A governing equation for web tension in a span considering thermal and viscoelastic effects is developed in this paper. The governing equation includes thermal strain induced by web temperature change and assumes viscoelastic material behavior. A closed-form expression for temperature distribution in the moving web is derived, which is utilized to obtain thermal strain. A model for web tension in a multispan roll-to-roll system can be developed using this governing equation. To evaluate the governing equation, measured data from an industrial web process line are compared with data from model simulations. Since the viscoelastic behavior of web materials is affected by the web temperature change, elevated temperature creep, and stress-relaxation experiments are conducted to determine the temperature-dependent viscoelastic parameters of the utilized viscoelastic model. Comparisons of the measured data with model simulation data are presented and discussed. An analysis of the web tension disturbance propagation behavior is also provided to compare transport behavior of elastic and viscoelastic materials.

## Introduction

The increased use of continuous processing of flexible materials by transporting them on rollers has resulted in considerable focus on investigating the encountered technical problems and providing suitable solutions to maintain productivity and good quality of products. The term web is used to refer to thin materials, which are manufactured and processed in continuous, flexible strip form, such as plastics, textiles, papers, and composites. Web tension and velocity are the two key transport variables that must be maintained at reference values to ensure the quality of web products. Tension variations in a web due to web treatment processes (such as heating, cooling, and printing) are encountered in many web handling industries; these processes create disturbances to web tension control.

Temperature in the moving web is one of the most important factors affecting the transport behavior as it changes both mechanical and physical properties [1]. In various web processes such as printing, coating, or other treatments, heat transfer between the web and environment or between the web and the contact area of a heated or cooled roller is encountered. Web tension is affected by the temperature change in the web span as it introduces thermal strain in the web.

Web tension behavior in the transport direction was investigated in Ref. [2]. Thermal effects from a heated roller and ambient air were included in web tension models. A dynamic model for web strain in a span that includes thermal and hygral effects was developed in Ref. [3]. A lumped capacitance model is considered to determine the heat transfer from the web surface in the free web span. A dynamic model for web tension behavior in a furnace of a continuous annealing line was developed in Ref. [4]. A new control scheme was proposed that considers thermal strain and the change in the Young's modulus. Research in utilizing models to design and develop roll-to-roll machines and design of model-based control systems has been active in the last several decades [59]. The model derived in this work can be used to design advanced model-based controllers to efficiently control web transport behavior.

Many web materials exhibit viscoelastic behavior when undergoing deformation during transport, especially in the presence of heating and cooling sources. In such situations, one needs to consider viscoelastic material properties in the derivation of the tension governing equation. Flexible and composite materials exhibit viscoelastic behavior that is directly attributed to the molecular structure of the underlying materials [10]. Viscoelastic behavior of web materials is considered in several papers [1113]. The existing work did not consider methods to determine governing equations for web tension behavior that includes the combined effects of both temperature distribution in the moving web and the viscoelastic material behavior.

In this paper, thermal and viscoelastic effects are included in the developed web tension governing equation. The induced thermal strain in a web span due to temperature change is determined by considering heat transfer in the region of wrap due to conduction from the roller and in the free web span due to convection to ambient air. Temperature-dependent viscoelastic behavior is investigated using the standard linear solid (SLS) model. The viscoelastic parameters of the utilized viscoelastic model are determined by conducting elevated creep and stress-relaxation experiments on several web composite materials. A section of an industrial web process line containing multiple spans is considered for validation of the proposed model. Model simulations are conducted with an existing control strategy implemented on the production web process line. The contributions of the paper lie in simultaneously considering the thermal and viscoelastic effects on web tension, determining the properties of viscoelastic model parameters, conducting model simulations by considering a section of an industrial web processing line, and comparison of the model simulation results with measurements obtained from an industrial processing line to validate the proposed model.

The remainder of the paper is organized as follows: Web tension governing equation is developed in Sec. 2 by considering thermal strain and a viscoelastic constitutive relation between mechanical strain and web tension. Viscoelastic model parameter identification for some composite materials transported in the industrial web processing line, model simulations, and comparison of simulation results with experimental results are described in Sec. 3. Disturbance propagation in a multispan web system is discussed in Sec. 4. Conclusions and potential topics for future work are provided in Sec. 5.

