Unsymmetrical Boundary Conditions Research Articles

On the Coupled Transient Hygrothermal Analysis in the Porous Cylindrical Panels

In this paper, transient coupled heat and moisture transfer in the cylinders and cylindrical panels of the porous medium are analyzed with three-dimensional finite element methods. Both Dofour and Soret effects are considered in the formulation, and the FEM-based weak forms of the equations are derived using Galerkin method. In the full cylinder cases, both symmetrical and unsymmetrical boundary conditions for heat and moisture are applied to cover three-dimensional and axisymmetric (two dimensional) analysis cases. The obtained systems of time-dependent differential equations are solved by Runge–Kutta method. The main objective of the present study is the analysis of fully coupled diffusion of heat and moisture in cylindrical coordinates using three-dimensional finite element methods. Also, this study addressed the ability of Runge–Kutta method to solve the coupled sets of differential equations. The results were compared with some analytical solutions in the literature, and a very good agreement was observed. The present formulation can be used for transient analysis of any cylindrical geometry with arbitrary geometrical dimensions and under desired boundary conditions.

Closed-form approximate solution for linear buckling of Mindlin plates with SRSR-boundary conditions

The present work deals with the buckling analysis of rectangular Mindlin plates consisting of laminated composites with symmetrical, balanced lay-up or isotropic materials. Along the longitudinal edges the plate is rotationally restrained by springs. The transversal edges are simply supported. In agreement with common notation, the boundary conditions are abbreviated as follows: simply supported (S), rotationally restrained (R), and fully clamped (C). As loading situation axial compression is considered. Aiming at high computational efficiency the problem is solved by the Rayleigh-Ritz-method. Since well suited shape functions for deflection and rotations with very few variables are used, closed-form approximate solutions for the buckling load can be obtained. For verification exact transcendental solutions and/or high fidelity finite element analyses are employed. Additionally, results are compared to those of existing closed-form approximate solutions.Apart from the special case of simply supported longitudinal edges where all methods yield exact or nearly exact results, the present method shows clear advantages: 1.Due to the type of shape functions it is able deal with unsymmetrical boundary conditions. 2.For the case of both longitudinal edges fully clamped where all closed-form approximate solutions show the greatest deviations the present method is significantly more accurate.

Development of a bi-directional standing wave linear piezoelectric actuator with four driving feet

A bi-directional standing wave linear piezoelectric ultrasonic actuator with four driving feet is proposed in this work. Two sandwich type transducers operated in longitudinal-bending hybrid modes are set parallelly. The working mode of the transducer is not simple hybrid vibrations of a longitudinal one and a bending one, but a special coupling vibration mode contained both longitudinal and bending components. Two transducers with the same structure and unsymmetrical boundary conditions are set parallelly to accomplish the bi-directional driving: the first transducer can push the runner forward, while the other one produces the backward driving. In the experiments, two voltages with different amplitudes are applied on the two transducers, respectively: the one with higher voltage serves as the actuator, whereas the other one applied with lower voltage is used to reduce the frictional force. The prototype achieves maximum no-load speed and thrust force of 244 mm/s and 9.8 N. This work gives a new idea for the construction of standing wave piezoelectric ultrasonic actuator with bi-directional driving ability.

Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

Governing equations of a compressed rotating rod with clamped–elastically clamped (hinged with a torsional spring) boundary conditions is derived. It is shown that the multiplicity of an eigenvalue of this system can be at most equal to two. The optimality conditions, via Pontryagin’s maximum principle, are derived in the case of bimodal optimization. When these conditions are used the problem of determining the optimal cross-sectional area function is reduced to the solution of a nonlinear boundary value problem. The problem treated here generalizes our earlier results presented in Atanackovic, 1997, Stability Theory of Elastic Rods, World Scientific, River Edge, NJ. The optimal shape of a rod is determined by numerical integration for several values of parameters.

Variation of Added Mass of the Centrifugal Impeller Due to the Direction of Vibration.

Values of added mass of a vibrating centrifugal impeller were analyzed by using unsteady potential theory. Impeller operating in diffusers with one and two vanes was modeled and the fluid forces were calculated for vibrations in the x and y directions covering various frequencies. Results of fluid forces were converted to mass, damping, stiffness matrices by the Fourier integral and the method of least squares. Mean values of added mass of the impeller in the diffusers which were obtained from mass matrix were larger than those in vaneless diffuser and different for vibrations of x and y directions. It was clarified that the prediction of radial force as the sum the circulation induced lift on the orbiting impeller, and the centrifugal forces on the apparent mass of the impeller was available for the unsymmetrical boundary conditions.

