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

Abstract

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.

Keywords

friction capstan textile fabric winding–unwinding fraction cycle traction cycle

The goal of this study was to investigate the rolling friction behavior of textile fabrics and to insert some tribological features into a handling system model. It is useful to know the friction coefficient evolution in order to improve simulators used in web handling, particularly in unwinding–winding operations.^1,2 These simulators are used for performance analysis, controller design, or optimization of the web handling process. The present study concerns the process between unwinding and winding.

The winding and unwinding phases have been modeled with consideration to nip load.^3,4 However, the friction between textile fabrics and rollers has been rarely reported in the scientific literature. The listed publications concerns essentially fiber, filament, or yarn.

In web/roll friction, the friction coefficient is commonly achieved using the capstan equation. This equation, known since the 18th century (Bernoulli and Euler), is used to calculate the friction coefficient μ between an inextensible rope and a roll.⁵ The rope is wrapped at an angle θ to the roll surface (Figure 1). A mass is fixed at a rope end, thus creating a tension T₁. Another tension T₂ has to be processed at the other end, maintaining the system balance. T₂ is higher than T₁ because of the friction between the rope and the roll. The tension T₂ is given by:

T_{2} = T_{1} e μ θ

(1)

where μ is the friction coefficient between the rope and the roll.

Figure 1.

Capstan principle.

This equation can be improved by including the web span bending rigidity and web elasticity.^6,7 However, to our knowledge, using the capstan equation to describe web elasticity without bending rigidity has not yet been done. The non-linearity of the friction coefficient relative to the normal load in the case of fiber friction has been previously introduced.⁸ Indeed, in the case of the polymer and then extended to the fiber, the friction coefficient is not independent of the normal force W.⁹ In fact, the friction force F can be expressed as:

F = k . W n with n \in [\frac{2}{3}, 1]

(2)

with the coefficients k depending on the material and n on the material and the surface roughness. n tends to be equal to 2/3 for a smooth surface and 1 for a rough surface,^10,11 and when the two first bodies in contact create adhesive junctions generating adhesive wear.¹²

Considering a fiber with a length equal to L, the normal load exerted on this fiber per unit length is equal to $w = W / L$ and the friction force per unit length is $f = F / L$ . The two previous equations lead to:⁸

f = k' . w n

(3)

with k′ in (N/m)^1- ⁿ .

Considering a fiber wrapped according an angle θ on a cylinder of radius R, the system is balanced if the load applied on an element $d s = R . d θ$ of the fibre is $d W = T . d θ$ because $d θ$ is small. Thus, $w = d W / d s = T / R$ and $f = d T / d s$ .

From equation (3):

\frac{d T}{T n} = k' \cdot R 1 - n \cdot d θ

(4)

The general solution is becomes:

(\frac{T_{2}}{T_{1}}) 1 - n = 1 + (1 - n) \cdot k' \cdot (\frac{R}{T_{1}}) 1 - n \cdot θ

(5)

n lies between 0.8 and 1 for fibers. In the limit near 0 ( $e x \approx 1 + x$ ), equation (5) presents a solution where n tends to 1:

\frac{T_{2}}{T_{1}} = \exp (k' \cdot (\frac{R}{T_{1}}) 1 - n \cdot θ)

(6)

where the following term, in equation (7) represents friction coefficient:

μ = k' \cdot (\frac{R}{T_{1}}) 1 - n

(7)

In the winding–unwinding context, equation (1) can be applied if no sliding between the roll and the web occurs, and if the web and roll velocities are equal, i.e. under steady state conditions. In the transient, starting and ending phases, for instance, sliding occurs between the web and the roll. The relationship between the tension at the two ends of the web portion is then:¹³

0 \leq T_{2} \leq T_{1} e μ θ

(8)

It can be seen that there are some differences between sliding, i.e. plane and rolling friction. In web/roll friction, velocities are higher and vary more widely than those usually encountered in sliding friction. Moreover, the normal load is currently lower than under plane friction conditions because it is a consequence of the traction applied on both ends of the material, while during plane friction, the normal load causes a compression of the material in the thickness direction. In rolling friction, the load is in the sliding direction, but in sliding friction, it is normal to the surface of the sliding direction.

