Circumferential Order Research Articles

Modelling of sound propagation in a non-uniform lined duct using a Multi-Modal Propagation Method

Liner non-uniformities, such as distributed impedances, may have a direct influence on the performance of turbofan engine liners. A relevant problem to study these effects is that of sound propagation in a hard-walled duct of circular cross-section, fitted with a region of non-uniform liner. Given the complex modal input amplitudes at one end of the hard-walled duct, the problem is to compute the complex modal output amplitudes at the other end. In the present paper, a Multi-Modal Propagation Method (MMPM) is proposed to solve this problem in the absence of mean flow. For simplicity, the liner impedance is set piecewise constant along the duct, while being arbitrarily variable along the circumference of each segment. The principle of the method is to expand the sound pressure and axial velocity into double infinite series using the rigid duct modal basis, and then to follow the projection coefficients evolution along the duct axis. Scattering matrices are obtained for individual segments and then combined to construct a global scattering matrix. It is numerically shown that the convergence rate of the infinite series is at least O ( M - 2 ) and O ( N - 1.5 ) , where M and N refer to the maximum circumferential and radial mode orders, respectively. Validation of the method is done in 2D by comparison with FEM. The present MMPM is shown to deal with realistic turbofan engine configurations with spliced liners, up to relatively high reduced wavenumbers K ∼ 50 .

Acoustic characteristics of a sound field inside a cylindrical structure with excited end plates: Influence of excitation position on vibro-acoustic coupling

Vibro-acoustic coupling between plate vibrations and the sound field inside a cylindrical structure is investigated. Each end plate of the cylindrical structure is excited by a point force. When the excitation position, which directly affects the vibration characteristics of the plates, is examined by shifting the point of application radially along the plate, the sound field is estimated based on the contribution defined as the ratio of acoustic energy stored in each acoustic mode to the total acoustic energy in the entire sound field. Coupling is intensified by the coincidence of a circumferential order with respect to the modes of plate vibration and of the sound field. Therefore, if the vibration modes at the two end plates have different orders due to the influence of the excitation position, then the sound field is composed of a number of acoustic modes. In particular, excitation at a position near the greatest flexural displacement causes the vibration energy to increase, and the contribution of the corresponding acoustic mode also increases. However, approaching the nodal circle of plate vibration, the excitation position develops coupling solely with the other plate vibration, because the occurrence of the vibration mode is restrained.

Acoustic characteristics inside cylindrical structure with end plates excited at different frequencies

In order to study coupling between vibration of a structure and a sound field in contact with the structure, a cavity surrounded by a rigid cylinder having thin elastic plates at both ends is adopted as an analytical model. When excitation forces of different frequencies are applied to the respective plates, the plate vibrations and the sound field inside the cavity become aperiodic, because of the coupling between the systems. In the present investigation, distribution of the sound pressure level inside the cavity is studied in detail in order to clarify the coupling behavior. The results show that when the respective plates vibrate at the same circumferential order, the vibration modes of the plates, which effect the coupling of the plate vibrations and the sound field, cause the aperiodic nature of each system to develop. In this case, since the dominant mode exists in formation of the sound field, it significantly influences the aperiodic nature of the coupling systems. In the case of vibration modes where the plates vibrate at different circumferential orders, the behaviors of the three systems, whose coupling has been restrained, approximate a steady state. Consequently, the dominant mode does not appear in the sound field.

Acoustic Characteristics of Sound Field in Cylindrical Enclosure with Exciting End Plates.

This paper describes the coupling of the vibration of exciting circular end plates and the sound field in a cylindrical enclosure. Assuming that the lateral wall of the cylindrical structure is rigid, a theoretical analysis is carried out for the relative amplitude and the phase difference as to both exciting forces, which influence the coupling of the vibration of the end plates and the sound field. If the vibration modes of both end plates are coincide with the different acoustic modes of the sound field with respect to the circumferential order, the distribution of the sound field has the complex nature of some acoustic modes. The distribution depends on the relative amplitude of both exciting forces. Coupling with the same vibration mode of both end plates, the acoustic modes with the cylindrical length orders are restrained from dominating the odd orders by the exciting forces of the same phase, while the inverse phase makes the even orders restrained.

