Acoustic Mode Shapes Research Articles

Mass-spring model for acoustic metamaterials consisting of a compact linear periodic array of dead-end resonators.

This paper presents a mass-spring model to predict the normal incidence acoustic response of a metamaterial composed of a compact linear periodic array of dead-end resonators. The dead-end resonators considered are ring-shaped Helmholtz resonators. The model is based on a mass-spring analogy and considers the thermoviscous losses in the metamaterial following an effective fluid approach. A matrix equation of acoustic motion is derived for the finite case of N-periodic arrays. Under external excitation, its direct solution predicts the sound absorption coefficient and transmission loss. Under the homogeneous case, the solution of its associated eigenvalue problem predicts the acoustic eigenfrequencies and mode shapes. The dispersion relation is also solved to predict the beginning of the first stopband, and a low frequency approximation allows development of a formula to estimate the first eigenfrequency. The results show that the system with N degrees of freedom has three stopbands over the frequency range studied, with zero sound absorption and transmission. The model also helps to understand how the acoustic dissipation, at a given resonant frequency, is affected by the position of the acoustic velocity nodes (eigenmodes) in the geometry of the metamaterial. Prototypes are designed, manufactured, and tested in an impedance tube to validate the model.

Source Characterization using Inverse Numerical Acoustics based on Pellicular Modes

An inverse method is used to characterize radiating noise sources. Measurements of the sound pressure in the field are used to reconstruct the vibration on the surface of a vibrating structure. The basis functions for this inverse calculation are pellicular modes. These modes are calculated by assuming that there is a thin air envelope around the vibrating structure. The acoustic modes are calculated using this air envelope and the acoustic mode shapes are then assumed to be structural modes. Though not the actual structural modes, these modes are easily calculated and form convenient basis functions. This procedure is demonstrated on two examples. The first is a small electric motor and the second is a generator set. It is demonstrated that the procedure can be used to accurately characterize the sound radiation from a source.

ACOUSTIC ADMITTANCE OF AN ALUMINIZED COMPOSITE SOLID PROPELLANT BY LASER DOPPLER VELOCIMETRY AT LOW PRESSURE

The present study involves the use of laser Doppler velocimetry (LDV) to determine the acoustic admittance of an aluminized composite solid propellant at low mean chamber pressure. The aim of this study is to measure the fluctuating velocity component perpendicular to the burning surface of the particles in the gaseous phase above the burning surface of the aluminized composite solid propellant for the imposed acoustic pressure oscillations. To check the accuracy and precision of the mini-LDV used in this study the acoustic mode shape was generated for the imposed acoustic oscillations with amplitudes of 215 and 6 Pa. The results obtained are very satisfying; they show that the predicted acoustic velocity mode shape matches with the generated mode shape using the LDV. The response function of a solid propellant indicates the inclination of the propellant to cause acoustic combustion instability. Both velocity fluctuations and imposed pressure fluctuations are measured experimentally and used to determine the acoustic admittance of the solid propellant and hence deducing the pressure coupled response function from acoustic admittance. The probe volume of the laser is maintained at a constant distance of 800-900 μm from the burning surface by a servomechanism using a linear actuator. The incorporation of the linear actuator dramatically increases the number of particles passing through the laser probe volume. The linear actuator is controlled using a microcontroller and the servomechanism is operated under an open loop system. A high-pressure window bomb is designed and fabricated. Low-frequency ranges (140-280 Hz) of the longitudinal acoustic modes are investigated in this paper. Tests were conducted at a mean chamber pressure of 0.5 MPa. To generate the acoustic pressure oscillations a rotary valve is designed and fabricated. The rotary valve generates a standing wave of the characterizing frequency depending upon the length of the chamber. Tests are conducted for two aluminized composite solid propellants. The results obtained are in good agreement with the literature for the pressure coupled response function at these mean chamber pressures.