In this study, the capstan equation was used and the bending rigidity of the material was neglected because the textile webs used were very thin and flexible. A friction measurement device was developed and described. Some experimental parameters were adjusted, such as initial web tension, sliding velocity and wrapping angle. The evolution of the friction coefficient relative to these parameters was studied and explained. The webs used were composed various materials, i.e. textile fabrics but also paper and polymer film. The objective of this study is not to analyze the results obtained for each fabric sample regarding its features, but rather to extract the overall friction behavior in rolling friction regarding the kind of fabric tested, i.e. from a qualitative point of view. In the last part of the study, a web transport simulator including friction between the web and roll is proposed. The ending and stating phases in a winding process were then considered with the friction coefficient taken into account.

Experimental method

Webs investigated

In this study, five different textile fabrics were used (Figure 2 and Table 1). Samples no. 1 and 3 were cotton plain woven fabrics like shirt cloth, and sample no. 2 is a knitted fabric with a side without a pile, i.e. a glabrous side (no. 2 G) while the other side had a pile (no. 2 P) due to the raising process. The sample no. 4 was a 2 and 1 lapping warp knitted fabric. Therefore, the samples included woven (samples no. 1 and 3) and knitted fabrics (no. 2 and 4) of various materials (cotton, polyethylene terephthalate, and polyamide 6.6). Moreover, two sides of the same fabric (no. 2), one with and the other without a pile, were compared. For comparison with textile fabrics, two other non-textile webs were used in some experiments: 70 g/m² writing paper and 120 µm-thick PVC film used in ink-jet printing.

Figure 2.

SEM pictures of woven fabric no. 1 (left) and warp knitted fabric no. 2 on the glabrous side (center) and pile side (right).

Table 1.

Fabric characteristics.

No.	Structure	Composition	Number of warp or wales per cm	Number of weft or courses per cm
1	Plain woven	100% cotton	28.5	26
2	Warp knitted sateen: 2 G: technical face (glabrous) 2 P: technical back (pile)	100% PET	20	22.5
3	Plain woven fabric	100% cotton	27	23
4	Warp knitted 2 and 1 lapping structure	100% PA 6.6	10	20

PET is polyethylene terephthalate and PA is polyamide.

Friction measurement

A specific friction measurement device was developed (Figure 3), composed of extruded aluminum structural elements to obtain a modular, rigid frame. The thickness of the friction roll was reduced to the minimum enabling low roll inertia (i.e. 10 mm). The shaft between roll and coupling device was machined steel to assure high torsion stiffness, supported on the frame by two self-aligning ball bearings in order to minimize shaft/bearing frictions. The shaft was coupled to a high efficiency planetary DC gearmotor (Maxon RE65) with a coupling device which allowed high torsion stiffness. A planetary gear reducer sustained low power loss under load. The fabric tension was obtained using two strain gauges (SM S-Type Load Cell 100 N). These gauges were mounted on a device permitting an easy change of the fabric sample, fabric tension or fabric/roll wrapping angle.

Figure 3.

Rolling friction bench test.

For the friction bench test, the following parameters were adjusted: the fabric sample, the nominal fabric tension (from 5 N to 20 N in this study), the fabric/roll wrapping angle varying from 60 to 100 degrees and the roll diameter (60 and 150 mm). The roll rotation velocity was calculated from a high resolution incremental encoder (Baumer Ivo GI342), mounted on the roll shaft, which gave the angular position with a precision of 0.044°. This high encoder resolution can resolve 57 µm on a 150 mm diameter roll (the thickness of a cotton fiber is about 20 µm). The roll velocity can reach up to 450 rpm, corresponding to a linear velocity (sliding velocity) of 3.5 m/s for the 150 mm diameter roll. The lowest reachable velocity was about 1 mm/s.

Figure 4 shows data recorded from the friction bench test. The torque pattern was transmitted to the servo amplifier, and the torque signal was monitored. The upstream and downstream web tensions, T₁ and T₂, respectively, were measured by the two strain gauges. The rotation angle was measured by the encoder. The sliding velocity was calculated from the angle measurement given by the incremental coder. The friction coefficient was calculated according to the capstan equation (equation (1)) from the measured fabric tensions at each end of the web.

Figure 4.