A classification of duct modes based on surface waves

For the relatively high frequencies relevant in a turbofan engine duct, the modes of a lined section may be classified in two categories: genuine acoustic 3D duct modes resulting from the finiteness of the duct geometry, and 2D surface waves that exist only near the wall surface in a way essentially independent of the rest of the duct. Per frequency and circumferential order there are at most four surface waves. They occur in two kinds: two acoustic surface waves that exist with and without mean flow, and two hydrodynamic surface waves that exist only with mean flow. The number and location of the surface waves depends on the wall impedance Z and mean flow Mach number. When Z is varied, an acoustic mode may change via small transition zones into a surface waves and vice versa. Compared to the acoustic modes, the surface waves behave—for example as a function of the wall impedance—rather differently as they have their own dynamics. They are therefore more difficult to find. A method is described to trace all modes by continuation in Z from the hard-wall values, by starting in an area of the complex Z-plane without surface waves.

The effect of internal point masses on the radiation of a ribbed cylindrical shell.

A finite element analysis is performed on a submerged cylindrical shell with internal frames and point masses attached to the frames. A point force on the shell is shown to excite resonances of the frames. Without the point masses, most of these resonances are evanescent and do not radiate to the far field. With the point masses, the high circumferential order resonances couple with those at low circumferential order with the result of increasing radiation to the far field. In this investigation, the radiation increase is about 10 dB over a broad frequency range (1.5 to 3 times the ring resonance).

624 両端板を有する円筒内部音場に及ぼす加振周波数の影響

This paper describes about the acoustic characteristics of the sound field in a cylindrical cavity coupled with the end plates excited by the different frequency. An analytical model is proposed, in which the elastic surround is represented by linear and torsional springs distributed uniformly around the exciting plate edge, and coupled vibration analysis is carried out for varying the frequency and the amplitude of the exciting force. The results show that the acoustic mode coupled with plate motion, which is coincident with the circumferential order, has an influence on the sound field, while the coupled behavior is relatively weak in case both the orders differ mutually.

Backward projection of harmonic pressure measurements using the generalized internal source density method and SVD

The Generalized Internal Source Density (GISD) method [P. R. Stepanishen, J. Acoust. Soc. Am. 102, 1955–1963 (1999)] is used to backward project acoustic harmonic pressure fields from an external measurement surface of revolution enclosing the finite three-dimensional vibrating body that generated the field. Both the normal velocity and pressure fields on the vibrating surface can be readily reconstructed using the method. The GISD method is based on first decomposing the acoustic field on the measurement surface into a modal summation of circumferential orders, where each modal pressure field is associated with an internal linear distribution of ring sources on the axis of revolution of the measurement surface. Each modal linear distribution of ring sources is evaluated using Singular Value Decomposition (SVD) methods. A brief review of the method is presented and discussed and then numerical results are presented to illustrate the accuracy of the method for a spherical shell and finite cylindrical shells with hemispherical end caps. The results show excellent agreement with the prescribed normal velocities and associated pressure fields over a wide frequency range.

A MODAL EXPANSION ANALYSIS OF NOISE TRANSMISSION THROUGH CIRCULAR CYLINDRICAL SHELL STRUCTURES WITH BLOCKING MASSES

This paper covers the development and application of a modal interaction analysis (MIA) to investigate the plane wave transmission characteristics of a circular cylindrical sandwich shell of the type used in the aerospace industry for satellite launch vehicles. The model is capable of handling many high order structural and acoustic modes, and can be used to investigate the sensitivity to different structural stiffness configurations, angles of incidence, damping and cavity absorption. The model has been developed to predict the structural response and transmitted noise when a number of discrete masses are applied to the shell. The study presented considers a set of cases where blocking masses, having a total weight equal to 8% of the cylinder weight, are attached to the cylinder. The simulations carried out show a substantial reduction of the sound transmission in many of the first 15 one-third octave frequency bands (frequency range 22·4–707 Hz). The blocking masses act on the shape of the cylinder normal modes and their orientations with respect to the plane of the incident wavenumber vector. In particular, the circumferential re-orientation reduces the coupling between the incident acoustic field and the structural modes of the cylinder. The modification of the structural mode shapes, both in axial and circumferential directions, also reduces the coupling between the cylinder modes and the acoustic modes of the interior.Simulations show the effect of the number of structural and acoustic modes included on the calculated frequency response, and indicate the number necessary for an accurate prediction of the resonant and non-resonant sound transmission through the structure. In particular, the effect of neglecting off-resonance acoustic and structural modes is investigated. It is shown that restricting the acoustic and structural modes to those having natural frequencies within an interval of ±40 and ±60 Hz, respectively, of the excitation frequency produces acceptably small errors in the transmission estimate. The simulations also show that in order to represent accurately the coupling effect between the structural and acoustic modes, for each acoustic mode of order ma,na , pa(axial, circumferential and radial order, respectively), it is necessary to account only for the structural modes with ns=naandms =ma±α with α=1, 3, 5,…, αmax. It is found that the time required to compute the sound transmission in a frequency range of 0–3123 Hz, using the minimum number of acoustic and structural modes required to compute an accurate response at each frequency, is 3% of that necessary for the computation of the full response using all the structural and acoustic modes with natural frequencies within the frequency range considered in the analysis.

