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

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.

1 Introduction

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 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] have developed an ALE 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 stick-and-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 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 3 summarizes the ALEM method for the dynamic analysis of reeving systems. Section 4 describes the new ALE contact elements that have been developed to discretize the rope–sheave contact interface. Section 5 explains 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 1 for the interested reader. Section 7 is devoted to the calculation of the tangential contact forces with the bristle model. Section 1 shows 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 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):

$$\begin{aligned} {T_{C}}\left ( \alpha \right ) &= \textstyle\begin{cases} {T_{1}}{e^{{\mu _{k}}\alpha }}&\text{if}\quad 0 \le \alpha < \beta _{C}, \\ {T_{2}}&\text{if}\quad \beta _{C} \le \alpha \le \pi , \end{cases}\displaystyle \quad \ \ {t_{C}}\left ( \alpha \right ) = \textstyle\begin{cases} \frac{\mu _{k}T_{1}}{R}{e^{\mu _{k}\alpha }}&\text{if}\quad 0 \le \alpha < \beta _{C}, \\ 0&\text{if}\quad \beta _{C} \le \alpha \le \pi , \end{cases}\displaystyle \end{aligned}$$

(1)

$$\begin{aligned} {T_{F}}\left ( \alpha \right ) &= \textstyle\begin{cases} T_{1}{e^{\mu _{k}\alpha }}&\text{if}\quad 0 \le \alpha < \beta _{F}, \\ {a_{1}}\alpha + {b_{1}}&\text{if}\quad \beta _{F} \le \alpha \le \pi , \end{cases}\displaystyle \quad {t_{F}}\left ( \alpha \right ) = \textstyle\begin{cases} \frac{\mu _{k}T_{1}}{R}{e^{\mu _{k}\alpha }}&\text{if}\quad 0 \le \alpha < \beta _{F}, \\ {a_{2}}\alpha + {b_{2}}&\text{if}\quad \beta _{F} \le \alpha \le \pi , \end{cases}\displaystyle \end{aligned}$$

(2)

where $\alpha$ is the angle along the sheave with origin at the low tension end, $T$ is tension (elastic axial force), $t$ is tangential contact force density (${\mathrm{{N}}}/{\mathrm{{m}}}$), subscripts $C$ and $F$ stand for Creep Theory and Firbank’s Theory, respectively, $T_{1}$ and $T_{2}$ are the low and high tensions at the two free spans of the slender body, respectively, $R$ is the radius of the sheave, $\mu _{k}$ is the kinetic coefficient of friction, $\beta _{C} = \frac{1}{{{\mu _{k}}}}\log \frac{{{T_{2}}}}{{{T_{1}}}}$ is the slip angle for Creep Theory, $\beta _{F}$ is the slip angle for Firbank’s Theory, that can be calculated as the solution of the nonlinear equation

$$ {e^{{\mu _{k}}{\beta _{F}}}}\left ( {1 + \frac{{\pi - {\beta _{F}}}}{2}{\mu _{s}}} \right ) - \frac{{{T_{2}}}}{{{T_{1}}}} = 0, $$

(3)

where $\mu _{s}$ is the static coefficient of friction, and the coefficients that define the linear evolution a the adherence angle are given by

$$ {a_{1}} = \frac{{{T_{2}} - {T_{\beta }}}}{{\pi - {\beta _{F}}}},\,\, \,\,\,{b_{1}} = {T_{\beta }} - {a_{1}}{\beta _{F}},\,\,\,\,\,{a_{2}} = \frac{{ - {\mu _{k}}{T_{\beta }}}}{{R\left ( {\pi - {\beta _{F}}} \right )}}, \,\,\,\,\,{b_{2}} = - \pi {a_{2}}, $$

(4)

with $T_{\beta}$ being the tension at $\alpha = {\beta _{F}}$. 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.

Fig. 1

Contact forces at rope–sheave interface

Figure 2 shows an example with $T_{1} = 1 \, \mathrm{{kN}}$, $T_{2} = 3 \, \mathrm{{kN}}$, $R = 1 \, \mathrm{{m}}$, $\mu _{k} = 0.4$, and $\mu _{s} = 0.5$. 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 $T$ and $t$ are 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 $\frac{{{T_{2}}}}{R}$ and 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 ($M_{dr} = (T_{2}-T_{1}) \times R$), 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.

Fig. 2

Contact Forces: comparison of Creep Theory and Firbank’s Theory

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 Bristle Theory, the axial load and the tangential contact forces are given by

$$ \begin{aligned} T\left ( \alpha \right ) &= \textstyle\begin{cases} T_{I}e^{r\alpha } + T_{II}e^{ - r\alpha }&\text{if}\quad 0 \le \alpha < \alpha _{b}, \\ {T_{b}}{e^{{\mu _{s}}\left ( {\alpha - {\alpha _{b}}} \right )}}& \text{if}\quad \alpha _{b} \le \alpha \le \pi , \end{cases}\displaystyle \\ t\left ( \alpha \right ) &= \textstyle\begin{cases} \frac{r}{R}\left ( T_{I}e^{r\alpha} - T_{II}e^{ - r\alpha } \right )& \text{if}\quad 0 \le \alpha < \alpha _{b}, \\ \frac{\mu}{R}T_{b}e^{{\mu _{s}}\left ( \alpha - \alpha _{b} \right )}& \text{if}\quad \alpha _{b} \le \alpha \le \pi , \end{cases}\displaystyle \end{aligned} $$

(5)

where $\alpha _{b}$ is the limit angle of the adherence arc ($\pi$ minus the angle of the slip arc), $T_{I}$, $T_{II}$, and $T_{b}$ are reference values of the tension that can be calculated as shown in [1], and the exponent $r$ is given by

$$ \begin{aligned} r &= \sqrt {\frac{{{k_{t}}{R^{2}}}}{{E{A^{*} }}}}, \\ E{A^{*} } &= \frac{{EA + {k_{n}}{R^{2}}}}{{EA \times {k_{n}}{R^{2}}}}, \end{aligned} $$

(6)

with $EA$ being the axial stiffness of the slender body, $k_{t}$ the transverse stiffness of the bristles (shear stiffness), and $k_{n}$ the normal stiffness of the bristles (stiffness of the cross-section to flattening). Figure 3 shows the normal and tangential contact forces obtained using the Bristle Theory with the same conditions used to create the plots in Fig. 2.

Fig. 3

Contact Forces: Bristle Theory

As mentioned, Eqs. (1)–(2) and Eq. (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, $T_{1}$ and $T_{2}$, the radius of the sheave, $R$, and the coefficients of friction, $\mu _{k}$ and $\mu _{s}$. The contact forces obtained with Bristle Theory also depend on the axial ($EA$), shear ($k_{t}$), and flattening ($k_{n}$) 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.

• 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 $r$ being a function of the different stiffness constants of the slender body. An exponent similar to $r$ was 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. 1 on 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, ${{\mathbf{{q}}}_{a}} = {\left [ { \begin{array}{c@{\quad}c} {{{\mathbf{{r}}}_{1}}}&{{{\mathbf{{r}}}_{2}}} \end{array} } \right ]^{T}}$;

  • 2. A set of modal coordinates to describe transverse and axial deformation in an element frame, ${{\mathbf{{q}}}_{m}} = {\left [ { \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} {{q_{x,1}}}& \ldots &{{q_{x,nx}}}&{{q_{y,1}}}& \ldots &{{q_{y,ny}}}&{{q_{z,1}}}& \ldots &{{q_{z,nz}}} \end{array} } \right ]^{T}}$;

  • 3. A set of twist angles to describe torsion, ${{\mathbf{{q}}}_{\theta }} = {\left [ { \begin{array}{c@{\quad}c} {{\theta _{1}}}&{{\theta _{2}}} \end{array} } \right ]^{T}}$;

  • 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, ${{\mathbf{{q}}}_{s}} = {\left [ { \begin{array}{c@{\quad}c} {{s_{1}}}&{{s_{2}}} \end{array} } \right ]^{T}}$.