Jet Velocity and Acoustic Excitation Characteristics of a Synthetic Jet Actuator

The effect of the excitation frequency of synthetic jet actuators on the mean jet velocity issuing from an array of circular orifices is investigated experimentally, focusing on the acoustic excitation characteristics of the actuator’s cavity. Two cavity configurations are considered. In the first configuration, synthetic jets are generated by exciting a single, large cavity having an array of sixteen orifices via sixteen piezoelectric elements. In the second configuration, the cavity volume of the first configuration is divided into eight isolated compartments, each with two orifices and two piezoelectric elements. Several distinct resonant peaks were observed in the frequency response of the synthetic jet actuator built with a single large-aspect-ratio cavity, whereas the case of compartmentalised cavities exhibited a single resonant peak. Acoustic simulations of the large-aspect-ratio-cavity volume showed that the multiple peaks in its frequency response correspond to the acoustic standing-wave mode shapes of the cavity. Due to its large aspect ratio, several acoustic mode shapes coexist in the excitation frequency range aside from the Helmholtz resonance frequency. When the actuator’s cavity volume is compartmentalised, only the Helmholtz resonance frequency is observed within the excitation frequency range.

Selection Rules for Resonant Longitudinal Injector-Coupling in Experimental Rocket Combustors

This paper investigates different types of longitudinal mode coupling in subscale rocket combustion chambers using experimental data and numerical simulations. Based on a one-dimensional planar wave acoustic model of coupled cavity resonators with two acoustic inlet boundary conditions, mode selection rules are derived, providing a simple way of predicting which injector and combustion chamber modes have matching frequencies. Longitudinal mode coupling of an injector with an acoustically open inlet boundary condition has been reported in the literature for the start-up transient of a research combustor experiment. In this experiment, every third injector mode couples to a corresponding chamber longitudinal mode, which is explained in terms of the selection rules derived in this paper. Numerical simulation results for a different combustor experiment show an unexpected mode coupling behavior when an acoustically closed injector inlet is used. Theoretical analysis by using the one-dimensional wave model and applying the derived selection rules shows that in this setup, the injector acoustic mode can accommodate two different acoustic boundary conditions at the injector-chamber interface simultaneously. This results in different acoustic mode shapes in the injector, explaining the unexpected behavior for the resonant coupling with an acoustically closed injector inlet.

Effect of Outlet Boundary Condition on the Acoustic Modeshape and Flame Dynamics of a Partially Premixed Swirl Stabilized Combustor

Abstract Industrial gas turbines commonly use lean premixed swirl stabilized combustors that are prone to thermo-acoustic instabilities. Combustor testing involves several steps ranging from atmospheric to high-pressure conditions. An open outlet boundary condition is maintained for atmospheric tests commonly, whereas high-pressure testing involves complex exit boundary conditions, which change the reflection coefficient and can affect the nature of instability. Current work aims at studying how the change in outlet boundary affects the nature of instability due to changes in the exit reflection coefficient and acoustic mode shape inside the combustor. A laboratory-scale industrial swirl burner that uses partially premixed methane and air at atmospheric conditions is analyzed for this purpose. A constant area contraction ratio of 6.5:1 is maintained at the exit of the combustor while varying the inlet Reynolds number at a constant global equivalence ratio. Flame dynamics based on conditional phase averaged OH* chemiluminescence images and spatial Rayleigh index maps were used to compare different flow rate and exit boundary cases. The outlet contraction affects both the frequency and amplitude of the dominant thermoacoustic mode. The orifice plate at the exit reduces the outlet reflection coefficient leading to a change in acoustic mode shape inside the combustor. Overall, the instability amplitude is reduced considerably for cases with an orifice plates at the exit boundary compared to open exit boundary cases. The results show the importance of defining outlet boundary conditions as a parameter in combustion instability studies. Care should be taken while comparing and interpreting results from different facilities where outlet boundary condition is different.

Optimum Multinozzle Configuration for Minimizing the Rayleigh Integral During High-Frequency Transverse Instabilities

Abstract This paper develops a formalism for optimizing nozzle location/configuration with respect to combustion stability of high-frequency transverse modes in a can combustor. The stability of these acoustically noncompact flames was assessed using the Rayleigh integral (RI). Several key control parameters influence RI—flame angle, swirling strength, nozzle location, as well as nozzle location with respect to the acoustic mode shape. In this study, we consider a N-around-1 configuration such as typically used in a multi-nozzle can system and study the overall stability of this system for different natural transverse modes. Typically, such nozzles are distributed in a uniformly circular manner for which we study the overall RI, and for cases where RI > 0, we optimize the nozzle distribution that can reduce and minimize RI. For a fixed geometry such as a circular configuration, the analysis shows how the flame's parameters must vary across the different nozzles, to result in a relatively stable system. Additionally, for a fixed set of flame parameters, the analysis also indicates the noncircular distribution of the N nozzles that minimizes RI. Overall, the analysis aims to provide insights on designing nozzle locations around the center nozzle for minimal amplification of a given transverse mode.

