Rope–sheave contact transient analysis in hoisting operations with a bristle model and an arbitrary Lagrangian–Eulerian approach

Multibody System Dynamics (2024) 63:477–509 https://doi.org/10.1007/s11044-024-10000-w RESEARCH Rope–sheave contact transient analysis in hoisting operations with a bristle model and an arbitrary Lagrangian–Eulerian approach José L. Escalona1 Received: 11 December 2023 / Accepted: 27 May 2024 / Published online: 18 June 2024 © The Author(s) 2024 Abstract This paper describes the development of a computational model for the rope–sheave contact interaction in reeving systems when the ropes are modeled with an arbitrary Lagrangian– Eulerian approach. This discretization approach has been developed in previous publications as a general and systematic method for the modeling and simulation of reeving systems. However, the rope–sheave contact model was avoided assuming the no-slip contact condition. The contact model developed in this paper introduces specialized ALE-ANCF-cubic rope contact elements that are used to discretize the rope segment winded at the sheave. The contact is modeled using a set of virtual discrete bristles attached to material points in the mid-line of the rope in one end and in contact with the sheave in the other end. Therefore, a second Lagrangian mesh, apart of the ALE mesh used to discretize the rope, is used to define the fixed ends of the bristles. The kinematics and dynamics used to calculate the normal and tangential contact forces are described in detail. The contact model is 3D and can be used to analyze the contact with a sheave groove with arbitrary shape. The tangential contact force model can be used to describe stick and slip contact conditions and, to improve the simulation performance of the model, an LuGre regularization tangential contact force model is used. The rope-sheave contact model is used to analyze the behavior of a simple elevator system. The numerical results show that the static rope-sheave contact interaction agrees well with an analytical solution of the problem. Finally, the same elevator system is analyzed dynamically for a cabin ride of 8 meters with a steady velocity of 1 m/s. Results show that the normal and tangential contact forces during the steady velocity period are not so different from the static solution, but very different from the classical Creep Theory and Firbank’s Theory. Keywords Contact mechanics ·Reeving systems ·Rope–sheave contact ·ALE method 1Introduction Rope–sheave and belt–pulley contact analyses are classical problems in mechanical engineering that have received the attention of important scientists since the 18th century like J.L. Escalona [email protected] 1Dept. of Mechanical and Manufacturing Engineering, University of Seville, Seville, Spain 478 J.L. Escalona Leonhard Euler. These contact analyses are fundamental for the design of machines that use long, highly flexible, slender bodies (cables, wire ropes, fiber ropes, tapes, belts, reinforced belts, etc.) that transmit movement and power through relatively long distances in an efficient and economical manner. During the 19th and 20th centuries, different studies were developed to improve the accuracy of the existing contact theories by considering different effects, like bending stiffness, effect of the radius of the sheave, centrifugal force, or by considering technological advances, like the use of fiber-reinforced belts in the industry. In the final part of the 20th century and the 21st century, highly detailed computational analyses were developed for the rope–sheave and belt–pulley contacts. A detailed literature description of this problem can be found in a recent and freely available paper by the author [1]. In the rest of this introduction, only some recent studies to this contact problem will the commented on. A fundamental part of the rope–sheave and belt–pulley contact analyses is the modeling and simulation of the dynamics of the slender body. Modeling of the slender body is challenging because it must account for high-speed large deformation, translation and rotation, and sudden changes in curvature and shape of the slender body. Standard methods in Flexible Multibody Dynamics and the Finite Element Method (FEM) are not well suited for an accurate and efficient solution of this problem. A recent trend combines the use of the Absolute Nodal Coordinate Formulation (ANCF) for the dynamic analysis of highly flexible slender bodies with the Arbitrary Lagrangian–Eulerian approach (ALE) for the dynamic description of the slender body. The recent and freely available paper of the author [2]includes a literature review of this method. A few recent papers related to the modeling of the dynamics of the slender body are described in the remainder of this introduction. Ntarladima et al. [3] have developed an ANCF model for the dynamic analysis of belt– pulley system in 2D. The contact model, that is based on a bristle model, is compatible with implicit time-integration and includes a specialized contact search method. The integration needed for the calculation