In this paper, a comprehensive continuous Cosserat rod model is developed to study the dynamics of a drill-string. The model presents the following features: (1) it can simulate the 3-D dynamics of the system, including lateral, axial and torsional motion; (2) it includes damping effects by adopting a Kelvin–Voigt material; (3) it uses a velocity-independent bit-rock interaction formulation in which the forces and torques at the bit are obtained by considering the dynamics of the cutting blades and the evolution of the soil profile, which is simulated with the help of an advection equation; (4) lateral contact is also included. In this paper, two application cases are analysed. The first one deals with a vertical borehole. It includes four simulations that are used to further investigate the hypothesis that low-dimensional lumped formulations are not able to capture the dynamics of a drill-string and, while doing so, evaluate the effect of the damping in the system's response. The predictions are compared with those given by two other models: the first one considers a 2-degrees-of-freedom (DOF) formulation, with 1-axial and 1-torsional DOF; the second one is a semi-discrete approach, with 1 DOF for the axial dynamics and a continuous formulation for the torsional one. Divergences between the predictions are found, associated with the presence of higher frequencies in the signals obtained with the Cosserat model, which reassures the hypothesis that the new continuous approach can capture aspects of the dynamics that cannot be modelled with low-dimensional representations. The efficiency of the drilling is also assessed for the simulations obtained with the new model. The highest performance was observed in those simulations where only axial oscillations occurred, while a performance drop was seen when those vibrations were accompanied by torsional stick–slip, suggesting that an optimum damping value associated with the best drilling performance could be found, were damping controllable. Finally, the second application case is used to illustrate the full potential of the new model to tackle the 3-D lateral, axial, and torsional dynamics in a curved borehole.

In this paper, a new model to study drill-string dynamics is developed. Some of the novelties of the paper are: (1) the use of a new bit-rock interaction relation that does not restrict backward rotation of the bit nor bit-bounce; (2) the use of an advection equation to avoid dealing with a system of delay-differential equations in the simulation of the cutting process; (3) the extension of the advection approach treated in previous works to allow rotation in both directions; (4) the combination of the previous items with a distributed approach for the torsional dynamics; (5) the inclusion of the 2-DOF model as a limiting case of the continuous model. For this purpose, a strategy considering an extra parameter α is used. All these aspects lead to a coupled non-linear formulation that can exhibit self-sustained vibrations associated with the regenerative cutting process. The new model is compared, from a numerical point of view, with an established 2-DOF approach. Five scenarios considering a 1200 m column with different friction parameters and top drive conditions are simulated. Dissimilar predictions are observed, showing a more complex response for the continuum model where higher frequencies are excited as well. These results indicate that the more sophisticated model could capture other aspects of the dynamics that are neglected in the 2-DOF approach.

Drill-strings employed in the oil extraction process present complex dynamics. Due to their slenderness (with length surpassing 1000 m), geometrical aspects of oil-wells, and contact forces, drill-strings exhibit a strong non-linear response. Many unwanted phenomena, like stick-slip oscillations, may occur under certain conditions. Moreover, there are several intrinsic uncertainties, e.g. in modelling the soil, which may lead to unwanted operation regimes and affect performance, i.e. degraded rate-of-penetration (ROP), and stick-slip. In this work, a stochastic approach to study drill-string dynamics is employed. The physical model is based on the Cosserat rod theory. Non-linearities of the drill-string problem may occur due to geometrical factors, such as finite displacements and rotations in deviated wells, as well as to contact and friction at the borehole wall and bit. In this particular paper, uncertainties are considered within the friction parameters. The random response is analysed through the trajectories of the drill-string axis and other quantities, such as the evolution of the angular velocities.

A cantilever beam modelled as a Cosserat rod, subjected to an external eccentric load is studied as a means to verify the capability of the current implementation to handle large displacement conditions.It is shown that the present strategy based on the Cosserat theory succesfully overcomes the known limitations of other implementations, such as the Modified Cosserat Rod Element (MCRE).

The trajectory of a ball impacting with an angle on a rigid boundary is recorded with a high-speed camera and the dynamics is reconstructed in a computer. Several experiments are carried out in order to obtain statistical distributions of the trajectory. On the other hand, a continuum model of a viscoelastic material ball simulates the experiment. If the values of the constitutive parameters (e.g. elastic and viscous modulus, friction coefficient, etc.) in the numerical model are correct, the simulated dynamics and the experimental data should match. In this study, the Bayesian inference is applied to identify two constitutive parameters (the friction and viscous coefficients) through statistical measures. The methodology shows to provide a useful tool to solve an inverse problem with a stochastic approach which allows to reference the results in a statistic frame starting from indirect and sparse information.

