Capstan Equation Research Articles

Drive Characteristics of Air-Cylinder-Type Artificial Muscle in Annular Bending

Air cylinders are actuators that slide a piston inside cylinders by applying air pressure. We propose an air-cylinder-type artificial muscle that can be flexibly bent by using a flexible tube for the cylindrical part. The actuator output was a string connected to a piston. When the air-cylinder-type artificial muscle bends, the inner wall of the tube and the string come into contact, causing output fluctuations owing to friction. In this study, we investigated the output when an artificial muscle was bent. After describing the structure of the air-cylinder-type artificial muscle, the measurement results of the resistance force at each part of the actuator are presented. A theoretical output inspired by the capstan equation was derived, and its validity was verified by comparison with experimental results.

An Improved Static Model for Bidirectional Notched Continuum Robot Considering the Cable Tension Loss

Abstract The notched continuum mechanism is particularly suitable for natural orifice transluminal surgery benefiting from its small size and hollow structure. However, the widely used kinematic model based on constant curvature assumption does not reveal the actual deformation of the continuum mechanism, and its control accuracy is unstable, while the general mechanics model has the problem that the tension of the distal driving cable is difficult to measure. In this paper, a nonconstant curvature static model for a bidirectional V-shaped notched continuum mechanism is presented. The deformation of each part of the continuum mechanism from the distal end to the proximal end is analyzed in turn. The tension loss of the driving cable caused by the contact with the continuum mechanism is modeled using the capstan equation. The recursive equation between the deformation of each part of the continuum mechanism from the proximal end is derived, which can be solved numerically. The bending state of the continuum mechanism can then be estimated when only the tension of the proximal flexible cable is known. The model is experimentally verified by driving the continuum mechanism to move at a very low speed. The experiment results show that the estimation effect of the proposed model is significantly improved compared with that of the constant curvature model.

Applicability of the Capstan Equation to Guitar Strings

The effects of friction were observed in electric guitar strings passing over an electric guitar saddle. The effects of changing the ratio of the diameter of the winding to the diameter of the core of the string, the angle through which the string is bent, and the length on either side of the saddle were measured. Relative tensions were deduced by plucking and measuring the frequencies of vibration of the two portions of string. Coefficients of friction consistent with the capstan equation were calculated and were found to be lower than 0.26 for wound strings (nickel plated steel windings on steel cores) and lower than 0.17 for unwound (tin plated steel) strings. The largest values of friction were associated with strings of narrower windings and wider cores and this may be due to the uneven nature of the contact between the string and saddle for wound strings or due the surface of the windings deforming more, encouraging fresh (and therefore higher friction) metal to metal contact. It is advised to apply lubrication under the saddle to string contact point after first bringing the string up to pitch rather than before in order to prevent this fresh metal to metal contact.

Controlling Surface of Rods With Entrained Particle as Asperities

Abstract Changing the surface properties (i.e., roughness or friction) can be instrumental for many applications but can be a complex and resource-intensive process. In this paper, we demonstrate a novel process of controlling the friction of a continuous rod by delivering inorganic microparticles. A standardized continuous particle transfer protocol has been developed in our laboratory for depositing particles from a liquid carrier system (LCS) to the cylindrical rod substrate. The particle transfer process can produce controllable and tunable surface properties. Polymeric binder is used to deliver the particles as asperities over the rod substrate and by controlling their size, shape, and distribution, the coefficient of friction of the rod is determined. Tabletop experiments are designed and performed to measure the friction coefficient following the Capstan equation. The entrained particles on the substrate will create size- and shape-based asperities, which will alter the surface morphology toward the desired direction. Both oblique and direct quantitative measurements are performed at different particles and binder concentrations. A systematic variation in the friction coefficient is observed and reported in the result section. It is observed from the capstan experiment that adding only 1% irregular shaped particles in the suspension changes the friction coefficient of the rods by almost 115%. The proposed friction control technique is a simple scale-up, low-cost, low-waste, and low-energy manufacturing method for controlling the surface morphology.