Analytical and empirical models for the acoustic dispersion relations in partially filled water pipes

Sound propagation in a partially filled pipe is a complex and not a well-researched topic. However, this class of pipes is of high importance because it is typical to millions of kilometres of buried sewers and drainage pipes laid across the world. Good knowledge of conditions in these pipes is critical to their successful operation and proactive rehabilitation. Acoustic methods are attractive means for their rapid condition assessment. Therefore, good understanding of sound propagation in this class of pipes is of importance. In response, this paper studies acoustic waves propagation in a pipe with different water filling ratio. Based on a 2-dimensional finite element method (FEM) study, an analytical approximation and empirical model are developed and proposed to predict the acoustic wave dispersion relations (curves) and mode shapes. These two models can be used to predict the acoustic frequency response function of a partially filled pipe with an accuracy of better than 3 dB below twice the first non-axisymmetric mode frequency, but at 1/2000 of computational cost when compared with the FEM. This work paves the way to the development of highly efficient, real-time robotic systems for autonomous inspection of partially filled pipes with acoustic means.

A Modified Fourier Series Solution for a Thermo-Acoustic Tube with Arbitrary Impedance Boundaries

Rijke tube is a benchmark model widely used in thermo-acoustic community. As an alternative to existing modeling methods, this work proposes a modified Fourier series solution for modal characteristic analyses of a one dimensional (1D) thermo-acoustic system. The proposed modeling framework allows the consideration of arbitrary impedance boundaries owing to the special feature of the Fourier expansion series enriched by boundary smoothing polynomial terms. Thermo-acoustic Helmholtz governing equation coupled with a first-order heat release model is discretized through Galerkin procedure. Thermo-acoustic modal parameters are obtained by solving a standard quartic matrix characteristic equation, different from conventionally used root searching based on a transcendental equation. Numerical examples are presented to validate the proposed model through comparisons with results reported in the literature. Influences of boundary impedance are analyzed. Results reveal a quantitative relationship between the thermo-acoustic instability and heat source position with respect to the acoustic mode shapes. Results also show the existence of a sensitive zone, in which the thermo-acoustic modal behavior of the impedance-ended (IE) tube shows drastic changes with the boundary impedance. Meanwhile, a stable zone can be achieved upon a proper setting of the boundary impedance through suitable combination of the real and imaginary parts of the impedance.

Physical model for acoustic resonance in annular cavity structure

A physical model for acoustic resonance in the annular cavity structure is developed to represent the typical characteristic when acoustic resonance occurs. Firstly, the measurement of sound pressure in the casing and rotor blades vibration is operated in a multistage high pressure compressor. The sharp peak frequency and discrete multi-tone occur in the frequency spectrum of sound pressure in the compressor, and the vibration of the first stage of rotor blades synchronously presents the high amplitude. The frequencies associated with rotor blades vibration can be calculated with rotating sound source theory. It is also confirmed that acoustic resonance occurs in the multistage compressor. With acoustic similarity principle, an annular cavity model is established to simulate the typical characteristics of acoustic resonance in the compressor based on Large Eddy Simulation (LES) and Lighthill acoustic analogy. The coupling relationship between cavity acoustic mode and disc vibration mode shape is expounded when acoustic resonance occurs in the model. And acoustic resonance will be locked in the certain flow rate range. All these characteristics match well with those occur in the multistage high pressure compressor.

Application of the Perforated Plate in Passive Control of the Nonpremixed Swirl Combustion Instability Under Acoustic Excitation

Perforated plates are widely used to attenuate noise emission and as acoustic liners in combustion chambers. In this study, the damping performance of the perforated plate located in the combustor inlet section is experimentally and numerically studied. The primary response of nonpremixed swirl flame under 30–400 Hz acoustic excitation with a 445 mm inlet length occurs at 134 Hz and 210 Hz modes. The perforated plate designed for 210 Hz sound absorption with a 328 mm cavity length and an 8.04% porosity is compared to plates with various cavity lengths and different orifice patterns. The acoustic absorption capability of perforated plates is evaluated by the Luong model and tested in an impedance tube. The acoustic measurements show that the sound absorption performance of each plate is strongly affected by the bias flow velocity and cavity length. The combustion results indicate that the installation of perforated plates at the inlet section has two effects: sound attenuation and redistribution of the pressure mode of the combustor. The acoustic mode analysis further demonstrated that, for damping the nonpremixed flame when the combustion instability is caused by the inlet pressure fluctuation, modification of the inlet acoustic mode shape is more efficient than the sound attenuation.

