Static Equilibrium Configuration Research Articles

Domain-wall resonance in exchange-coupled magnetic films

Exchange coupling between magnetic films in multilayer geometries can strongly influence magnetization behavior and spin-wave energies by correlating the motion of spins in one film with the motion of spins in adjacent films. In a similar fashion correlations can be expected between domain walls existing in neighboring exchange coupled films. In this paper we show that the motion of domain walls in neighboring films can depend strongly on interfilm exchange coupling. A static equilibrium configuration exists for the domain walls, and small amplitude oscillations about equilibrium can result in domain-wall resonances that involve interfilm exchange energies. Frequencies for optic- and acoustic-type domain-wall resonances are calculated and effects of a small static applied magnetic field are examined, revealing distinctly different behaviors for the acoustic and optic domain-wall resonances. The possibility for sensitive measurements of the local coupling by studying domain-wall resonance is discussed. Resonances with parallel and antiparallel coupling are considered and the response to a small in-plane driving field is calculated.

Head-Disk Interface Performance With a Starved Liquid Bearing—Analytical Solution to Reynolds Equation in the Near-Contact Regime

Abstract Further increase in magnetic recording density requires a reduction in slider flying height. The current study employs the short-bearing approximation to determine analytically the static equilibrium configuration of a slider supported by a starved liquid bearing that operates between ideally smooth surfaces. The solution incorporates rheological behavior based on previously acquired data. The accuracy of the short-bearing approximation is assessed by determining how well the resulting solution satisfies the Reynolds equation. The analysis suggests a means of designing a slider to achieve head/medium spacings in the neighborhood of 20 nm.

Free-Vibration Analysis of Compressed Clamped Circular Plates

Small free vibrations of a thin elastic clamped circular plate under a compressive thrust applied at its edges is studied near the nonlinear static deformation state. Based on the solution of von Karman's plate equations for the static state of circular plates, the dynamic-eigenvalue or free-vibration problem is reduced to one of solving three algebraic equations leading to an analytic solution. This is achieved by introducing a coefficient of transformation related to the eigenfunctions and presenting the algebraic homogeneous equations in terms of the eigenvalues. The natural frequency versus the applied load is derived and plotted for various values of the load. The frequency of the small vibrations vanishes at the limit load. The advantage of the proposed analytic method lies in its ability to predict and describe the stability of the static equilibrium configuration and the postbuckling path.

Theory of vibrating diaphragm-type pressure sensor

Starting with the von Karman equation, the series solution of the problem of nonlinear bending of circular thin plates under the action of uniform pressure and uniform initial edge tension (preloaded) at the periphery is obtained. Based on this solution, small vibrations in the vicinity of the static equilibrium configuration are discussed. Here, a series method is suggested in which the characteristic value problem is reduced to a third-order algebraic characteristic value problem with explicit expression, and thus the characteristic value problem of axisymmetric free vibration is solved precisely

Small Vibrations of Flexible Bars by Using the Finite Element Method with Equivalent Uniform Stiffness and Mass Methodology

The object of this research is to calculate natural frequencies and mode shapes of a small oscillation vibration, superimposed on large static displacement, by using the finite element method in conjunction with the equivalent uniform stiffness and mass method. The equivalent uniform stiffness and mass matrices are determined from the differential equation by using Galerkin's method. The analysis herein is three-fold. Firstly, the large static equilibrium configuration of the deformed flexible beam is established. This is achieved by using the Euler-Bernoulli equation in conjunction with equivalent pseudo-linear and/or equivalent non-linear systems of constant stiffness. This concept and methodology were used quite extensively by the authors to study large deformation characteristics of various flexible beam problems of uniform and variable stiffness and of arbitrary loading conditions (see, for example, the work of Fertis and Afonta [1, 2], and Fertis and Lee [3, 4]). Secondly, the differential equation of motion for small amplitude vibrations from the large static equilibrium configuration is derived. Galerkin's technique is used to approximate the differential equation of motion. Finally, the natural frequencies of vibration and the associated mode shapes are determined by solving the resulting eigenvalue problem. A canned eigensolver from the University of Akron IMSL is used to extract the eigenvalues and eigenvectors. The concept of equivalent pseudo-linear systems of constant stiffness is used extensively in this research, in order to simplify the solution of these complex non-linear problems.

