Ribbed Cylindrical Shell Research Articles

On Stability Analysis of Ribbed Cylindrical and Conical Shells under Axial Compressive Forces

An experimental study into the stability of ribbed cylindrical and conical shells is described. It is shown that when the equivalent rigidity introduced earlier is chosen as an argument in the formula for calculation of the critical stresses at the stage of shell design, this formula has the form of a linear function.

The formation of energy pass bands from periodically spaced point masses on a ribbed cylindrical shell

An infinite cylindrical shell has bending energy that is able to propagate down the shell at all frequencies with little attenuation other than from material damping and the surrounding medium. In contrast, making the shell finite causes bending resonances to occur only at discrete frequencies. The addition of periodically spaced circumferential stiffeners causes the bending wave dispersion curves to be seen at additional axial modes, which is referred to as Brillouin folding. The stiffeners also cause energy stop bands to occur when the load is a single Fourier mode. In this presentation, periodically spaced point masses in the circumferential direction are added to a ribbed finite cylindrical shell and analyzed via the finite element method. It is seen that Brillouin folding occurs circumferentially. Also, energy pass bands are created in a certain pattern. Analysis of the axisymmetric version of this shell combined with information on the type of aperiodicity yields a prediction of the energy pass band behavior. This provides a qualitative approach toward understanding the motion of nonaxisymmetric shell structures. [Work supported by ONR.]

A numerical technique for computation of flow-induced vibration and noise

This paper discusses the use of commercially available structural acoustics solvers for computation the flow-induced vibration and noise at low speed. The major issue of this problem is the modeling turbulent boundary layer (TBL) forcing function for finite-element models of structures. Predictions of structural vibrations using the TBL cross-spectral density function directly is not computationally tractable for large complex structures, because extremely fine finite-element meshes had to be used to resolve the TBL pressure fluctuating scales. This paper discusses a more practical approximated discrete forcing function model, which represents only the low-wave number region of the TBL excitation in terms of the cross-spectral function of grid forces. The high-wave number (convective ridge) content of the excitation, which is insignificant for large structures with fixed boundaries, is thus ignored. This method is more feasible in practice because the finite-element meshes now need only resolve the structural wave scale instead of the convective turbulence scale. Numerical examples on the flow-induced vibration of a clamped plate, and the flow-induced noise radiated from a ribbed cylindrical shell will be discussed. [Work supported by ONR, Code 333.]

Homogenization of a ribbed cylindrical shell over the full frequency range for efficient calculation of structural-acoustic behavior

A homogenization method for complex structures, applicable for all frequencies, is described for a cylindrical shell with periodically spaced ribs. This approach has application to naval and aerospace structures. The homogenization method utilizes a local–global decomposition facilitated by adding and subtracting canceling smooth forces. The smooth global problem has an infinite-order structural operator, and periodic discontinuities are replaced by equivalent distributed suspension terms. The global problem can be solved very efficiently since all rapidly varying scales have been removed. The local problem, which provides transfer function information for the global problem, is solved separately and independently, except for amplitude information from the global problem. Once formulated, the self-contained global problem is solved first, and the local solution can be reconstructed afterwards. In the present work, the ribs are modeled as annular plates periodically attached to the cylindrical shell. The coupled shell equations are re-cast using the method of local–global homogenization, and are solved efficiently using a combination of symbolic manipulation and numerical methods. Comparisons withother solution methods are presented, and the extension to fluid loading is discussed. [Work sponsored by ONR.]

On the use of the sonar equation approximation when computing the response of a target in shallow water

The sonar equation may be used to compute the scattering response of a target in shallow water by approximating the target to be a point scatterer. The approximation significantly simplifies the interaction of the scatterer with the medium. In the more general case, since the field produced by a source arrives at the scatterer through multipaths, it is incident on the target at various directions. The response of the scatterer must be computed at all of the incident angles to obtain the correct scattered field in shallow water. Scattering results computed using the sonar equation are compared to the more correct solution for a ribbed cylindrical shell placed in shallow water for various monostatic and bistatic geometries to examine the validity of the use of the sonar equation at various frequencies and geometries. [Work supported by ONR.]

Multiple scattering effects produced by a cylindrical shell in shallow water

Multiple scattering effects produced by a finite ribbed cylindrical shell placed in a shallow-water environment are evaluated for the monostatic geometry. A combination of finite elements/infinite elements methods and the method of superposition is applied to compute first-order multiple scattering effects produced by a cylinder placed close to the shallow-water surface. Results are shown for a cylinder having a length to a diameter ratio of 6.8 at a ka=10, where k is the wave number in the fluid at the location of the cylinder and a is the radius of the cylinder. [Work supported by ONR.]

