Wave based method for flexural vibration of thin plate with general elastically restrained edges

Application of the R-function method to nonlinear vibrations of thin plates of arbitrary shape

The boundary conditions of basic load support structures formed by beam elements are neither purely simply supported nor clamped that can be termed as conventional. It has certain finite rotational and translational stiffness representing an intermediate state (a welded end/riveted structures to name a few). The conventional methods of separation of variables for beam vibration analysis have not found substantial applications for boundary conditions which are not conventional. The paper attempts to bring out the effect of variation of the end supports, in terms of variation in translational and rotational stiffness, on the vibration characteristics of a beam element in terms of frequency and mode shapes. Fourier sine series approximation has been used for theoretical evaluation. Numerical assessment has been undertaken on an I-beam. Variable stiffness has been accounted for by using variable elastic end conditions and results on frequencies and mode shapes of the I-beam have been presented.

The compressive behaviors of delaminated orthotropic plates with three simply supported and free edges, which are simplified models of stiffener plates, are studied experimentally and analytically. In the analysis, Rayleigh-Ritz approximation method is adopt,ed to obtain an estimate of buckling load. The compressive buckling load reduction is found to be significant and almost proportional to the delamination width. Local delamination buckling is found to occur when the delaminated portion is thin and large. The analytical results agree well with the experimental ones. Key Word : Composite laminate, Free edge, Interlaminar delamination, Compressive buckling. Fiber reinforced coinposite 1aminat.e~ attract intensive attentions as a prommising candidate of a future material not only in aerospace engineering but also various engineering fields owing to its remarkable properties such as high strength and high stiffness ratio, good fatigue resistance, design-ability, etc. Interlaminar delamination, which is suspected to occur owing to impact load, fatigue, etc., IS one of typical damage patterns of laminated structures. Its introduction causes sig~iificant reduct,ion of compressive strength of the plates. Therefore, a lot of works have been performed to disclose the effects of the delaminat,ion(s), particularly on one dimensional model1-lo) and have shown possibility of significant loss of compressive buckling and postbuckling properties. Works on the effect of delaminations on the compressive behaviors of two dimensional plate are limited only on special The delamination is also thought to appear at a free edge of the stiffener plate of stiffened panels and to reduce significantly its compressive strengt,h. In the present paper, the stiffener portion is modeled as a plate with three simply-supported and one free edges. The eflect of a delamination on compressive buckling properties of the orthotropic plate is analytically and experimentally studied. 'Associate Professor, Mechanical Engineering, Member AIAA 'Member AIAA 2 Theory A plat,e with three simply-supported and one free edges showu ill Fig.1 is considered. It is assumed to be uniform in thickness direction. that is, no bending-st,retchillg coupling is considered even in the delaminated area. A delamination is assunled t o locate along the free edge. The boundary conditions are where the values u , v, w denote, respectively, displacem e ~ k s ill c, y dricl : directions. The values a and b are the width and length of the plate. The term rZy is a Co~vright @ 1993 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 485 shear stress in x y plane. The values Q,, M, and My are a transverse shear force and bending moments in x and y directions. The value EO is the normalized displacement of the loading edge. When the transverse shear deformation is neglected, the conlpl~sblve bucklillg bllebb u,, of the prcvnt plate wlthout a delam~nat~on 1s where h is the plate thickness, X = a lb is aspect ratio of the panel and the coefficients DZ2, D66 are the components of bending stiffness matrix of the laminate. The component Dl l does not appear in the expression of the buckling stress of Eq.5. The Kirchhoff's hypothesis is assumed t o hold even in the neighborhood of the delamination tiofit. The continuity condition of the displacements nt Ll~e edge of the delamination ;L. = a c can be written simply ~ i t h the displacements of the midplanes of each delaminated and undelaminated portions. U I + h k ~ w o ~ ~ = uo V I + h k I w O l y = v0 (7)

A new method of experimental buckling of circular cylindrical roof shell models with free edges subjected to external pressure

