## Abstract

The prediction and measurement of vibrations of the low-frequency transverse modes of tensioned webs are of increasing interest for process monitoring, quality control, and process stability in roll-to-roll flexible hybrid and stretchable electronics manufacturing, nanomanufacturing, coated layer patterning, and other continuous manufacturing technologies. Acting as distributed added mass, the surrounding air significantly affects the frequency responses of taut thin webs in ambient roll-to-roll processes in comparison with those in vacuo. In this paper, we present closed-form, semi-analytical, universal hydrodynamic functions used to accurately predict the lowest symmetric and anti-symmetric transverse frequency response for any uniaxially tensioned web of arbitrary material and aspect ratio used in roll-to-roll processes. Experimental validation is carried out by using pointwise laser measurements of acoustically excited webs with different pre-tensions, web materials, and aspect ratios. These closed-form hydrodynamic functions provide roll-to-roll process designers a convenient way to predict the lowest frequencies of such web systems without the need to resort to computationally intensive methods; alternately, these functions allow for the quick identification of conditions when air-coupling is important to determine the web’s vibration response. The results presented herein are expected to help ongoing efforts to improve process monitoring and control in a variety of roll-to-roll continuous manufacturing technologies.

## 1 Introduction

Flexible hybrid electronics and stretchable electronics are rapidly growing technology sector with a global market expected to exceed \$75 billion the early 2020s as reported by the National Academies [1]. Because of the diverse applications of flexible electronics to photovoltaics, batteries, sensors for internet of things (IoT), human wearable devices, flexible lighting, and flexible displays [25], the growth of flexible electronics coincides with the digital transformation of many industry sectors.

Roll-to-roll (R2R) processes offer generic technology platforms for scaleup of flexible electronics manufacturing to reduce the cost and increase the volume throughput [69]. Figure 1 shows a flexible thin web cross a span between two rollers in a typical R2R manufacturing setup for flexible electronics manufacturing. There is increasing interest in the thermomechanics [8,9] and vibrations of tensioned webs in R2R processes for process monitoring, quality control, and process stability [1019].

Fig. 1
Fig. 1
Close modal

Web vibration can be detrimental to process control; thus, several groups have worked on web vibration control [12,14,15]. Web system designers are interested in ensuring that the web frequencies are tuned away from several external excitation sources in the R2R process. For example, in printing applications, the web vibration frequency will need to be tuned away from the firing frequency of printer heads. In addition, web frequencies need to be tuned away from roller-eccentricity-induced boundary excitation frequencies in processes.

On the other hand, intentional excitation and measurement of vibration modes may serve as a way for generating non-uniform distributed functional patterns (such as Chladni patterns) in a wet coated layer on flexible thin substrates [2023] or to measure web pre-tension instead of using load cells [16,17]. Intentional modal vibrations might also be used for rapid inline reliability monitoring of R2R products, including thin-film delamination and cracking, instead of traditional offline cyclic bending and torsion tests [6,24]. Characterization of web vibration may also enable new ways of strain distribution monitoring or damage detection in R2R flexible electronics manufacturing.

The vibrations of thin flexible webs in R2R processes in flexible electronics manufacturing are significantly affected by the surrounding air, which can reduce the frequencies and change the mode shapes in comparison with those in vacuo. This is primarily because the mass density of thin polymer or paper typically used as substrates in flexible electronics manufacturing is much smaller compared to that of the sheet metal. As such the surrounding air couples strongly with the transverse vibration of such taut webs over their large area and must be considered to accurately predict web frequencies, mode shapes, or web pre-tension. Pramila et al. [25,26] first investigated this using potential flow theory for the surrounding air and a uniaxially tensioned membrane model for the web. However, the linear eigenvalue problem for out-of-plane vibration of a uniaxially tensioned membrane is ill-posed, in that the spatial dependence of any eigenmode in a direction transverse to the tension cannot be defined. The work is followed by Chang and Moretti [27], Raman et al. [10], Vaughan and Raman [11], Hara and Watanabe [28,29], Yao and Zhang [30] who used a uniaxially tensioned Kirchhoff plate model and investigated free edge flutter and web stability caused by crossflow. Most of these works assume the air is incompressible, inviscid, and irrotational. Kulachenko et al. [31] used the Helmholtz equation since they assume the surrounding air is compressible. Noting that potential flow theory is not suitable for edge wake phenomena when airflow occurs transverse to the tensioned direction, Bidkar et al. [32] used a vortex-lattice method for inviscid and incompressible flows. However, all these previous works require significant computational work to determine the effect of surrounding air on the web vibration. As such these computational approaches are not easily accessible to web system designers and controllers.

One way to make aerodynamic/hydrodynamic coupling easily accessible is by generating easy-to-use semi-analytical expressions for the coupling based on detailed computations. The trick here is to make the hydrodynamic functions as “Universal” as possible, so that they represent a wide range of operating conditions, properties, and geometries relevant to the application. One such example of accessible and thus highly utilized hydrodynamic functions are Sader’s functions [33,34] that can predict the added mass and viscous damping of multiple modes of microcantilevers immersed in fluids for use in the Atomic Force Microscope. Sader’s hydrodynamic functions are based on two-dimensional computational solutions of linearized, unsteady, incompressible Navier–Stokes equations and are valid over a wide range of unsteady Reynold’s numbers, microcatilever geometries, and surrounding fluid properties.