## Governing Equation for Web Tension Considering Thermal and Viscoelastic Effects

To develop a governing equation for web tension by considering both thermal and viscoelastic effects, the following procedure is employed. First, a governing equation for longitudinal strain in the web is developed by utilizing the law of conservation of mass in a control volume encompassing a span (web between two adjacent rollers); the longitudinal strain is a combination of both mechanical strain and thermal strain. Second, to determine thermal strain due to heating/cooling of the web, one requires the temperature distribution in the moving web. Heat transfer due to conduction is assumed in the region of web wrap on the roller and due to convection in the free span between rollers to determine the temperature distribution. Third, to determine a constitutive law that relates web mechanical strain and tension, the web is assumed to be linearly viscoelastic. These three aspects are combined to develop a governing equation for web tension.

### Governing Equation for Web Strain.

Application of the law of conservation of mass for the control volume shown in Fig. 1 gives the following relation between web strains and web transport velocities:
$ddt∫0Liρ(x,t)A(x,t) dx=ρi−1Ai−1vi−1−ρiAivi$
(1)

where the following notation is used: x, transport direction coordinate, t: time, $ρ(x,t)$, material density; A, web cross section area; Li, length of the ith web span; εi, web strain in the ith web span; ti, web tension in the ith web span; vi, web transport velocity on the ith roller; ωi, angular velocity of the ith roller. Note that t without a subscript is used for time and with an i is used to denote web tension in the ith span. Also, note that small strain assumption is utilized in subsequent derivations, which is true for many flexible materials (such as metals, polymers, and paper) processed using roll-to-roll methods.

Since the mass of a representative volume element of the web in the stretched state is equal to the mass in the unstretched state, the following relationship can be written for an elemental mass dm:
$dm=ρAdx=ρuAudxu$
(2)
where the subscript u denotes the unstretched state and dx denotes the length of the elemental mass dm. Assuming small strain, the relationship between the stretched length and the unstretched length is given by
$dx=(1+εx)dxu$
(3)
Combining Eqs. (2) and (3) gives
$ρ(x,t)A(x,t)ρu(x,t)Au(x,t)=dxudx=11+εx(x,t)$
(4)
Note that in the above equations, the subscript x in εx is used to indicate that this is longitudinal (or transport) direction strain and depends on the transport direction coordinate x. Substitution of Eq. (4) into Eq. (1) for the ith span results in
$ddt∫0Liρu(x,t)Au(x,t)dx1+εi(x,t)=ρu,i−1Au,i−1vi−11+εi−1(Li−1,t)−ρu,iAu,ivi1+εi(Li,t)$
(5)
where $εi−1(Li−1,t)$ is the strain at the entering point of the control volume and $εi(Li,t)$ is the strain at the exiting point of the control volume. Assuming the cross-sectional area and the density of the web in the unstretched state are constant, Eq. (5) can be written as
$ddt∫0Lidx1+εi(x,t)=vi−11+εi−1(Li−1,t)−vi1+εi(Li,t)$
(6)
Using the small strain assumption ($ε≪1$), further simplification can be made to the last equation by considering that $1/(1+ε)=(1−ε)/(1−ε2)≈(1−ε)$. Thus, Eq. (6) can be written as
$ddt∫0Li(1−εx,i(x,t))dx=vi−1(1−εi−1(Li−1,t))−vi(1−εi(Li,t))$
(7)
Since the adjacent rollers are stationary ($L˙=0$), Eq. (7) can be simplified as
$ddt∫0Liεi(x,t)dx=vi(1−εi(Li,t))−vi−1(1−εi−1(Li−1,t))$
(8)
Equation (8) shows that the time rate of change of the integral of the strain in the ith span is a function of the web velocities on the adjacent rollers and the strains in the ith and $(i−1)$th spans. The total strain in the ith span is assumed to be given by strain due to mechanical stress and strain due to temperature distribution in the moving web as follows:
$εi(x,t)=εt,i(t)+εϑ,i(x,t)$
(9)
where $εt,i(t)$ is the average mechanical strain in the span and $εϑ,i(x,t)$ is the thermal strain, which is dependent on both the spatial coordinate and time. Substituting Eq. (9) into Eq. (8) and simplifying results in
$Lidεt,i(t)dt=(vi−vi−1)+(vi−1εt,i−1(t)−viεt,i(t))+(vi−1εϑ,i−1(Li−1,t)−viεϑ,i(Li,t))−ddt∫0Liεϑ,i(x,t)dx$
(10)
The thermal strain is assumed to depend linearly on the change in temperature as follows:
$εϑi(x,t)=α(ϑi(x,t)−ϑ0)$
(11)

where α is the coefficient of thermal expansion, $ϑi(x,t)$ is the temperature distribution in the moving web, and $ϑ0$ is the reference temperature. In the following, a closed-form expression for temperature distribution in a moving web will be derived.

