The receptance method applied to ring-stiffened cylindrical shells: Analysis of modal characteristics

The receptance method is applied to determine the natural frequencies and mode shapes of circular cylindrical shells stiffened by rings. The receptances of cylindrical shell and of a ring to forces in the radial and circumferential directions are derived in terms of the modal characteristics of each. A matrix equation of free vibration, which must be solved by an iterative technique, results by eliminating the angular variable. An iterative solution is practical, since the size of the matrices remains at two times the number of stiffening rings, regardless of the number of modes of the unstiffened cylinder and rings included in the solution. The validity of the method is demonstrated by comparing results for specific cases with the results obtained theoretically and experimentally by others. When various stiffener configurations are being considered for a given cylindrical shell, the modal characteristics of the shell without stiffeners may be calculated once and used repeatedly to calculate the frequencies of the stiffened shell configurations. The form of the results offers potential for simplifications which are presented in a companion paper.     

Vibration of simply supported cylindrical shells with longitudinal stiffeners

The vibration of simply supported cylindrical shells stiffened by discrete longitudinal stiffeners is investigated by using an energy method. Vlasov's thin walled beam theory is used for stringers. Shell theories based on Donnell's approximate theory and Flügge's more exact theory are used for the skin and numerical results indicate that Donnell's approximate theory gives excellent results for the stiffened shells. Sinusoidal wave form is considered in the longitudinal direction, and mode shapes in the circumferential direction are represented by Fourier series. Numerical results on frequencies and mode shapes computed for a shell stiffened by various number of stiffeners are presented and compared favorably with existing experimental results and other analytical solutions.     

ON THE FREE VIBRATION OF STIFFENED SHALLOW SHELLS

A finite element analysis for free vibration behaviour of doubly curved stiffened shallow shells is presented. The stiffened shell element is obtained by the appropriate combinations of the eight-/nine-node doubly curved isoparametric thin shallow shell element with the three-node curved isoparametric beam element. The shell types examined are the elliptic and hyperbolic paraboloids, the hypar and the conoidal shells. The accuracy of the formulation is established by comparing some of the authors' results of specific problems with those available in the literature. Numerical results of additional stiffened shells are also presented to study the effects of various parameters of shells and stiffeners such as orientation (i.e., along x -/y -/both x and y directions), type (concentric, eccentric at top and eccentric at bottom) and number of stiffeners, stiffener depth to shell thickness ratio, and aspect ratio, shallowness and boundary conditions of shells on free vibration characteristics.     

TRANSIENT ANALYSIS OF RING-STIFFENED COMPOSITE CYLINDRICAL SHELLS WITH BOTH EDGES CLAMPED

A theoretical method is developed to investigate the effects of ring stiffeners on vibration characteristics and transient responses for the ring-stiffened composite cylindrical shells subjected to the step pulse loading. Love's thin shell theory combined with the discrete stiffener theory to consider the ring stiffening effect is adopted to formulate the theoretical model. The ring stiffeners are laminated with a composite material and have a uniform rectangular cross-section. The Rayleigh-Ritz procedure is applied to obtain the frequency equation. The modal analysis technique is used to develop the analytical solutions of the transient response. The analysis is based on an expansion of the loads, displacements in the double Fourier series that satisfy the boundary conditions. The effect of stiffener's eccentricity, number, size, and position on transient response of the shells is examined. The theoretical results are verified by comparison with FEM results.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.     

