Development of a hybrid SmEdA/SEA model for predicting the power exchanged between low and high modal density subsystems

Statistical modal Energy distribution Analysis (SmEdA) was developed from classical Statistical Energy Analysis (SEA). It allows computing power flow between coupled subsystems from the deterministic modes of uncoupled subsystems without assuming the SEA modal energy equipartition. SmEdA is well adapted in mid-frequency when the subsystems have not a very high modal density. However, for some systems e.g. the plate-cavity system, one subsystem can exhibit a low modal density while the other one a high one. The goal of the paper is then to propose an extension of SmEdA formulation that allows describing one subsystem by its deterministic modes, and the other one as a diffuse field statistically supposing modal energy equipartition. The uncertain subsystem is then characterized by sets of natural frequencies and mode shapes constructed based on Gaussian Orthogonal Ensemble matrix and the cross-spectrum density of a diffuse field, respectively. This formulation permits not only the computation of mean noise response but also the variance generated by the uncertainties and furthermore without bringing in much computation. It is demonstrated that the obtained analytical results from the proposed hybrid SmEdA/SEA are consistent with numerical results computed by FEM with an appropriate degree of uncertainty.

Mode count and modal density of nonsymmetric cross-ply laminated composite beams

Fiber-reinforced composites are used in many weight critical applications owing to their high strength-to-weight and stiffness-to-weight ratios. In certain applications, fiber-reinforced composites are subjected to broadband excitation sources that act over a significant portion of the audible frequency range leading to the response of a large number of higher order structural modes. In predicting the response levels of such systems, regardless of whether it is modeled in isolation or using a statistical energy analysis framework, it becomes necessary to quantify the number of resonant modes available to receive and store energy within a frequency band. Conventionally, the mode count and modal density are two parameters used for this purpose. Generally, the analysis of the mode count and modal density of anisotropic fiber-reinforced composite structures have received considerably less attention compared to their isotropic metallic counterparts, and as a result a number of key analytical formulations are yet to be derived and investigated. In this work, the modal distribution and density of nonsymmetric cross-ply laminated composite beams coupled in bending and longitudinal extension are analyzed. A wave approach is used to derive an expression for the mode count of the beam having generalized boundary conditions. Using numerical examples and nonlinear regression analysis, simplified expressions are then obtained for the average mode count function of the beam for different boundary conditions. An analytical expression for the modal density is obtained by taking the differential of the average mode count function with respect to frequency. The wave approach employed in this study is validated based on comparison with results from past literature in addition to finite element simulations. The expression for the modal density is also validated using a finite element model and is shown to be independent of boundary conditions.

On the efficient evaluation of modal density and damping of one-dimensional periodic structures using a dynamic homogenization method

Modal density and damping are key parameters for structural vibration analysis. However, the current methods for calculating these properties of periodic structures in high-frequency environment are still insufficient in accuracy and efficiency. In this article, a semi-analytical form for the modal density and damping of longitudinal vibration of one-dimensional periodic structures is proposed based on a dynamic homogenization method. By virtue of asymptotic perturbation expansion, explicit expression is obtained for the dispersion relation. And then, the modal density is obtained in terms of the semi-analytical form by differentiating frequency with respect to wave number. By noting that the high-frequency homogenization method is valid only in the neighborhood of the standing wave frequencies, a weighted technique is introduced to compensate this deficiency. Based on the mode strain energy method, the damping loss factor is also obtained using the high-frequency homogenization results. Because explicit expression can be obtained analytically, the shortage of low computational efficiency and accuracy faced by traditional analysis is significantly made up by the proposed method, which is confirmed by numerical examples.

Acoustics of the banjo: theoretical and numerical modelling

A previous paper [Woodhouse et al., Acta Acustica 5, 15 (2021) https://doi.org/10.1051/aacus/2021009] showed acoustical measurements of an American 5-string banjo alongside similar measurements on a guitar, revealing a strong contrast in bridge admittance. Theoretical and numerical modelling is now presented to probe the physics behind this contrast. Without the bridge and strings, the banjo membrane has a rising trend of admittance associated with its modal density, and it has a distinctive pattern of sound radiation because an ideal membrane has no critical frequency. When the bridge and strings are added to the banjo, three formants shape the amplitude envelope of the admittance. One is associated with local effects of mass and stiffness near the bridge, and is sensitive to bridge mass and the break angle of the strings over the bridge. The other two formants are associated with dynamical behaviour of the bridge, analogous to the “bridge hill” in the violin.

Improved Sound Radiation of Flat Panel Loudspeakers Using the Local Air Spring Effect

The low-frequency performance of exciter-driven flat-panel loudspeakers is technically challenging. The lower modal density results in high deviations in the frequency response, and dips of more than 20 dB are possible. This paper presents an alternative approach for optimizing the modal behavior through the additional air spring effect of an irregular shaped enclosure. The additional mode-dependent air compliance suppresses the panel’s anti-phase components, which minimizes dips in the frequency response and improves the response without adding mass to the system. The approach is analyzed with the measured and simulated results of a prototype. Furthermore, additional enclosure changes were made to visualize the influence of the air spring improved system.

An Experimental Investigation of Modal Densities of Composite Honeycomb Sandwich Cylindrical Shells

Honeycomb sandwich composite cylindrical shells are widely used in aerospace structures. Experimentally observed modal densities of such shells are not reported. In this work, modal densities of a typical honeycomb sandwich composite cylinder are obtained experimentally by measuring the drive point admittance. The results are in good agreement with those estimated theoretically that incorporated transverse shear deformation. Its limitations at higher frequencies are investigated and the frequency beyond which the estimation is in error is determined. The results provide an example to prove the need for measuring the imaginary part of the driving point admittance and using it in the determination of the modal densities of honeycomb sandwich-type structures. Experiments are carried out with two boundary conditions for the cylinder and the results provide experimental evidence for the fact that the modal densities at high frequencies do not depend on the boundary conditions. At higher frequencies, it is expected that both of the face sheets vibrate independently. This frequency can be approximately estimated as the fundamental bending mode frequency of the wall of the honeycomb core. The modal density determined through the measured driving point admittance will have a sharp reduction at this frequency and this feature can be used in identifying this phenomenon. The experimental results are in very good agreement with the above results.

Modal density and critical frequency of composite panels considering transverse shear deformation and rotary inertia

In this work, expressions for estimating the modal density, speed of the bending wave, critical frequency and coincidence frequency of panels are derived considering orthotropic properties of the face sheets, transverse shear deformation and the rotary inertia. Presence of rotary inertia results in an increase in the modal density and a reduction in the speed of the bending waves. The influence is significant at higher frequencies. The critical and coincidence frequencies increase due to rotary inertia. Results for a typical equipment panel of spacecraft are presented and they show the need for incorporating rotary inertia while determining these parameters.