A Study on Interactions of Thin Flexible Media with Grooved Rollers

Home , Bending

A STUDY ON INTERACTIONS OF THIN FLEXIBLE MEDIA WITH GROOVED ROLLERS

A Dissertation Presented

Tuğçe Kaşıkcı to

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Mechanical Engineering

Northeastern University Boston, Massachusetts

August 2016

ABSTRACT Tape to grooved roller interaction is modelled and analyzed by means of numerical and experimental methods. In general, air enters the tape-roller interface as the tape is transported between the reels. This phenomenon, known as air entrainment, causes air lubrication under the tape, and it could eventually lead to reduction of rigid body contact pressure. However, traction in the tape roller interface is critical for the tape to stay on a prescribed path during transport. Contact mechanics in wrapping a thin-shell (tape/web) around a grooved cylindrical surface (roller) under tension is investigated in the slow and operational tape transport speed limits.

In the slow tape transport speed limit, the effects of air lubrication in the tape-roller interface can be neglected. In this case the equations of equilibrium lead to analytical solutions, however; the problem is nonlinear due to the unknown nature of the contact area and location. It is shown that in case of no air entrainment, three distinct contact cases describe the interaction of the tape/web with respect to the lands. Non-dimensional analysis shows that contact state depends on the width of the groove and the land, and the non-dimensional belt-wrap pressure only modulates the amplitude of the deflected profile.

In the operational tape transport speed limit, the tape deflection, air and contact pressures are coupled to one another. The tape is modeled as a thin, translating, tensioned shell. The effects of air lubrication are modeled by using the Reynolds lubrication equation. The coupled problem, generally known as the foil bearing problem, is solved numerically by using a transient coupling algorithm. The traction characteristics of the tape-roller interface are investigated as functions of tape tension, tape transport speed and groove geometry. It was shown that air lubrication effects reduce the contact force; however the underlying effects of tape mechanics are not entirely eliminated. Unlike the classical foil bearing solution, bending of the tape into the grooves dominates the mechanics of the coupled air-lubrication/tape-deflection problem.

Tape clearance over the grooved rollers is measured experimentally by using a modified, commercially available tape transport system. A novel method to measure this clearance over rollers with helical grooves is described. Experiments show that tape-to-roller clearance increases with increasing tension and tape transport velocity. Analysis shows that the air entrainment is responsible for the velocity dependence, whereas the tension dependence is due to

II the tape bending into the grooves. Thus, it has been shown that the tape deflection in the lateral direction is critical for thin tapes, in order to accurately assess the contact over grooved rollers.

This work contributes to our understanding of traction mechanics of thin webs over grooved rollers, which has been understudied in the past, and helps in selecting design parameters for improved traction.

III

ACKNOWLEDGMENTS

Here I would like to pay my gratitude to all people who have touched my life with their unlimited support.

First and foremost, I would like to thank my adviser, Professor Sinan Müftü for his invaluable help, guidance and patience throughout this study. Without his valuable mentoring, ideas, this work would never be possible. His guidance gave me enthusiasm to work more diligently every day.

I would like to thank members of my committee, Professors Hamid Nayeb-Hashemi and Carol Livermore, for their insightful comments and valuable feedback and advice on my research.

I would like to dedicate this study in the memory of late Professor Yaman Yener for his unconditional support and mentorship. He was like a second ‘father’ to me, as he was to many other students. I miss him every day. In addition, I would like to thank Ms. Demet Yener, Ms. Zeynep Yener for their unconditional support and I would like to thank friends of the Yaman Yener Memorial Fund for their financial support during my studies.

I also would like to thank my lab members, and especially Dr. Hankang Yang, for his valuable comments, support, and friendship throughout these years. Moreover, I would like to thank Mr. Soroush Irandoust, Mr. Qiyong Chen, Ms. Ara Kim, Ms. Tinting Zhu and last but not least Mr. Runyang Zhang to carry on this research onto the next step.

I gratefully acknowledge the financial support provided in support of my work by the International Storage Industry Consortium and their sponsor companies and the Oracle Corporation; and also the summer internship at Quantum Research. I am grateful for the technical assistance I received from many colleagues at Quantum and Oracle. Especially, I would like to thank Drs. Turguy Goker, Ming Chih-Weng, and Ash Nayak at Quantum Research, Dr. Mark Watson, Clark Jenson, Peter Coburn, and Kathryn Barnes of Oracle Corporation, and Drs. Mark Lantz and Johan Engelen at IBM Zurich, for the research motivation throughout my studies.

Moreover, I would like to thank my parents, Ayla and Bülent Kaşıkcı for their unconditional love and support during my studies. They always supported and encouraged me to dream big, and achieve even bigger.

I would like to thank my friends especially to Tugbay Sahin, Aysun Demircan, Selen Uguroglu, Gonca Aydogdu, Engin Pehlivanoglu, Levent Akkok, Oguz Barin, Emre Guler, Drs. Melda Ulusoy, Alphan Ulusoy, Baran Yıldırım, Onur Arslan and Ozgur Keles for their insightful discussions and support in every part of my life.

Last but not least, I would like to thank Mr. Stephen Lafaille, for his unconditional support in all aspects of life from start to finish of this study. If he had been the only person left in my life, I could still achieve a lot of things.

The roots of education are bitter, but the fruit is sweet

-Aristotle

CONTENTS

ABSTRACT ...... I

ACKNOWLEDGMENTS ...... III

CONTENTS ...... V

LIST OF FIGURES ...... VIII

LIST OF TABLES ...... XII

1 Introduction ...... 13

1.1 Historical Perspective on Data/Information Storage on Flexible Media ...... 17

1.2 Mechanics of Tape Moving in Contact with a Roller ...... 20

1.3 Foil Bearing Literature Survey ...... 23

1.4 Traction Literature Survey ...... 25

1.5 Traction in Grooved Rollers ...... 27

1.6 Research Objective ...... 31

2 Governing Equations ...... 33

2.1 Equation of the Motion of the Tape ...... 33

2.1.1 Assumptions ...... 33

2.1.2 Force and Moment Resultants ...... 34

2.2 Boundary Conditions and Initial Conditions ...... 41

2.3 Tape Geometry ...... 43

2.4 Equation of Air Lubrication ...... 44

2.5 Contact Models ...... 49

2.6 Tape-Guide Spacing ...... 52

3 Wrap Pressure ...... 53

3.1 Introduction ...... 53

3.2 Boundary Conditions...... 56

3.3 Solution of Tape Pressure for Contact States ...... 60

3.4 Summary and Conclusion ...... 68

4 Mechanics of Tape Grooved Roller Interface with Air Lubrication Effects: Part-1 ...... 70

4.1 Introduction ...... 70

4.1.1 Traction ...... 70

4.2 Mathematical Model of the Tape-Roller Interactions with Air Lubrication ...... 71

4.3 Contact Analysis for a Full Roller with Air-lubrication Effects ...... 74

4.4 Contact analysis for a single land with air-lubrication effects ...... 79

4.5 Summary and Conclusions ...... 85

5 Mechanics of Tape Grooved Roller Interface with Air Lubrication Effects: Part-2 ...... 86

5.1 Introduction ...... 86

5.2 Theory ...... 88

5.3 Experiments ...... 92

5.4 Results and Discussion ...... 96

5.4.1 Experimental results and comparison with theory ...... 96

VII

5.4.2 Contact mechanics of non-translating tape ...... 98

5.4.3 Contact mechanics of translating tape ...... 100

5.5 Summary and Conclusions ...... 106

6 Summary and Conclusions ...... 107

6.1 Wrap Pressure ...... 107

6.2 Traction Modelling ...... 108

6.2.1 Conclusion of Traction Modelling ...... 108

6.3 Air Lubrication Effects on Contact of Tape with Grooved Roller ...... 108

6.3.1 Conclusion of Air Lubrication Effects on Contact of Tape with Grooved Roller 108

7 Future Work ...... 110

APPENDIX ...... 111

REFERENCES ...... 113

VIII

LIST OF FIGURES

Figure 1.1 Essential parts of a magnetic tape drive ...... 14

Figure 1.2 Demonstration of layers of a magnetic tape (Source: Sony Corporation [1]) ...... 14

Figure 1.3 A grooved roller and grooves in details (Source: Oracle Corporation) ...... 16

Figure 1.4 a) Total worldwide digitial archieve capacity on tape 2010-2015 b) Total worldwide digitial archieve capacity by media type [11]...... 19

