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.     

A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks

We present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling – the heterogeneous multiscale method (HMM). We derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples.     

Corotational Finite Element Formulation for Virtual-Reality Based Surgery Simulators

Surgical simulation provides a means for trainees to develop surgical competence that encompasses requisite knowledge, technical and cognitive skills and decision-making ability. Considering virtual-reality based surgery simulators, the key requirement is sufficiently accurate and numerically efficient computation of deformation behavior of soft tissues, which is highly nonlinear. The paper offers a simplified geometrically nonlinear corotational finite element formulation to meet the imposed requirements. The approach is used in combination with a rather simple type of finite element and an appropriate solver is chosen for fast computation of dynamical behavior. The finite element formulation is enriched with a coupled-mesh technique to enable modelling of complex geometries by relatively simple computational models. A few examples of models of internal organs are provided to discuss the aspects of the developed tools.     

Nonlinear dynamic modeling of surface defects in rolling element bearing systems

In this paper an analytical model is proposed to study the nonlinear dynamic behavior of rolling element bearing systems including surface defects. Various surface defects due to local imperfections on raceways and rolling elements are introduced to the proposed model. The contact force of each rolling element described according to nonlinear Hertzian contact deformation and the effect of internal radial clearance has been taken into account. Mathematical expressions were derived for inner race, outer race and rolling element local defects. To overcome the strong nonlinearity of the governing equations of motion, a modified Newmark time integration technique was used to solve the equations of motion numerically. The results were obtained in the form of time series, frequency responses and phase trajectories. The validity of the proposed model verified by comparison of frequency components of the system response with those obtained from experiments. The classical Floquet theory has been applied to the proposed model to investigate the linear stability of the defective bearing rotor systems as the parameters of the system changes. The peak-to-peak frequency response of the system for each case is obtained and the basic routes to periodic, quasi-periodic and chaotic motions for different internal radial clearances are determined. The current study provides a powerful tool for design and health monitoring of machine systems.     

Finite element modelling for fatigue stress analysis of large suspension bridges

Fatigue is an important failure mode for large suspension bridges under traffic loadings. However, large suspension bridges have so many attributes that it is difficult to analyze their fatigue damage using experimental measurement methods. Numerical simulation is a feasible method of studying such fatigue damage. In British standards, the finite element method is recommended as a rigorous method for steel bridge fatigue analysis. This paper aims at developing a finite element (FE) model of a large suspension steel bridge for fatigue stress analysis. As a case study, a FE model of the Tsing Ma Bridge is presented. The verification of the model is carried out with the help of the measured bridge modal characteristics and the online data measured by the structural health monitoring system installed on the bridge. The results show that the constructed FE model is efficient for bridge dynamic analysis. Global structural analyses using the developed FE model are presented to determine the components of the nominal stress generated by railway loadings and some typical highway loadings. The critical locations in the bridge main span are also identified with the numerical results of the global FE stress analysis. Local stress analysis of a typical weld connection is carried out to obtain the hot-spot stresses in the region. These results provide a basis for evaluating fatigue damage and predicting the remaining life of the bridge.     

Damage identification in plates using finite element model updating in time domain

A response sensitivity-based approach is presented for identifying the local damages in isotropic plate structures from the measured structural dynamic responses. The local damage is simulated by a reduction in the elemental Young's modulus of the plate. In the forward analysis, the forced vibration responses of the plate under external force are obtained from Newmark direct integration. In the inverse analysis, a response sensitivity-based finite element model updating approach is used to identify local damages of the plate in time domain. The damage identification results are obtained iteratively with the penalty function method with Tikhonov regularization using the measured structural dynamic responses. Two numerical examples are investigated to illustrate the correctness and efficiency of the proposed method. Both single damage and multiple damages cases are studied. The effects of measurement noise and measurement point on the identification results are investigated. Studies in this paper indicate that the proposed method is efficient and robust for both single and multiple damages for plate structures. Good identified results can be obtained from the short time histories of a few number of measurement points.     

A Green-Naghdi Model in a 2D Problem of a Mode I Crack in an Isotropic Thermoelastic Plate

In this article, the generalized thermoelastic theory under Green and Naghdi models are used to study the thermoelastic interaction in an isotropic material containing a finite crack inside the material. The crack boundary is due to a prescribed temperature and stress distribution. Based on the Green-Naghdi type II and type III models, the formulation is applied to generalized thermoelasticity with an appropriate choice of parameters. Numerical solutions of the displacement components, temperature, and stress components are obtained using the finite element method. The results have been verified numerically and are represented graphically. Comparisons were made with expected results from Green and Naghdi model of type III and Green and Naghdi model of type II.     

