Improvements of the smearing technique for cross-stiffened thin rectangular plates

New developments in the simplified smearing technique for modeling vibrations of cross-stiffened, thin rectangular plates are presented. The computationally efficient smearing technique has been known for many years, but so far the accuracy of, say, predicted natural frequencies has been inadequate. The reason is that only the stiffeners at a right angle to the axis of angular motion are taken into account when calculating the bending stiffness, whereas the stiffeners that are parallel to this axis of angular motion are neglected. To improve predictions, the parallel stiffeners are taken into account in this paper. The improved smearing technique results in better accuracy for predicted natural frequencies of flat stiffened plates, as demonstrated for both simply supported and clamped boundary conditions. The improved prediction accuracy is demonstrated by comparing results from a numerical model based on the current development with results from finite element (FE) simulations that include the exact cross-sectional geometries of the stiffened panel. In order to demonstrate applications of the improved smearing technique, the predicted forced response is compared with both experimental and FE results. Another improvement concerns the orientation of the stiffeners. The original smearing technique presupposes that the stiffeners are parallel to the edges of the plate, but simple considerations make it possible to relax this requirement. To test the validity of the resulting technique a series of plates are examined for stiffeners angled relative to the plate edges.     

流体加载下加肋板结构的声辐射特性研究

本文采用有限元和边界元方法对加肋板结构的声辐射进行了计算分析,研究分析了加肋板结构的板厚、板面积、板边长比、肋骨惯性矩和间距、边界条件以及板材和流体介质等对结构辐射声功率的影响,得到了一些有意义的结论,从而为揭示加板结构声辐射规律,降低结构的声辐射提供了一定的依据。     

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.     

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.     

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.     

超高速撞击声发射信号在载人航天器加筋结构的传播特性实验研究

为研究超高速撞击声发射信号经过载人航天器加筋结构后的传播规律,分别在平板结构和加筋结构上模拟高速撞击实验,利用传感器阵列采集高速撞击产生的声发射信号。结合小波和傅里叶分析方法从板波模态、频域以及时域三方面分析加筋结构对声发射信号传播特性的影响,并研究成坑和穿孔损伤模式下声发射信号的传播规律。结果表明:加筋板中的信号高频部分比平板中高频部分能量少,筋体对信号高频部分有滤波效果。加筋结构受高速撞击产生穿孔损伤时,S0模态声波的能量增多。研究成果可为载人航天器结构的高速撞击感知与定位技术提供有利参考。      

Dynamic instability of stiffened plates subjected to non-uniform harmonic in-plane edge loading

The dynamic instability characteristics of stiffened plates subjected to in-plane partial and concentrated edge loadings are studied using finite element analysis. In the structural modelling, the plate and the stiffeners are treated as separate elements where the compatibility between these two types of elements is maintained. The method of Hill's infinite determinants is applied to determine the dynamic instability regions. Numerical results are presented to study the effects of various parameters, such as static load factor, aspect ratio, boundary conditions, stiffening scheme and load parameters on the principal instability regions of stiffened plates using Bolotin's method. The results show that location, size and number of stiffeners have a significant effect on the location of the boundaries of the principal instability region.     

Transmission loss of periodically stiffened laminate composite panels: shear deformation and in-plane interaction effects

This paper investigates the transmission loss of symmetric and asymmetric laminate composite panels periodically reinforced by composite stiffeners. A comprehensive model based on periodic structure theory is developed. First order shear deformation theory is used and the coupling of the in-plane motion of the panel with its out-of-plane motion is taken into account. Stiffeners interact with the panel through three forces (two in-plane, one out-of-plane) and a torsion moment. Three types of cross sections are investigated for the composite stiffeners: I-shaped, C-shaped, and omega-shaped cross-sections. The model is validated numerically by comparison with the finite element/boundary element method. Experimental validations are also conducted. In both cases, excellent agreement is obtained. Numerical results show that the in-plane coupling effect is increased by increasing the panel thickness and the stiffener's eccentricity. The in-plane coupling effect is also found to increase with frequency.     

