Nonlinear motions of a flexible rotor with a drill bit: stick-slip and delay effects

In this article, a discrete model of a drill-string system is developed taking into account stick-slip and time-delay aspects, and this model is used to study the nonlinear motions of this system. The model has eight degrees-of-freedom and allows for axial, torsional, and lateral dynamics of both the drill pipes and the bottom-hole assembly. Nonlinearities that arise due to dry friction, loss of contact, and collisions are considered in the development. State variable dependent time delays associated with axial and lateral cutting actions of the drill bit are introduced in the model. Based on this original model, numerical studies are carried out for different drilling operations. The results show that the motions can be self-exited through stick-slip friction and time-delay effects. Parametric studies are carried out for different ranges of friction and simulations reveal that when the drill pipe undergoes relative sticking motion phases, the drill-bit motion is suppressed by absolute sticking. Furthermore, the sticking phases observed in this work are longer than those reported in previous studies and the whirling state of the drill pipe periodically alternates between the sticking and slipping phases. When the drive speed is used as a control parameter, it is observed that the system exhibits aperiodic dynamics. The system response stability is seen to be largely dependent upon the driving speed. The discretized model presented here along with the related studies on nonlinear motions of the system can serve as a basis for choosing operational parameters in practical drilling operations.     

Coupled axial/torsional vibrations of drill-strings by means of non-linear model

In the present work a geometrically non-linear model is presented to study the coupling of axial and torsional vibrations on a drill-string, which is described as a vertical slender beam under axial rotation. It is known that the geometrical non-linearities play an important role in the stiffening of a beam. Here, the geometrical stiffening is analyzed using a non-linear finite element approximation, in which large rotations and non-linear strain-displacements are taken into account. The effect of structural damping is also included in the model. To help to understand these effects comparisons of the present model with linear ones were simulated. The preliminary analysis shows that linear and non-linear models differ considerably after the first periods of stick-slip. The behavior is more evident with the increase of the friction in the lower part of the drill.     

Stochastic torsional stability of an oil drill-string

The aim of this paper is to investigate the influence of uncertainties on the torsional vibration of drill-strings, in order to find out which uncertainty affects most significantly the torsional stability. The unstable torsional behavior is commonly associated to polycrystalline diamond compact bits, and manifests itself in the form of stick-slip oscillations. The stick-slip is a severe type of self-excited vibration characterized by large fluctuations in the rotation of the bit. It not only increases the bit wear, but also can cause drill-string failures. The analysis were done using a mathematical model of the drill-string based on classical torsion theory discretized by means of the finite element method. The bit-rock torque was included in the model as a nonlinear boundary condition at the bottom end of the drill-string. The values of the model parameters are typical values of a real drilling situation, which are subject to a high degree of uncertainty, what justifies a stochastic analysis. We have built probability distributions for the uncertain parameters and used Monte Carlo method to obtain the stochastic stability maps.     

Measuring the efficiency of vertical drill-strings: A vibration perspective

A drill-string is a slender structure with nonlinear dynamics; it is an equipment used in the oil industry to drill the rock in the search of oil and gas. The aim of this paper is to investigate the efficiency of the drilling process in terms of input/output power. The continuum system is linearized about a prestressed configuration, the finite element method is applied to discretize the linear system, then a reduced-order model is constructed using the normal modes of the linear system; only torsional and axial vibrations are considered in the analysis. Uncertainties related to the speed imposed at the top are also included in the analysis. The rotational top speed is modeled using two different stochastic processes and the Monte Carlo Method is employed to approximate the statistics of the response of the resulting stochastic differential equations.     

RESEARCH ON MODELING AND NUMERICAL METHOD OF FREE STANDING BODY ON PLANAR RIGID MULTIBODY DYNAMICS

采用非光滑多体系统动力学的方法研究浮放物体与基础平台组成的多体系统,建立其非光滑接触的动力学方程与数值算法.浮放物体由主体部分和支撑腿组成,其间通过含黏弹性阻力偶的转动铰连接.支撑腿与基础平台间的接触力简化为接触点的法向接触力和摩擦力,采用扩展的赫兹接触力模型描述接触点的法向接触力,采用库伦干摩擦模型描述其摩擦力.采用笛卡尔坐标系下的位形坐标作为系统的广义坐标.首先,将基础平台运动看作非定常约束,用第一类拉格朗日方程建立系统的动力学方程,并采用鲍姆加藤约束稳定化的方法解决违约问题.随后给出基于事件驱动法和线性互补方法的数值算法.当相对切向速度为零时,构造静滑动摩擦力的正负余量和正、负向加速度的互补关系,从而将接触点黏滞——滑移切换的判断以及静滑动摩擦力的计算转化为线性互补问题进行求解,并采用Lemke算法求解线性互补问题.最后,通过数值仿真选择合适的步长;通过仿真结果说明浮放物体运动中存在的黏滞-滑移切换现象以及基础平台运动、质心位置对浮放物体运动的影响.     