Fig. 4

Example of a reeving system

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. 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 5 shows 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 a and b model the free-spans of the rope next to the sheave. In the zoomed view of the sheave, the set of ALE rope-contact elements $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ can be observed. This section describes the properties of these elements, that are specially designed to model the rope–sheave contact.

Fig. 5

Contact rope elements

Regarding the rigid body, a set of coordinates ${{\mathbf{{q}}}^{2}}$ is 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

$$ {{\mathbf{{q}}}^{2}} = \left [ { \textstyle\begin{array}{c} {{{\mathbf{{r}}}^{2}}} \\ \boldsymbol{\Theta }^{2} \end{array}\displaystyle } \right ] ,$$

(7)

where ${{{\mathbf{{r}}}^{2}}}$ are the global components of the position vector of the origin of the body frame and $\boldsymbol{\Theta }^{2}$ is 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 $\theta _{s}^{2}$ in a body-fixed direction $Y_{s}^{2}$. It is assumed that the plane of the sheave is perpendicular to the axis $Y_{s}^{2}$. This means that the absolute rotation matrix of the sheave frame $\left \langle {{O^{2}};X_{s}^{2},Y_{s}^{2},Z_{s}^{2}} \right \rangle$ can be obtained as

$$ {{\mathbf{{A}}}^{s}} = {{\mathbf{{A}}}^{2}}{\mathbf{{A}}}_{0}^{s}{{\mathbf{{A}}}_{\theta }} ,$$

(8)

where ${{\mathbf{{A}}}^{2}}$ is the absolute rotation matrix of the frame $\left \langle {{O^{2}};{X^{2}},{Y^{2}},{Z^{2}}} \right \rangle$, ${\mathbf{{A}}}_{0}^{s}$ is the constant rotation matrix of the sheave frame with respect to the Body 2 frame when $\theta _{s}^{2} = 0$, and ${{\mathbf{{A}}}_{\theta }}$ is the rotation matrix due to the sheave rotation given by

$$ {{\mathbf{{A}}}_{\theta }} = \left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c} {\cos \theta _{s}^{2}}&0&{\sin \theta _{s}^{2}} \\ 0&1&0 \\ { - \sin \theta _{s}^{2}}&0&{\cos \theta _{s}^{2}} \end{array}\displaystyle } \right ] .$$

(9)

The frame of the sheave when $\theta _{s}^{2} = 0$, whose absolute rotation matrix is given by ${{\mathbf{{A}}}^{si}} = {{\mathbf{{A}}}^{2}}{\mathbf{{A}}}_{0}^{s}$, is called here sheave-intermediate frame. For the full geometric definition of the sheave, in addition to the rotation matrix ${\mathbf{{A}}}_{0}^{s}$, the constant local position vector of the sheave center in the Body 2 frame, ${\mathbf{{\bar{u}}}}_{s}^{2}$, must be specified. The absolute position vector of the center of the sheave is given by

$$ {\mathbf{{r}}}_{s}^{2} = {{\mathbf{{r}}}^{2}} + {{\mathbf{{A}}}^{2}}{\mathbf{{\bar{u}}}}_{s}^{2} .$$

(10)

Other geometric vectors that define the rope to sheave tangent points will be defined in Sect. 4.3 of this document.

4.2 ALE contact elements

The rope-contact elements $c_{i}$ are equivalent to the cubic ALE elements defined in [17, 18]. The nodal coordinates of this ALE element are

$$ {\mathbf{{q}}} = \left [ { \textstyle\begin{array}{c} {{{\mathbf{{q}}}_{a}}} \\ {{{\mathbf{{q}}}_{\theta }}} \\ {{{\mathbf{{q}}}_{s}}} \end{array}\displaystyle } \right ] ,$$

(11)

where ${{{\mathbf{{q}}}_{\theta }}}$ and ${{{\mathbf{{q}}}_{s}}}$ are the same twist and arc-length nodal coordinates used in the ALEM-elements $a$ and $b$, but the absolute coordinates ${{{\mathbf{{q}}}_{a}}}$ include position vectors ${{\mathbf{{r}}}}_{i}$ and slopes ${{\mathbf{{r'}}}}_{i}$, as follows:

$$ \begin{aligned} {{\mathbf{{q}}}_{a}} &= {\left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} {{{\mathbf{{r}}}_{1}}^{T}}&{{{{\mathbf{{r'}}}}_{1}}^{T}}&{{{\mathbf{{r}}}_{2}}^{T}}&{{{{ \mathbf{{r'}}}}_{2}}^{T}} \end{array}\displaystyle } \right ]}^{T}, \\ {{{\mathbf{{r'}}}}_{i}} &= \frac{{\partial {{\mathbf{{r}}}_{i}}}}{{\partial s}},{ \mathrm{{ }}}\quad i = {\mathrm{{1}}}{\mathrm{{,2}}}{\mathrm{{.}}} \end{aligned} $$

(12)

However, the ALE rope-contact element does not include modal coordinates because they are considered needless. Therefore, the ALE rope-contact element includes 12 + 2 + 2 = 16 nodal coordinates.

The absolute position vector of an arbitrary point in the center line of the element is given by

$$ \begin{aligned} {\mathbf{{r}}} &= {\mathbf{{N}}}{{\mathbf{{q}}}_{a}} \\ {\mathbf{{N}}} &= \left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} {{N_{1}}{{\mathbf{{I}}}_{3 \times 3}}}&{{N_{2}}{{\mathbf{{I}}}_{3 \times 3}}}&{{N_{3}}{{ \mathbf{{I}}}_{3 \times 3}}}&{{N_{4}}{{\mathbf{{I}}}_{3 \times 3}}} \end{array}\displaystyle } \right ] \\ {N_{1}} &= \frac{1}{4}{\left ( {\xi - 1} \right )^{2}}\left ( {2 + \xi } \right ){\mathrm{{, }}}\quad {N_{2}} = \frac{{{s_{2}} - {s_{1}}}}{8}{ \left ( {\xi - 1} \right )^{2}}\left ( {\xi + 1} \right ), \\ {N_{3}} &= \frac{1}{4}{\left ( {\xi + 1} \right )^{2}}\left ( {2 - \xi } \right ){\mathrm{{, }}}\quad {N_{4}} = \frac{{{s_{2}} - {s_{1}}}}{8}{ \left ( {\xi + 1} \right )^{2}}\left ( {\xi - 1} \right ), \\ \xi &= \frac{{2s - {s_{1}} - {s_{2}}}}{{{s_{2}} - {s_{1}}}}{\mathrm{{ }}}, \end{aligned} $$

(13)

where ${\mathbf{{N}}}$ is the cubic shape function matrix.

4.3 Arc-length coordinates of ALE rope-contact element

In the original ALEM formulation in which the rope to sheave contact was not modeled, the arc-length coordinates of the elements $a$ and $b$ at the nodes tangent of the sheave, that is, $s_{2}^{a}$ and $s_{1}^{b}$ were calculated using the no-slip and continuity conditions:

$$ \begin{aligned} s_{2}^{a} &= s_{2}^{a}\left ( 0 \right ) - \theta _{s}^{2}R, \\ s_{1}^{b} &= s_{2}^{a}, \end{aligned} $$

(14)

where $s_{2}^{a}\left ( 0 \right )$ is the initial value of those nodal coordinates and $R$ is the sheave radius. The second equation in Eq. (14) is a connectivity condition implying that the rope segment in contact with the sheave does not exist, that is, it is not modeled.