Smearing technique for vibration analysis of simply supported cross-stiffened and doubly curved thin rectangular shells

Plates stiffened with ribs can be modeled as equivalent homogeneous isotropic or orthotropic plates. Modeling such an equivalent smeared plate numerically, say, with the finite element method requires far less computer resources than modeling the complete stiffened plate. This may be important when a number of stiffened plates are combined in a complicated assembly composed of many plate panels. However, whereas the equivalent smeared plate technique is well established and recently improved for flat panels, there is no similar established technique for doubly curved stiffened shells. In this paper the improved smeared plate technique is combined with the equation of motion for a doubly curved thin rectangular shell, and a solution is offered for using the smearing technique for stiffened shell structures. The developed prediction technique is validated by comparing natural frequencies and mode shapes as well as forced responses from simulations based on the smeared theory with results from experiments with a doubly curved cross-stiffened shell. Moreover, natural frequencies of cross-stiffened panels determined by finite element simulations that include the exact cross-sectional geometries of panels with cross-stiffeners are compared with predictions based on the smeared theory for a range of different panel curvatures. Good agreement is found.     

Free vibration of non-circular cylindrical shells with variable circumferential profile

An analysis is presented of the free vibration of non-circular cylindrical shells with a variable circumferential profile expressed as an arbitrary function. The applicability of thin-shell theory is assumed and the governing equations of vibration of a non-circular cylindrical shell are written in a matrix differential equation by using the transfer matrix of the shell. Once the transfer matrix has been determined by numerical integration of the matrix equation, the natural frequencies and mode shapes of vibration are calculated numerically in terms of the matrix elements. The method is applied to cylindrical shells of three or four-lobed cross-section, and the effects of the length of the shell and the radius at the lobed corners on the vibration are studied.     

Large amplitude flexural vibration of stiffened plates

The large amplitude free flexural vibration of thin, elastic orthotropic stiffened plates is studied. The boundary conditions considered are either simply supported on all edges or clamped on all edges and the in-plane edge conditions are either immovable or movable. The governing dynamic equations are derived in terms of non-dimensional parameters describing the stiffening achieved, and the solutions are obtained on the basis of an assumed one-term vibration mode shape for various stiffener combinations. In all cases, the non-linearity is found to be of the hardening type (i.e., the period of non-linear vibration decreases with increasing amplitude). Some interesting conclusions are drawn as to the effect of the stiffening parameters on the non-linear behaviour. A simple method of predicting the postbuckling and static large deflection behaviour from the results obtained in this analysis is included.     

Radial vibrations of orthotropic laminated hollow spheres

The three-dimensional elasticity problem of the radial vibrations of a composite hollow spherical shell laminated of spherically orthotropic layers is considered. After formulating the equations, the exact determinantal equation from which the frequencies of vibration can be extracted is developed. Some calculated results for combinations of isotropic and orthotropic materials indicate the sensitivity of the frequencies to the geometry and material make up of the shells.     

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.     

The flexural vibration of a line-stiffened plate with fluid loading,part I: Analysis

This paper considers the vibration of a symmetrical system consisting of an infinite fluid loaded plate bearing a finite number of parallel stiffeners. The system is driven at the central stiffener by a travelling wave line force. Formal solutions for the equations of motion are found in terms of cosine transforms. Manipulation of the equations allows the problem to be reduced to the solution of a set of linear algebraic equations in the vibration amplitudes at the stiffeners. The coefficients in these equations depend in a simple way upon the stiffener parameters, and upon particular values of the cosine transform of a function which depends only on the plate and fluid parameters, and the stiffener positions.     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method

This paper presents a mesh-free Galerkin method for the free vibration and stability analyses of stiffened plates via the first-order shear deformable theory (FSDT). The model of a stiffened plate is formed by (1) regarding the plate and the stiffener separately, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, and (3) superimposing the strain energy of plate and stiffener. Because there are no meshes used in this method, the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines. Several numerical examples are computed by this method to show its accuracy and convergence. The present results demonstrate good agreement with the existing solutions given by other researchers and the ANSYS. Influences of support size and order of the complete basis functions on the numerical accuracy are also investigated.     