Global active control of harmonic noise in a vibro-acoustic cavity using Modal FxLMS algorithm

The Modal FxLMS algorithm is recently proposed for reduction of global level of noise in vibro-acoustic cavities. The algorithm minimizes acoustic potential energy expressed in modal domain which allows a choice of acoustic modes whose modal amplitudes are desired to be minimized. The working of the algorithm and its effectiveness has been demonstrated previously through a numerical study and an experimental study is required to test the effectiveness of the algorithm in practice. To meet this objective this paper presents an experimental study of this algorithm for global active noise control in a rectangular box cavity. The algorithm utilizes modal filtered reference signals and modal amplitudes of the selected acoustic modes of the cavity. To identify modal filtered reference signals, modal secondary paths are utilized. In the present study, modal secondary paths are identified using experimentally identified physical secondary paths and acoustic mode shapes. Modal amplitudes of the acoustic modes are identified on the basis of acoustic pressure measurements from eight microphones suitably placed throughout the cavity. The performance of the algorithm is studied under the presence of acoustic and structural disturbances using one or two control loudspeakers. The active control is carried out at harmonic frequencies coinciding with cavity-controlled resonances and panel-controlled resonances. It is found from the experimental results that the Modal FxLMS algorithm is successful in reducing modal amplitudes of the selected acoustic modes. When all the significantly contributing acoustic modes are chosen, a reduction in global level of noise is observed at cavity as well as panel controlled resonances.

Numerical study on the influence of acoustic natural frequencies on the dynamic behaviour of submerged and confined disk-like structures

The dynamic response of disks has been deeply studied in the last years given that their dynamic characteristics present similarities with more complex disk-like structures used in real engineering applications, such as hydraulic turbine runners. Because of disk-like structures could present fatigue damage or critical failures as a result of resonance conditions, it is of paramount importance to determine their natural frequencies.The dynamic response of disk-like structures is heavily affected by the added mass effect when they are surrounded by a heavy fluid. This added mass is greatly affected by the proximity of walls. Furthermore, the surrounding fluid cavity has its own natural frequencies and mode shapes, called acoustic natural frequencies and acoustic mode-shapes. All studies of submerged and confined disks have been carried out considering that the acoustic natural frequencies of the surrounding fluid cavity are much higher than the natural frequencies of the disk, so they do not affect each other. However, in some cases the acoustic natural frequencies are close to the natural frequencies of the submerged structure, which can be affected considerably. This case has not been deeply discussed yet.In this paper, the influence of the acoustic natural frequencies of a cylindrical fluid cavity on the natural frequencies of a disk has been analysed numerically. First, the effect of the added mass of the fluid has been estimated when the acoustic natural frequencies of the fluid cavity are much higher than the natural frequencies of the disk. For this case, different geometrical and material parameters have been considered. Then, the parameters that affect the acoustical natural frequencies of the fluid cavity have been identified. Finally, the case with acoustic natural frequencies close to the structural natural frequencies is studied in detail and the affectation between both is discussed. All the results presented in this paper have been dimensionless in order to be used for a wide range of disk-like structures.Therefore, with this study it is possible to identify for which conditions the dynamic response of a generic disk-like structure will be affected by the acoustic natural frequencies of its surrounding fluid cylindrical cavity.

A model combustor for studying a reacting jet in an oscillating crossflow.

This paper discusses a novel model combustion experiment that was built for studying the structure and dynamics of a reacting jet in an unsteady crossflow. A natural-gas-fired dump combustor is used to generate and sustain an acoustically oscillating vitiated flow that serves as the crossflow for transverse jet injection. Unlike most other techniques that are limited in operating pressure or acoustic amplitude, this method of generating an unsteady flow field is demonstrated at a pressure of 10 atm with peak-to-peak oscillation amplitudes approaching 20% of the mean pressure. An optically accessible test section designed for these conditions provides access for advanced laser and optical diagnostic measurements. Detailed measurements provide insight into the complex acoustic-hydrodynamic-combustion coupling processes and offer high-quality, high-resolution validation data for numerical simulations. Careful instrumentation port design considerations for the higher amplitude acoustics are detailed. As a whole, this paper focuses on select representative segments of the experiment operational space that highlight our strategy of providing an oscillatory flowfield. This includes presenting the acoustic operational space such as acoustic amplitudes, frequencies, and mode shapes. Select imaging results are then reported to support our strategies capability to produce high-fidelity measurements.

