensor location assessment with accurate surface mapping, while the lack of mass burden and tight measured data offered the author the chance to avoid dynamics’ distortions and area averaging respectively, with a clear benefit for the assessment of new quantities, like surface rotations and strains as in [7], and the tuning of numerical models as in [57].
This paper wants to detail the methodology behind the high quality results obtained about rotational FRFs’ estimation and related rotational Coherence function maps, and, further, about strain FRFs, in the framework of a pointwise comparative approach, focused on the derivative results from SLDV, DIC and ESPI measurement technologies, at the same (numerically best obtained) physical location of the test specimen. After Section 2 devoted to briefly describe the testing sessions, Section 3 is therefore dedicated to the methods at the basis of this work: it focuses on the novel and extensive usage of robust numerical differentiations (detailed in Appendix A) on the high quality experimental Receptance maps previously obtained (see [28], [29], [30], [31]), defining the logic to extract refined maps of both dynamic rotation and surface strain FRFs. Section 4 depicts the proposed pointwise comparative framework to have meaningful matchings among the techniques. Section 5 addresses in detail the first pointwise comparison of the evaluated results (rotational FRFs, rotational Coherence functions and strain FRFs), with the systematic usage of the FRAC and MAC metrics on the whole datasets (described in Appendix B), permitting the direct matching of figures coming from competing optical instruments, with related comments, before drawing the final conclusions in Section 6.
Section snippets
The testing activity in brief
In Fig. 1 an overview of the test session arrangement in the common set-up at TU-Wien is available. The three measurement technologies of Table 1 were focused on the same portion of the specimen surface to sense the same dynamic behaviour of the restrained plate, as the latter was chosen to express a high modal density inside the frequency range of interest, with a complex mixing of different shape patterns.
The specimen was an aluminium thin rectangular plate (250 × 236 × 1.5 mm, slightly
Experimental Impedances for rotation & strain distributions
As the vibration data are sampled on a finer grid by optical techniques, a clearer map of the deforming pattern becomes available, with much more freedom in choosing better sensor locations (see [67]), keeping in mind that full-field technologies do not need any FE model. The scanning vibrometry (like stepped grid SLDV and its continuous scanning variants, see [43], [44], [68], [69], [70], [71]) can challenge any accelerometric measurement chain in the reachable sight, especially for light
Details on the pointwise comparative approach proposed
When the modal density is high and near or repeated poles are present, the dynamic behaviour of a structure may markedly change along the sampled locations with the blending of shapes, as was clearly faced during the tests at the basis of the TEFFMA project [7], [28], [29], [31], [54], making the comparison among different tools quite prohibitive, if not meaningless. A precise common reference for all the three techniques was therefore needed, to be sure that the data in the comparisons were
Results of the comparisons in the pointwise matching
The details on the Receptance FRF maps acquisition can be found in Section 2 and extensively in [29], [30], [31]. The basic features of the datasets here retained are the restricted geometry grid (57 × 51 dofs) of the reference based on SLDV, and the common range of [20.312–1023.438] Hz, with the SLDV native spectral even spacing of 0.78125 Hz among the 1285 frequency lines. By means of these choices, the SLDV dataset was only restricted (by dropping the outlier grid points), but not
Conclusions
This paper has highlighted the relevance that native full-field measurements are gaining in deriving novel advanced physical features, here rotational- and strain-FRFs from experiment-based Receptances, which are becoming strategical for complex design procedures, in particular showing the relevant advantage of consistent and almost continuous data fields when numerically processed. The chasing of accurate mapping of dynamic rotational and strains FRFs faced the first thorough, methodologically
CRediT authorship contribution statement
Alessandro Zanarini: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Software and computing tools for the paper, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualization, Supervision, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This activity is part of the Project TEFFMA - Towards Experimental Full Field Modal Analysis, funded by the European Commission at the Technische Universitaet Wien , through the Marie Curie FP7-PEOPLE-IEF-2011 PIEF-GA-2011-298543 grant, for which the Research Executive Agency is greatly acknowledged. TU-Wien, in the person of Prof. Wassermann and his staff, are kindly acknowledged for hosting the TEFFMA project of the author at the Schwingungs- und Strukturanalyse/Optical Vibration
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