In this paper, we present closed-form, semi-analytical hydrodynamic functions for the accurate prediction of air-coupled vibration response of uniaxially tensioned webs valid over two orders of magnitude of range of web aspect ratio and for arbitrary web material mass density and tension. The taut webs are considered stationary because many flexible electronics manufacturing operations work at low speeds (less than 3.5 m/min [3,6,9]). The primary interest is on the added mass effect of air-coupling. Thus, the hydrodynamic functions presented here are based on three-dimensional computations of webs coupled with the incompressible, inviscid, irrotational surrounding fluid. Specifically, we present hydrodynamic functions for the two lowest frequency eigenmodes of the web. The hydrodynamic functions for other eigenmodes can be obtained following our method shown in Sec. 2.4. The hydrodynamic functions are validated in experiments using acoustically excited and laser-measured web vibrations with three different types of web materials with various in-span length-to-width ratios and pre-tensions. The hydrodynamic functions for air-coupled web vibrations derived in this article will allow R2R web system designers and R2R system dynamics and control engineers to estimate the transverse web frequencies using simple formulas rather than using expensive computational models. The only data needed to determine the air-added effects for lowest eigenmodes are the uniaxial tension, length, aspect ratio, and mass areal density of any rectangular web, as well as the density of surrounding air (or other fluid).

## 2 Mathematical Models of Air-Coupled Web Systems

### 2.1 Field Equations for the Coupled System and Their Discretization.

Inspired by prior works on vibration of metal plates submerged in water with applications to ship research [3537], in this section we develop a solution method to couple a two-dimensional (2D) tensioned web model with a three-dimensional (3D) potential flow model for surrounding air. The ratio of air mass density to web material mass density is an important indicator of the strength of the air-coupling effect on web vibrations and is much greater for typical web materials (plastics, papers) used in R2R manufacturing than for metal webs. As a result, the effect of surrounding air on the vibrations of flexible taut webs in R2R processes is expected to be significant in a similar manner as the effect of surrounding water on the vibration of submerged metal plates.

We assume the rectangular web cross a span between two rollers in R2R with 2D isotropic, linearly elastic, uniaxially tensioned Kirchhoff plate. A linear membrane model for a uniaxially tensioned web leads to non-unique eigenfunction dependence along the direction transverse to the tension. Hence, we choose to include very small but finite bending stiffness of a uniaxially tensioned plate in the model for the vibrating web. In-plane oscillations and out-of-plane oscillations are uncoupled in this flat tensioned plate model. The equation of motion for out-of-plane vibrations of an axially stationary web can be derived from Hamilton’s principle as [10,11,38]
$ρareaw¨(x1,x3,τ)+D∇4w(x1,x3,τ)−N11w,11(x1,x3,τ)=p$
(1)
where x1, x2, and x3 are the coordinates along longitudinal (in the plane of the web along the direction of applied tension), transverse (normal to web surface), and lateral directions (in the plane of the web), respectively, as shown in Fig. 1; τ is the time, w(x1, x3, τ) is the web deflection transverse to the plane of the web; ρarea is the web areal mass density; D = Eh3/[12(1 − υ2)] is the web bending stiffness, with E, h, and υ denoting Young’s modulus, thickness, and Poisson’s ratio of the web, respectively; $∇4$ is the biharmonic operator; N11 is the web’s uniaxial pre-tension per unit width; p is the aerodynamic pressure differential between the bottom surface and upper surface of the vibrating web.
We choose a 3D potential flow model for the surrounding air assuming the surrounding air is initially quiescent (no bulk velocity or transport) and is incompressible, inviscid, and irrotational. This is because compared to broad R2R applications, the R2R processes for flexible electronics are typical with low speed (less than 3.5 m/min [3,6,9]) with quiet manufacturing environments and thus near-quiescent conditions. In addition, the flow can be considered incompressible since the transverse vibration frequencies are very slow. Since the focus of the work is on the added mass effect of the air, we assume that the surrounding fluid is inviscid. The assumption of irrotational flow follows from the fact that we are not considering steady air flows over free edges which would lead to vortices [32]. Then, the aerodynamics of surrounding air is coupled to the web vibration through the surface pressure differential as [10,11,25]
$p=2ρairϕ˙(x1,0+,x3,τ)$
(2)
where the factor “2” comes from the anti-symmetric aerodynamic pressure on the two surfaces of the web; ρair is the density of surrounding air; ϕ(x1, 0+, x3, τ) is the velocity potential of air on the web; and $ϕ˙(x1,0+,x3,τ)=∂ϕ(x1,0+,x3,τ)∂τ$. Substituting Eq. (1) into Eq. (2), we obtain the governing equation of the vibrating web coupled to surrounding air as
$ρareaw¨(x1,x3,τ)+D∇4w(x1,x3,τ)−N11w,11(x1,x3,τ)=2ρairϕ˙(x1,0+,x3,τ)$
(3)
The continuity equation of the surrounding potential flow can be written as a Laplace’s equation
$∇2ϕ(x1,x2,x3,τ)=∂2ϕ(x1,x2,x3,τ)∂x12+∂2ϕ(x1,x2,x3,τ)∂x22+∂2ϕ(x1,x2,x3,τ)∂x32=0$
(4)
where $∇2$ is the Laplace operator and ϕ(x1, x2, x3, τ) is the velocity potential of air. The gradient of the velocity potential gives component of velocity of air along its direction. Thus, the velocity of air normal to the web equals the normal vibrating velocity of the web, i.e.
$−ϕ,2(x1,0+,x3,τ)=w˙(x1,x3,τ)$
(5)
Next, we nondimensionalize Eqs. (3)(5) as [10,11]
$x1′=x1L,x2′=x2L,w′=wL,x3′=x3Lε=DL2N11,Λ=Lρairρarea,τ′=τLN11ρarea,ϕ′=ϕLρareaN11$
(6)
where L is the in-span length of the web in the longitudinal direction, ɛ is the bending stiffness-to-tension ratio, and Λ is some measure of air density to areal mass density of web. The primes denote the nondimensional quantities. ɛ is normally very small, ranging from 10−6 to 10−4, but non-negligible for taut webs used in flexible electronics applications. It distinguishes the spatial dependence of eigenmodes in the lateral direction. Using Eq. (6), we can rewrite Eqs. (3)(5) in nondimensional form as
$w¨′(x1′,x3′,τ′)+ε∇4w′(x1′,x3′,τ′)−w,11′(x1′,x3′,τ′)=2Λϕ˙′(x1′,0+,x3′,τ′)$
(7)
$∇2ϕ′(x1′,x2′,x3′,τ′)=0$
(8)
$−ϕ,2′(x1′,0+,x3′,τ′)=w˙′(x1′,x3′,τ′)$
(9)