Viscous effects on the acoustics and stability of a shear layer over an impedance wall

The effect of viscosity and thermal conduction on the acoustics in a shear layer above an impedance wall is investigated numerically and asymptotically by solving the linearised compressible Navier–Stokes equations (LNSE). It is found that viscothermal effects can be as important as shear, and therefore including shear while neglecting viscothermal effects by solving the linearised Euler equations (LEE) is questionable. In particular, the damping rate of upstream-propagating waves is found to be under-predicted by the LEE, and dramatically so in certain instances. The effects of viscosity on stability are also found to be important. Short wavelength disturbances are stabilised by viscosity, greatly altering the characteristic wavelength and maximum growth rate of instability. For the parameters considered here (chosen to be typical of aeroacoustic situations), the Reynolds number below which the flow stabilises ranges from$10^{5}$to$10^{7}$. By assuming a thin but non-zero-thickness boundary layer, asymptotic analysis leads to a system of boundary layer governing equations for the acoustics. This system may be solved numerically to produce an effective impedance boundary condition, applicable at the wall of a uniform inviscid flow, that accounts for both the shear and viscosity within the boundary layer. An alternative asymptotic analysis in the high-frequency limit yields a different set of boundary layer equations, which are solved to yield analytic solutions. The acoustic mode shapes and axial wavenumbers from both asymptotic analyses compare well with numerical solutions of the full LNSE. A closed-form effective impedance boundary condition is derived from the high-frequency asymptotics, suitable for application in frequency domain numerical simulations. Finally, surface waves are considered, and it is shown that a viscous flow over an impedance lining supports a greater number of surface wave modes than an inviscid flow.

Geoacoustic inversion using low frequency broadband acoustic measurements from L-shaped arrays in the Shallow Water 2006 Experiment.

When using geoacoustic inversion methods, one objective function may not result in a unique solution of the inversion problem because of the ambiguity among the unknown parameters. This paper utilizes acoustic normal mode dispersion curves, mode shapes, and modal-based longitudinal horizontal coherence to define a three-objective optimization problem for geoacoustic parameter estimation. This inversion scheme is applied to long-range combustive sound source data obtained from L-shaped arrays deployed on the New Jersey continental shelf in the summer of 2006. Based on the sub-bottom layering structure from the Compressed High-Intensity Radiated Pulse reflection survey at the experimental site, a two-layer (sand ridge overlaying a half-space basement) range-independent sediment model is utilized. The ambiguities of the sound speed, density, and depth of the sand ridge layer are partially removed by minimizing these objective functions. The inverted seabed sound speed over a frequency range of 15-170 Hz is comparable to the ones from direct measurements and other inversion methods in the same general area. The inverted seabed attenuation shows a nonlinear frequency dependence expressed as αb=0.26f1.55(dB/m) from 50 to 500 Hz or αb=0.32f1.65(dB/m) from 50 to 250 Hz, where f is in kHz.

A numerical method for seeking the relationship between structural modes and acoustic radiation modes of complicated structures

Both structural modes and acoustic radiation modes play important roles in the investigation of structure-borne sound. However, little work has been done for inherent relations between these two kinds of modes. Previous work has mainly dealt with simple and regular structures such as rectangular plates and single-layer cylindrical shells. Therefore, the relationship between structural modes and acoustic radiation modes of complicated structures which has great theory significance and utility value is an important problem that must be studied. This paper presents a numerical method for seeking the relationship between structural modes and acoustic radiation modes of complicated structures. First, a governing equation for relating these two kinds of modes is given based on the characteristics of the modes. Then, substitute the normal structural mode shape matrix and the acoustic radiation mode shape matrix which are obtained by FEM into the governing equation, the modal participating coefficients can be solved, thus we can get the corresponding relations between these two kinds of modes. Using the model of a simply supported truncated conical shell, a numerical example is presented with the numerical method which this paper has proposed. And then, the radiated sound power is calculated to verify the effectiveness of this method and the correctness of this conclusion. The results show that the numerical method proposed in this paper is feasible.

