An Approximate Analytic Solution of the One-Dimensional Foil Bearing Problem

1999 ◽  
Vol 66 (4) ◽  
pp. 1028-1032 ◽  
Author(s):  
A. A. Renshaw
Keyword(s):  
Iterative Solution ◽  
Approximate Solutions ◽  
Approximate Analytic Solution ◽  
One Dimensional ◽  
Solution Algorithms ◽  
Foil Bearing ◽  
Governing Differential Equation ◽  
Exponential Power ◽  
The One ◽  
Entrained Air

Approximate solutions are found to the one-dimensional steady-state foil bearing problem in which a longitudinally translating, flexible tensioned sheet of material rides on a thin film of entrained air as it wraps around a fixed cylinder. The approximate solutions use the first few terms of a scaled, exponential power series and minimize the squared residual in the governing differential equation. The solutions are simple, compare well with more exact solutions, and provide excellent starting points for more sophisticated, iterative solution algorithms.

Simple Approximate Solutions for Dynamic Response of Suspension System

2022 ◽  
Vol 21 ◽  
pp. 20-31
Author(s):  
Jacob Nagler
Keyword(s):  
Analytic Solution ◽  
Approximate Solutions ◽  
Quantitative Agreement ◽  
Gravitational Energy ◽  
Suspension System ◽  
Approximate Analytic Solution ◽  
Stiffness Properties ◽  
Coupling Case ◽  
The One ◽  
The Impact

An approximate simplified analytic solution is proposed for the one DOF (degree of freedom) static and dynamic displacements alongside the stiffness (dynamic and static) and damping coefficients (minimum and maximum/critical values) of a parallel spring-damper suspension system connected to a solid mass-body gaining its energy by falling from height h. The analytic solution for the prescribed system is based on energy conservation equilibrium, considering the impact by a special G parameter. The formulation is based on the works performed by Timoshenko (1928), Mindlin (1945), and the U. S. army-engineering handbook (1975, 1982). A comparison between the prescribed studies formulations and current development has led to qualitative agreement. Moreover, quantitative agreement was found between the current prescribed suspension properties approximate value - results and the traditionally time dependent (transient, frequency) parameter properties. Also, coupling models that concerns the linkage between different work and energy terms, e.g., the damping energy, friction work, spring potential energy and gravitational energy model was performed. Moreover, approximate analytic solution was proposed for both cases (friction and coupling case), whereas the uncoupling and the coupling cases were found to agree qualitatively with the literature studies. Both coupling and uncoupling solutions were found to complete each other, explaining different literature attitudes and assumptions. In addition, some design points were clarified about the wire mounting isolators stiffness properties dependent on their physical behavior (compression, shear tension), based on Cavoflex catalog. Finally, the current study aims to continue and contribute the suspension, package cushioning and containers studies by using an initial simple pre – design analytic evaluation of falling mass- body (like cushion, containers, etc.).

Analysis of One-Dimensional Consolidation of Non-Homogeneous Layer with Exponential Flow Law

2014 ◽  
Vol 638-640 ◽  
pp. 374-379 ◽  
Author(s):  
Feng Xi Zhou ◽  
Yi Ming He ◽  
Ying Xin
Keyword(s):  
Numerical Solution ◽  
Pore Water Pressure ◽  
Water Pressure ◽  
Decisive Role ◽  
Homogeneous Layer ◽  
One Dimensional ◽  
Governing Differential Equation ◽  
Degree Of Consolidation ◽  
Flow Law ◽  
The One

Based on exponential flow law, the analytical solution to the one-dimension consolidation governing differential equation was deduced when the laws of permeability and compressibility coefficients with depth can be expressed as exponential function. By finite difference method, the numerical solution of excess pore water pressure and degree of consolidation was obtained, then the reliability of numerical solution is verified by comparing numerical results with analytical results, and consolidation behavior of non-homogeneous layer with exponential flow law under various parameters is analyzed. The results showed that under the condition of the two-sided drainage, the heterogeneity of foundation consolidation of index of seepage speed depends on the index of the size and the size of the non-uniform parameters. That is when the index m is bigger, increase the permeability coefficient, reduce the compression coefficient, the consolidation is faster, but the inhomogeneous parameters are still play a decisive role.

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S175-S186 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Gabriela Melo ◽  
Jörg Schleicher ◽  
Amélia Novais ◽  
...  
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Synthetic Data ◽  
Approximate Solutions ◽  
Vertical Gradient ◽  
Correct Description ◽  
Lateral Velocity ◽  
One Dimensional ◽  
Velocity Variations ◽  
The One

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

scholarly journals Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

2017 ◽  
Vol 8 (1-2) ◽  
pp. 52
Author(s):  
Samir Karasuljic ◽  
Enes Duvnjakovic ◽  
Vedad Pasic ◽  
Elvis Barakovic
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Uniform Convergence ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
One Dimensional ◽  
The One ◽  
Theoretical Results

We consider an approximate solution for the one--dimensional semilinear singularly--perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.

scholarly journals Power1D: a Python toolbox for numerical power estimates in experiments involving one-dimensional continua

2017 ◽  
Vol 3 ◽  
pp. e125 ◽  
Author(s):  
Todd C. Pataky
Keyword(s):  
Analytical Solutions ◽  
Approximate Solutions ◽  
Random Field Theory ◽  
One Dimensional ◽  
Signal Noise ◽  
Noise Modeling ◽  
Flexible Modeling ◽  
Broad Signal ◽  
The One ◽  
High Level

The unit of experimental measurement in a variety of scientific applications is the one-dimensional (1D) continuum: a dependent variable whose value is measured repeatedly, often at regular intervals, in time or space. A variety of software packages exist for computing continuum-level descriptive statistics and also for conducting continuum-level hypothesis testing, but very few offer power computing capabilities, where ‘power’ is the probability that an experiment will detect a true continuum signal given experimental noise. Moreover, no software package yet exists for arbitrary continuum-level signal/noise modeling. This paper describes a package called power1d which implements (a) two analytical 1D power solutions based on random field theory (RFT) and (b) a high-level framework for computational power analysis using arbitrary continuum-level signal/noise modeling. First power1d’s two RFT-based analytical solutions are numerically validated using its random continuum generators. Second arbitrary signal/noise modeling is demonstrated to show how power1d can be used for flexible modeling well beyond the assumptions of RFT-based analytical solutions. Its computational demands are non-excessive, requiring on the order of only 30 s to execute on standard desktop computers, but with approximate solutions available much more rapidly. Its broad signal/noise modeling capabilities along with relatively rapid computations imply that power1d may be a useful tool for guiding experimentation involving multiple measurements of similar 1D continua, and in particular to ensure that an adequate number of measurements is made to detect assumed continuum signals.

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