Torsional Vibration Control and Cosserat Dynamics of a Drill-Rig Assembly

Aspects of drill-string vibrations in the context of a recently developed integrated model of a drill-rig assembly based on the Cosserat theory of rods are discussed. Computer simulations are used to compare existing rotary feedback strategies currently in use to optimise drilling performance where torsional slip-stick vibrations are a hazard. Guided by the wave nature of axial and torsional vibrations in axially symmetric drill-string configurations, we present a new control mechanism, torsional rectification, and compare its performance with existing controllers within the context of the model. The practical guidelines for the improvement of drilling rates in a wide variety of circumstances are discussed.     

具有刚性间隙约束输流管的碰撞振动

管道与间隙约束间的碰撞振动是工程输流管结构的一个重要动力学现象. 迄今,人们通常采用光滑的非线性弹簧来模拟管道与间隙约束之间的碰撞力,但这种光滑的碰撞力无法真实反映碰撞前后管道状态向量的非光滑传递特征. 本文基于非光滑理论建立了具有刚性间隙约束简支输流管的非线性碰撞振动模型,利用 Galerkin 法离散了无穷维的管道模型, 并引入恢复系数构造了碰撞前后管道各处状态向量的传递矩阵,运用四阶龙格库塔法分析了脉动内流激励下管道与刚性间隙约束的非光滑碰撞振动现象,着重讨论了刚性间隙约束参数对管道动态响应随流速脉动频率变化的影响规律,特别是碰撞振动的周期性运动规律. 研究结果表明,刚性约束间隙值对管道碰撞振动行为的影响较大,在某些脉动频率下管道会出现多周期和非周期性的运动形态,还可出现非光滑系统特有的黏滑现象. 此外,碰撞恢复系数对管道振动的影响也比较显著,较小的恢复系数值更容易使管道在大范围脉动频率区间出现复杂的非周期碰撞振动.      

VIBRO-IMPACT DYNAMICS OF PIPE CONVEYING FLUID SUBJECTED TO RIGID CLEARANCE CONSTRAINT 1)

管道与间隙约束间的碰撞振动是工程输流管结构的一个重要动力学现象. 迄今,人们通常采用光滑的非线性弹簧来模拟管道与间隙约束之间的碰撞力,但这种光滑的碰撞力无法真实反映碰撞前后管道状态向量的非光滑传递特征. 本文基于非光滑理论建立了具有刚性间隙约束简支输流管的非线性碰撞振动模型,利用 Galerkin 法离散了无穷维的管道模型, 并引入恢复系数构造了碰撞前后管道各处状态向量的传递矩阵,运用四阶龙格库塔法分析了脉动内流激励下管道与刚性间隙约束的非光滑碰撞振动现象,着重讨论了刚性间隙约束参数对管道动态响应随流速脉动频率变化的影响规律,特别是碰撞振动的周期性运动规律. 研究结果表明,刚性约束间隙值对管道碰撞振动行为的影响较大,在某些脉动频率下管道会出现多周期和非周期性的运动形态,还可出现非光滑系统特有的黏滑现象. 此外,碰撞恢复系数对管道振动的影响也比较显著,较小的恢复系数值更容易使管道在大范围脉动频率区间出现复杂的非周期碰撞振动.     

Criticality of bifurcation in the tuned axial–torsional rotary drilling model

We study the bifurcation characteristics of a lumped-parameter model of rotary drilling with 1:1 internal resonance between the axial and the torsional modes which leads to the largest stability thresholds. For this special case, the two-degree-of-freedom model for the drill-string reduces to an effectively single-degree-of-freedom system facilitating further analysis. The regenerative effect of the cutting action due to the axial vibrations is incorporated through a delayed term in the cutting force with the delay depending on the torsional oscillations. This state dependency of the delay introduces nonlinearity in the current model. Steady drilling loses stability via a Hopf bifurcation, and the nature of the bifurcation is determined by an analytical study using the method of multiple scales. We find that both subcritical and supercritical Hopf bifurcations are present in this system depending on the choice of operating parameters. Hence, the nonlinearity due to the state-dependent delay term could both be stabilizing or destabilizing in nature, and the self-interruption nonlinearity is essential to capture the global behavior. Numerical bifurcation analysis of a global axial–torsional model of rotary drilling further confirms the analytical results from the method of multiple scales. Further exploration of the rotary drilling dynamics unravels more complex phenomena including grazing bifurcations and possibly chaotic solutions.     