Figure 1.5 Areal density of hard disk and tape products [11]...... 20

Figure 1.6 Representation of tape wrapped around a roller. Note that tape is under tension T at

each side, and moving in the direction shown with a velocity Vx...... 20

Figure 1.7 Stribeck curve for different lubrication regimes ...... 22

Figure 1.8 Tape-roller configuration for a) x-z direction b) y-z direction c) close-up for y-z direction...... 28

Figure 2.1 A schematic of the tape and guide...... 34

Figure 2.2 Force distribution on a cylindrical shell element ...... 37

Figure 2.3 Moment distribution on cylindrical shell element ...... 37

Figure 2.4 Schematic of symmetric analysis around a single land ...... 42

Figure 2.5 Geometrical regions of tape...... 43

Figure 3.1 y-z cross section of tape and roller configuration ...... 54

Figure 3.2 Tape-roller contact cases a) case-1 b) case-2 c) case-3 ...... 56

Figure 3.3 Forces applied on a tape-roller system ...... 59

Figure 3.4 Deflection profiles for the same belt-wrap pressure values Γ = 0.005, 0.05, 0.11, 0.16, and 0.2 for different land and groove width ( LLLG, ) values, a) (0.5, 1), b) (1, 1.5), c) (1, 0.5). .. 62

Figure 3.5 Dry contact cases for LL = 0.5different LG values for a) case-1, b) case-2, c) case-3 63

Figure 3.6 Effects of different groove ( LG ) and land width ( LL ) values on (a) half-contact width

a , (b) inner shear force Vi , (c) corner shear force Vc , (d) contact force per length q and pull- down force per length Γa . The results are obtained for Γ=0.1. Note that black, red and blue lines indicate CS-3, CS-2 and CS-1, respectively...... 66

Figure 3.7 Predicted deflection profiles for dimensional roller parameters ...... 67

Figure 4.1 A schematic of the tape and guide...... 71

Figure 4.2 Guide shape δ y ( y) ...... 72

Figure 4.3 Air pressure and contact pressure distributions for three different groove types: (LL,

LG) a,b) (0.3, 0.5) mm, c,d) (0.5, 0.3) mm, e,f) (0.25, 0.25) mm. Tape running direction is along

the + x-axis. Tape velocity Vt = 6 m/s and tape tension Tx = 0.7 N. Other variables are reported in the text...... 77

Figure 4.4 Lateral (y-)direction cross-sections of the tape deflection, air pressure and contact

pressure distributions obtained at the center of wrap, for three different groove types: (LL, LG)

a,b) (0.3, 0.5) mm, c,d) (0.5, 0.3) mm, e,f) (0.25, 0.25) mm. Tape velocity Vt = 6 m/s and tape tension Tx = 0.7 N. Other variables are reported in the text...... 78

Figure 4.5 Tape-land clearance h, contact pressure pc and air pressure p variations for tension T =

0.5 N, land width 2LL =50 m and groove width 2LG = 50 m, tape speeds a) 0.5 m/s, b) 2 m/s, c) 6 m/s...... 휇 ...... 휇 ...... 82

Figure 4.6 Tape-land clearance h, contact pressure pc and air pressure p variations for tension T =

0.5 N, land width 2LL =50 m and groove width 2LG = 450 m, tape speeds a) 0.5 m/s, b) 2 m/s,

c) 6 m/s...... 휇 ...... 휇 ...... 83

Figure 4.7 Tape-land clearance h, contact pressure pc and air pressure p variations for tension T =

0.5 N, land width 2LL =600 m and groove width 2LG = 50 m, tape speeds a) 0.5 m/s, b) 2 m/s, c) 6 m/s...... 휇...... 휇 ...... 84

Figure 5.1 Schematic depiction of a grooved roller and the details of the area considered in analysis...... 89

Figure 5.2 Description of the three possible contact states when the tape is “pushed” against the grooved roller by the belt-wrap pressure...... 90

Figure 5.3 Test setup showing a flanged roller and the position of the displacement sensor. Note that the spin-sensor is not shown in this figure...... 93

Figure 5.4 Close up SEM picture of grooved roller surface. Note that measurements with laser and Fotonic sensor are taken on this roller ...... 94

Figure 5.5 Tape roller and sensors configuration on the tape path ...... 94

Figure 5.6 Fotonic Sensor(top) and laser sensor (bottom) data. On top graph, gray lines show the raw data where blue line shows the cleaned data ...... 94

Figure 5.7 Comparison of numerical and experimental flying height values for tension values of a) T= 0.3, 0.5, 0.7 and 0.9 N analyses and b) T= 0.4, 0.6 and 0.8 N...... 97

Figure 5.8 Variation of tape deflection predicted by closed form analysis for 2LL = 137 m and

2LG = 245 m for T = 0.3 – 0.8 N...... 휇 99 휇 Figure 5.9 Variation of tape deflection and contact force predicted by closed form analysis for T = 0.5 N and different land and groove width combinations...... 99

Figure 5.10 Tape deflection and corresponding air and contact pressure around the land at steady

state for (2LL, 2LG) = (50, 300) µm, T = 0.5N at Vt = 6 m/s...... 102

Figure 5.11 Lateral (y-) direction cross section of tape deflection w for different land and groove width combinations...... 103

Figure 5.12 Lateral (y-) direction cross section of air pressure p for different land and groove width combinations...... 103

Figure 5.13 Lateral (y-) direction cross section of contact pressure pc for different land and groove width combinations...... 104

Figure 5.14 Axial (x-) direction cross section of tape deflection w and air pressure p variations for tension T = 0.5 N, for different land and groove width land width...... 104

Figure 5.15 Traction coefficients for land width 2LL and groove width 2LG combinations for all tension values. Blue lines represent T=0.3 N, red lines represent T=0.5 N and green lines represent T=0.7 N...... 105

XII

LIST OF TABLES

Table 3.1 An example of dimensional and non-dimensional parameter comparison for the studies on tape and the web...... 60

Table 3.2 Contact force per unit width for two typical cases of applications. Related parameters are: a) E = 9 GPa, ν =0.3, c = 5 µm, R = 7.5 mm, T = 0.25 N, b = 12.7 mm; and for b) E = 4 GPa, ν =0.3, c = 100 µm, R = 7.5 cm, T/b = 44 N/m...... 68

Table 4.1 Parameters used in the study presented in Section 4.3...... 76

Table 4.2 Parameters used in Section 4.4...... 81

Table 5.1 Parameters used in this chapter...... 95

1 Introduction

While data storage on magnetic tape is a mature technology, its efficiency for archival storage is unparalleled. Specifically tape recording provides the lowest cost/MB stored among the other data storage technologies, in part due to its low operational cost, high data reliability, portability and longevity of stored data. From the storage industry point of view, tape storage is still highly significant and preferred as the tier 2 or 3 source for backup, recovery and active archive. Tape also has a very important role in cloud storage. Most of the data stored in the cloud has a copy or have migrated to tape, for backup and recovery purposes.

There are two major categories of tape drive configurations, namely linear and scanning drives. In linear tape recording technology, group of heads write data in long parallel tracks along the entire length of the tape. The head can be positioned in the lateral direction to read or write additional tracks. In scanning technology, heads write short dense tracks across the width of the tape. In this method, the read-write heads are placed on a rapidly spinning drum, as the tape moves slowly over the head. In the most used technology from this method; helical scan recording, data is written in short tracks on diagonal tracks across the tape. This is the format of most videotape systems.

A linear tape drive is presented in Figure 1. When a cartridge is inserted in the tape drive, a leader mechanism starts the auto threading operation which takes the tape from the supply pack and pulls it over the guiding elements, heads and into the take-up pack, until tape is ready for the read/write operations. The firmware on the tape drives are installed such that, the head-stack on which the read/write heads are located, can track the across the width (lateral direction) of the tape. Recent demands on data storage volume necessitate use of tape drive libraries which include multiple drives and a robot system that moves cartridges. Arguably, the most important part in a tape drive is the read/write heads. The function of read/write head-stack is to exchange data between tape and the outside world. In modern tape drives, the data is digitized and sent to the write head as a stream of 1s and 0s. Recording happens when the rings of ferromagnetic material on the head is magnetized where it contacts the tape. A coil of wire on the head carries a current that is proportional to the signal that would be recorded. Recorded data can be retrieved

by the current induced on the head by the magnetic flux on the tape. Thus, in principal the same head can be used for both recording and playing. Another advantage of tape drives is that they are mostly isolated from the environment. However, since they can be partially exposed to airborne particles from cartridge insertion, they should be operated in a clean environment.

