Assessment of the apparent bending stiffness and damping of multilayer plates; modelling and experiment
Introduction
The combined high stiffness and light weight of sandwich composites make them increasingly used by today's transportation and construction industries for example. In this context, the design of light multilayer plates with optimized damping and isolation performances, for given frequency bands, remains a major industrial challenge and a hot topic of research. Thus, this article concerns the vibrational behavior of such lightweight composite plates in a broad-band frequency range.
Three analytical and four experimental vibroacoustics methods identifying the equivalent complex bending stiffness (or flexural rigidity) and equivalent loss factor of a three-layered plate are compared. These equivalent parameters, also known in the literature as apparent stiffness and apparent loss factor (see for example the studies from Nilsson [1] or Backström [2]), are of course frequency dependent. The purpose of this work is then to identify them up to the high-frequency domain where most of the experimental methods meet their limits in terms of precision and resolution (frequency or spatially). Several techniques are reported in the literature to handle such limitations, in particular wave number fitting approaches (see for example the work of Berthaut et al. [3] on ribbed structures and the one of Cherif et al. [4] on composite panels with honey comb core), image source methodologies (see for example the method of Cuenca et al. for estimating material properties of plates [5,6]) or the recently proposed Virtual Fields Method combined with Laser Doppler Vibrometry or optical deflectometry by Berry et al. [7,8]. Here we made the choice to compare four experimental approaches: traditional modal analysis [9], high-resolution modal analysis [10], CFAT methodology that uses a corrected finite differences scheme [11], and a wave correlation technique comprising an image source model that uses Hankel's functions [12].
Concerning the comparison with predictions, different approaches exist to model the vibrational behavior of multilayer plates (see for example a synthesis done by Carrera [13] or more recently the work of Shorter [14], Manconi and Mace [15], and Ghinet and Atalla [16], for thick composite laminated panels). Here three models are presented, compared and discussed in the first part of the manuscript - a) model of Guyader (traveling wave approach) [17,18] b) model of Ross, Kerwin and Ungar (strain energy) [[19], [20], [21]] c) Lamb waves model [22]. The plate under study (a steel/polymer/steel sandwich system) is then presented, and the experimental protocols and assessment procedures of the four experimental techniques considered are detailed in section 3. Using traditional modal analysis in the low frequency domain and finite-element model (FEM) calculations, the frequency dependent equivalent Young's modulus (derived from the equivalent bending stiffness) is identified up to 800 Hz. Then, a high-resolution modal analysis method allows identifying equivalent loss factor as function of frequency up to 2.5–3 kHz where modal overlap is high. Using two different methodologies (CFAT methodology and Hankel's functions image source model), contactless measurements of the velocity field extend the identification of the frequency dependent complex Young's modulus up to the very high frequency domain (20 kHz) and confirm both previous modal analyses estimations. Measurements and predictions are then compared and discussed in section 4. The article ends with the identification of the polymer core complex modulus that is finally compared with DMA measurements/extrapolations.
The chosen analytical and experimental methods do not limit this work to a benchmark study but rather aims at the discussion of measurement approaches that cover a wide frequency range. These measurements are validated with predictions, focusing on damping identification which is not straightforward in cases of high modal overlaps. Hence - extending a previous oral contribution [23] where only preliminary experimental results were given without theoretical discussions and fair comparisons - the principal novelties of this work can be summarized as follows:
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comparing for a given three-layered plate experimental modal analyses techniques (in the low/mid-frequency domains) to wave number fitting approaches (up to very high frequencies) with a focus on accurate damping characterization
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obtaining - over the whole [40 Hz–20 kHz] frequency range - overlapping experimental results in good agreement with an equivalent single layer plate model with frequency dependent apparent bending stiffness and damping
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using modelling predictions to identify through an inverse vibratory problem the missing mechanical properties of a given layer (here the polymeric core complex Young's modulus) and confirming this results with DMA measurements/extrapolations.
Section snippets
Equivalent single layer modelling
In this section three analytical modelling of multilayer are presented, compared and discussed on two examples of sandwich plates. Only main principles of the methods are recalled in this first section; more details on each of the techniques are given in appendices and corresponding cited references.
Experimental characterization of material properties
Four experimental procedures have been used to estimate the material properties of a 300 × 400 × 1.05 mm3 SPS multilayer plate sample. The SPS plate is presented in Fig. 3. The layer thicknesses reported (and also listed in Table 1) correspond to average values determined using optical microscope images of the plate's cross section. Also, sheets of the polymer layer alone were available (see Fig. 3b left) so that its density could be determined by measuring and weighting large specimens. Before
Comparing experimental and analytical results
In this last section, we compare the experimental results to the analytical predictions allowing the identification of the polymer core complex modulus. Finally the article ends with the comparison of these identified polymer characteristics with DMA measurements/extrapolations based on the time-temperature superposition principle.
Conclusions
This paper is focused on the vibrational behavior of sandwich composite plates. Frequency dependent equivalent Young's modulus and equivalent loss factor of a steel-polymer-steel sandwich system are estimated in a broad-band frequency range using different experimental methods. The results are in very good agreement with an equivalent single layer analytical modelling (model of Guyader) based on wave propagation analysis. The comparison with this theoretical approach allows the identification
Acknowledgments
This work was performed within the framework of the Labex CeLyA of Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-11-IDEX-0007), It was partly funded by INSA-Lyon (BQR VIVARIUM project). The authors would like to thank Céline Sandier (engineer at LVA), Youssef Gerges (postdoctoral fellow at LVA), and master's students Valentin Henry, Eddy Fasana and Nhu Hai Nguyen for their precious help during measurements. We also express our gratitude to Jean-Louis
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