A Sub-band Filter Design Approach for Sound Field Reproduction

The purpose of sound field reproduction is to use loudspeakers to produce desired sound at particular locations in a given environment, which has a wide range of applications such as virtual reality, etc. The computational load required to design and implement filters involved in sound reproduction systems can be significant, especially when the desired sound has rich information over a wide frequency band. To reduce the computational load, sub-band filtering approaches are usually used in sound reproduction systems. In the present work, an approach is proposed to design the sub-band filters used in sound reproduction systems in a more convenient way, where the filter design problem is formulated into a convex optimization problem. Detailed analysis has been conducted on how to specify the response characteristics of each sub-band and how different sub-band filters can be combined into one full band filter in the design and implementation of the system. Results also show that even if the sub-band filter structure is not necessary, this approach can also be applied to reduce the computational load in designing inverse filters when the plant responses involve relatively large differences in delay time among different frequency bands

A Transfer-Matrix-Based Approach to Predicting Acoustic Properties of a Layered System in a General, Efficient, and Stable Way

Layered materials are one of the most commonly used acoustical treatments in the automotive industry, and have gained increased attention, especially owing to the popularity of electric vehicles. Here, a method to model and couple layered systems with various layer types (i.e., poro-elastic layers, solid-elastic layers, stiff panels, and fluid layers) is derived that makes it possible to stably predict their acoustical properties. In contrast with most existing methods, in which an equation system is constructed for the whole structure, the present method involves only the topmost layer and its boundary conditions at two interfaces at a time, which are further simplified into an equivalent interface. As a result, for a multi-layered system, the proposed method splits a complicated system into several smaller systems and so becomes computationally less expensive. Moreover, traditional modeling methods can lose stability when there is a large disparity between the magnitudes of the waves within the layers (e.g., at higher frequencies, for a thick layer, or for extreme parameter values). In those situations, the contribution of the most attenuated wave can be masked by numerical errors, hence inducing instability when inverting the system. Here, the accuracy of the wave attenuation terms is ensured by decomposing each layer’s transfer matrix analytically and reformulating the equation system. Therefore, this method can produce a stable prediction of acoustical properties over a large frequency and parameter region. The fact that the proposed method can couple different layer types in a general, efficient, convenient, and stable way is beneficial, for example, when numerically optimizing the design of the acoustical treatments. The predicted acoustic properties of layered systems calculated using the proposed method have been validated by comparison with those predicted by previously existing methods. Further, an optimal design exercise is performed to find a lightweight layered dash panel treatment

A General Stable Approach to Modeling and Coupling Multilayered Systems with Various Types of Layers

In this article, a general method is proposed to model layered systems with two-by-two transfer matrices, and further, to solve for the acoustic absorption, reflection, and transmission coefficients. Since the proposed method uses the matrix representation of various layers and interfaces from the Transfer Matrix Method (TMM), the equation system can be established efficiently. However, the traditional TMM can lose stability when there is a large disparity between the magnitudes of the waves traveling in opposite directions within the layers (i.e., at higher frequencies, for a thick layer, or for extreme parameter values). In such cases, the contribution of the most attenuated wave can be masked by numerical errors and can induce instability when solving the system. Therefore, in the proposed method, to stabilize the calculated acoustic properties of the system, the principle is to ensure the accuracy of the wave attenuation terms by decomposing each layer’s transfer matrix and reformulating the equation system. This method can couple different layer types in a general way and is easy to assemble and implement with numerical code. The predicted acoustic properties of layered systems calculated using the proposed method have been validated by comparison with those predicted by other existing methods

Study of the Impact of Boundary Conditions on Acoustical Behavior of Granular Materials and their Implementation in the Finite Difference Method

Granular materials display significant differences in their acoustical response when tested in a standing wave tube, compared with the behavior of more traditional sound absorbing materials such as fibrous webs and foams. The latter materials can often be modeled as an equivalent fluid with the further assumption that the material properties do not depend on the input signal level. In contrast, the level dependence of the acoustical behavior of granular materials has been observed in measurements of glass bubbles, as reported in previous studies, for example. When the input level is low, the absorption coefficient of the glass bubble stack shows solid-like behavior with multiple peaks associated with modal response of the stack. On the other hand, when the input level is high, glass bubble stacks show fluid-like behavior, with the quarter wavelength resonance in the direction of the tube axis dominating the response. In the current work, the boundary conditions at the air/granule interface and the granule/tube wall boundary are studied, as is the mechanism causing the variation of the apparent stiffness of the granule stack. The proposed model is implemented with a finite difference approach, and the model predictions are compared with acoustic measurements of granule stacks

Modeling of a Flexible Perforated Membrane Backed by Granular Materials

It has recently been shown that adding activated carbon particles to the interior of a Helmholtz resonator can improve the resonator’s absorption at low frequencies. A similar improvement has been found when a layer of activated carbon partially fills the space behind a finite, tensioned impermeable membrane. In that case, absorption peaks due to the modal response of the tensioned membrane were found to be significantly enhanced in the low frequency range. In the present work, the modeling aspects of the latter work are extended in a number of ways. First, the circular membrane is considered to be both tensioned and flexurally stiff, and further, it is micro-perforated: i.e., it has a finite flow resistance. Secondly, the particle layer behind the membrane is modeled by using a finite difference procedure that accounts for interaction of the particle layer and walls that contain it: i.e., the particle stack itself is allowed to exhibit modal behavior in the radial direction. Finally, the interaction of the membrane nearfield and the particle stack is fully accounted for. It is shown that previously observed behavior can be reproduced, and further that the modal behavior of the particle stack may also enhance the system’s absorption