Wave localization on a submerged cylindrical shell with rib aperiodicity.

The results of a numerical study of vibration localization due to stiffener variability in a framed shell are reported. An axisymmetric finite element (FE)-infinite element model is used to obtain predictions in good general agreement with previously reported experimental results. Over the frequency band of this study, up to three times the ring frequency, two structural resonances dominate the vibratory response of the shell for high circumferential orders (n > 10). Localization is shown to be linked to the sensitivity of the local resonance frequencies of the system to specific geometrical parameters. Specifically, rib thickness variations strongly affect the first pass band, while rib spacing variations strongly affect the second pass band.

The radiation and vibratory response of a fluid-loaded structure with high internal complexity

Recently, experiments were reported in which complex internal structure was shown to dramatically influence the scattering cross section of fluid-loaded shells [Photiadis et al., J. Acoust. Soc. Am. 101, 895–899 (1997)] and where spatially varying local admittances were proposed to explain the observed scattering details [Bucaro et al., J. Acoust. Soc. Am. 100, 2721 (1996)]. Here, measurements of the vibratory response of this same structure [a ribbed shell with a large number (∼1000) of internal oscillators] are shown. Comparisons are made to the response of an identical shell with no internal oscillators. For the complex structure, no distinct pass and stop bands for any of the circumferential orders is seen, and a high degree of localization at all circumferential orders and frequencies is observed. Generally speaking, the wave-number-frequency plots associated with the complex structure are significantly different than those for the simple framed cylinder at all but the highest azimuthal components (n>17). Moreover, the wave-number decompositions of these data reveal a strong dominance of local bending in the response of the complex structure and evidence that strong coupling between the circumferential orders contributes to the significantly enhanced radiation levels. [Work supported by the Office of Naval Research.]

The forward projection of harmonic pressure fields using the generalized internal source density method

A previously developed generalized internal source density (GISD) method is extended to address the forward projection of harmonic pressure fields from normal velocity fields on a closed surface of revolution in a fluid. The GISD approach is based on decomposing the normal velocity field on the closed surface into a summation of circumferential orders where the pressure field for each order is associated with an internal linear source distribution of ring sources on the axis of revolution of the surface. The linear source distributions are formulated as integral equations of the first kind which are then solved using singular value decomposition methods. Far-field pressures are simply related to the Fourier transforms of the linear source distributions. Numerical results are presented to illustrate the forward projection of harmonic fields using the GISD method for both Dirichlet and Neumann problems. The numerical results for spherical and finite cylindrical vibrators illustrate the usefulness of the GISD method over a wide range of frequencies for both acoustic radiation and scattering problems.

A generalized internal source density method for the forward and backward projection of harmonic pressure fields from complex bodies

A generalized internal source density (GISD) method is presented to address the forward and backward projection of harmonic pressure fields from complex three-dimensional bodies. The GISD approach is based on decomposing the field on a closed surface of revolution in the fluid into a summation of circumferential orders where the pressure field for each order is associated with an internal linear distribution of ring sources on the axis of revolution of the surface. Both Dirichlet and Neumann boundary value problems may be addressed using the approach in which the axial variation of each ring source distribution is formulated as the solution of a minimum mean square error problem which is subsequently solved using SVD methods. The resulting linear distributions of ring sources can be simply used to determine the entire pressure and velocity fields of a complex harmonically vibrating body. In particular, the pressure and/or normal velocity on a vibrating body can be readily obtained from the pressure field over a closed surface. Local intensity and radiated power can thus be simply obtained. Far field pressures are simply obtained from the axial Fourier transforms of the ring source distributions. Several examples are presented to illustrate the behaviour of the source distributions for spherical geometries. The analytical results obtained via the GISD method agree with those obtained via classical methods.

