Simple Support Boundary Conditions Research Articles

Natural frequencies of nonlinear vibration of axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.

Exact solutions for the free in-plane vibrations of rectangular plates

All classical boundary conditions including two distinct types of simple support boundary conditions are formulated by using the Rayleigh quotient variational principle for rectangular plates undergoing in-plane free vibrations. The direct separation of variables is employed to obtain the exact solutions for all possible cases. It is shown that the exact solutions of natural frequencies and mode shapes can be obtained when at least two opposite plate edges have either type of the simply-supported conditions, and some of the exact solutions were not available before. The present results agree well with FEM results, which show that the present solutions are correct and the direct separation of variables is practical. The exact solutions can be taken as the benchmarks for the validation of approximate methods.

Flexural Vibration Analysis for Symmetric Honeycomb Panels of Simple Support Boundary Conditions

The flexural vibration of symmetric honeycomb panels having simple support boundary conditions is investigated by using the classical plate theory,first=order shear deformation plate theory,third=order shear deformation plate theory.The honeycomb core of cells is modeled as a layer of orthotropic material whose physical and mechanical properties are determined by using the corrected Gibson's formula.The comparative studies conducted on aluminum honeycomb panels indicate that both the classical and first-order shear deformation plate theories are inadequate for the flexural vibration of honeycomb panels.Using the Reddy third order shear deformation plate theory,the effects of panel thickness,core thickness and cell angle on flexural vibration of symmetric honeycomb panels are investigated and the corresponding curve figure is drawn.The curve figure is instructive to application of honeycomb panels in engineering.

КОЛЕБАНИЯ ТОНКОСТЕННОЙ УПРУГОЙ КОНСТРУКЦИИ ИЗ НЕЗАМКНУТЫХ ОРТОТРОПНЫХ ЦИЛИНДРИЧЕСКИХ ОБОЛОЧЕК СО СВОБОДНОЙ И ШАРНИРНО ЗАКРЕПЛЕННОЙ ГРАНИЧНЫМИ ОБРАЗУЮЩИМИ

The problem of existence of natural vibrations of an elastic orthotropic thin-walled structure constructed from circular non-closed infinite cylindrical shells with free and simple support boundary conditions on the edge generators and an elastic orthotropic thin-walled structure constructed from circular non-closed finite cylindrical shells with free and simple support Navier’s boundary conditions at the edge generators, which have a simple support on the boundary directional curves, are studied. Using the system of equations of the related classical theory of orthotropic cylindrical shells dispersion equations and asymptotic formulas are obtained for determining the eigenvalue frequencies of possible vibration types for the corresponding thin-walled structures. An algorithm for separating the possible vibrations is presented. Approximate values of dimensionless characteristics of eigenvalue frequencies and fading characteristics of the related vibration forms are given for the cases of orthotropic thin-walled structures constructed from various numbers of identical circular non-closed finite cylindrical shells.

Analysis of anisotropic plate using multiquadric radial basis function

In this paper, multiquadric radial basis function is used for dynamic and static analysis of anisotropic plates. Multiquadric radial basis function is applied for spatial discretization and Newmark implicit scheme is used for temporal discretization. The spatial discretization of the differential equations generates greater number of algebraic equations than the unknown coefficients. The multiple linear regression analysis, which is based on the least squares error norm, is employed to obtain the coefficients. Considering simple supported boundary conditions, an analogy between isotropic skew plates and rectangular anisotropic plates is used for solutions. The effect of fiber orientation is observed in clamped square and rectangular anisotropic plates. The results obtained by this method are compared with those obtained by other analytical methods.

Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Two distinct types of simple support boundary conditions are formulated for rectangular plates undergoing in-plane free vibration. Each type of simple support is shown to be analogous to the well known simple support edge conditions encountered in the free transverse vibration analysis of rectangular plates. It is shown that exact solutions may be obtained for plate free vibration eigenvalues and mode shapes when two opposite plate edges are given either type of simple support, the other two edges being given any combination of classical edge conditions. A complete analysis is provided for plate in-plane vibration with pairs of clamped or free edge conditions imposed on the non-simply supported boundaries and exact eigenvalues are tabulated for a large range of plate aspect ratios. This appears to be the first thorough study of these in-plane vibration problems with exact solutions.

