Generalized asymptotic expansions for the wavenumbers in infinite flexible in vacuo orthotropic cylindrical shells

Analytical expressions are found for the wavenumbers and resonance frequencies in flexible, orthotropic shells using the asymptotic methods. These expressions are valid for arbitrary circumferential orders n. The Donnell-Mushtari shell theory is used to model the dynamics of the cylindrical shell. Initially, an in vacuo cylindrical isotropic shell is considered and expressions for all the wavenumbers (bending, near-field bending, longitudinal and torsional) are found. Subsequently, defining a suitable orthotropy parameter ?, the problem of wave propagation in an orthotropic shell is posed as a perturbation on the corresponding problem for an isotropic shell. Asymptotic expressions for the wavenumbers in the in vacuo orthotropic shell are then obtained by treating ? as an expansion parameter. In both cases (isotropy and orthotropy), a frequency-scaling parameter (η) and Poisson's ratio (ν) are used to find elegant expansions in the different frequency regimes. The asymptotic expansions are compared with numerical solutions in each of the cases and the match is found to be good. The main contribution of this work lies in the extension of the existing literature by developing closed-form expressions for wavenumbers with arbitrary circumferential orders n in the case of both, isotropic and orthotropic shells. Finally, we present natural frequency expressions in finite shells (isotropic and orthotropic) for the axisymmetric mode and compare them with numerical and ANSYS results. Here also, the comparison is found to be good.     

Existence of axisymmetric modes of thin cylindrical shells with axial displacement restraint

A necessary and sufficient condition is derived for the existence of the axisymmetric mode of cylindrical shells with a radial displacement having one half wave along the axis and axial displacement restrained at both ends. The condition is purely geometric and consists of an upper bound on the mean radius to thickness ratio for a given fixed value of the length to mean radius ratio, above which the mode with one half wave in radial displacement along the axis ceases to exist. The proof is based on enforcing two basic lemmas concerned with the simplicity of the eigenvalues of the shell and the uniform ordering of these eigenvalues with the number of nodes of their corresponding radial displacement eigenfunctions.     

扬声器辐射体旋转薄壳的非线性振动方程

推导了扬声器辐射体旋转薄壳的离散非线性振动方程。从虚功原理出发,选用扬声器辐射体旋转薄壳的本征模态对连续体进行离散。薄壳的几何非线性采用Sanders非线性薄壳理论的应变一位移关系。方程系数由有限元方法确定。方程表明轴对称模态由驱动力直接激励,非轴对称模态由轴对称模态非线性耦合激励,该耦合激励表现为参数激励。方程揭示了扬声器非线性失真的机制,可用于分析扬声器辐射体薄壳非线性引起的谐波失真、分谐波失真和互调失真。      

Damping properties of layered cylindrical shells,vibrating in axially symmetric modes

The damping of cylindrical shells coated with unconstrained layers of viscoelastic material either on one side of the shell (inside or outside) or on both sides is estimated. The basic equations of motion are derived which describe harmonic forced flexural damped vibrations in axisymmetric modes. For pure sinusoidal modes expressions for the overall loss factors are given. The damping properties of cylindrical shells of finite length, coated on the inside or outside, or on both sides (symmetrically or unsymmetrically) are compared. Classical thin shell theory is used for the analysis. It is shown how two-layered damped shells differ from two-layered damped beams. The extent of damping reduction in shells resulting from the fact that the shell cross-section is closed is discussed.     