The associated boundary conditions are as follows:

1. Turnbull et al. [39] proved that simply supported boundary conditions are adequately accurate assumptions to predict linear vibrations of pre-tensioned webs across finite radius rollers
$w′(0,x3′,τ′)=w′(1,x3′,τ′)=0,w,11′(0,x3′,τ′)=w,11′(1,x3′,τ′)=0$
(10)
2. There is no shear force or bending moment for the free edges
$w,33′(x1′,±1/(2κ),τ′)+υw,11′(x1′,±1/(2κ),τ)=0,w,333′(x1′,±1/(2κ),τ′)+(2−υ)w,311′(x1′,±1/(2κ),τ′)=0$
(11)
3. The fluid is allowed to exchange between the upper and lower half space; i.e., the web is unbaffled
$ϕ′(x1′,0,x3′,τ′)=0,outsidetheareaofweb$
(12)
4. The fluid is stationary in the far field
$limx1′2+x2′2+x3′2→∞ϕ,normal′(x1′,x2′,x3′,τ′)=0$
(13)

To solve the coupled eigenvalue problem underlying the free vibrations of the air-coupled web (i.e., Eqs. (7)(9)), we separate the time and space variables for both the web deflection and the velocity potential of air as
$w′(x1′,x3′,τ′)=∑m=1∞∑n=1∞qmn′(τ′)Wmn′(x1′,x3′)$
(14)
$ϕ′(x1′,x2′,x3′,τ′)=∑m=1∞∑n=1∞Amn2′(τ′)φmn2′(x1′,x2′,x3′)$
(15)
where $Wmn′(x1′,x3′)$ represents a nondimensional admissible set of basis functions for web vibration, $qmn′(τ′)$ is the generalized coordinate. Basis functions with m and n as odd numbers are symmetric with respect to $x1′=0.5$ and $x3′=0$, respectively; while those with even-numbered m and n values are anti-symmetric with respect to $x1′=0.5$ and $x3′=0$, respectively. $φmn2′$ is the three dimensional aerodynamic function corresponding to web vibration in the corresponding basis function $Wmn′(x1′,x3′)$. The 2’s in the subscripts of $Amn2′$ and $φmn2′$ signify that the components are caused by vibration in the direction normal to the plane of the web. There is no initial crossflow along the longitudinal or lateral direction, so no additional terms due to flow transport appear in the equations [11]. Substituting Eqs. (14) and (15) into Eq. (9) gives
$Amn2′(τ′)=−q˙mn′(τ′)$
(16)
$φmn2,2′(x1′,0+,x3′)=Wmn′(x1′,x3′)$
(17)
We normalize the web vibration basis functions $Wmn′(x1′,x3′)$ as
$∫−1/(2κ)1/(2κ)∫01Wmn′2(x1′,x3′)dx1′dx3′=1$
(18)
where κL/b is the in-span length-to-width ratio (i.e., aspect ratio) of web, b is the width of web in the lateral direction. Combining Eq. (7) and Eqs. (14)(18) and applying assumed modes method (AMM) with inner products with $Wmn′$ yields
$(I+2ΛMair′)q′¨+K′q′=0$
(19)
where
$(K′)ij;mn=∫−1/(2κ)1/(2κ)∫01ε[Wij,11′(x1′,x3′)Wmn,11′(x1′,x3′)+Wij,33′(x1′,x3′)Wmn,33′(x1′,x3′)+2(1−υ)Wij,13′(x1′,x3′)Wmn,13′(x1′,x3′)+υWij,11′(x1′,x3′)Wmn,33′(x1′,x3′)+υWij,33′(x1′,x3′)Wmn,11′(x1′,x3′)]+Wij,1′(x1′,x3′)Wmn,1′(x1′,x3′)dx1′dx3′$
(20)
$(Mair′)ij;mn=∫−1/(2κ)1/(2κ)∫01Wij′(x1′,x3′)φmn2′(x1′,0+,x3′)dx1′dx3′$
(21)

$q′$ is the vector of the generalized coordinates $qmn′(τ′)$. The stiffness matrix $K′$ is symmetric and diagonal. The added air mass matrix $Mair′$ is symmetric but with off-diagonal terms, since the inner products of any two symmetric $Wmn′(x1′,x3′)$ and $φmn2′(x1′,0+,x3′)$ or any two anti-symmetric $Wmn′(x1′,x3′)$ and $φmn2′(x1′,0+,x3′)$ are not necessarily zero. $M′=I+2ΛMair′$ gives the new mass matrix for the coupled system.

### 2.2 Exact Solutions to the In Vacuo Eigenvalue Problem.

In the absence of air-coupling (Λ = 0), exact solutions to the eigenvalue problem can be found conveniently. The in vacuo eigenmode $Wmn′$ that satisfies the simply supported boundary conditions in Eq. (10) can be shown to be
$Wmn′(x1′,x3′)=sin(mπx1′)Ymn′(x3′)$
(22)
where
$Ymn′(x3′)=C1′coshγ1mn′x3′+C2′sinhγ1mn′x3′+C3′cosγ2mn′x3′+C4′sinγ2mn′x3′$
(23)
$γ1mn′=ωmn′2−m2π2ε+m2π2,γ2mn′=ωmn′2−m2π2ε−m2π2$
(24)
Additionally, the boundary conditions in Eq. (11) can be simplified as
$[B1′coshγ1mn′2κ−B2′cosγ2mn′2κB3′sinhγ1mn′2κB4′sinγ2mn′2κ]{C1′C3′}=0$
(25)
or
$[B1′sinhγ1mn′2κ−B2′sinγ2mn′2κB3′coshγ1mn′2κ−B4′cosγ2mn′2κ]{C2′C4′}=0$
(26)
where $B1′=γ1mn′2−υm2π2,B2′=γ2mn′2+υm2π2,B3′=γ1mn′3−2m2π2γ1mn′$$+υm2π2γ1mn′,$$B4′=γ2mn′3+2m2π2γ2mn′−υm2π2γ2mn′.$ The in vacuo mode frequency $ωmn′$ can be solved by letting
$|B1′coshγ1mn′2κ−B2′cosγ2mn′2κB3′sinhγ1mn′2κB4′sinγ2mn′2κ|=0,or|B1′sinhγ1mn′2κ−B2′sinγ2mn′2κB3′coshγ1mn′2κ−B4′cosγ2mn′2κ|=0$
(27)

