Traction Between a Web and a Smooth Roller

Web traction over rollers is known to deteriorate due to air entrainment at high web speeds. In this paper, a general model is presented to predict the traction capability of an impermeable web over a smooth roller. The model considers the effects of the web and roller speeds, roller radius, combined roughness of the two surfaces, web tension and thickness, friction coefficient, and compressible air bearing. The change of tension N ∆ due to mechanical slip between the roller and the web is calculated by a simultaneous solution of the in-plane and out-of-plane equilibrium of the web. The problem is nondimensionalized and the effects of nine of the eleven non-dimensional parameters on traction are investigated for a range of values. Formulas involving the non-dimensional parameters for the traction capability are presented in two variable polynomial forms. Traction Between a Web and a Smooth Roller 10/15/02 2 Introduction Thin, flexible, continuous sheets such as magnetic recording tape, paper, textiles and various forms of film are generally called webs. Web handling operations involve translation of a tensioned web over and under rollers, driers, coating and calendering stations, depending on the application. In web handling systems, rollers are one of the most common guiding elements. The function of a roller is to guide the web without changing its tension. This is possible if good traction exists between the web and the roller. However, web traction over rollers is known to deteriorate due to air entrainment at high web speeds [1,2]. In this paper, a general model is presented to predict the traction capability of an impermeable web over a smooth roller. A web guided by an ideal roller without any bearing resistance, operating in vacuum should not experience any slip over the roller, if the friction coefficient between the web and the roller is greater than zero [3]. However, some amount of air entrainment in the web roller interface is unavoidable in practice. The entrained air layer causes mixed lubrication condition to develop in the web-roller interface, where the web is supported partially by the rigid body contact of the asperities, and partially by the air layer which has a super ambient pressure level. The mechanics of a web interacting with a self-acting air lubrication layer is described by the foil bearing problem, which excludes the contact conditions [4-8]. Mixed lubrication in foil bearing problem has been investigated by [912]. In the case of mixed lubrication in the web roller interface, the belt-wrap pressure, T/Rr, is balanced partially by rigid body contact between the asperities and partially by air pressure. With increasing web speed, the amount of entrained air and consequently the magnitude of air pressure increases. This results in a smaller portion of the belt-wrap pressure to be shared by the rigid body contact pressure. Thus one can see, by considering Coulomb's friction law, f n F F μ = , that the frictional force required to sustain high traction can be effectively reduced, without altering the value of the coefficient of static friction, μ. It is possible for the web to lose contact with the roller as a result of air entrainment. However, it should be noted that, in practice, this is unlikely; as the web starts to fly, the contribution of the roller to entrain air will decrease; and the web will contact the roller, again. If the web operating conditions are not adjusted, web will chatter over the roller, causing transient tension changes and web scratches. Ducotey and Good developed an algorithm for predicting web traction over smooth rollers [1]. They added modifications to the classical foil bearing theory to accommodate the effects of web permeability, side air leakage and surface roughness. Results were favorable when compared to their experiments. They concluded that an increase in surface roughness significantly increases traction. Ducotey and Good also modeled grooved rollers where they showed that using circumferentially placed groves on a roller can improve the traction characteristics [13]. Rice et al. introduced a model for determining the effective asperity height for a variety of surface characteristics in the web-roller interface [2]. They modeled the web as an Euler-Bernoulli beam; used the compressible Reynolds lubrication equation to model the air entrainment effects; and included an asperity compliance model for contact between the web and the roller. Experimental measurements of the traction capability of a given web-roller interface involves inducing slip at different transport speeds, by applying a breaking Traction Between a Web and a Smooth Roller 10/15/02 3 torque on the roller [1,2]. At different web speeds, different breaking torque values cause slip between the web and the roller. These experiments, thus establish a functional relationship between the web speed and slipping torque value. When slip occurs, the web tension increases in the sliding direction by ∆N, which is related to the breaking torque value. The results of the experiments are typically reported by calculating an equivalent coefficient of friction e μ or traction coefficient with the capstan formula, 1 1 ln e N T μ β ∆ = + (1) where the inlet tension is T and the wrap angle is β [1,2]. Note that in these experiments, a small ∆N indicates weak traction capability of the interface. Traction loss primarily depends on the web speed. However, roller radius, combined roughness of