Flexural gravity wave scattering by two semi-infinite non-identical ice sheets, which are separated by a clean water surface, is investigated in the presence of a step bottom topography. The problem is investigated in the cases of (i) intermediate water depth and (ii) shallow water. The effect of two edge conditions, (i) simply supported edge and (ii) free edge, on wave scattering is also analyzed. Employing linear velocity potential theory, the problem is studied in the frequency domain. The physical phenomenon is modeled as a boundary value problem having the Laplace equation as the governing equation, and it is solved using the eigenfunction expansion method. Considering the bottom topography and upper boundary, the fluid domain is divided into three regions, and in each region, velocity potential is expressed in terms of infinite Fourier series. Velocity and pressure are matched at the intermediate surface of two regions, and a system of algebraic equations with unknown coefficients is obtained. The complete solution of the present problem is recognized by solving the system of equations numerically. The energy relation is derived using Green's theorem. The reflection and transmission coefficients are computed and compared with the energy relation to check the accuracy of the present method. For different parameters (depth ratio, clean waterway, and different ice properties), reflection coefficient and transmission coefficient are computed and presented as a function of angular frequency. The value of the reflection coefficient ranges from zero to unity, whereas at certain frequencies, the transmission coefficient attends value more than unity.

Determination of resonance frequencies and vibration modes of mechanical structures is one of the most important tasks in the product design procedure. The main goal of this paper is to describe a pioneering application of the solution structure method (SSM) to 2D structural natural vibration analysis problems and investigate the numerical properties of the method. SSM is a meshfree method which enables construction of the solutions to the engineering problems that satisfy exactly all prescribed boundary conditions. This method is capable of using spatial meshes that do not conform to the shape of a geometric model. Instead of using the grid nodes to enforce boundary conditions, it employs distance fields to the geometric boundaries and combines them with the basis functions and prescribed boundary conditions at run time. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure. In the paper we will explain the key points of the SSM as well as investigate the accuracy and convergence of the proposed approach by comparing our results with the ones obtained using analytical methods or traditional finite element analysis. Despite in this paper we are dealing with 2D in-plane vibrations, the proposed approach has a straightforward generalization to model vibrations of 3D structures.

A general analytical method, referred to as the Fourier spectral element method, is presented for the dynamic analysis of plate structures consisting of any number of arbitrarily oriented rectangular plates. The compatibility conditions between any two adjacent plates are generally described in terms of three-dimensional elastic couplers with both translational and rotational stiffnesses. More importantly, all plates involved can be arbitrarily restrained along any edges in contrast to the commonly imposed condition: each plate has to be simply supported along, at least, one pair of parallel edges. Thus, plate structures here are not limited to Levy-type plates as typically assumed in other techniques. The flexural and in-plane displacement fields on each plate are analytically expressed as accelerated Fourier series expansions and the expansion coefficients are considered as the generalized coordinates to be determined using the familiar Rayleigh–Ritz technique. The accuracy and reliability of the present method are validated by both finite element analysis (FEA) and experimental data for box structures under various boundary conditions.

Local FEM Analysis of Composite Beams and Plates: Free-Edge effect and Incompatible Kinematics Coupling

The problem of cantilever deflection is a classic worked example in most elasticity textbooks. Due to the inability for the displacement field to completely satisfy the boundary condition of zero displacement at the entire built-in end from the top surface to the bottom surface, the best solution so far is attained from weak boundary conditions. This paper proposes a modified problem in which the boundary is partially built-in, i.e. the cantilever is clamped on the top and bottom but free on the other two sides. Therefore, the cantilever’s displacement is prohibited only at the top and bottom surfaces at the partially built-in end. By implementing zero displacement therein, the displacement solution can be obtained based on strong boundary conditions. Furthermore, the condition of zero horizontal displacement at the mid-point of the partially built-in end is also achieved through the prescribed boundary conditions. Results also reveal that the displacement at the partially built-in end can be reduced by selecting materials with negative Poisson’s ratio. As an alternative to being a worked example, the proposed problem can be used as an assessment question.

Admissible boundary conditions and solutions to internally pressurized thin arbitrarily laminated cylindrical shell boundary-value problems

