Elastic response of an orthogonally reinforced plate

This paper develops a three-dimensional fully elastic analytical model of a solid plate that has two sets of embedded, equally spaced stiffeners that are orthogonal to each other. The dynamics of the solid plate are based on the Navier–Cauchy equations of motion of an elastic body. This equation is solved with unknown wave propagation coefficients at two locations, one solution for the volume above the stiffeners and the second solution for the volume below the stiffeners. The forces that the stiffeners exert on the solid body are derived using beam and bar equations of motion. Stress and continuity equations are then written at the boundaries and these include the stiffener forces acting on the solid. A two-dimensional orthognalization procedure is developed and this produces an infinite number of double indexed algebraic equations. These are all written together as a global system matrix. This matrix can be truncated and solved resulting in a solution to the wave propagation coefficients which allows the systems displacements to be determined. The model is verified by comparison to thin plate theory and finite element analysis. An example problem is formulated. Convergence of the series solution is discussed. The frequency limitations of the model are examined.     

Sound radiation from fluid loaded orthogonally stiffened plates

The radiation of sound from infinite fluid loaded plates is examined when the plates are reinforced with two sets of orthogonal line stiffeners. The stiffeners are assumed to be equally spaced and exert only forces on the plate. The response to a convected harmonic pressure is found by using Fourier transforms and is given in terms of the harmonic amplitudes of the stiffener forces. These forces satisfy an infinite set of simultaneous equations to which a numerical solution must be found. An expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined with particular reference to point force excitation.     

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.     

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.     

板状材料上加强筋的几何形状对兰姆波透射特性的影响

为研究板状材料上加强筋结构对兰姆波透射特性的影响,通过有限元仿真和实验的方法,结合兰姆波频散曲线和二维傅里叶变换分析方法,研究了不同几何形状的加强筋结构对兰姆波垂直入射时的透射特性。仿真与实验的结果表明:加强筋结构会对兰姆波产生梳状滤波器效应,即同一几何形状的加强筋对不同频率的兰姆波透射程度不同;梳状滤波器的通带带宽和通带个数受到加强筋几何形状的影响;随着加强筋高度的增加,梳状滤波器的通带带宽变小且通带个数增加;随着加强筋厚度的增加,梳状滤波器的通带带宽和个数并不呈现单调变化;对于同一高厚比的加强筋,随着加强筋体积的变大,梳状滤波器的通带宽变小且通带个数增加。该研究结果可用于指导利用兰姆波对具有加强筋结构的板状材料进行结构健康监测和无损检测时兰姆波有效频率的选择,从而提高兰姆波能量的利用率和信号的信噪比。      

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.     

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.     

Input power flow in a submerged infinite cylindrical shell with doubly periodic supports

A submerged cylindrical shell reinforced by supports of rings and bulkheads is the primary structure of submarine, torpedo and all kinds of submerged aircrafts, so it is significant to study its characteristics of structure-borne sound. By means of periodic structure theory, the input power flow from a cosine harmonic line force into a submerged infinite cylindrical shell, reinforced by doubly periodic supports of rings and bulkheads, is studied analytically. The harmonic motion of the shell and the sound pressure field in the fluid are described by Flügge shell equations and Helmholtz equation, respectively. Since the fluid radical velocity and the shell radical velocity must be equal at the interface of the outer shell wall and the fluid, the motion equations of this coupled system are obtained. Both four kinds of forces (moments) between rings and shell and four kinds of forces (moments) between bulkheads and shell are considered. The solution is obtained in series form by expanding the system responses in terms of the space harmonics of the spacings of both stiffeners and bulkheads. The input vibrational power flow into the structure is obtained and the influences of different structural parameters on the results are analyzed. The analytic model is close to engineering practice, and it will give some guidelines for noise reduction of this kind of shell.     

Guided plate wave scattering at vertical stiffeners and its effect on source location

This paper examines guided wave transmission characteristics of plate stiffeners and their influence on the performance of acoustic noise source location. The motivation for this work is the detection of air leaks in manned spacecraft. The leaking air is turbulent and generates noise that can be detected by a contact-coupled acoustic array to perform source location and find the air leak. Transmission characteristics of individual integral stiffeners are measured across a frequency range of 50-400 kHz for both high and low aspect-ratio rectangular stiffeners, and comparisons are made to model predictions which display generally good agreement. It is demonstrated that operating in frequency ranges of high plate wave stiffener transmission significantly improves the reliability of noise source location in the plate. A protocol is presented to enable the selection of an optimal frequency range for leak location.     

An experimental study of the effects of resilient bars on plate vibration

Resilient bars can provide a low-cost, effective improvement to sound insulation performance. They are commonly used in timber-framed floor/ceiling assemblies in North America and Europe. Resilient bars are often modelled as springs isolating the two connected plates thereby forming a mass-spring-mass system. However, as a furring system of plates, resilient bars may modify the vibration energy distribution across a connected plate by acting as stiffeners. The authors investigate this issue by measuring acceleration levels at different locations relative to the fixing positions and thereby derive vibration waveforms for the connected plate in a small-scale structural simulation of a floor-ceiling system. The results were compared with timber-joist-ribbed, and timber-brander-ribbed, structures. The vibration modes of a suspended plate were also measured for comparative purposes. The results indicated that resilient bars did not perform as stiffeners whereas joists and timber branders did effectively stiffen their connected plates. Resilient bars neither forced orthotropic plate behaviour at low frequencies, nor separated the plate into sub-plates at higher frequencies. Resilient-bar-ribbed plates may also differ from independent plates. The modal behaviour of resilient-bar-ribbed plates is more complex and their effect on modal density and radiation efficiency are worthy of further research.     

