Vibroacoustic Performance of Plane Plate with Random Stiffness

Two decades ago, big emphasis was given in the transportation industry when announcing new structural solutions able to solve a series of engineering problems; among them the noise and pollutant emissions have to be remembered, for example. The increased attention was paid to look for passive/active control solution able to extend the functionality of each given ‘standard’ design: one can define this approach as top-down. With the time passing, several steps were performed backward and specifically for the noise and vibration problems, the research programmes were directed to investigate the new possibilities offered by the emerging new materials and configurations: a classical bottom-up approach. The functionally graded materials and multifunctional materials are good examples of this new possibilities still to be deeply investigated. It has been recognized for some time that sandwich panels offer potential benefits for tailoring of transmission loss (TL). As early as 1959, Kurtze and Watters [1] studied the application of sandwich panels to increase the sound insulation between adjoining spaces. They investigated the relation between bending and shear waves and TL characteristics, and proposed laminated plates with a soft but incompressible core to increase the coincidence frequency by several octaves. Their results hence suggest how such a panel can be designed to approximate the mass law in levels over a wide frequency range. Lang and Dym [2] optimized the design of a sandwich panel with a goal to exceed the TL values predicted by the mass law by at least 20 dB in a selected frequency range. Starting from nine design variables (density, Young’s modulus, shear modulus, Poisson’s ratio and thickness of the core, and density, and the Young’s modulus, Poisson’s ratio and thickness of the skins), a number of assumptions and linked constraints reduced the design variables to just the density and thickness for the core and the skins. The results indicated two basic means for improving the TL of sandwich panels. The first is to use panels that follow mass law (more massive, less stiff) over a large frequency range. This solution has obvious drawbacks for weight-constrained and load-limiting applications. The second is to increase the core stiffness in order to increase the symmetric coincidence frequency, while maintaining the anti-symmetric coincidence at low frequencies. Barton [3] and Grosveld [4] considered an aeronautical application of honeycomb panels for improving sidewall attenuation in a light twin-engine aircraft, with respect to propeller tonal noise and its five main harmonic components. In the last decade other authors have analyzed the dynamic behaviour and the TL properties of sandwich panels [5-9]. Most of the research has focused on developing and testing new numerical tools for describing the dynamic behaviour of 2-D sandwich panels (i.e. transfer matrix approach, spectral finite elements). Thamburaj and Sun [9] demonstrate again that an anisotropic core can lead to higher TL and that the proper design of face sheet thickness can further improve the performance. Additional indication that structural acoustic optimization has the potential to achieve significant gains for reducing interior noise levels in aerospace structures may be found in the work of Cunefare et al. [10-11]. The results indicated that spatial variations of the design parameters would yield improved performance as compared to uniform properties. With the assumed disturbance, spatial orientation and directionality must be considered to reach an optimal design. A recent development in sandwich structures is the use of a core made of a lattice of truss elements. The layout of the core may be periodic or it can have a random topology. Manufacturing these panels by casting makes them a feasible option. There has been some NASA testing of the performance of these structures [12]. Casting the core provides the opportunity to readily implement various core configurations through proper design of the mould. The design freedom of a truss provides a great deal of latitude in the investigation of a truss core’s influence on the structural acoustic performance of panels that incorporate them. In a periodic structure, as sandwich one, the impedance mismatch generated by periodic discontinuities in the geometry and/or in the constituent material cause destructive wave interference phenomena over specific frequency bands called stop bands. The location and the extent of these band gaps depend on the nature of wave propagation [13]. Recent studies [14] have shown that the capability of classical periodic structures to attenuate the propagation of waves can be enhanced by using shunted piezoelectric materials ([15], [16]). Periodic arrays of shunted piezoelectric patches can be employed even for broad band vibrations attenuation in plate structures [17]. Shunted piezoelectric patches act as sources of impedance mismatch, which yield interference phenomena resulting from the interaction between incident, reflected and transmitted waves produced by the mismatch. The impedance mismatch corresponding to the shunted piezo-electrics can be tuned to achieve strong attenuation over frequency bands that are defined by the shunting circuit connected to the patches. Hence the activity to be presented here is the straight continuation of a previous work completely devoted to the optimisation of sandwich plates [18]. Here, the target is the same but it was studied how and what a random distribution of the stiffness can influence the target of vibroacoustic performances. Some of the present analysis can be classified in the studies concerning the master-fuzzy systems, [19-22]. The investigated models are based on standard Finite Element Approach: two simple plates connected trough point-to-point stiffnesses, two simple plate with an intermediate core. The Vibroacoustic target is the acoustic radiated power and the distribution of the natural frequencies. The final goal of the present work is the comparison between the optimised configuration, developed with a multi-objectives and multi-constraints framework [19], and configurations having a core with a random distribution of stiffness. These master-fuzzy systems can represent the first approximation of a structure with embed multifunctional or functionally graded materials randomly activated. References 1. Kurtze, G., and Watters, B. G., 1959, “New wall design for high transmission loss or high damping,” J. Acoust. Soc. Am. 31, pp. 739-748. 2. Lang, M. A., and Dym, C. L., 1975, “Optimal acoustic design of sandwich panels,” J. Acoust. Soc. Am. 57(6) Part II, pp. 1481-1487. 3. Barton, C. K., and Mixson, J. S., 1981, “Noise transmission and control for light twin-engine aircraft,” J. of Aircraft 18(7), pp. 570-575. 4. Grosveld, F. W., and Mixson, J. S., 1985, “Noise transmission though an acoustically treated and honeycomb-stiffened aircraft sidewall,” J. of Aircraft 22(5), pp. 434-440. 5. El-Raheb, M., 1997, “Frequency response of a two-dimensional truss-like periodic panel,” J. Acoust. Soc. Am. 101(6), pp. 3457-3465. 6. El-Raheb, M., and Wagner, P., 1997, “Transmission of sound across a truss-like periodic panel: 2-D analysis,” J. Acoust. Soc. Am. 102(4), pp. 2176-2183. 7. El-Raheb, M., and Wagner, P., 2002, “Effects of end cap and aspect ratio on transmission sound across a truss-like periodic double panel,” J. Sound Vib. 250(2), pp. 299-322. 8. Thamburaj, P., and Sun, Q., 2002, “Optimization of anisotropic sandwich beams for higher sound transmission loss” J. Sound Vib., 254(1), pp.23-36. 9. 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