Data recorded with the friction bench test (from top to bottom): the motor torque reference, the measured motor torque, the tension on the two webs ends T₁ and T₂, and the rotation angle of the roll.

Measurement under drive motor torque control was chosen rather than velocity control because the duration of the test was shorter, so the fabric was less exposed to thermal stress and abrasive wear due to friction. Nevertheless, the results obtained from both methods were correlated (Figure 5). The coefficient of friction is presented relative to sliding velocity. The standard deviation of the friction coefficient values was low and varied from 0.002 to 0.01, depending on the fabric tested.

Figure 5.

Comparison of the results obtained between torque and velocity driving control.

Experimental results and discussion

Influence of the fabric tension

The friction coefficient μ decreased while the fabric tension, related to the normal force, increased. This phenomenon is illustrated in Figure 6 for sample no. 1. The same phenomenon was observed for the other textile samples tested. It has also been observed for nylon fibers,¹⁴ yarn,¹⁵ wool,¹⁶ non-woven materials,¹⁷ and plastic webs.¹⁸ This phenomenon is consistent with equation (7). As a consequence, the friction coefficient decreased as the normal force increased. In web/rolling friction, the normal force is produced by web tension.

Figure 6.

Coefficient of friction relative to fabric tension using fabric no. 1.

Influence of the wrap angle

The friction coefficient μ slightly increased with the wrap angle between the fabric and the roll (Figure 7). According to equations (1), (6), and (7), the coefficient is independent of the wrap angle. Nevertheless, the dependence of a high range of wrap angles, i.e. from 90 to 450°, on the coefficient of friction calculated according equation (1) has been shown in the literature.¹⁹ In this study, the range of variations in the wrap angle was quite small, which can explain the very small influence of the wrap angle on the friction coefficient.

Figure 7.

Coefficient of friction relative to wrap angle using fabric no. 1.

Influence of the roll radius

The roll radius had little influence on the friction coefficient (variations in the range 3–6%). This result is coherent with equation (1). In fact, this equation does not take into account the roll radius. Nevertheless, according to equation (7), the ratio of the friction coefficients obtained with large and the small rolls, 75 and 30 mm in radius, respectively, should be between 1 (for n = 1) and 1.36 (for n = 2/3). In this study, the friction coefficient ratio was low with a maximum value of 1.06. According to equation (2), n is influenced by the roughness of the materials in contact. The roughness of the two rolls was not identical: Ra was 0.26 µm for the small roll and 1.90 µm for the large roll. This difference in roughness probably influenced the results.

Influence of the sliding velocity

As shown in Figure 8, when the sliding velocity increased, the friction coefficient increased until it became almost stable. In order to compare the results obtained with the friction bench used in this study and a linear tribometer, some extra measurements were taken. The measurement conditions taken as the reference are: the large roll, i.e. 75 mm in diameter, a warp angle of 100° and an initial tension T_initial on each web end of 10 N; sample no. 1 was used, and was 90 mm in width. The tribometer used for this experiment was a linear plane-pin tribometer.²⁰ The length of the textile sample in contact with the roll was 130 mm. The surface area A was then 117 cm². The average pressure P applied on the rolling friction bench was:

P = \frac{2 \cdot T_{initial}}{A} \int_{0}^{θ / 2} \cos (θ) \cdot d θ

(9)

Figure 8.

Friction coefficient relative to sliding velocity for all the webs tested. The half-lines with a slop closed to zero are the results of linear regression (the equations of these half-lines are given in the legend). The plain dots correspond to the stabilization of the coefficient of friction given by the half-lines.

In this case, the pressure P was 1.3 kPa.

The contact web/roll was approximately planar; therefore, a flat slider with a rectangular sole was used. This slider was made of aluminum and the borders had chamfers in order to avoid any sharp contact. The sole area of the slider was 10 cm² (2 × 5 cm²) and the normal load was 1.3 N.

The results obtained with this tribometer under the conditions described above were compared to the results obtained with the rolling friction bench in Figure 9. The evolution of the coefficient of friction was the same with both measurements. Figure 8 shows that, regardless of the sliding velocity, the pile side no. 2 P gave a higher friction coefficient than the glabrous side of the same sample 2 G for the same fabric. Both plain woven fabrics no. 1 and 3 showed the same kind of behavior. The warp knitted fabric no. 4 presented a peculiar evolution relative to the sliding speed.