PREDICTION OF DYNAMIC CHARACTERISTICS USING UPDATED FINITE ELEMENT MODELS

Model updating techniques are used to update a finite element model of a structure so that an updated model predicts the dynamics of a structure more accurately. The application of such an updated model in dynamic design demands that it also predict the effects of structural modifications with a reasonable accuracy. This paper deals with updating of a finite element model of a structure and its subsequent use for predicting the effects of structural modifications. Updated models have been obtained by a direct model updating method and by an iterative method of model updating based on the frequency response function (FRF) data. The suitability of updated models for predicting the effect of structural modifications is evaluated by some computer and laboratory experiments. First a study is performed using a simulated fixed-fixed beam. Cases of complete, incomplete and noisy data are considered. Updated models are obtained by the direct and the FRF-based method in each of these cases. These models are then used for predicting the changes in the dynamic characteristics brought about due to a mass and a beam modification. The simulated study is followed by a study involving actual measured data for the case of an F-shape test structure. The updated finite element models for this structure are obtained again by the direct and the FRF-based method. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design. It is found that the predictions based on the iterative method based updated model are reasonably accurate and, therefore, this updated model can be used with reasonable accuracy to perform dynamic design. The predictions on the basis of the direct method based updated model are found to be reasonably accurate in the lower portion of the updating frequency range but the predictions are in a significant error in the remaining portion of the updating frequency range. It is concluded that the updated models that are closer to the structure physically are likely to perform better in predicting the effects of structural modification.     

Shallow rectangular TLD for structural control implementation

A simple and practical model for the application of shallow rectangular tuned liquid damper (TLD) in structural vibration control is presented in this paper. The dynamic properties of shallow liquid in rectangular containers subjected to forced horizontal oscillation are analysed directly from the continuity and momentum equations of fluids. Following some practical assumptions, the nonlinear partial differential equations describing the wave movement of shallow liquid in rectangular containers are established and a numerical procedure for the solutions of these equations is proposed based on the finite element method. The formula for determining the control force provided by a shallow rectangular TLD is presented. The advantage of the proposed approach for the modeling of shallow rectangular TLD is that it simplifies a three-dimensional problem into a one-dimensional problem and therefore reduces the computation efforts significantly. The whole process forms a solid foundation and provides a simplified procedure for the design and analysis of shallow rectangular TLD.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

Galloping of iced quad-conductors bundles based on curved beam theory

Galloping refers to wind-induced, low-frequency, large-amplitude oscillations that have been more frequently observed for a bundle conductor than for a single conductor. In the present work two different models are built to investigate the galloping of a bundle conductor: (1) a finite curved beam element method and (2) a hybrid model based on curved beam element theory. The finite curved beam element model is effective in dealing with the spacers between the bundled conductors and the joint between the conductors and spacers that can be simulated as a rigid joint or a hinge. Furthermore, the finite curved beam element model can be used to deal with large deformation. The hybrid model invokes the small deformation hypothesis and has a high computational efficiency. A hybrid model based on conventional cable element theory is also programmed to be compared with the aforementioned models based on curved beam element theory. Numerical examples are presented to assess the accuracy of the different models in predicting the equilibrium conductor position, natural frequencies and galloping amplitude. The results show that the curved beam element models, involving more degrees of freedom and coupling of translational and torsional motion, are more accurate at simulating the static and dynamic characters of an iced quad-conductor bundle. The use of hinges, rather than rigid connections, reduces the structural response amplitudes of a galloping conductor bundle.     

Nonlinear nonuniform torsional vibrations of bars by the boundary element method

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross-section taking into account the effect of geometrical nonlinearity. The bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional boundary conditions. The transverse displacement components are expressed so as to be valid for large twisting rotations (finite displacement-small strain theory), thus the arising governing differential equations and boundary conditions are in general nonlinear. The resulting coupling effect between twisting and axial displacement components is considered and torsional vibration analysis is performed in both the torsional pre- or post-buckled state. A distributed mass model system is employed, taking into account the warping, rotatory and axial inertia, leading to the formulation of a coupled nonlinear initial boundary value problem with respect to the variable along the bar angle of twist and to an “average” axial displacement of the cross-section of the bar. The numerical solution of the aforementioned initial boundary value problem is performed using the analog equation method, a BEM based method, leading to a system of nonlinear differential-algebraic equations (DAE), which is solved using an efficient time discretization scheme. Additionally, for the free vibrations case, a nonlinear generalized eigenvalue problem is formulated with respect to the fundamental mode shape at the points of reversal of motion after ignoring the axial inertia to verify the accuracy of the proposed method. The problem is solved using the direct iteration technique (DIT), with a geometrically linear fundamental mode shape as a starting vector. The validity of negligible axial inertia assumption is examined for the problem at hand.     

Reliable reduced-order models for time-dependent linearized Euler equations

Development of optimal reduced-order models for linearized Euler equations is investigated. Recent methods based on proper orthogonal decomposition (POD), applicable for high-order systems, are presented and compared. Particular attention is paid to the link between the choice of the projection and the efficiency of the reduced model. A stabilizing projection is introduced to induce a stable reduced-order model at finite time even if the energy of the physical model is growing. The proposed method is particularly well adapted for time-dependent hyperbolic systems and intrinsically skew-symmetric models. This paper also provides a common methodology to reliably reduce very large nonsymmetric physical problems.     