of the generalized elastic forces is done with specific quadratures that avoid membrane locking [4] in the belt. Resulting belt–pulley contact forces agree well with classical Euler–Eytelwein formula. Devigne et al. [5]havedevelopedanALE cable formulation for multibody systems (MBS) applications. The model is applicable to quasistatic analysis, with only axial strains in the slender body, and can be used to analyze a cable–pulley contact problem with and without friction and a constraint-based normal contact analysis. Their results show contact forces that also agree well with the classical Euler–Eytelwein formula. The ALE formulation described in that paper is fully general for quasistatic applications. In the work of Eliseev and Vetyukov [6], the dynamics of belt drives was studied using the continuum mechanics string theory using a combination of Lagrangian and Eulerian coordinates and using the concept of axially moving continua. This work analytically describes the appearance of concentrated contact forces at the edges of belt–pulley contact. Oborin and Vetyukov [7] found the semianalytical solution of the contact between a flexible slender body and a rigid body, both with a constant translational velocity. This problem resembles the belt–pulley contact. The frictional contact model accounts for stickand-slip relative motion. The calculated contact forces vary exponentially with the distance, being the exponent proportional to a constant that is a function of the shear, bending and axial stiffness constants. Again, results predicted the appearance of concentrated contact forces at the edges of the contact. Vetyukov et al. analyzed in [8] the belt–pulley contact problem with a nonmaterial kinematic description under static and frictionless conditions. They describe a semianalytical solution using a nonlinear rod theory and a nonmaterial finite element discretization (similar concept as ALE). As in previous studies, they found concentrated contact forces at the edges of the contact in the semianalytical solution that turned into Rope–sheave contact transient analysis in hoisting operations... 479 short regions with local increase of the normal contact forces when the FEM was applied. As in the work of Ntarladima et al. [3], they also described unrealistically space-oscillatory behavior of the axial strain computed with the FEM that was blamed on membrane locking [4]. This effect disappeared with a refined mesh. Scheidl et al. [9] developed a nonmaterial shell-based FEM for the modeling of a steel belt and to study the lateral run-off due to geometric imperfections. The contact model was kind of a bristle model with no-slip condition. The model was validated with experimental results showing a good agreement. Peng et al. [10] used the ALE-ANCF model developed by Hong and Ren [11] to study the dynamics of reeving systems. The contact model avoids the discretization of the cable segments in contact with the pulleys. For the tangential contact forces, integrated values of the assumed tangential contact force densities are applied on the tangent points of the cables on the pulleys. These forces depend on transmission factors that in turn depend on the type of pulley. With this approach, the details at the contact interface is not simulated, but the overall dynamics of the system can be studied. Zheng et al. [12] have developed a similar work as Peng et al. [10], but considering the contact of the cables that move through holes. In this case, contact is assumed to happen at single points where nodal points are permanently located thanks to the ALE approach. Lee et al. [13] studied the dynamics of a reeving system using a viscoelastic model of the rope an a simplified rope–sheave contact method based on Firbank’s Theory [14]. Their purpose is the accurate positioning of the payload. Simulation results are compared with experimental results with good agreement. In this paper, the next section describes the two main rope–sheave contact theories for steady hoisting operations and a recently proposed one that is valid only for static analysis. Section 3summarizes the ALEM method for the dynamic analysis of reeving systems. Section 4describes the new ALE contact elements that have been developed to discretize the rope–sheave contact interface. Section 5explains the bristle model used for the calculation of the rope–sheave contact forces. In Sect. 6, all the details about the contact geometry are explained. The kinematic calculations that are needed to get the contact geometry are cumbersome and therefore given in detail in the Annex 1for the interested reader. Section 7 is devoted to the calculation of the tangential contact forces with the bristle model. Section 1shows simulation results of a simple elevator system, including a parametric analysis of the static configuration and a transient analysis of a ride of the elevator. Summary and conclusions are taken in Sect. 9. 