Hydrodynamic and semikinetic treatments of plasma flow along closed geomagnetic field lines are compared. The hydrodynamic treatment is based on a simplified 16‐moment set of transport equations as the equations for the heat flows are not solved; the hear flows are treated heuristically. The semikinetic treatment is based on a particle code. The comparison deals with the distributions of the plasma density, flow velocity, and parallel and perpendicular temperatures as obtained from the two treatments during the various stages of the flow subject to certain assumed boundary conditions. In the kinetic treatment, the appropriate boundary condition is the prescription of the velocity distribution functions for the particles entering the flux tubes at the ionospheric boundaries; those particles leaving the system are determined by the processes occurring in the flux tube. The prescribed distributions are half‐Maxwellian with temperature To and density no. In the hydrodynamic model, the prescribed boundary conditions are placed on density (no), flow velocity (Vo), and temperature (To). We found that results from the hydrodynamic treatment critically depend on Vo; for early stages of the flow this treatment yields results in good agreement with those from the kinetic treatment, when Vo = (kTo/2πm)1/2, which is the average velocity of particles moving in a given direction for a Maxwellian distribution. During this early stage, the flows developing from the conjugate ionospheres show some distinct transitions. For the first hour or so, the flows are highly supersonic and penetrate deep into the opposite hemispheres, and both hydrodynamics and kinetic treatments yield almost similar features. It is found that during this period heat flow effects are negligibly small. When a flow penetrates deep into the opposite hemisphere, the kinetic treatment predicts reflection and setting up of counterstreaming. In contrast, the hydrodynamic treatment yields a shock in the flow. The reasons for this difference in the two treatments is discussed, showing that in view of the relatively warm ions, the coupling of ion beams and the consequent shock formation in the off‐equatorial region are not likely due to the enhancements in the beam temperatures. The counterstreaming in the kinetic treatment and the shock in the hydrodynamic treatment first advance upward to the equator and then downward to the ionospheric boundary from where the flow originated. The transit time for this advancement is found to be about 1 hour for the perspective models. After 2 hours or so, both models predict that the flows from the ionospheric boundaries are generally subsonic with respect to the local ion‐sound speed. At late stages of the flow, when a substantial fraction of ions entering the flux tube begin to return back in the kinetic treatment, the hydrodynamic treatment with the boundary condition Vo = (kTo/2πm)1/2 yields an overrefilling, and the choice of Vo becomes uncertain.

A broad choice of numerical schemes and methods currently exists for blood flow simulations. The results rely critically on the prescription of boundary conditions. The outflow boundary condition for a one-dimensional (1D) flow solver is usually prescribed via a Windkessel or lumped parameter model. The weakness of such an approach is the determination of the parameters. In the present work, we use an alternative approach, that is, a reflection coefficient (RC), to lumped parameter models for distal boundary conditions. With such a RC, the number of parameters required is reduced to one. We derive the theoretical foundation for the RC. Specifically, we couple a transmission line theory for peripheral resistance with a 1D arterial flow solver. We apply this method to a healthy and a stenosed virtual aorta, and show this method can reproduce some subtle features in arterial pressure propagation, such as the steepened pressure waveform and the reflection from the stenosed site. In summary, the RC parameter has strong physical implications in the theory of wave propagation and may be used in flow simulations where reflections need to be explicitly modeled. NOVELTY STATEMENT: A novel coupled one-dimensional-transimission line model has been developed in this work with detailed implementations. Only one outflow boundary condition, that is, the refection coefficient is required in the model. Reflections for a pulse wave from aortic terminals as well as from a stenotic site are numerically simulated.

Finite element modeling of cold-formed steel channels with solid and slotted webs in shear

A general formulation of boundary conditions for semiconductor-metal contacts follows from a phenomenological procedure sketched here. The resulting boundary conditions, which incorporate only physically well-defined parameters, are used to study the classical unipolar drift-diffusion model for the Gunn effect. The analysis of its stationary solutions reveals the presence of bistability and hysteresis for a certain range of contact parameters. Several types of Gunn effect are predicted to occur in the model, when no stable stationary solution exists, depending on the value of the parameters of the injecting contact appearing in the boundary condition. In this way, the critical role played by contacts in the Gunn effect is clearly established.

A previously established model of steady one-dimensional two-ion electrodiffusion across a liquid junction is reconsidered. It involves three coupled first-order nonlinear ordinary differential equations and has the second-order Painlevé II equation at its core. Solutions are now grouped by Bäcklund transformations into infinite sequences, partially labelled by two Bäcklund invariants. Each sequence is characterized by evenly-spaced quantized fluxes of the two ionic species, and hence evenly-spaced quantization of the electric current density. Finite subsequences of exact solutions are identified, with positive ionic concentrations and quantized fluxes, starting from a solution with zero electric field found by Planck, and suggesting an interpretation as a ground state plus excited states of the system. Positivity of ionic concentrations is established whenever Planck’s charge-neutral boundary conditions apply. Exact solutions are obtained for the electric field and ionic concentrations in well-stirred reservoirs outside each face of the junction, enabling the formulation of more realistic boundary conditions. In an approximate form, these lead to radiation boundary conditions for Painlevé II. Illustrative numerical solutions are presented, and the problem of establishing compatibility of boundary conditions with the structure of flux-quantizing sequences is discussed.