井下钻柱纵向横向耦合振动模型建立与数值分析

针对井下钻柱运动的复杂性,基于动力学理论,建立了井下钻柱纵向和横向耦合振动的数学模型,并进行数值求解及分析.根据井下钻柱的实际工况,以整个井下钻柱为研究对象,提出了钻柱纵向和横向耦合振动的动力方程,并利用解析法和无量纲法分别求解出其动刚度和动阻尼的表达式,以及钻柱前两阶振动的固有频率.分析结果表明:当井下钻柱振动频率增大时,其动刚度呈幅值衰减的周期性变化,而其动阻尼呈幅值增强的周期性变化;井下钻柱长度和横截面面积越大,其动刚度和动阻尼的幅值越小;井下钻柱的Poisson(泊松)比对其振动的动刚度、动阻尼和前两阶固有频率没有影响;同时,井下钻柱的第二阶固有频率始终大于第一阶固有频率.该文的研究方法和模型为井下钻柱钻具分析和结果优化提供了理论参考和实际意义.     

大挠度钻柱强度分析的平衡微分方程法

对小井眼、大曲率井中钻柱强度问题,以井轴为基准轴,在对井轴弯挠描述和钻柱微段三维受力变形分析的基础上,建立大位移钻柱平衡微分方程,采用Longe-Kutta法解之求内力,并依此求应力和建立强度条件.对H767侧钻水平井施工中钻柱应力计算分析,结果说明与有限元模型和弹性化软绳模型比较相吻合,该模型比有限元模型计算简捷方便;比弹性化软绳模型更完善可信;该井钻柱破坏事故愿因在于井眼曲率过大,兼有应力集中.     

On the Wellbore Contact of Drill Strings in a Finite Element Model

In drill string dynamics the Finite Element Method is usually applied to models of very long drill strings in a wellbore with arbitrary curvature. Taking account of geometrical constraint between the drill string and the wellbore, a high density of nodes is necessary. This density is much higher than the one needed to describe the natural vibrations properly, so this firstly leads to an extension of the computing time. A penalty function is frequently utilized to describe the contact problem between the drill string and the wellbore where the contact normal force acts only on the nodal points of the drill string. It was recognized that only node-to-surface contact models cannot fulfill this geometrical constraint, because the segment between two nodal points deeply penetrates the wellbore wall in some cases. A process with Gaussian points along the segment in time domain will be introduced, so that the drill string will be described according to this geometrical constraint with good accuracy but with a smaller density of nodes and less computing time. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Creep Stability of Thin-Walled Composite Structures

The studies on the long-term stability of composite plates and shells under limited creep carried out mainly by the research associates of the Institute of Polymer Mechanics are reviewed. The statement of the stability problems is discussed, according to which a viscoelastic structural member can be regarded as stable if a disturbance in the form of a small initial deflection asymptotically tends with time to a small constant value. In the case of stability, as evidenced by experiments, the increase in the axisymmetric components of the initial deflection, dominating in the early stage, die down with time. On the contrary, the amplitudes of nonaxisymmetric initial imperfections grow at an increasing velocity. Analytical investigations show that the initial imperfections, when expanded into Fourier series, have a spectrum of short- and long-term critical forces. The deflection components having a critical force exceeding the external load are damped out, whereas those having a smaller critical force increase infinitely. The accelerated growth in the deflection, after a time, leads to transient buckling of the shell into a new stable equilibrium form. The problems of optimization of the structure and geometry of thin-walled composite constructions, with constraints on their long-term stability and critical time, are discussed.     

A contribution to efficient calculation of complex drill string dynamics for deep hole drilling

This paper presents a hybrid model to describe drill string dynamics for deep hole drilling. Generally, a typical rotary drill string has a length of several kilometers, but the diameter is less than half a meter. Due to the large ratio of length to diameter, a drill string is a very flexible system. Consequently, an operating drill string is always affected by axial, torsional and lateral vibrations, which potentially induce serious failures. In order to avoid fatal defects, simulations to forecast vibrations are necessary. The simulation should be capable to exhibit the complex dynamical phenomena, e.g. sick-slip, forward whirl and backward whirl, and interactions between drill string and borehole. Usually, these simulations are very time-consuming. In this work, a hybrid model consisting of lumped masses connected with weightless beam elements representing the drill string is developed. The interaction between the drill string and the borehole is implemented by unilateral constraints to describe the nonlinear contact behavior. It was shown that accuracy and simulating time were improved by this model with respect to classical finite-element models. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear multivariable modeling and stability analysis of vibrations in horizontal drill string

Nowadays, economical extraction of the oil and gas, and their products is an important problem. Up to now, vertical wells have been used for oil extraction but some wells exist that they cannot be reached by using the vertical drilling. Horizontal and directional drillings are some techniques to reach these wells. In this article, dynamics of the horizontal drill string is studied. According to the forces that affect the dynamics of drilling, its longitudinal vibration is analyzed. After determination of the boundary conditions and normal modes of the system with using mode summation method, displacement of the bit is obtained. Finally, dynamics of the horizontal drill string is simulated and the effect of increasing the number of modes and the mode convergence phenomenon is discussed. Through this study, while it is shown that four modes are sufficient for a reliable prediction of the stable and unstable conditions; five modes are included for the system analysis. Then, the effects of other parameters such as the mud frequency and constants of bit/rock interaction model are studied. Through the proposed parametric study, the stability of the system is investigated. Moreover, the dominant modes which cause the instability of the horizontal drilling process are determined.     