2 Rope–sheave contact theories The contact interaction of a long slender body wound on a circular surface and subjected to tensile forces is a classical problem in mechanics that has its application in machinery to the belt–pulley and rope–sheave contacts. Many research works have studied this problem. There are two theories that provide simple closed-form solutions to this contact problem, namely: 1. Creep Theory, developed after the seminal works of Euler (capstan formula) and Eytelwein, later refined by Reynolds, Grashof, and Swift [15]. 2. Firbank’s Theory, also called Shear Theory, developed by Firbank [14]. Both theories are compared in [16]. Creep Theory assumes that the slender body is flexible and extensible, linear elastic in the axial direction, and with zero bending stiffness. This theory is applicable to quasistatic (effect of centrifugal forces is neglected) analysis with 480 J.L. Escalona Fig. 1 Contact forces at rope–sheave interface constant angular velocity of the sheave and constant axial loads at the free spans of the slender body. Firbank’s Theory was developed in the 1960s, when rubber belts reinforced with steel wires became popular. Firbank’s Theory assumes that the slender body is flexible but inextensible, and the rubber material shows shear deformation that uncouples the axial motion of the sheave and the steel wires, and with zero bending stiffness. Like Creep Theory, Firbank’s Theory is applicable to quasistatic analysis with constant angular velocity of the sheave. Essentially, the main difference between these two theories is that Creep Theory considers that the load is transmitted at the contact interface through the axial deformation in the slender body while Firbank’s Theory considers that load is transmitted through shear deformation. Using these theories, the space-evolution of the tension in the slender body and the tangential contact force distributions are simply given by (see Fig. 1): TC(α)=T1eμkαif 0 ≤α<β C, T2if βC≤α≤π, tC(α)=μkT1 Reμkαif 0 ≤α<β C, 0ifβC≤α≤π, (1) TF(α)=T1eμkαif 0 ≤α<β F, a1α+b1if βF≤α≤π, tF(α)=μkT1 Reμkαif 0 ≤α<β F, a2α+b2if βF≤α≤π, (2) where αis the angle along the sheave with origin at the low tension end, Tis tension (elastic axial force), tis tangential contact force density (N/m), subscripts Cand Fstand for Creep Theory and Firbank’s Theory, respectively, T1and T2are the low and high tensions at the two free spans of the slender body, respectively, Ris the radius of the sheave, μkis the kinetic coefficient of friction, βC=1 μklog T2 T1is the slip angle for Creep Theory, βFis the slip angle for Firbank’s Theory, that can be calculated as the solution of the nonlinear equation eμkβF1+π−βF 2μs−T2 T1=0,(3) where μsis the static coefficient of friction, and the coefficients that define the linear evolution a the adherence angle are given by a1=T2−Tβ π−βF ,b 1=Tβ−a1βF,a 2=−μkTβ R(π−βF),b 2=−πa2,(4) Rope–sheave contact transient analysis in hoisting operations... 481 Fig. 2 Contact Forces: comparison of Creep Theory and Firbank’s Theory with Tβbeing the tension at α=βF. In these equations the slip angle is the limit of the contact zone where there is relative slip between the rope and the sheave, and kinetic friction applies. Figure 2shows an example with T1=1kN, T2=3kN, R=1m, μk=0.4, and μs= 0.5. In both theories, in the slip arc the tangential contact force equals the normal contact force times the kinetic coefficient of friction, and in the adherence arc, where static friction applies, the tangential contact force is smaller than the normal contact force times the static coefficient of friction. The expressions of Tand tare the same in the slip arc, however, the size of the slip arcs differs, as shown in the equations above, and it can be observed in Fig. 2. In the adherence arc, both theories describe totally different situations. In Creep Theory, the normal contact pressure is constant and equal to T2 Rand the tangential contact force is zero. In Firbank’s Theory, both normal and tangential contact forces evolve linearly, as shown in the figure. Note that, because the integral of the tangential contact forces equals the drive torque (Mdr =(T2−T1)×R), the area under the green curves in the plots must be the same. Therefore, the slip arc in Creep Theory must be larger than that in Firbank’s Theory. Because all the power from the sheave to the rope (in case it is a drive sheave) is transmitted through tangential contact forces (normal contact forces do not contribute to the torque), in Creep Theory all the power is transmitted at the slip arc while in Firbank’s Theory the power is transmitted in both the slip and the adherence arcs. Creep Theory accounts for axial extension but does not account for shear deformation. Firbank’s Theory accounts for shear theory but does not account for axial extension. There is a simple method that considers both axial extension and shear deformation. Besides, it accounts also for the flattening of the cross-section of the slender body due to the normal contact forces. This method is based in the bristle contact method. In a recent paper by the author [1], the bristle contact method was used to find a simple analytical solution of the rope–sheave contact. It will be called here Bristle