Stability of ribbed cylindrical shells with a nonideal shape

A finite-difference method was proposed and used for analysis of critical loads of shells that have initial imperfections of different types, both regular and local. Ribbed shells were analyzed with allowance for the discreteness in the arrangement of the ribs.

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

The radiation associated with complexity is examined by use of finite element–infinite element models of framed fluid loaded shells for frequencies up to ka=10. The vibro-acoustic response of the shell with no internal complexity is compared to an identical structure with point masses attached to the frames. The simple ribbed shell is seen to have pass and stop band structure with radiation physics that is dominated by supersonic components. The addition of point masses to the structure gives rise to major changes in the response that include a coupling of radiating and otherwise nonradiating circumferential orders, resulting in a significant increase in radiation (about 10 dB) over a broad frequency range. These results are in general agreement with nearfield acoustical holography (NAH) experimental results. [Work supported by ONR.]

Analysis of static problems for smooth and ribbed cylindrical shells under complex mechanical loads

Static problems for thin isotropic smooth and locally reinforced cylindrical shells subjected to local loads with complex boundary conditions are examined using the linear theory. The stress-strain state of a casing is determined by the technical theory of shells, while the stress-strain state of its ribs is determined by the Kirchhoff-Klebsch theory of rods. Stiffening elements are applied eccentrically and may have the same or different stiffnesses. Stiffness may also change along a given element Regular and irregular arrangements of stiffening elements are used. Shells of minimum weight are also examined. The problems are solved by the finite differences method. The reliability of the results is demonstrated by comparison with experimental data and numerical results obtained with a denser grid. The goal of these investigations is to determine qualitative and quantitative characteristics of the stress and strain distributions, as well as to establish corresponding distribution laws that could be used to predict solutions for other shells based on assigned parameters without the need for calculations.

Influence of oscillation form on the amplitude-frequency relationship for ribbed cylindrical shells

Influence of oscillation form on the amplitude-frequency relationship for ribbed cylindrical shells

The effect of rib resonances on the vibration and wave scattering of a ribbed cylindrical shell

Wave phenomena have previously been observed for cylindrical shells with periodically spaced ribs. The understanding of these waves, such as Bragg and Bloch waves, requires knowledge of the periodicity of the ribs and how they interact with each other. In this paper, the understanding of the individual rib resonances gives explanation to the resonance bands that are seen in both experiment and numerical simulations. For these resonances, periodicity becomes less important in computing the frequencies of resonances on the shell and in the far field. However, lack of periodicity in rib spacing and size affects the width and strength of these resonance bands. [Work supported by ONR.]

Imaging of supersonic surface fields on a cylindrical shell

Conventional imaging techniques represent a radiating body as a distribution of equivalent monopole sources. This paper presents a more rigorous approach for imaging the supersonic components of both surface pressure and normal velocity produced by scattering or radiation from a nominally cylindrical body. The technique employs the Helmholtz integral equation to relate monopole and dipole strengths. A modal expansion of the surface pressure on a baffled cylinder forms the basis for determining an array-weighting function used to estimate the surface fields. The technique makes use of acoustic pressure data measured in a horizontal plane at ranges where the evanescent field is insignificant. No far-field constraints are imposed in the derivation of the method, allowing the measurements to be conducted within the geometric near-field of the body. When the array is located in the far field, this method is shown to yield similar array-weighting functions to a method derived by Williams [E. G. Williams, J. Acoust. Soc. Am. 99, 2022 (1996)]. Mid-frequency, 2≤ka≤20, scattering and radiation data from measurements of a ribbed cylindrical shell are used to illustrate the technique. The intensity images reveal membrane wave and rib scattering processes in both the time and frequency domains. [Work supported by NSWC and ONR.]

The extraction of a target scattering response from measurements made in a range-dependent shallow-water environment

The free-field scattering response of a target may be extracted from measurements made over long ranges in a range-independent shallow-water environment by the use of vertical array of sources and vertical array of receivers. The procedure requires the generation of a low-order shallow-water mode by the source array and the extraction from the received echo at the receiver array of a low-order shallow-water mode. The algorithm is generalized to range-dependent environments where additional complications exist due to the mode coupling produced by the range dependence in the shallow-water environment, which exists in addition to the mode coupling produced by the scatterer. Results of numerical simulations are presented for a ribbed cylindrical shell with hemispherical end caps. The target scattering response is computed using the finite elements and infinite elements method, and the propagation is computed using the parabolic equations method. [Work supported by ONR.]

