Fully coupled vibrations of actively controlled drillstrings

A fully coupled model for axial, lateral, and torsional vibrations of actively controlled drillstrings is presented. The proposed model includes the mutual dependence of these vibrations, which arises due to bit/formation and drillstring/borehole wall interactions as well as other geometric and dynamic non-linearities. The active control strategy is based on optimal state feedback control designed to control the drillstring rotational motion. It is demonstrated by simulation results that bit motion causes torsional vibrations, which in turn excite axial and lateral vibrations resulting in bit bounce and impacts with the borehole wall. It is also shown that the results are in close qualitative agreement with field observations regarding stick-slip and axial vibrations and that the proposed control is effective in suppressing them. However, care must be taken in selecting a set of operating parameters to avoid transient instabilities in the axial and lateral motions.     

Modeling and analysis of stick-slip and bit bounce in oil well drillstrings equipped with drag bits

Rotary drilling systems equipped with drag bits or fixed cutter bits (also called PDC), used for drilling deep boreholes for the production and the exploration of oil and natural gas, often suffer from severe vibrations. These vibrations are detrimental to the bit and the drillstring causing different failures of equipment (e.g., twist-off, abrasive wear of tubulars, bit damage), and inefficiencies in the drilling operation (reduction of the rate of penetration (ROP)). Despite extensive research conducted in the last several decades, there is still a need to develop a consistent model that adequately captures all phenomena related to drillstring vibrations such as nonlinear cutting and friction forces at the bit/rock formation interface, drive system characteristics and coupling between various motions. In this work, a physically consistent nonlinear model for the axial and torsional motions of a rotating drillstring equipped with a drag bit is proposed. A more realistic cutting and contact model is used to represent bit/rock formation interaction at the bit. The dynamics of both drive systems for rotary and translational motions of the drillstring, including the hoisting system are also considered. In this model, the rotational and translational motions of the bit are obtained as a result of the overall dynamic behavior rather than prescribed functions or constants. The dynamic behavior predicted by the proposed model qualitatively agree well with field observations and published theoretical results. The effects of various operational parameters on the dynamic behavior are investigated with the objective of achieving a smooth and efficient drilling. The results show that with proper choice of operational parameters, it may be possible to minimize the effects of stick-slip and bit-bounce and increase the ROP. Therefore, it is expected that the results will help reduce the time spent in drilling process and costs incurred due to severe vibrations and consequent damage to equipment.     

Vibration analysis of drillstrings with self-excited stick-slip oscillations

Drillstring vibration is one of the major causes for a deteriorated drilling performance. Field experience revealed that it is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. Sick-slip oscillations due to contact between the drilling bit and formation is known to excite severe torsional and axial vibrations in the drillstring. A dynamic model of the drillstring including the drillpipes and drillcollars is formulated. The equation of motion of the rotating drillstring is derived using Lagrangian approach in conjunction with the finite element method. The model accounts for the torsional-bending inertia coupling and the axial-bending geometric nonlinear coupling. In addition, the model accounts for the gyroscopic effect, the effect of the gravitational force field, and the stick-slip interaction forces. Explicit expressions of the finite element coefficient matrices are derived using a consistent mass formulation. The generalized eigenvalue problem is solved to determine modal transformations, which are invoked to obtain the reduced-order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of stick-slip excitations.     

Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

COUPLED AXIAL-LATERAL-TORSIONAL DYNAMICS OF A ROTOR-BEARING SYSTEM GEARED BY SPUR BEVEL GEARS

The coupled lateral-torsional dynamics of parallel rotor-bearing systems has been intensively investigated. However, little attention has been paid to the analysis of coupled vibrations of angled rotor-bearing systems so that the torsional and the lateral vibrations of those systems are usually analyzed separately. In this paper, the coupled axial-lateral-torsional dynamics of a rotor-bearing system geared by bevel gears is studied. The meshing of two spur bevel gears is analyzed on the basis of a pair of virtual cylindrical gears, and thereafter the constraint condition describing the relationship between the generalized displacements of bevel gears is derived under some assumptions. The coupled dynamic model is established by using Lagrange's equation under this constraint condition. The numerical results of a number of case studies show that the critical speeds of the coupled model are different from those of the uncoupled model both in values and modes, and the threshold speed of stability is fairly less than that of the uncoupled model. The effects of system parameters, such as the pitch cone angles, on the coupling behavior are also discussed.     