Theory. This solution is not comparable with Creep Theory and Firbank’s Theory because it applies to static conditions only. In 482 J.L. Escalona Fig. 3 Contact Forces: Bristle Theory Bristle Theory, the axial load and the tangential contact forces are given by T(α)=TIerα +TIIe−rα if 0 ≤α<α b, Tbeμs(α−αb)if αb≤α≤π, t(α)=r RTIerα −TIIe−rαif 0 ≤α<α b, μ RTbeμs(α−αb)if αb≤α≤π, (5) where αbis the limit angle of the adherence arc (πminus the angle of the slip arc), TI,TII, and Tbare reference values of the tension that can be calculated as shown in [1], and the exponent ris given by r=ktR2 EA∗, EA∗=EA +knR2 EA ×knR2, (6) with EA being the axial stiffness of the slender body, ktthe transverse stiffness of the bristles (shear stiffness), and knthe normal stiffness of the bristles (stiffness of the cross-section to flattening). Figure 3shows the normal and tangential contact forces obtained using the Bristle Theory with the same conditions used to create the plots in Fig. 2. As mentioned, Eqs. (1)–(2)andEq.(5) are not comparable because the former are valid for steady-state dynamics and the later is valid for statics. Keeping that in mind, there are a few interesting aspects to describe: 1. The contact forces obtained with Creep Theory and Firbank’s Theory only depend on the tensions at the free spans, T1and T2, the radius of the sheave, R, and the coefficients of friction, μkand μs. The contact forces obtained with Bristle Theory also depend on the axial (EA), shear (kt), and flattening (kn) stiffness of the slender body. 2. Creep Theory and Firbank’s Theory locate the slip arc at the low-tension end of the sheave. However, Bristle Theory locates the slip arc at the high-tension end of the sheave. In Creep Theory, there is always a slip arc, where all the power is transmitted, however, the slip arc does not necessarily appear in Firbank’s Theory and Bristle Theory. Rope–sheave contact transient analysis in hoisting operations... 483 3. In the adherence arc, Creep Theory and Firbank’s Theory result in constant or linear normal contact forces. Bristle Theory results in an exponential evolution of the normal contact force in the adherence arc, with the exponential coefficient rbeing a function of the different stiffness constants of the slender body. An exponent similar to rwas found in the paper [7]. 4. In contrast to the results of Creep Theory and Firbank’s Theory, Bristle Theory predicts tangential contact forces that can change sign in the adherence arc. See Fig. 1on the right. Think now about a hoisting machine, like an elevator, that is at rest, with a cabin at one end of the rope and a counterweight at the other. In this case, the assumptions of Bristle Theory apply. When a ride starts, the brake opens, the drive applies motor torque and the elevator system starts moving until reaching a constant velocity. If this operation is analyzed with a bristle contact model adapted to transient dynamic analysis, the contact force distribution has to turn from the that given in Eq. (5) to that which is directly comparable with Eqs. (1)–(2) obtained with Creep Theory and Firbank’s Theory. One important objective of this paper is to study if the mentioned features 1–4 of the different contact forces described above remain valid under dynamic-steady conditions. 3 The ALEM method for the modeling and simulation of reeving systems The ALEM (Arbitrary Lagrangian–Eulerian–Modal) method is a specialized FEM formulation to model and discretize wire ropes in reeving systems, like the tower crane shown in Fig. 4. The elements used to discretize the wire ropes have the following properties: •They are defined under the ALE approach. The nodes are not necessarily fixed in material points. There can be material flow within the element. •The set of nodal coordinates includes: 1. A set of absolute position coordinates of the nodal points, qa=r1r2T; 2. A set of modal coordinates to describe transverse and axial deformation in an element frame, qm=qx,1... qx,nx qy,1... qy,ny qz,1... qz,nz T; 3. A set of twist angles to describe torsion, qθ=θ1θ2T; 4. A set of two arc-length coordinates (also called material coordinates) that define the position of the nodes in the reference configuration of the rope, qs=s1s2T. With these features, the ALEM method was applied to model reeving systems is a systematic manner. The properties of the resulting model are: •It requires a minimum set of elements to model the reeving system. Just one element is needed to model each free span of the rope. •It can be used to represent the tensile force, transverse vibration, and torsion of the wire ropes. •The equations of motion (EOM) are systematically obtained in a preprocessing stage using symbolic computation. In the ALEM formulation described in [17,18], the rope–sheave contact is not modeled. The no-slip condition and the torque balance in the sheave are used to model the dynamics of the reeving systems without entering the detail of the rope–sheave contact conditions. 