Another important contact assumption made in the original ALEM formulation is that the position of the two tangent points of the rope to the sheave are known at each time instant. Of course, this is not accurate in reality because the total contact-arc in the sheave can vary with time at both ends of the contact region, but it is a reasonable assumption in most reeving systems used in real life. In the formulation based on ALE rope-contact elements, no assumption is made about the position of the contact points. This is implemented by using contact elements at both ends of the rope segment in contact with the sheave that overflow the theoretical contact arc. In other words, the new ALE rope-contact elements guarantee that the “entry” element $c_{1}$ and the “exit” element $c_{nc}$, with $nc$ being the number of contact elements, are partially in contact with the sheave. This condition is imposed using constraints on the value of $s_{1}^{c1}$ and $s_{2}^{cnc}$ that are explained next.

In reeving systems, the approximate position of the tangent points of the rope to the sheave, $t1$ and $t2$ shown in blue in Fig. 6, are assumed to be known in the design phase, as is the contact arc $D_{s}$. For each sheave in the reeving system, the following constant vectors can be defined:

$$ {{\mathbf{{\bar{u}}}}_{ti}},\,\,{{\mathbf{{\bar{t}}}}_{ti}},\quad i = 1,2, $$

(15)

where ${{\mathbf{{\bar{u}}}}_{ti}}$ and ${{\mathbf{{\bar{t}}}}_{ti}}$ are the position vector of tangent point $i$ with respect to the center of the sheave and the tangent vector to the tangent point $i$. Both vectors are projected to the sheave-intermediate frame. The following nonlinear constraints are used to calculate the value of $s_{1}^{c1}$ and $s_{2}^{cnc}$:

$$ \begin{aligned} {\left ( {{\mathbf{{u}}}_{1}^{c1} - {{\mathbf{{A}}}^{si}}{{{\mathbf{{ \bar{u}}}}}_{t1}}} \right )^{T}}\left ( {{{\mathbf{{A}}}^{si}}{{{\mathbf{{\bar{t}}}}}_{t1}}} \right ) + {d_{s}} &= 0, \\ {\left ( {{\mathbf{{u}}}_{2}^{cnc} - {{\mathbf{{A}}}^{si}}{{{\mathbf{{\bar{u}}}}}_{t2}}} \right )^{T}}\left ( {{{\mathbf{{A}}}^{si}}{{{\mathbf{{\bar{t}}}}}_{t2}}} \right ) - {d_{s}} &= 0. \end{aligned} $$

(16)

Fig. 6

Modeling of the rope–sheave contact

These constraints mean that the dot product of the red vectors in Fig. 6 ($\vec{u}_{1}^{c1} - {\vec{u}_{t1}}$ and $\vec{u}_{2}^{cnc} - {\vec{u}_{t2}}$) times the tangents at the end-contact points equals $-d_{s}$ or $d_{s}$, respectively. Distance $d_{s}$ is selected by the user to guarantee that the end elements completely cover the rope–sheave contact zone. It is clear that these two scalar constraints, which can be used to find $s_{1}^{c1}$ and $s_{2}^{cnc}$ at each instant of time, guarantee that the distance from the nodal points of the contact elements to the tangent points to the sheave is approximately equal to $d_{s}$. In these equations, vectors ${\mathbf{{u}}}_{1}^{c1}$ and ${\mathbf{{u}}}_{2}^{cnc}$ are calculated as

$$ \begin{aligned} {\mathbf{{u}}}_{1}^{c1} &= {\mathbf{{r}}}_{1}^{c1} - {\mathbf{{r}}}_{s}^{2}, \\ {\mathbf{{u}}}_{2}^{cnc} &= {\mathbf{{r}}}_{2}^{cnc} - {\mathbf{{r}}}_{s}^{2}, \end{aligned} $$

(17)

where ${\mathbf{{r}}}_{1}^{c1}$ and ${\mathbf{{r}}}_{2}^{cnc}$ are the absolute position vectors of the contact-rope end nodes, which are functions of the nodal coordinates, and ${\mathbf{{r}}}_{s}^{2} = {{\mathbf{{r}}}^{2}} + {{\mathbf{{A}}}^{2}}{\mathbf{{\bar{u}}}}_{s}^{2}$ is the absolute position of the sheave center.

Once $s_{1}^{c1}$ and $s_{2}^{cnc}$ are calculated, assuming equal-length contact elements, the following equation can be used to find the value of the initial arc-length coordinate for all contact elements:

$$ s_{1}^{ci} = s_{1}^{c1} + (i - 1) \times \frac{{s_{2}^{cnc} - s_{1}^{c1}}}{{nc}},\quad i = 1,2,\dots ,nc. $$

(18)

4.4 Connectivity conditions of contact elements

Connectivity conditions at the contact element nodal points are:

$$ \begin{aligned} {\mathbf{{r}}}_{2}^{ci} &= {\mathbf{{r}}}_{1}^{c(i + 1)}, \\ {\mathbf{{r'}}}_{2}^{ci} &= {\mathbf{{r'}}}_{1}^{c(i + 1)}, \\ \theta _{2}^{ci} &= \theta _{1}^{c\left ( {i + 1} \right )}, \\ s_{2}^{ci} &= s_{1}^{c\left ( {i + 1} \right )}. \end{aligned} $$

(19)

These conditions guarantee continuity in absolute position, slope, twist angle, and arc-length coordinate.

The connectivity conditions between elements $a$ and $c_{1}$, or $c_{2}$ and $b$, are not that obvious. The reason is that elements $a$ and $b$ do not have the slope coordinates that elements $c_{1}$ or $c_{2}$ do. On top of that, elements $a$ and $b$ include transverse modal coordinates that change the direction of the deformed shape of the element central line with respect to the slope of the contact elements at the nodal points. In addition to the following connectivity conditions:

$$ \begin{aligned} {\mathbf{{r}}}_{2}^{a} &= {\mathbf{{r}}}_{1}^{c1},\quad {\mathbf{{r}}}_{2}^{cnc} = {\mathbf{{r}}}_{1}^{b}, \\ \theta _{2}^{a} &= \theta _{1}^{c1},\quad \theta _{2}^{cnc} = \theta _{1}^{b}, \\ s_{2}^{a} &= s_{1}^{c1},\quad s_{2}^{cnc} = s_{1}^{b}, \end{aligned} $$

(20)

more complex equations must be accounted for to guarantee a common tangent to the central line at the nodal elements, thus avoiding the existence of a “kink” which is physically inadmissible. These equations are explained next.