### Temperature Distribution in the Moving Web.

The web temperature distribution $ϑ(x,t)$ in the moving web can be determined by considering heat transfer in the region of wrap on the roller and in the free web span [1416]. Figure 2 shows the temperature notation in the region of roller wrap, free web span, and the ambient air. In the region of wrap on the roller, heat transfer takes place primarily due to heat conduction over a heated/chilled roller in the transverse direction (along web thickness). Therefore, it is modeled using the following one-dimensional heat equation in the transverse direction:
$Kϑ∂2ϑ∂y2=∂ϑ∂t$
(12)

where $Kϑ$ is the thermal conductivity constant and y is the transverse direction coordinate (along web thickness). The web is treated as a slender rectangular bar within the region of wrap. Since the wrap region is small compared to the length of the web span, the temperature along the transport direction within the wrap region is assumed to be the same. If $ϕ$ is the angle of wrap, R is the roller radius, and $v¯$ is the average web transport velocity on the roller, then the resident time of a representative volume element is given by $Rϕ/v¯$; in general, this resident time is small because the average transport speed is much larger than the product of roller radius and wrap angle.

The initial conditions for Eq. (12) are given by
$ϑ(y,0)=θw,i−1, for 00$

where h is the web thickness, $θw,i−1$ is the temperature of the web prior to engaging the (i − 1)th roller, $θR,i−1$ is the temperature of the roller, and $θU,i−1$ is the ambient temperature in the region of wrap. Note that ϑ is used to denote temperature that is a function on both the spatial coordinate and time, whereas θ is used for a boundary condition or a temperature of the web for a fixed spatial location.

The solution of Eq. (12) with the above initial conditions is given by
$ϑN,i−1(y,t)=θU,i−1+θR,i−1−θU,i−1hy+2π∑n=1∞(θw,i−1−θU,i−1+(−1)n(θR,i−1−θw,i−1)n) sin nπyhexp(−n2π2Kϑth2)$
(13)
where the subscript N denotes the temperature state at the end of the web wrap region. The temperature of the web at the beginning of the ith web span can be written as
$θN,i(t)=1h∫0hϑN,i−1(y,t)dy$
(14)
Heat transfer in the free web span occurs primarily due to convection to/from the ambient air, and modeled using a lumped capacitance model given by
$dϑi(x,t)dt=∂ϑi∂t+v(x,t)∂ϑi∂x=−1Tϑ(ϑi(x,t)−ϑU,i(x,t))$
(15)
where $Tϑ=ρhc/γ*$ is the thermal time constant, ρ is the web density, c is the specific heat of the web material, and $γ*$ is the coefficient of heat transmission, which represents the heat flux for a unit difference in temperature between the web surface and the surrounding ambient air. To simplify the solution of Eq. (15), it is assumed that the web transport velocity is equal to the average web velocity, i.e., $v(x,t)=vr$ and the ambient temperature $ϑU,i$ is a function of time only. The temperature in the web with these assumptions is given by
$ϑi(x,t)=ϑi(0,t)exp(−xvrTϑ)+(1−exp(−xvrTϑ))ϑU,i(t)$
(16)

where $ϑi(0,t)=θN,i(t)$, which is provided by Eq. (14). In the following, Eq. (16) is utilized in the derivation of the mechanical strain governing equation.

### Mechanical Strain Governing Equation.

Using Eqs. (11) and (16), the last term in the right-hand side of Eq. (10) can be written as
$ddt∫0Liεϑ,i(x,t)dx=αddt∫0Li(ϑi(x,t)−ϑ0)dx=αvrTϑ(1−δi)dϑi(0,t)dt+α(Li−vrTϑ(1−δi))dϑU,i(t)dt$
(17)
where $δi=exp(−τi/Tϑ)$ and $τi=Li/vr$. Now, substituting Eq. (17) into Eq. (10) and linearizing the resulting equation around a reference web velocity, vr, and reference strain, εr, by setting $vi(t)=vr+Vi(t)$ and $εt,i(t)=εr+Si(t)$ (where $Vi(t)$ and $Si(t)$ are the velocity and strain variations, respectively), and ignoring the second-order terms, results in the following governing equation for variation in strain:
$LiSi˙=(Vi(t)−Vi−1(t))+vr(Si−1(t)−Si(t))+vrα(ϑi−1(Li−1,t)−ϑi(Li,t))−αvrTϑ(1−δi)dϑi(0,t)dt−α(Li−vrTϑ(1−δi))dϑU,i(t)dt$
(18)
Taking the Laplace transform of Eq. (18) and assuming the constant ambient temperature around the web span, i.e., $((dϑU,i(t)/dt)=0)$, yields the following equation:
$Si(s)=(Vi(s)−Vi−1(s))(τis+1)+Si−1(s)(τis+1)+α(ϑi−1(Li−1,s)−ϑi(Li,s))(τis+1)−αTϑ(1−δi)(sτis+1)ϑi(0,s)$
(19)