An elastic rod in frictional contact with a rigid cylinder

We investigate the load transmission along an elastic rod of finite cross-section in contact with a rigid cylinder, as system often referred to as the generalized capstan problem. In the presence of friction, the idealized classic capstan equation predicts that the tension along a perfectly thin and flexible filament increases exponentially along the contact region. In practical applications, however, the validity of the idealized capstan equation is compromised due to the interplay between finite rod thickness, bending stiffness, and the forces applied at the rod extremities. Here, we combine precision model experiments, finite element simulations, and theoretical modeling to investigate the contact mechanics and the force transmission along an elastic rod in frictional contact with a rigid cylinder. We study two cases when the rod is either static or sliding. First, we focus on the static case, in the absence of friction, by considering equal loads at both extremities of the rod. We show that as the loading force is increased, the nature of contact transitions from a localized region to an extended band at the surface of the cylinder. The latter is characterized by double-peaked contact force distribution. In the sliding case, friction is activated by inducing a relative motion between the rod and the cylinder. We applied a fixed loading force at one rod extremity while pulling the other extremity at a constant velocity. The driving force is monitored during sliding. For increasing loading forces, we find that the force ratio is non-monotonic and displays a local minimum, in contradiction with the constant ideal capstan prediction. This minimum force ratio coincides with the transition from a single contact point to an extended contact region. A theoretical analysis based on Euler’s elastica serves to rationalize the results from the physical and numerical experiments. In addition to predicting the nature of the contact region (single point versus extended line), our model provides quantitative predictions for the wrapping angle and the driving-to-loading force ratio. Finally, we leverage our mechanics-based framework to predictively understand the force ratio at the ends of two commercially available engineering belts (spring-steel and polyurethane) in sliding contact with a steel cylinder.

A large displacement model for superelastic material side-notched tube instruments

Side-notched tube structures made of superelastic material are increasingly being studied in surgical robotics as they can easily be integrated into surgical instruments and offer the surgeon enhanced dexterity. Their simple design and ability to achieve large bending angles and small bending radii make them particularly suited for use in constrained workspace surgery. Up to now, however, no model has been able to accurately predict the behavior of such types of structures for large bending angles. This study, therefore, proposes a novel approach to model large displacements of side-notched tubes taking into account the superelasticity of the material, the non-constant curvature of the side-notched structure, as well as the friction induced by the actuating wire. More specifically, an equivalent, strain dependent, linear elastic modulus is used to model the superelastic behavior of the instrument’s material. The deformation of each separate section of the side-notched tube is calculated successively to capture the non-constant curvature of the NiTi backbone. The capstan equation is used to model cable friction. The model was tested on four samples of 2.3 mm diameter and was able to predict the bending angle of the side-notched tubes with a root mean square error (RMSE) of as low as 5.4∘ (3.1%) and the position of each notch with an RMSE of as low as 0.33 mm (2.5%) across its entire bending range (0∘ to 180∘). The model was demonstrated for both tension and relaxation of the actuating wire.

A Study of Transmission Error Modeling and Preload Compensation for the Cable-Driven Sheaves Used in Space Docking Locks

ABSTRACTThe transmission error of cable-driven sheaves (CDS) used in space docking locks directly affects the synchronous docking of two spacecraft, which is guaranteed mainly by the preload applied to their serial cables. But it is difficult controlled precisely because of the complicated cable deformation and operating conditions. The synchronous testing efficiency of the docking locks is inevitably influenced, correspondingly. This paper proposes a prediction model for the transmission error of CDS based on their cable deformation. In this model, the deformations of non- and free sectional cables are both modified on finite element analysis, which are respectively derived from classical Capstan equation and Hooke’s law for them without considering the effects of the friction coefficient between wire strands. Based on the proposed model, the relationships between the transmission error and dominating factors are analyzed. Then the preload compensation for transmission error is obtained at the engaging and locking angles of the docking locks, respectively. Experiments validate the model. This can provide a valuable reference in controlling the transmission error of CDS and improving the assembly efficiency of docking locks.

