A mathematical analysis of wind effects on a long-jumper

1991 ◽  
Vol 33 (1) ◽  
pp. 65-76 ◽  
Author(s):  
Neville de Mestre
Keyword(s):  
Mathematical Analysis ◽  
Order Approximation ◽  
Wind Effects ◽  
Zeroth Order ◽  
Perturbation Model ◽  
First Order ◽  
Zeroth Order Approximation ◽  
The Difference ◽  
Drag And Lift ◽  
Run Up

AbstractA perturbation model is used to predict the distance jumped by a long-jumper for a range of tailwinds and headwinds. The zeroth-order approximation is based on gravity being the only force present, the effects of drag and lift only being included in the first-order corrections. The difference in predicted distances produced by the zeroth and first-order approximations is less than 2% for headwinds or tailwinds upto 4 ms−1. Most increases or decreases due to wind are caused by changes in the run-up speed, and consequently the take-off angle and speed.

TOWARDS (DE)CONSTRUCTING 4D YANG-MILLS THEORY

2008 ◽  
Vol 23 (14n15) ◽  
pp. 2099-2106
Author(s):  
MASAFUMI FUKUMA ◽  
KEN-ICHI KATAYAMA
Keyword(s):  
Order Approximation ◽  
Zeroth Order ◽  
Yang Mills ◽  
Zeroth Order Approximation ◽  
Glueball Spectrum

We investigate 3D Yang-Mills theory coupled to an adjoint scalar, which can be regarded as a zeroth order approximation in (de)constructing 4D Yang-Mills theory. We develop a new algorithm to obtain the renormalized Hamiltonian with the Karabali-Nair variable, by carefully identifying finite local counterterms. We also discuss how this formalism can be applied to obtain glueball spectrum.

Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. D37-D41 ◽  
Author(s):  
Yves Rogister ◽  
Michael A. Slawinski
Keyword(s):  
Ray Tracing ◽  
Order Approximation ◽  
Anisotropy Parameter ◽  
Field Measurements ◽  
Velocity Model ◽  
Western Canada ◽  
First Order ◽  
Approximation Linear ◽  
The Difference ◽  
First Order Approximation

We study wave propagation in anisotropic inhomogeneous media. Specifically, we formulate and analytically solve the ray-tracing equations for the factorized model with wavefront velocity increasing linearly with depth and depending elliptically on direction. We obtain explicit expressions for traveltime, wavefront (phase) angle, and ray (group) velocity and angle, and study these seismological quantities for a model that successfully describes field measurements in the Western Canada Basin. By considering numerical examples, we also show that the difference between the wavefront and ray velocities depends only slightly on the anisotropy parameter, whereas the difference between the wavefront and ray angles is, in a first-order approximation, linear in the anisotropy parameter.

Flame acceleration due to wall friction: Accuracy and intrinsic limitations of the formulations

2015 ◽  
Vol 29 (32) ◽  
pp. 1550205 ◽  
Author(s):  
Berk Demirgok ◽  
Hayri Sezer ◽  
V’yacheslav Akkerman
Keyword(s):  
Reynolds Number ◽  
Thermal Expansion ◽  
Order Approximation ◽  
Combustion Process ◽  
Wall Friction ◽  
Zeroth Order ◽  
First Order ◽  
Flame Acceleration ◽  
Wide Range ◽  
Cylindrical Tubes

The analytical formulations on the premixed flame acceleration induced by wall friction in two-dimensional (2D) channels [Bychkov et al., Phys. Rev. E 72 (2005) 046307] and cylindrical tubes [Akkerman et al., Combust. Flame 145 (2006) 206] are revisited. Specifically, pipes with one end closed are considered, with a flame front propagating from the closed pipe end to the open one. The original studies provide the analytical formulas for the basic flame and fluid characteristics such as the flame acceleration rate, the flame shape and its propagation speed, as well as the flame-generated flow velocity profile. In the present work, the accuracy of these approaches is verified, computationally, and the intrinsic limitations and validity domains of the formulations are identified. Specifically, the error diagrams are presented to demonstrate how the accuracy of the formulations depends on the thermal expansion in the combustion process and the Reynolds number associated with the flame propagation. It is shown that the 2D theory is accurate enough for a wide range of parameters. In contrast, the zeroth-order approximation for the cylindrical configuration appeared to be quite inaccurate and had to be revisited. It is subsequently demonstrated that the first-order approximation for the cylindrical geometry is very accurate for realistically large thermal expansions and Reynolds numbers. Consequently, unlike the zeroth-order approach, the first-order formulation can constitute a backbone for the comprehensive theory of the flame acceleration and detonation initiation in cylindrical tubes. Cumulatively, the accuracy of the formulations deteriorates with the reduction of the Reynolds number and thermal expansion.