A model for drill-string dynamics in arbitrary borehole geometries based on the Cosserat rods theory is presented. The objective of this work is to exploit the capabilities of the Cosserat rods to capture the full range of possible dynamics. The continuous model is implemented in a finite element environment and verified with related benchmarks reported in the literature. Non-linearities of the drill-string problem occur due to geometrical factors such as finite displacements and rotations as well as to contact and friction at the borehole wall. A contact and friction model is introduced. An application is solved and compared against an approach where the friction effect is introduced through an external load (soft-string model). All possible translations and rotations in the space are considered in the Cosserat formulation of the 1-D continuum. Singularities that may arise due to the rotations description are successfully dealt with using a formulation based on quaternions. Thus, the advantage of capturing the whole dynamics with an intrinsic contact and friction model is apparent. The solution herein obtained can address more complex phenomena, which results in an improvement over the commonly used linear beam-like models where only torsional or axial-torsional effects are considered. Moreover, the beam-like models are not appropriate for non straight borehole geometries, whereas the Cosserat formulation overcomes this limitation. To sum up, a dynamic, continuous and non-linear model that considers the torsional, axial and bending effects in arbitrary borehole geometry is proposed. The formulation intrinsically includes contact and friction at the wellbore wall and bottom. A close agreement between the solution for the present Cosserat rod model and a soft-string model is found for the case under study.

In recent years, disciplines such as transport, space, and civil engineering are using lighter structural members to withstand the action of forces. On the one hand, this characteristic of structures results advantageous as smaller cross-sections are required, and therefore less material is used. On the other hand, the reduction of self-weight added into the design load-state makes these elements prone to suffering from strong undesired vibrations. Many different methods are addressed in the literature to mitigate the effect of unwanted vibrations: the installation of mass-tuned-dampers, the addition of viscoelastic materiales in the contact regions, friction dampers, or piezoelectric devices, among others. In this work the coupling mechanism for a passive vibration system in a beam-like structure modelled via the special theory of Cosserat rods is studied. The addition of a piezoelectric device in the mechanical structure is considered as a means to reducing the unwanted vibration phenomena and leads to a coupled electromechanical system. A procedure to derive the constitutive laws required for rod elements with mixed elastic material and piezoelectric devices is herein discussed. The present methodology could also be used to explore constitutive laws for different piezoelectric configurations, which could be of interest to control unwanted torsional vibrations in rotating structures such as drill-strings.

A theoretical and numerical framework to evaluate rolling contact using a Lagrangian formulation in the sense of continuum mechanics is established. The treatment corresponds to a non-stationary and large deformation model that takes into account all inertial effects. A finite element formulation is implemented featuring cylinder plate contact. The unilateral contact of hyperelastic and viscoelastic rolling wheel on a rough foundation on which friction laws hold is characterized by a highly nonlinear variational inequality. The solutions might predict the dependence of stress distribution with viscous properties. A parametric study shows the maximum power dissipation as a function of velocity, coefficient of friction and viscosity parameter. (C) 2018 Elsevier Ltd. All rights reserved.

In this work, the influence of different crack arrangements in the stress distribution of hard chromium (HC) coatings was determined. Three parameters for position and length of the cracks for two different types of coatings were probabilistically modeled based on measured scanning electron microscopy (SEM) images. Probability density functions (PDF) for those parameters were obtained to characterize each kind of coating. A two-dimensional finite element (FE) model of the coating in contact with a rigid disk was developed, modeling cracks with elliptical shapes. A Monte Carlo method was used to simulate different crack distributions for each kind of coating, and values of stress and strains in the domain were obtained. Both the J-integral and the stress intensity factors (SIFs) were taken as comparative parameters of the results. Coatings which statistically present larger quantities of shorter cracks have lower values of J-integral and SIFs, and, therefore, distribute stresses better than those with low density of longer cracks.

The objective of this work was to evaluate the influence of martensite fraction on the wear mode and the energy dissipation by friction of dual phase (DP) steel tested under reciprocating sliding conditions. For this purpose, a Ti-Nb microalloyed steel was heat treated in a conventional furnace at temperatures between 780 and 880°C (intercritical annealing temperature) for 3 min to obtain DP microstructures with volume fractions of martensite between 25 and 90%. Wear tests were carried out in both DP and as-received samples, using a reciprocating tribometer with ball-on-flat geometry, at two constant applied loads, 2.5 and 4 N. The wear damage of each sample was measured through volume loss and the dissipated energy during the test. The obtained results evidenced a significant influence of the contact load over the wear mode, because at low load the DP wear was reduced with increased hardness but just up to 75% of martensite. At high load, the sliding process promotes an oxide mixture in the ferritic microstructure that acts as a factor in wear reduction.