环空钻柱结构三维非线性分析

应用有限元理论和牛顿－拉裴逊法对弯曲井眼中环空钻柱结构进行非线性分析．根据变形特点，提出了对不同参量采用不同形式的单元的描述计算分析方法。采用罚函数法处理待定边界问题。计算表明了井眼曲率对钻头侧向力的非线性效应。     

Regularized adelic formulas for string and superstring amplitudes in one-class quadratic fields

We obtain regularized adelic formulas for gamma and beta functions for fields of rational numbers and the one-class quadratic fields and arbitrary quasicharacters (ramified or not). We consider applications to four-tachyon tree string amplitudes, generalized Veneziano amplitudes (open string), perturbed Virasoro amplitudes (closed string), massless four-particle tree open and closed superstring amplitudes, Ramond-Neveu-Schwarz superstring amplitudes, and charged heterotic superstring amplitudes. We establish certain relations between different string and superstring amplitudes.     

气体钻井钻柱气固耦合横向振动数学模型的建立与控制

根据钻柱力学和动量守恒基本理论,建立了考虑气体钻井液对钻柱内外耦合影响时的钻柱横向振动模型,此模型与一般的梁式结构的振动模型不同,它包含了钻柱轴力、钻柱内注入压力、环空压力和钻柱内气体对钻柱振动的影响.同时给出边界条件和初始条件.通过把系统外激励函数当作控制变量,利用Banach空间几何性质证明了此系统存在唯一最优控制元.     

Buckling delamination of a sandwich plate-strip with piezoelectric face and elastic core layers

In this study, the buckling delamination problem of a sandwich plate-strip with a piezoelectric face and elastic core layers is studied. It is assumed that the plate-strip is simply supported and grounded along its two parallel ends and is subjected to uniformly-distributed compressive forces on these ends. Moreover, we suppose that the plate-strip has two interface inner cracks between the face and the core layers and it is also supposed that before the plate-strip is loaded (i.e. in the natural state), the surfaces of these cracks have insignificant initial imperfections. Due to compressive forces acting along the cracks we investigate the evolution of the initial imperfections of the cracks’ surfaces. Hence, the values of the critical buckling delamination force of the considered plate-strip are determined from the criteria, according to which, the considered initial imperfections of the cracks’ surfaces grow indefinitely by the compressive forces. Mathematical modeling of the considered problem is formulated within the scope of the exact nonlinear equations of electro-elasticity in the framework of the piecewise homogeneous body model, the solution of which is found numerically by employing the finite elements method. Numerical results showing the influences of the geometrical and material parameters as well as the coupling of the electrical and mechanical fields on the values of the critical force are presented and analyzed.     

一种新的大位移井钻柱几何非线性分析方法

提出了基于实测的井深及相应的井斜角和方位角来获得确保井内钻柱参考构形长度不变的井眼轴线插值方法．当以空间大位移井的井眼轴线为钻柱的参考构形时,钻柱内的初始内力可以由井眼轴线的曲率和挠率确定．利用基于在空间自然坐标系下的包含所有单元刚体位移和常应变模式的位移函数,严格地按虚功原理推出了具有初始曲率和挠率的钻柱单元内由初始内力所引起的等效节点力计算公式,为大位移井钻柱的几何非线性处理提供了理论依据．澄清了钻柱有限元分析中的若干基本概念．为随后进行的以井眼轴线为参考构形的小变形分析,计算钻柱的自重和基于自然坐标系下的线性刚度矩阵及一致载荷列阵提供了保证．     

Analysis of the Dynamics of Slender Continua Using Karhunen-Loève-Transformation

The dynamics of continua with very small diameter–to–length ratio, like bridges or drill–strings, has been object of mechanical analysis for a long time. While it is often possible to create a well–suited mechanical model, it is difficult to determine the exact current loads and the exact operational state. For drill–strings, the load of the drill–bit depends on the material of the rock, but also on other unknown disturbances like differential sticking which can occur along the drill–string without being noticed directly. Karhunen–Loève–Transformation (KLT) provides a possibility to describe the dynamics of a continuous system with few Characteristic Functions (CF), as long as the motion of the system is stationary. On the other hand, the resulting CFs of the KLT are sensitive to changes in the dynamic system behavior. These changes can result e. g. from the occurrence of stick–slip of the bit or differential sticking of the string. On the basis of a simple model, we show that this sensitivity can be used to detect and characterize such changes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of the dynamic phenomena “stick-slip” and “whirl” of a drill string demonstrator