### Linear Viscoelastic Model Relating Mechanical Strain and Web Tension.

Many physical models have been used to describe the time-dependent behavior of viscoelastic materials. A viscoelastic solid can be described in terms of a three-element viscoelastic model, such as the SLS model. Figure 3 shows the SLS model consisting of a Maxwell element (linear spring and dashpot in series) and a linear spring in parallel. This model is selected because it can describe the stress-relaxation and creep behaviors [17], and reasonably reflects the linear viscoelastic behavior of web materials transported under small strains. The relationship between stress and strain in this model can be written as [10]
$σibi+σi˙E2,i=(E1,ibi)εi+(1+E1,iE2,i)εi˙$
(20)
where $E1,i$ and $E2,i$ denote the stiffness of the two springs and bi denotes the damping coefficient of the damper; the subscript i denotes that these parameters correspond to the ith span. The sum $E1,i+E2,i$ is called the initial modulus of the SLS model and $E1,i$ is called the equilibrium modulus. By taking Laplace transform of Eq. (20), the following relationship between the stress and strain in the frequency domain can be obtained:
$Si(s)=σi(s)(biE2,is+1)E1,i(bi(E1,i+E2,i)E1,iE2,is+1)$
(21)
where $σi(s)$ is the Laplace transform of $σi(t)$. Define the tension variation around the reference value tr as $Ti(t)=ti(t)−tr$. Let $Ti(s)$ be the Laplace transform of $Ti(t)$. Then, since the stress is given by $σi(s)=Ti(s)/A$, Eq. (21) can be written as
$Si(s)=Ti(s)(τr,is+1)AE1,i(τc,is+1)$
(22)
where $τr,i=bi/E2,i$ and $τc,i=bi(E1,i+E2,i)/(E1,iE2,i)=τr,i(E1,i+E2,i)/E1,i$. Substituting Eq. (22) into Eq. (19) results in the following governing equation for web tension:
$Ti(s)=E1,i(τc,is+1)(τr,i−1s+1)E1,i−1(τc,i−1s+1)(τr,is+1)(τis+1)Ti−1(s)+(AE1,i(τc,is+1)vr(τr,is+1)(τis+1))[(Vi(s)−Vi−1(s))+αvr(ϑi−1(s)−ϑi(s))−αvrTϑ(1−δi)(sϑi(0,s))]$
(23)

Equation (23) provides the tension behavior of a moving web in the ith span considering the effect of temperature change and viscoelastic material behavior. The following key assumptions were utilized to derive this governing equation for web span tension: (1) strain is assumed to be small, (2) only heat conduction in the thickness direction is assumed between the web and the roller surface in the region of wrap, (3) only convection between the web surface and ambient air is assumed in the free span, and (4) the web material is assumed to be linearly viscoelastic.

It is evident that there is transport of tension from upstream spans ($Ti−1$) to the downstream spans. The first term of the equation includes the viscoelastic effects of the upstream and downstream spans. Web tension is affected by temperature difference between adjacent web spans ($ϑi−1−ϑi$), and the temperature of the web at the beginning of the web span ($ϑi(0,t)$); these are obtained from Eqs. (13), (14), and (16). This span governing equation allows (1) analysis of web transported through heating/cooling sections and (2) the control system designed to test the robustness of controllers applied to web handling processes by addressing viscoelastic and thermal effects on web tension.

## Model Validation

To evaluate the developed model, experiments and model simulations were conducted for a section of an industrial web process line used for manufacturing flooring materials in web form. To conduct model simulations, one requires temperature-dependent viscoelastic parameters of the SLS model. In the following, we first provide a discussion on the stress-relaxation and creep experiments that were conducted to determine the SLS model parameters, followed by a description of the experimental process line and a discussion of the model validation results.