Figure 9.

Friction coefficient relative to the sliding velocity obtained from the rolling friction bench and the linear tribometer.

An increase in the friction coefficient with sliding velocity has been reported several times in the literature,^8,14,21 and various explanations have been given for this trend. For Howell et al.,⁸ the friction coefficient between an acetate yarn and a lubricated chrome roller increased with the sliding velocity (up to 7 m/s) because of the heat generated by friction. For Taylor and Pollet,²² the coefficient of friction increased and then stabilized because the entrapped air acted as a lubricant. In this study, the coefficient of friction reached a steady state at a sliding velocity of 1.6 cm/s. This value of velocity is much lower than that used in the present study. Budinski highlighted the fact that the effect of the velocity on the friction coefficient depends on the material in the case of a plastic web (PTFE or PVC).¹⁸ In fact, the friction coefficient can decrease, remain stable, or increase.

The influence of entrapped air as a lubricant was evaluated. In the first experiment, a powerful air jet (8 bars and 100 liters per minute) was oriented in two configurations: first, parallel to the roll axis between the roll and the fabric, creating a depression and suppressing the air entrapped between the fabric and the roll. In the second experiment, the air jet was oriented perpendicular to the roll axis, thereby increasing the quantity of air at the interface between the roll and the fabric by creating an air cushion. Under both conditions, the friction coefficient μ varied only slightly from 0.40 to 0.42, i.e. by 5% (using fabric sample no. 1). Moreover, the evolution of the friction coefficient with the sliding velocity was not modified. These results show air entrapped between the roll and the fabric cannot explain the friction coefficient evolution relative to sliding velocity.

The influence of the temperature increase due to friction has been considered. For homogeneous materials, equation (10) (Fourier’s equation) shows that, with a given temperature gradient, the heat flow increases with the thermal conductivity of the material.

\overset{⇀}{ϕ} = - λ \overset{⇀}{\nabla} T

(10)

where

\overset{⇀}{ϕ}

is the heat flow density in the direction normal to the surface expressed in W/m², λ is the thermal conductivity of the structure of the material in the heat flow direction expressed in W/m/K, and T the temperature field depending on the time given in K.

Therefore, the more thermal energy a material absorbs, the more it is a thermal conductor. For a fibrous material, the thermal conductivity is a combination of the thermal conductivity of the air and of the fibers (weighted respectively by the fraction of the volume taken up by each component). The thermal conductivity for pure aluminum is 237 W/m/K, less than 1 W/m/K for polyester, 6.6 W/m/K for polyamide or cotton, and about 0.03 W/m/K for air at 20℃. Moreover, the roll is much thicker than the fabric, 10 mm against a maximum of 1 mm for the fabrics used; moreover, during the rotation of the roll, the fabric friction zone changes. In order to evaluate the increase in fabric temperature due to friction during measurement, it would be better to measure the interface temperature during the test, but the results could be influenced by the measurement itself. Because of all the considerations described above, the relative thicknesses of the two bodies, i.e. the roll and the fabric, and their respective thermal conductivity, it is assumed the increase in temperature of the interface fabric/roll during friction with a view to thermal point can be quantified on the fabric surface with an infra-red camera (Flir Thermacam™ SC3000). The results show that the increase in the fabric temperature was 0.1℃ during a prolonged friction test of 35 s, while the natural duration of the test is 5 s (Figure 10). This low variation has no significant impact on friction. In fact, during friction between a textile fabric and the aluminum roll, heat flows from the fabric to the aluminum and the fabric does not heat up. This heat absorption from the roller was also observed in.¹⁴ Previous investigations have shown that the evolution of the friction coefficient relative to the sliding velocity is not due to the parameters of the measurement method. Therefore, this friction behavior is likely due to textile fabric properties alone.

Figure 10.

Fabric temperature at the beginning (left) and the end of the friction experiment (right).

During friction on the friction bench, the fabric is in traction. A study of fabric tensile behavior was done under the same conditions as on the friction bench. On the rolling friction measurement bench, the nominal textile web tension was 10 N on both ends at the beginning of the measurement. During the friction test, these tensions can vary from 0.6 to 1.4 N per centimeter width. The tensile test was performed after providing an initial preload of 1 N/cm over a load range of 0.6 N/cm to 1.4 N/cm.