During operation, the tape moves between two reels with a transport speed of Vx , under tension of T. In modern tape drives, tape’s speed and tension are controlled simultaneously by a closed- loop control system. In the current generation of tape drives, the tape transport speed during read/write operations is 2 − 6 m/s, and it can reach up to 10 m/s during seeking and winding operations. Applied tension on the tape is usually between 0.2 − 0.6 N.

Guiding rollers Takeup reel

Read/write head Supply reel cartridge

Figure 1.1 Essential parts of a magnetic tape drive

Figure 1.2 Demonstration of layers of a magnetic tape (Source: Sony Corporation [1])

A magnetic tape is a composite (plate) composed of 4 layers: back coat layer, base film, non- magnetic layer and magnetic layer (Figure 1.2). A typical cartridge holds 900 − 1200 m of tape.

Base layer of the tape is coated with a coating from both sides with back coat layer and non- magnetic layer. In this layer, there are also linking and curing agents and lubricants. In some media, carbon is added for antistatic protection. Base layer is the part where most of the electrostatic is held and it’s where the most of the tape thickness is coming from. Typical base layer thickness is on the order of 3 – 5 µm. Base film materials are bi-axially oriented polyethylene terephthalate (PET), PEN or Aramid. For the magnetic layer of tapes, the most

commonly used magnetic particles are -Fe2O3 and CrO2 [2].

훾 Proper positioning of the tape in the width (lateral) direction, also referred as lateral tape guiding, is very important for the read/write operation. Tape guiding in a recorder is typically provided by rolling guides. Rollers with flanges are placed near the supply and take-up packs to attenuate excessive lateral tape motion (vibration) due pack motion. Basically, they work as dampers for the tape, so high frequency vibrations coming from supply reel will not be transferred to rest of the tape path and the read/write heads. On the downstream side of the head, the vibration that could occur on the tape path is reduced with a flanged roller in order to prevent excessive scatter wind in the take-up pack. This helps reduce stresses that could come from uneven winding in the packed tape. However, while reducing the vibration, these rollers can also introduce vibration to the system due to flange contact. As the tape approaches the head, this vibration could spoil tracking of the system. Thus, in the recent tape drive designs, at least one non-flanged roller is used. Non-flanged rollers are arranged as the closest ones to the head, and they are responsible for passing the tape to the head with the least disturbance.

Besides flanges, the guides can also be with or without surface texture. In recent tape drives, mostly, grooves with surface texture are preferred due to their ability to keep traction (Figure 1.3). For smooth rollers, as tape starts moving, air starts accumulating between the smooth roller and the tape, and eventually causes the tape to lose contact with the roller. However, in the case of grooved rollers, instead of building up, air flows through the grooves and tape still keeps contact with the grooves. Thus, grooved rollers are preferred in most contemporary tape drives. We will present more details on how grooved rollers affect the tape flying height in the upcoming chapters.

Figure 1.3 A grooved roller and grooves in details (Source: Oracle Corporation)

As indicated above the tape vibrates in the lateral and out-of-plane directions during transport. Excessive lateral tape motion (LTM) can cause problems with track finding and following. Due to a variety of reasons LTM has a wide frequency band [3]. In general, the read/write head can be actuated to follow the LTM up to ~0.5 kHz. When the frequency exceeds this range, due to the limited bandwidth of the track servo heads, tape movement cannot be followed. This could eventually result in read and write errors. Some of the sources for LTM are reel motor instabilities, tape edge contacts, friction on the head/tape interface, tape edge imperfections, roller dynamics, roller bearing friction, roller tilt, tape waviness, tension fluctuations, parametric excitation, friction induced vibration from the head and the reels, and surface friction between different tape layers in the packs [4].

Traction between the surface of the rollers and tape is another factor that affects LTM. Traction is defined as the ratio of the contact force applied by the tape on the roller to the total applied belt wrap pressure by the tensioned tape. Fundamentally, increased traction is associated with increased friction between the tape and roller, thus also with reducing the LTM. However, most of the rollers in tape drives have imperfections and this causes axial movement of the roller along the axis. Thus, increased friction in the imperfect rollers will increase the LTM on the tape [5].

Roller surface characteristics affect traction and LTM. Smooth rollers develop an elastohydrodynamicaly activated air layer between the roller and the tape, thus, they provide less traction. Grooved rollers provide relatively better traction than smooth rollers, thus they help to reduce LTM.

1.1 Historical Perspective on Data/Information Storage on Flexible Media

First model of a magnetic recording device was published by Oberlin Smith in 1888 based on his visit to Edison’s lab. The audio words were transformed into electrical sound and recorded by magnetization. Later, in 1894, Valdemar Poulsen discovered magnetic recording, and patented telegraphone in 1898, which is considered as the first successful magnetic recording device. In this device, a wire was wrapped around a drum and a write/read head was moved by a screw bead on top. After Pulsen’s patent expired in 1918, Curt Stille developed the Dailygraph magnetic recorder as a dictating machine, in Germany [6].

During 1930s, wire tape was replaced by thin plastic tape. In 1930, AEG Belin started collaboration with BASF, to develop a magnetic tape. In 1934, BASF shipped the first 50,000 meters of magnetic tape. This tape was made of a foil of cellulose acetate as the carrier and it was coated with a lacquer of iron oxide as the magnetizing material. During 1935, in Radio Fair in Berlin, the magnetophone and magnetic tape were first introduced to public. In 1944, 3M started experiments with tape coating [7].

In 1949, with the latest improvements in technology, and to address the needs of the US Social Security Administration, IBM decided to move the data from punched cards to magnetic tape. The motivation behind this development was the fact that punch cards were taking too much space, and the only option was to go for magnetic tape recording [8].

Video recording was first demonstrated in 1951 by 3M. In 1953, IBM marketed the first magnetic drive, IBM 726, and the data density for this product was 4 bytes/mm. In 1957, IBM introduced the first magnetic rigid disk drive, IBM 305 or RAMAC (random access memory accounting matching) with a storage capacity of 3 bits/mm2. In 1970, IBM introduced the floppy disk with a capacity of few hundred kilobytes. The sizes of these disks were very small compared to other machines. They were 200 mm, round, flexible disks in a rectangular protective shell. In 1976, these disks were followed by 133.2 mm diameter disks. Microfloppy disks with 85.8 mm 75 mm diameter were introduced in 1980 and 1987, respectively [9]. 18

In 2007, 2% of 2.64 billion MB capacity of stored information was digital whereas in 2012, out of 295 billion MB, 94% of the data was digital. The data growth over the span of 5 years has grown more than 11,000%. A study by IDC forecasted that from 2009 to 2020, the total amount of digital data will grow about 44%, leading to 35 ZB by 2020 (1 ZB = 1021 bytes) [10]. In addition, Enterprise Strategy Group projects that the digital archive capacity will grow by 303,000 PB (1 PB = 1015 bytes) from 2008 to the end of 2015 [11]. With such numbers, it’s clear that there’s a massive data growth. Tape recording industry as a whole has a significant share of the storage capacity, with the growing data and the worldwide digital archive capacity. It is estimated for tape archive capacity to have a compound annual growth rate of 45% (See Figure 1.4) although, the overall amount of data that was stored in tape have increased, the percentage has eroded to 27% from 38% (Figure 1.5). Tape has the advantage over cost per GB over the disk drives, and this cost decreases as the data amount increase. It is also projected that the capacity of a tape cartridge will increase 40% per year within the years 2012-2022. INSIC technology Roadmap predicts digital tape capacity to reach 128 TB by year 2022 [12]. The current generation, LTO-6 has a storage capacity of 2.5 TB. Latest updates on the technology include Oracle presenting StorageTek T10000D with 8.5 TB in 2013, IBM TS1150 at 10 TB in 2014 and LTO-9 and LTO-10 are projected to have capacities of 62.8 and 120 TB per cartridge, respectively [13, 14]. Recently, Sony and Fujifilm projected 185 TB and 220 TB per storage space, respectively [15, 16]. By 2015, the projections made by Technology Roadmap have almost doubled.

Figure 1.4 a) Total worldwide digitial archieve capacity on tape 2010-2015 b) Total worldwide digitial archieve capacity by media type [11].

Figure 1.5 Areal density of hard disk and tape products [11].

1.2 Mechanics of Tape Moving in Contact with a Roller

x = L1 x = L2 θw Rg Vt

z x

x = 0 x = Lx

Figure 1.6 Representation of tape wrapped around a roller. Note that tape is under tension T at each side, and moving in the direction shown with a velocity Vx.