### Determination of Viscoelastic Model Parameters.

The viscoelastic parameters of the SLS model were determined by conducting elevated temperature creep and stress-relaxation experiments. These experiments were conducted on a tensile testing machine equipped with an environmental chamber to allow for testing at elevated temperatures; test specimens were prepared according to the ASTM D638-08 [18]. Experiments were conducted for two composite web materials, Royelle and Stratamax Better, that are manufactured by Armstrong World Industries; we will refer to these materials as web 1 (Royelle) and web 2 (Stratamax Better). Both materials are made of felt and gel (with different felt combinations). Felt is made of cellulose, limestone, and styrene-butadiene-rubber latex, and gel is made of polyvinyl chloride, di-iso phenyl phthalate, limestone, titanium dioxide, and other process aids. Experiments were conducted at different levels of temperature (i.e., 18 °C, 66 °C, 110 °C, and 154 °C). These temperature levels were chosen as they are close to the operating temperatures in the industrial web processing line. Since the procedure is the same, we will show the experimental findings for only one web 1.

Stress-relaxation tests were carried out by deforming the specimens to a given amount and the decrease in stress was recorded for a duration of 1000 s. Specimens were deformed gradually until they reached the desired strain of 1%. After that, strain was maintained constant for 1000 s and the decrease in stress with respect to time was recorded. Data from these tests are provided in Fig. 4.

The solid lines indicate exponential curve fitting, whereas the discrete marks indicate the experimental data. It is clear that the stress-relaxation decreases with increasing temperature. The results obtained from the stress-relaxation experiments were used to determine the initial modulus of elasticity (E) for the web materials and the linear spring constant (E1). The initial modulus of elasticity is the sum of the moduli of the two linear springs in the SLS model, which can be determined from the relaxation data as the modulus at the initial time using the following equation:
$E(t)=E1+E2e−(t/τr)$
(24)

The modulus E1 represents the steady-state response of the stress-relaxation function as shown in Eq. (24). Therefore, E1 was determined from the stress-relaxation curves after large time. E2 was determined as the difference between the initial modulus E and E1.

Creep experiments were conducted by applying a constant load of 1 MPa for a duration of 1000 s. After the desired loading was achieved, data were collected during the unloading time of 60 s. Prior to conducting the creep experiments, the time rate of temperature increase in the environmental chamber was 2 °C/min to ensure a uniform temperature distribution across the specimen. Data indicate that the creep strain increases with increasing temperature for both web materials. The results of the creep experiments were used to determine the viscosity parameter b. The creep function is given by
$J(t)=1E1−E2E1(E1+E2)e−(t/τc)$
(25)

where $τc=τr((E1+E2)/E1)$ is the creep or retardation time and τr is the relaxation time. The viscosity parameter b was evaluated from the regression analysis of the creep data using Eq. (25).

Figure 5 shows the temperature-dependent modulus for web 1 and web 2. It can be seen that the Young's modulus decreases with increasing temperature for both web materials. The cross marks in the graph represent the experimental data, whereas the solid lines are for curve fitting. The temperature-dependent moduli determined are $362 exp(−0.0025θ)$ for web 1 and $463 exp(−0.0029θ)$ for web 2.

Figure 6 shows the data corresponding to the temperature-dependent modulus E1 of the SLS model, which can be computed at a given temperature using: ($312−1.633 θ0.854$) for web 1 and ($368−0.0064 θ1.97$) for web 2. The modulus E2 is obtained by subtracting E1 from E for each web material.

The viscosity parameter b was determined using the creep data and based on the methods provided in Refs. [19] and [20]. The resulting expressions are: $7.58(θr/θ)0.34$ for web 1 and $5.27(θr/θ)0.37$ for web 2, where θr and θ are the room temperature and operating temperature, respectively.

### Web Process Line Description and Model Validation Results.

A section of the coating and fusion line at Armstrong World Industries is considered for model simulations; a line schematic of the section is provided in Fig. 7. In this section, the web is coated by a clear coat and transported through heating and cooling rolls to modify web properties. Heating the web melts and cures the coating to get a uniform coat thickness on the surface of the web material. The cooling process is necessary after heating to relieve internal stresses that are induced by the heating process. This section was selected as it involves thermal process as the web is transported over heated/chilled rolls. Measured data from this section are compared with data from the model simulations. In model simulations, we employed the same control strategies that are used in the industrial process line. Simulations were conducted for the two web flooring products, Royelle (web 1) and Stratamax Better (web 2).