484 J.L. Escalona Fig. 4 Example of a reeving system In this paper, the ALEM method is extended to account for the rope–sheave contact, thus requiring the discretization of the rope segment in contact with the sheave under the ALE description. 4 Reeving systems with rope–contact elements 4.1 Kinematics of the reeving system Reeving systems can be modeled as a multibody system including a set of rigid bodies, to which a set of sheaves or reels are attached, and a set of wire ropes wound on them. There are drive sheaves or reels (winches) that introduce power into the system, and deviation sheaves that are passive elements. Figure 5shows just part of an arbitrary reeving system. It includes just one rigid body, Body 2, to which a sheave is attached. A rope segment is wound to that sheave. The ALEM elements aand bmodel the free-spans of the rope next to the sheave. In the zoomed view of the sheave, the set of ALE rope-contact elements c1,c2, c3,andc4can be observed. This section describes the properties of these elements, that are specially designed to model the rope–sheave contact. Regarding the rigid body, a set of coordinates q2is used to describe the absolute position and orientation of the body frame with respect to the global frame. These coordinates can be any set used in rigid multibody dynamics. As an example, one can take the reference coordinates q2=r2 2,(7) Rope–sheave contact transient analysis in hoisting operations... 485 Fig. 5 Contact rope elements where r2are the global components of the position vector of the origin of the body frame and 2is a set of Euler angles used to describe the orientation of the body frame with respect to the global frame. The sheave is assumed to rotate with respect to Body 2 an angle θ2 sin a body-fixed direction Y2 s. It is assumed that the plane of the sheave is perpendicular to the axis Y2 s.This means that the absolute rotation matrix of the sheave frame O2;X2 s,Y2 s,Z2 scan be obtained as As=A2As 0Aθ,(8) where A2is the absolute rotation matrix of the frame O2;X2,Y2,Z2,As 0is the constant rotation matrix of the sheave frame with respect to the Body 2 frame when θ2 s=0, and Aθ is the rotation matrix due to the sheave rotation given by Aθ=⎡ ⎣ cosθ2 s0sinθ2 s 010 −sinθ2 s0cosθ2 s ⎤ ⎦.(9) The frame of the sheave when θ2 s=0, whose absolute rotation matrix is given by Asi = A2As 0, is called here sheave-intermediate frame. For the full geometric definition of the sheave, in addition to the rotation matrix As 0, the constant local position vector of the sheave center in the Body 2 frame, ¯ u2 s, must be specified. The absolute position vector of the center of the sheave is given by r2 s=r2+A2¯ u2 s.(10) Other geometric vectors that define the rope to sheave tangent points will be defined in Sect. 4.3 of this document. 492 J.L. Escalona The number of interpenetration areas is called nia. For example, in the example shown in Fig. 9, where the rope does not touch the bottom of the sheave groove, nia =2, which in the figure are identified as left land right rareas. It can be deduced that with this groove geometry it is also possible to have a third interpenetration area at the bottom of the groove. In the simple but not realistic case of Fig. 10, there is just one interpenetration area, namely nia =1 (by the way, this is the geometry that is analyzed in most papers on rope–sheave contact). Finding the points of maximum penetration is an optimization procedure that is explained in [19]. With this method, the solution of a set of two nonlinear equations provides the position vector of the points of maximum indentation at each interpenetration area. 2. Calculation of normal contact forces. At each interpenetration area, the normal contact force is calculated using a Hunt–Crossley force model, as follows: fi n=fi nnii=1,...,nia, fi n=Knδi nnn +cdδi n˙ δi nif δi n>0, 0ifδi n≤0, (24) where niis the normal vector to the rope cross-section and δi nis the penetration value at the interpenetration area i,Knis a normal stiffness coefficient, nn is the penetration exponent, and cdis a damping coefficient. The normal contact forces are assumed to be applied at the points of maximum penetration. The stiffness coefficient Knequals the coefficient knused in Eq. (6) divided by the distance between bristles. 3. Calculation of tangential contact forces. At each interpenetration area, the tangential contact force is calculated using a bristle force model, as follows: fi t=⎧ ⎪ ⎨ ⎪ ⎩−μf i n˙ δi t ˙ δi tif bristle end slips, −Ktδi tif bristle end sticks, i=1,...,nia, (25) where δi tis the deformation vector of the bristle in the tangential contact direction, μ is the friction coefficient (at this point, no different is made between static and kinetic coefficients of friction), and Ktis the bristle stiffness in the lateral direction. The stiffness coefficient Ktequals the coefficient ktused in Eq. (6) divided by the distance between bristles. Since the contact forces fi nand fi tgenerated by the bristles are concentrated forces applied at the Lagrangian knots (bristle attachment points to the rope), the calculation of the generalized contact forces that enter the equations of motion does not require the use of quadrature, but the addition of the generalized forces due to all fi nand fi tforces that are applied at varying position within the ALE contact elements. 