Modeling of mechanisms of Formation of Quasi-Epitaxial Organic Interfaces

ABSTRACTWe introduce a model to calculate the static equilibrium configuration between two monolayers of large, planar organic molecules bonded by van der Waals (vdW) forces. The Model significantly simplifies analysis by replacing the conventional atom-atom potential summation with a single ellipsoidal potential centered in the molecular plane. Our results indicate that recent observations of crystalline “quasi-epitaxial” growth of these planar molecular films in ultrahigh vacuum results from the relatively large intralayer elasticity as compared to the small interlayer stress. Good agreement between calculated and observed structures is achieved using no adjustable parameters. The Model is used to predict molecular structures which are likely to form quasi-epitaxial layers. Quasi-epitaxy is a general feature of large planar molecules of similar shape and size in adlayers and substrates where there is comparatively weak vdW bonding between layers.

Free vibration of variable stiffness flexible bars

The research work here deals with the free undamped vibration analysis of flexible nonprismatic members. During vibration, the beam carries its own weight, as well as other weights that are attached to the member and participate in its vibrational motion. The variation at the mass can be of any arbitrary nature, and its moment of inertia, or stiffness EI, where I is the cross-sectional moment of inertia and E is the Young's modulus, may also vary in an arbitrary manner. Since vibrations of flexible members are taking place with respect to a large static configuration position, it becomes important to locate this position as accurately as possible. The method of the equivalent systems as developed by Fertis and co-workers can be used for this purpose. Therefore, the free vibration analysis here is done in two steps. The first step involves the solution of the Euler-Bernoulli nonlinear differential equation, in conjunction with the method of the equivalent systems, in order to establish the static equilibrium configuration and position. From this position, the small vibrations of the member can be determined by using appropriate differential equations that incorporate the effect of the large static deformation. The results are compared by using more than one method to calculate the frequencies.

Alfven wave transmission through the solar atmosphere

The transmission and reflection coefficients of shear Alfven waves in an open magnetic structure emerging from the Sun are determined. Both two-dimensional slab and axisymmetric static equilibrium configurations are solved numerically on a curvilinear coordinate grid adapted to the magnetic field geometry and to the density stratification. The equilibrium structure obtained numerically extends from the photosphere up to an altitude of 20,000 km, spanning 14 density scale heights and allowing for a magnetic flux tube expansion by a factor 40-400, corresponding to a filling factor of 2.5% to 0.25%. From 20,000 km up to 4 R ⊙ a magnetic monopole model is adopted

Effect of Static Offset on TLP Modeling

A tension leg platform (TLP) is a compliant offshore structure that experiences large horizontal displacements under the action of wind and current loads. The presence of a static offset does affect the dynamic characteristics of the TLP in that the geometry of the structure changes, the tendon loads increase due to increased buoyancy associated with platform setdown, and the individual tendons tend to sag under the action of their own weight. An accurate prediction of the static offset requires that the catenary behavior of the tendons and risers be accounted for in the analysis. Presented in this paper is an effective analytical tool for the analysis of the static equilibrium configuration of a TLP under the action of prescribed wind and current loads acting on the platform and for the determination of the dynamic characteristics of the TLP for small dynamic oscillations about a large static offset configuration. Once the static or dynamic response of a TLP has been determined, localized bending stresses near the supports of tendons and risers can be determined routinely from simple asymptotic formulas developed by the writer.

Nonlinear Dynamic Behavior of the Human Knee Joint—Part I: Postmortem Frequency Domain Analyses

Characteristics results of postmortem experiments on five knee-joint specimens are reported. The experiments were performed to investigate the applicability of a local linearization technique that would make it possible to describe the dynamic behavior of the joint in terms of transfer functions. The results indicate that the stiffness of the bracing wires, attached to muscle tendons to create a static equilibrium position, can be accounted for when determining the stiffness of the joint. Besides the static equilibrium configuration, the magnitude of the dynamic load and the type of dynamic load applied to the joint can be shown to have their influence. As the influence of the dynamic load is significant, it has to be concluded that in essence the knee joint has to be regarded as a nonlinear system, making application of a Local Linearization Technique questionable. However, when the magnitude of the dynamic load is included as an additional measurement parameter, an indication can be obtained about the behavior of the joint and the degree of nonlinearity.