We obtain $γ1mn′$ and $γ2mn′$ by substituting $ωmn′$ into Eq. (24) and solve for $C1′,C2′,C3′,andC4′$ by Eqs. (25) and (26) with the normalization of $Wmn′$ as in Eq. (18).

The lowest symmetric and anti-symmetric in vacuo eigenmodes corresponding to m = 1, n = 1 and m = 1, n = 2 respectively can be seen in the inset of Fig. 2. They can be understood as the lowest bending and torsion dominated modes of the web, respectively. The air-coupling significantly changes both the natural frequencies and the shapes of these eigenfunctions as will be discussed in subsequent sections.

Fig. 2
Fig. 2
Close modal

The in vacuo mode frequencies are clustered tightly about the equivalent frequencies of a tensioned string and are not easy to distinguish by varying either the bending stiffness-to-tension ratio ɛ (usually very small but finite, see in Ref. [11]) or web in-span length-to-width ratio κ (as shown in Fig. 2). As shown in Fig. 2, all mode frequencies with the same m cluster close together when κ is small. The mode frequencies separate when κ increases; however, the lowest symmetric and anti-symmetric mode frequencies (i.e., n = 1, 2) with the same m remain tightly clustered.

### 2.3 Discretized System Analysis of Coupled Eigenvalue Problem.

In the presence of air-coupling (Λ ≠ 0), analytical solutions to the eigenvalue problem are not available. Rather, the discretized system (Eq. (19)) needs to be solved. We choose an admissible function basis (normalized as Eq. (18)) as follows:
$Wmn′(x1′,x3′)=sin(mπx1′)Ymn′(x3′),Ymn′(x3′)=Cmn′(x3′)n−1,Cm1′=2κ,Cm2′=24κ3,Cm3′=160κ5,Cm4′=896κ7,Cm5′=4608κ9,Cm6′=22528κ11,Cm7′=106496κ13,Cm8′=491520κ15$
(28)

While the exact in vacuo eigenfunctions can also be used as a comparison function basis for this discretization, the exact in vacuo eigenfunctions depend on the web material and aspect ratio. As our goal is to develop “Universal” hydrodynamic functions, the computational effort required to calculate the discretization basis for every web material and aspect ratio considered is substantial. Instead, the use of a polynomial basis in the x3 direction allows the same basis function to be used for all the web aspect ratios and materials used, thus substantially reducing the computational effort with little effect on the accuracy of prediction.

To solve for the corresponding aerodynamic potentials $φmn2′$, we apply abaqus [40], a 3D finite element solver, to Eqs. (8) and (15) with boundary conditions (12), (13), and (17). Figure 3 shows the admissible function basis $Wmn′(x1′,x3′),$ the corresponding 2D on-web air velocity potentials $φmn2′(x1′,0+,x3′),$ and 3D air velocity potentials $φmn2′(x1′,x2′,x3′)$ with cross-sectional views. The infinite fluid boundary is truncated to an inner domain with higher mesh density and an outer domain with lower mesh density to reduce the computational cost. The inner domain is a uniform cube, and the outer domain is the volume between the inner domain and a larger cube containing the inner domain.

Fig. 3
Fig. 3
Close modal

A convergence study determines the mesh density and size of each domain. We take κ = 1 for the study and use basis functions $W11′,W12′,W13′,W14′$ to check all diagonal entries in $Mair′$, as discussed in Sec. 2.1. We first fix the inner and outer domain cubic side lengths to be one time and two times of the largest dimension of the web, respectively. Then, the mesh density convergence is determined when the diagonal entries in $Mair′$ change by less than 1% in progressive computations where the mesh density is doubled in each computation. With mesh density converged, we increase the cubic domain side length by twice the largest dimension of the web in progressive computations until the diagonal entries in $Mair′$ change by less than 1%. Based on the convergence study, we use 512,000 elements with 531,441 nodes in the inner domain cube of side length equaling twice the largest dimension of the web, and we use 3250 elements with 4016 nodes in the outer domain cube whose extent ranges from two times to six times of the largest dimension of the web. After obtaining $Wmn′$ and $φmn2′$, $(K′)ij;mn$ and $(Mair′)ij;mn$ can be solved by numerical integration of Eqs. (20) and (21). Substituting $(K′)ij;mn$ and $(Mair′)ij;mn$ into Eq. (19), we will obtain the air-coupled natural frequencies for the webs.

Using this approach, we compare in Table 1 the two lowest frequencies of an air-coupled web as predicted by (1) using a comparison function basis, the exact in vacuo eigenfunctions and (2) using the polynomial admissible function basis in Eq. (28) for a web with parameters chosen as ɛ = 1.6 × 10−5, Λ = 5.7, κ = 2.5. We increase the sizes of $M′$ and $K′$ matrices from 1 × 1 to 4 × 4. The polynomial admissible function basis approximates well both two lowest frequencies of air-coupled web with less than 6% error compared to when the in vacuo eigenfunctions are used as a basis.