Figure 8 shows that the friction coefficient stabilized for each textile sample. The dotted lines are fitted to the friction coefficient curve with a linear regression and have a slope close to zero. The plain dots on these lines correspond to the beginning of stabilization. Figure 11 shows the correlation between the specific elastic modulus and web stabilization velocity at which sliding friction achieves steady state. The tendency was for the stabilization velocity to increase with the specific elastic modulus. The sliding velocity at which the friction coefficient became constant was correlated with the specific elasticity modulus, measured during the tensile test. The friction coefficient was calculated according to the capstan equation (equation (1)); therefore, the results show the difference in tension at each end of the fabric induced by the roll rotation, which became constant at a lower velocity when the fabric was less rigid. Therefore, the evolution of the friction coefficient with sliding velocity was due to an increase in the difference in tension between the two web ends mediated by roll rotation and leading to an elongation of the fabric. The elongation increased with sliding velocity up to a sliding limit, depending on the specific elastic modulus of the fabric. After this limit, the friction coefficient remained constant.

Figure 11.

Stabilization velocity of the coefficient of friction relative to the specific elasticity modulus for the different webs tested.

Influence of repeated friction cycles

The aim here was to study the evolution of the friction coefficient relative to several increasing and decreasing cycles of sliding velocity. At least five acceleration-deceleration cycles were performed. In Figure 12, the friction coefficient was higher when the sliding velocity increased than when it decreased. This phenomenon was verified for all the textile fabrics studied. Moreover, the first cycle was similar or gave a higher coefficient of friction than the others. However, for the non-textile webs, i.e. the PVC and the paper webs, the influence of the acceleration-deceleration cycles was the opposite: the friction coefficient was higher for the deceleration period than for the acceleration period (Figure 13).

Figure 12.

Repetitive friction cycles using fabric no. 1.

Figure 13.

Acceleration–deceleration friction cycle for the textile web no. 1 (left) and the PVC film web (right).

Some investigations were performed in order to understand these results. First, it was verified by SEM observations for all the fabrics tested that there was no visible wear after the cyclic measurement process. Additionally, because the fabric was stressed in traction during the friction test, this traction was reproduced on a tensile machine in order to study the behavior of the fabric in traction without any friction stress. The effect of repeated friction cycles on the elongation of the material was tested on a traction machine. Each fabric web was tested with at least eight consecutive traction cycles, reproducing the tension variations during the friction test: an initial preload of 1 N/cm and the tensile cycles over a load range from 0.6 N/cm to 1.4 N/cm. Between the two cycles, there was a waiting time of two seconds, corresponding to the duration between two acceleration-deceleration cycles in the friction tests.

The evolution of the tensile force during several tensile cycles is illustrated in Figure 14. The results were similar for the different fabrics studied. It can be seen in Figure 14 that the deformation was plastic. This plasticity was greater for the first and the second cycles than for the following cycles. This deformation was due to the rearrangement of the yarns inside the fabric. The same phenomenon was observed in the friction measurements (Figure 12).

Figure 14.

Repetitive traction cycles using fabric no. 1.

Figure 15 shows the evolution of the specific elasticity modulus of the fabrics relative to the number of cycles. Each cross represents five measurements. These results show that there was a significant difference between the first and second cycles for the textile webs, corresponding to textile structural rearrangement. For the paper web, the evolution was similar between the first and the second cycles, but this material was more rigid. Therefore, the cyclic traction behavior of these materials cannot explain the differences observed in friction behavior during the acceleration and deceleration phases between fabrics and paper or plastic webs (Figure 13). The authors assumed this phenomenon was due to the differences in the surface state, e.g. roughness (Figure 16) and/or air permeability between textile fabrics and paper and plastic webs. Further investigations must to be done to confirm this.

Figure 15.

Specific elasticity modulus for the webs investigated relative to the number of traction cycles. The PVC film modulus is 3300 N/cm regardless of the traction cycle.

Figure 16.

SEM photographs of the textile web no. 1 (left) and the PVC film web (right).