Geometrically nonlinear dynamic response of stiffened plates with moving boundary conditions

An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The first approximation solution of the non-resonance is obtained by means of the method of multiple scales. The primary parametric resonance and primary resonance of the stiffened plate are studied by using the same method. The accuracy of the method is validated by comparing the results with those of finite element analysis via ANSYS. Numerical examples for different stiffened plates are presented to discuss the steady response of the non-resonance and the amplitude-frequency relationship of the primary parametric resonance and primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude-frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.     

Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations

Free and forced vibrations of triangular plate are investigated. Diverse types of stiffeners were attached onto the plate to suppress the undesirable large-amplitude oscillations. The governing equation of motion for a triangular plate, based on the von Kármán theory, is developed and the nonlinear ordinary differential equation of the system using Galerkin approach is obtained. Closed-form expressions for the free undamped and large-amplitude vibration of an orthotropic triangular elastic plate are presented using the two well-known analytical methods, namely, the energy balance method and the variational approach. The frequency responses in the closed-form are presented and their sensitivities with respect to the initial amplitudes are studied. An error analysis is performed and the vibration behavior, as well as the accuracy of the solution methods, is evaluated. Different types of the stiffened triangular plates are considered in order to cover a wide range of practical applications. Numerical simulations are carried out and the validity of the solution procedure is explored. It is demonstrated that the two methods of energy balance and variational approach have been quite straightforward and reliable techniques to solve those nonlinear differential equations. Subsequently, due to the importance of multiple resonant responses in engineering design, multi-frequency excitations are considered. It is assumed that three periodic forces are applied to the plate in three specific positions. The multiple time scaling method is utilized to obtain approximate solutions for the frequency resonance cases. Influences of different parameters, namely, the position of applied forces, geometry and the number of stiffeners on the frequency response of the triangular plates are examined.     

Elastic response of a cylinder containing longitudinal stiffeners

This paper develops a three-dimensional analytical model of a cylinder that contains a longitudinal stiffener. The model begins with the equations of motion for a fully elastic solid that produces displacement fields with unknown wave propagation coefficients. These are inserted into stress and displacement equations at the cylinder boundaries and at the location of the stiffener. Orthogonalization of these equations produces an infinite number of indexed algebraic equations that can be truncated and incorporated into a global matrix equation. Solving this equation yields the solution to the wave propagation coefficients and allows the system's displacements and stresses to be calculated. The model is verified by comparison of the results of a plane strain analysis example to a solution generated using finite element theory. A three-dimensional example problem is formulated and the displacement results are illustrated. The inclusion of multiple stiffeners is discussed.     

Space harmonic analysis of sound radiation from a submerged periodic ring-stiffened cylindrical shell

An analytical method is developed to study radiated sound power characteristics from an infinite submerged periodically stiffened cylindrical shell excited by a radial cosine harmonic line force. The harmonic motion of the shell and the pressure field in the fluid are described by Flügge shell equations and Helmholtz equation, respectively. By using periodic theory of space harmonic analysis, the response of the periodic structure to harmonic excitations has been obtained by expanding it in terms of a series of space harmonics. Radiated sound power on the shell wall along the axial direction and the influence of different parameters on the results are studied, respectively. A conclusion is drawn that the stiffeners have a great influence at low and high frequencies while have a slight influence at intermediate frequencies for low circumferential mode orders. The work will give some guidelines for noise reduction of this kind of shell.     

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.     

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。      

SOUND TRANSMISSION THROUGH PERIODICALLY STIFFENED CYLINDRICAL SHELLS

A relatively long circular cylindrical shell stiffened by periodically deployed stiffeners is found in many practical applications such as the aircraft fuselage. For the sound transmission analysis of such a system, not only structural interactions between the shell and stiffeners but also vibro-acoustic interactions between the structure and acoustic media have to be considered. By idealizing the system as an infinitely long cylinder subjected to a plane wave incidence, an exact solution is obtained for the first time for this type of problem. In the analysis, the solution is obtained in a series form by expanding the system responses in terms of the space harmonics of the stiffener spacing. Characteristics of the system responses and effects of important design parameters are studied using the transmission losses calculated from the analysis.     

NON-LINEAR VIBRATION AND THERMAL BUCKLING OF AN ORTHOTROPIC ANNULAR PLATE WITH A CENTRIC RIGID MASS

A computational analysis of the non-linear vibration and thermal post-buckling of a heated orthotropic annular plate with a central rigid mass is examined for the cases of immovably hinged as well as clamped constraint conditions of the outer edge. First, based on von Karman's plate theory and Hamilton's principles, the governing equations, in terms of the displacements of the middle plane, of the problem are derived. Then, upon assuming that harmonic responses of the system exist, the non-linear partial differential equations are converted into the corresponding non-linear ordinary differential equations through elimination of the time variable by using the Kantorovich time-averaging method. Finally, by applying a shooting method, the fundamental responses of the non-linear vibration and thermal post-buckling of the plate are numerically obtained. For some prescribed values of the parameters, such as the material rigidity ratio, temperature rise and so on, the curves of the fundamental frequency versus specified amplitude and the thermal post-buckled equilibrium paths of the plate are numerically presented.     

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.     

Influence of stiffeners on plate vibration and radiated noise excited by turbulent boundary layers

The influence of stiffeners on plate vibration and noise radiation induced by turbulent boundary layers is investigated by wind tunnel measurements. Plates with and without stiffeners are tested under the flow speed of 60 m/s, 71 m/s and 86 m/s, respectively. The stiffeners are set either perpendicular or parallel to the direction of the free stream. Measured vibration and noise levels are compared with theoretical calculations, where wall pressure cross-spectra are described by the Corcos model. For the plates tested, it is evident that stiffeners perpendicular to the direction of the free stream could increase noise radiation, but have almost no influence on vibration level of plates.