Table 1

The two lowest frequencies solved by AMM with exact in vacuo eigenfunctions and estimated admissible polynomial function basis (Eq. (28)), ɛ = 1.6 × 10−5, Λ = 5.7, κ = 2.5

$ω11′$$ω12′$
Sizes of $M′$ and $K′$In vacuo eigenfunctionsAdmissible function basisIn vacuo eigenfunctionsAdmissible function basis
In vacuo3.141823.141843.144453.14445
1 × 12.004852.000182.498052.49914
2 × 21.830171.938062.429582.45271
3 × 31.825561.936912.429042.42984
4 × 41.824261.936902.429012.42902
$ω11′$$ω12′$
Sizes of $M′$ and $K′$In vacuo eigenfunctionsAdmissible function basisIn vacuo eigenfunctionsAdmissible function basis
In vacuo3.141823.141843.144453.14445
1 × 12.004852.000182.498052.49914
2 × 21.830171.938062.429582.45271
3 × 31.825561.936912.429042.42984
4 × 41.824261.936902.429012.42902

The computational approach above, which has also been described in prior works (see in Ref. [11]), has some disadvantages: (a) it is computationally expensive, (b) it requires a new computation for every aspect ratio and web material properties, (c) it requires expertise in fluid–structure interaction computations, and (d) it has not been validated experimentally. In what follows we present an approach to overcome these disadvantages using hydrodynamic functions and validate it in experiments.

### 2.4 Hydrodynamic Function.

Here, we develop closed-form, semi-analytical, hydrodynamic functions for the first symmetric and anti-symmetric basis functions of the pre-tensioned web as a way to estimate the air-coupling on the lowest two eigenmodes of tensioned webs. The method outlined can be easily extended to compute hydrodynamic functions for other basis functions.

With the basis in place, we solve the air velocity potentials with different aspect ratios. So that we can use the same integration domain for different κ's, we rescale the coordinates $x1*=x1′,x3*=κx3′;$ all the properties with κ = 1 are denoted with *. The hydrodynamic functions then are multiplicative functions that depend on κ's that allow the added air mass entries computed for any aspect ratio to be expressed in terms of the “reference” added air mass entries when κ = 1. Thus, we have
$(Mair)ij;mn=L(Mair′)ij;mn=LFijmn2(κ)(Mair*)ij;mn,Fijmn2(1)=1$
(29)
Fijmn2(κ)'s are functions relating the magnitudes of $(Mair′)ij;mn$ with respect to web aspect ratio for different basis functions $Wij′$ and different air velocity potentials on the surface of web $φmn2′(x1′,0+,x3′)$. Specifically, we use the basis functions (m, 1) and (m, 2) in Eq. (28) to fit F11112(κ) and F12122(κ), the hydrodynamic functions for the lowest symmetric and anti-symmetric basis functions. They are fitted to computed values with κ from 0.1 to 10 using a functional form in terms of a polynomial in terms of log10(κ) along the lines of Sader’s hydrodynamic functions [33,34], since it will balance the fitting range for both κ > 1 and κ < 1. The fitted hydrodynamic functions are determined to be
$F11112(κ)=1−1.17557[log10(κ)]−0.10578[log10(κ)]2+0.54019[log10(κ)]3+0.16507[log10(κ)]4−0.25226[log10(κ)]5−0.00822[log10(κ)]6;F12122(κ)=1−1.91774[log10(κ)]+1.16202[log10(κ)]2+0.46179[log10(κ)]3−0.82329[log10(κ)]4−0.19040[log10(κ)]5+0.38791[log10(κ)]6$
(30)

Figure 4 shows a comparison of nondimensional added mass from finite element method and fitted functions with admissible function basis $W11′=2κsin(πx1′),$ and $W12′=24κ3sin(πx1′)x3′$. As can be seen, in the fitting range, the maximum absolute related errors between the semi-analytical hydrodynamic functions (i.e., Eq. (30)) and the finite element computed solutions are 0.197% and 0.644% for ij = mn = 11 and ij = mn = 12, respectively.

Fig. 4
Fig. 4
Close modal
With the 1 term approximation the lowest symmetric and anti-symmetric frequencies of an air-coupled web can be determined using the hydrodynamic function as follows:
$(fair)11=(ωair)112π=12DL4π2+N11L2ρarea+0.51048ρairLF11112(κ)(Hz)(fair)12=(ωair)122π=12DL4[π2+24(1−υ)κ2]+N11L2ρarea+0.26150ρairLF12122(κ)(Hz)$
(31)
Recall since ɛ = D/L2N11 is very small for industrial flexible electronics applications, typically from 10−6 to 10−4, we can simplify Eq. (31) as
$(fair)11≈(ωair)112π≈12LN11ρarea+0.51048ρairLF11112(κ)(Hz)(fair)12≈(ωair)122π≈12LN11ρarea+0.26150ρairLF12122(κ)(Hz)$
(32)

Thus, R2R web designers and R2R system dynamics and control researchers only need to substitute physical properties into Eq. (32) to find the mode frequencies. These hydrodynamic functions can be used for a broad range of in-span length-to-width ratios from 0.1 to 10.

The analysis above assumes no air-coupling between different basis functions in the mass matrix. However, in principle, different basis functions couple due to the off-diagonal terms in the added air mass matrix $Mair′$. Although there is no closed-form solution for the correct eigenmodes, they can be calculated computationally.

We use AMM to estimate the mode shapes for air-coupled webs. We return to Eq. (19) and rewrite as $M′q′¨+K′q′=0,$ and the corresponding eigenvectors of the matrix $(M′)−1K′$ determine the contribution of each basis function to the air-coupled eigenmodes. This is discussed later in more detail in the context of the experimental measurements in Sec. 4.