Modeling and verification of multi-winding rope winch for facade operation

This paper presents the modeling and verification of a multi-winding rope winch for facade operation. The winch enables a robot to ascend and descend high-rise buildings using synthetic fiber ropes and can easily be installed on buildings without a gantry. To improve the positioning accuracy for facade operation, a rope-winch position model that accurately tracks the position of the winch is derived. This model comprises deformation from the rope tension and displacement from slip between the rope and winch. Owing to the nonlinearity of the synthetic-fiber-rope characteristics, the stiffness levels with and without preload are derived separately. Based on the rope stiffness, deformation from the elasticity is analytically derived. The slip length is estimated using the capstan equation, and the total slip through the differential gear mechanism is estimated. To verify the model, experiments are conducted to track the positional profile through the application of open-loop control and feedforward control on a test bench. The prediction model can maintain the root mean square error of the position tracking within 0.8%.

Mechanical and Empirical Parameter Design on a Multi-wound Differential Pulley Winch for a Wall-Climbing Robot

This paper proposes a novel multi-wound differential pulley winch (MWDPW) component for assisting the ascending/descending operations of a wall climbing robot. The robotic platform enabling rope access in dangerous environments (ROPE RIDE) climbs vertical walls using a rope and embedded winch. The original winch installed on the ROPE RIDE was a single wound winch, which had problems such as rope slip, velocity ripple during descending motion, resulting the bad cleaning performance and unstable motion on the wall. These problems are mainly due to the concentration of traction force on the rope because a small portion of the winch pulley holds the entire weight of the robot. Therefore, we have developed a new winch, MWDPW, by using multi-wound differential traction pulley and pressure rollers to solve the traction force concentration problem by distributing the traction force along the entire wrapping angle. Compared to other multi-wound winches, the MWDPW has the special feature of a differential gear and pressure roller to distribute the traction force and minimize the rope slip. The tension of the MWDPW is analyzed using the basic capstan equation, and empirical results to minimize the rope slip are presented by varying design parameters such as the winding method and presence of the pressure rollers. We expect the proposed mechanism to improve the safety of a wall-climbing robot for wall-cleaning operations.

Highly Repeatable Rope Winch Design With Multiple Windings and Differential Gear Mechanism

This paper presents a rope winch with high position repeatability using a differential gear mechanism. This winch is easy to apply to existing buildings using synthetic fiber ropes, which are generally used by building exterior maintenance workers, and can be widely utilized as a winch of a gondola or climbing robot. Multiple pulleys are used to achieve high payloads, and differential gear mechanisms are used to distribute the torque across each pulley evenly. Uniform torque distribution minimizes slipping on each pulley, which reduces the power loss due to friction. This is proven through an analysis using the capstan equation and an experiment on a test bench. In addition, a criterion for the position repeatability of the winch is designed, and its results show that one pressure roller achieves optimal performance when the roller at the free end is pressed to 37 N. Extensive experiments were carried out on the final model of the winch, and position repeatability of under 50 mm deviation per 10 m was achieved through stable torque distribution under various payloads.

Frictional coefficient during flossing of teeth

ObjectiveThis in-vitro study aimed to develop a technique to measure the frictional forces and determine the frictional coefficient (μ) associated with the rubbing of dental floss against teeth. Incorrect flossing technique and the etiology of grooves at the cementoenamel junction (CEJ) of proximal area of teeth has long been a controversial topic. We hypothesized that the μ between teeth surfaces and dental floss is affected by contact angulation. Materials and methodsTests were conducted using two different types of dental floss (waxed and unwaxed nylon) on different surfaces (enamel, dentine, smooth and rough glass rods) under different moisture conditions (dry and wet). The μ generated by performing C-shape flossing was measured, using the Capstan equation, at constant load (100g) over different flossing contact sliding angulations. In addition, the surface characteristics of intact and used nylon flosses were compared using a scanning electron microscope (SEM). ResultsThe mean μ was highest with a smooth glass rod (0.42±0.11), followed by rough glass rods (0.30±0.07), dry enamel (0.27±0.08), wet enamel (0.23±0.06), then dentine (0.18±0.04). Moreover, higher μ was associated with waxed floss when used against dry enamel, smooth and rough glass rods (P<0.001). At different moisture conditions, waxed floss demonstrated greater μ with dry surfaces (P<0.03). No association was found between μ and dental floss contact angulation. Both floss types showed deterioration after usage; although waxed type exhibited markedly greater deterioration when used on dry surfaces. In conclusion, the magnitude of the μ was found to be influenced by surface roughness, moisture condition, and independent of the contact angulation area during sliding of dental floss. SignificanceIt is important to consider the potential side effects of frictional forces on both tooth surface and dental floss during clinical application.