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC SHELLS. PART II: A NEW APPROACH FOR FREE TRANSVERSE CONSTRAINED VIBRATION OF CYLINDRICAL SHELLS

The non-linear dynamic behaviour of infinitely long circular cylindrical shells in the case of plane strains is examined and results are compared with previous studies. A theoretical model based on Hamilton's principle and spectral analysis previously developed for non-linear vibration of thin straight structures (beams and plates) is extended here to shell-type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. In the present work, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions corresponding to the constrained vibrations, which exclude the axisymmetric displacements. The non-linear strain energy is expressed by taking into account the non-linear terms due to the considerable stretching of the shell middle surface induced by large deflections. It has been shown that the model presented here gives new results for infinitely long circular cylindrical shells and can lead to a good approximation for determining the fundamental longitudinal mode shape and the associated higher circumferencial mode shapes (n>3) of simply supported circular cylindrical shells of finite length. The non-linear results at small vibration amplitudes are compared with linear experimental and theoretical results obtained by several authors for simply supported shells. Numerical results (non-linear frequencies, vibration amplitudes and basic function contributions) of infinite shells associated to the first four mode shapes of free vibrations, are obtained, using a multi-mode approach and are summarized in tables. Good agreement is found with results from previous studies for both small and large amplitudes of vibration. The non-linear mode shapes are plotted and discussed for different thickness to radius ratios. The distributions of the bending stresses associated with the mode shapes are given and compared with those obtained via the linear theory.     

VIBRATION ANALYSIS OF TWISTED CANTILEVERED CONICAL COMPOSITE SHELLS

The finite element method based on the Hellinger-Reissner principle with independent strain is applied to the vibration problem of cantilevered twisted plates and cylindrical, conical laminated shells. With a small number of elements, the present assumed strain finite element method is validated by convergence tests and numerical tests, comparing with the previous published vibration results for cantilevered conical shell. Computational effort and virtual storage reduce significantly due to good convergence. This study presents the twisting angle effect on vibration characteristics of conical laminated shells. Parameter studies with varying shallowness of cylindrical and conical shells are carried out. As the curvature increases, the fundamental mode shape changes from twisting mode to bending mode. For shells with a large curvature, the fundamental frequency, which is always characterized to bending mode, is almost constant independent of twisting angle. The twisting angle affects greatly twisting frequency and mode shape.     

Influence of initial geometrical imperfections on vibrations of axially compressed stiffened cylindrical shells

The vibrations of stiffened cylindrical shells having axisymmetric or asymmetric initial geometrical imperfections and axial preload are analyzed. The analysis is based on a solution of the von Kárman-Donnell non-linear shell equations, an “exact” solution of the compatibility equation, and a first order approximation by the Galerkin method of the equilibrium equation. The stiffeners are closely spaced and “smeared” stiffener theory is employed. The results of an extensive parametric study carried out on shells similar to those used in vibration and buckling tests at the Technion show that stiffening of the shell will lower the imperfection-sensitivity of its free vibrations, but the decrease depends on the type of stiffening (stringers or rings), the mode shapes of the vibration and the imperfection, the stiffener strength and eccentricity. The imperfection-sensitivity decrease, caused by the stiffeners, is greater for vibration mode shapes with high imperfection-sensitivity than for other vibration mode shapes. The sensitivity differences between stringer and ring-stiffened shells depend especially on the vibration and the imperfection mode shapes, and on their coupling. Small imperfections change the natural frequencies of stiffened shells in the same directions as for isotropic shells, but to a smaller extent. The frequency dependence on the external load is also strongly affected by the imperfection mode shape. The results correlate well with earlier ones for isotropic shells.     

Vibrations of segmented cylindrical shells by a fourier series component mode method

The vibrations of a multi-segment cylindrical shell with a common mean radius are studied. The shell is uniform for any segment but the material and geometric properties may vary from segment to segment. The solution is based on the component mode method coupled with Fourier series and Lagrange multipliers. It is shown that a single segment shell with boundary conditions of free support without tangential constraint is sufficient for an arbitrary shell with arbitrary boundary conditions. Results are presented for simply supported shells and clamped-free shells for two segments with different length and thickness.     

