Vibrations Of Spherical Shells Research Articles

A unified Jacobi-Ritz-spectral BEM for vibro-acoustic behavior of spherical shell

In order to solve the vibration and acoustic characteristics of spherical shell in light fluid environment, based on Jacobi-Ritz-spectral BEM, a unified analysis formula for acoustic vibration of spherical shell under arbitrary boundary conditions is established. Based on the First-order shear deformation theory (FSDT) and domain decomposition method (DDM), the theoretical model of spherical shell structure is established. The improved Jacobi polynomial is innovatively used to construct the displacement tolerance function of the spherical shell. Based on the spectral Kirchhoff-Helmholtz integral formula, the theoretical model of the acoustic fluid outside the spherical shell is established. The special form of Jacobi polynomial Chebyshev polynomial is used to describe the excitation sound pressure on the spherical shell segment, so as to ensure that the generalized coordinates of the structure can be perfectly matched with the acoustic boundary element nodes. In addition, the integration along the meridian of the shell is used to control the movement of the fluid, which simplifies the surface integration. The CHIEF method is used to solve the problem of non-unique solution of acoustic variables of rotary structure. Compared with the published literature, numerical simulation results and experimental results, the proposed method has higher calculation accuracy. In addition, based on this method, the influence of boundary conditions, geometric dimensions and other factors on the acoustic and vibration characteristics of spherical shells is discussed, which accumulates data for analyzing the acoustic and vibration behavior of spherical shells.

Thermally Induced Vibrations of Functionally Graded Shallow Spherical Shells Under Cooling Shock

Based on the first-order shear deformation theory (FSDT), the large-amplitude vibration behavior of shallow spherical shells made of functionally graded materials (FGMs) due to the rapid decrease of temperature (cooling shock) is investigated. FGM is assumed to be a mixture of stainless steel and low-carbon steel whose properties are considered temperature dependent and are estimated based on the power-law model. According to FSDT and von Kármán assumptions, the governing equations of motion in conjunction with corresponding boundary conditions are obtained using Hamilton’s principle. The temperature distribution is also achieved by means of the 1D Fourier transient heat conduction equation, considering two time-dependent thermal loading scenarios. The considered thermal boundary conditions allow that a temperature difference is created with a time delay for the structure under cooling shock. Also, the results of recent experiments are employed to estimate the temperature-dependent properties of structures. In the solution approach, the generalized differential quadrature method and the Newmark-beta integration scheme are utilized. Selected numerical results are presented to study the effects of thermal load rapidity time, geometry, magnitude of thermal load, and material gradient index on the thermally induced vibrations of spherical shells. Stresses generated in the shell due to the thermally induced vibrations are also investigated.

Dynamic analysis of spherical shell partially filled with fluid

In present study, a hybrid finite element method is applied to investigate the free vibration of spherical shell filled with fluid. The structural model is based on a combination of thin shell theory and the classical finite element method. It is assumed that the fluid is incompressible and has no free-surface effect. Fluid is considered as a velocity potential variable at each node of the shell element where its motion is expressed in terms of nodal elastic displacement at the fluid–structure interface. Numerical simulation is done and vibration frequencies for different filling ratios are obtained and compared with existing experimental and theoretical results. The dynamic behavior for different shell geometries, filling ratios and boundary conditions with different radius to thickness ratios is summarized. This proposed hybrid finite element method can be used efficiently for analyzing the dynamic behavior of aerospace structures at less computational cost than other commercial FEM software.

Nonlocal elasticity theory for radial vibration of nanoscale spherical shells

This paper presents the radial vibration of nanoscale spherical shells based on the nonlocal elasticity theory . The shell is considered elastic, homogeneous and isotropic. The nonlocal differential equation of radial motion is derived in terms of radial displacement . The relation between the nonlocal and local frequencies is also investigated. Considering the small-scale effect, the general characteristic equation for radial vibration of spherical shell is obtained by applying boundary conditions. Moreover, the characteristic equations for two special cases are presented. To demonstrate the accuracy of the present formulation, theoretical calculations of the fundamental frequency have been compared with those available in the literature and a good agreement is achieved. The variations of the frequencies with the nonlocal parameter, radius ratio and Poisson's ratio are also examined. It is observed that the frequencies are affected when the size effect is taken into consideration. • Propose the nonlocal shell model for predicting the radial vibration of nanoscale spherical shells. • The influence of small scale on the frequencies of the nanoscale spherical shells is investigated. • The relation between the nonlocal and local frequencies is developed.