5.3 Contact geometry Figures 11 and 12 show some details of the contact geometry. The drawing on the left of Fig. 11 shows a global-frontal view of the sheave–rope while the drawing on the right shows a frontal and a lateral-detailed view of a segment of the rope for the case of a sheave with planar profile (without groove). Figure 12 shows the lateral view of a sheave with a V-shape groove. At each point Pof the contact rope and for each interpenetration area i (i=1,...,nia), there are three associated points: Rope–sheave contact transient analysis in hoisting operations... 493 Fig. 10 Rope-to-sheave contact forces with simplified geometry Fig. 11 Details of the contact geometry Fig. 12 Rope contact in a sheave with V-shape groove. Geometry of the right surface contact 1. Point CR is the assumed contact point in the rope. 2. Point CS is the assumed contact point in the sheave. 3. Point BE is the bristle-end point that is in contact with the sheave. Once these three points are identified, the normal penetration δi nis calculated as the (scalar) distance between points CR and CS. The deformation vector of the bristle δi tis the vector that connects points CS and BE. 494 J.L. Escalona The details of the calculation of δi nand δi tand their time derivatives, that are needed to find the contact forces using Eqs. (24)and(25), are given in Annex 1. 5.4 Persistent variables As described in the previous two subsections, the solution of the tangential stick–slip contact problem requires the use of the following set of persistent variables for each interpenetration area iof each contact point P: •A boolean variable “stick” that equals 1 in the stick phase and 0 in the slip phase. •The value of ˘ δi tand ˙ ˘ δi t. They will be needed for the calculation of the tangential contact force during slip phase. •The value of us BE. It will be needed for the calculation of the tangential contact force during the stick phase. •Although the relative velocity vrel BE can be computed as a function of the mentioned persistent variables, to reduce the number of computations, it is convenient to treat it as a persistent variable for the evaluation of Eq. (48) during the slip phase. 5.5 Regularization of the stick–slip contact with the LuGre contact model The stick–slip model presented in the previous subsection has two drawbacks: 1. The 2-states’ contact (stick and slip contact) creates numerical difficulties since it is a nonsmooth dynamic model with unilateral constraints. The reason is that the constraint in the position of the bristle ends us BE that appear in the stick phase do not apply during the slip phase. 2. The so-called static and kinetic coefficients of friction, μsand μk, that are known to be different, are treated as equal. This is because otherwise a time-discontinuous tangential contact force would be obtained that, in turn, would create additional numerical difficulties. These difficulties are avoided if the LuGre tangential contact model is used [20]. The first drawback is avoided because the stick or slip states are not considered. Instead, each bristle is assumed to be permanently in an intermediate state, not totally sticking, not totally slipping, that varies continuously with the relative velocity of the bristle ends BE.Inthe LuGre model, the friction coefficients μsand μkare different. However, this approach does not create a time-discontinuous tangential contact force. Instead, the tangential contact force decreases for increasing relative velocities when the relative velocity is low. In fact, this phenomenon is real and called the Stribeck effect. Thus, the LuGre model also avoids the second drawback of the stick–slip model. On the other hand, the LuGre model has other drawbacks, like the appearance of new parameters, like the so-called Stribeck velocity,that are difficult to obtain in practice but with a very important influence in the simulation results. Besides, the LuGre friction model can produce tangential contact forces that are larger than the saturation friction, which is physically inadmissible. With this model, it is assumed that Eqs. (42)–(45) are always applicable. The magnitude of the tangential contact force is obtained as fi∗ t=fi sl +fi st −fi sle−vrel BE vstr α ,(26) Rope–sheave contact transient analysis in hoisting operations... 495 Fig. 13 Elevator system Table 1 Elevator parameters Cabin and payload mass 1000 kg Cabin initial height 12.0 m Counterweight mass 700 kg Counterweight initial height 8.0 m Radius of drive sheave 300 mm Height of drive sheave 20 m Ride displacement 8 m Nominal velocity 1.0 m/s Number of ropes 5 Rope diameter 10 mm Linear density of all ropes 2.0 kg/m Axial stiffness (EA) of all ropes 40 MN where fi∗ tis a regularized tangential contact force that averages the “stick” tangential force fi st,giveninEq.(42), and the “slip” tangential force fi sl,giveninEq.