Dynamics And Control Of A Slender,Flexible Rotor With Time-Delay Effects

The motion of a drill string,which is one component of a rotary drill rig used for oil and gas exploration and production,is an extremely slender and flexible rotor system with complex dynamic behavior.During the drilling operations,the drill string is suffering nonlinearities such as geometric nonlinearity,dry friction,collisions between pipe and wellbore,variable time delays,and loss of contact of the drill bit,etc..Although the drill-string dynamics has been studied extensively,the time delay influenced axial-torsional-lateral dynamics,non-smooth dynamics,stability analysis,and control of the drill string with delay,have not yet been considered.Thus,this thesis is mainly focusing on the modeling of the drill-string dynamics,the discerption of the non-smooth bit-rock interface,the stability analysis of the axial-torsional coupled dynamics,the active control of the time-delay system,as well as the experimental study of drill-string dynamics.Based on these studies in this thesis,some research findings and conclusions are obtained as the following.Firstly,an improved discrete model of a drill-string system is developed taking into account axial-torsional-lateral coupling and time-delay aspects,and this model is used to studied the nonlinear motion of both the drill pipe and the bottom-hole assembly.Three contact interfaces namely drill pipe-wellbore interaction,drill bit-rock interaction,and drill bit-wellbore interaction are considered in the model development.Nonlinearities that arise due to dry friction,loss of contact,collisions,and state variable dependent time delays of the drill bit are introduced into the model.Based on this 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 frictions and simulations reveal that the drill pipe undergoes forward synchronous whirling at low friction coefficient while undergoes pure rolling backward whirling at higher friction coefficient.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.Secondly,an ODE-PDE method is presented for the modelling of the multiple regenerative effects,which comes from the loss of contact effects of the drill bit-rock interaction.Instead of using a delay differential equation,the ODE-PDE method uses an ordinary differential equation?ODE?for the cutter which is modeled as discrete parameter element,and uses a partial differential equation?PDE?for the updating of the workpiece surface which is considered to be a spatially continuous.Thus,the equivalent time delay in the ODE-PDE model is allowed to be any integer multiple of the tooth-pass period,and this present an exact solution for the effects of loss of contact.Analysis of this system reveals that the loss of contact between the workpiece and the cutter results in two principal features,namely,a non-smooth cutting force and multiple regenerative effects.The model of the spatially continuous workpiece is cast into a system of ODEs through the semi-discretization method.Subsequent analysis results in a non-smooth discrete-time map,which can be mathematically regarded as a high-dimensional tent map.Iterations of this mapping show that the time delay can vary in a wide range,and due to the multiple regenerative effect,this delay can be as high as ten times the constant delay value.Through parametric studies,it is learned that the system can exhibit stable cutting behavior,as well as periodic,quasi-periodic,chaotic,and hyperchaotic behavior.An accelerated algorithm for the calculation of the Lyapunov exponents is presented based on economy-size orthogonal-triangular decomposition for the high-dimensional tent map.By this accelerated algorithm,the Lyapunov spectrum and Kaplan-Yorke dimension are computed and they show the limit cycle,limit torus,chaotic,and hyperchaotic attractors of the multiple regenerative affected drilling system.At the same time,the Lyapunov spectrum indicates that there is no zero Lyapunov exponent at some hyperchaotic dynamics,this finding denied the claims that“At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point”^[121].Thirdly,stability analyses are carried out for both the reduced order model and the multiple-segments model with taking account of axial-torsional coupling and state-dependent delay.For the 2-DOF reduced order model,the linear stability analysis is conducted using both analytical method and semi-discretization method.The stability charts are obtained in the plane of penetration rates and rotary speeds.Expanding the 2-DOF model by lumped mass method,a multi-segment model is presented taking into account state-dependent time delay and nonlinearities.Bit bounce is observed through time histories of axial vibrations,while stick-slip phenomenon is noted in the torsion response.The normal strain contours of this spatial-temporal system demonstrate the existence of strain wave propagation along the drill string.The shear strain wave exhibits features of wave nodes and wave loops along the drill string,which indicate that the torsional motion has the properties of a standing wave.When the penetration rate is varied,qualitative changes are observed in the system response.The observed behavior includes chaotic and hyperchaotic dynamics.By semi-discretization method,stability analysis reveals a stable region for the degenerate one-segment model.This stable region becomes infinitesimally small,as the resolution of spatial discretization is increased to infinity.This finding suggests that drill-string motions have a high likelihood of being self-exited in practical drilling operations.The using of a simplistic model?model with 2-DOF?to study the complex dynamics of a drill string could lead to inaccurate predictions.Fourthly,the response of a single-DOF linear time-delay system is studied analytically.The frequency response of the delay system at external excitation is obtained at varying range of parameters for time delay and delayed stiffness.The amplitude-frequency shows the system undergoes multi-peak resonance vibration.The amplitude of the sub-resonance is dependent on the delayed stiffness,while the number of the sub-resonance is dependent on the time delay.The ridgeline cluster of the multiple formant of the frequency response in the parametric space is deduced analytically.The ridge-line-cluster equation shows the exact trend of the formants in parametric space.At the same time,the number of formants in the parametric plane is obtained.Base on the response of the delay system,a control strategy using delayed feedback is proposed and is applied to the PID control as well as the LMS adaptive filter.An improved PID controller is obtained by adding a feedback of delayed state variables.For the active control of torsional vibration in a coupled axial-torsional drilling system,the stability region is enlarged greatly by the improved PID controlled strategy with delayed feedback.At the same time,an improved LMS adaptive filter is obtained by using multiple groups of delayed feedback.For the adaptive system identification purpose,the learning curve convergence,the mean square error,and the computational consumption are compared between the improved LMS and the standard LMS algorithm.The results show a great advantage in the mean square error of the improved LMS algorithm with delayed feedback,without loss any computational efficiency.Finally,some experiments are carried out to verify some theoretical results proposed above.The first experiment built an axial-torsional-lateral coupled scaled drilling rig based on the dynamic similarity.The experiment verified the nonlinear phenomena including torsional stick-slip vibration,forward whirling,backward whirling,bifurcation and chaos in drilling processes.The second experiment focus on the multiple regenerative effect and the improved PID controller.The quasi-periodic and chaotic beat vibration predicted by the ODE-PDE shows a good agreement with the experimental results.The active vibration control shows the advantage of the improved PID controller with delayed feedback.The third experiment built a drill-string rig based on the reduced order model.The experiment results indicate the complex dynamic behavior at low spinning speed of the drilling rig.The stability chart is also obtained and some comparison are made between the experiment and model predicted stability.Using the improve LMS adaptive control with multiple groups of delayed feedback,the active control of torsional vibration shows that the improve LMS algorithm enlarged the stability region by 115%,revels that the LMS control algorithm can benefit by the delayed feedback.