Experimental Study of Granular Activated Carbon Stacks’ Level- and Time-Dependent Behavior

Granular materials, such as activated carbon, have shown advantages in absorbing low-frequency sounds, but they also exhibit significant nonlinearities. In this article, three unique types of granular particle stacks\u27 acoustical behavior are described and experimentally analyzed. First, when measuring the sound absorption spectra of particle stacks with varying depths and sample holder sizes, it was found that the frictional interaction with the tube circumferential wall can significantly affect the measurements, and a 1-D plane-wave model is only valid when the sample is thin. Secondly, when granular samples were tested with increasing excitation levels, the results indicated a decreased modulus and an increased damping. By further varying excitation level and bandwidth, a displacement- related metric was identified to describe this behavior. Thirdly, when measuring particle stacks over a longer period, the stack\u27s resonance peak gradually shifts to a higher frequency as the sample consolidates. This shift in frequency was proportional to log(Time)

Prediction of Acoustical Behavior of Granular Material Stacks as Measured in a Standing Wave Tube by using a Biot Theory-based Model

The acoustical behavior of granular materials, such as activated carbon and silica gel, has drawn attention in recent studies, due to their favorable properties such as good low frequency sound absorption. Like other more traditional porous materials, granular materials can also be tested in a standing wave tube for a convenient assessment of their acoustical properties. However, the behavior of granular materials stacked in a standing tube is more complex than that of traditional materials. For example, the response of lightweight glass bubbles reveals a clear dependence on the sound pressure level of the input signal. Also, when tested in standing wave tubes of different diameters, the same type of granular materials displays differences in their behavior. The apparent stiffness of granule pack is also related to the depth of the stack. In the present work, a model based on Biot theory is proposed, together with a consideration of the effect of the change of boundary conditions and the granule stack stiffness in different test configurations. The model is realized by using a finite difference method, and the simulation results are compared with measurements of different types of granular materials

A Finite Difference Approach to Study the Impact of Boundary Conditions on the Acoustical Behavior of Particle Stacks

The responses of particle stacks to incident sound waves show interesting features that are very different from those of a homogeneous continuum. Further studies of the acoustical performance of particle stacks can help both to discover potential noise control applications of these types of materials, and to help provide better insight into the internal status of the particle stacks. In the current study, a finite difference (FD) model for a particle stack was built based on the Biot poro-elastic theory. The intention in developing this model is to describe the acoustical behavior of particle stacks with consideration of not only the finite stiffness of the particles, but also the influence of gravity and friction between the particles and the inner wall of their enclosure: i.e., the cylindrical sample holder of a standing wave tube, in this work. The derivation of governing equations and boundary conditions is introduced, together with acoustic measurement results of particles stacks consisting of micron-scale glass bubbles, including absorption coefficients and surface impedances that are compared with the theoretical predictions. The possible application scenarios of such materials, and potential developments that will further improve the FD model will also be discussed

Frequency Reduction and Attenuation of the Tire Air Cavity Mode due to a Porous Lining

The tire air cavity mode is known to be a significant source of vehicle structure-borne road noise near 200 Hz. A porous lining placed on the inner surface of a tire is an effective countermeasure to attenuate that resonance. The two noticeable effects of such a lining are the reduction in the cavity resonance frequency and the attenuation of the air cavity mode. In this paper, through both theoretical and numerical analysis, the mechanisms underlyiing the effects of a porous lining were studied. A two-dimensional duct-shaped theoretical model and a torus-shaped numerical model were created to investigate the lined tire in conjunction with the Johnson-Champoux-Allard model describing the viscous and thermal dissipative effects of the porous material. Design parameters of the porous lining were varied to study their impact and optimal ranges of the design parameters were identified. Finally, in an experimental analysis, the sound attenuation and the frequency drop were observed in measurements of force, acceleration, and sound pressure. In conclusion, it was demonstrated that the suggested theoretical and numerical models successfully predict the effects of porous linings and that the frequency reduction results from the decreased sound speed within the tire owing to the presence of the liner

Predicting Acoustical Performance of High Surface Area Particle Stacks with a Poro-Elastic Model

Because of the high sound absorption they offer at low frequencies, there is a growing interest in high surface area particles and how they might be applied in noise control. Therefore, a model that can accurately predict the acoustic behavior of this type of materials will be useful in relevant applications. A poro-elastic model based on a combination of Biot theory and an existing rigid model of granular activated carbon (GAC) is introduced in the current work. The input parameters for this model consist of a certain number of properties that are known by measurement, and a set of values obtained by matching the model prediction with acoustic measurements. Measured absorption coefficients and surface impedance of stacks of several types of different activated carbon particles are shown in this paper. A fitting procedure that determines the unknown parameters is also described. It is shown that the model is able to predict the acoustic behavior of the particle stacks, and especially to capture the frame resonances at low frequencies, thus, validating the proposed model. Beyond the activated carbon used in the present tests, it is reasonable to generalize this model to stacks of other high surface area particles