Dynamic Stability of Spinning Beams with an Unsymmetrical Cross-Section and Distinct Boundary Conditions Subjected to Time-Dependent Spin Speed*

ABSTRACT The equations of motion of a spinning beam with a rectangular cross-section are formulated using the Euler beam theory and the assumed mode method. The spin speed consists of steady-state and time-dependent portions. The resulting equations of motion are not in standard Mathieu-Hill's equation form, due to the time-dependent coefficient of the gyroscopic term. These equations of motion are then reduced to a set of first-order differential equations with time-dependent coefficients. The regions of instability due to parametric excitations are determined using the multiple scale method. Numerical results are presented for a spinning beam subjected to combinations of end conditions in the two orthogonal planes of transverse vibration. Widths of the unstable regions are found to decrease as the aspect ratio of the rectangular cross-section approaches unity for spinning beams with an identical set of end conditions in both transverse vibration planes.These regions vanish when the aspect ratio becomes one. However, this is not the case when the beam is subjected to distinct end conditions in the two planes. For a given aspect ratio, interesting changes in the unstable regions are observed as the spin speed varies within, as well as across, critical spin speed zones.

Postbuckling behavior of rectangular moderately thick plates and sandwich plates

Postbuckling behavior is investigated for Rectangular Reissner's moderately thick plates and sandwich plates. The fundamental equations and boundary conditions are expressed in unified dimensionless form for rectangular moderately thick plates and sandwich plates. Exact solutions of series form with a number of different boundary conditions, especially with unsymmetrical boundary conditions, are obtained by developing a new technique of mixed Fourier series in nonlinear analysis. The nonlinear partial differential equations are reduced to an infinite set of simultaneous nonlinear algebraic equations, which are truncated by iteration in numerical computations.

Nonlinear flexural vibration of rectangular moderately thick plates and sandwich plates

Nonlinear flexural vibration is investigated for rectangular Reissner moderately thick plates and sandwich plates. The fundamental equations and boundary conditions are expressed in unified dimensionless form for rectangular moderately thick plates and sandwich plates. Highly accurate solutions of series form with many different movable and immovable boundary conditions, especially with unsymmetrical boundary conditions, are obtained by means of the method of harmonic balance and by developing a new technique of mixed Fourier series in nonlinear analysis. The nonlinear partial differential equations are reduced to an infinite set of simultaneous nonlinear algebraic equations, which are truncated in numerical computations. Solutions of the nonlinear fundamental frequency of rectangular plates are obtained by iteration. The multimode approach includes not only the influences of transverse shearing deformation and rotatory inertia, but also the coupling effect of vibrating modes on the nonlinear fundamental frequency. The present solutions are satisfactory in comparison with other available results.

Non-isothermal Crystallization of a Polymer Film on Chilling Rolls

Abstract The problem of heat exchange and non-isothermal crystallization of a polymer film (leaving an extruder) on the surface of chilled rolls is discussed. Factors such as enthalpy of crystallization, unsymmetric boundary conditions and dependences of thermal properties of a polymer on temperature and degree of crystallinity are taken into account. The statement of the problem is based, first, on joint analysis of non-stationary energy balance equation and kinetics of non-isothermal crystallization and, second, on inclusion of dependences of thermal properties of a polymer on temperature and crystallinity into a mathematical model under discussion. The influence of technological and engineering factors on the quality of film is discussed.

Boundary element crack closure calculation of three-dimensional stress intensity factors

A general method for boundary element-crack closure integral calculation of three-dimensional stress intensity factors is presented. An equation for the strain energy release rate in terms of products of nodal values of tractions and displacements is obtained. Embedded and surface cracks of modes I, II, and III are analyzed using the proposed method. The multidomain boundary element technique is introduced so that the crack surface geometry is correctly modeled and the unsymmetrical boundary conditions for mode's II and III crack analysis are handled conveniently. Conventional quadrilateral elements are sufficient for this method and the selection of the size of the crack front elements is independent of the crack mode and geometry. For all of the examples demonstrated in this paper, 54 boundary elements are used, and the most suitable ratio of the width of the crack front elements to the crack depth is 1/10 and the calculation error is kept within ±1.5 percent. Compared to existing analytical and finite element solutions the boundary element-crack closure integral method is very efficient and accurate and it can be easily applied to general three-dimensional crack problems.

A unified approach to free vibration of locally damaged beams having various homogeneous boundary conditions

The problem of free vibration of uniform beams containing a local material damage has been studied with the aim of arriving at accurate closed form analytical expressions for the natural frequency for various homogeneous boundary conditions. Elementary beam theory is used, with the local material damage modelled as an effective reduction in Young's modulus, and the exact solution of the transcendental equations for the frequency parameter is obtained for various symmetric and unsymmetric boundary conditions in terms of the identified damage parameters. The numerical results for the natural frequency show that it is possible to arrive at a simple polynomial type of representation which is the same for various boundary conditions and mode numbers. The results also show that the nature of changes in frequency with respect to the damage location is of the same type as the local curvature in the undamaged beams. Finally, the study demonstrates that the analytical expressions derived for representing the damage dependence can also be used, with slight modification, in situations where there is local stiffening.

