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A NONSMOOTH APPROACH for the MODELLING of a MECHANICAL ROTARY DRILLING SYSTEM with FRICTION Samir Adly, Daniel Goeleven

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A NONSMOOTH APPROACH for the MODELLING of a MECHANICAL ROTARY DRILLING SYSTEM with FRICTION Samir Adly, Daniel Goeleven A NONSMOOTH APPROACH FOR THE MODELLING OF A MECHANICAL ROTARY DRILLING SYSTEM WITH FRICTION Samir Adly, Daniel Goeleven To cite this version: Samir Adly, Daniel Goeleven. A NONSMOOTH APPROACH FOR THE MODELLING OF A ME- CHANICAL ROTARY DRILLING SYSTEM WITH FRICTION. Evolution Equations and Control Theory, AIMS, In press, X, 10.3934/xx.xx.xx.xx. hal-02433100 HAL Id: hal-02433100 https://hal.archives-ouvertes.fr/hal-02433100 Submitted on 8 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Manuscript submitted to doi:10.3934/xx.xx.xx.xx AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX A NONSMOOTH APPROACH FOR THE MODELLING OF A MECHANICAL ROTARY DRILLING SYSTEM WITH FRICTION SAMIR ADLY∗ Laboratoire XLIM, Universite´ de Limoges 87060 Limoges, France. DANIEL GOELEVEN Laboratoire PIMENT, Universite´ de La Reunion´ 97400 Saint-Denis, France Dedicated to 70th birthday of Professor Meir Shillor. ABSTRACT. In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the modelling of a rotary oil drilling system with friction. We study different kinds of frictions and analyse the mathematical properties of the involved dynamical systems. We show that using a general Stribeck model for the frictional contact, we can formulate the rotary drilling system as a well-posed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results. 1. Introduction. The first systematic study of friction is due to Leonardo da Vinci (1452-1519) (see e.g. [12]). The famous italian scientific discovered indeed that the friction force is proportional to load, opposes the motion and is independent of the contact area. These fundamental results have been rediscovered by Guillaume Amontons (1663-1705) and developped by Charles-Augustin de Coulomb (1736-1806) ([4], [11]). The Coulomb friction force F is a function of the load and direction of the sliding velocity v. Arthur Morin (1795-1880) found that the friction at zero sliding speed (static friction) is larger than the Coulomb friction (dynamic friction) [26]. Osborne Reynolds (1842-1912) introduced the concept of viscous friction in relation to lubricated contact [34] and Richard Stribeck (1861-1950) observed that friction force decreases with the increase of the sliding speed from the static friction to the Coulomb friction [35]. All these fundamental discoveries on the friction phenomena have since been the subject of much research by the engineering community (see e.g. [3], [6], [7], [17], [21], [27], [28], [32]). Most models of friction are nonsmooth in the sense that the function involved in the model v 7! F (v) is not continuous at v = 0. The pioneering works of Jean-Jacques Moreau (1923-2014) and Panagiotis D. Panagiotopoulos (1950-1998) initiated the development of a mathematical framework applicable to the study of nonsmooth mechanical problems in using advanced tools from modern convex analysis and set-valued analysis (see e.g. [1], [2], 2010 Mathematics Subject Classification. 34A60, 46N10, 49J52, 70F40. Key words and phrases. Coulomb friction, mechanical system with friction, evolution variational inequalities, nonsmooth dynamics. ∗ Corresponding author: xxxx. 1 2 SAMIR ADLY AND DANIEL GOELEVEN [14], [15], [20], [23], [24], [25], [29], [30]). The approach of Moreau and Panagiotopoulos can in particular be used to write a precise and rigorous mathematical model describing the friction force and the stick-slip phenomena. This approach using set-valued functions left aside the complicated transition processes between ”stick” and ”slip” but lead however to rigorous mathematical models like differential inclusions, variational inequalities and hemivariational inequalities. There is an abundant literature for the mathematical modelling and analysis of friction problems in the context of contact mechanics. For a comprehensive reference on this subject, we refer to the book by F. Shillor, M. Sofonea and J. Telega [33]. In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the