Free flexural vibration analysis of symmetric honeycomb panels

The free flexural vibration of symmetric rectangular honeycomb panels having simple support boundary conditions is investigated in this paper using the classical plate theory, Mindlin's improved plate theory, and Reddy's third-order plate theory. The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose physical and mechanical properties are determined using the Gibson and Ashby correlations. The comparative studies conducted on aluminum honeycomb panels indicate that both the classical and improved plate theories are inadequate for the flexural vibration of honeycomb panels.

The finite element analysis of composite laminates and shell structures subjected to low velocity impact

In this investigation, the composite laminate and shell structures subjected to low velocity impact are studied by the ANSYS/LS-DYNA finite element software. The contact force is calculated by the modified Hertz contact law in conjunction with the loading and unloading processes. In the case of composite laminate, the impact-induced damage including matrix cracking and delamination are predicted by the appropriated failure criteria and the damaged area are plotted. Two types of shell structure, cylindrical and spherical shells, are considered in this paper. The effects of various parameters, such as shell curvature, clamped or simple supported boundary conditions and impactor velocity are examined through the parametric study. Numerical results show that structures with greater stiffness, such as smaller curvature and clamped boundary condition, result to a larger contact force and a smaller deflection. The impact response of the structure is proportional to the impactor velocity.

Effect of symmetrical boundary conditions on the vibration of thick hollow cylinders

A finite element for axisymmetric elasticity is formulated directly in cylindrical coordinates. The element is used to study the vibration of hollow, isotropic finite length cylinders. The ratio of the inside radius to the outside radius is assumed as 0.2, 0.5 and 0.9 for a given cylinder length. Frequencies are computed for free–free end boundary conditions that can be compared with the existing literature. Additionally, fixed–fixed boundary conditions are studied. Five different configurations for simple supported boundary conditions are analyzed and compared. Numerical results are given in tabular form and some fundamental mode shapes are compared graphically.

Maxwell and Lamé eigenvalues on polyhedra

In a convex polyhedron, a part of the Lame eigenvalues with hard simple support boundary conditions does not depend on the Lame coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lame coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non-convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non-H2 singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non-convex polyhedron, the spectrum cannot be approximated by finite element methods using H1 elements. Similar properties hold in polygons. We give numerical results for two L-shaped domains. Copyright © 1999 John Wiley & Sons, Ltd.

Computer simulation of impact damage on thin-skinned carbon fibre composite panels

An analytical study was carried out using the 3-D explicit finite element (FE) code MSC/DYTRAN to analyse geometric nonlinear impact problems. The panel under investigation has an impact window of 90 x 80 mm with simple support boundary condition. T300/914 preimpregnated unidirectional carbon fibre/epoxy composite material was used and the lay-up sequence was [(0/90/45/45){3}]{s}. The nominal skin thickness was 3 mm. The impactor was modelled using solid elements and was excited by assigning an initial velocity. Damage analyses were undertaken using the Chang-Chang failure criteria implemented within MSC/DYTRAN together with degradation models. The progressive impact damage scenario was simulated, enabling the prediction of matrix and fibre failure. Low and high velocity cases were studied under low energy impact, significantly different behaviours were observed. It is concluded that MSC/DYTRAN is an appropriate tool for analysing impact in thin-skinned carbon fibre materials.

A triangular conforming element for laminated shells

A 30 degree of freedom (DOF) conforming shell element is developed for laminated composite materials. This element is 10-noded and has a triangular shape. Sander's thin shell theory is used. The element is used to perform both static and dynamic analysis on a composite cylindrical shell. Results obtained by the present element are compared with those available in the literature (exact, experimental, and numerical) for simple support and cantilever boundary conditions. These comparisons show that one can get reasonably accurate results with the present element and good convergence characteristics.

Creep buckling of cylindrical panels under multiaxial loading

This work presents an analysis of the creep buckling problem of geometrically imperfect circular cylindrical panels under multiaxial loading with simple support boundary conditions. The analysis is based on a nondimensional form of Donnell-type equations for a slightly imperfect cylindrical panel. The elastic constitutive equations for a thin panel are employed. The basic elastic equilibrium equations in the middle surface displacement components are derived through the employment of the principle of virtual displacements. For creep deformations, Odqvist's constitutive equations for steady creep are employed. In the present analysis, creep buckling is characterized by either reaching a maximum deflection limit or experiencing a bifurcational buckling state. Based on the present analysis, a computer program has been developed for the creep buckling analysis of cylindrical panels. The panel ends are assumed to be simply supported. The applied loading is assumed to be axial compression and lateral pressure. Numerical results are presented for imperfect isotropic panels under both uniaxial and multiaxial loading. For the case of uniaxially compressed plates, the present results are generally in good agreement with previous experimental and analytical results. These results suggest that each of the level of the axial compressive load, the level of the lateral pressure, the amplitude and direction of the initial imperfection, and the panel curvature greatly affect the creep buckling times of cylindrical panels. The results presented indicate that avoiding negative (toward the centre of curvature) initial imperfections would help to suppress the creep buckling process for cylindrical panels under multiaxial loading conditions. It would also help to eliminate the possibility of a bifurcational buckling state to exist. This can be achieved by intentionally introducing a small positive initial imperfection in the form of a simple out of roundness.

