A dual modal formulation for multiple flexural subsystems connected at a junction in energy-based models

Abstract

Statistical energy methods in vibroacoustics, like the statistical energy analysis (SEA) or the statistical modal energy distribution analysis (SmEdA), rely on specific modal coupling assumptions (MCAs) between subsystem modes. These methods assume that the behavior of subsystem mode amplitudes mimic that of oscillators, that the modes within a subsystem are uncoupled, and that the coupling between two different subsystems only takes place through the interaction of resonant modes. In the case of more than two subsystems being connected at a junction, however, it becomes difficult to establish a modal interaction scheme for them. In SEA, the problem is avoided by resorting to the travelling wave approach instead of the modal one. Nevertheless, there is a need for other energy-based methods, like SmEdA, to deal with such a situation. In this work it is proposed to extend the displacement-stress dual formulation, originally intended for two subsystems, to the case of multiple flexural waveguides connected at a junction. Numerical results are presented for a test case consisting of a floor coupled to two walls at right angle. The fulfillment of the MCAs by the dual modal formulation is examined in terms of the impedance mismatch between the floor and the walls.

Energy-based reformulated Craig-Bampton method for multiple flexural subsystems connected at a junction with low impedance mismatch

“…The developments concerning the CB approaches in the forthcoming sections will be applied to the same benchmark problem in [7]. This will ease comparison with the results reported in that previous work.…”

“…In test case #1 the floor is made of concrete and it is 20 cm in thickness, the walls being brickwork constructed and 5 cm in thickness. It was shown in [7] that in such situation the structure could be divided into three subsystems: a single subsystem for the floor (panels P1 + P2) and two more subsystems for each wall (see Figure 2a). The DMF performed well in that case and provided a modal coupling scheme in accordance with the MCAs.…”

“…However, in a second test case #2, all panels were assumed to be made of concrete and of 20 cm in thickness. The DMF failed in that occasion, and it was argued in [7] that a sub-structuring scheme in which every panel was identified with a subsystem, having clamped conditions at the junction, would be more appropriate for test case #2 (see Figure 2b). This is so because all panels having the same stiffness, their motion gets blocked at the junction by the three remaining ones.…”

“…SEA circumvents the problem by resorting to the wave approach to characterize that situation (see e.g., [4][5][6]). Yet in a recent work [7], it was shown that it was also possible to establish a modal scheme satisfying the MCAs, whenever there was a strong impedance mismatch between the subsystems sharing the joint. The dual modal stress-displacement formulation was used for that purpose [8][9][10][11][12].…”

“…The dual modal stress-displacement formulation was used for that purpose [8][9][10][11][12]. The current work is to be viewed as a follow-up of [7], where we now address the case of all connected subsystems having similar stiffness i.e, when there is a low impedance mismatch at the joint.…”