modelling of a rotary oil drilling system with friction. The drilling technique is used in the oil industry. The mechanical system involves a rotating bit to produce a bore-hole in the rock, a drill-string to rotate the bit and a rotary drive at surface to rotate the drill-string. In this paper, we consider the model of M. Kidouche and R. Riane [19] where the interaction between the bit and the rock is considered as an combination of cutting of the rock and frictional contact. We show that in using a general Stribeck model for the frictional contact, we can formulate the rotary drilling system as a well-posed evolution variational inequality. We analyse the mathematical properties of the variational inequality and give some numerical simulations to illustrate the theory. 2. Mathematical preliminaries. Let (H; h·; ·i; k · k) be a real Hilbert space. Here h·; ·i denotes the scalar product on H and k · k is the associated norm: (8x 2 H): kxk = p hx; xi: A set-valued map F : H ⇒ H is a multifunction that associates to any x 2 H a subset F(x) ⊂ H. Given a set-valued map F : H ⇒ H. The domain D(F) of F is defined by : D(F) = fx 2 H : F(x) 6= ;g: The range R(F) of F is given by: R(F) = S x2H F(x): The graph G(F) of F is defined by : G(F) = f(x; y) 2 H × H : y 2 F(x)g: The inverse F −1 of F is given by the relation: y 2 F(x) () x 2 F −1(y): One says that F is monotone if and only if hx∗ − y∗; x − yi ≥ 0; 8(x; x∗) 2 G(F); 8(y; y∗) 2 G(F): One says that F is maximal monotone if and only if it is monotone and its graph is maximal in the sense of inclusion, i.e., G(F) is not properly contained in the graph of any other monotone operator. One extensively-used maximal monotone operator is the subdifferential of a proper con- vex and lower semicontinuous function. Let Φ: H ! R [ f+1g; x 7! Φ(x) be a proper convex and lower semicontinuous function. The effective domain of Φ is defined by: dom(Φ) = fx 2 H : Φ(x) < +1g: We say that w 2 H is a subgradient of Φ at x 2 H if Φ(v) − Φ(x) ≥ hw; v − xi; 8v 2 H: The set of subgradients of Φ at x, denoted by @Φ(x), is called the subdifferential of Φ at x: @Φ(x) = fw 2 H : Φ(v) − Φ(x) ≥ hw; v − xi; 8v 2 Hg: The set-valued function: @Φ: H ⇒ H; x 7! @Φ(x) is a maximal monotone operator. The domain of @Φ is defined by D(@Φ) = f x 2 H : @Φ(x) 6= ;g: It is clear that D(@Φ) ⊂ dom(Φ). Remark 1. If Φ: Rn ! R; x 7! Φ(x) is a convex and Gateaux differentiable, then n (8x 2 R ): @Φ(x) = frΦ(x)g: Remark 2. Let Φ: R ! R; x 7! Φ(x) be a convex function. Then 0 0 (8x 2 R): @Φ(x) = [Φ−(x); Φ+(x)] 0 0 where Φ−(x) and Φ+(x) denote the left and the right derivative of Φ at x, respectively. NONSMOOTH APPROACH FOR DRILLING SYSTEMS 3 Example 2.1. For example, if Φ: R ! R; x 7! Φ(x) = jxj, then 8 < −1 if x < 0; @Φ(x) = [−1; +1] if x = 0; : +1 if x > 0: If Φ: R ! R; x 7! Φ(x) = maxf0; x2 − 4g, then 8 > 2x if x < −2; > <> [−4; 0] if x = −2; @Φ(x) = 0 if x 2] − 2; +2[; > > [0; +4] if x = +2; :> 2x if x > +2: Let us here recall the basic rules of subdifferential calculus. Let Φ: H ! R[f+1g; x 7! Φ(x) be a proper convex and lower semicontinuous function and λ > 0. Then for every x 2 D(@Φ) : @(λΦ)(x) = λ∂Φ(x): Let Φ1 : H ! R[f+1g; x 7! Φ1(x) and Φ2 : H ! R [ f+1g; x 7! Φ2(x) be two proper convex and lower semicontinuous functions. Then for every x 2 D(@Φ1) \ D(@Φ2), we have : @Φ1(x) + @Φ2(x) ⊂ @(Φ1 + Φ2)(x): Let Φ1 : H ! R [ f+1g; x 7! Φ1(x) be a proper convex and lower semicontinuous function and Φ2 : H ! R [ f+1g; x 7! Φ2(x) a Gateaux differentiable function. Then for every x 2 D(@Φ1), we have @Φ1(x) + @Φ2(x) = @Φ1(x) + rΦ2(x): Let us now recall an existence and uniqueness result, due to Kato [10, 18], for a general nonlinear Cauchy problem involving a maximal monotone operator in a Hilbert space. Theorem 2.1. Let (H; h·; ·i; k · k) be a real Hilbert space and let F : D(F) ⊂ H ⇒ H be a maximal monotone operator. Let t0 2 R, α 2 R and x0 2 D(F) be given and suppose that f :[t0; +1[! H satisfies df f 2 C0([t ; +1[; H); 2 L1 ([t ; +1[; H): 0 dt loc 0 0 Then there exists a unique x 2 C ([t0; +1[; H) satisfying dx 2 L1 ([t ; +1[; H); dt loc 0 x is right-differentiable on [t0; +1[; x(t) 2 D(F); t ≥ t0; x(t0) = x0; dx α x(t) + f(t) 2 (t) + F(x(t)); a.e.
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