Non‐linear analysis of symmetric structures with unsymmetric boundary conditions

Two efficient computational procedures are presented for generating the global approximation vectors used in conjunction with the reduction methods for the large‐deflection non‐linear analysis of symmetric structures with unsymmetric boundary conditions. Both procedures are based on restructuring the governing equations for each of the unsymmetric global approximation vectors to delineate the different contributions to the symmetric and antisymmetric components of this vector. In the first procedure the unsymmetric global approximation vectors are approximated by linear combinations of symmetric and antisymmetric modes, which are generated by using the finite element method. The amplitudes of these modes are computed by using the classical Rayleigh‐Ritz technique. The second procedure is based on using a preconditioned conjugate gradient (PCG) technique for generating the global approximation vectors, and selecting the preconditioning matrix to be the matrix associated with the symmetric response. In both procedures the size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions. The similarities between the two procedures are identified, and their effectiveness is demonstrated by means of two numerical examples of large‐deflection, non‐linear static problems of shells.

Spatial and temporal temperature distribution in the healthy and locally diseased wall of the heart

The radial and circumferential temperatures in the wall of left ventricle (LV) of the heart are obtained by solving the bio-heat energy equation for cases of regional death of the muscular tissue (local infarction) or local deprivation of oxygen supply (hypoxia) to the muscle. The model is an extension of our one-dimensional solution for the radial instantaneous temperatures in the wall, based on our mechanical model of the LV function. A similar approach yields the effects of the nonsymmetric boundary conditions imposed by the right ventricle (RV) on the temperature distribution in the LV wall. The results indicate very pronounced local temperature gradients due to local myocardial dysfunctions or unsymmetric boundary conditions on the wall.

Hypervirial analysis of enclosed quantum mechanical systems. III. Unsymmetrical boundary conditions

AbstractNew theoretical developments based upon hypervirial analysis of enclosed systems are given. Systems which obey unsymmetrical boundary conditions are analyzed. Several results which were given by the authors in two previous communications are generalized.

Catalyst particle effectiveness with unsymmetrical boundary conditions

In trickle-bed reactors part of the surface of the catalyst particles may be covered by liquid and part by gas. Under such conditions conventional expressions for effectiveness factors are not applicable because the surface concentration is not uniform. In this paper more general equations for both particle and overall effectiveness factors are derived. The first development is for slab geometry where the two plane surfaces of equal areas have different reactant concentrations. This is followed by treatment of a cubic particle where one or more faces are covered by liquid. Approximate methods are then presented for evaluating the effectiveness factor for any fractional liquid coverage, f. Comparison of the results at permissible values of f indicates that a simple weighting-factor method gives good results. The method is illustrated by comparing predicted and experimental reaction rates for a case where experimental results are available—the hydrogenation of α-methyl styrene.

Spin-wave resonance spectrum of a ferromagnetic boundary layer with unsymmetrical boundary conditions

A study is presented on the effects of unsymmetrical (d1 =−d2) attachment of surface spins and regards the excitation spectrum for a conducting ferromagnetic layer with spin-system damping. The conductivity causes nonuniformity in the high-frequency field in the film, which alters the shape of the spin-wave spectrum.

Scale of the convection cell structure in a horizontal layer of liquid with unsymmetrical boundary conditions for heat transfer

An interpretation is given of some experimental results on the effects of initial conditions on the cellular structure of convective motion.

Comments on the effect of axially varying and unsymmetrical boundary conditions on heat transfer with turbulent flow between parallel plates: A.P. Hatton and A. Quarmby, Int. J. Heat Mass Transfer6, 903–914 (1963)

Comments on the effect of axially varying and unsymmetrical boundary conditions on heat transfer with turbulent flow between parallel plates: A.P. Hatton and A. Quarmby, Int. J. Heat Mass Transfer6, 903–914 (1963)

Heat transfer with unsymmetrical thermal boundary conditions

Heat transfer with unsymmetrical thermal boundary conditions

The effect of axially varying and unsymmetrical boundary conditions on heat transfer with turbulent flow between parallel plates

The solution is presented to the heat-transfer problem with turbulent flow between parallel plates with heating on one side only. The configuration has some practical significance when considered as a limiting case of the annulus with a heated core. The velocity and eddy diffusivity variations due to Deissler are used but are modified in the central region of the passage by assuming a constant eddy diffusivity. In solving the energy equation the eddy diffusivities for heat and momentum are taken equal. The eigenvalues and functions necessary to calculate the variation of Nusselt number with distance along the passages are tabulated for three Reynolds and three Prandtl numbers. From the two basic solutions of uniform temperature and uniform heat flux the analysis is extended to certain axial variations which are of practical interest and finally, by using superposition, the result is obtained for unequal uniform heat fluxes on each side of the passage.