scholarly journals Some Aspects of Chemical Abundance Determinations in Planetary Nebulae

1978 ◽  
Vol 76 ◽  
pp. 225-233 ◽  
Author(s):  
Lawrence H. Aller
Keyword(s):  
Order Approximation ◽  
Planetary Nebulae ◽  
Chemical Compositions ◽  
Zeroth Order ◽  
Chemical Abundance ◽  
Zeroth Order Approximation ◽  
Gaseous Nebulae ◽  
First Approximation

The determination of the chemical compositions of gaseous nebulae in general and of planetary nebulae in particular is a difficult undertaking. The zeroth-order approximation is straightforward, the first approximation is challenging, and the second approximation is almost intractable.

Experimental study of stability prediction for high-speed robotic milling of aluminum

2019 ◽  
Vol 26 (7-8) ◽  
pp. 387-398 ◽  
Author(s):  
Daxian Hao ◽  
Wei Wang ◽  
Zhaoheng Liu ◽  
Chao Yun
Keyword(s):  
High Speed ◽  
Order Approximation ◽  
Low Frequency ◽  
Computer Numerical Control ◽  
Numerical Control ◽  
Zeroth Order ◽  
Regenerative Chatter ◽  
High Speed Milling ◽  
Zeroth Order Approximation ◽  
Robotic Milling

It has been fully demonstrated that the regenerative chatter theory is applicable for predicting chatter-free milling parameters for computer numerical control machine tools, but researchers are still arguing whether it is effective for robotic milling processes. The main reason is that the robot’s modes greatly shift, depending on its varying dynamic parameters and joint configurations. More experimental investigations are required to study and better understand the mechanism of vibration in robotic machining. The present paper is focusing on finding experimental support for chatter-free prediction in robot high-speed milling by the regenerative chatter theory. Modal tests are first conducted on a milling robot and used to predict stability lobes by zeroth order approximation. A number of high-speed slotting tests are then carried out to verify the prediction results. Thus, the regenerative chatter theory is proved to be also applicable to robotic high-speed milling. Furthermore, low-frequency modes of the robot structure are investigated by more modal experiments involving a laser tracker and a displacement sensor. The low-frequency modes are identified as the main part of the prediction error of the zeroth order approximation method, which could also be dominant in low-speed robotic milling processes. In addition, robots are different from computer numerical control machines in terms of stiffness, trajectory following error, forced vibration, and motion coupling. These long-period trend terms have to be carefully taken into account in the regenerative chatter theory for robotic high-speed milling.

scholarly journals On the Renormalization of the Covariance Operators

2012 ◽  
Vol 140 (2) ◽  
pp. 637-649 ◽  
Author(s):  
Max Yaremchuk ◽  
Matthew Carrier
Keyword(s):  
Numerical Approximation ◽  
Hadamard Matrix ◽  
Order Approximation ◽  
Computational Cost ◽  
Three Dimensions ◽  
Moderate Increase ◽  
Zeroth Order ◽  
Background Error ◽  
First Order ◽  
Covariance Operators

Many background error correlation (BEC) models in data assimilation are formulated in terms of a smoothing operator [Formula: see text], which simulates the action of the correlation matrix on a state vector normalized by respective BE variances. Under such formulation, [Formula: see text] has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of [Formula: see text]) is a computationally expensive procedure, which needs an efficient numerical approximation. In this study approximate renormalization techniques based on the Monte Carlo (MC) and Hadamard matrix (HM) methods and on the analytic approximations derived under the assumption of the local homogeneity (LHA) of [Formula: see text] are compared using realistic BEC models designed for oceanographic applications. It is shown that although the accuracy of the MC and HM methods can be improved by additional smoothing, their computational cost remains significantly higher than the LHA method, which is shown to be effective even in the zeroth-order approximation. The next approximation improves the accuracy 1.5–2 times at a moderate increase of CPU time. A heuristic relationship for the smoothing scale in two and three dimensions is proposed for the first-order LHA approximation.

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