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Mathematical models play an increasing role in understanding and predicting machining processes, in particular milling. However, despite the considerable efforts that have been dedicated to this problem, a majority of milling models still rely on simplifying assumptions to calculate the chip thickness. In this paper, the chip thickness is determined without these simplifications, based on a surface function that describes the milled surface and on information about the workpiece boundary. By combining the partial differential equation (PDE) governing the evolution of this surface function with the ordinary differential equations (ODE) governing the tool/machine dynamics, a mixed PDE-ODE formulation is proposed to describe the dynamics of the milling process. The coupled system of differential equations is solved using an algorithm that combines finite difference (ODE) and finite volume (PDE) methods. A case study is presented to compare the proposed approach with the classical delay differential equations (DDE) model formulation for milling processes based on a simplified chip thickness model. The PDE-ODE formulation represents an explicit mathematical model for milling process dynamics; it yields a theoretically exact chip thickness and offers a means to assess the validity of models based on DDE formulation. Moreover, the proposed formulation is capable of simulating transient tool behaviors when the tool is milling the outer region of the workpiece, which is in general neglected by the DDE-based models.

In models for the dynamics of the milling process, the characterization of the chip thickness plays a central role. In this abstract, we develop an alternative model for the computation of the chip thickness. The chip thickness is computed using (i) a surface function describing the evolution of the milled surface and (ii) the information about the shape of the workpiece. Combining a partial differential equation (PDE) governing the surface function with the ordinary differential equations (ODE) governing the tool dynamics, a mixed PDE-ODE model formulation is proposed to describe the milling process dynamics. Compared with the classical delay differential equation (DDE) formulation for milling dynamics, this novel formulation is more accurate because less assumptions have been made to compute the chip thickness. Case studies show that the two formulations generally agree well with each other. However, a sizable difference on predicted limit cycles, characterizing the periodic tool motion, occurs when the axial depth of cut is relatively large. This PDE-ODE formulation brings a novel perspective for analyzing milling processes as well as a means to assess the validity of models that based on DDE formulation.

This paper extends bit/rock interface laws for drag (PDC) bits, originally formulated for homogeneous rocks, to the transition between two rock layers with distinct mechanical properties. It formulates a set of relations between the weight-on-bit, the torque-on-bit, the depth-of-cut per bit revolution, and the engagement parameter of the bit in the lower rock layer. This model enables us to extend the 2D E−S diagram for the homogeneous case to a 3D E−S diagram for the transitional case, where the third dimension is related to the engagement parameter. Moreover, this model is used to derive an expression for the drilling efficiency for the transitional phase. Examples are provided for describing the 3D E−S diagram and drilling efficiency under the condition of quasi-stationary drilling (i.e., constant angular velocity, constant weight-on-bit). These examples show that the drilling efficiency depends nonlinearly on the bit engagement between the two rock layers. This intrinsic dependency is closely related to the bit shape.

An extension of the drill-string dynamics model for drilling in heterogeneous rock formations is presented to study the heterogeneity effect on both the axial and torsional dynamics.

An extension of the drill-string dynamics model for drilling in heterogeneous rock formations is presented to study the heterogeneity effect on both the axial and torsional dynamics.

This paper extends bit/rock interface laws for drag (PDC) bits, originally formulated for homogeneous rocks, to the transition between two rock layers with distinct mechanical properties. It formulates a set of relations between the weight-on-bit, the torque-on-bit, the depth-of-cut per bit revolution, and the engagement parameter of the bit in the lower rock layer. This model enables us to extend the 2D E−S diagram for the homogeneous case to a 3D E−S diagram for the transitional case, where the third dimension is related to the engagement parameter. Moreover, this model is used to derive an expression for the drilling efficiency for the transitional phase. Examples are provided for describing the 3D E−S diagram and drilling efficiency under the condition of quasi-stationary drilling (i.e., constant angular velocity, constant weight-on-bit). These examples show that the drilling efficiency depends nonlinearly on the bit engagement between the two rock layers. This intrinsic dependency is closely related to the bit shape.

The loads acting on Polycrystalline-Diamond-Compact (PDC) bits in rotary drilling systems (see Figure 1-left) change abruptly when drilling into interbedded rock formations consisting of hard and soft layers. Such fast load changes due to formation heterogeneity can significantly affect the dynamic response of the system and therewith the vibrational signature at the bit, which in turn may influence the durability of the bit and drilling efficiency. Therefore in this abstract we develop a dynamic model for drill-string systems for such drilling scenarios.

The loads acting on Polycrystalline-Diamond-Compact (PDC) bits in rotary drilling systems (see Figure 1-left) change abruptly when drilling into interbedded rock formations consisting of hard and soft layers. Such fast load changes due to formation heterogeneity can significantly affect the dynamic response of the system and therewith the vibrational signature at the bit, which in turn may influence the durability of the bit and drilling efficiency. Therefore in this abstract we develop a dynamic model for drill-string systems for such drilling scenarios.