Stability of self-gravitating systems of fermions

We discuss static equilibrium and infinitesimal radial motion of self-gravitating systems consisting of two kinds of fermions with different masses. Calculations shows that static equilibrium configurations occur in a wide domain of values of central mass density. The domains, in which all equilibrium configurations are stable against radial perturbations, are obtained.

The Effect of Tether Damage on Tension Leg Platform Dynamics

The design of a tension leg platform (TLP) provides for tethers of equal stiffness at each corner. Damage to the tethers results in a loss of stiffness and consequently a change in the static equilibrium configurations and dynamic response of the platform and also affects the tether dynamics. Computer algorithms for the linear and nonlinear analysis of a tension leg platform subjected to regular waves have been developed assuming that the platform is modeled as a rigid body. In the linear case the tethers are modeled as one-degree of freedom springs whereas in the nonlinear study the tethers are modeled as a network of beam elements having bending and axial stiffness and also geometric stiffness because of the large displacements assumed. It is assumed that the hydrodynamic forces on the assembled structure equal the sum of the hydrodynamic forces on the component bodies while their interaction is ignored. A parametric study of the platform motion responses and tension variations has been presented for cases involving the loss of tether stiffness. Motion responses of the platform and the tension at each leg have been computed for a regular wave traveling from aft to forward for cases involving intact tethers and with a certain amount of stiffness lost, in one instance at both aft tethers, and in another case at only the aft starboard tether. A regular wave of 10.0-m height and 8.0-s period in 160-m water depth has been used for calculation purposes. The total mean pretension is assumed to remain constant. It is seen that the lateral motions of the platform (such as surge) are not affected by the loss of tether stiffness while the motions in the vertical plane (such as heave and pitch) increase with reduction in stiffness. In all cases the dynamic tension amplitude increases with loss of tether stiffness. For a given configuration, the nonlinear analysis yielded larger dynamic tension amplitudes than those obtained from linear analysis because of the inclusion of bending stiffness in the tethers and also because large displacements were taken into account.

Force Coefficients for Open-Ended Squeeze-Film Dampers Executing Small-Amplitude Motions About an Off-Center Equilibrium Position

Fluid-film forces generated by squeeze-film dampers (SFD) in response to small-amplitude motions about an off-center equilibrium configuration are of considerable importance for stability analyses of rotating machinery employing SFDs with weak or no centering springs. This approach is the basis for calculation of linearized force coefficients obtained by subjecting the journal to small perturbations in velocity and acceleration. The analysis considers the fluid flow in an open-ended SFD due to small-amplitude motions of the journal center about the static equilibrium configuration. Simplified governing equations are derived from the full Navier-Stokes equations by order of magnitude tests. Approximate damping and inertia force coefficients for finite-length SFDs are derived by implementing a correction factor to the long SFD model solution. This finite-length correction is of extreme simplicity and shows excellent agreement with a numerical solution to the differential equations for the pressure including fluid inertia effects. The correction presented tends asymptotically, as the SFD L/D ratio approaches zero, to the existing formulae for the short SFD model.

Dynamical conductivity of soliton lattice and polaron lattice in the continuum model of polyacetylene.

The dynamical conductivity due to phonons around the soliton lattice and around a polaron lattice, and the creation energy of these two structural configurations were calculated in a continuum model of polyacetylene. The crossover from the soliton lattice to the polaronlike distortion proposed by Kivelson and Heeger to explain the abrupt onset of the magnetic susceptibility around 5% doping is not found in a strictly one-dimensional calculation. The harmonic lattice fluctuations around the static equilibrium configurations were obtained and used to calculate and compare the absorption spectra for the various defect configurations.