车辆纵向非光滑多体动力学建模与数值算法研究

在建立车辆纵向多体系统的动力学模型中, 将车身与车轮视为刚体, 两者通过减振器链接; 将传动系统视为一个圆盘通过扭簧和阻尼器与驱动轮连接; 将车轮与路面间的接触力简化为法向约束力、摩擦力和滚阻力偶,其中摩擦力的模型采用库仑干摩擦模型. 采用笛卡尔坐标作为该系统的广义坐标用于描述该系统的位形, 给出系统单双边的约束方程, 应用第一类拉格朗日方法建立了系统的动力学方程. 由于摩擦与滚阻的非光滑性, 使得该系统动力学方程不连续. 为便于计算, 建立了车轮与路面接触点的相对切向加速度与摩擦力余量的互补条件、车轮角加速度与滚阻力偶余量的互补条件, 以及车轮轮心法向加速度与路面法向约束力的互补条件. 将接触—分离、黏滞—滑移的判断问题转化成线性互补问题的求解, 并给出了具有约束稳定化的基于事件驱动法的数值计算方法. 最后, 应用该方法对车辆纵向多体系统进行了仿真, 分析了输出扭矩、摩擦及滚阻系数对其动力学行为的影响.      

Dynamics of windscreen wiper blades: Squeal noise,reversal noise and chattering

The motion of a windshield wiper blade is modelled by a mass-spring-damper system on a moving frictional surface. The system dynamics is time-varying, since three different regimes of motion, characterized by different degrees of freedom, are possible. Indeed the system, which schematizes a blade cross-section, can experience stick and slip motions when it is in contact with the glass surface, and free-flight motion when it is detached. The contact between the system and the surface is governed by Stribeck׳s friction law and Poisson׳s impact law, which make the dynamics non-smooth. The model is numerically implemented in an event-driven code, and simulations are performed which reproduce the three basic classes of undesired oscillations observed in the motion of real windscreen wipers, i.e., squeal, reversal and chattering noises. Attention is focused on the causes of these vibrations, and remedies for reducing or avoiding them are proposed.     

Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction

Nonlinear Dynamics - In this paper, the stick-slip vibrations of an archetypal self-excited smooth and discontinuous (SD) oscillator are investigated. The mathematical model of the self-excited SD...     

Coupled lateral-torsional-axial vibrations of a helical gear-rotor-bearing system

Considering the axial and radial loads, a math- ematical model of angular contact ball bearing is deduced with Hertz contact theory. With the coupling effects of lateral, torsional and axial vibrations taken into account, a lumped-parameter nonlinear dynamic model of helical gearrotor-bearing system （HGRBS） is established to obtain the transmission system dynamic response to the changes of dif- ferent parameters. The vibration differential equations of the drive system are derived through the Lagrange equation, which considers the kinetic and potential energies, the dis- sipative function and the internal/external excitation. Based on the Runge-Kutta numerical method, the dynamics of the HGRBS is investigated, which describes vibration properties of HGRBS more comprehensively. The results show that the vibration amplitudes have obvious fluctuation, and the frequency multiplication and random frequency components become increasingly obvious with changing rotational speed and eccentricity at gear and bearing positions. Axial vibration of the HGRBS also has some fluctuations. The bearing has self-variable stiffness frequency, which should be avoided in engineering design. In addition, the bearing clearance needs little attention due to its slightly discernible effect on vibration response. It is suggested that a careful examination should be made in modelling the nonlinear dynamic behavior of a helical gear-rotor-bearing system.     

Nonlinear lateral-torsional coupled motion of a rotor contacting a viscoelastically suspended stator

Interaction of a rotor with a stationary part is a kind of serious malfunction that could result in a catastrophic failure if remained undetected. Past analytical and numerical simulation work on rotor?Cstator interactions mainly focus on the vibrations along the lateral directions. The torsional degree of freedom (dof) is usually ignored. The present work is aimed to study the influence of a rotor to stator contact on the lateral-torsional coupled vibrations. A mathematical model consisting of interacting vibratory systems of rotor and stator is presented. The contact is modeled using contact stiffness, damping and Coulomb friction. Equations derived for kinetic, potential and dissipation energies and non-conservative external forces are used in the Langrange??s equations for deriving the motion equations for the rotor?Cstator system. Equations revealed that the lateral-torsional motion coupling exists twofold for the rotor. The unbalance couples lateral-torsional motion of rotor through inertia and damping matrices. Coupling due to the rotor?Cstator friction occurs through a force vector. The nonlinear equations are solved using a Runge?CKutta fourth-order numerical integration scheme using relatively small time step. Results obtained through the proposed model are compared with the identical rotor?Cstator system without torsional dof and differences are identified. Effect of several parameters such as speed, relative inertia, coefficient of friction and contact damping on the bifurcation behavior of the rotor?Cstator motion has been investigated. Vibration motions presented in the forms of spectrum cascade of the coast-up response, and orbit and Poincaré plots of the steady-state response are exhibiting rich dynamic behavior of the system.     