In what follows we study the use of these hydrodynamic functions in the analysis of web vibration characteristics. Figure 5 shows the ratios of air-coupled to in vacuo mode frequencies computed using Eq. (32) for the lowest symmetric and anti-symmetric modes for three materials as a function of in-span length-to-width ratio. All these webs are with fixed width = 215.9 mm and pre-tension N11 = 200 N/m, in the typical range of industrial usage [4143], with related web areal mass densities as ρarea = 116.01 g/m2, 71.55 g/m2, and 42.58 g/m2, air density ρair = 1.225 kg/m3 [10]. From Fig. 5 we can see that air-coupling reduces the lowest natural frequencies from 20% to 60% compared to the in vacuo values in the range we investigated. Air-coupling effect on web natural frequencies is more pronounced for webs with large in-span length-to-width ratio and for webs with lower areal mass density. The latter observation is in line with Eq. (7) which suggests that air-coupling is modulated by the magnitude of Λ = air/ρarea, which implies that lighter webs (low areal mass density) are more sensitive to air-coupling than heavier webs. Air-coupling also separates the clustered frequencies since the mode frequency of lowest symmetric mode reduces more than that of lowest anti-symmetric mode.

Fig. 5
Fig. 5
Close modal
Table 2

Properties of the webs

PropertyDuPont Nomex 410 paperPolyimide filmUnit
Young’s modulusa2.752.752.50GPa
Poisson’s ratioa0.300.300.34
Web areal densityb42.58 ± 0.653$116.01±2.024$$71.55±0.676$$g/m2$
Thickness50.8127.050.8μm
Width215.9215.9215.9mm
In-span length-to-width ratios1.5:1, 2:1, 2.5:1, and 3:1
PropertyDuPont Nomex 410 paperPolyimide filmUnit
Young’s modulusa2.752.752.50GPa
Poisson’s ratioa0.300.300.34
Web areal densityb42.58 ± 0.653$116.01±2.024$$71.55±0.676$$g/m2$
Thickness50.8127.050.8μm
Width215.9215.9215.9mm
In-span length-to-width ratios1.5:1, 2:1, 2.5:1, and 3:1
a

Given in Refs. [44,45].

b

Measured by averaging the weights of six area-known rectangular films. Means and standard deviation values are listed.

## 3 Experimental Setup

To experimentally validate the hydrodynamic functions, we use acoustically excited and laser-measured web vibrations with three different types of web materials with various in-span length-to-width ratios and pre-tensions. We conduct tests on rectangular webs with geometric dimensions and material properties given in Table 2. Each web is wrapped onto four live rollers (Hex-axls 1.9″ diameter conveyor rollers with bearing-in, McMaster-Carr) which are mounted on an aluminum frame via brackets (Conveyor roller mounting brackets for hex axle, McMaster-Carr) as shown in Fig. 6(a). The aluminum frame is built with 25.4 mm × 25.4 mm framing rails (T-Slotted Framing Rails, McMaster-Carr) and fastened by associated fasteners and brackets (McMaster-Carr). Two rollers in intermediate height are used for adjusting κ. We load in-span tension to the web by hanging a weight-known dumbbell on one end of web wrapping over the lowest roller. A small portion of web is wrapped on the dumbbell bar and taped uniformly afterward. The other end of the web is wound onto the small upper roller which has a pin hole for rotation lockage. Pre-tensions in our experiments are 111.64 N/m and 163.33 N/m. These values are industrially relevant [4143]. Due to the frictional interactions with the rollers, low pre-tension cannot stretch the web evenly and high pre-tension can lead to wrinkles. We mount a speaker (5 Watts, Creative A220, Creative Technology Ltd.) on a boom stand which is separated from the aluminum frame for vibration isolation and place it close to one free edge of the web as Fig. 6 shows. We measure frequency response function (FRF) of the speaker before we use it as the excitation source. Swept sine waves are sent to the speaker from Dynamics Signal Analyzer (DSA) (HP 35670A, The Keysight Technologies, Inc., Santa Rosa, CA) with an increment of 0.1 Hz. The speaker cone displacement is sensed by a laser-based triangulation measurement system (Microtrak 7000, MTI Instruments) whose sensor head is fixed to a tripod for non-contact measurement. The cone displacement-to-the input voltage ratio is found fairly constant in the frequency range up to 100 Hz and phase response drops from 0 deg smoothly. We measure FRFs of 15 points for κ = 1.5, and κ = 2, and 25 points for κ = 2.5, and κ = 3 where these points are equally distributed in the x1x3 plane as shown in Fig. 6(b). We choose the swept upper cutoff frequency to be slightly higher than the estimated frequencies (fair)12 and the lower cutoff frequency to be lower than the first resonant frequency for time-saving. We average the frequencies from FRFs of these points to attain web measured resonant frequencies. The associated mode shapes are approximated subsequently by fitting the amplitudes of these measured points at the resonant frequencies. We fit the measured mode shapes to obtain smooth surfaces as
$W1n(measured)(x1,x3)=sin(πx1/L)(a1+a2x3+a3x32+a4x33+a5x34)$
(33)
where $a1,a2,a3,a4,anda5$ are multiplicative constants.
Fig. 6
Fig. 6
Close modal

## 4 Results and Discussions

We experimentally validate the hydrodynamic functions through their effects on both the natural frequencies and mode shapes. As an example, Fig. 7 shows one FRF gain and phase plots of a $127μm$ DuPont Nomex 410 Paper with κ = 2.5, and pre-tension $N11=111.64N/m$ excited acoustically by the author described earlier. Two amplitude peaks are distinct at resonant frequencies, $17.6Hz$ and $22.4Hz$, and nearly 180-deg phase across the peaks. The FRF gain and phase plots of other scenarios are very similar and not presented here; rather, the measured resonance frequencies from those measurements are recorded.

