Inertia Effects on the Dynamics of a Disk Levitated by Incompressible Laminar Fluid Flow

1983 ◽  
Vol 105 (3) ◽  
pp. 643-653
Author(s):  
D. K. Warinner ◽  
J. T. Pearson
Keyword(s):  
Film Thickness ◽  
Closed Form ◽  
Mass Flow ◽  
Transient Flow ◽  
Closed Form Solution ◽  
Form Solution ◽  
High Mass ◽  
Squeeze Film ◽  
Navier Stokes Equation ◽  
Inertia Effects

This paper develops a nonlinear ordinary differential equation (O.D.E.) of motion for a disk parallel to a flat plate and levitated by incompressible laminar flow of fluid supplied from a central orifice. The fluid’s inertia, reflected in high mass flow rates, is accounted for. The transient flow velocity and pressure field are found by iterative integration of the Navier-Stokes equation to determine the O.D.E. for the time-dependent height of the disk (or fluid film thickness). The film thickness is found by not only numerically integrating the O.D.E., but also by linearizing the equation to obtain a closed-form solution. The results of this combined squeeze-film, source-flow case compare favorably with experimental data presented which span cases from negligible inertia (viscous dominance) to cases of inertia dominance. Fortunately, the closed-form solution differs only slightly from the numerical solution; this provides relatively accurate expressions for the frequencies and damping coefficients in terms of the geometry, load (or weight of disk), mass flow rate, and the fluid properties.

Squeeze Film Characteristics of Circular Contacts at Constant Squeeze Velocity Motion with Non-Newtonian Lubricants

2011 ◽  
Vol 63-64 ◽  
pp. 147-151
Author(s):  
Li Ming Chu ◽  
Wang Long Li ◽  
Hsiang Chen Hsu
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Hydrodynamic Lubrication ◽  
Closed Form Solution ◽  
Elastohydrodynamic Lubrication ◽  
Form Solution ◽  
Squeeze Film ◽  
High Pressure Stage ◽  
Time Calculation ◽  
Film Characteristics

In this paper, the numerical solutions in pure squeeze motion are explored by using hydrodynamic lubrication (HL) and elastohydrodynamic lubrication (EHL) models at constant squeeze velocity with power law lubricants. This paper also proposes a closed form solution to calculate the relationship between central pressure and central film thickness under HL condition. In order to save time calculation, the present closed form solution can be used as the initial condition for analysis of EHL at the high-pressure stage. In addition, this paper also discussed the HL and EHL squeeze film characteristics.

Modeling Finite Squeeze Film Dampers

2002 ◽  
Author(s):  
A. El-Shafei
Keyword(s):  
Closed Form ◽  
Reynolds Equation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Perturbation Solution ◽  
Squeeze Film ◽  
Closed Form Solutions ◽  
Special Cases ◽  
Squeeze Film Dampers ◽  
Finite Difference Solution

Most closed form solutions of Reynolds’ equation assume either a short bearing approximation or a long bearing approximation. These closed form approximations are used in rotordynamic simulation applications, otherwise a Finite Difference solution of Reynolds’ equation would be prohibitively time consuming. Recently, there have been proposed series solutions for Reynolds’ equation for special cases. In this paper, a perturbation solution to the governing equations is proposed to obtain a closed form solution of Reynolds’ equation for a finite squeeze film damper executing a circular centered orbit. The pressure field and velocity profiles are obtained. It is shown that in the limit the finite damper solution approaches either the appropriate short or long damper. This perturbation solution can be used with appropriate boundary conditions, for various damper sealing configurations, and provides insight into the damper performance.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

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