Vibrations of drill strings in oil and gas well bores can reduce drilling performance or lead to damage of drill string components. These vibrations are neither optically observable nor measurable because of the geological formation. Hence usually simulations are used to understand the dynamic behavior. The test stand “OSTrator” adds more insight to this problem. The demonstrator represents a scaled drill rig granting direct access to the complex dynamic behavior of the real drill string. Two phenomena are of special interest: the stick-slip effect, a torsional oscillation, and the whirl effect, a periodic torsional-lateral movement with permanent wall contact. Since the OSTrator was built for showcase purposes, it has to be piloted in and out Stick-Slip and Whirl domains by a control algorithm. Therefore efficient multi-body-systems able to represent these two effects in faster than real time are demanded. In this presentation, the modeling of the stick-slip and the whirl effect are discussed. The models should be able to predict these effects and supply data to avoid or confront them. Key questions are the modeling of the wall contact and the resistance at the drill bit. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interaction of π and K mesons and nucleons in the model of composite superconformal strings

We construct a model of composite superconformal strings to describe hadron interaction amplitudes and the hadron spectrum. It belongs to a new class of dual (string) models and falls outside the scope of the usual approaches. In the framework of this model, the ordinary hadron scale of the order of 1GeV^?2 is regarded as a string scale α′. The supersymmetry requirements are satisfied only on the two-dimensional world surface. We calculate interaction amplitudes of π and K mesons in the framework of the theory. We find a method for removing the baryon parity degeneracy.     

Motion of a vibrating string in the presence of a convex obstacle: A free boundary problem

Here we study the motion of a vibrating string in the presence of an arbitrary obstacle. We show that if the string always rebounds on the concave parts of the obstacle, it can either rebound or roll on the convex parts. The latter is the case if the velocity of the string is null at the contact point just before contact, or if the contact point propagates at a characteristic speed. Four examples are given. The three first correspond to the same obstacle, a sinusoidal arc, but with different initial conditions. In the first case, the string rebounds on the whole of the obstacle and the motion is explicitly determined when it is periodic. In the second case, the string rolls on the convex part of the obstacle up to the inflexion point and then rebounds on the concave part and unwinds on the convex part. In the third case, the string is initially at rest on the obstacle; then it instantaneously leaves the concave part while it unwinds progressively on the convex part. The fourth case is similar to the third but with a different obstacle; the motion, which is periodic, is determined explicitly.     

The local boundedness of the perturbed motions of an imperfect gyroscope in gimbals with dissipative and accelerating forces

An unbalanced dynamically symmetrical gyroscope in gimbals with constructive imperfections is considered in a central Newtonian field of forces. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings of the suspension and an accelerating (electromagnetic) moment applied to the axis of rotation of another ring. The equations of motion have a partial solution for which the basic plane of the frame is perpendicular to the direction from the specified fixed point of the frame to the centre of gravitation, the basic plane of the mantle is parallel to this direction and the rotor rotates with an arbitrary constant angular velocity.

The equations of perturbed motions of the reduced system with two degrees of freedom are obtained to within third-order terms at the corresponding position of equilibrium. In the domain of admissible values of the parameters F_o the characteristic equation of the system is considered and its coefficients are written down. A domain in F_o is specified in which complex conjugate pairs of the eigenvalues have small moduli of the real parts but the absolute values of the second- to fourth-order off-resonance mistuning between the imaginary parts are not small. For an imperfect gyroscope in gimbals with dissipative and accelerating forces the sufficient conditions of the local uniform boundedness of motions perturbed with respect to the specified partial solution are obtained in this domain. The conditions found provide the local uniform boundedness of solutions irrespective of the forms of higher than the third order in the equations of perturbed motions. These conditions are obtained in the form of constraints for the coefficients of the normal form and, finally, for the original parameters of the system and the real and imaginary parts of the eigenvalues. To provide a clear interpretation of the results, special cases when all but two parameters are fixed are analysed. The domains of local uniform boundedness are constructed in the two-dimensional domains F_o using a personal computer.     

Model Order Reduction of a Drill-String-Model with Self-Excited Stick-Slip Vibrations

The dynamical behavior of a drill-string is defined by its small diameter-to-length ratio, which makes the string vulnerable to torsional vibrations. In combination with the nonlinear friction characteristic at the drill bit, this can lead to self-excited stick-slip vibrations which are detrimental to the drilling process. The string can be modeled by the Finite Element Method or as a Multi-Body system to represent the distributed character of the system. The analysis of the resulting high-dimensional model is, however, elaborate and time-consuming. We show that through Galerkin Projection onto the first two Characteristic Functions gained from Karhunen-Loève-Transformation, a reduced system can be obtained which reproduces the essential dynamical properties of the original system, e.g. the stick-slip motion. With the reduced system, the linear stability of the drill-string can be analyzed. We show that by reducing the inertia of the rotary table the system can be stabilized. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Paradox of Nicolai and related effects

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler’s paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form.     

Paradox of Nicolai and related effects

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler’s paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form.