Sliding occurs at higher web velocity when there is greater adhesion between the roll and the web. Moreover, because of adhesion, during the deceleration phase, web elongation is kept constant over a greater speed range.

Web/roll simulation with a variable friction coefficient

The experimental results presented in the previous section show that the friction coefficient strongly depends on the sliding velocity, particularly when the sliding velocity is several centimeters per second. Therefore, the coefficient of friction has to be taken into account in the winding-unwinding control process. Thus, the coefficient of friction was implemented in the web/roll simulator presented below by including a variable friction coefficient.

In a such system, the classic relation coupling web tension and roll velocity can be adapted to highlight the friction coefficient μ (Figure 17), details of which can be found in Koç:¹

\frac{d}{d t} (\frac{a (t)}{1 + ɛ_{k - 1}} + l_{a} - a (t) + \frac{1}{μ} \ln (\frac{1 + ɛ_{k - 1}}{1 + ɛ_{k}}) + \frac{L}{1 + ɛ_{k}}) = - \frac{V_{k + 1}}{1 + ɛ_{k}} + \frac{V_{k}}{1 + ɛ_{k - 1}}

(11)

where a(t) is the length of the portion of web adhering to the roll, ɛ_k and L are the web elongation and the web length between the considered roll and the next one, respectively, and V_k is the linear velocity of the considered roll.

Figure 17.

Adherence and sliding zone.

This relation can be modified in order to isolate the web elongation dynamics ɛ_k:

\frac{d}{d t} (\frac{L}{1 + ɛ_{k}}) = - \frac{V_{k + 1}}{1 + ɛ_{k}} + \frac{V_{k}}{1 + ɛ_{k - 1}} - \frac{d}{d t} (- \frac{ɛ_{k - 1} a (t)}{1 + ɛ_{k - 1}}) - \frac{d}{d t} (l_{a} + \frac{1}{μ} \ln (\frac{1 + ɛ_{k - 1}}{1 + ɛ_{k}}))

(12)

with

ɛ_{k}, ɛ_{k - 1} \neq - 1

To improve the convergence of the simulator algorithm, it is useful to calculate the derivative terms on the right side of equation (12):

\frac{d}{d t} (l_{a} + \frac{1}{μ} \ln (\frac{1 + ɛ_{k - 1}}{1 + ɛ_{k}})) = \frac{d}{d t} (\frac{1}{μ}) \ln (\frac{1 + ɛ_{k - 1}}{1 + ɛ_{k}}) + \frac{1}{μ} \frac{1 + ɛ_{k}}{1 + ɛ_{k - 1}} \frac{\frac{d ɛ_{k - 1}}{d t} (1 + ɛ_{k}) - \frac{d ɛ_{k}}{d t} (1 + ɛ_{k - 1})}{(1 + ɛ_{k}) 2}

(13)

\frac{d}{d t} (- \frac{ɛ_{k - 1} a (t)}{1 + ɛ_{k - 1}}) = - \frac{ɛ_{k - 1}}{1 + ɛ_{k - 1}} \frac{d a (t)}{d t} + a (t) (\frac{- \frac{d ɛ_{k - 1}}{d t}}{(1 + ɛ_{k - 1}) 2})

(14)

However, equation (12) implies knowledge of adherence and sliding zone evolution on the roller. In order to calculate the length of the adhesion zone, the torque balance applied to the roller is considered (Figure 17):

M = (T_{2} - T_{1}) R + J \frac{d ω}{d t} = (T_{1} \sin (\frac{θ}{2}) + T_{2} \sin (\frac{θ}{2})) d

(15)

where R is the roll radius, J is the moment of inertia in rotation of the considered roll, ω is the roll angular velocity and d is the distance between the roll center and the application point of the normal force.

