A Power-Based Framework for Quantifying Parameter Uncertainties in Finite Vibroacoustic Metamaterial Plates

Abstract

Vibroacoustic metamaterials (VAMMs) are artificial materials that are specifically designed to control, direct, and manipulate sound waves by creating a frequency gap, known as the stop band, which blocks free wave propagation. In this paper, a new power-based approach that relies on the active structural intensity (STI) for predicting the stop band behavior of finite VAMM structures is presented. The proposed method quantifies the power loss in a locally resonant finite VAMM plate in terms of percentage, such as STI_99% and STI_90%, for stop band prediction. This allows for the quantitative analysis of the vibration attenuation capabilities of a VAMM structure. This study is presented in the context of a two-dimensional VAMM plate with 25 resonators mounted in the middle section of the plate. It has been demonstrated that this method can predict the stop band limits of a finite VAMM plate more accurately than using negative effective mass, unit cell dispersion analysis, or the frequency response function methods. The proposed approach is then implemented to establish a framework for investigating the influence of parameter uncertainties on the stop band behavior of the VAMM plate. Based on the STI_99% method, which aims for significant vibration reduction, stricter tolerances in the mass fabrication process are required to ensure the robustness of VAMM. Conversely, the STI_90% method suggests that larger fabrication tolerances can be leveraged to achieve a broader stop band range while still meeting the desired performance level, leading to cost savings in manufacturing.

Keywords: active structural intensity; finite VAMM; power loss; stop band; uncertainty.

Figure 1

Example of FRF curves of the host structure and VAMM for (a) an ideal system and (b) a system with a resonator mass reduction of 21%. Two possible approaches, namely the intersection point method (black dashed lines) and the 20 dB reduction method (maroon dashed lines) are shown for predicting the stop band width. The gray region represents the stop band.

Figure 2

Definition of internal forces and moments as well as the rotational and translational vibrational velocities [45].

Figure 3

Illustration of Equation (11).

Figure 4

Schematic representation of the VAMM plate and its UC with its dimensions.

Figure 5

Effective mass of a single UC shown in Figure 4.

Figure 6

(a) FE mesh of a 1D periodic system based on [49]. (b) A UC of a rectangular lattice exhibiting periodicity along the directions, $a_i$. The master DOFs are marked in maroon. The indices describe the positions, such as left, right, top, and bottom.

Figure 7

(a) Dispersion relation for wave propagation in the x-direction (0 – $\pi$) using the ${UCDA}_{\mathrm{inv}}$ method for the undamped infinite VAMM plate and (b) dispersion relation for wave propagation in the x-direction using ${UCDA}_{\mathrm{dir}}$ for the damped system. The gray region represents the stop band.

Figure 8

(a) Amplitude reduction, $A_{\mathrm{red}}$, and (b) energy reduction, $E_{\mathrm{red}}$, of the wave propagating through a single-damped UC.

Figure 9

(a) Amplitude reduction, $A_{\mathrm{red}}$, and (b) energy reduction, $E_{\mathrm{red}}$, of the wave propagating through five damped UCs. The gray region represents the stop band.

Figure 10

Illustration of the VAMM plate model with boundary condition (BC) nodes, input nodes, output nodes, and resonators.

Figure 11

Comparison of the host structure’s FRF with the finite VAMM plate using different stop band prediction methods. (a) Undamped case: $m_{\mathrm{eff}}$ and ${UCDA}_{\mathrm{inv}}$. (b) Damped case: ${UCDA}_{\mathrm{dir}}$.

Figure 12

Power flow on the bare plate and VAMM plate along the length of the plate for a frequency of 131 Hz.

Figure 13

(a) Illustration of the VAMM plate with a contoured integral line and normal vectors defined for the calculation of the power loss and (b) the power loss of the damped VAMM plate.

Figure 14

Illustration of the VAMM plate with the contoured integral line and normal vectors defined for the calculation of power loss.

Figure 15

Comparison of the power loss in (a) undamped and (b) damped VAMM plates. Comparison of stop bands predicted by ${STI}_{99\%}$ and ${STI}_{90\%}$ with (c) $P_{\mathrm{loss},\mathrm{Rel}}$, and (d) with FRFs. The gray region represents the stop band.

Figure 16

Comparison of the host structure’s FRF with the finite VAMM plate using different stop band prediction methods. (a) Undamped case: ${STI}_{99\%}$, $m_{\mathrm{eff}}$, and ${UCDA}_{\mathrm{inv}}$. (b) Damped case: ${STI}_{99\%}$, ${STI}_{90\%}$, $UCD{A}_{\mathrm{dir}-99\%}$, and $UCD{A}_{\mathrm{dir}-90\%}$.

Figure 17

Simulation flow in uncertainty analysis.

Figure 18

Stopband width as a function of CV for two different threshold values: (a) $90\%$ and (b) $99\%$.