207 球形シェルの減衰評価における解析法の検討

An analysis for vibration of spherical shells with a constrained viscoelastic layer and metal layers is presented. In this paper, vibration damping properties of three layered spherical shells with viscoelastic material are analyzed from both of SEM (Strain energy method) and CEM (Complex eigenvalue method). Eigenvalue and eigenvector (displacement functions) are analyzed based on the Ritz method, when natural frequency, modal loss factor and strain energy are obtained. In CEM, the elastic modulus of viscoelastic material is dealt with complex quantity considering material loss factor. In SEM, modal loss factor is defined as a ratio of damping energy (strain energy consumed in viscoelastic layer) dissipated during a vibration cycle and total strain energy of the shell. The accuracy and validity of the present results from two methods are illustrated through investigation of convergence and comparison with the established results from the literature. Numerical results are presented for different parameters, such as the opening-angles, material loss factors and circumferential wave numbers.

Free vibration analysis of thick spherical shells

The axisymmetric and non-symmetric vibrations of spherical shells are analysed using the thick shell theory. The semi-analytical method is used to reduce the dimension of the problem. A thick shell finite element is developed for the analysis of general shells of revolution and used for this purpose. The element can be used for the analysis of thick and thin shells of revolution. Effects of transverse shear, normal strain and rotary inertia are included in the formulation. Both shallow and deep shells are considered for the analysis. Frequencies are obtained for spherical caps with and without centre cutout.

Some Results on Axisymmetric Vibration of Spherical Shells of Variable Thickness

Natural frequencies and mode shapes are presented for the free axisymmetric vibration of spherical shells with linearly varying thickness along the meridian. Clamped and hinged edges corresponding to opening angles 30, 45, 60 and 90 deg have been considered in this technical brief to cover a wide range from shallow to deep spherical shells. Variations in thickness are seen to have very pronounced effects on the frequencies and mode shapes.

Natural frequencies and modes of vibration of spherical shells with attached masses

Natural frequencies and modes of vibration of spherical shells with attached masses

An analysis of natural vibration of spherical shells.

An analysis of natural vibration of spherical shells.

Asymmetric Modes and Associated Eigenvalues for Spherical Shells

Free asymmetric vibration of spherical shells with clamped and hinged boundary conditions are analyzed using the finite element method. Element stiffness and consistent mass matrices are derived using the improved shell theory, which takes into account the effects of shear deformation and rotary inertia. Natural frequencies for a wide spectrum of shell geometry ranging from shallow cap to hemispherical shell have been computed and are found to be in close agreement with the available data in the literature.

Free vibrations of thin elastic spherical shells

This paper is devoted to the problem of free, harmonic vibrations of thin, elastic, spherical shells. The differential equations are rederived in an invariant form together with the appropriate kinematic and static boundary conditions.The complete solution is presented for axisymmetric and non-axisymmetric vibrational modes for a shell in the shape of a spherical zone with two boundaries. The hitherto unsolved problem of non-axisymmetric vibrations of a spherical shell with a circular opening is included as a special case.Numerical examples are given covering a variety of boundary conditions and a wide range of the geometrical parameters.

Comparison of human skull and spherical shell vibrations—implications for head injury modeling

Spherical shell models have been formulated as geometric approximations of the human head. By insuring geometrical and material similarity between the model and human head, impact response and skull fracture studies were expected to yield results in close agreement with model predictions. This, however, was not the case. Model predictions of skull fracture loads were typically twice the observed level. A comparison was made between the resonant frequencies of two dry human skulls and corresponding spherical shell models. Poor agreement was observed. A vibrational analysis of the model revealed that the uniformity of the spherical shell approximmately doubles the effective stiffness and resonant frequencies as compared with the dry human skull. The elastic modulus of the model was adjusted to bring its resonances into closer agreement with those of the skull. The synthesized“effective” modulus was 50% lower than the material property of human cranial bone. Interestingly, this adjustment also brought model predictions of skull fracture into closer agreement with available human cadaver data.