The position vector of an arbitrary point in the ALEM elements ($a$ or $b$) is given by

$$ {\mathbf{{r}}} = {\mathbf{{N}}}{{\mathbf{{q}}}_{a}} + {{\mathbf{{A}}}^{e}}{\mathbf{{S}}}{{\mathbf{{q}}}_{m}} ,$$

(21)

where $\mathbf{{N}}$ is a shape function matrix that just contains linear polynomials, ${\mathbf{{A}}}^{e}$ is the element rotation matrix, and ${\mathbf{{S}}}$ is the matrix that contain the modal sine-functions. Therefore, the tangent vector to the centerline at an arbitrary point is given by

$$ {\mathbf{{t}}} = \frac{d}{{ds}}{\mathbf{{r}}} = \frac{{d{\mathbf{{N}}}}}{{ds}}{{\mathbf{{q}}}_{a}} + {{\mathbf{{A}}}^{e}}\frac{{d{\mathbf{{S}}}}}{{ds}}{{\mathbf{{q}}}_{m}} = {\mathbf{{N'}}}{{ \mathbf{{q}}}_{a}} + {{\mathbf{{A}}}^{e}}{\mathbf{{S'}}}{{\mathbf{{q}}}_{m}} .$$

(22)

Therefore, the slope vectors at the entrance and exit contact rope elements can be calculated as a function of the wire rope ALEM elements using the following formulas:

$$ \begin{aligned} {\mathbf{{r'}}}_{1}^{c1} &= {\mathbf{{N'}}}\big( {s = s_{2}^{a}} \big){\mathbf{{q}}}_{a}^{a} + {{\mathbf{{A}}}^{e}}{\mathbf{{S'}}}\big( {s = s_{2}^{a}} \big){\mathbf{{q}}}_{m}^{a}, \\ {\mathbf{{r'}}}_{2}^{cnc} &= {\mathbf{{N'}}}\big( {s = s_{1}^{b}} \big){\mathbf{{q}}}_{a}^{b} + {{\mathbf{{A}}}^{e}}{\mathbf{{S'}}}\big( {s = s_{1}^{b}} \big){\mathbf{{q}}}_{m}^{b}. \end{aligned} $$

(23)

As explained in Sect. 3, the work presented in this paper is an attempt to perform a detailed analysis of the rope–sheave contact interaction within the context of the ALEM method for modeling and simulation of reeving systems. The ALEM method is very efficient. One reason is that only one ALEM element is required to discretize an entire free rope span thanks to the use of modal amplitudes as element coordinates. However, these ALE elements are not appropriate for contact modeling because the mode shapes are not appropriate to describe the quasicircular shape of the rope segment in contact with the sheave. Of course, the question of connectivity conditions or nodal coordinate constraints will be much simpler if the entire reeving system were discretized with a single element type. However, this approach will ruin all the benefits of the ALEM method. Furthermore, as described in [17], ALE-ANCF cubic elements are not appropriate for modeling transverse string vibrations when a small number of elements is used.

5 Rope–sheave contact model

This section describes the complete modeling technique used to analyze the rope–sheave contact interaction in dynamic simulations.

5.1 Bristle comb

The rope–sheave contact model that is going to be used in this work is based on the bristle model described in [1]. Since an ALE approach is used in this work to discretize the ropes, the nodal points cannot be used to locate the assumed bristles attached to the rope. The reason is that nodes are not necessarily attached to material points in the rope. However, the bristle root point at the rope centerline and the bristle end at the sheave groove need to be related to material points. Geometric points do not work for “carrying” the bristles. Understanding this fact is simple when thinking, for example, about stick contact. When stick contact happens, material points in one of the bodies have the same position and velocity as other material points in the other contacting body. Geometric points that move along the rope centerline are not valid to describe stick contact. Therefore, describing the rope–sheave contact requires the use of a double mesh. One mesh to define the rope elements (free rope-span elements and contact elements) and another mesh that defines the points of attachment of the bristles.

Figure 7 shows a simple elevator-like mechanism that includes a cabin and a counterweight connected with a rope that is wound in a drive sheave. For convenience, the rope is represented very thick. The mechanism is represented in two different positions in order to understand the configuration of the two meshes. The nodes of the ALE mesh are represented with black circles while the knots used to attach the bristles are represented with blue circles.

Fig. 7

ALE and bristle meshes in a rope wound on a sheave. Physical space

The ALE mesh includes the two ALEM (free rope-span) elements $a$ and $b$, and four ALE contact elements $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$. The nodes that belong to contact elements remains approximately fixed in space (Eulerian nodes) while the nodes connected to ALEM elements and to the cabin and counterweight are linked to material points (Lagrangian nodes). All the knots used to attach the bristles are linked to material points (Lagrangian knots).

Figure 8 shows the two meshes in an undeformed-reference configuration of the rope in the same two positions of the system represented in Fig. 7. As it can be observed, the arc-length coordinates associated with the first node of $c_{1}$ and the second node of $c_{2}$ move along the rope centerline. It can also be observed that only the bristles that instantaneously belong to contact elements are deformed (are “active”). Clearly, when the system moves, bristles change the element to which they belong. During dynamic simulation, one has to keep track of the bristles that instantaneously belong to the contact elements for the calculation of the normal and tangential contact forces.

Fig. 8

ALE and bristle meshes in a rope wound on a sheave. Reference-undeformed space

The ALE mesh and the bristle mesh cover the whole rope, but they are independent. The Lagrangian knots, or points of attachment of the bristles to the rope, can change the ALE elements to which they belong. During simulation, they eventually cross the ALE nodes as the ALE elements vary their position along the rope. Only when the Lagrangian knots lie within any contact element $c_{i}$, they are active, this is, they produce contact forces if rope–sheave penetration exists. It is important to highlight that the ALE nodal coordinates are the generalized coordinates of the system that appear as integration variables in the equations of motion. The position of the Lagrangian knots does not require generalized coordinates for their definition, since they occupy fixed positions within the rope that is defined just using a constant distance between the bristles.

5.2 Rope-to-sheave contact forces

The interaction between the rope contact elements and the sheaves is due to the contact forces. In this investigation, an elastic or penalty approach is used. This means that the volume of the rope is allowed to penetrate the volume of the sheave and the normal contact forces are calculated as a function of the penetration and, possibly, the velocity of penetration.

Figure 9 shows a rope contact element wound on a sheave. In this element, an interior point in the centerline of the rope is highlighted together with the normal and tangential contact forces, $f_{n}$ and $f_{t}$, respectively, applied on it. These are forces per unit length, such that the total contact forces are obtained as the line integral along the element centerline of these forces. The forces $f_{n}$ and $f_{t}$ can be considered as the integrals in the cross-section of the rope of the stresses that appear in the contact areas. In fact, in addition to these forces, a resultant torque due to the contact forces needs to be considered as well. As it can be observed in the side view of the sheave in Fig. 9, the forces in the contact area can appear in separated regions that depend mainly on the geometry of the sheave groove.

Fig. 9

Rope-to-sheave contact forces

The generalized rope–sheave normal and tangential contact forces ${\mathbf{{Q}}}_{n}^{ci}$ and ${\mathbf{{Q}}}_{t}^{ci}$, respectively, are computed in this investigation using quadrature. This means that the forces $f_{n}$ and $f_{t}$ must be calculated for each time instant at a set of grid points in the element centerline to calculate the generalized contact forces applied in the element. The calculation of the normal and tangential contact forces in the grid points follows the sequence:

• 1. Contact search. (1) Find the interpenetration areas between the rope cross-section and the sheave groove and (2) find the points of maximum penetration at each of these areas. 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 $l$ and right $r$ areas. 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.