The angle of the sliding zone θ′ can be deduced from the distance d:

d = R \cos (θ') = \frac{(T_{2} - T_{1}) R + J \frac{d ω}{d t}}{(T_{1} \cdot \sin (\frac{θ}{2}) + T_{2} \cdot \sin (\frac{θ}{2}))}

(16)

From equation (16), it is easy to show, from the sliding distance, the adhesion zone between the web and the roller (Figure 17):

a (t) = R (θ - a \cos (\frac{(T_{2} - T_{1}) + \frac{J}{R} \frac{d ω}{d t}}{((T_{1} + T_{2}) \sin (\frac{θ}{2}))})) with (T_{1} + T_{2}) \neq 0

(17)

Web elongation is then calculated by applying the inverse of Hooke’s law to equation (17):

a (t) = R (θ - a \cos (\frac{(ɛ_{2} - ɛ_{1}) + \frac{J}{ESR} \frac{d ω}{dt}}{((ɛ_{1} + ɛ_{2}) \sin (\frac{θ}{2}))})) with (ɛ_{1} + ɛ_{2}) \neq 0

(18)

where ɛ₁ and ɛ₂ are the web elongations induced by the tensions T₁ and T₂, respectively, E is the web elasticity modulus, and S is the web cross-section. In a textile material, E is not constant with deformation but it can be considered to be constant in a small range of deformation; therefore, it is possible to define a local elasticity modulus (Figure 14).

The behavior of the friction coefficient was implemented in the simulator according to the experimental results obtained in the previous section. The friction coefficient value depends on the web tension and the sliding velocity (for a given web and roller). Generally, in web transport systems, sliding between the web and the roller occurs during transient phases (when the velocity increases or decreases) and the sliding velocity remains at relatively low values:

μ (t) = f (T, V_{g})

(19)

where T is the web tension and V_g is the sliding velocity.

In order to calculate the web velocity, the main difficulty is in dealing with the case when the roll loses contact with the roller. In fact, when the adherence zone between the web and the roller exists, it can be assumed that the web velocity is equal to the roller velocity. If the web totally loss adherence, the web velocity is supposedly equal to the velocity of the next adhering roller, therefore:

\frac{d V_{web}}{d t} = \frac{d V_{roll}}{d t} = - \frac{R 2}{J} T_{1} - \frac{F_{rs}}{J} V_{roll} + \frac{R 2}{J} T_{2} if a (t) \neq 0 \frac{d V_{web}}{d t} = \frac{d V_{next adhering roll}}{d t} if a (t) = 0 \frac{d V_{roll}}{d t} = - \frac{F_{rs}}{J} V_{roll} if a (t) = 0

(20)

where F_rs is the friction force between the roll and its shaft.

Figure 18 shows the simulation results for a variable friction coefficient compared to the results obtained without the friction coefficient taken into account. In a web fabric transport system, the transient phases are critical and the adherence zone varies. For example, at 20 s a web velocity step occurs and the web loses contact with the roller. In the case of a simulator that includes the friction coefficient and the capability of the web to lose contact with the roll, the impact of velocity variation on web tension is much lower because of the temporary loss of contact. These simulations show that the inclusion of the friction coefficient improves the simulation.

Figure 18.

Simulation results obtained with or without including the friction coefficient and the zone of adherence variation.

Conclusion

The rolling friction of fabrics using the capstan method was studied with regard to the experimental conditions: web tension, wrap angle, roll diameter and sliding velocity. The results are coherent with the literature. Focus was placed on the evolution of the friction coefficient between the fabric and roll relative to the sliding velocity. In fact, the friction coefficient increased with sliding velocity and stabilized at a specific velocity, depending on the fabric. The various arguments presented in the literature, i.e. air lubrication and temperature increase, were investigated and were found to be non-influential in the context of this study. An explanation for this phenomenon was given and argued, by the comparison between the friction and traction cycle experiments, to be a consequence of fabric deformation due to traction occurring in the capstan experiment. Moreover, the stabilization sliding velocity was shown to be correlated to the specific elastic modulus of the web. The evolution of the friction coefficient relative to sliding velocity and to web tension was introduced into a transport simulator. The simulation results obtained with and without the friction coefficient taken into account clearly show the improvement of the simulator due to the consideration of the friction coefficient. In this study, the bending rigidity of the material was neglected because the fabrics used were thin. This study should be continued with fabric rigidity taken into account.

Footnotes

Funding

This work was partly funded by Région Alsace.

Acknowledgments

The authors wish to thank the Web Handling Research Center of Strasbourg for the design and manufacture of the mechanical part composing the friction measurement device. The authors also thank Daniel Mathieu, assistant professor at the University of Haute Alsace, for the thermal measurements.

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