Motion modelling and error compensation of a cable-driven continuum robot for applications to minimally invasive surgery.

The continuum robot is widely used in minimally invasive surgery (MIS) because of its flexibility, dexterity and safety. However, because of the friction in the transmission system, backlash, and shape error from nominal kinematics, the tracking accuracy of the continuum robot is low, which may damage the tissues and organs during surgery. A novel error compensation method is presented to improve control. Considering the nonlinear friction and coupling effects, the static model of a robot is derived based on the principle of virtual work. A modified Capstan equation including bending rigidity and nonlinear friction is established to analyze the transmission characteristics of a cable-pulley system. The results of the experiments conducted with a continuum robot indicate the validity of the proposed compensation method. The established model and the proposed compensation method can be used for the cable-driven continuum robot to guarantee safety and stability in MIS.

A cable-pulley system modeling based position compensation control for a laparoscope surgical robot

Due to nonlinear friction and backlash characteristics of cable-driven mechanism (CDM) in laparoscope surgical robot end-effector, it is difficult to predict force and control position precisely during the surgical procedure. Current studies of transmission characteristics of CDM mainly focus on the tendon-sheath system (TSS) with the classical Capstan equation which neglects the effect of bending rigidity of the cable. However, analysis for transmission characteristics of a cable-pulley system (CPS) is rare. This paper presents a tension and displacement transmission model of a CPS in the end-effector of laparoscope surgical robot. The model considers the bending rigidity of the cable and spatial location of the pulleys. Experiments validate the proposed model. Based on the model, a position compensation algorithm of feedforward control is presented to reduce the tracking position errors. The experimental validations show substantial improvements on performance of position tracking errors for the use of the proposed algorithm.

Quantifying the Frictional Forces between Skin and Nonwoven Fabrics.

When a compliant sheet of material is dragged over a curved surface of a body, the frictional forces generated can be many times greater than they would be for a planar interface. This phenomenon is known to contribute to the abrasion damage to skin often suffered by wearers of incontinence pads and bed/chairbound people susceptible to pressure sores. Experiments that attempt to quantify these forces often use a simple capstan-type equation to obtain a characteristic coefficient of friction. In general, the capstan approach assumes the ratio of applied tensions depends only on the arc of contact and the coefficient of friction, and ignores other geometric and physical considerations; this approach makes it straightforward to obtain explicitly a coefficient of friction from the tensions measured. In this paper, two mathematical models are presented that compute the material displacements and surface forces generated by, firstly, a membrane under tension in moving contact with a rigid obstacle and, secondly, a shell-membrane under tension in contact with a deformable substrate. The results show that, while the use of a capstan equation remains fairly robust in some cases, effects such as the curvature and flaccidness of the underlying body, and the mass density of the fabric can lead to significant variations in stresses generated in the contact region. Thus, the coefficient of friction determined by a capstan model may not be an accurate reflection of the true frictional behavior of the contact region.

A one‐dimensional computational model for hyperelastic string structures with Coulomb friction

In this paper, we consider a new model for the simulation of textiles with frictional contact between fibers and no bending resistance. In the model, one‐dimensional hyperelasticity and the Capstan equation are combined, and its connection with conventional hyperelasticity and Coulomb friction models is shown. Then, the model is formulated as a problem with the rate‐independent dissipation, and we prove that the problem possesses proper convexity and continuity properties. The article concludes with a numerical algorithm and provides numerical experiments along with a comparison of the results with a real measurement. Copyright © 2016 John Wiley &amp; Sons, Ltd.