Vibro-acoustic response of orthogonally stiffened panels: The effects of finite dimensions

This paper investigates the effects of finite dimensions on the vibro-acoustic response of orthogonally stiffened panels. Three types of excitations are considered: acoustical excitation, point force excitation and random excitation by a turbulent boundary layer. In each case, a spatially windowed periodic model is compared with a Rayleigh-Ritz model where the modes of the un-stiffened panel are used as the basis functions. The latter model accounts for the reflected wave field generated at the boundaries by assuming that the panel is simply supported. On the contrary, the windowed periodic model only accounts for finiteness on sound radiation (the assumption of an infinite periodic structure is used to calculate the panel response). Numerical studies show that when the bending wavelength becomes comparable or smaller than the stiffener spacing, the periodic model is able to reproduce the results obtained with the Rayleigh-Ritz model. To complement the study, the developed models are compared with numerical simulations (finite element method) and with experimental results.     

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.     

含损伤黏弹性阻尼加筋层合板声功率和指向性变异*

该文旨在研究损伤位置和程度对自由阻尼加筋层合板声辐射功率和指向性的影响。基于Mindlin和Timoshenko梁理论,建立了自由阻尼层合板-梁组合结构有限元模型。数值求解四边简支边界条件自由阻尼加筋层合板振动响应,继而通过Rayleigh积分计算加筋层合板辐射声功率和指向性。将计算得到的前4阶模态固有频率、声辐射功率与指向性与已有文献进行了对比基本一致,验证了数值模型的正确性。最后,详细讨论了损伤位置和程度对自由阻尼加筋层合板固有频率、振型、声辐射功率和指向性的影响,结果表明：随着结构损伤程度的增大,声辐射功率峰值向低频移动,在更多角度上出现明显的指向性;声辐射功率和指向性对损伤位置比损伤程度更加敏感。     

基于波数域直接求解法的正交加筋板声辐射研究

为了研究正交加筋板的声辐射问题,基于波数域直接求解法,建立了研究正交加筋板声辐射特性的理论模型。先利用傅里叶变换法求解周期结构的声振理论模型,得到波数域中关于结构响应的无限大耦合代数方程组,采用数值方法将其截断成有限项求解,结合稳相法便可快速获得远场辐射声压。该方法对单向和正交加筋板的预测结果与现有文献中的理论结果取得了良好的吻合,验证了理论模型的准确性和可靠性;并进一步通过数值算例研究了作用点位置,加强筋间距及平板厚度对结构声辐射特性的影响。      

Assessing the effect of a barrier between two rooms subjected to low frequency sound using the boundary element method

The boundary element method (BEM) has been used to compute the acoustic wave propagation through a single vertical panel, which separates two rooms, made of concrete, when one of the rooms is excited by a steady-state, spatially sinusoidal, harmonic line load pressure at low frequencies. This work focuses on how the connection of the panel to the ceiling affects the acoustic insulation provided by the wall. Perfect double-fixed partitions and acoustic barrier-type structures with differently-sized gaps between the ceiling and the barrier are studied. The BEM model is formulated in the frequency domain and takes the air-solid interaction fully into account. Insulation dips are localised in the frequency domain and identified with dips associated with both the wall's natural dynamic vibration modes and with those associated with the air in the rooms. The influence of the wall's thickness on acoustic insulation is also analysed. The computed results obtained with the acoustic barrier type structure are compared with those obtained by a rigid model. The importance of the rooms' surface conditions is assessed, modelling the rooms with cork.     

Structural and acoustic responses of a fluid-loaded cylindrical hull with structural discontinuities

This paper studies the low frequency vibrational behaviour and radiated sound of a submarine hull under axial excitation. The submarine is modelled as a fluid-loaded cylindrical shell with internal bulkheads and ring-stiffeners and closed at each end by circular plates. A smeared approach is used to model the ring stiffeners. The external pressure acting on the hull due to the fluid loading is calculated using an infinite model and is shown to be a good approximation at low frequencies. The radiated sound pressure is obtained by considering the finite cylindrical hull to be extended by two semi-infinite rigid baffles. The sound pressure is then only due to the radial displacement of the cylindrical shell, without taking into account the scattering at the finite ends. The main aim of this paper is to observe the influence of the various complicating effects such as the bulkheads, ring-stiffeners and fluid loading on the structural and acoustic responses of the finite cylindrical shell. Results from the analytical models presented in this paper are compared to the computational results from finite element and boundary element models.     

Free vibration of plated structures by grillage method

Free vibration of plated structures such as plates, stiffened plates and cellular structures are analysed by the grillage method. The accuracy of the method is assessed by comparing with published results and those of the finite element method. Reasonably good results are obtained by the grillage method, which is less demanding on computer time and thus reduces the cost of analysis substantially. By using the grillage method the influence of the number of stiffeners in a stiffened plate and the number of webs in a cellular structure on the vibration characteristics of plated structures is investigated.     