Fig. 7
Fig. 7
Close modal
Tables 35 present comparisons between the estimated frequencies by Eq. (32) and the measured resonant frequencies of the 50.8 μm DuPont Nomex 410 Paper, the 127 μm DuPont Nomex 410 Paper, and the 50.8 μm DuPont polyimide film under two pre-tensions and four in-span length-to-width ratios, respectively. The hydrodynamic functions allow the prediction of the first two web vibration frequencies to within 10% across all the measurements. Factors contributing to the discrepancies can be categorized into two aspects: web material anisotropy, non-uniform distribution of pre-tension, initial out-of-flatness of web, that are all effects not included in the mathematical model; and measurement, fitting, and parameter value uncertainties. For example, a 1% variation of air density, a one-σ variance of web areal mass density in Table 2, and a 2% fitting error in Fmn2(κ) can cause up to 1.2% difference calculated by error propagation function
$Δ(fair)mn=(∂(fair)mn∂ρareaΔρarea)2+(∂(fair)mn∂ρairΔρair)2+(∂(fair)mn∂Fmn2(κ)ΔFmn2(κ))2(Hz)$
(34)
where Δ denotes the variation of a variable. The assumption of uniform pre-tension is also often hart to ensure in practice. Misalignment of rollers, friction characteristics between web and roller, anisotropic and inhomogeneous web material, residual stress in the web, and non-uniformity in applied pre-tension often produce in-plane strain energy variations crossing the web span.
Table 3

The estimated and measured resonant frequencies for 50.8 μm DuPont Nomex 410 paper, ρarea = 42.58 g/m2, b = 215.9 mm

(fair)11 (Hz)(fair)12 (Hz)
κEstimationExperimentDiscrepancyEstimationExperimentDiscrepancy
Pre-tension N11 = 111.64 N/m
1.5/136.2033.16 ± 0.038.39%48.0645.59 ± 0.125.14%
2/126.1723.88 ± 0.038.75%35.8633.97 ± 0.075.27%
2.5/120.4718.88 ± 0.047.77%28.6527.77 ± 0.073.07%
3/116.8115.59 ± 0.037.26%23.8822.83 ± 0.074.40%
Pre-tension N11 = 163.33 N/m
1.5/143.7839.77 ± 0.029.16%58.1354.75 ± 0.035.81%
2/131.6529.29 ± 0.027.46%43.3839.33 ± 0.149.34%
2.5/124.7622.42 ± 0.059.45%34.6531.54 ± 0.108.98%
3/120.3418.95 ± 0.026.83%28.8926.26 ± 0.319.10%
(fair)11 (Hz)(fair)12 (Hz)
κEstimationExperimentDiscrepancyEstimationExperimentDiscrepancy
Pre-tension N11 = 111.64 N/m
1.5/136.2033.16 ± 0.038.39%48.0645.59 ± 0.125.14%
2/126.1723.88 ± 0.038.75%35.8633.97 ± 0.075.27%
2.5/120.4718.88 ± 0.047.77%28.6527.77 ± 0.073.07%
3/116.8115.59 ± 0.037.26%23.8822.83 ± 0.074.40%
Pre-tension N11 = 163.33 N/m
1.5/143.7839.77 ± 0.029.16%58.1354.75 ± 0.035.81%
2/131.6529.29 ± 0.027.46%43.3839.33 ± 0.149.34%
2.5/124.7622.42 ± 0.059.45%34.6531.54 ± 0.108.98%
3/120.3418.95 ± 0.026.83%28.8926.26 ± 0.319.10%
Table 4

The estimated and measured resonant frequencies for 50.8 μm DuPont Polyimide film, ρarea = 71.55 g/m2, b = 215.9 mm

(fair)11 (Hz)(fair)12 (Hz)
κEstimationExperimentDiscrepancyEstimationExperimentDiscrepancy
Pre-tension N11 = 111.64 N/m
1.5/133.8632.47 ± 0.034.11%42.9641.28 ± 0.043.91%
2/124.5924.06 ± 0.032.16%32.0929.88 ± 0.066.89%
2.5/119.2918.84 ± 0.032.33%25.6423.79 ± 0.047.22%
3/115.8715.83 ± 0.060.25%21.3819.58 ± 0.048.40%
Pre-tension N11 = 163.33 N/m
1.5/140.9638.23 ± 0.076.67%51.9749.02 ± 0.115.67%
2/129.7429.35 ± 0.331.31%38.8137.56 ± 0.073.23%
2.5/123.3321.59 ± 0.067.46%31.0130.08 ± 0.113.01%
3/119.1918.30 ± 0.034.64%25.8624.35 ± 0.065.82%
(fair)11 (Hz)(fair)12 (Hz)
κEstimationExperimentDiscrepancyEstimationExperimentDiscrepancy
Pre-tension N11 = 111.64 N/m
1.5/133.8632.47 ± 0.034.11%42.9641.28 ± 0.043.91%
2/124.5924.06 ± 0.032.16%32.0929.88 ± 0.066.89%
2.5/119.2918.84 ± 0.032.33%25.6423.79 ± 0.047.22%
3/115.8715.83 ± 0.060.25%21.3819.58 ± 0.048.40%
Pre-tension N11 = 163.33 N/m
1.5/140.9638.23 ± 0.076.67%51.9749.02 ± 0.115.67%
2/129.7429.35 ± 0.331.31%38.8137.56 ± 0.073.23%
2.5/123.3321.59 ± 0.067.46%31.0130.08 ± 0.113.01%
3/119.1918.30 ± 0.034.64%25.8624.35 ± 0.065.82%
Table 5

The estimated and measured resonant frequencies for 127 μm DuPont Nomex 410 paper, ρarea = 116.01 g/m2, b = 215.9 mm