  Fig. 10

  

  Rope-to-sheave contact forces with simplified geometry

• 2. Calculation of normal contact forces. At each interpenetration area, the normal contact force is calculated using a Hunt–Crossley force model, as follows:

  $$ \begin{aligned} {\mathbf{{f}}}_{n}^{i} &= f_{n}^{i}{{\mathbf{{n}}}^{i}}\,\,\,\,\,i = 1, \dots ,nia, \\ f_{n}^{i} &= \textstyle\begin{cases} {{K_{n}}{{\left ( {\delta _{n}^{i}} \right )}^{nn}} + {c_{d}}\delta _{n}^{i} \dot{\delta}_{n}^{i}}&{{\text{if}}\,\,\,\,\delta _{n}^{i} > 0,} \\ 0&{{\text{if}}\,\,\,\,\delta _{n}^{i} \le 0,} \end{cases}\displaystyle \end{aligned} $$

  (24)

  where $\mathbf{{n}^{i}}$ is the normal vector to the rope cross-section and $\delta _{n}^{i}$ is the penetration value at the interpenetration area $i$, $K_{n}$ is a normal stiffness coefficient, $nn$ is the penetration exponent, and $c_{d}$ is a damping coefficient. The normal contact forces are assumed to be applied at the points of maximum penetration. The stiffness coefficient $K_{n}$ equals the coefficient $k_{n}$ used 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:

  $$ {\mathbf{{f}}}_{t}^{i} = \textstyle\begin{cases} - \mu f_{n}^{i} \dfrac{{{\mathbf{{\dot{\delta}}}}_{t}^{i}}}{{\big| {{\mathbf{{\dot{\delta}}}}_{t}^{i}} \big|}}& \text{if bristle end slips}, \\ - {K_{t}}\boldsymbol{\delta}_{t}^{i}&\text{if bristle end sticks,} \end{cases}\displaystyle \quad i = 1,\dots ,nia, $$

  (25)

  where $\boldsymbol{\delta}_{t}^{i}$ is the deformation vector of the bristle in the tangential contact direction, $\mu$ is the friction coefficient (at this point, no different is made between static and kinetic coefficients of friction), and $K_{t}$ is the bristle stiffness in the lateral direction. The stiffness coefficient $K_{t}$ equals the coefficient $k_{t}$ used in Eq. (6) divided by the distance between bristles.

Since the contact forces ${\mathbf{{f}}}_{n}^{i}$ and ${\mathbf{{f}}}_{t}^{i}$ generated 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 ${\mathbf{{f}}}_{n}^{i}$ and ${\mathbf{{f}}}_{t}^{i}$ forces 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 $P$ of the contact rope and for each interpenetration area $i$ ($i = 1, \ldots ,nia$), there are three associated points:

• 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.

Fig. 11

Details of the contact geometry

Fig. 12

Rope contact in a sheave with V-shape groove. Geometry of the right surface contact

Once these three points are identified, the normal penetration $\delta _{n}^{i}$ is calculated as the (scalar) distance between points $CR$ and $CS$. The deformation vector of the bristle $\boldsymbol{\delta}_{t}^{i}$ is the vector that connects points $CS$ and $BE$.

The details of the calculation of ${\delta}_{n}^{i}$ and $\boldsymbol{\delta}_{t}^{i}$ and 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 $i$ of each contact point $P$:

• A boolean variable “stick” that equals 1 in the stick phase and 0 in the slip phase.

• The value of $\breve{\boldsymbol{\delta}} _{t}^{i}$ and $\dot{\breve{\boldsymbol{\delta}}} _{t}^{i}$. They will be needed for the calculation of the tangential contact force during slip phase.

• The value of ${\mathbf{{u}}}_{BE}^{s}$. It will be needed for the calculation of the tangential contact force during the stick phase.

• Although the relative velocity ${\mathbf{{v}}}_{BE}^{rel}$ 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 ${\mathbf{{u}}}_{BE}^{s}$ that appear in the stick phase do not apply during the slip phase.

• 2. The so-called static and kinetic coefficients of friction, ${\mu _{s}}$ and ${\mu _{k}}$, 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 $BE$. In the LuGre model, the friction coefficients ${\mu _{s}}$ and ${\mu _{k}}$ are 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

$$ f_{t}^{i*} = f_{sl}^{i} + \left ( {f_{st}^{i} - f_{sl}^{i}} \right ){e^{ - {{\left ( { \frac{{\left | {{\mathbf{{v}}}_{BE}^{rel}} \right |}}{{{v_{str}}}}} \right )}^{ \alpha }}}} ,$$

(26)

where $f_{t}^{i*}$ is a regularized tangential contact force that averages the “stick” tangential force $f_{st}^{i}$, given in Eq. (42), and the “slip” tangential force $f_{sl}^{i}$, given in Eq. (45), in such a way that, when the relative velocity $\big| {{\mathbf{{v}}}_{BE}^{rel}} \big|$ is small, the tangential contact force is close to $f_{st}^{i}$, and when $\big| {{\mathbf{{v}}}_{BE}^{rel}} \big|$ is large, the tangential contact force is close to $f_{sl}^{i}$. The so-called Stribeck velocity $v_{str}$ is a velocity parameter that controls the stick–slip transition. The exponent $\alpha$ is a constant that takes values between 0.5 and 2.

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).

Fig. 13

Elevator system

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.

Table 1 Elevator parameters

Table 2 Rope–sheave contact parameters

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 on sheaves with realistic V- or 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 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.

Fig. 14

Contact forces. Numerical versus analytical results

Fig. 15

Contact forces. Effect of 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 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.

Fig. 16

Tension along wound rope. Numerical versus analytical results

Fig. 17

Tension along wound rope. Effect of bending stiffness

Fig. 18

Tension along wound rope. Effect of bending stiffness, zoomed view

6.2 Transient dynamic simulation

This subsection analyzes the rope–sheave contact forces during an 8-m cabin ride with a steady velocity of $V = 1 \, \mathrm{{m/s}}$. 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 $EI = 10 \, \mathrm{{Nm^{2}}}$. Four time instants are selected to analyze the rope sheave contact forces:

• 1. $t = 1.2 \, \mathrm{{s}}$, when the drive torque is a minimum.

• 2. $t = 5.0 \, \mathrm{{s}}$, when the velocity is steady.

• 3. $t = 9.2 \, \mathrm{{s}}$, when the drive torque is a maximum.

• 4. $t = 12.0 \, \mathrm{{s}}$, when the system is at rest after the ride.

Fig. 19

Time history of cabin velocity and drive torque

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 space-distribution. 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 $t = 12.0 \, \mathrm{{s}}$ is very similar to the situation at the initial instant $t = 0.0 \, \mathrm{{s}}$, however, the contact force distributions, particularly 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.

Fig. 20

Contact forces at $t = 1.2 \, \mathrm{{s}}$

Fig. 21

Contact forces at $t = 5.0 \, \mathrm{{s}}$

Fig. 22

Contact forces at $t = 9.2 \, \mathrm{{s}}$

Fig. 23

Contact forces at $t = 12.0 \, \mathrm{{s}}$

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 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 $t = 3 \, \mathrm{{s}}$, 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.

Fig. 24

Comparison of simulated contact forces at $t = 3 \, \mathrm{{s}}$ with Creep Theory and Firbank’s Theory

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 torque-balance at the sheaves allowed studying the overall dynamics of the reeving system without entering in the details of the contact. In this paper, a contact model based on the modeling of virtual bristles that are attached to the rope centerline was used to simulate the rope–sheave contact. After summarizing the ALEM method, new cubic ANCF-ALE contact elements were defined specifically to model the rope–sheave contact area. The contact model based on bristles requires the use of a double mesh: one ALE mesh for the global deformation analysis of the rope and one bristle-mesh for the contact analysis that in practice is just active in the contact arc with the sheaves. The contact model is based on Coulomb friction and it can be used to analyze the stick–slip contact of a rope wound on a sheave with an arbitrary groove geometry. The concept of interpenetration areas at the cross-section of the rope, each of which has an associated independent bristle, allows for the 3D analysis of the contact. The kinematics analysis of the contact interaction, that is needed to calculate the normal and tangential contact forces, is quite cumbersome. An LuGre regularization procedure has been used to facilitate the numerical integration of the resulting equations of motion.