The forward and backward projection of harmonic pressure fields from complex three-dimensional bodies

The forward and backward projection of pressure fields from complex three-dimensional harmonically vibrating bodies is an important problem in noise control. A new generalized internal source density (GISD) method is presented to address the projection problem. The GISD approach is based on decomposing the pressure field on a closed surface of revolution in the fluid into a summation of circumferential orders where the pressure field for each order is associated with an internal linear distribution of ring sources on the axis of revolution. The axial variation of each source distribution is formulated as the solution of a mean square error problem and the resulting distributions can then be simply used to determine the entire external and/or surface pressure and velocity fields of the vibrating body. Analytical and numerical results are presented for several examples to illustrate the basic approach for bodies with different aspect ratios. Exact expressions for the source strength distributions for a sphere are developed and shown to have a vanishingly small region of support about the center of the sphere. For cylindrically shaped bodies with large aspect ratios, the spatial characteristics of the source strength distributions more closely match the normal velocity distribution.

Sound radiation from a cylindrical duct. Part 2. Source modelling, nil-shielding directions, and the open-to-ducted transfer function

This paper analyses the sound radiated from the front face of a hard-walled circular cylindrical duct in a subsonic mean flow when the duct contains acoustic sources typical of those in a ducted-fan aeroengine. Two main results are established for modes of any given frequency and circumferential order. The first result is that in certain easily calculated directions, called here the nil-shielding directions, the sound radiated by ducted sources is the same as the sound radiated by the corresponding open sources, i.e. by unducted sources of the same distribution and strength radiating into free space. Thus in these special directions the duct has no noise-shielding effect. The second result is that, in the Kirchhoff approximation, the sound radiated by the open sources in the nil-shielding directions determines the sound radiated by the ducted sources in all directions; i.e. the sound fields radiated by open and ducted sources are related by an open-to-ducted transfer function. This function is such that the sound radiated by the ducted sources is a linear combination of certain diffraction functions, in which the coefficients are given by the sound radiated by the open sources in the nil-shielding directions. The diffraction functions do not depend on the sources and are here calculated explicitly in terms of Bessel functions. The method used in the paper is Kirchhoff's approximation; within linear theory this gives the nil-shielding directions exactly, i.e. in agreement with the Wiener—Hopf solution, and gives the main beam of the radiated field, including the major side-lobes, to good accuracy. The results are relevant to the sound radiated into the forward arc by a ducted turbofan aeroengine.

Sound radiation from finite cylindrical shells, partially covered with longitudinal strips of compliant layer

A vibro-acoustic model of a finite cylindrical shell partially covered with a compliant material layer is presented. The finite shell is terminated by a cylindrical rigid baffle and is covered by strips of compliant material lying parallel to the shell axis. The support shell motion is obtained by using Flügge’s operator and the layer is taken to be a locally reacting material having dissipative properties but without thickness. The whole system is excited by mean of a harmonic driving force applied to the supporting shell. In the infinite exterior medium, the acoustic pressure satisfies the Helmholtz equation. The acoustic pressure and the shell displacement are expanded in the eigenfunctions of the finite, non-covered, in vacuo,cylindrical shell. This leads to the construction of linear systems whose unknowns are the modal amplitudes of the pressure and of the shell displacement. If the cross-section of the partially covered shell has a plane of symmetry, the linear system splits into two independent, symmetric and antisymmetric parts. Nevertheless, each system displays coupling of circumferential order, meaning that the problem is no longer axisymmetric, as the totally covered problem was. Numerical results concerning acoustic radiation in water (radiated power) and shell vibration (radial quadratic velocity of the shell) are presented. They exhibit a surprising influence of the partial covering of a shell by a compliant layer: namely, that partially covering a shell in water can considerably increase the radiation compared to that from the bare cylinder, in a frequency range located well below the coincidence frequency.