Method of Green’s function of nonlinear vibration of corrugated shallow shells

Based on the dynamic equations of nonlinear large deflection of axisymmetric shallow shells of revolution, the nonlinear free vibration and forced vibration of a corrugated shallow shell under concentrated load acting at the center have been investigated. The nonlinear partial differential equations of shallow shell were reduced to the nonlinear integral-differential equations by using the method of Green’s function. To solve the integral-differential equations, the expansion method was used to obtain Green’s function. Then the integral-differential equations were reduced to the form with a degenerate core by expanding Green’s function as a series of characteristic function. Therefore, the integral-differential equations became nonlinear ordinary differential equations with regard to time. The amplitude-frequency relation, with respect to the natural frequency of the lowest order and the amplitude-frequency response under harmonic force, were obtained by considering single mode vibration. As a numerical example, nonlinear free and forced vibration phenomena of shallow spherical shells with sinusoidal corrugation were studied. The obtained solutions are available for reference to the design of corrugated shells.     

Predictions and measurements of sound transmission through a periodic array of elastic shells in air

Analytical and numerical approaches have been made to the problems of (a) propagation through a doubly periodic array of elastic shells in air, (b) scattering by a single elastic shell in air, and (c) scattering by a finite periodic array of elastic shells in air. Using the Rayleigh identity and the Kirchhoff-Love approximations, a relationship is found between the elastic material parameters and the size of the bandgap below the first Bragg frequency, which results from the axisymmetric resonance of the shells in an array. Predictions and laboratory data confirm that use of a suitably "soft" non-vulcanized rubber results in substantial insertion loss peaks related to the resonances of the shells. Inclusion of viscoelasticity is found to improve the correspondence between predictions and data. In addition the possible influences of inhomogeneity due to the manufacturing of the elastic shells (i.e., the effects of gluing sheet edges together) and of departures from circular cylindrical cross-sections are considered by means of numerical methods.     

扬声器辐射体旋转薄壳的谐波失真

研究了扬声器辐射体旋转薄壳几何非线性引起的谐波失真。采用摄动法和有限元法确定了2次和3次谐波振型,计算了薄壳材料参数和几何参数对谐波失真的影响。结果表明扬声器薄壳谐波失真的机制是分割振动模态的基波共振和超谐波共振,以基波共振为主。采用高阻尼、大杨氏模量和低密度的振膜材料可以降低振膜谐波失真;厚度对谐波失真的峰值影响不大;锥壳半顶角过大,可使3次谐波明显增大。      

Resonant acoustic scattering by a finite linear grating of elastic shells

Theoretical and experimental studies of the acoustic scattering by a finite linear grating of elastic cylindrical shells are performed. It is observed that a resonant interaction takes place at low frequency when the shells are very close to each other. This phenomenon can be clearly associated to the Scholte-Stoneley wave that propagates around a single shell. It is shown that each resonance of the Scholte-Stoneley wave is split up into N resonances when N shells compose the grating.     

无内共振时的扬声器分谐波

研究了驱动频率低于扬声器辐射体薄壳轴对称模态最低固有频率的扬声器1/2分谐波失真。采用以位敏探测器作为光电传感器的激光三角法实验观测扬声器振膜的振动位移和模态,确定了参与的非轴对称模态的周向波数。采用多尺度法求解了扬声器的非线性模态方程。给出了扬声器分谐波的阈值电压公式。结果表明扬声器辐射体薄壳的分谐波失真源于直接激励的轴对称模态耦合激发了非轴对称模态的振动,这种耦合激励表现为参数激励。增大扬声器振膜材料的损耗因子、杨氏模量和厚度可提高产生分谐波的阈值电压。      

考虑铺层角的复合材料圆柱壳自由振动三维弹性准确解

为了获得任意角度铺层的多层复合材料圆柱壳的自由振动准确解,在三维弹性理论的基础上,结合分层理论和状态空间法,建立横向位移和应力的传递矩阵,轴向和环向位移采用双螺旋模式的位移函数,对任意角度铺层复合材料圆柱壳简支边界条件下的自由振动进行了理论推导,得到了自由振动方程的精确形式。与文献理论解和有限元计算结果对比,结果表明,关注频率在2倍的环频率以下时,薄壳的固有频率计算精度能控制在1%以内,厚壳的固有频率计算精度能控制在2%以内。对于厚壳的计算可将壳体沿厚度方向划分为多层来处理,这样能有效提高计算精度。计算分析了铺层角对壳体固有频率的影响,环向模态数较低时,固有频率随着铺层角的增加呈抛物线变化趋势;环向模态数较高时,固有频率随着铺层角的增大单调递增。该理论方法同样适用于均质各向同性壳和正交各向异性圆柱壳。      

