Preliminary Study on Stick-Slip in Drillstring With Analytical Model Expressed With Neutral Delay Differential Equation

2014 ◽  
Author(s):  
Tomoya Inoue ◽  
Tokihiro Katsui ◽  
Chang-Kyu Rheem ◽  
Zengo Yoshida ◽  
Miki Y. Matsuo
Keyword(s):  
Differential Equation ◽  
Analytical Model ◽  
Delay Differential Equation ◽  
Drill Bit ◽  
Neutral Delay ◽  
Stick Slip ◽  
Scientific Drilling ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Slip Phenomenon

Stick-slip is a major problem in offshore drilling because it may cause damage to the drill bit as well as crushing or grinding the sediment layer, which is crucial problem in scientific drilling because the purpose of the scientific drilling is to recover core samples from the layers. To mitigate stick-slip, first of all it is necessary to establish a model of the torsional motion of the drill bit and express the stick-slip phenomenon. Toward this end, the present study proposes a model of torsional waves propagating in a drillstring. An analytical model is developed and used to derive a neutral delay differential equation (NDDE), a special type of equation that requires time history, and an analytical model of stick-slip is derived for friction models between the drill bit and the layer as well as the rotation speed applied to the uppermost part of the drill string. In this study, the stick-slip model is numerically analyzed for several conditions and a time series of the bit motions is obtained. Based on the analytical results, the appearance of stick-slip and its severity are discussed. A small-scale model experiment was conducted in a water tank to observe the stick-slip phenomenon, and the result is discussed with numerical analysis. In addition, utilizing surface drilling data acquired from the actual drilling operations of the scientific drillship Chikyu, occurrence of stick-slip phenomenon is discussed.

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

scholarly journals Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-28 ◽  
Author(s):  
Zeqing Liu ◽  
Ling Guan ◽  
Sunhong Lee ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Higher Order ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper is concerned with the higher order nonlinear neutral delay differential equation[a(t)(x(t)+b(t)x(t-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

Oscillations of Neutral Delay Differential Equations

1986 ◽  
Vol 29 (4) ◽  
pp. 438-445 ◽  
Author(s):  
G. Ladas ◽  
Y. G. Sficas
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Neutral Delay Differential Equations ◽  
Delay Differential

AbstractThe oscillatory behavior of the solutions of the neutral delay differential equationwhere p, τ, and a are positive constants and Q ∊ C([t0, ∞), ℝ+), are studied.

scholarly journals Oscillatory and asymptotic behaviour of a nonlinear second order neutral differential equation

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
R. Rath ◽  
N. Misra ◽  
L. Padhy
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
T Tau

AbstractIn this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE) $$\left[ {r(t)(y(t) - p(t)y(t - \tau ))'} \right]^\prime + q(t)G(y(h(t))) = 0$$ are obtained, where q, h ∈ C([0, ∞), ℝ) such that q(t) ≥ 0, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C ([0, ∞), ℝ), G ∈ C (ℝ, ℝ) and τ ∈ ℝ+. Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.

scholarly journals Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Zeqing Liu ◽  
Jingjing Zhu ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Approximate Solutions ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

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