Utilization of guided elastic waves for the characterization of circumferential cracks in hollow cylinders

The scattering of guided elastic waves from a circumferential crack in an isotropic hollow cylinder is investigated analytically using the S-parameter formalism. A general formula, analogous to that of [B. A. Auld, Acoustic Fields and Waves in Solids (Krieger, Malabar, FL, 1990), 2nd ed., Vol. II, p. 200], is derived for the scattered amplitude of an arbitrary mode due to the incidence of an arbitrary mode. The amplitudes are expressed as an integral of the inner product of the crack opening displacement caused by the incident mode and the traction field of the scattered mode. Using a simple approximate form of known elastostatic solutions for the crack opening displacements due to pure tension, a formula is derived for the scattering of guided wave modes of arbitrary circumferential order due to the incidence of axisymmetric wave modes with large wavelength and uniform stress distribution. These results are analyzed and conclusions drawn on the possibility of crack sizing using relatively simple scattering amplitude information.

A numerically stable global matrix method for cylindrically layered shells excited by ring forces

This paper defines a numerically stable method for modeling cylindrical shells that can have multiple viscoelastic layers and external compliant coatings. The method is numerically stable over a wide range of axial wavenumbers and circumferential orders because it uses a ‘‘global matrix’’ approach with appropriately defined coefficients. The excitations used here are time-harmonic ring forces that can push on the shell in the radial, circumferential, and axial directions. The ring forces can have a linear phase shift around the circumference of the shell, so helical waves can be excited and studied for any circumferential order. Analytically, the order can be complex, although the software is presently implemented for real orders. The method is verified by modeling a thin cylindrical shell surrounded by fluid. To provide a very demanding test of the numerical stability, the drive point response of the shell is computed. This computation demonstrates the numerical stability of the method over a wide range of axial wavenumbers and circumferential orders.

Improved methods for prediction of electromagnetic noise radiated by electrical machines

Improved methods for predicting the electromagnetic noise radiated by electrical machines are presented. The preferred method is based on the calculation of the sound-pressure and stator-vibration velocity distributions on a finite-cylindrical acoustic model. It results in a simple formula for calculating the acoustic power from the relative sound intensity and the stator-surface vibration, and is therefore eminently suitable for use during routine design. The method eliminates the need to calculate the sound-pressure and air-velocity distributions around a machine, as is required for existing methods. It can also account for the effect of an axial variation of the radial vibration. In an alternative method the relative sound intensity is calculated from a spherical acoustic model in which the radial vibration of a machine is expressed as a function of the circumferential mode order in the form of a Fourier series rather than a Legendre series, as in previous analyses. This makes it possible to use Fourier series throughout the various stages leading up to the prediction of electromagnetic noise. Predictions of the relative sound intensity from the different models are compared, and the sound-pressure distribution and the acoustic power are investigated both theoretically and experimentally on a TEFC 3-phase induction motor. It is shown that methods based on a spherical acoustic model are only geometrical and physical approximations, whilst a finite-cylindrical acoustic model is more generally applicable.

Active control of coupled wave propagation in fluid-filled elastic cylindrical shells

A control approach to reduce the total power propagating along fluid-filled elastic cylinders is analytically investigated. The motion of the cylinder is described by the Kennard shell equations fully coupled to the interior acoustic field. The vibration disturbance source is a pre-determined free propagating wave of either n=0 or n=1 circumferential order and the control forces considered are appropriate harmonic line forces radially applied to the structure. The radial displacement of the shell wall at discrete locations downstream of control forces is minimized using feedforward quadratic optimal theory. The difference of total power flow through the system before and after control is then used to evaluate the impact of the fluid on the performance of the control approach. For the breathing circumferential mode (n=0), owing to the coupling between the two media, the fluid decreases the control performance when the disturbance is a structural-type incident wave. When the disturbance is a fluid-type incident wave, with a pressure near field concentrated at the shell wall, significant reductions of the transmitted power flow can be achieved. For the beam mode (n=1), even though the control is applied to the structure, the fluid increases the control performances below the first acoustic cut-off frequency and decreases it above this frequency.