Rotationally symmetric vibrations of orthotropic layered cylindrical shells

The derivation of the general equations of motion for the analysis of laminated cylindrical shells consisting of layers of orthotropic laminae, and the equations of motion for rotationally symmetric deformation made previously by the authors are used in this study. The three coupled differential equations governing the rotationally symmetric motion of each layer of a cylindrical shell with rotary inertia neglected are replaced by another set of three differential equations where the solutions can be obtained systematically. General solutions for laminated cylindrical shells of finite length are presented. Coupled frequencies and several mode shapes for a fixed-end cylindrical shell with one and two orthotropic layers of various geometric dimensions are calculated for illustrative purposes. The results based on the present analysis for a single layered shell are compared to the results obtained according to the classical analysis.     

Solitary waves in an inhomogeneous cylindrical shell interacting with an elastic medium

A new generalized sixth-order nonintegrable equation is derived to model axisymmetric longitudinal wave propagation in an inhomogeneous cylindrical shell interacting with a nonlinear elastic medium. Exact soliton-like solutions to this equation are constructed with allowance for geometric and physical nonlinearities, both individually and in combination.     

Meridional ray backscattering enhancements for empty truncated tilted cylindrical shells: measurements,ray model,and effects of a mode threshold

Narrow-band backscattering experiments are used to characterize a meridional ray enhancement on a tilted, finite empty cylindrical shell having a blunt truncation. The meridional ray of the lowest order flexural leaky Lamb wave is examined, which has previously been shown to lead to large backscattering enhancements for excitation frequencies near and above the shell's coincidence frequency. The measurements are used to validate a convolution formulation ray theory describing the far-field backscattered amplitudes. Comparisons are also made with an approximate partial wave series solution for the finite cylindrical shell. The amplitude of the meridional ray enhancement is dependent on the nature of the reflection of the leaky wave from the shell truncation. While the peak measured amplitude agrees with predictions at low frequencies, experiments indicate the enhancement is degraded at high frequencies and exhibits an abrupt drop near the frequency of the mode threshold (cutoff) for the next-highest flexural mode. The nature of the leaky wave end reflection is examined using an approximate calculation of the energy reflection coefficient for leaky waves on a semi-infinite free plate. Results suggest the observed degradation is the result of mode conversion effects.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

Wave interpretation of numerical results for the vibration in thin conical shells

The dynamic behaviour of thin conical shells can be analysed using a number of numerical methods. Although the overall vibration response of shells has been thoroughly studied using such methods, their physical insight is limited. The purpose of this paper is to interpret some of these numerical results in terms of waves, using the wave finite element, WFE, method. The forced response of a thin conical shell at different frequencies is first calculated using the dynamic stiffness matrix method. Then, a wave finite element analysis is used to calculate the wave properties of the shell, in terms of wave type and wavenumber, as a function of position along it. By decomposing the overall results from the dynamic stiffness matrix analysis, the responses of the shell can then be interpreted in terms of wave propagation. A simplified theoretical analysis of the waves in the thin conical shell is also presented in terms of the spatially-varying ring frequency, which provides a straightforward interpretation of the wave approach. The WFE method provides a way to study the types of wave that travel in thin conical shell structures and to decompose the response of the numerical models into the components due to each of these waves. In this way the insight provided by the wave approach allows us to analyse the significance of different waves in the overall response and study how they interact, in particular illustrating the conversion of one wave type into another along the length of the conical shell.