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.     

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.     

Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation

The paper describes a method for the prediction of dynamic characteristics of rectangular plates with cutouts. The method is based on the use of variational principles in conjunction with finite difference technique. A concept of interlacing grids has been developed to express the strain energy of nodal subdomains into which the plate is divided. The use of this idea has been demonstrated in relation to internal and boundary nodes. Natural frequencies and corresponding mode shapes of rectangular plates with one and two cutouts have been predicted and experimentally verified.     

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.     

A note on vibration of a cantilever plate immersed in water

The added mass of the fluid surrounding it plats an important role in the dynamic behaviour of a submerged structure. The first few mode shapes and the respective natural frequencies of a submerged cantilever plate are found by using a finite element procedure, eigenvalues being obtained by a simultaneous iteration technique. The influence of the water depth below the plate and also of the water's lateral extent is considered, in order to test the convergency of the results. Results on the effects of the depth of immersion on the natural frequencies and mode shapes of the cantilever plate for different aspect ratios are presented.     

Radiation efficiency of submerged rectangular plates

The average radiation efficiency of point-excited submerged rectangular plates is investigated in two methods, deterministic analysis and statistical approach, respectively. In the deterministic analysis, the effect of mutual impedance by water loading on the velocity of the plate is illustrated analytically by using a modal summation method. The cross-modal contributions to the average radiation efficiency are averaged to zero by averaging over all possible excitation positions on the plate. In the statistical approach, by analyzing the engineering formulae of the average radiation efficiency in air, this paper modifies the formulae to be applicable in water. The numerical comparisons show that the modified formulae reflect the average level in the frequency region controlled by corner modes and are accurate enough in the region controlled by monopole and edge modes. On this basis, approximate expressions for predicting the average radiation efficiency of the submerged beam-stiffened rectangular plates are proposed. The main differences between air loading and water loading are considered. Firstly, as dry modes taken in analysis instead of real vibration modes in water, the vibration of the stiffened plate is not only determined by the first mode but by several modes simultaneously at low frequencies. Secondly, the “corner mode region” becomes inappropriate as the plate is stiffened. The proposed formulae are validated numerically in different size, thickness, and stiffener amount of the stiffened plate.     

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.     

Vibration analysis of circular segment shaped plates

Circular segment shaped plates are analyzed to determine their natural frequencies and mode shapes of vibration. The analysis is based on the finite element approach. The curved sided triangular plate bending element is used for solving the problem. The effect of variation of the size of the plate on the vibrational characteristics is studied and several important conclusions are made.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

NATURAL VIBRATIONS OF LAMINATED COMPOSITE BEAMS BY USING MIXED FINITE ELEMENT MODELLING

A six-node, plane-stress mixed finite element model has been developed by using Hamilton's energy principle for the natural vibrations of laminated composite beams. Continuity of the transverse stress and displacement fields has been enforced through the thickness of the laminated beam in the formulation for proper modelling. The transverse stress components have been invoked as the nodal degrees of freedom by applying elasticity relations. Natural frequencies of laminated composite beams obtained through the present formulation have been shown to be in good agreement with the data available in the literature. Various mode shapes have also been presented as benchmark solutions.     

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.     

Vibration of moderately thick square orthotropic stepped thickness plates

Numerous studies that address the vibration of stepped thickness plates are reported in the literature. Predominately, classical plate theory has been used to formulate studies for both isotropic and anisotropic stepped plates. Mindlin plate theory has been employed to obtain results for thick isotropic stepped thickness plates. Exact solutions, Rayleigh-Ritz, differential quadrature and finite element methods have been employed to compute results for frequency of vibration. Results for frequency of vibration for thick orthotropic stepped thickness plates are presented here using orthorhombic material properties of aragonite. The finite element method has been used to compute frequencies and determine mode shapes for simply supported and clamped square Mindlin plates.     

Natural frequencies and vibration modes of laminated composite plates reinforced with arbitrary curvilinear fiber shape paths

The development of tow-placement technology has made it possible to control fiber tows individually and place fibers in curvilinear distinct paths in each layer of a laminated plate. This paper presents an analytical method for determining natural frequencies and vibration modes of laminated plates having such curvilinear reinforcing fibers. Spline functions are employed to represent arbitrarily shaped fibers, and Ritz solutions are used to derive frequency equations using series type shape functions. The strain energy is evaluated by numerical integration involving the fiber orientation angle, and is calculated using the derivative of the spline function in minute intervals. The results show that the natural frequencies obtained by the present method agree well with results from finite element analyses. The vibration mode shape contour plots of the plates are seen to reflect clear influences of the fiber shapes.     