Rib resonances present in the scattering response of a ribbed cylindrical shell

The presence of ribs in an evacuated, finite cylindrical shell, placed in a fluid medium, produces highlights in its scattering response at certain resonant frequencies. These resonances are identified from the end-incident bistatic response of a ribbed cylindrical shell and are shown to be produced by the flexural vibrations of individual ribs satisfying the boundary conditions of an annular plate.

Vibrations of a ribbed cylindrical shell with elastically fastened faces

We study the free vibrations of a reinforced cylindrical shell of revolution with elastically fastened faces. The computational model presented makes it possible to take account of various structural characteristics, including reinforcing ribs and attached rigid bodies. The solution is constructed using the linear theory of thin elastic shells and rods and the Ritz method. Three figures.

The structural acoustics and active control of interior noise in a ribbed cylindrical shell

Numerical studies are carried out regarding the structural acoustics of a ribbed aluminum cylindrical shell structure intended to represent the essential structural features of a small aircraft fuselage. Calculations are made to determine both the wall normal displacements and the interior acoustic pressures for the case in which the shell wall is forced dynamically at a point. The structural responses are further decomposed into their frequency-wave-number components. Through a series of comparisons between those responses for an infinite shell, the ribbed vacuum-filled shell, and the ribbed air-filled shell, the relevant structural acoustic mechanisms are interpreted. The frequencies at which interior acoustic ‘‘resonances’’ are observed are connected to specific mechanisms, including cavity responses driven by structural modes, cavity modes forcing the structure, and mixed structure/air-cavity modes. Numerically based active control experiments are carried out using end actuation, and the relative performance of this control ‘‘system’’ is compared when operating on the various mode types.

Internal–external measurement of three‐dimensional velocities of a submerged shell by optical vibrometry.

A unique experimental capability for measuring both the interior and exterior three‐axis surface motions of a completely submerged ribbed cylindrical shell has been developed. This system is composed of two three‐dimensional laser Doppler vibrometers positioned by a pair of high‐precision robots. The two vibrometers work together to measure the three‐components of velocity up to 50 kHz on both the interior and exterior cylinder surfaces. For the interior, a compact three‐dimensional vibrometer was designed and constructed to operate in conjunction with a cylindrical‐based scanner mounted to the stiff endcaps of the shell. This scanning system is capable of positioning the laser vibrometer with a precision of +/− 3 mil over 100% of the interior cylinder surface. A second three‐dimensional laser vibrometer designed for exterior (in‐water) operation was constructed and integrated into the NRL near‐field acoustic holography system. This large workspace Cartesian‐based scanner is capable of collocating the foc...

On the scattering of sound from a submerged thin walled cylinder coated with a compliant layer

There is currently an abundance of both mathematical and finite-element models of the scattering of underwater sound from cylindrical bodies treated with compliant coatings. However, there is a significant lack of corresponding scattering measurements. This paper presents the results of a set of simple measurements of the monostatic backscatter from a coated, thin, ribbed cylindrical shell with flat endcaps for ka of 2–8. The coating layer had an input impedance approximately one-fifth that of water, providing a near pressure-release boundary condition. Of particular interest is a minimum in the measured target strength which results from the resonant interaction of the coating and the shell. Results from finite-element and modal expansion models of the coated shell are also presented with the aim of providing a physical understanding of the resonance. [Work supported by ONR.]

Vibrational response of a simple oscillator attached to an elastic structure

The effect of many internal degrees of freedom on the vibration of elastic structures has been of great recent interest; several theories have predicted damping effects arising from associated internal resonances. As an initial effort to explore this phenomena experimentally, the behavior of a single such oscillator attached to ribbed cylindrical shell section has been measured. The single oscillator has been found to give rise to a fairly broadband resonance structure, with a bandwidth that paradoxically increases with decreasing frequency. These results, along with the physical interpretation and modeling, will be discussed.

Extraction of target characteristics from the response of a ribbed cylindrical shell in a shallow water environment

Bistatic and monostatic target strength values of a ribbed cylindrical shell contain various scattering phenomena such as Bloch waves, rib resonances, specular reflections and other effects due to waves traveling on the surface of the shell. When the target is placed in a shallow water environment, the response is further cluttered by the presence of the various multipaths that a signal travels in a bounded medium. By using a vertical line array of sources and receivers, the target response may be extracted. A vertical array of sources is used to suppress certain shallow water modes in the incident field. A receiver array is used to remove certain shallow water modes from the scattered field. Time-frequency analysis is a final filter for selecting the modes of interest for which the target response is extracted. The extracted target response is correlated against the free field target characteristics previously determined by time-frequency synthesis techniques.