Another rational analytical approximation to the Thomas-Fermi equation

2018 ◽  
Vol 5 (1) ◽  
pp. 7-13 ◽  
Author(s):  
W. Robin
Keyword(s):  
Analytical Approximation ◽  
Thomas Fermi

An Analytical Thermodynamic Approach to Friction of Rubber on Ice

2012 ◽  
Vol 40 (2) ◽  
pp. 124-150
Author(s):  
Klaus Wiese ◽  
Thiemo M. Kessel ◽  
Reinhard Mundl ◽  
Burkhard Wies
Keyword(s):  
Coefficient Of Friction ◽  
Liquid Layer ◽  
Contact Area ◽  
Leading Edge ◽  
Viscous Friction ◽  
Analytical Approximation ◽  
Real Contact Area ◽  
Length Scales ◽  
Real Contact ◽  
Ice Surface

ABSTRACT The presented investigation is motivated by the need for performance improvement in winter tires, based on the idea of innovative “functional” surfaces. Current tread design features focus on macroscopic length scales. The potential of microscopic surface effects for friction on wintery roads has not been considered extensively yet. We limit our considerations to length scales for which rubber is rough, in contrast to a perfectly smooth ice surface. Therefore we assume that the only source of frictional forces is the viscosity of a sheared intermediate thin liquid layer of melted ice. Rubber hysteresis and adhesion effects are considered to be negligible. The height of the liquid layer is driven by an equilibrium between the heat built up by viscous friction, energy consumption for phase transition between ice and water, and heat flow into the cold underlying ice. In addition, the microscopic “squeeze-out” phenomena of melted water resulting from rubber asperities are also taken into consideration. The size and microscopic real contact area of these asperities are derived from roughness parameters of the free rubber surface using Greenwood-Williamson contact theory and compared with the measured real contact area. The derived one-dimensional differential equation for the height of an averaged liquid layer is solved for stationary sliding by a piecewise analytical approximation. The frictional shear forces are deduced and integrated over the whole macroscopic contact area to result in a global coefficient of friction. The boundary condition at the leading edge of the contact area is prescribed by the height of a “quasi-liquid layer,” which already exists on the “free” ice surface. It turns out that this approach meets the measured coefficient of friction in the laboratory. More precisely, the calculated dependencies of the friction coefficient on ice temperature, sliding speed, and contact pressure are confirmed by measurements of a simple rubber block sample on artificial ice in the laboratory.

An analytical approximation for the GARCH option pricing model

1999 ◽  
Vol 2 (4) ◽  
pp. 75-116 ◽  
Author(s):  
Jin-Chuan Duan ◽  
Geneviève Gauthier ◽  
Jean-Guy Simonato
Keyword(s):  
Option Pricing ◽  
Analytical Approximation ◽  
Pricing Model ◽  
Option Pricing Model

scholarly journals Multi-fidelity black-box optimization for time-optimal quadrotor maneuvers

2021 ◽  
pp. 027836492110333
Author(s):  
Gilhyun Ryou ◽  
Ezra Tal ◽  
Sertac Karaman
Keyword(s):  
Real World ◽  
High Speed ◽  
Analytical Approximation ◽  
Black Box ◽  
Bayesian Optimization ◽  
Optimization Framework ◽  
Quadrotor Aircraft ◽  
Time Optimal ◽  
Feasibility Constraints ◽  
Generation Problem

We consider the problem of generating a time-optimal quadrotor trajectory for highly maneuverable vehicles, such as quadrotor aircraft. The problem is challenging because the optimal trajectory is located on the boundary of the set of dynamically feasible trajectories. This boundary is hard to model as it involves limitations of the entire system, including complex aerodynamic and electromechanical phenomena, in agile high-speed flight. In this work, we propose a multi-fidelity Bayesian optimization framework that models the feasibility constraints based on analytical approximation, numerical simulation, and real-world flight experiments. By combining evaluations at different fidelities, trajectory time is optimized while the number of costly flight experiments is kept to a minimum. The algorithm is thoroughly evaluated for the trajectory generation problem in two different scenarios: (1) connecting predetermined waypoints; (2) planning in obstacle-rich environments. For each scenario, we conduct both simulation and real-world flight experiments at speeds up to 11 m/s. Resulting trajectories were found to be significantly faster than those obtained through minimum-snap trajectory planning.

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