The rope–sheave contact theories that are used in the industry are mainly based on two closed-form solutions, Creep Theory and Firbank’s Theory. These theories are both valid for steady-state dynamic analysis. A recently proposed contact theory developed by the author, called here Bristle Theory, is just valid for static analysis. A comparison of the normal and tangential contact force distributions that these three theories provide shows very important fundamental differences. Bristle Theory just adds a new parameter, the exponent $r$, to the closed-form solution. This parameter is a function of the axial stiffness, the shear stiffness, and the flattening stiffness of the rope.

Simulation results show that the ALEM method with the bristle contact model is able to accurately simulate the contact forces in static conditions with a relatively small number of elements. The tension along the rope suffers of membrane locking that needs to be fixed in the future. The normal contact forces are strongly influenced by the bending stiffness of the rope at the edges of the contact. Transient dynamic simulation shows that the normal contact forces are almost equal to those that appear in static analysis with Bristle Theory. The tangential contact forces remain linear during transient dynamics along the contact arc, adapting the shape of the line to the drive torque that is instantaneously transmitted at the sheave. Simulation results show that contact force distributions obtained with Creep Theory and Firbank’s Theory are completely different from those obtained with the ALEM method and the bristle contact model, but not so different from the results obtained with the static Bristle Model. The differences are particularly important in the tangential direction.

A very important conclusion of this paper, from the design point of view of reeving systems, is that the classical rope–sheave contact theories must be revised. In the future, the author will search for an extension of Bristle Model to dynamic steady-state conditions. For practical applications, a new theory could be more accurate and it would probably require the experimental calculation of just one additional parameter, the exponent $r$, that is known to be a function of the relative stiffness of the ropes under different modes of deformation.

Appendix 1: Calculations in contact geometry

1.1 1.1 Contact kinematics

At each point $P$ of the centerline of the rope, two new frames are defined. The P-sheave frame, $\left \langle {P\,;\,\,{X^{P}},{Y^{P}},\,\,{Z^{P}}} \right \rangle$, shown in Fig. 12, is a frame that results from the rotation of the sheave frame about the axis $Y_{s}^{2}$ an angle ${\alpha ^{P}}$ such that the point $P$ is contained in the plane $\left \langle {{Y^{P}},{Z^{P}}} \right \rangle$. The contact frame $\left \langle {P\,;\,\,{\mathbf{{t}}}_{1}^{Pi},\,\,{\mathbf{{t}}}_{2}^{Pi},{{\mathbf{{n}}}^{Pi}}} \right \rangle$, $i = 1,\dots ,nia$ is defined such that the normal to the contact surface ${\mathbf{{n}}}^{Pi}$ makes an angle ${\beta ^{i}}$ with the axis ${Z^{P}}$, as shown in Fig. 12 for the left contact (superscript $l$ in the figure stands for left) of a sheave with a V-shaped groove. Angle ${\beta ^{i}}$ defines the orientation of the contact surface in the diametric section of the sheave. Note that only in the case of a planar sheave, like that shown in Fig. 11, vector ${\mathbf{{t}}}_{2}^{Pi}$ is parallel to axis $Y_{s}^{2}$, however, vector ${ \mathbf{{t}}}_{1}^{Pi}$ is parallel to axis regardless the geometry of the sheave groove. Therefore, at each point $P$ of the rope centerline, there are as many contact frames as interpenetration areas, $nia$. The normal contact force is aligned with ${\mathbf{{n}}}^{Pi}$ and the tangential contact force is contained in the tangent plane $\left \langle {{\mathbf{{t}}}_{1}^{Pi},\,\,{\mathbf{{t}}}_{2}^{Pi}} \right \rangle$, as it is the deformation vector of the bristle $\boldsymbol{\delta} _{t}^{i}$. This means that the bristle is assumed to show a 3D deformation that is not contained in the plane $\left \langle {X_{s}^{2},Z_{s}^{2}} \right \rangle$ of the sheave.

The calculation of the contact frame at each time instant starts with the calculation of the position of the rope point $P$ with respect to the sheave center, as follows:

$$ {\mathbf{{u}}}_{P}^{s} = {{\mathbf{{r}}}^{P}} - {\mathbf{{r}}}_{s}^{2} ,$$

(27)

where ${\mathbf{{r}}}^{P}$ and ${\mathbf{{r}}}_{s}^{2}$ are the absolute position vectors of the point $P$ and the sheave center, given by Eqs. (13) and (10), respectively. That vector is projected to the sheave frame (the “hat” ˆ means the projection of the vector in the sheave frame):

$$ {\mathbf{{\hat{u}}}}_{P}^{s} = {\left ( {{{\mathbf{{A}}}^{s}}} \right )^{T}}{\mathbf{{u}}}_{P}^{s} .$$

(28)

The contact frame can be obtained using the angle ${\alpha ^{P}}$ that makes vector ${\mathbf{{u}}}_{P}^{s}$ with the sheave frame shown in Fig. 6, as follows:

$$ {\alpha ^{P}} = {\tan ^{ - 1}}\left ( { \frac{{{{\left [ {{\mathbf{{\hat{u}}}}_{P}^{s}} \right ]}_{x}}}}{{{{\left [ {{\mathbf{{\hat{u}}}}_{P}^{s}} \right ]}_{z}}}}} \right ). $$

(29)

Assuming that the angle ${\beta ^{i}}$ of the interpenetration area $i$ is known, the transformation matrix from the contact frame to the global frame is given by:

$$ \begin{aligned} {{\mathbf{{A}}}^{Pi}} &= \left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c} {{\mathbf{{t}}}_{1}^{Pi}}&{{\mathbf{{t}}}_{2}^{Pi}}&{{\mathbf{{n}}}{}^{Pi}} \end{array}\displaystyle } \right ] = {{\mathbf{{A}}}^{s}}{{\mathbf{{A}}}_{\alpha P}}{{\mathbf{{A}}}_{\beta i}}, \,\,\,\,i = 1,\dots ,nia, \\ {{\mathbf{{A}}}_{\alpha P}} &= \left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c} {\cos {\alpha ^{P}}}&0&{\sin {\alpha ^{P}}} \\ 0&1&0 \\ { - \sin {\alpha ^{P}}}&0&{\cos {\alpha ^{P}}} \end{array}\displaystyle } \right ], \\ {{\mathbf{{A}}}_{\beta i}} &= \left [ { \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1&0&0 \\ 0&{\cos \beta {}^{i}}&{ - \sin \beta {}^{i}} \\ 0&{\sin \beta {}^{i}}&{\cos \beta {}^{i}} \end{array}\displaystyle } \right ]. \end{aligned} $$

(30)

If the surface of the groove is not conical (curved line in the diametric section instead of a straight line as in Fig. 7) then the angle ${\beta ^{i}}$ varies with the position of the contact point and the calculation of the contact frame is more involved.