Nonlinear response and sonic fatigue of National Aerospace Space Plane surface panels

T is becoming increasingly apparent that the development of supersonic/hypersonic vehicles such as the NASP cannot become a reality until the fatigue problem under severe aerodynamic, acoustic, and thermal environment is solved. Furthermore, with an increasing use of light-weight materials such as fiber-reinforced composites, metal matrix composites, and various intermetallic compounds, there is an urgent need for improved analytical methods for nonlinear response and sonic fatigue predictions. The proposed NASP concept involves operations at subsonic, supersonic, and hypersonic speeds. Thus, aerodynamic, acoustic, and thermal loadings should be considered for various phases of flight. In the early stages of the takeoff, the aft surface thermal protection systems will be in the near field of the engine exhaust noise. As the speed of the vehicle increases, the effect of engine exhaust noise (except in the near vicinity of exhaust nozzles) will decrease, and at Mach 1 and higher speeds the acoustic loads are expected to become negligible. However, at supersonic and hypersonic speeds, the fluctuating surface pressures due to convecting turbulent boundary layer will become significant. In addition, local impinging shocks on the structural surface could induce severe dynamic loads. Supersonic/hypersonic vehicles will be subjected to severe surface temperatures exceeding SCOOT.1'4 Structural surface components and control devices can be expected to behave in a highly nonlinear fashion with respect to both geometry and materials.4^8 No reliable analytical procedures have yet been developed which could handle these complexities and give meaningful results that are useful for the design of these vehicles. The thermal surface protection systems of aircraft structures are usually constructed from discretely stiffened panels and stiffened shells. A typical discretely stiffened panel is shown in Fig. 1. Due to anticipated high-surface temperatures of advanced supersonic/hypersonic vehicles, multi-wall and/ or multilayer constructions might need to be utilized for the design of thermal surfaces. High-cycle fatigue failures have occurred in discretely stiffened surface panels with the majority of cracks appearing in the near vicinity of the stiffening element.5'13 Thus, proper dynamic interaction between the panel and various stiffening elements should be taken into account when calculating the nonlinear response of the surface panels. A single bay panel with clamped or simple support boundary conditions might provide reasonable estimates on deformations,14'23 but the predictions of principal stresses of a discretely stiffened panel by a single panel model could be unrealistic. Since fatigue damage is controlled by local stress conditions, accurate stress predictions are essential for fatigue life estimates. A time domain Monte Carlo-type approach has been successfully applied to a variety of problems of a linear and nonlinear nature with complex random inputs.24'33 Utilizing the nonlinear time domain stress response solutions and fatigue information from constant amplitude coupon tests, preliminary fatigue damage models have been constructed. The adverse thermal conditions could result in degradation of strength, stiffness, and fatigue life. In addition, thermal inplane loads could induce buckling and snap through type vibrations of surface panels. The time domain procedures could account for these effects in the general formulation of the total response problem. General Features of Time-Domain Method For structural dynamic problems of nonlinear/probabilistic nature where close form or effective approximate solutions are not possible, the time domain Monte Carlo method could provide a feasible approach to construct practical solutions. The recent advent of high-speed digital computers has made this procedure a useful and effective method. To illustrate the basic features of the time domain approach, consider a generalized form of a second-order nonlinear equation