General Method for Pullback Force Estimation for Polyethylene Pipes in Horizontal Directional Drilling

AbstractThis paper proposes a general method for estimating pullback force applied on polyethylene (PE) pipes during horizontal directional drilling (HDD) installations based on adapting the Capstan equation to a PE pipe negotiating a curved bore. Compared to current standards, this method can be implemented in any installation regardless of the bore geometry, such as ground surface grade or number of segments composing the bore path. To verify the proposed method, pullback data collected from an HDD installation project is compared to forces estimated by the proposed method as well as standard methods. This paper demonstrates the proposed method’s ability in estimating a more-accurate pullback load.

Transmission capability of precise cable drive including bending rigidity

The precise cable drive has been widely used in all types of industrial, commercial, aerospace, military, medical and research applications as an alternative to gearboxes, belts and pulleys, chains and sprockets and other currently available transmission. Classical capstan equation is usually used to investigate the transmission capability of precise cable drive. However, the classical capstan equation cannot hold for precise cable drive because of the existing bending stiffness and elastic elongation of the steel cable. This paper develops analytical methods for predicting the transmission capability of precise cable drive including bending rigidity. This work illustrates interconnections between the transmission capability and design parameters, including preload force, wrapped angle, capstan radius ratio, bending stiffness and friction coefficient. The result is useful to predict the transmission capability of precise cable drive in practical applications.

An improved Capstan equation including power-law friction and bending rigidity for high performance yarn

The tension of high performance yarn in the weaving region has influence on uniformity and pattern effect of fabric, which is difficult to be predicted in weaving process. In this paper, a modified Capstan equation including bending rigidity and power-law friction effects was developed, and solved numerically. The effects of the bending rigidity and power-law friction on tension ratio were analyzed. The experimental results were compared with the classical Capstan equation and modified one, showing that there was an approximate agreement between the modified tension ratio and experimental values. And therefore, the modified tension ratio could be used to characterize the tension variation and provide theoretical support for precisely controlling of yarn tension.

Direct Evaluation of Coulomb Friction Coefficient from Sheet Strip Stretch Test on a Cylinder Surface

Knowing the flow curve of a sheet metal, strain distribution on a sheet strip may be used to roughly but quickly evaluate the Coulomb friction coefficient. Strain distribution on the strip being stretched on a cylindrical surface of interest may be measured by an optical strain measurement system. This could be used to estimate the stresses on the specimen. The capstan equation is then used to roughly evaluate the coefficient of friction acting between the sheet strip and the cylindrical surface. Validation of the approach is done using the simulation of the process. The corresponding experiments can be performed easily on a sheet metal testing device equipped with an optical strain measurement system, which is commonly used for the experimental evaluation of FLCs.

Fabric friction behavior: study using capstan equation and introduction into a fabric transport simulator

All thin materials, such as textiles, papers, polymers, or metals, are handled on rollers during manufacture and subsequent use. This requires several unwinding and winding processes. The goal of this study was to investigate the friction behavior of fabrics relative to sliding velocity and to introduce friction coefficient evolution in a fabric transport model. For the experimental part, a specific fabric/roll friction bench is presented. The friction coefficient was calculated from the capstan equation. The evolution of the friction coefficient was quantified relative to the sliding velocity for different textile fabrics and also for a polymer and a paper web. The influence of some measurement process parameters was studied: web tension, roll diameter, and wrap angle. The friction coefficient initially increased with sliding velocity and then became constant. This phenomenon can be explained by the deformation of the fabric due to friction, thereby inducing an increase in web tension with the sliding velocity. The relationship between tension and rolling friction behavior of the fabric was investigated. The variability of the friction coefficient was introduced to web/roll simulator, and improvements to the simulator are shown.