the two surfaces, web tension and thickness and permeability of the web are also factors that affect traction. Effects of web or roller permeability have been studied by various authors [1,14,15]; these effects are not considered in this work. In this paper, a model for the traction measurements as described in the above paragraph is introduced. The change of tension N ∆ due to slip is calculated by considering the equilibrium of the in-plane stress resultant N. This is in contrast to earlier models, where N ∆ is calculated from ( ) / s n r N F T R μ ∆ = [2,13]. Nine non-dimensional parameters are identified and their effects on traction are investigated for a range of values. Formulas involving the non-dimensional parameters for the traction capability are also presented. Model of the web-roller interface The equations of equilibrium for a web travelling over a cylindrical surface at steady state as shown in Figure 1 are given by the following coupled set of equations [15,17], ( ) ( ) 4 ' 2 ' 2 4 2 ( ) s a c d w d w dN dw N D D w cv N p p p dx dx dx dx R x ρ + + − − = − + − (2a) 0 dN dx τ + = (2b) where ' r w w w = − is the web deflection measured with respect to the equilibrium spacing wr calculated for a stationary web [15]. Other variables are listed in the nomenclature. Equation (2a) represents the combined out-of-plane and moment equilibria. In particular on the left hand side of this equation, the first term represents the bending stiffness of the web; the second term represents the shell-stiffness due to the wrapping the web around a cylindrical surface; and the third and the fourth terms represent the effects of in-plane stress resultant N on the out-of-plane equilibrium. Note that the third term of the right hand side has the only remaining term of the gyroscopic accelerations at steady state ρcv(dw/dx). The right hand side of equation (2a) shows the effect of air pressure p measured with respect to the ambient pressure pa, the rigid body contact pressure pc, and the belt-wrap pressure N/R(x). The curvature of the web is given by, ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 2 2 1 1 e e ( ) L x x L b b r H x H x L H x L H x L H x L R x R − − − + − + = − − + − − − + − (3) Traction Between a Web and a Smooth Roller 10/15/02 4 where Rr is the roller radius and ( ) 2 / 1 /T D b = is the characteristic bending length of the web [17]. Tangency points of the web L1 and L2 are ordered as follows 1 2 0 L L L < < < . Equation (2b) represents the in-plane equilibrium. In this problem, the in-plane traction τ occurs due to sliding contact. Coulomb’s friction law is used to find τ as follows, 1 if 0 1 if 0 s c v p v τ μ − > = < (4) where μs is the dynamic or the “slow-speed” friction described in reference [2]. The rigid body contact pressure is modeled using a non-linear empirical relation [2,9,18], 2 0 0 1 c h p P σ = − . (5) The web-roller clearance h is calculated by, h w δ = + (6) where δ(x) is the initial clearance between the web and the roller measured normal to the roller surface as shown in Figure 1. The web is assumed to have simple supports on it boundaries, at L x , 0 = : 0 2 2 = = dx w d D w . (7) The upstream tension T is specified as the boundary condition of the in-plane equilibrium equation (3b). For v > 0 this becomes, at 0 = x : N = T . (8) The air lubrication in the interface is modeled by using the Reynolds lubrication equation with first-order slip flow corrections. The corrections for slip flow become necessary when the web-roller clearance is in the order of the length of molecular mean free path of air λ. This equation for a compressible, ideal gas is [19], ( ) 3 2 6 6 d dp dph ph h V dx dx dx λ μ + = . (9) On the boundaries the air pressure is set to the ambient pressure pa. Equations (2-9) are coupled, non-linear equations. Their solution is obtained numerically as described in [14]. Non-dimensional Form of the Governing Equations The set of equations (2-9) has one independent variable x, four dependent variables, w, p, pc, and N and sixteen parameters that affect the dependent variables. In order to reduce the number of parameters that control the problem the following nondimensional independent variables are defined, w x x L = , c w w = , 0 σ h h = , a p p p = , a c c p p p = , T N N = (10) where Lw = βRr is the wrap length. Substitution of these variables in the governing equations results in following eleven non-dimensional control parameters, ( ) 2 1 υ − = T Ec S , w c C L = , a r p R P T = , / v V T cρ = , r w R L = β , w L L L x X = , w R R L x X = , c 0 σ = Σ , a p p P 0 0 = , 0 σ λ = Λ , 2 0 6 a w a vL B p μ σ = . (11) Traction Between a Web and a Smooth Roller 10/15/02 5 The coefficient of friction μs remains unchanged. With these non-dimensional variables the governing equations become, ( ) ( ) 2 4 ' 2 2 ' 2 4 2 1 12 c SC d w d w dN dw S w V N P p p N dx dx dx dx C β β + + − − = + − − (12a) 0 dN dx τ + = (12b) ( ) 3 2 6 d dp dph ph h B dx dx dx + Λ = (13) h w δ Σ = + (14) ( )2 0 1 c p P h = − (15) ' r w w w = − (16) 1 if 0 1 if 0 s c V PCp V τ μ − > = < (17) Results In this work, the effects of nine of the eleven non-dimensional parameters described above on the traction between a roller and a flexible web are investigated. The effect of XL and XR are not considered; it is reasonable to assume that by specifying these two parameters much greater than the non-dimensional bending length, b/Lw, their effect can be neglected. The change in tension at the onset of sliding N ∆ strongly depends on air