Dynamics of skin-stringer panels using modified wave methods

The response of a damped, periodic, simply supported skin-stringer structure is studied by two variants of the decaying wave method. The proposed method is based on a combination of a wave propagation method with transfer matrices. One variant of the decaying wave method is based on an exact solution transfer matrix; the other is based on a finite element formulation. It is shown that both variants give response curves that compare very well with previous results for the undamped structure, which illustrates both the accuracy and the numerical stability of the decaying wave method variants. Numerical results, including deflections at various points at various frequencies as well as mode shapes, are presented, and comparisons are offered. The numerical results demonstrate the accuracy and reliability of both variants of the decaying wave method, and this is because advantages of three methods are combined in each variant. admittedly difficult to model, but much of the fuselage's dynamic behavior can be captured by a much simpler struc- ture: a flat, periodic, single row, simply supported skin- stringer structure of finite length. This one row structure, seen in Fig. 1, is designed to emulate somewhat a long circumferen- tial strip of the fuselage structure between two supporting circumferential beams, whose effects are somewhat simulated by a simple support boundary condition. The assumptions of a flat and periodic structure, while designed to make the calculation easier, are not completely unreasonable. Thus, the simpler structure should contain much of the same dynamic behavior as sections of fuselage skin would. The single row simply supported skin-stringer structure has been analyzed by many different researchers using several different methods. Lin1 found the dynamic behavior of this structure analytically, using the bending and torsional charac- teristics of the stringers. His analysis gives some important results, such as the existence and location of frequency filter bands in the infinite structure, but it does not provide a simple way to get complete response curves for the finite or infinite structure. Further, were the boundary conditions at the top or bottom of the structure to change, this type of analysis be- comes extremely difficult. Thus, it became necessary to look at this problem using a numerical method. The first choice of numerical method for many engineers is the finite element method, and finite element codes are com- monplace throughout industry. With regard to skin-stringer structures, Mei2 obtained the finite element matrices for a thin-walled open cross-sectional stringer, and he takes into account coupled bending and torsional vibrations that com- monly occur in this type of beam. Bogner et al. 3 derived a panel element using Hermite cubic polynomials for displace- ment patterns, which ensure continuity of displacement and slope at the element edges. The combination of these element types has led to finite element analysis of skin-stringer struc-

Acoustic propagation through baffles with rectangular compliant tubes

Compliant tube baffles have been designed as effective barriers for acoustic propagation underwater over wide bands of frequencies. Acoustic energy reflection occurs due to compliant cross-sectional resonances in the tubes that exhibit significant volume deformation. Additional bandwidth is achieved with multiple layers of compliant tubes tuned to different frequencies. The design challenge is to provide sufficient baffle insertion loss levels and bandwidth while limiting adverse transmission peaks associated with interactions between noncompliant tube resonances and, tube and fluid layer system resonances. This paper discusses the acoustic performance of baffles with compliant tubes of rectangular cross section as opposed to the conventional designs with elliptical cross section. The tubes consist of flat face plates held apart along their edges by spacers. The knifelike edge support constitutes a simple support boundary condition for bending deformation of the face plate. This compliant tube configuration allows added design flexibility and degrees of freedom in minimizing the adverse effects of transmission peaks in a baffle’s insertion loss. Unequal face plate thicknesses and varied mass of the edge spacer have been effectively utilized in this regard.

Buckling of Circular Cylindrical Shells Using a Displacement Function

The buckling problem of circular cylindrical shells under axial compression, external pressure, and torsion is investigated using a displacement function φ. A governing differential equation for the stability of thin cylindrical shells under combined loading of axial compression, external pressure, and torsion is derived. A method for the solutions of this equation is also presented. The advantage in using the present equation over the customary three differential equations for displacements is that only one trial solution is needed in solving the buckling problems as shown in the paper. Four possible combinations of boundary conditions for a simply supported edge are treated. The case of a cylinder under axial compression is carried out in detail. For two types of simple supported boundary conditions, SS1 and SS2, the minimum critical axial buckling stress is found to be 43.5 percent of the well-known classical value Eh/R3(1−ν2) against the 50 percent of the classical value presently known.

Analysis of unbalanced angle-ply rectangular plates

Unbalanced angle-ply laminated rectangular plates under transverse normal loading are analyzed. By expanding the load and the displacements in single Fourier series the governing partial differential equations are reduced to ordinary differential equations for simple support boundary conditions on two opposite edges and arbitrary boundary conditions on the other two edges. The closed form solution for the ordinary differential equations is obtained for 45° angle-ply laminates. Several numerical examples are solved and their results are presented in graphs. The degree of coupling between stretching and bending depends on the transverse loading, the edge boundary conditions and the aspect ratio of the plate, as well as on the degree of anisotropy of the layers, the angle of orientation and the number of plies in the composite. The type of in-plane boundary condition (movable or immovable) has a significant effect on bending for simply supported plates but not for clamped plates. The Fourier series method discussed in this paper can be used with the method of superposition to solve problems having other edge conditions, as well as transverse loading conditions.