STRUCTURAL DAMAGE DETECTION BASED ON A MICRO-GENETIC ALGORITHM USING INCOMPLETE AND NOISY MODAL TEST DATA

This paper describes a procedure for detecting structural damage based on a micro-genetic algorithm using incomplete and noisy modal test data. As the number of sensors used to measure modal data is normally small when compared with the degrees of freedom of the finite element model of the structure, the incomplete mode shape data are first expanded to match with all degrees of freedom of the finite element model under consideration. The elemental energy quotient difference is then employed to locate the damage domain approximately. Finally, a micro-genetic algorithm is used to quantify the damage extent by minimizing the errors between the measured data and numerical results. The process may be either of single-level or implemented through two-level search strategies. The study has covered the use of frequencies only and the combined use of both frequencies and mode shapes. The proposed method is applied to a single-span simply supported beam and a three-span continuous beam with multiple damage locations. In the study, the modal test data are simulated numerically using the finite element method. The measurement errors of modal data are simulated by superimposing random noise with appropriate magnitudes. The effectiveness of using frequencies and both frequencies and mode shapes as the data for quantification of damage extent are examined. The effects of incomplete and noisy modal test data on the accuracy of damage detection are also discussed.     

Nodal line optimization and its application to violin top plate design

In the literature, most problems of structural vibration have been formulated to adjust a specific natural frequency: for example, to maximize the first natural frequency. In musical instruments like a violin; however, mode shapes are equally important because they are related to sound quality in the way that natural frequencies are related to the octave. The shapes of nodal lines, which represent the natural mode shapes, are generally known to have a unique feature for good violins. Among the few studies on mode shape optimization, one typical study addresses the optimization of nodal point location for reducing vibration in a one-dimensional beam structure. However, nodal line optimization, which is required in violin plate design, has not yet been considered. In this paper, the central idea of controlling the shape of the nodal lines is proposed and then applied to violin top plate design. Finite element model for a violin top plate was constructed using shell elements. Then, optimization was performed to minimize the square sum of the displacement of selected nodes located along the target nodal lines by varying the thicknesses of the top plate. We conducted nodal line optimization for the second and the fifth modes together at the same time, and the results showed that the nodal lines obtained match well with the target nodal lines. The information on plate thickness distribution from nodal line optimization would be valuable for tailored trimming of a violin top plate for the given performances.     

Dynamic behaviour of isotropic plates under combined acoustic excitation and static in-plane compression

A study is described which was carried out to investigate the dynamic response of an aircraft-type aluminium alloy plate, with fully clamped boundaries, subjected to combined acoustic excitation and uniaxial in-plane compression. Experiments were conducted in an acoustic tunnel with sound pressure levels up to 150 dB. The non-linear characteristics of the plate have been examined under sinusoidal and broad band random excitation and a study made of the statistical and spectral properties of the response. Theoretical and experimental estimates of the plate natural frequencies and mode shapes are described.     

Free vibration of a circular plate elastically restrained along some radial segments

An analysis is presented for the free vibration of a circular plate restrained against deflection along radial segments. With the reaction forces acting on the segments regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force distributions along the segments are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the supports. The natural frequencies and the mode shapes of the plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to circular plates supported along several radial segments located at equal angular intervals, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the supports is discussed.     

Determination of the steady state response of viscoelastically point-supported rectangular specially orthotropic plates by an energy-based finite difference method

A method based on a variational procedure in conjunction with a finite difference method is used to examine the free vibration characteristics and steady state response to a sinusoidally varying force applied at the center of a viscoelastically point-supported orthotropic elastic plate of rectangular shape. Using the energy-based finite difference method, the problem is reduced to the solution of a system of algebraic equations. The influence of the mechanical properties, and of the damping of the supports to the mode shapes and to the steady state response of viscoelastically point-supported rectangular plates is investigated numerically for a concentrated load at the center for various values of the mechanical properties characterizing the anisotropy of the plate material and for various damping ratios. The results are given for the frequencies and mode shapes of the first three symmetrical modes. Convergence studies are made. The validity of the present approach is demonstrated by comparing the results with other solutions based on the Kirchhoff-Love plate theory.     

Free vibrations of a vessel consisting of circular plates and a shell of revolution having varying meridional curvature

An analytical solution procedure is presented for the free vibration of vessels consisting of a shell of revolution having varying meridional curvature and circular plate lids. The Lagrangian of vibration of the combined system is obtained in quadratic forms of boundary values. The frequency equation and the relations among the boundary values are obtained from minimizing conditions of the Lagrangian with respect to the unknown boundary values. The natural frequencies and the mode shapes of vessels having elliptical and hyperbolical meridians have been obtained by carrying out numerical calculations. Effects of various parameters upon natural frequencies and mode shapes are illustrated in discussions of numerical results.     

Vibrations of heated plates with two opposite edges simply supported

Vibration analysis of the family of rectangular plates with two opposite sides simply supported can be simplified by assuming mode shapes. In the present paper a vibration analysis of such plates which are heated so as to have a temperature varying in the direction parallel to these sides is presented. A steady state temperature which satisfies the Laplace equation is considered. Due to the assumption of mode shapes the governing plate differential equation, which in general is a function of the x and y co-ordinates, becomes a function of one co-ordinate. This equation is analyzed by a finite difference method and solved by a standard simultaneous iteration technique. The accuracy of the method is ascertained by comparing the results for some well known boundary conditions when there is no temperature effect with the standard solutions available in the literature. From the results an attempt has been made to correlate the natural frequency with the temperature. Plates of uniform thickness with different length to breadth ratios have been analyzed. The assumed linear temperature distribution satisfies the Laplace equation and the plate is free to expand in its plane at its edges so that no thermal stresses will be induced.