(45), in such a way that, when the relative velocity vrel BEis small, the tangential contact force is close to fi st, and when vrel BEis large, the tangential contact force is close to fi sl. The so-called Stribeck velocity vstr is a velocity parameter that controls the stick–slip transition. The exponent αis a constant that takes values between 0.5and2. 6 Simulation results The simulation of the simple elevator system shown in Fig. 13 is used to demonstrate the results of the ALEM discretization method and the rope–sheave contact model developed in this investigation. The next subsection shows results in a static position. The following subsection shows the simulation results of an 8-m ride of the system (hoisting operation). The main parameters of the elevator system are shown in Table 1. The main parameters of the rope–sheave contact model are given in Table 2. During the ride, the cabin moves downwards. Although the contact model presented in this paper is fully 3D, and it is capable of accounting for a realistic geometry of the sheave groove, in the results presented in this paper the unrealistic planar groove shown in Fig. 10 has been used. That is, the number of interpenetration areas in the rope cross-section is one, nia =1. The simulation of ropes wound 496 J.L. Escalona Table 2 Rope–sheave contact parameters Number of contact elements 20 Coefficient of friction 0.4 Distance between bristles 5 mm Stribeck velocity 0.01 m/s Normal stiffness/unit length 2.5e8 N/m Normal viscous damping 100 N/(m/s) Tangential stiffness/unit length 3e7 N/m Tangential viscous damping 10 N/m/s Fig. 14 Contact forces. Numerical versus analytical results on sheaves with realistic Vor U-shaped grooves is the subject of a future work. The 3D contact simulation requires a detailed analysis for which there is no room left in this paper. Instead, the results presented in this paper have been selected to analyze the rope–sheave contact theories presented in Sect. 2. To this end, the sheave with a planar groove is an adequate option. Regarding contact parameters given in Table 2, they have been selected using practical considerations: getting a physically acceptable rope–sheave penetration (much smaller than the rope diameter), sufficiently smooth distribution of normal and tangential contact forces, and fast time integration. This means that these parameters have not been experimentally identified. Although parameter identification is needed when using the method described in this paper to industrial applications, this task is considered out of the scope of this paper. 6.1 Static results Figure 14 shows the normal and tangential contact forces in the rope–sheave interface when the rope is considered with zero bending stiffness. Results are compared with the analytical solution presented in [1]. An excellent agreement can be observed. Figure 15 shows the Rope–sheave contact transient analysis in hoisting operations... 497 Fig. 15 Contact forces. Effect of bending stiffness same results for varying bending stiffness of the rope. The following conclusions can be drawn: 1. The higher the bending stiffness, the shorter the contact angle. This is an effect that can be easily understood intuitively. 2. For nonzero bending stiffness, the normal contact forces increase significantly in the entrance and in the exit of the drive sheave. This effect is well documented in the literature, like in the book of Feyrer [21] or the paper by Vetyukov et al. [8]. 3. Tangential contact forces are almost unaffected by the bending stiffness. Figure 16 shows the space evolution of the tension along the segment of the rope wound on the sheave. The plot shows the numerical results and the analytical results presented in [1] in the case of zero bending stiffness of the rope. In general, there is a good agreement, but edge effects can be observed. Figue 17 is the same plot but showing the effect of the bending stiffness of the rope. The most prominent feature is the highly oscillatory behavior of the tension at the edges of the contact area. This effect is documented in the papers by Ntarladima et al. [3] and Vetyukov et al. [8]. The membrane locking effect [4] seems to be responsible of these unrealistic oscillations. Vetyukov et al. [8] showed that this effect can be avoided refining the finite element mesh. The same conclusion is given by Ntarladima et al. [3]. In fact, in their results this effect does not appear when using a minimum of 60 finite elements. Recall that the simulation presented here uses 20 finite element in the contact zone. However, Ntarladima et al. also showed that a specific reduced-integration quadrature can help avoid membrane locking. This quadrature has been used in the present work, with some benefits to avoid membrane locking, but with adverse effects in the accuracy of the 498 J.L. Escalona Fig. 16 Tension along wound rope. Numerical versus analytical results Fig. 17 Tension along wound rope. Effect of bending stiffness solution. Figure 18 is a zoomed view of the previous one in the central part of the contact area. It can be observed that the tension level increases slightly with the bending stiffness. 