Figure 1.6 shows a schematic depiction of a tape wrapped around a cylindrical roller, under uniform tension T. When the tape is stationary and if the roller surface doesn’t have any grooves, the applied contact pressure pc between the tape and the roller is found as follows,

T pc = (1.2.1) RLgy

where Rg is the radius of the guide, and b is the width of the tape. In fact, roller and tape surfaces are not ideally smooth and contact takes place on the peaks of the asperities [17]. Therefore the (local) real contact pressure can be different than the overall average contact pressure.

In the case of a tape translating in the longitudinal (x-) direction, with transport velocity Vx, the air surrounding the tape is also dragged in a small boundary layer around the tape. This boundary layer will eventually create a lubrication effect in the tape-roller interface. At sufficiently high tape velocities the air pressure in the interface could rise to the same order magnitude as the

contact pressure. This has a significant consequence, as the belt pressure (T/RgLy) is now shared between air the contact pressures,

T ppc+= air (1.2.2) RLgy

In response to increased air pressure, tape deflects away from the roller surface and if pair is large enough, contact can be lost and the tape could be supported by the air over the roller. In any case, when the tape is moving, the deflection of the tape and the interfacial air and contact pressures are significantly coupled. The mechanics of this problem is known as the foil bearing problem.

As stated above, air entrainment introduces lubrication between the roller and tape and thus it could have detrimental effects on the tractive capacity of a rolling-guide (roller). The condition where the thin air film could support most of the load but is not thick enough to prevent the contact between the surfaces is called partial hydrodynamic lubrication. The limits for hydrodynamic lubrication is first studied by Stribeck [18] and Gumbel [19]. Figure 1.7 explains the lubrication regimes for the hydrodynamic lubrication. As shown in this figure, friction is proportional to the non-dimensional Stribeck number ( V/W) where represents the viscosity of

air, V represent relative fluid velocity and W represents휂 the load which휂 in this case is the belt- wrap pressure. Note that for small Stribeck numbers (e.g. at slow transport speeds), friction rapidly rises, indicating the termination of hydrodynamic lubrication. In this regime, the friction force is dominated by the properties of the surfaces, whereas in the high Stribeck number limit it is dominated by the fluid film viscosity. The mixed film lubrication usually encompasses the elastohydrodynamic conditions as well the transition to both lubrication states. In fluid-film

lubrication state, the lubricant acts like a thick surface (more than the combination of asperities of both surfaces) and its generation usually relies on the motion and geometry of both surfaces. Traction curves mainly tell us how the interaction between air and the guide work in this system [20-22].

Boundary Lubrication

Mixed Film Lubrication

Fluid-film Lubrication

Vx/W

휂 Figure 1.7 Stribeck curve for different lubrication regimes

When two rough surfaces brought together, contact occurs at the peaks of their asperities, giving

rise to asperity contact pressure pc [17]. The apparent contact pressure in the tape-guide

interface is pc =T/RgLy. As mentioned before, the total contact pressure can be portioned to have air and asperity contact components. If we assume that shear stress due to contact is due to

Coulomb friction, then the total tractive force Ff can be obtained by integrating the asperity

contact pressure pc and a friction coefficient over the contact area. Note that a traction coefficient can be found by taking the ratio of the frictional force to the normal force.

= FNc∫ p dA (a) A = µ Ffc∫ p dA (b) (1.3.1) A

Ff µT = (c) FN

1.3 Foil Bearing Literature Survey

Growth of the tape recording industry in 1950s and 60s demanded research in the area of flexible media and guiding systems. Blok and VanRossum [23] were the first to investigate experimental and theoretical aspects of foil bearings. In their work, tape was assumed to be perfectly straight in the free span and becomes a perfect cylinder with a thin air gap. Later, Baumeister [24] formulated the foil bearing, by assuming incompressible fluid, infinitely wide and perfectly flexible foil and by neglecting the fluid friction and foil inertia effects. Eshel and Elrod [25] used the asymptotic behavior of the film thickness to numerically solve the entrance and exit regions of the problem. They used the same assumptions used in the previously mentioned papers. In 1968, Licht experimentally verified the analytical solution of the foil bearing gap [26].

The effects of the assumptions of the foil bearing formulation have been tested by many authors, analytically. Barlow [27] and Eshel [28] studied the effects of incompressibility on foil bearing. Eshel found that the predicted nominal clearance decreases by including the effects of air’s compressibility. Eshel and Elrod [29] (1967), Barlow [27] and Norwood [30] studied the effects of tape stiffness on foil bearing solution. Finite width effects were studied by Eshel and Elrod [31]. Barlow [27] investigated the effects of small and large wrap angles. Eshel [32] studied the effects of fluid inertia in foil bearings. Elrod [33] studied the effects of Reynolds roughness and White, in his papers on 1980 [34] and 1983 [35], studied the effects of surface roughness and two sided surface roughness respectively. Patir and Cheng [36] modified Reynolds equation for surfaces with isotropic roughness and surfaces with directional patterns. White [37] introduced computer simulations for solving the air pressure between head/disk interfaces for floppy disks. Lacey and Talke [38] presented numerical methods to solve for partial contact in tape/head spacing mainly due to asperities and further verified their model with interferometric measurements. Ducotey and Good [39] modelled the effect of web permeability on traction of a roller-web system. In 1999, Hashimoto [40] modelled the air thickness over a guide roller using finite width compressible foil bearing theory. Based on the discretized numerical results by Hashimoto [41], he set up a single formula based on the web width, tension and velocity for air film thickness. To verify the model, he carried experiments by measuring the clearance between the web and the guide with his model parameters. The results showed that the model works at a reasonable level but their biggest accomplishment was to add the finite width in the model. In

2000, Müftü and Altan [42] modelled a porous paper web over a cylindrical guide. Ducotey and Good [43] investigated modelling air entrainment on a web roller system by using circumferentially grooved rollers. This is the first study for using grooved rollers for higher web roller traction. In 2002, Rice et al. [44] modelled traction over a smooth roller by using a two parameter model which involves contact pressure and clearance of the web and the roller. They also proposed a new model on calculating the asperity engagement height from surface roughness. Traction loss experiments carried over 8 webs with varying surface roughness were used to validate their models and the test methods.

As the foil bearing problem consists of a coupled nonlinear partial differential equations, numerical solutions to the simulation of this phenomenon have gained importance in the recent years. Earliest work of using numerical techniques in solution is by Stahl et al. [45]. Their method of solving was different from the conventional foil bearing problems in 3 ways: They wrote the equations of motion according to the natural position of the tape, they kept the inertial terms from axial motion and also they assumed the tape has a finite segment where they apply the boundary conditions. However, their solution assumed very small time steps, thus required long time to solution. Granzow and Lebeck [46] published another paper solving Reynolds and foil equations simultaneously, by using the finite difference method and reduced the time required to one-tenth. Heinrich and Wadhwa [47] and Hendriks [48] used the finite element method to solve the 1D foil bearing problem. Lacey and Talke [49] proposed a linearized stiffness approach where they couple the stiffness between air bearing and tape and the elastic foundation on the tape is updated with air pressure and spacing between the tape and the head. With this formulation, they eliminated the need for a good guess at the start of numerical analyses and they reduced the time further, where the robustness of the solution is also improved.

Modeling the tape behavior in different conditions became easier with the improvements made to 2D solutions. Numerical solution to 2D foil bearing problem was first discussed by Greenberg [50]. He solved the Reynolds equation along with the tape equation using linear shell analysis. Following Lacey and Talke’s [49] 1D solution to contact pressure, Müftü and Benson [51] solved the contact pressure in a 2D model. Later, Ducotey and Good [43] solved the 2D foil bearing problem for grooved rollers. Arregui et al. [52] modelled the air pressure in the head tape interface and further modified the Reynolds equation with first-order slip flow terms. They

25 solved the 2D foil bearing problem in head/tape interface, but they used Koiter shell equation for the elastic tape behavior. Lopez and Müftü [53] studied the fluid structure interaction between a web and flotation guide. They modelled the web by using the elastica formulation. They used two different kinds of formulations for the air flotation. For their 2D model, they used Navier- Stokes equations and for 1D model, they used the height averaged version of Navier-Stokes equations. They used finite volume numerical approach to solve for their coupled problem. In 2013, air entrainment between a web and a convex roller was studied by Jeenkour et al. [54]. They investigated the change in air film thickness under convex roller shapes. They concluded that the minimum and central air film thickness can be reduced with a convex roller which increases the traction. However, the change is highly dependent on the shape parameter of the roller. In 2014, air entrainment between a web and a roller was studied by Ma et al. [55] by using computational fluid dynamics commercial codes. They set up an initial film thickness between the web and the roller and investigated the distribution of pressure and velocity along the wrap region. They concluded that CFD is also a good way to study air entrainment in this problem.