It is common practice in web handling to divide the web process line into tension zones, where a tension zone is defined as the web section between two independent driven rollers [21]. Since a typical web process line contains a significantly more number of idle rollers when compared to the number of driven rollers, it considerably simplifies both the analysis of the web process line and the design of control systems. This simplification means that the dynamics of the idle rollers are treated as perturbations or disturbances to the tension zone control systems. In the case of Fig. 7, there is a tension zone from the coating roller to the heating roller and a tension zone from the heating roller to the set of four cooling rollers. Note that the cooling rollers are either mechanically slaved with a single motor shaft driving the four roller shafts using a transmission or electronically slaved with a single controller driving all four motors, and so these are treated as a single drive unit. With the definition of the tension zones, the coating and fusion line web section shown in Fig. 7 can be simplified to the figure shown in Fig. 8. In Fig. 8, M0 is the coating roller, M1 is the heating roller, and M2 denotes the integration of the four cooling rollers into one unit. We will refer to the tension zone between M0 and M1 as zone 1 and the one between M1 and M2 as zone 2. Note that the span tension equation is applied to the length of the web in each tension zone; in this regard, one can consider the tension in each of these zones as an average value. We will denote tension in zone 1 as t1 and in zone 2 as t2. It is noted that both the span lengths between the driven rolls and the wrap angle of the web around the rolls will be maintained even though this is not apparent from the simplified line sketch shown in Fig. 8.

The transport velocity of the web on the ith roller is denoted by $vi$ and the input torque from the ith motor is denoted by $ui$. Coating, heating, and cooling rollers are all driven and coating and heating rollers are under pure speed control, whereas cooling roller is under speed and tension control; the speed-based tension control strategy that is utilized is provided in Fig. 9.

Since there is no heating or cooling process in zone 1, the governing equation for web tension variation in this zone is given by
$T1(s)=(AE1,1(τc,1s+1)vr(τr,1s+1)(τ1s+1))[(V1(s)−V0(s))]+1(τ1s+1)T0(s)$
(26)
In zone 2, since the web is heated and subsequently cooled, the governing equation for web tension variation considering the effect of the thermal strain and the viscoelastic behavior is given by
$T2(s)=E1,2(τc,2s+1)(τr,1s+1)E1,1(τc,1s+1)(τr,2s+1)(τ2s+1)T1(s)+(AE1,2(τc,2s+1)vr(τr,2s+1)(τ2s+1))[(V2(s)−V1(s))+αvr(ϑ1(s)−ϑ2(s))−αvrTϑ(1−δ2)(sϑ2(0,s))]$
(27)
Velocity variations dynamics ($Vi(s)$) shown in Eqs. (26) and (27) are determined as follows: The roller angular velocity dynamics of fixed radius is given by
$Jiω˙i=(ti+1−ti)Ri+niui−bfiωi$
(28)
where ui is the input torque applied on the roller i, ni is the gear ratio of the motor to roller i, and bfi is the coefficient of the friction in the roller shaft bearings. The web velocity is related to the roller angular velocity by the relation, $vi=Riωi$, assuming there is no slip between the web and the roller. The governing equation for the dynamics of web velocity variation is given by
$v˙i=Ri2Ji(ti+1−ti)+niRiJiui−bfiJivi$
(29)
Define $ti=Ti+tr$ and $ui=Ui+uoi$ where Ti and Ui are the variations in tension and control input, respectively, uoi is the control input that maintains the forced equilibrium at the reference values [21], and tr is the reference tension. Thus, the dynamics for web velocity variations in the time domain can be written as
$V˙i=Ri2Ji(Ti+1−Ti)+niRiJiUi−bfiJiVi$
(30)
In the frequency domain, with $Vi(s)$ as the Laplace transform of $Vi(t)$, the dynamics for web velocity variations can be written as
$Vi(s)=Ri2/bfi(τvis+1)(Ti+1(s)−Ti(s))+niRi/bfi(τvis+1)Ui(s)$
(31)

where $τvi=Ji/bfi$ and $Ui=ui−uoi$. Equation (31) represents the deviation (variation) in velocity from its reference value ($Vi=vi−vr$, where vi is the actual velocity and vr is the reference velocity).