Bifurcation phenomena in non-smooth dynamical systems

The aim of the paper is to give an overview of bifurcation phenomena which are typical for non-smooth dynamical systems. A small number of well-chosen examples of various kinds of non-smooth systems will be presented, followed by a discussion of the bifurcation phenomena in hand and a brief introduction to the mathematical tools which have been developed to study these phenomena. The bifurcations of equilibria in two planar non-smooth continuous systems are analysed by using a generalised Jacobian matrix. A mechanical example of a non-autonomous Filippov system, belonging to the class of differential inclusions, is studied and shows a number of remarkable discontinuous bifurcations of periodic solutions. A generalisation of the Floquet theory is introduced which explains bifurcation phenomena in differential inclusions. Lastly, the dynamics of the Woodpecker Toy is analysed with a one-dimensional Poincaré map method. The dynamics is greatly influenced by simultaneous impacts which cause discontinuous bifurcations.     

Non-smooth dynamics and stability of mechanical systems featuring a freeplay non-linearity

This paper concerns modelling and time-domain simulation of the dynamics of systems featuring a freeplay non-linearity. The topic of stability is also addressed. We propose to develop a reduced order model based on mode synthesis where the DOF that might have contact are separated from the regular substructures. This formulation allows for a direct application of Moreau's theory for non-smooth dynamics of contact problems. The methodologies developed are applied to a fluid-structure coupled system as given by an aircraft wing with freeplay in the control surface connection.     

On the global dynamics of chatter in the orthogonal cuttingmodel

The large-amplitude motions of a one degree-of-freedom model of orthogonal cutting are analysed. The model takes the form of a delay differential equation which is non-smooth at the instant at which the tool loses contact with the workpiece, and which is coupled to an algebraic equation that stores the profile of the cut surface whilst the tool is not in contact. This system is approximated by a smooth delay differential equation without algebraic effects which is analysed with numerical continuation software. The grazing bifurcation that defines the onset of chattering motion is thus analysed as are secondary (period-doubling, etc.) bifurcations of chattering orbits, and convergence of the bifurcation diagrams is established in the vanishing limit of the smoothing parameters. The bifurcation diagrams of the smoothed system are then compared with initial value simulations of the full non-smooth delay differential algebraic equation. These simulations mostly validate the smoothing technique and show in detail how chaotic chattering dynamics emerge from the non-smooth bifurcations of periodic orbits.     

On the dynamics of tapping mode atomic force microscope probes

A?mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. A?multimode Galerkin approximation is utilized to discretize the nonlinear partial differential equation of motion governing the cantilever response and associated boundary conditions and obtain a set of nonlinearly coupled ordinary differential equations governing the time evolution of the system dynamics. A?comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A?higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors.     

Micro-rheology of dense particulate flows: Application to immersed avalanches

We rely here on a non-smooth contact dynamics (NSCD) approach to treat particle collisions in a direct numerical simulation of a dense particulate flow. Interactions between particles are considered by a non-smooth formulation of particle dynamics at the microscopic scale, which enables one to straightforwardly implement complex contact laws. The hydrodynamic coupling is achieved by a distributed Lagrange multiplier/fictitious domain (DLM/FD) method. As a preliminary step, the relevance of our NSCD-DLM/FD method is assessed by comparing results of 2D sedimentation simulations with those obtained with a usual molecular dynamics collision model. Then, we use it to investigate how a fully immersed granular packing collapses depending on its initial particle volume fraction, providing clues on the micro-rheology of dense particulate flows.     

Mechanical energy transductions in standing wave ultrasonic motors: Analytical modelling and experimental investigations

The present work deals with experimental and theoretical analyses of mechanical energy transductions in standing wave ultrasonic motors. A piezoelectric translator prototype previously developed is tested with regard to both out-of-plane and tangential mechanical behaviours. Influences of the vibration amplitude, the normal pre-load and the dynamic friction coefficient at the stator/frame interface are pointed out through the acquisition of speed-driving force characteristics. In the main part of the article, theoretical approaches assuming the decoupling of the out-of-plane and tangential behaviours are proposed: the `complete' model takes into account transient phenomena and tangential inertia effects, and the `simplified' model supposes that the steady state is achieved. In both models, equivalent mass-spring systems allow the intermittent stator/frame contact to be characterized with regard to the vibration amplitude and the normal pre-load. Successive contact and flight periods are clearly shown. During contact periods, the sequences of stick-slip phases are at the origin of the driving mechanism. They are theoretically discriminated through the study of their behaviour equations. Finally, experimental and theoretical data fitting proves the validation of analytical analyses and allows the future optimization of standing wave ultrasonic motors to be envisaged.