1.4 Traction Literature Survey

Starting with Knox and Sweeney [56] in 1971, scientists were interested to measure the air film thickness in between the rollers and the web. Most comprehensive study done on traction is conducted by Ducotey and Good [57] in 1995, where they investigated traction coefficient change on a non-grooved roller with respect to different factors that affect contact. Some of these factors are tension, velocity, wrap angle, web width, surface roughness and roller diameter. They reported that traction coefficient depends primarily on the air film thickness and surface roughness. Higher surface roughness prevents the formation of a thick boundary beyond a certain value, thus increasing the roughness will not have any effect on increasing the traction.

Müftü and Benson [58] implemented a one dimensional finite difference code to study the traction of a porous paper web on a nonrotating guide with the effects of air lubrication and by using a parabolic asperity compliance model. Their work included the numerical solution to air lubrication and tape deflection equations coupled in the transient domain. They are the first to include air lubrication, contact pressure and web deflection in their solution. In their paper, they

concluded that the webs with higher porosity were less prone to lose traction due to air entrainment.

In 1999, Ducotey and Good [59] used a variety of experimental methods to verify their one dimensional prediction of traction coefficient. They also included the effects of permeability, side leakage and roller roughness in their models. Their models matched well with their experiments. Müftü and Altan [42] used a steady state model to investigate the traction of a permeable web on a non-grooved roller. They characterized traction by looking into web’s separation on the roller depending mainly on the permeability of the web. In 2004, Müftü and Jagodnik [60] performed a study on the traction between a smooth roller and a web. They solved the change in tension by predicting the slip using both in-plane and out-of-plane equilibrium. Their air lubrication equation consists of Reynolds equation with the correction of first order slip flow. Their formulation considers the effects of the web and roller in terms of speed, radius and uses a model that combines roughness of roller and tape, tension, thickness and friction coefficient. They also looked in the effects of different parameters on traction on the interface. In 2012, Hashimoto [61] studied a similar problem by investigating the friction characteristics between uncoated paper and steel roller. He presented three different lubrication regimes (dry, mixed and fluid lubrication), depending on the clearance and asperity engagement height. Demarcation boundaries between the regimes were based on the combined roughness of the web and the roller. He presented a local effective coefficient friction for the three different lubrication regimes. In his experimental studies, his figures were very similar to Stribek curves and his theoretical local effective coefficient model agreed very well with his results for a wide range of speed, tension, and wrap angle and web widths.

In 2014, Sunami et al. studied the friction coefficient between a plastic film and steel roller by experimentally measuring the traction coefficient between films with different thicknesses and a steel roller. They concluded that the static coefficient of friction has increased with decreasing film thickness. However, kinetic friction coefficient has decreased with the decrease in film thickness.

1.5 Traction in Grooved Rollers

It is crucial to keep the tape in contact with the rollers during operation on a tape path. Tape’s alignment on the tape path between the supply and take up packs is very important to eliminate the lateral tape motion, which causes misalignment between tape and head, which in turns leads to reduced read/write performance. There are several methods to keep the tape in contact with the rollers and inserting grooved rollers in the tape path is one of these methods. A typical grooved roller is described in Figure 1.8. Grooved rollers were first reported by Daly and Peterson [1] in 1968. The purpose of inventing these rollers was to reduce the air cushion caused by speed of the tape in the presence of light tension, which causes tape to lose contact with the rollers and slip of the roller surface. Hourticolon et al. [62] , Lioy et al.[63], Stewart and Rice [64] and VanNoy et al. [65] introduced random shaped land and groove combinations and manufacturing processes for these rollers for venting the super ambient air pressure. Later, Poorman [66] introduced measured groove spacing in between lands with a spiral arrangement of grooves where the land was 2.4-3.7 times larger than the grooves. On 2004, Coburn [67] introduced grooved rollers with different first and second angles to the axis of the roller. Tanaka [68] took the grooved rollers to a new level by introducing different clockwise and counterclockwise grooves that start from the opposite ends of a roller and meets at the center.

In the literature, grooved bearings are analyzed by several authors. Reddi [69] analyzed a conical grooved roller using the transient incompressible Reynolds equation. He used the finite element model to solve the governing equations. White [37] analyzed a slotted spherical head configuration using finite difference approach. Berger et al. [70] used the finite element method to solve the Reynolds equation to modelling a single grooved wet clutch.

Most notable starting work for a comprehensive grooved roller air entrainment study is done by Ducotey and Good [43]. They modelled the web in the wrap region as a cylindrical shell, and everywhere out of wrap angle, the web is modelled as a plate. The air entrainment between the web and the roller is modelled as a two dimensional steady state Reynolds equation and air behavior is assumed compressible. They reported their results by using the traction coefficient as the operation metric. The maximum value of the traction coefficient is 1, when the tape makes

full contact with the guide, and there’s no air entrainment; and, the minimum value is 0, which occurs when the tape has absolutely no contact with the roller.

Figure 1.8 Tape-roller configuration for a) x-z direction b) y-z direction c) close-up for y-z direction.

The load a cylindrical roller can support when air entrainment is not present depends on the wrap

angle (θ ), static coefficient of friction ( µs ) and pressure between tape and the roller (in this case

belt-wrap pressure Pw ). As describes previously, traction is defined as the friction force between roller and tape.

In 2005, Rice and Gans [67] studied the effects of grooving on traction. They presented two methods to experimentally characterize traction. These methods include introducing a very small amount of slip into the system. The first method involves measuring the maximum tension

differential between upstream and downstream of the roller ( ∆=TThigh00 − T low ), where Thigh0 is

the tension per unit width at the upstream of the roller, and Tlow0 is the tension in the downstream

of the roller. Second method for reporting traction is in terms of effective friction coefficient (fe).

fe is measured by Rice [71] and Rice and Gans [72] after many experiments using a capstan friction tester. They derived the effective friction coefficient equation as follows,

1 T f = ln high0 (1.5.1) e θ −∆ TThigh0

The capstan equation (Equation (1.3.1) only accounts for the tension variation in the circumferential direction, but the loss of wrap due to air entrainment is not considered. In their paper, Rice and Gans modified the capstan equation as follows,

T− PR θ high a = e f eff (1.5.2) Tlow− PR a

where

2 2 Thigh= T high0 − cVρ w and Tlow= T low0 − cVρ w (1.5.3)

where Thigh and Tlow are the effective up/downstream tensions that include the effects of

centripetal acceleration due to web speed, Vw, c is the web thickness and ρ is the mass density of 3 the web per unit volume (kg/m ). The steady state air pressure Pa and the effective web to roller spacing heff are described as follows,

3/2 3.094µVR = (1.5.4) Pa  Rheff

1 L = 3 heff ∫ h( y) dy (1.5.5) L 0

The effective spacing between the web and roller, for grooved rollers becomes,

1/3 Gw hGeff= D ) (1.5.6) GLww+

In order to determine effective wrap angle θeff in Equation(1.5.2), Rice and Gans started with the standard Reynolds equation for squeeze films and they solved for the time that would take the air

to move from an initial spacing hi to the top of the asperities α for grooved rollers. This equation is expressed as,

µLL2 1 ∆= WF t . 2 (1.5.7) 2(TRP /− a ) a

LW where LF = is the fraction of the lands to the total area of the roller, with LW and GW LGWW+ denoting the land and groove widths, respectively.

The effective wrap angle is directly related to the term θ * which is given as the ratio of the time to squeeze the air into the grooves to the time it takes tape to wrap around the roller, and it is expressed with the equation,

∆⋅tV θ * = W (1.5.8) Rθ

Then, they introduced the effective wrap angle θeff as:

(1−⋅θθ** ) if θ <1 θ = eff  * (1.5.9)  0 if θ ≥1

If θ * = 0 , there is no reduction in the wrap angle and is θ * ≥1, there’s complete loss of wrap. They tested 14 impermeable webs and 19 grooved and non-grooved rollers by using the test method presented in Rice and Gans [72]. They showed that non-grooved rollers have much less traction compared to grooved rollers. Moreover, they report that surface roughness is very important to hold traction. Rice [71] states one constraint for Equation (1.5.8), where the cross- width bending stiffness (B) has to be smaller than a 38.4 for their equation to be applicable. The cross bending stiffness, B, is defined as follows,

(TR / )(1 / N ) 4 B = < 38.4 (1.5.10) cD

where D is the bending rigidity of the web,

Ec3 D = (1.5.11) 12(1−ν 2 )

E is the elastic modulus ν is the Poisson’s ratio, c is the thickness of the web and N is the number of grooves on the roller.