Boreholes with complex trajectories are drilled with the help of downhole rotary steerable systems. These robotic actuators, which are embedded in the drillstring, are used to steer the bit in the desired direction. This paper presents a dynamic non-smooth borehole propagation model for planar directional drilling. Essential nonlinearities, induced by the saturation of the bit tilt and by non-ideal (undergauged) stabilizers, are modeled using complementarity conditions, leading to a closed-form analytical description of the model in terms of a so-called delay complementarity system. The analytical form of the model allows for a comprehensive dynamic and parametric analysis. Firstly, (quasi-)stationary solutions generated by constant actuator forces are analyzed parametrically as a function of the actuation force. Secondly, an analysis of the local stability of these solutions shows the coexistence of multiple (stable and unstable) solutions and their dependency on key system parameters, such as the weight-on-bit and bit characteristics. Thirdly, a numerical simulation study shows the existence of steady-state oscillations, which are a consequence of the non-smooth characteristics of the bit tilt saturation and the stabilizers. Such limit cycles represent borehole rippling, which is the planar equivalent of the highly detrimental borehole spiraling observed in practice. The constructed model and the pursued analysis provide essential insights in the effects causing undesired borehole rippling. Herewith, the presented results can be used to support improved directional drilling system design and to form the basis for further work on automation techniques for the downhole robotic actuator to mitigate spiraled boreholes.

Boreholes with complex trajectories are drilled with the help of downhole rotary steerable systems. These robotic actuators, which are embedded in the drillstring, are used to steer the bit in the desired direction. This paper presents a dynamic non-smooth borehole propagation model for planar directional drilling. Essential nonlinearities, induced by the saturation of the bit tilt and by non-ideal (undergauged) stabilizers, are modeled using complementarity conditions, leading to a closed-form analytical description of the model in terms of a so-called delay complementarity system. The analytical form of the model allows for a comprehensive dynamic and parametric analysis. Firstly, (quasi-)stationary solutions generated by constant actuator forces are analyzed parametrically as a function of the actuation force. Secondly, an analysis of the local stability of these solutions shows the coexistence of multiple (stable and unstable) solutions and their dependency on key system parameters, such as the weight-on-bit and bit characteristics. Thirdly, a numerical simulation study shows the existence of steady-state oscillations, which are a consequence of the non-smooth characteristics of the bit tilt saturation and the stabilizers. Such limit cycles represent borehole rippling, which is the planar equivalent of the highly detrimental borehole spiraling observed in practice. The constructed model and the pursued analysis provide essential insights in the effects causing undesired borehole rippling. Herewith, the presented results can be used to support improved directional drilling system design and to form the basis for further work on automation techniques for the downhole robotic actuator to mitigate spiraled boreholes.

This paper investigates the effects of a down-hole anti-stall tool (AST) in deviated wells on the drilling performance of a rotary drilling system. Deviated wells typically induce frictional contact between the drill-string and the borehole, which affects the drill-string dynamics. In order to study the influence of such frictional effects on the effectiveness of the AST in improving the rate-of-penetration and drilling efficiency, a model-based approach is proposed. A dynamic model with coupled axial and torsional dynamics of a drilling system including the down-hole tool in an inclined well is constructed. Furthermore, the frictional contact between the drill-string and the borehole is modelled by a set-valued spatial Coulomb friction law affecting both the axial and torsional dynamics. These dynamics are described by state-dependent delay differential inclusions. Numerical analysis of this model shows that the rate-of-penetration and drilling efficiency increases by inclusion of the AST, both in the case with and without spatial Coulomb friction. Furthermore, a parametric design study of the AST in different inclined drilling scenarios is performed. This study reveals a design for the AST, which gives optimal drilling efficiency, robustly over a broad range of inclined drilling scenarios.

This report documents the results of an investigation aimed at developing analytical models for calculating closure and failure resistance of deep tunnels, based on an improved dilatation model for the rock. The phenomenon of rock dilatancy is first reviewed, and an improved dilation model - in which the rate of dilation progressively vanishes with the plastic deformation - is described. The improved dilatation model involves the introduction of a single parameter: the maximum inelastic volume increase that the material can experience. The new dilatation model was then used for the development of two mathematical models of a deep cylindrical tunnel, one for hydrostatic, the other one for nonhydrostatic loading. For the hydrostatic loading, it was shown that closure of the tunnel requires the solution of a nonlinear ordinary differential equation and, for the nonhydrostatic loading, a system of nonlinear partial differential equations of the hyperbolic type is involved. In both cases, the numerical procedures are discussed in detail: Runge-Kutta for the hydrostatic loading, and the method of characteristics for the nonhydrostatic case. Model test experiments were then reviewed in an attempt to validate the improved dilatation model. Keywords: Rock dilatancy; Elastoplastic models; Hydrostatic loading; Nonhydrostatic loading; Laboratory experiments; Variable dilatancy.