Vibration and stress analysis of thin rotating discs using annular finite elements

The finite element method is applied to the stress and vibration analysis of thin rotating discs. By making use of the axisymmetric properties, annular finite elements are derived which describe the bending and stretching of such discs and are characterized by having only four degrees of freedom. These elements incorporate the desired number of diametral nodes in their dynamic deflection functions, and allow for any specified thickness variation in the radial direction. The resulting mathematical model is thus particularly efficient for numerical computation. The accuracy and convergence of the method is demonstrated by numerical comparison with both exact and experimental data.     

A finite element analysis of the harmonic response of damped three-layer plates

Solutions have been obtained for the vibration response under harmonic excitation of three-layer plates with a constrained viscoelastic layer (e.g., plates with two metallic outer layers and a viscoelastic core) by means of a finite element method. Damping has been introduced by replacing the real modulus of the viscoelastic material by a complex equivalent which accounts for the phase difference between strain and stress. Triangular finite elements were used with different numbers of degrees of freedom and the dynamic stiffness of the overall structure was calculated. The present method allows for the nonlinear stress-strain behaviour of the viscoelastic material, the effects of the rotatory inertia and the extension within the viscoelastic core. In addition, the use of triangular elements allows for a great variety of shapes and boundary conditions. The finite element computation has been verified by comparison with experimental results for circular three-layer plates and for sandwich beams.     

基于空域分解的非线性有限声束快速计算

研究非线性有限声束的一种快速数值计算算法.理论研究表明非线性有限声束存在多种耦合:谐波之间的全耦合,各场点沿轴向的递推耦合以及沿径向的局部耦合,因此可以通过声场的径向空间区域分割提高计算效率,采用多线程实现并行计算.对非线性高斯聚焦波的计算结果表明,当声场计算规模与分割线程数合理匹配时,算法能够显著提高计算速度同时保证计算精度,计算结果与理论分析相符.     

Dynamic holographic modal analysis for NDT

This paper describes crack and defect detection in structures through modification of the vibrational modal patterns and surface responses to stress. Features are made visible with dynamic holographic interferometery combined with parameter estimation. The procedure involves an unconventional, optimized, laser-illumination method. The methods are especially applicable to large structures and could prove pivotal to improved designs, monitoring and maintenance. Components and structures could be designed to better withstand operating stresses, and existing structures could be analysed to predict their response to stress. Since the modal characterization of a structure can act as a type of fingerprint, holographic interferometry can also be used to monitor structural degradation due to operating and aging. Modal characterization includes identification of resonant frequencies and also the corresponding mode shapes. Holographic interferometry provides for direct modal characterization of a structure as well as measuring its small loading dynamic response. The project demonstrated that a wide variety of defects can be located in structural components, vessels and pipes. An analytical exercise also demonstrates the ability to use global modal characteristics to determine the presence of local corrosion and erosion.     

Prediction of displacement and stress fields of a notched panel with geometric nonlinearity by reduced order modeling

The focus of this investigation is on a first assessment of the predictive capabilities of nonlinear geometric reduced order models for the prediction of the large displacement and stress fields of panels with localized geometric defects, the case of a notch serving to exemplify the analysis. It is first demonstrated that the reduced order model of the notched panel does indeed provide a close match of the displacement and stress fields obtained from full finite element analyses for moderately large static and dynamic responses (peak displacement of 2 and 4 thicknesses). As might be expected, the reduced order model of the virgin panel would also yield a close approximation of the displacement field but not of the stress one. These observations then lead to two “enrichment” techniques seeking to superpose the notch effects on the virgin panel stress field so that a reduced order model of the latter can be used. A very good prediction of the full finite element stresses, for both static and dynamic analyses, is achieved with both enrichments.     

Dynamic characteristics of launch vehicle and spacecraft connected by clamp band

Spacecraft is usually fastened to the launch vehicle by clamp band in the aerospace industry. The application of clamp band joint brings local stiffness variation to the launch vehicle and spacecraft (LV/SC) system and affects the dynamic characteristics of the system. In this paper, the dynamic responses of the LV/SC system to the vibration and impact excitations were studied, where the effect of the clamp band joint was taken into account. Firstly, the mathematical model of the axial stiffness of the clamp band joint was derived. In the model, contact and slippage between the components were accommodated. Then the stiffness model was employed to construct the coupling dynamic model for the LV/SC system using the finite element software ANSYS. Finally, modal analysis and response analysis were carried out on the coupling dynamic model to investigate the dynamic characteristics of the LV/SC system; the simulation results were compared with those based on the dynamic model where the launch vehicle and the spacecraft were considered to be fixed together to explore the effect of the clamp band joint on the LV/SC system.     

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter ν = 10^?3, the two-dimensional flow past a cylinder at Reynolds number Re = 200, and the three-dimensional flow past a cylinder at Reynolds number Re = 1000.