An uncertainty quantification study is carried out for the problem of the frontal collision of two elastic bodies. The time of contact and the resultant force function involved during the collision are the quantities of interest. If the initial conditions and the mechanical and geometrical properties were known, the response prediction would be deterministic. However, if the data contains any uncertainty, a stochastic approach becomes appropriate. Based on the Principle of Maximum Entropy (PME), and under certain restrictions on the parameter values, we derive the probability density function (PDF) for each of the stochastic parameters to construct a probabilistic model. Two cases are dealt with: one of a collision involving two spheres and another of a collision of two discs. In the first case, a parameter involving geometry and material properties is assumed stochastic. Since a functional relationship exists, the propagation of the uncertainty of the time of contact can be done symbolically. However, the interaction force function can only be computed from the solution of a nonlinear ordinary differential equation. Given the PDF of the parameter, the problem of uncertainty propagation is tackled using Monte Carlo simulations. The comparison of both approaches yields an excellent agreement. With respect to the collision of two discs, first the small deformation problem, within the Hertz theory, is addressed with a Monte Carlo method. When the discs undergo large deformations, the problem is approximated using the equations of Finite Elasticity discretized by the finite element method (FEM) and combined with Monte Carlo simulations. In a first illustration, the modulus of elasticity is assumed stochastic with a gamma PDF. Further, the disc collision problem is analyzed when two parameters are stochastic: the modulus of elasticity and the Poisson's ratio. It is shown that under certain dispersion ranges, the PDF of the interaction force function undergoes a qualitatively change exhibiting bimodality.

The application that inspires this work is the percussion drilling. This problem has impacts and presents uncertainties. In this first analysis the focus is on the construction of an efficient reduced-order model to deal with the nonlinear dynamics due to the impacts. It is important to have an efficient reduced-order model to perform the stochastic analysis. The simplified full model is constructed using the finite element method, and three different bases are used to construct the reduced-order models: LIN-basis (composed by the normal modes of the associated linear problems), PODdir-basis (obtained through proper orthogonal decomposition -direct method) and PODsnap-basis (obtained through proper orthogonal decomposition -snapshot method). The shapes of the elements of LIN-basis, PODdir-basis, and PODsnap-basis are compared. One important conclusion is that the information necessary to represent the details of a vibroimpact dynamics, measured by the proper orthogonal values, is more than the usual 99% recommended.

The dynamics of a flexible beam forced by a prescribed rotation around an axis perpendicular to its plane is addressed. Three approaches are considered, two of them related with simplified theories, within Strength of Materials, and the third one using Finite Elasticity. In the Strength of Materials approaches, the governing equations of motion are derived by superposing the deformations and the rigid motion in the first model, and in the second by stating the stationarity of the Lagrangian (including first- and second-order effects in order to capture the stiffening due to the centrifugal forces) through Hamilton's principle. Two actions are considered: gravity forces (pendulum) and prescribed rotation. Comparison of the two Strength of Materials models with the model derived from Finite Elasticity is carried out. Predictions for the same problems, interpreted in the context of the specific model, are compared and it was found that sometimes they give rather different results, both in the results and in the computational cost. Energy analyses are performed in order to obtain information about the quality of the numerical solutions. The paper ends with an example of a pendulum with a finite pivot including friction and flexibility. When the structural elements are sufficiently slender and the rotational speeds are low, so that the resulting deformations are small, the Strength of Material model that includes the load stiffening and the Finite Elasticity approach, lead to similar results. It can be concluded that the stiffening phenomenon is appropriately considered in the first model. On the contrary, when the Strength of Material hypothesis are not fulfilled, the problem should be addressed via the Finite Elasticity model. Additionally, cases with complexities such as friction at a finite pivot can only be addressed by Finite Elasticity.

The aim of this paper is to propose a new way to measure the efficiency of the proper orthogonal decomposition (POD) to construct a reduced-order model in Structural Dynamics. It investigates the efficiency of three reduced-order models for a vibroimpact problem: (1) PODdir-basis, which is the basis constructed using the direct method of the proper orthogonal decomposition; (2) PODsnap-basis, which is the basis constructed using the snapshot method of the POD; and (3) LIN-basis, which is the basis composed by the normal modes of the associated linear (LIN) conservative system. The efficiency is measured in terms of (1) number of elements to represent the dynamics with a given precision and (2) computational cost to simulate the time response within a given precision.