Model simulations were conducted with the same controllers (proportional–integral controllers) and gains that were used in the industrial process line; proportional and integral gain values of 6.3 and 1 for the velocity controller, and 7.8 and 0.1 for the tension controller are used. The velocity reference used in the simulations is 0.5 m/s for both web 1 and web 2 to match that of the industrial web processing line. The web tension reference in the industrial web processing line was 978 N (220 lb) for both web 1 and web 2, which was also used in the model simulations. It is assumed that the reference velocity is identical for all tension zones in the simulations. The other web line parameters used in the model simulations are shown in Table 1. The web material parameters used in the model simulations are provided in Table 2. The average web temperature in span L1 is 26 °C, whereas the average temperature in web span L2 is 125 °C. These temperatures were measured in the industrial web processing line using a laser temperature sensing device. This average temperature was used only for the determination of the temperature-dependent viscoelastic parameters discussed in Sec. 3.1. The model simulations use Eqs. (27) and (31) for each span of the given web section, and as such consider the evolution of temperature distribution as given in Sec. 2.2. An idle roller mounted on load cells is used to measure web tension in web span L2, which is fed back to the tension controller that gives a speed correction to the inner speed loop. The inner speed-loop controls the speed of the cooling rollers (see Fig. 9).

Data from model simulations were compared with measured data from the industrial process line and are shown in Fig. 10 for web 1. Figure 11 shows the fast Fourier transform for the model tension output results and the measured data for web 1. The frequency content of both data is also correlated well. Similar trends are obtained for web 2.

Tension zone 1 between the coating and heating rollers is not equipped with a load cell roller for measuring tension in the industrial process line. Thus, web tension was determined using only model simulations and the data are shown in Fig. 12 for web 1. This tension zone does not have any cooling/heating areas as the coating process is implemented at room temperature, but its behavior is affected by its adjacent web spans and transport of disturbances from the upstream spans. Similar trends were obtained for web 2. In Sec. 4, we provide a discussion on how to study propagation of tension disturbances in a multispan system using the governing equations derived in this paper.

## Disturbance Propagation in Multispan System

There are many sources of both tension and speed disturbances in roll-to-roll systems. For example, due to eccentric or out-of-round rollers, machine-induced vibrations, nonideal material behavior, process-induced disturbances, etc. [22]. These disturbances are propagated from upstream spans to downstream streams as a consequence of transporting the web from the unwind roll to the rewind roll. In the following, we provide a discussion of the differences between the behavior of the tension model for a multispan system when elastic and viscoelastic material behaviors are considered. Figure 13 shows three-span system in which the upstream velocity and tension variations, V0 and T0, are considered as disturbances to the multispan system. In the figure, T1, T2, and T3 are the tension variations in web spans L1, L2, and L3, respectively, and V0, V1, V2, and V3 are the velocity variations of the rollers Ro, R1, R2, and R3, respectively. Define the following variables: $Kve,i=AE1,i/vr, τc,i=bi(E1,i+E2,i)/(E1,iE2,i), τr,i=bi/E2,i, τi=Li/vr$ and $Kth=αvr$, i = 1, 2, and 3. Also, define the following transfer functions:
$G1,i(s)=E1,i(τc,is+1)(τr,i−1s+1)E1,i−1(τc,i−1s+1)(τr,is+1)(τis+1); G2,i(s)=Kve,i(τc,is+1)(τr,is+1)(τis+1)$
Assuming a constant temperature at the beginning of a free web span (i.e., $dϑU,i(t)/dt=0$), the tension variation dynamics in the ith span are given by
$Ti(s)=G1,i(s)Ti−1(s)+G2,i(s)[(Vi(s)−Vi−1(s))+Kth(ϑi−1(s)−ϑi(s))]$
(32)
Sequential substitution of Eq. (32) for n spans provides the following generalized transfer function from V0 to tension variation in the nth zone, Tn as:
$Gvn(s)=Tn(s)V0(s)={−G2,1(s) n=1−G2,1(s)∏i=2nG1,i(s) n>1$
(33)
The transfer function from the disturbance $T0(s)$ to web tension in zone n is given by
$Gtn(s)=Tn(s)T0(s)=∏i=1nG1,i(s), n≥1$
(34)

We provide a brief discussion on these disturbance propagation effects by considering a web section with three tension zones as shown in Fig. 13, where with web lengths in the three tension zones, L1, L2, and L3, are at 26 °C, 70 °C, and 125 °C, respectively. The temperature of the upstream span to the roller R0 is considered the same as the downstream roller (tension zone L1). The viscoelastic parameters of the SLS model at these temperatures are determined using the equations derived in Sec. 3. For this study, we considered all web span lengths to be equal to 3 m and the web reference velocity to be 0.5 m/s.