The position vector of the contact point in the rope with respect to the sheave center is given by

$$ {\mathbf{{u}}}_{CR}^{s} = {{\mathbf{{A}}}^{Pi}}{\breve{\mathbf{u}}}_{CR}^{s} = {{ \mathbf{{A}}}^{Pi}}\left ( {\breve{\mathbf{u}}}_{P}^{s}- \begin{bmatrix} 0 \\ 0 \\ r \end{bmatrix} \right ) ,$$

(31)

where $r$ is the radius of the rope and symbol $\breve{\mathbf{u}}$ includes the components of $\mathbf{{u}}$ in the $P_{i}$ contact frame. In the case of a V-shape groove, the position vector of the contact point in the sheave ${\mathbf{{u}}}_{CS}^{s}$ is obtained after the following set of calculations:

$$ \begin{aligned} {\mathbf{{u}}}_{Q}^{s} &= {{\mathbf{{A}}}^{s}}{{\mathbf{{A}}}_{\alpha P}} \left [ { \textstyle\begin{array}{c} {R - b\tan {\beta ^{i}}} \\ 0 \\ 0 \end{array}\displaystyle } \right ],\,\,\,{{\mathbf{{u}}}_{PQ}} = {\mathbf{{u}}}_{Q}^{s} - {\mathbf{{u}}}_{P}^{s}, \\ {\breve{\mathbf{u}}}_{PQ} &= {\left ( {{{\mathbf{{A}}}^{Pi}}} \right )^{T}}{{ \mathbf{{u}}}_{PQ}},\,\,\,{\breve{\mathbf{u}}}_{CS}^{P} = \left [ { \textstyle\begin{array}{c} 0 \\ 0 \\ {{{\left [ {\breve{\mathbf{u}}}_{PQ} \right ]}_{z}}} \end{array}\displaystyle } \right ], \\ {\mathbf{{u}}}_{CS}^{s} &= {\mathbf{{u}}}_{P}^{s} + {{\mathbf{{A}}}^{Pi}}{ \breve{\mathbf{u}}}_{CS}^{P}, \end{aligned} $$

(32)

where $Q$ is the point centered in the sheave midplane shown in Fig. 12, $R$ is radius of the sheave at the bottom of the groove, and $b$ is half-width of that bottom part. The normal indentation of the bristle is given by

$$ \delta _{n}^{i} = \left | {{\mathbf{{u}}}_{CR}^{s} - {\mathbf{{u}}}_{CS}^{s}} \right | .$$

(33)

In the case of a planar sheave as in Fig. 11, this expression reduces to

$$ \delta _{n}^{i} = R - {\left [ {\breve{\mathbf{u}}}_{CR}^{s} \right ]_{z}} .$$

(34)

Equations (27)–(34) provide the values of the position of points $CR$ and $CS$, the contact frames $\left \langle {P\,;\,\,{\mathbf{{t}}}_{1}^{Pi},\,\,{\mathbf{{t}}}_{2}^{Pi},{{\mathbf{{n}}}^{Pi}}} \right \rangle$, $i = 1,\dots ,nia$, and the normal indentation $\delta _{n}^{i}$ for each point $P$ in the mid-line of the contact rope as a function of the generalized coordinates. Nevertheless, the position of the bristle end $BE$ cannot be obtained using the value of the generalized coordinates only. It depends on the time-history of the rope–sheave contact interaction. The calculation of the position of $BE$ is detailed next.

It is important to observe that there are not one, but two $BE$ points, one on the bristle end that moves with the rope and the other on the sheave. The position vectors of both points instantaneously coincide. However, their absolute velocity may or may not be the same. This is due to the stick–slip friction. If stick friction applies, both velocities are equal. If slip friction applies, both velocities are different.

The position vector of the $BE$ point with respect to the sheave center is given by

$$ \begin{aligned} {\mathbf{{u}}}_{BE}^{s} &= {\mathbf{{u}}}_{CS}^{s} + { \boldsymbol{\delta}}_{t}^{i} = {\mathbf{{u}}}_{CS}^{s} + {{\mathbf{{A}}}^{Pi}} \breve{\boldsymbol{\delta}}_{t}^{i} , \\ \breve{\boldsymbol{\delta}}_{t}^{i} &= \left [ { \textstyle\begin{array}{c} {\delta _{t1}^{i}} \\ {\delta _{t2}^{i}} \\ 0 \end{array}\displaystyle } \right ], \end{aligned} $$

(35)

where ${\delta _{t1}^{i}}$ and ${\delta _{t2}^{i}}$ are the bristle-end lateral deformations along the directions ${\mathbf{{t}}}_{1}^{Pi}$ and ${\mathbf{{t}}}_{2}^{Pi}$, respectively. These two components of the lateral deformations must be treated as nongeneralized coordinates. The dynamic equations of ${\delta _{t1}^{i}}$ and ${\delta _{t2}^{i}}$ are developed next.

The absolute velocity of the $BE$ point that belongs to the sheave is given by

$$ {\mathbf{{\dot{r}}}}_{BE}^{s} = {{\mathbf{{\dot{r}}}}^{s}} + {{\mathbf{{\dot{A}}}}^{s}}{ \mathbf{{\hat{u}}}}_{BE}^{s} ,$$

(36)

where ${\mathbf{{\hat{u}}}}_{BE}^{s}$ is considered as constant because the sheave is treated as a rigid body. The absolute velocity of the $BE$ point that belongs to the rope is given by

$$ \dot{\mathbf{r}}_{BE}^{P} = \dot{\mathbf{r}}^{P} + \mathbf{A}^{Pi} \dot{\breve{\mathbf{u}}}_{BE}^{P}+\dot{\mathbf{A}}^{Pi}{ \breve{\mathbf{u}}}_{BE}^{P} ,$$

(37)

where

$$ \begin{aligned} {\breve{\mathbf{u}}}_{BE}^{P} &= {\breve{\mathbf{u}}}_{CR}^{P} + \breve{\boldsymbol{\delta}}_{n}^{i} + \breve{\boldsymbol{\delta}}_{t}^{i} = {\breve{\mathbf{u}}}_{CR}^{P} + \breve{\boldsymbol{\delta}}^{i} = \left [ { \textstyle\begin{array}{c} 0 \\ 0 \\ { - r} \end{array}\displaystyle } \right ] + \left [ { \textstyle\begin{array}{c} {\delta _{t1}^{i}} \\ {\delta _{t2}^{i}} \\ {\delta _{n}^{i}} \end{array}\displaystyle } \right ], \\ \dot{\breve{\mathbf{u}}}_{BE}^{P} &= \dot{\breve{\boldsymbol{\delta}}}^{i} = \left [ { \textstyle\begin{array}{c} {\dot{\delta}_{t1}^{i}} \\ {\dot{\delta}_{t2}^{i}} \\ {\dot{\delta}_{n}^{i}} \end{array}\displaystyle } \right ], \\ {{{\mathbf{{\dot{A}}}}}^{Pi}} &= {{{\mathbf{{\dot{A}}}}}^{s}}{{\mathbf{{A}}}_{\alpha P}}{{ \mathbf{{A}}}_{\beta i}}, \end{aligned} $$

(38)

where ${\breve{\boldsymbol{\delta}}}^{i}$ is the total deformation vector of the bristle-end, being the sum of ${\breve{\boldsymbol{\delta}}}_{n}^{i}$ and ${\breve{\boldsymbol{\delta}}}_{t}^{i}$.

1.2 1.2 Stick–slip contact friction

The contact kinematics developed in the previous subsection can be used to define the stick–slip contact conditions.