Cleavage, dislocation emission, and shielding for cracks under general loading

The purpose of this paper is to consider the conditions for the existence of a cleavage crack in a lattice and its response to all types of external loads when shielded by neighboring dislocations. The theory will be constrained to cleavage on one plane and crack branching is not permitted. In the first part of the paper, general relations are derived for the elastic interactions between a cleavage crack and a dislocation and between pairs of dislocations in the presence of the crack. These results are expressed in terms of the local k-field of the crack tip, the shielding contributions from the dislocations, and the complete specification of the forces on the two types of defects. In the second part, criteria for crack cleavage and emission under static conditions are developed for general loading. It is found that for reasons having to do with the basic nature of a crack on the atomic level, that the condition for equilibrium of a cleavage crack involves primarily only the mode I part of the local k-field, and that smaller contributions from modes II and III depend on subtle details of the atomic bonding in the crack core. On the other hand, dislocation emission depends on the full specification of all three components of the local k-field. The complete description of the crack tip conditions is then found to be possible in terms of a simple crack stability diagram in the three dimensional local k-space. The primary result is that combined cleavage and emission states are possible under mixed loading wherein a cleavage crack can generate significant plasticity in mode II and III dislocation configurations. In the third part, the overall static equilibrium configuration of the shielded core crack and its dislocation cloud is considered. General theorems developed show that for symmetric dislocation configurations, mode I configurations only shield the k 1 loading at the crack tip etc, and that the slip force mutually exerted on two symmetrically placed dislocations is exactly zero. Simplified fracture toughness relations are also derived. Finally, the overall picture developed for a shielded cleavage crack under general loading is summarized.

Generalized co‐ordinate partitioning in static equilibrium analysis of large‐scale mechanical systems

AbstractThis paper presents a computer‐based method for automatic formulation and efficient numerical solution of static equilibrium equations for nonlinear constrained mechanical systems with conservative forces. Nonlinear holonomic constraint equations and a potential energy function are written in terms of a maximal set of Cartesian generalized co‐ordinates. A stable static equilibrium configuration is found by minimizing the potential energy of the system, subject to the kinematic constraint equations, i.e. constrained optimization. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent and independent co‐ordinates and construct an influence coefficient matrix that relates variations in dependent and independent co‐ordinates. This information is employed to convert the constrained optimization problem to an unconstrained optimization problem. A simple example is presented to illustrate the method. An algorithm that may be used in analysis of large‐scale systems is presented.

Self‐consistent magnetotail theory: Equilibrium structures including arbitrary variation along the tail axis

The self‐consistent theory of the quiet magnetotail is extended to two‐dimensional static equilibrium configurations with arbitrary variations in two cartesian space dimensions. For the case of negligible pressure anisotropy and small plasma flow velocities, self‐consistent theory leads to a nonlinear equation which is solved numerically and analytically for several special cases. To define boundary conditions we use the weakly two‐dimensional magnetotail theory by Birn et al. [1975] to fit the tail region to the entire magnetosphere. We investigate the influence of boundary conditions and find that small changes of boundary values cause the equilibrium to form oscillatory structures with magnetic islands in the plasma sheet. This model is able to explain structures observed in the quiet magnetotail as equilibrium phenomena. Besides these taillike solutions, there exists another class of equilibria which is stable against all two‐dimensional perturbations. These solutions have one neutral point; they contain less energy than the taillike solution, the difference of energy being of the same order as the energy release in a magnetospheric substorm.

Nonconservative Buckling of Precompressed Columns

The influence of precompression on the stability of a cantilever column under the action of a subtangential force is investigated. Included also in the results is a limit on the position of the applied subtangential force (for a given precompression) at which a static equilibrium configuration is possible as opposed to a kinetic or flutter mode of stability.

Doubly diffusive magnetic buoyancy instability in the solar interior

An investigation of the buoyancy of diffuse magnetic fields has shown that in the presence of rotation, static equilibrium configurations of the toroidal magnetic field and ambient plasma can exist. In that case, the escape of toroidal magnetic flux from the solar interior may be determined by the growth of instabilities which the equilibrium configuration may be subject to. In connection with the present investigation, it is assumed that in the region of toroidal magnetic flux amplification, the magnetic field has not as yet filamented into flux ropes, and is therefore 'diffuse'. A study is conducted of the MHD stability of an electrically conducting and differentially rotating gas in the presence of a toroidal magnetic field, an external constant gravitational field, and radiance pressure. The full dispersion relation for the magnetic buoyancy problem is developed, and the solutions of the dispersion relation are discussed.

Gravitational Field Equations with Cross‐terms and Static Equilibrium Configurations

AbstractConsidering Einstein's equations modified by additional cross‐terms we calculate spherically symmetric equilibrium configurations of cold degenerated matter. The effects of such modifications proposed by TREDER [5] to describe properties of a unitary field theory depend on the sign of the coupling and on some critical energy density. If this density is higher than the nuclear density, we do not find significantly larger equilibrium masses than in general relativity. However the modifications of the gravitional dynamics resulting from the cross‐terms become physically significant.