Flow-Excited Acoustic Resonance of Trapped Modes of a Ducted Rectangular Cavity

Flow over ducted cavities can lead to strong resonances of the trapped acoustic modes due to the presence of the cavity within the duct. Aly and Ziada (2010, “Flow-Excited Resonance of Trapped Modes of Ducted Shallow Cavities,” J. Fluids Struct., 26(1), pp. 92–120; 2011, “Azimuthal Behaviour of Flow-Excited Diametral Modes of Internal Shallow Cavities,” J. Sound Vib., 330(15), pp. 3666–3683; and 2012, “Effect of Mean Flow on the Trapped Modes of Internal Cavities,” J. Fluids Struct., 33, pp. 70–84) investigated the excitation mechanism of acoustic trapped modes in axisymmetric cavities. These trapped modes in axisymmetric cavities tend to spin because they do not have preferred orientation. The present paper investigates rectangular cross-sectional cavities as this cavity geometry introduces an orientation preference to the excited acoustic mode. Three cavities are investigated, one of which is square while the other two are rectangular. In each case, numerical simulations are performed to characterize the acoustic mode shapes and the associated acoustic particle velocity fields. The test results show the existence of stationary modes, being excited either consecutively or simultaneously, and a particular spinning mode for the cavity with square cross section. The computed acoustic pressure and particle velocity fields of the excited modes suggest complex oscillation patterns of the cavity shear layer because it is excited, at the upstream corner, by periodic distributions of the particle velocity along the shear layer circumference.

A numerically efficient damping model for acoustic resonances in microfluidic cavities

Bulk acoustic wave devices are typically operated in a resonant state to achieve enhanced acoustic amplitudes and high acoustofluidic forces for the manipulation of microparticles. Among other loss mechanisms related to the structural parts of acoustofluidic devices, damping in the fluidic cavity is a crucial factor that limits the attainable acoustic amplitudes. In the analytical part of this study, we quantify all relevant loss mechanisms related to the fluid inside acoustofluidic micro-devices. Subsequently, a numerical analysis of the time-harmonic visco-acoustic and thermo-visco-acoustic equations is carried out to verify the analytical results for 2D and 3D examples. The damping results are fitted into the framework of classical linear acoustics to set up a numerically efficient device model. For this purpose, all damping effects are combined into an acoustofluidic loss factor. Since some components of the acoustofluidic loss factor depend on the acoustic mode shape in the fluid cavity, we propose a two-step simulation procedure. In the first step, the loss factors are deduced from the simulated mode shape. Subsequently, a second simulation is invoked, taking all losses into account. Owing to its computational efficiency, the presented numerical device model is of great relevance for the simulation of acoustofluidic particle manipulation by means of acoustic radiation forces or acoustic streaming. For the first time, accurate 3D simulations of realistic micro-devices for the quantitative prediction of pressure amplitudes and the related acoustofluidic forces become feasible.

構造音響連成系における伝達率を用いた連成固有モードの主体性の判別

With the development of box structure like a car or plane, noise reduction technology is required. For interior noise reduction, it is necessary to identify a resonance mode which is affected by the acoustic characteristics. Meanwhile, noise problem with structural vibration has been studied as coupled structural-acoustic system. In case of coupled system, it is useful to understand a resonance mode as an acoustic dominant coupled mode for noise reduction. Hence, it is necessary to develop a method to identify the acoustic dominant coupled mode. The purpose of this study is ”identifying the acoustic dominant coupled mode by proposing the coupled structural-acoustic transmissibility”. The dominant characteristics of system in the coupled system and the coupled transmissibility are mentioned in a theory section. The proposed theory is verified by FEM simulation. It is possible to classify the dominant characteristics by analysing single system using the coupled transmissibility. For verifying the practicality of the method, experiments using a car are carried out. The coupled modes are identified by modal property identification experiment. Two coupled modes having an acoustic longitudinal 1st mode shape are identified. The acoustic dominant coupled mode is identified by measuring the coupled transmissibility. Hence, it is concluded that the acoustic dominant coupled mode can be identified by the coupled transmissibility.