This paper deals with the crack detection in structural elements by means of a genetic algorithm optimization method. The crack model takes into account the existence of contact between the interfaces of the crack. Many of the methods to detect a crack in beam-like structures are based on linear one dimensional models and are not straightforwardly applicable to structures such as beams or arcs with a breathing crack with or without contact. The present study addresses bi- and three-dimensional models to handle the dynamics of a structural element with a transverse breathing crack. The methodology is not restricted to beam-like structures since it can be applied to any arbitrary shaped 3D element. The crack is simulated as a notch or a wedge with a unilateral Signorini contact model. The contact can be partial or total. All the simulations are carried out using the general purpose partial differential solver FlexPDE, a finite element (FE) code. A genetic algorithm (GA) optimization method is successfully employed for the crack detection. The dynamic response at some points of the damaged structures are compared with the solution of the computational (FE) model using least squares for each proposed crack depth and location. An objective function arises which is then optimized to obtain an estimate of both parameters. Physical experiments were performed with a cantilever damaged beam and the resulting data used as input in the detection algorithm.

The dynamic of a flexible beam forced by an imposed rotating around an axis perpendicular to its plane is addressed. Three approaches are dealt with, two related with simplified theories, belonging to the Strength of Materials and the third one using Finite Elasticity. Within the Strength of Materials approaches, the governing equations are derived by superposing the deformations and the rigid motion in the first model, and by stating the stationarity of the Lagrangian (including first and second order effects in order to capture the stiffening due to the centrifugal forces) through Hamilton’s principle in the second one. Two actions are considered: gravity forces (pendulum) and prescribed rotation. The stiffening effect due to the centrifugal forces is considered only for the beam rotating at high speeds. Comparison of this too models with the equation of Finite Elasticity is carried out. Energy analysis are performed in order to obtain information about the quality of the numerical solution. Mecánica Computacional Vol XXIX, págs. 4153-4167 (artículo completo) Eduardo Dvorkin, Marcela Goldschmit, Mario Storti (Eds.) Buenos Aires, Argentina, 15-18 Noviembre 2010 Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar

The study of the interaction between two deformable bodies that collide is of great interest since desired effects, as a stable contact, or undesired ones, as the failure of a mechanical part, can be predicted. In this work, a Signorini type contact model with impact considering large rotations and deformations is addressed. The problem is stated using the Continuum Mechanics formulation in two and three dimensions with the Lagrangian description employing the two Piola–Kirchhoff stress tensors with linear or non-linear constitutive equations. The governing equations are solved through a general purpose software oriented to solve partial differential equations by means of a finite element discretization. The impact of a deformable body over a rigid boundary or over other deformable body is tackled. Also a three-dimensional problem is addressed (i.e. a spheric ball impacting on a rigid surface). Numerical illustrations include parametric studies on the energy, the impulse forces and the time of contact for different initial conditions and materials. A comparison among the models is shown. Additionally the problem of the impact of two deformable solid balls is solved and contrasted with simpler models developed by other authors.

The nonlinear bending problem of a beam embedded in a Winkler soil with clearances is addressed by means of an analytical approach. This problem is of interest, for instance, in the study of the drillstrings behavior under certain load conditions. Usually, a drillstring is modeled as a bar inside an outer cylinder (bore-hole wall) with clearances that add strong nonlinearities. This work is part of a wider studyon drillstringsand a paper on thenonlinearvibrationof thistypeof structurewaspresented in ENIEF 2006. Here the title problem is simplified to a plane bar making contact with a Winkler-type soil. The governing differential problem is derived using a minimal energy principle. The unknowns are the lateral displacement due to bending (as usual) and the length of contact. The consideration of the latter unknown leads to special restrictions among the admissible directions within the Calculus of Variation deriving in a particular statement of the problem. Once the differential problem is fully established, a solutionof pairs of load-lengthof contact is found. Two particular examples are worked out: a cantilever beam with a lateral tip load and a simply supported beam subjected to external end bending moments, in both cases with several soil stiffness values. The numerical results are compared with a finite element model. The availability of an analytical approach permits the calibration of other numerical solutions, e.g. the study of convergences issues. On the other hand other complexities, such as the consideration of axial loads including self-weight that leads to the inclusion of the second order effect, are under study at present.

Sensibility analysis of experimentally measured frequencies as a criterion for crack detection has been extensively used in the last decades due to its simplicity. However the inverse problem of the crack parameters (location and depth) determination is not straightforward. An efficient numerical technique is necessary to obtain significant results. Two approaches are herein presented: The solution of the inverse problem with a power series technique (PST) and the use of artificial neural networks (ANNs). Cracks in a cantilever Bernoulli–Euler (BE) beam and a rotating beam are detected by means of an algorithm that solves the governing vibration problem of the beam with the PST. The ANNs technique does not need a previous model, but a training set of data is required. It is applied to the crack detection in the cantilever beam with a transverse crack. The first methodology is very simple and straightforward, though no optimization is included. It yields relative small errors in both the location and depth detection. When using one network for the detection of the two parameters, the ANNs behave adequately. However better results are found when one ANN is used for each parameter. Finally, a combination between the two techniques is suggested.