Stick contact

During stick contact, the position of the $BE$ remains constant at the sheave. Therefore, at each time step, the tangential deformation of the bristle can be obtained from Eq. (35) as

$$ {\breve{\boldsymbol{\delta}}}_{t}^{i} = {\left ( {{{\mathbf{{A}}}^{Pi}}} \right )^{T}}\left ( {{\mathbf{{u}}}_{BE}^{s} - {\mathbf{{u}}}_{CS}^{s}} \right ), $$

(39)

where ${\mathbf{{u}}}_{BE}^{s}$ has the same value as the in previous time step. The time derivative of the tangential deformation is obtained using the fact that the absolute velocity of points $BE$ belonging to the sheave and the rope coincide, that is,

$$ \dot{\mathbf{r}}_{BE}^{s} = \dot{\mathbf{r}}_{BE}^{P}\: \Rightarrow \: \dot{\mathbf{r}}^{s} + \dot{\mathbf{A}}^{s}\hat{{\mathbf{u}}}_{BE}^{s} \left ( \breve{\delta}_{i}^{t} \right ) = \dot{\mathbf{r}}^{P} + \mathbf{A}^{Pi}\dot{\breve{\mathbf{u}}}_{BE}^{P}+\dot{\mathbf{A}}^{Pi}{ \breve{\mathbf{u}}}_{BE}^{P}\left ( \breve{\delta}_{i}^{t} \right ) .$$

(40)

In this equation brackets mean functional dependency. Equation (40) can be used to find the velocity of deformation of the bristles, as follows:

$$ \dot{\breve{\boldsymbol{\delta}}}^{i} = \dot{\breve{\boldsymbol{\delta}}}_{n}^{i} + \dot{\breve{\boldsymbol{\delta}}}_{t}^{i} = \dot{\breve{\mathbf{u}}}_{BE}^{P} = \left (\mathbf{A}^{Pi} \right )^{T}\left [ \dot{\mathbf{r}}^{s}+ \dot{\mathbf{A}}^{s}\hat{{\mathbf{u}}}_{BE}^{s}\left ( \breve{\boldsymbol{\delta}}_{i}^{t} \right )- \dot{\mathbf{r}}^{P} + \dot{\mathbf{A}}^{Pi}{\breve{\mathbf{u}}}_{BE}^{P}\left ( \breve{\boldsymbol{\delta}}_{i}^{t} \right ) \right ] .$$

(41)

Once ${\breve{\boldsymbol{\delta}}}_{t}^{i}$ is calculated using Eqs. (31) and (32), its value is used to find $\dot{\breve{\boldsymbol{\delta}}}_{n}^{i}$ and $\dot{\breve{\boldsymbol{\delta}}}_{t}^{i}$ using Eq. (41). The tangential contact force in stick contact applied on the rope is given by

$$ {\mathbf{{f}}}_{t}^{i} = {{\mathbf{{A}}}^{Pi}}\left ( k_{t} \breve{\boldsymbol{\delta}}_{i}^{t}+c_{t} \dot{\breve{\boldsymbol{\delta}}}_{i}^{t} \right ) $$

(42)

while the tangential contact force applied in the sheave is equal in magnitude and opposite direction.

Stick–slip transition happens in a particular time step if the magnitude of the tangential contact force exceeds the saturation value, as follows:

$$ \left | {{\mathbf{{f}}}_{t}^{i}} \right | > \mu f_{n}^{i} .$$

(43)

In such a case, the value given in Eq. (42) is invalid and the alternative equation is used to calculate the force:

$$ {\mathbf{{f}}}_{t}^{i} = \mu f_{n}^{i}{{\mathbf{{i}}}_{ft}} ,$$

(44)

where ${\mathbf{{i}}}_{ft}$ is a unitary vector aligned with the force that results from Eq. (42).

Slip contact

During slip contact, the magnitude of the force equals the saturation value, and its direction is opposite to the slip velocity, as follows:

$$ {\mathbf{{f}}}_{t}^{i} = - \mu f_{n}^{i} \frac{{{\mathbf{{v}}}_{BE}^{rel}}}{{\left | {{\mathbf{{v}}}_{BE}^{rel}} \right |}} ,$$

(45)

where the relative velocity at the bristle end ${\mathbf{{v}}}_{BE}^{rel}$ is given by

$$ {\mathbf{{v}}}_{BE}^{rel}\left ( {\breve{\boldsymbol{\delta}}}_{t}^{i}, \dot{\breve{\boldsymbol{\delta}}}_{t}^{i} \right ) = {\mathbf{{\dot{r}}}}_{BE}^{P} - {\mathbf{{\dot{r}}}}_{BE}^{s} ,$$

(46)

where, as shown in this equation, the relative velocity at the bristle end is a function of the tangential deflection of the bristle and its time derivative. However, at this point, ${\breve{\boldsymbol{\delta}}}_{t}^{i}$ and $\dot{\breve{\boldsymbol{\delta}}}_{t}^{i}$ are unknown. Because the bristle viscoelastic model applies in the slip phase too, manipulating Eq. (42) yields

$$ k_{t}\breve{\boldsymbol{\delta}}_{i}^{t}+c_{t} \dot{\breve{\boldsymbol{\delta}}}_{i}^{t} = \left (\mathbf{A}^{Pi} \right )^{T}\mathbf{f}_{t}^{i} .$$

(47)

If Eqs. (45)–(46) are substituted into Eq. (47), the result is a first-order ordinary differential equation in terms of ${\breve{\boldsymbol{\delta}}}_{t}^{i}$. Strictly speaking, the solution of the tangential contact problem in the slip phase requires the addition of two new state variables, $\delta _{t1}^{i}$ and $\delta _{t2}^{i}$, per contact point $P$ per interpenetration area $i$. The associated differential equation Eq. (47) must be integrated forward in time together with the equations of motion of the system during simulation. These new state variables and differential equations should be cancelled every time a stick phase starts and set on again every time a stick–slip transition happens. Clearly, this solution is computationally very costly and difficult to implement in the application at hand, in which a large and variable number of $P$ points must be used. Nevertheless, this “exact” procedure can be substituted with the following easy-to-implement scheme:

• 1. Calculate the relative velocity ${\mathbf{{v}}}_{BE}^{rel}$ and the tangential contact force ${\mathbf{{f}}}_{t}^{i}$ using the values of ${\breve{\boldsymbol{\delta}}}_{t}^{i}$ and $\dot{\breve{\boldsymbol{\delta}}}_{t}^{i}$ of the previous time step. Note that ${\breve{\boldsymbol{\delta}}}_{t}^{i}$ and $\dot{\breve{\boldsymbol{\delta}}}_{t}^{i}$ only affect the direction of ${\mathbf{{f}}}_{t}^{i}$ but not its magnitude.

• 2. Using a backward-difference formula, assume that

  $$ {\left [ \dot{\breve{\boldsymbol{\delta}}}_{t}^{i} \right ]_{j}} = \frac{{{{\left [ {\breve{\boldsymbol{\delta}}}_{t}^{i} \right ]}_{j}} - {{\left [ {\breve{\boldsymbol{\delta}}}_{t}^{i} \right ]}_{j - 1}}}}{{\Delta t}} ,$$

  (48)

  where $j$ is the actual time-step and $j - 1$ is the previous time-step.

• 3. Substitute Eq. (48) into Eq. (47) and solve the resulting linear equation to get ${\left [ {\breve{\boldsymbol{\delta}}}_{t}^{i} \right ]_{j}}$.

The slip–stick transition occurs when the relative velocity ${\mathbf{{v}}}_{BE}^{rel}$ reaches a zero value. The time instant in which this happens is difficult to identify due to the time-discretization. Therefore, a slip–stick transition is considered to occur anytime the following condition is fulfilled:

$$ \left [ {{\mathbf{{v}}}_{BE}^{rel}} \right ]_{j}^{T}{\left [ {{\mathbf{{v}}}_{BE}^{rel}} \right ]_{j - 1}} < 0, $$

(49)

that is, when the dot product of the relative velocity in this time step multiplied by the relative velocity in the previous time step is negative. This is equivalent to assuming that a discrete variable has taken a zero value anytime its sign changes from one step to the next. When this condition is fulfilled, the value of ${\mathbf{{u}}}_{BE}^{s}$ is calculated using Eq. (35) and stored, and it will remain constant throughout the next stick contact period.