Comparison of structural response and fatigue endurance of aircraft flap-like box structures subjected to acoustic loading

The results of an extensive test program to characterize the behavior of typical aircraft structures under acoustic loading and to establish their fatigue endurance are presented. The structures tested were the three flap-like box-type of structures. Each structure consisted of one flat (bottom) and one curved (top) stiffener stiffened skin panel, front, and rear spars, and ribs that divided the structures into three bays. The three structures, constructed from three different materials (aircraft standard aluminum alloy, Carbon Fibre Reinforced Plastic, and a Glass Fibre Metal Laminate, i.e., GLARE) had the same size and configuration, with only minor differences due to the use of different materials. A first set of acoustic tests with excitations of intensity ranging from 140 to 160 dB were carried out to obtain detailed data on the dynamic response of the three structures. The FE analysis of the structures is also briefly described and the results compared with the experimental data. The fatigue endurance of the structures was then determined using random acoustic excitation with an overall sound pressure level of 161 dB, and details of crack propagation are reported.     

Noise reduction analysis for a stiffened finite plate

This paper presents an analytical solution for the vibration and acoustic responses of a finite stiffened plate that is covered with decoupling layers and subjected to external excitation. The theory of elasticity is used for the decoupling layer, and the stiffened plate is modeled by the plate theory and Euler–Bernoulli beam equation. Equations are constructed by the boundary conditions at the plate/coating and coating/fluid interfaces. The problem can be solved by the proposed method in this paper. Test verification shows that a good correlation exists between theoretical and test results. Thus, the theoretical study in this paper is correct. Numerical results show that shear waves insignificantly affect the structural vibration level difference (VR) under low frequencies. The noise reduction of the stiffened plate covered with decoupling layers is greatly influenced by the decoupling layer loss factor. A failure region of the vibration level difference is present in the low frequency band of the decoupling layer. Furthermore, the thickness of the decoupling layer significantly affects noise reduction.     

Vibro-acoustic design sensitivity analysis using the wave-based method

Conventional element-based methods, such as the finite element method (FEM) and boundary element method (BEM), require mesh refinements at higher frequencies in order to converge. Therefore, their applications are limited to low frequencies. Compared to element-based methods, the wave-based method (WBM) adopts exact solutions of the governing differential equation instead of simple polynomials to describe the dynamic response variables within the subdomains. As such, the WBM does not require a finer division of subdomains as the frequency increases in order to exhibit high computational efficiency. In this paper, the design sensitivity formulation of a semi-coupled structural-acoustic problem is implemented using the WBM. Here, the results of structural harmonic analyses are imported as the boundary conditions for the acoustic domain, which consists of multiple wave-based subdomains. The cross-sectional area of each beam element is considered as a sizing design variable. Then, the adjoint variable method (AVM) is used to efficiently compute the sensitivity. The adjoint variable is obtained from structural reanalysis using an adjoint load composed of the system matrix and an evaluation of the wave functions of each boundary. The proposed sensitivity formulation is subsequently applied to a two-dimensional (2D) vehicle model. Finally, the sensitivity results are compared to the finite difference sensitivity results, which show good agreement.     

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.     

Scattering model of a cylindrical shell with internal axisymmetric frames by using the Circumferential Admittance Approach

A numerical model is proposed for predicting scattering pressure by a fluid-loaded cylindrical shell stiffened by axisymmetric internal frames and impacted by an acoustic plane wave. The proposed developments are based on the Circumferential Admittance Approach (CAA) which allows us assembling a numerical model of the fluid loaded shell with finite element models of the internal frames. The scattering pressure model deduced with the CAA can then take into account: (a) internal frames having a cross section with a complex geometry and thickness variations (like T-shaped stiffeners, bulkheads, and hemispherical end caps); (b) variations of frame spacings; and (c) frame-shell coupling in the three translational directions and tangential rotation. Comparison with the numerical and experimental results of the literature for a periodic stiffened shell shows that the scattering from Bragg, Bloch–Floquet, and Helical waves is correctly predicted. The effects on the backscattering pressure of axial and tangential coupling forces are highlighted. Finally, an example of a non-periodically stiffened shell is presented to highlight the versatility of the approach proposed.