Sensitivity investigation of the different direction vibrations of a SNOM probe

In this article, we analyze the sensitivity of lateral, axial, and torsional vibration of a scanning near-field optical microscope (SNOM) probe. An exact solution for the title problem is obtained using the theory of beam and torsional vibrations. Sensitivities are obtained for a wide range of the intervening physical parameters. This paper discusses the relationship between the sensitivity of a SNOM probe and its geometry. The analysis is presented for a probe in three different cases lateral vibration, axial vibration, and torsional vibration, for a fixed-free configuration. To derive the mathematical relation between the probe and sample (by utilizing springs and damping to simulate different environments) through the process of theory analysis for different environments. Thus a universal touchless SNOM probe sensitivity controller is developed for appropriate control and adjustment of the probe sensitivity suitable under different environments. Such controller helps the SNOM probe achieve the nanoscale resolution through proper control and adjustment probe sensitivity in accordance to changes to different environments and reduce the chance of being damaged during the scan, which in the end further advances the sample inspection technique.     

Laser measurement of torsional vibrations/longitudinal creepage of a railway wheel set on a scaled test bench

Since more than one century, test benches remain an essential tool to study various aspects of the railway dynamics such as for instance running stability, safety or even ride comfort. For each of these applications, the knowledge of the contact conditions (forces and relative displacements) between the wheel and the rail is a necessary condition to develop a sound understanding of the physical phenomena. More specifically, as soon as the longitudinal dynamics of the vehicle-track system is involved in the study (like for the performance of a locomotive, the rolling noise or rail corrugation), a precise measure of the longitudinal creepage between the wheel and the rail is needed to verify numerical predictions from theoretical models. In this paper, we focus on the measurement of torsional vibrations of a scaled wheel set which is rolling on a roller (representing infinite rails). First, a theoretical overview of the conditions under which these torsional vibrations are excited is given. Then, the experimental set-up used to study the phenomenon is presented. During the experiment, the wheel set torsional vibrations are measured using the rotational laser Doppler vibrometer, and the measure is used to calculate the longitudinal creepage of the wheel. Results are compared with outputs of a multi-body model of the test bench.     

Chaotic torsional vibration of imbalanced shaft driven by a limited power supply

In this paper, torsional vibrations of imbalanced shaft driven by a limited power supply are studied. It is shown that mutual interaction of shaft and power supply may in particular result in chaotic self-oscillations that correspond to the strange attractors in the phase space of the coupled dynamical system “shaft–power supply”. In this particular model, strange attractors represent classical Lorenz and Feigenbaum attractors. Rotation characteristic of the power supply and resonance characteristic of the shaft rotational motion in one of the resonance zones are studied. It is shown that at certain intervals, these characteristics may be non-unique, which corresponds to the case of chaotic dynamics. Such non-trivial properties of the coupled system “shaft–power supply” could be used for a better understanding of complex vibrational phenomena in real applied systems such as problems related to the damping of the torsional vibrations.     

Instability regimes and self-excited vibrations in deep drilling systems

This paper analyzes the stability of the discrete model proposed by Richard et al. (2004 [1], 2007 [2]) to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω₀ of the drillstring imposed at the rig. On the one hand, if Ω₀ is larger than a critical velocity Ω_c, the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω₀ is smaller than Ω_c, the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω_c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.     