In 2007, Tran et al. [73] investigated the traction coefficient of spirally grooved rollers. Compared to circumferentially grooved rollers, traction coefficient varies with the roller surface tape contacts with. Moreover, the average traction coefficient for a spirally grooved roller turned out to be less than the one for circumferentially grooved rollers. In their analyses, they used quasi-static numerical simulations by using the finite difference method. In 2010, Hikita and Hashimoto [74] studied the effects of triangular shaped micro-grooves, on cylindrical rollers in LCD film transportation environments. They concluded that by using optimized microgrooves in production line, they can reduce the amount of slippage even at lower tensions; however, they also brought attention to the possible occurrence of wrinkles at lower tensions. Thus, they encouraged the method of designing optimized micro grooves in mass production environments. They also proposed a concave roller with micro-grooves, and they claimed that there is no slippage in between roller and the web. To support this claim, in 2011, Hashimoto [75] did experiments on concave micro-grooved rollers and they confirmed that indeed concaved grooved rollers prevented slippage more than cylindrically grooved rollers.

1.6 Research Objective

As stated in the 2022 Tape Technology Roadmap [11], the main problems that need to be eliminated from tape path include, but not limited to out-of-plane motion, air bleeding/entrainment from the packs, and roller traction issues. The roadmap also states that a 3D model is needed to see how the tape dips into the grooves and how the tape mechanics influences traction between tape and grooved rollers. In this work, we address this problem and model the tape behavior on a grooved roller.

In Chapter 2, we introduce the governing equations for the 2D foil bearing problem. In Chapter 3, we analyze the contact mechanics of the tape with a grooved roller, in the case of dry contact, where the effects of tape transport are excluded from the study. In Chapter 4, we include the effects of air entrainment and describe the tape mechanics over grooved rollers. In Chapter 5, we

32 investigate the factors that influence the traction of a thin tape that is being transported over a grooved roller.

2 Governing Equations

Axially translating material systems can be found at a variety of applications such as magnetic tape drives, cable systems or roll-to-roll manufacturing systems. The common properties of all systems are the fact that they require various components in the system such as rotating and nonrotating guides to support the system components so that the translation between two points on the system could be completed smoothly. Especially, in magnetic tape drives, there are many other components in the system such as read/write head, supply and take-up reels which introduces extra disturbances to the system.

In this chapter, we start by presenting the equation of motion of a tape which is modelled as a shell wrapped around a cylindrical surface. After, we introduce the Boundary conditions of the tape along with the tape geometry. Equation of the air lubrication will be introduced in Chapter 2.4, and it follows the derivation of Modified Reynolds equation which was first introduced by Burgdorfer [76]. The contact model assumed in this study will be introduced in 2.5 and tape- guide spacing assumption is given in 2.6.

2.1 Equation of the Motion of the Tape

Schematic of a tape wrapped around a cylindrical surface is shown in Figure 2.1. The tape’s

wrap angle around the cylinder isθ , the radius of the guide is Rg . The tape is under tension T and

is moving along the x-direction with transport velocity Vx. The tape consists of two flat segments connected by a cylindrical segment over the guide. The tape shown in the figure is assumed to be simply supported at its ends x=0 and x=Lx.

2.1.1 Assumptions Tape is assumed to be a cylindrical shell in the wrap region and a flat plate between the simple supports and the roller. The transition between the cylinder and the flat plates is handled by using a piecewise continuous radius of curvature which becomes infinity over the flat sections. In the derivation of the equation of motion, the following assumptions, first stated by Love, are made (Flügge [77] and Timoshenko [78]),

Figure 2.1 A schematic of the tape and guide.

1. The transverse normal stress component (σ z ) in the shell is negligible as compared to the

normal stress components in the in-plane directions of the shell (σ x , σ y ) 2. Normals to the undeflected mid-surface remain normal after the deflection. 3. All displacements and strains are sufficiently small, so that the second and higher order magnitude terms are neglected in strain displacement equations. 4. The thickness of the shell is very small compared to the other dimensions of the shell. Moreover, the tape is considered to be modelled as a curved beam. Curved beam assumption is valid only when Rg/h<10. However, for our case Rg/h=1195 which is at least 100 times bigger than the assumption states.

2.1.2 Force and Moment Resultants Force resultants acting on the mid-surface of the tape are obtained by integrating the stresses through the thickness of the tape as follows:

c/2 = σ Nxx∫ dz (2.1.1) −c/2

c/2 z =σ + N yy∫ 1 dz (2.1.2) −c/2 R

c/2 = τ Nxy ∫ xy dz (2.1.3) −c/2

c/2 z =τ + N yx ∫ yx 1 dz (2.1.4) −c/2 R

The shear force resultants acting per unit length are found as follows:

c/2 = − τ Qx ∫ xz dz (2.1.5) −c/2

c/2 z =−+τ Qy ∫ yz 1 dz (2.1.6) −c/2 R

The moment resultants are found as follows:

c/2 = − σ Mxx∫ zdz (2.1.7) −c/2

c/2 z =−+σ M yy∫ 1 zdz (2.1.8) −c/2 R

c/2 = − τ Mxy ∫ xy zdz (2.1.9) −c/2

c/2 z =−+τ M xy ∫ yx 1 zdz (2.1.10) −c/2 R

where σ x and σ y are the normal stresses and τ xy and τ yx are the shear stresses acting in the x-y plane.

Equilibrium equations are obtained by the summation of forces and moments in x, y, z directions

on an infinitesimal cylindrical shell element. From Figure 2.2, we can tell that N x has also a component in z-direction, which should be included in the summation of forces. Similarly, the same consideration has to be made for Qx in the x-direction force equilibrium. However, at the end of calculations all these terms are multiplied with the derivatives of displacements, and since derivatives of displacements are neglected, these additional terms end up having no effect in the final equations.

P Ny+

Nyx+ Qx Q + Nx y Qy

z Nxy Nyx Nxy+ y Nx+ Q + Ny x x R

Figure 2.2 Force distribution on a cylindrical shell element

Myx+

My+

Mxy z Mx+

y My

Mxy+ Myx x R

Figure 2.3 Moment distribution on cylindrical shell element

Force equilibrium equations are: x-direction:

∂NQ∂N xx+yx −=0 (2.1.11) ∂∂x yR y-direction:

∂∂NN y+= xy 0 (2.1.12) ∂∂yx z-direction:

∂Q ∂Q 11∂∂2wu x ++y ++ = Npx 2 (2.1.13) ∂∂x y R ∂ x Rx ∂ where p is the applied pressure. Similarly, the moment equilibrium could be written as:

about the x-axis:

∂∂MM y+ xy −=Q 0 (2.1.14) ∂∂yxy

about the y-axis:

∂M ∂M x +yx −=Q 0 (2.1.15) ∂∂xyx

about the z-axis:

M NN−−xy =0 (2.1.16) yx xy R

We can eliminate Qx and Qy by using (2.1.14), (2.1.15) in (2.1.13):

∂2M ∂∂∂222MMM11∂∂2wu x +xy +yx +y + ++ = 2 22Npx  (2.1.17) ∂x ∂∂ xy ∂∂ yx ∂ y R∂ x R ∂ x

Stresses and strains in an element can be related by using the Hooke’s law for plane stress condition:

E σx=( ε xy + νε ) (2.1.18) 1−ν 2

E σy=( ε yx + νε ) (2.1.19) 1−ν 2

E τγ= (2.1.20) xy 21( +ν ) xy

where E is the elastic modulus and ν is the Poisson’s ratio, ε x and ε y are the normal strains in

the x- and y-directions, respectively, and γ xy is the shear strain. Strain-displacement relations for a cylindrical shell are represented as follows [77]:

∂∂uz1 2 w w ε =−+ (2.1.21) x ∂x RRz +∂θ 2 Rz +

∂∂vw2 ε = − z (2.1.22) y ∂∂yy2

1 ∂v R +∂ zu ∂2 w z z γ xy = +−+ (2.1.23) Rz+ ∂θθ R ∂ x ∂∂ yR RZ +

where the circumferential coordinate is xR= θ . However, since the thickness of the tape, z is

much less than its radius, R the variations through the thickness can be neglected. Moreover Nxy