6.2 Transient dynamic simulation This subsection analyzes the rope–sheave contact forces during an 8-m cabin ride with a steady velocity of V=1m/s. Figure 19 shows the time-history of the cabin velocity and the absolute value of the applied drive torque. Selected bending stiffness of the ropes is EI =10Nm2. Four time instants are selected to analyze the rope sheave contact forces: Rope–sheave contact transient analysis in hoisting operations... 499 Fig. 18 Tension along wound rope. Effect of bending stiffness, zoomed view Fig. 19 Time history of cabin velocity and drive torque 1. t=1.2s, when the drive torque is a minimum. 2. t=5.0s, when the velocity is steady. 3. t=9.2s, when the drive torque is a maximum. 4. t=12.0s, when the system is at rest after the ride. In all the plots shown in Figs. 20–23, the contact forces at the initial instant obtained with static analysis are shown in light gray. In all these figures, it can be observed that the normal contact force distribution changes very little with respect to the static solution. However, the tangential contact forces vary significantly, keeping in all instants a quasilinear spacedistribution. Keeping in mind that the integral under the tangential contact forces (green curves) equals the instantaneous drive torque, the resulting quasilines make sense physically. It is worth highlighting that the situation at the final instant t=12.0s is very similar to the situation at the initial instant t=0.0s, however, the contact force distributions, particularly 500 J.L. Escalona Fig. 20 Contact forces at t=1.2s Fig. 21 Contact forces at t=5.0s Fig. 22 Contact forces at t=9.2s the tangential one, is very different. This result suggests that the static solution presented in Eq. (5) is just one possible static equilibrium contact force distribution. For the transient dynamic simulation, a Matlab code has been developed. The code is not optimized for computational efficiency. The equations of motion were integrated forward in time using the generalized-alpha method with a constant time step of 1 ms. Using a 12th Rope–sheave contact transient analysis in hoisting operations... 501 Fig. 23 Contact forces at t=12.0s Gen Intel Core i7-1260P CPU @ 2.1 GHz processor, the simulation of 12 seconds takes an average of 2 hours and 21 minutes. Clearly, the real-time simulation capability of the ALEM method for the simulation of reeving systems is ruined when modeling the rope– sheave contact with the method described in this work. 6.3 Discussion of the results In Sect. 2, Creep Theory and Firbank’s Theory that are applicable to the analysis of the steady-state rope-sheave contact were compared with Bristle Theory that is applicable to static analysis, with the caveat that they are valid theories in different conditions. Nevertheless, simulation results have shown that in transient dynamics the rope–sheave contact force distributions do not differ much from the static results provided by Bristle Theory. These results suggest that it is worth investigating a rope–sheave contact theory in closed-form that is valid in steady dynamics and takes into account the axial stiffness, shear stiffness, and flattening stiffness of the ropes. Figure 24 shows the contact forces at t=3s, during the steady-state period. Plots on the left are simulation results. Plots on the right are the result of the application of Creep Theory and Firbank’s Theory. In this case, simulation results, and also under Firbank’s Theory, the whole rope–sheave contact segment is in adherence condition. The main difference in the normal contact forces is due to the edge effects introduced by the bending stiffness of the rope. Tangential contact forces are totally different. It is important highlight that in Europe the safety calculations of elevators that are related to the rope–sheave contact follow the standard EN 81-1 [22] that is based on Creep Theory. 7 Summary and conclusions The purpose of this paper was twofold. The first goal was to present a computational model for the dynamic simulation of reeving systems including the detailed analysis of the rope– sheave contact interaction. The second goal was to study and put in question the rope–sheave contact theories that are nowadays used in the industry. The computational model, based on a FEM technique called ALEM, was proposed by the author with coauthors in a few papers in the past. However, in previous works, the rope– sheave contact interaction was not analyzed in detail. The no-slip condition and the torquebalance at the sheaves allowed studying the overall dynamics of the reeving system without 508 J.L. Escalona Declarations Disclaimer The present paper only reflects the author’s view. The European Commission and its Research Executive Agency (REA) are not responsible for any use that may be made of the information it contains. Competing interests The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Escalona, J.L.: An analytical solution of the rope–sheave contact in static conditions based on a bristle model. Mech. Mach. 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