Analytical solutions for open nonshallow spherical shell vibrations

This paper is concerned with axisymmetric as well as nonsymmetric vibrations of open nonshallow thin elastic spherical shells. Without employing the usual auxiliary variables for reduction of shell motion equations introduced by Van der Neut and Berry, independent analytical solutions for middle‐surface displacements are obtained and explicitly expressed in terms of associated Legendre functions. In order to gain physical insights into the free‐vibration characteristics of an open nonshallow shell, theoretical calculations together with asymptotic descriptions are made of natural frequencies and mode shapes of a hemispherical shell with a free edge. The numerical predictions obtained herein compare excellently against the experimental results obtained previously by Hwang and recently in the David W. Taylor Naval Ship R & D Center, Bethesda, Maryland. Essential features of shell dynamics are ultimately displayed by normalized frequency (Ω) and nondimensional shell thickness parameter (β). Five families of natural frequencies, i.e., low Rayleigh bending, mixed bending‐membrane, torsional, bending, and membrane frequencies, are found. The corresponding mode shapes exhibit distinctive displacement patterns. [Work supported by Naval Sea Systems Command.]

Improved design of spherical multimode hydrophone

An improved design is made for a spherical multimode hydrophone by dividing each hemisphere into two zones and selecting the position of the line of division between them, so that the resonances associated with higher modes of the spherical shell vibrations, specifically n=3 and n=5 modes in the present case, are eliminated. The elimination of these resonances has led to the provision of a frequency-independent dipole pattern over a wide range of frequencies. The present theory includes the effect of spherical shell bending, since it is important at higher modes. Accordingly, the resonant frequencies are clearly identified by the associated mode numbers. In both theory and experiment, these resonant effects were eliminated by employing the newly designed hydrophone.

Investigation of the vibrations of spherical shells in a nonclassical formulation

Investigation of the vibrations of spherical shells in a nonclassical formulation

Some inverse problems in the theory of spherical shell vibrations

Some inverse problems in the theory of spherical shell vibrations

On logarithmic derivatives of the associated legendre functions of arbitrary complex degree: PMM vol. 38, n≗ 3, 1974, pp. 575–576

In solving certain problems of the theory of vibrations of spherical shells it is more convenient to calculate not the associated Legendre functions P m n ( cos θ) and their derivatives themselves but rather the logarithmic derivatives

Asymmetric vibrations of spherical shells

Asymmetric vibrations of spherical shells

Closed-Form Solution of Forced Vibrations of Spherical Shells with Attachments

An asymptotic high-frequency closed-form solution of the point-excited spherical shell [D. Feit et al., J. Appl. Mech. 36, 859 (1969)] is extended here to nonaxisymmetrically point-excited shells with point attachments characterized by an arbitrary impedance. The structure-borne sound field near the attachment displays the characteristics of the diffracted acoustic field near an infinite cylinder. A dashpot located diametrically opposite the drive point drastically reduces the shell response if its impedance is matched to the drive-point impedance of the shell.

Closed-Form Solution of Forced Vibrations of Spherical Shells with Attachments

The forced vibrations of a finite elastic system can be analyzed as a superposition of normal modes or as waves in an infinite waveguide with reflecting discontinuities. The latter leads to a closed-form display of the solution [e.g., for strings, P. M. Morse, Vibration and Sound (McGraw-Hill Book Company, New York, 1948), 2nd ed., pp. 94–95]. This technique is applicable to systems whose normal modes are not known [e.g., coupled beams, J. C. Snowdon, Vibration and Shock in Damped Mechanical Systems (John Wiley & Sons, Inc., New York, 1969), pp. 230–239]. This waveguide formulation is illustrated here for spherical shells coupled to arbitrarily located spring-mass systems. The response of this composite system is constructed by matching the oscillator impedance to the shell impedance formulated in terms of the closed-form solution of the homogeneous shell [D. Feit et al., J. Appl. Mech. 36, 859 (1969)]. Because conventional methods (modal expansions, computer methods) are laborious for structures measuring many flexural wavelengths, the waveguide solution is specialized to these conditions, i.e., to the high-frequency range. [Supported by Office of Naval Research, Structural Mechanics Program.]