On dynamics of elastic rod based on exact Cosserat model

The analysis of kinematics and dynamics of an elastic rod with circular cross section is studied on the basis of exact Cosserat model under consideration of the tension and shear deformation of the rod. The dynamical equations of a rod with arbitrary initial shape are established in general form. The dynamics of a straight rod under axial tension and torsion is discussed as an example. In discussion of static stability in the space domain the Greenhill criteria of stability and the Euler load are corrected by the influence of tension and shear strain. In analysis of dynamical stability in the time domain it is shown that the Lyapunov and Euler stability conditions of the rod in space domain are the necessary conditions of Lyapunov's stability in the time domain. The longitudinal, torsional and lateral vibrations of a straight rod based on exact model are discussed, and an exact formula of free frequency of lateral vibration is obtained. The free frequency formulas of various simplified models, such as the Rayleigh beam, the Kirchhoff rod, and the Timoshenko beam, can be seen as special cases of the exact formula under different conditions of simplification.     

Vibration response of misaligned rotors

Misalignment is one of the common faults observed in rotors. Effect of misalignment on vibration response of coupled rotors is investigated in the present study. The coupled rotor system is modelled using Timoshenko beam elements with all six dof. An experimental approach is proposed for the first time for determination of magnitude and harmonic nature of the misalignment excitation. Misalignment effect at coupling location of rotor FE model is simulated using nodal force vector. The force vector is found using misalignment coupling stiffness matrix, derived from experimental data and applied misalignment between the two rotors. Steady-state vibration response is studied for sub-critical speeds. Effect of the types of misalignment (parallel and angular) on the vibration behaviour of the coupled rotor is examined. Along with lateral vibrations, axial and torsional vibrations are also investigated and nature of the vibration response is also examined. It has been found that the misalignment couples vibrations in bending, longitudinal and torsional modes. Some diagnostic features in the fast Fourier transform (FFT) of torsional and longitudinal response related to parallel and angular misalignment have been revealed. Full spectra and orbit plots are effectively used to reveal the unique nature of misalignment fault leading to reliable misalignment diagnostic information, not clearly brought out by earlier studies.     

Flow-induced vibrations of long circular cylinders modeled by coupled nonlinear oscillators

The dynamics of long slender cylinders undergoing vortex-induced vibrations (VIV) is studied in this work. Long slender cylinders such as risers or tension legs are widely used in the field of ocean engineering. When the sea current flows past a cylinder, it will be excited due to vortex shedding. A three-dimensional time domain model is formulated to describe the response of the cylinder, in which the in-line (IL) and cross-flow (CF) deflections are coupled. The wake dynamics, including in-line and cross-flow vibrations, is represented using a pair of non-linear oscillators distributed along the cylinder. The wake oscillators are coupled to the dynamics of the long cylinder with the acceleration coupling term. A non-linear fluid force model is accounted for to reflect the relative motion of cylinder to current. The model is validated against the published data from a tank experiment with the free span riser. The comparisons show that some aspects due to VIV of long flexible cylinders can be reproduced by the proposed model, such as vibrating frequency, dominant mode number, occurrence and transition of the standing or traveling waves. In the case study, the simulations show that the IL curvature is not smaller than CF curvature, which indicates that both IL and CF vibrations are important for the structural fatigue damage. Supported by the National Natural Science Foundation of China (Grant No. 10532070), the Knowledge Innovation Program of Chinese Academy of Sciences (Grant No. KJCX2-YW-L07), and the LNM Initial Funding for Young Investigators     

Optimal control of structures with semiactive-tuned mass dampers

In this paper, the optimal performance of a magnetorheological (MR) damper which is used in a tuned mass damper in reducing the peak responses of a single-degree-of-freedom structure subjected to a broad class of seismic inputs including the harmonic, pulse, artificially generated and recorded earthquake excitations are studied. The optimal semiactive control strategy minimizes an integral norm of the main structure squared absolute accelerations subject to the constraint that the non-linear equations of motion are satisfied and is determined through a numerical solution to the Euler-Lagrange equations. The optimal performance evaluated for an MR damper is compared to an equivalent passive-tuned mass damper with optimized stiffness and damping coefficients. It is shown numerically that the optimal performance of the MR damper is always better than the equivalent passive-tuned mass damper for all the investigated cases and the MR damper has a great potential in suppressing structural vibrations over a wide range of seismic inputs.     