Free vibration study of layered cylindrical shells by collocation with splines

The free vibration of circular cylindrical thin shells, made up of uniform layers of isotropic or specially orthotropic materials, is studied using point collocation method and employing spline function approximations. The equations of motion for the shell are derived by extending Love's first approximation theory. Assuming the solution in a separable form a system of coupled differential equations, in the longitudinal, circumferential and transverse displacement functions, is obtained. These functions are approximated by Bickley-type splines of suitable orders. The process of point collocation with suitable boundary conditions results in a generalized eigenvalue problem from which the values of a frequency parameter and the corresponding mode shapes of vibration, for specified values of the other parameters, are obtained. Two types of boundary conditions and four types of layers are considered. The effect of neglecting the coupling between the flexural and extensional displacements is analysed. The influences of the relative layer thickness, a length parameter and a total thickness parameter on the frequencies are studied. Both axisymmetric and asymmetric vibrations are investigated. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed.     

Free vibration of joined conical-cylindrical shells

An analysis is presented for the free vibration of joined conical-cylindrical shells. The governing equations of vibration of a conical shell, including a cylindrical shell as a special case, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices of the shells and the point matrix at the joint, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method has been applied to a joined truncated conical-cylindrical shell and an annular plate-cylindrical shell system, and the natural frequencies and the mode shapes of vibration calculated numerically. The results are presented.     

A GROUP THEORETIC APPROACH TO THE LINEAR FREE VIBRATION ANALYSIS OF SHELLS WITH DIHEDRAL SYMMETRY

This paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry. Examples of such shell structures are cylindrical shells, conical shells, shells with circumferential stiffeners, corrugated shells, spherical shells, etc. The group theoretic approach is used to exploit the inherent symmetry in the problem. For vibration analysis, the group theoretic results give the correct symmetry-adapted basis for the displacement field. The stiffness matrix K and the mass matrix M are identically block diagonalized in this basis. The generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization. The Simo element is used in the finite element formulation of the shell equilibrium equations. Numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented. The results are shown to be in very good agreement with those reported in the literature. The computational advantages and physical insights due to the group theoretic approach are also discussed.     

湍流激励下柔性层贴敷加筋板自噪声的特征机理

为了研究湍流激励下柔性层贴敷加筋板自噪声的特征机理,基于湍流边界层壁面脉动压力功率谱模型、周期加筋板弯曲运动方程和固体波动方程以及力平衡与位移连续边界条件,建立了湍流边界层壁面脉动压力激励下柔性层贴敷单向周期加筋板的振动及内噪声物理模型.研究发现:橡胶外贴时对湍流激励下壳板的减振降噪主要依靠橡胶的厚度模态振动.无筋时,...     

正交加筋板声透射的板梁理论模型研究

为了研究正交加筋板的声透射问题,基于经典薄板和梁振动理论,建立了正交加筋板声透射的板梁理论模型。首先通过分析加强筋的受迫弯曲和扭转运动,求得了平板和加强筋线接触之间的反力和反力矩,然后将其引入到平板振动控制方程中,得到了正交加筋板声振方程,最后采用空间谐波展开法求解该方程得到了传声损失的表达式;在此基础上,首先研究了无限大平板和单向加筋的隔声性能,通过与解析解及两种简化模型的计算结果作对比,验证了所建理论模型的有效性;并进一步研究了加筋形式对正交加筋板隔声性能的影响。结果表明:选择合适的加筋形式可以有效避开结构的隔声波谷。      

On sound transmission into a stiffened cylindrical shell with rings and stringers treated as discrete elements

In the context of the transmission of airborne noise into an aircaft fuselage, a mathematical model is presented for the transmission of an oblique plane sound wave into a finite cylindrical shell stiffened by stringers and ring frames. The rings and stringers are modeled as discrete structural elements. The numerical case studies was typical of a narrow-bodied jet transport fuselage. The numerical results show that the ring-frequency dip in the transmission loss curve that is present for a monocoque shell is still present in the case of a stiffened shell. The ring frequency effect is a result of the cylindrical geometry of the shell. Below the ring frequency, stiffening does not appear to have any significant effect on transmission loss, but above the ring frequency, stiffeners can enhance the transmission loss of a cylindrical shell.