Figure 14 shows the bode magnitude plot for tensions in three successive spans (with lengths L1, L2, and L3) in response to the upstream tension disturbance T0, when considering web 1 to be viscoelastic. It can be observed that the disturbance T0 affects web tension dynamics at low frequencies as the temperature increases in successive spans in the transport direction. The viscoelastic parameters $E1,i, E2,i$, and bi decrease with increase in web temperature. As a result, the creep time $τc,i$ and the relaxation time $τr,i$ are decreased. For the viscoelastic case, the magnitude of the transfer function $Gtn(s)$ decreases with increase in web temperature in the successive spans in the transport direction. This behavior can be attributed to the fact that the viscoelastic parameters of web materials decrease with increasing temperature. The magnitude of these transfer functions is proportional to the temperature difference in the adjacent web spans. At frequencies greater than $1/τi$, attenuation of the disturbances V0 and T0 is observed as the web travels into subsequent spans.

Figure 15 shows the effect of propagation of disturbance T0 on web tensions, $T1, T2, and T3$, with increasing temperature in the same three tension zone section by considering elastic behavior for the web 1 material. The evolution of web tension the elastic web was obtained by setting the viscoelastic parameter b = 0; this means that $τci=0$ and $τri=0$. It can be seen that the magnitude of the transfer functions from the input T0 to Tn is constant at low frequencies, although there is a temperature change in successive web spans. A comparison of the elastic and viscoelastic cases indicates that the low frequency behavior is different for the parameters considered. The developed governing equations will assist in such disturbance propagation studies with various viscoelastic material parameters. We have also studied the effect of the upstream velocity disturbance V0 via the transfer function $Gvn(s)$ on downstream span tensions; similar disturbance propagation magnitudes were observed for both viscoelastic and elastic cases.

## Conclusions

In this paper, we have derived a governing equation for web span tension by considering heating/cooling of moving webs and viscoelastic behavior for the web material. A simple heat transfer model that provides a closed-form equation for temperature distribution and a first-order viscoelastic material model are employed in the derivation of the web span tension governing equation. Although model simulations and experimental results were provided for specific flooring materials that are processed and manufactured in the industrial web processing line, the models are applicable to other web materials if the viscoelastic parameters of those webs are determined by appropriate stress-relaxation and creep tests. The advantage of the developed governing equation for web tension lies in the development of advanced model-based tension control systems to improve transport behavior. Development of advanced tension control systems based on this tension equation will be considered as part of a future study.

## Funding Data

• U.S. National Science Foundation (Grant Nos. 0854612 and 1246854).

## Nomenclature

• A =

cross-sectional area of web

•
• b =

viscosity of the web

•
• bfi =

coefficient of friction of roller i

•
• c =

specific heat of the web material

•
• E =

modulus of elasticity (Young's modulus)

•
• E1, E2 =

springs moduli of the standard linear solid model

•
• h =

web thickness

•
• Ji =

inertia of roller i

•
• $Kϑ$ =

thermal conductivity constant

•
• Li =

length of span i

•
• m =

mass of the web

•
• ni =

gear ratio of the motor to roller i for driven rollers

•
• Ri =

•
• Si =

strain variation

•
• t =

time

•
• ti =

web tension in the span i

•
• Ti =

web tension variation in the span

•
• $Tw,i$ =

span time constant

•
• $Tϑ$ =

thermal time constant

•
• ui =

control input to roller i

•
• vi =

surface velocity of roller

•
• Vi =

velocity variation in the roller

•
• α =

coefficient of thermal expansion

•
• $γ*$ =

coefficient of heat transmission

•
• δ =

exponential function of the ratio of time constants

•
• εi =

strain in span i

•
• $ε˙i$ =

strain rate in span i

•
• θ =

temperature, function of t only

•
• ϑ =

temperature, function of x and t

•
• $v¯$ =

average web velocity

•
• ρ =

web density

•
• σ =

applied stress

•
• $σ˙$ =

stress rate

•
• τc =

creep time

•
• τi =

time constant

•
• τr =

relaxation time

•
• $ϕ$ =

angle of wrap

•
• ω =

roller angular velocity

### Subscripts

Subscripts

• i =

span or roller number

•
• n =

index used in infinite sums

•
• R =

pertaining to the roller

•
• u =

unstretched state

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