A reduced dimensionality model of torsional vibrations in star molecules

The torsional vibrations of star molecules are studied with a reduced dimensionality model. In this model, the molecule is described by two equivalent sets of lumped inertial cylinders and vibrational frequencies are predicted by solution of the coupled equations of motion. Force constants are determined by including them as free parameters in the model and fitting the computed frequencies to their analogs as determined using full normal coordinate analysis at the HFSCF level of theory. Best agreement between the methods occurs when torsional force constants are included for the first two layers of the molecule. This reveals that non-bonded torsional interactions are important in the vibrational dynamics of these systems. Further insight is afforded by an analysis of why simple harmonic oscillator models are sufficient for modeling some related systems but fail to reproduce the trend in global mode frequencies for saturated aliphatic star molecules. The analysis reveals that the origin of this failure lies in backbone flexibility in these branched polymeric systems.     

扭摆振动实验

为了丰富“非线性系统实验”课程的教学内容,开设了扭摆振动实验,将力学中的非线性现象引入到教学中.在实验内容的设计上,采取线性振动和非线性振动相结合的方式,即:首先研究扭摆的线性振动,如阻尼振动、受迫振动等,先对振动问题有基本的理解,然后再研究扭摆的非线性振动,分别固定驱动频率、改变励磁电流和固定励磁电流、改变驱动频率,观察非线性摆的运动情况.     

Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.     

NON-LINEAR FORCED VIBRATIONS OF PLATES BY AN ASYMPTOTIC-NUMERICAL METHOD

Non-linear forced vibrations of thin elastic plates have been investigated by an asymptotic-numerical method (ANM). Various types of harmonic excitation forces such as distributed and concentrated are considered. Using the harmonic balance method and Hamilton's principle, the equation of motion is converted into an operational formulation. Based on the finite element method a starting point corresponding to a non-linear solution associated to a given frequency and amplitude of excitation is computed. Applying perturbation techniques in the vicinity of this solution, the non-linear governing equation obtained is transformed into a sequence of linear problems having the same stiffness matrix. Employing one matrix inversion, a large number of terms of the perturbation series of the displacement and frequency can be easily computed with a small computation time. Iterations of this method lead to a powerful path-following technique. Comprehensive numerical tests for forced vibrations of plates subjected to time-harmonic lateral excitations are reported.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

Effects of large amplitude and transverse shear on vibrations of triangular plates

In this paper an analytical investigation of large amplitude free flexural vibrations of isotropic and orthotropic moderately thick triangular plates is carried out. The governing equations are expressed in terms of the lateral displacement, w, and the stress function, F, and are based on an improved non-linear vibration theory which accounts for the effects of transverse shear deformation and rotatory inertia. Solutions to the governing equations are obtained by using a single-mode approximation for w, Galerkin's method and a numerical integration procedure. Numerical results are presented in terms of variations of non-linear frequency ratios with amplitudes of vibrations. The effects of transverse shear, rotatory inertia, material properties, aspect ratios, and thickness parameters are studied and compared with available solutions wherever possible. Present results are in close agreement with those reported for thin plates. It is believed that all of the results reported here that are applicable for moderately thick plates are new and therefore, no comparison is possible.     

Study on the prestressed sandwich piezoelectric ceramic ultrasonic transducer of torsional-flexural composite vibrational mode

Based on the classical torsional and flexural vibrational theory of a slender rod, the prestressed sandwich torsional-flexural composite mode piezoelectric ceramic ultrasonic transducer is studied. This type of transducer consists of the slender metal rods and the longitudinally and tangentially polarized piezoelectric ceramic rings. The resonance frequency equations for the torsional and flexural vibrations in the transducers are derived. The simultaneous resonance of the torsional and flexural vibrations in the transducer is acquired by correcting the length of the metal slender rods resulting from the piezoelectric ceramic elements. The experimental results show that the measured resonance frequencies of the transducers are in good agreement with the computed ones, and the measured resonance frequencies of the torsional and the flexural vibrations in the composite transducers are also in good agreement with each other.