(fair)11 (Hz)(fair)12 (Hz)
κEstimationExperimentDiscrepancyEstimationExperimentDiscrepancy
Pre-tension N11 = 111.64 N/m
1.5/131.0229.48 ± 0.174.96%37.5637.16 ± 0.221.06%
2/122.6421.65 ± 0.004.37%28.0827.69 ± 0.001.39%
2.5/117.8117.49 ± 0.061.80%22.4422.22 ± 0.070.98%
3/114.6814.69 ± 0.020.07%18.7118.55 ± 0.070.86%
Pre-tension N11 = 163.33 N/m
1.5/137.5235.15 ± 0.036.32%45.4342.54 ± 0.276.36%
2/127.3825.84 ± 0.035.62%33.9733.38 ± 0.041.74%
2.5/121.5419.80 ± 0.008.08%27.1526.12 ± 0.013.79%
3/117.7517.55 ± 0.051.13%22.6321.32 ± 0.375.79%
(fair)11 (Hz)(fair)12 (Hz)
κEstimationExperimentDiscrepancyEstimationExperimentDiscrepancy
Pre-tension N11 = 111.64 N/m
1.5/131.0229.48 ± 0.174.96%37.5637.16 ± 0.221.06%
2/122.6421.65 ± 0.004.37%28.0827.69 ± 0.001.39%
2.5/117.8117.49 ± 0.061.80%22.4422.22 ± 0.070.98%
3/114.6814.69 ± 0.020.07%18.7118.55 ± 0.070.86%
Pre-tension N11 = 163.33 N/m
1.5/137.5235.15 ± 0.036.32%45.4342.54 ± 0.276.36%
2/127.3825.84 ± 0.035.62%33.9733.38 ± 0.041.74%
2.5/121.5419.80 ± 0.008.08%27.1526.12 ± 0.013.79%
3/117.7517.55 ± 0.051.13%22.6321.32 ± 0.375.79%

As seen in the experimental results, the air loading significantly changes both the natural frequencies and the mode shapes of the measured webs. Figure 8 illustrates the lowest symmetric and anti-symmetric experimental mode shapes for the associated resonant frequencies in Fig. 7. Figures 8(a)8(d) show the exact mode shapes, with (a) and (b) for the in vacuo case. In-air-coupled eigenmodes are computed using the AMM described in Sec. 2.1. Table 6 shows a convergence study of the order required in AMM. The computed in-air eigenmodes are shown in Figs. 8(c) and 8(d). We plot the mode shapes with 93 × 38(3534) nodes.

Fig. 8
Fig. 8
Close modal
Table 6

L-2 norm of mode shapes from different AMM orders, 3534 points, $127μm$ DuPont Nomex 410 Paper with κ = 2.5, and pre-tension N11 = 111.64 N/m

AMM order nIn air mode shape 11 (approximated with symmetric in vacuo basis functions)In air mode shape 12 (approximated with anti-symmetric in vacuo basis functions)
n = 141.9624.83
n = 234.6932.23
n = 334.0031.56
AMM order nIn air mode shape 11 (approximated with symmetric in vacuo basis functions)In air mode shape 12 (approximated with anti-symmetric in vacuo basis functions)
n = 141.9624.83
n = 234.6932.23
n = 334.0031.56

The following are the key observations in comparing the computed and experimental in-air eigenmodes:

1. The experimental and predicted in-air eigenmodes clearly couple the basis functions. The corresponding in vacuo eigenmodes contribute the major characteristic to the in-air eigenmodes, but the effects of other coupled basis functions reduce the amplitude at the free edges. The off-diagonal terms in the added air mass matrix cause the coupling.

2. The AMM with three basis functions is able to closely predict the measured mode shapes. For symmetric case, the L-2 norms for the computed and experimental in air modes are 34.00 and 33.99 respectively; for the anti-symmetric case, the L-2 norms for the computed and experimental in air modes are 31.56 and 29.13, respectively.

3. The experimentally measured mode shapes are slightly asymmetric about the x3 = 0 axis. The asymmetry in the contour plots of mode shapes shown in Figs. 8(g) and 8(h) indicate non-uniform distribution of pre-tension exists in the experiments. Because of the misalignment of rollers, friction characteristics between web and roller, anisotropic and inhomogeneous web material, the distributed pre-tension is always slightly non-uniform. The mode shapes of other scenarios are similar to these in Fig. 8 in terms of web in-span deformation; thus, they are not repetitively provided in this paper.

## 5 Conclusions

Accurate prediction of transverse web vibrations in R2R manufacturing for flexible hybrid electronics and stretchable electronics can help improve process control and stability. We study in detail the discretized models of uniaxially tensioned Kirchoff plates surrounded by three-dimensional potential flow for different aspect ratios and web materials commonly used in flexible electronics manufacturing. We derive closed-form, semi-analytical hydrodynamic functions for the lowest two frequencies that are valid for arbitrary material and aspect ratio used in R2R processes. We experimentally validated both the predicted web frequencies from hydrodynamic functions and the corresponding mode shapes from potential flow theory using pointwise laser measurements of acoustically excited webs with different pre-tensions, web materials, and aspect ratios. The main results are as follows:

1. The hydrodynamic functions allow the prediction of the symmetric and anti-symmetric mode frequencies to within 10% across all the measurements. They provide R2R process designers a convenient way to predict the lowest frequencies of air-coupled web systems without need to resort to computationally intensive methods.

2. Based on computations, air-coupling reduces the lowest symmetric and anti-symmetric mode frequencies of webs from 20% to 60% compared to the in vacuo values in the range we investigated and separates the clustered frequencies.

3. Air-coupling changes the eigenmodes of webs by cross coupling various in vacuo modes. These are caused by the off-diagonal terms in the added air mass matrix.

4. The lowest symmetric and anti-symmetric mode frequencies using hydrodynamic functions are experimentally validated with three different materials (two DuPont Nomex 410 Papers with different thickness and one polyimide film), two different pre-tensions, and four different aspect ratios representing materials, tensions, and aspect ratios commonly found in R2R flexible electronics manufacturing.

5. The lowest symmetric and anti-symmetric mode shapes are experimentally validated. The differences between L-2 norms for the computed and experimental in air modes are 0.03% for symmetric case and 7.70% for the anti-symmetric case.

We expect that the semi-analytical solutions to air-coupled web vibrations will find use among web system designers who are interested in avoiding the detrimental effects of web vibration on process quality and stability in R2R flexible electronics lines and/or in innovations that might exploit web vibrations for value-added processes.

## Acknowledgment

This work was supported by the National Science Foundation under Grant No. CMMI-1344654: “Scalable Nanomanufacturing: Large scale manufacturing of low-cost functionalized carbon nanomaterials for energy storage and biosensor applications.”

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

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