entrainment; and the bearing number B controls the pressure generation to a great extent. Therefore, in investigating the effects of the non-dimensional parameters on N ∆ we paired each parameter with B, while keeping all the rest of the parameters constant. For example, we investigated ( , ) N f B C ∆ = and ( , ) N f B ∆ = Σ , etc. Values of the parameters for a typical tape transport are given in Table 1; the corresponding non-dimensional parameters are given in Table 2. Table 3 lists the ranges of the non-dimensional parameters used in this work. When, for example, the effects of B and S are investigated simultaneously, their values are taken from Table 3, (14 ≤ B ≤ 570 and 148 ≤ S ≤ 2470), and all the rest of the parameters are kept at their base values, given in Table 2. In the numerical simulations, the B range is divided into 40 equal parts and the range of the second parameter is divided into 25 equal parts. This results in a total of one thousand parameters, to obtain, for example the ( , ) N f B S ∆ = variation. Effects of sliding friction to steady state conditions in the web-roller interface Before presenting the effects of non-dimensional parameters on traction, the steady state conditions at the tape-roller interface for a typical mixed lubrication case is presented. For this case the following values of the non-dimensional parameters were used: B = 113.5, C = 1.061×10, P = 3.8, V = 1.94×10, β = π/2, Σ = 2×10, Po = 10, XL = XR = 1.061, Λ = 1.27. Figure 2 shows the steady state solution of equations (2-9) represented in terms of the non-dimensional parameters defined in equation (11). In particular, the variation of in plane stress resultant (tension) N , tape-roller spacing h , contact pressure c p and air Traction Between a Web and a Smooth Roller 10/15/02 6 pressure p are presented in the wrap region which spans 1 0 1 x L ≤ − ≤ . For this case the tape-roller spacing is less than one in the wrap region, which indicates full rigid body contact; h varies from approximately 0.86 on the entry side ( 0 x = ) to 0.8 on the exit side. Thus it is seen that when the effects of asperity compression and in-plane stress resultant are considered no constant gap region exists in the interface. The in-plane tension N increases approximately 40% along the interface due to sliding friction causing a subsequent increase in the belt-wrap pressure. Increasing belt-wrap pressure applies higher compression on the asperities along the interface, resulting in the variable gap height indicated above. The same effect is also responsible for the slight linear increase in contact pressure c p and air pressure p . On the exit side of the interface the air pressure shows the ondulation characteristic of the elastohydrodynamic lubrication [15,20]. This change in air pressure results in a sudden increase of the contact pressure and a sudden dip in web-roller spacing. Special care is given to the numerical resolution of the entry and exit regions: the spatial grid used in the solution has x ∆ =2×10 which results in many grid points in the transition regions. Curve fitting In this work two variable functions, such as 1 2 ( , ) f V V , are sought to represent the variation of N ∆ , where V1 = B and V2 ∈ {S, C, P, P0, V, β, Σ, Λ, μs}. A two variable curve fitting method is developed for this. Polynomials of the following form were used, 2 1 0 0 I J j i ij i j N D V V = = ∆ = (18) The order of polynomials I and J has an upper limit of three. Curve fits for N ∆ as a function of B and V2 are considered acceptable over a given range, provided that the maximum error does not exceed ten percent. For this reason, curve fitting is conducted using the proportional least squares regression method, which minimizes the sum of the squared percent errors. For all cases, curve fits were obtained using I = 1, 2 and 3 for N ∆ as a function of B. This curve fitting is conducted for each of the 25 values of the second variable V2. This gives 25 different values for each of the constants Dij of the polynomials. Regression is then performed on each Dij, by setting J = 1, 2 and 3 and by using Dij as a function of the second variable, V2. Trying different combinations of polynomials allowed flexibility in curve fitting options. In this work, most of the curve fits are reported using 3 order polynomials in B and V2, i.e., I = J = 3. The ranges for which the maximum error is less than ten percent are reported as well as the average and maximum percent errors across the acceptance range. The success of each curve fit was evaluated based on maximum error εmax and average error εave calculated between the curve fit and the original solution for N ∆ . Results are reported in V1 and V2 ranges where εmax and εmin are less than 10%. Traction results Variation of tension change N ∆ as a function of the bearing number B and the other nine parameters are given in Figure 3. Note again that in these plots a high value of N ∆ indicates a high traction interface. Parts a and b of Figure 3 shows that parameters V Traction Between a Web and a Smooth Roller 10/15/02 7 and S do not significantly affect the change of traction N ∆ in the range of values considered. On the other hand N ∆ depends with varying degrees of strength on C, P, β, μs, Σ, Λ. Figure 3a shows that in the range of V and B values considered here, N ∆ variation is independent of the non-dimensional tape speed V. A third order curve represents the data, 1 3 6 2 6 3 3 2 5.81 10 1.19 10 1.01 10 4.40 10 for 5 10 5 10 , 14 570 and 0.1 N B B B