= Nyx and Mxy = Myx respectively. Therefore, we can express the stress resultants in terms of displacement as:

Ec∂∂ u w v N x = ++ν (2.1.24) 1−∂ν 2 xR ∂ y

Ec∂∂ v u w N y = ++ν (2.1.25) 1−ν 2  ∂∂yx R

Ec∂∂ v u NNxy= yx = + (2.1.26) 21( +ν )  ∂∂xy

∂∂22ww = +ν MDx 22 (2.1.27) ∂∂xy

∂∂22ww = +ν MDy 22 (2.1.28) ∂∂yx

∂2w MMD= =(1 −ν ) (2.1.29) xy yx ∂∂xy

where the bending rigidity is D= Ec3212( 1 −ν ) . In this work, we are only interested in the radial displacement of the shell. Thus the effect of in-plane displacements u and v could be neglected. Substitution of equations (2.1.27) - (2.1.29) in equation (2.1.17) yields:

∂∂22ww ∂ 2 wN D∇−4 wN − N −2 N =− p x (2.1.30) x ∂x22y ∂ yxy ∂∂ xy R

The inertial forces are added by using D’Almbert’s principle. When the tape is in motion, the Dw2 inertial forces are represented with the term ρ , where ρ is the mass density of tape per A Dt 2 A D unit area and is the material time derivative which accounts for the transport effects of tape Dt ∂ velocity [79]. For the case of zero tape velocity where this term reduces to differential , the ∂t material time derivative of w could be written as follows:

Dw∂∂∂∂ w w w w =+++VVV (2.1.31) Dt∂∂∂∂ txyz x y z

where Vx, Vy and Vz are velocities in x, y and z directions respectively. Since the tape only moves

in the axial direction, Vy and Vz are zero. Thus the equation of motion (2.1.30) becomes:

∂2w ∂∂ 22 ww ∂2w ∂ 2 w ∂ 2 w N ρ 24+ + +∇ − − − =− x AVV t 222t 2D wNx 22 Ny Nxy p (2.1.32) ∂x ∂∂ xt ∂ t ∂x ∂ y ∂∂ xy R

The tape has an initial tension ofT . The in plane-stress resultant N x can be modified to include this tension as follows, by using equation (2.1.24) and neglecting the in-plane deformation components:

T Ec w = + N x 2 (2.1.33) LRy 1−ν

Thus, (2.1.32) can be written as:

∂2w ∂∂ 22 w w Tw∂2 ∂ 2 w ∂ 2 w T ρ 24+ + +∇ + − − − =− AVV t 2222t D w Dws 2 Nxy Ny 2 p (2.1.34) ∂x ∂∂ xt ∂ t Lyy∂ x ∂∂ xy ∂ y LR

22 where the shell stiffness Ds = Ec R (1 −ν ) is only defined in the curved section of the tape.

2.2 Boundary Conditions and Initial Conditions

For both full-width and single land analyses, the tape is assumed to be simply supported along its upstream and downstream edges. The simple support boundary conditions are modelled as follows,

Zero moment for x=0, Lx and 0 ≤≤yLy

∂∂22ww = +=ν MDx 220 (2.2.1) ∂∂xx

Zero displacement for x=0, Lx and 0 ≤≤yLy

w = 0 (2.2.2)

Along the lateral edges of the tape we use two different types of boundaries. First one is for the full width analysis (See Figure 2.1 for a depiction of the coordinate system). In this case, the tape

has free edges such that the bending moment My and the shear force Vy disappear.

Zero moment for y=0, Ly and 0 ≤≤xLx

∂∂22ww = +=ν MDy 220 (2.2.3) ∂∂yx

Zero shear force for y=0, Ly and 0 ≤≤xLx

∂∂33ww = +−ν = VDy 32(20) (2.2.4) ∂y ∂∂ xy

(a)

(b) (c)

Figure 2.4 Schematic of symmetric analysis around a single land

The second one is for the symmetric analysis around a single land (Figure 2.4). The tape has symmetry boundary conditions, where the shear force and the slope of the tape is zero.

Zero shear force for −LG ≤≤ yL GL +2 Land 0 ≤≤xLx

∂∂33ww = +−ν = VDy 32(20) (2.2.5) ∂y ∂∂ xy

Zero slope for −LG ≤≤ yL GL +2 Land 0 ≤≤xLx

∂w = 0 (2.2.6) ∂y

Initial displacement at t = 0 for any x,y

w0 = 0 (2.2.7)

Initial displacement velocity for t = 0

w 0 = 0 (2.2.8)

2.3 Tape Geometry

Figure 2.5 Geometrical regions of tape

The tape is divided into 3 regions as shown in Figure 2.5. In regions 1and 3, the tape is straight and in region 2, the tape is wrapped around the cylindrical guide with a wrap angle θ .

The curvature of the tape is different in different regions. As can be seen in Figure 2.5, in regions 1 and 3, the tape is flat, and thus the radius of curvature is in infinite. In region 2, since the tape follows the contour of the cylindrical guide, its curvature is assumed to be equal to that of the guide.

∞ 0 ≤≤xL  x1 R= Rgx L12 ≤≤ xL x (2.3.1)  ∞ Lxx2 ≤≤ xL

Tape is modelled as a cylindrical shell in region 2 and in regions 1 and 3, it is modelled as a flat plate. Thus, the shell stiffness Ds of the cylindrical shell is:

0 0 ≤≤xLx1   Ec = ≤≤ Ds 22 Lxx12 xL (2.3.2)  R (1−ν )  0 Lxx2 ≤≤ xL

The belt wrap pressure PBW is zero on the flat parts of the tape, and equal to T/RLY in region-2,

0 0 ≤≤xL  x1  T PBW = Lx12≤≤ xLx (2.3.3)  RLgy  0 Lxx2 ≤≤ xL

2.4 Equation of Air Lubrication

In this section, we present the derivation of the Reynolds equation of lubrication. The Reynolds number represents the ratio of the inertial forces to the viscous forces in the flow, and it is defined as follows,

ρUh2 Re* = (2.4.1) µ B where U is the bearing velocity, is the fluid density, B is the characteristic bearing length (in this case radius of our grooved roller)휌 , h is the clearance and is the viscosity of the fluid, which in this case, the viscosity of air. For our analyses, it can be휇 shown that the modified Reynolds number is in the range 2.2× 10−−21 −× 2.6 10 . Therefore, we conclude that the flow in the tap- roller interface is dominated by viscous forces.

When the lubricating film conditions are steady, the momentum equations can be combined with the continuity equations to form the Reynolds equation. It is a single differential equation that relates the pressure, the fluid density, the surface velocities and the film thickness. In the derivation of the Reynolds equation the flow velocity on the moving (bearing) surfaces are assumed to be the same as that of the moving surfaces. This assumption is known as the no-slip condition [80]. In case the bearing spacing become on the same order as the molecular mean free path length then the no-slip condition no longer holds. In this case, slip occurs between the bearing surfaces and the fluid film. Therefore, the Reynolds equation should be modified for the slip effects. The resulting form of the Reynolds equation is called the modified Reynolds equation. This phenomenon was first modeled by Burgdorfer [76] and later modified by Hsia λ [81] and Mitsuya [82]. The Knudsen number, Kn = where is mean molecular free path and h h is the film thickness, is a measure of the molecular collisions.휆 When Kn < 0.01, unmodified Reynolds equation could be used, and for higher Kn, modified Reynolds equations should be used. For our studies, Kn is approximately 0.06. Therefore we use the modified Reynolds equation in this work. The assumptions for the modified Reynolds equation are as follows:

1- The fluid is Newtonian. 2- The film between the shell and guide is isothermal 3- Viscous terms dominate the momentum equations, i.e. inertial terms are negligible.

* = ρµ2 Re( BU /) ( h / B)  1 4- Air behaves as an ideal gas ρ ∝ p . 5- The entrance and exit flow effects in the fluid film are neglected.

For a lubricating fluid in a given control volume, the momentum balance is presented with Navier-Stokes equation. Since we are interested in the fluid being in steady-state, all inertial terms could be neglected. Also, the thickness of the lubricating film is very small compared to its length and width, thus it can be neglected as well. Lastly, by neglecting the body forces acting on the fluid we end up with the reduced form of the Navier-Stokes equation:

∂∂∂pu = µ (2.4.2) ∂∂xz ∂ z

∂∂∂pv = µ (2.4.3) ∂∂yz ∂ z

The mass conservation equation is given as follows:

∂∂∂ρρρuvw ∂ ρ + + +=0 (2.4.4) ∂∂xy ∂∂ zt

where u, v and w are the fluid velocity components in the x-, y- and z-directions. Please note the distinction between these variables and the tape deflection components.

In the air lubrication problem, Knudsen number is considerably high, especially due to low flying height. Since we mentioned above, when Kn is higher than 0.01, Reynolds equation requires slip flow corrections. In this study, we considered first order slip corrections introduced by Burgdorfer [76]. According to his studies, the boundary conditions for the corrections are:

∂u uxy( , ,0) = Vx + λ ∂z z=0 ∂v vxy( , ,0) = V + λ y ∂z z=0 (2.4.5) ∂u u( xyh,,) = Vx − λ ∂z zh= ∂v v( xyh,,) = Vy − λ ∂z zh=

For fixed guide systems, the surface on the guide is stationary, however for roller-tape systems both surfaces are moving. Therefore, we keep the all surfaces in analyses. Considering the velocity of lower surface defines the velocities at z = 0 we can rewrite Eq. (2.4.5) again:

G ∂u uxy( , ,0) = Vx + λ ∂z z=0 ∂v vxy( , ,0) = VG + λ y ∂z z=0 (2.4.6) ∂u u( xyh,,) = Vx − λ ∂z zh= ∂v v( xyh,,) = Vy − λ ∂z zh=

G G where Vx and Vy represent the tape velocity components and Vx and Vy represent the roller velocity components along the x- and y-axes, respectively.

Momentum equations (2.4.2) and (2.4.3) can be integrated through the thickness by using the boundary conditions given in Equation (2.4.6) to obtain the flow velocities in x and y directions:

1 ∂+pzλ =2 −−λ + −GG + u z zh h( Vxx V ) Vx (2.4.7) 22µλ∂+xh

1 ∂+pzλ =2 −−λ + −GG + v z zh h( Vyy V ) Vy (2.4.8) 22µλ∂+yh

Since p is proportional to , we can write express equation (2.4.4) as:

휌 ∂∂∂pu pv pw ∂ p + + +=0 (2.4.9) ∂∂xy ∂∂ zt

Equations (2.4.7), (2.4.8) are inserted in (2.4.9) and the result is integrated in the z direction:

h hh h 2∂∂∂pu 22 pv pw 2 ∂ p dz+ dz + dz += dz 0 (2.4.10) ∫∫∫∫∂∂xy ∂∂ zt h1 hh 11 h 1

For first and second term we get,

hh 22∂∂ dh pudz= pudz − pu dz (2.4.11) ∫∫∂∂x x h dx hh11

hh 22∂∂ dh pvdz= pvdz − pv dz (2.4.12) ∫∫∂∂y y h dy hh11

For the third term we get:

h2 ∂pw w dz= pw 2 (2.4.13) ∫ ∂z w1 h1

For a general z location along the film that are bounded between locations h1 and h2, continuity equation (2.4.9) becomes:

h2 z2 h − λz 32 2 2 ∂∂1 p z z ∂p h ∂p h2 dh GG1 p −−hλ z +( Vxx − V ) +Vx[ z] − puh ∂x2µλ ∂ x  32 ∂x (2 +∂h) x h1 dx h1

h2 z2 h − λz 32 2 2 ∂∂1 p z z ∂p h ∂p h2 dh GG1 + p −−hλ z +( Vyy − V ) +Vy[ z] − pvh(2.4.14) ∂y2µλ ∂ y  32 ∂y (2 +∂h) y h1 dy h1

wh∂p + pw 22+=z 0 wh11∂t

When h1=0 and h2 = h and = a we get,

휆 휆 VV+ G ∂∂1 p 3 6λa ∂ph ( xx) ∂h p(−+ h )1 + −puh ∂∂x12µ x hx ∂2 ∂ x VV+ G ∂∂1 p 3 6λa ∂ph ( yy) ∂h p(−+ h )1 + −pvh (2.4.15) ∂∂y12µ y hy ∂2 ∂ y ∂p +pw( −+ w) h =0 h 0 ∂t

We impose w0 = 0 . The wh term represents the rate of change of clearance due to both tangential and axial motion. Thus wh can be written as:

∂∂hh wu= + v (2.4.16) hh∂∂xy h

49 when (2.4.16) is inserted (2.4.15) in, the modified Reynolds equation is obtained as follows,

 ∂∂33p 66λλaa ∂∂p  ∂ph GG ∂ph ∂ph  ph 1+ +ph 1 +  = 62µa +(VVVVxx +) ++( yy)  ∂∂x xh ∂∂ y yh  ∂ t ∂ x ∂ y (2.4.17)

Two terms on left hand side represent the Poiseuille flow. They describe the flow due to pressure gradients in the lubrication area. The first term on the right hand side is the squeeze film term which represents the local rate of pressure change, as well as the effect of bearing motion in the normal direction. Second and third terms on the right side of the equation are the wedge terms. They stand for the change of shape of fluid film. Since clearance varies along the film, there will be a different flow rate at every section.

The modified Reynolds equation requires a pressure boundary condition. The pressure outside the tape region and at the edge of the tape is ambient pressure.

pypLypxpxLP(0,) =( x ,) =( ,0) =( , ya) = (2.4.18)

When roller used as a guide element, the groove is sufficiently deep to assume that the pressure inside the grooves are equal to the ambient pressure Pa ,

PP= a (2.4.19) around the edges of the lands where N is the number of grooves, and y-coordinate changes with every groove.

2.5 Contact Models

In this section we introduce various contact pressure models for contact between rough surfaces. First, we introduce the Greenwood-Williamson rough contact model [17] which gives the following equation for the contact pressure:

∞ = − 3/2φ Pc (, x y ) p0 ∫ ( z h ) ( z) dz (2.5.1) h

50 where z represents the roughness variation, h is the separation between the guide and the web and φ ( z) denotes the distribution of the asperity peaks on the surface. p0 denotes the pressure required to force the h to be zero and it is given as:

4 σ t p0 = (σηtp RE) c (2.5.2) 3 Rp

where σ t is the mean surface roughness height, Rp is the radius of the asperity peaks, η is the density of the asperities and Ec is the composite modulus of the guide and tape surfaces [83]:

11−ν 2 1−ν 2 = + G (2.5.3) EEEcG

where ν G and EG represent the Poisson’s ratio and elastic modulus for the guide, respectively.

Lacey and Talke [38] proposed an experimental way to measure the contact pressure directly between a tape and a guide surface.

2 h Pc =−βα11 −−Hh( ) (2.5.4) α where β denotes the compliance parameter, α denotes the asperity engagement height, h denotes the film gap and H is the Heaviside step function where

Hh( −=α ) 1 h > α when (2.5.5) Hh( −=α ) 0 h ≤ α

Rice et al. [44] presented a method to determine the asperity engagement height by combining roughness of the roller and tape surfaces. They developed an individual engagement height prediction method based on the surface topography:

(RRz− pm ) αi=R pm + (2.5.6) (RRz/ pm )

where Rpm is the average of five highest peaks in the sample and Rz is the average of five highest and five lowest valleys in the sample. Furthermore, they presented a composite engagement height αc which presented the roughness of two surfaces in contact. They proposed 3 different models for 3 different contact cases.

For the sum model case (case-1)

αααc= Gt + (2.5.7)

For the root mean square method (case-2)

22 αc= αα Gt + (2.5.8)

For the maximum model (case-3)

αc= max( αα Gt , ) (2.5.9)

In this work, we will only use roughness of the tape since the relative roughness of the guides are very small regarding to the roughness of the back side coating of tape.

2.6 Tape-Guide Spacing

The clearance h between the tape and the guide depends on the tape deflection w and the shape of the guide and it is defined as follows,

훿 hxyt( ,,) = wxyt( ,,) +δ ( xyt ,,) (2.6.1)

3 Wrap Pressure

3.1 Introduction

In this chapter, the contact of a thin flexible tensioned tape with a grooved cylindrical roller is investigated. In order to provide a baseline for the contact conditions, the effects of air lubrication in the tape-roller interface are neglected. In this work, an analytical solution is presented to predict the contact states of a thin, flexible, tensioned tape/web wrapped around a grooved roller. The contact surface of the roller consists of circumferentially oriented lands which are separated by grooves. The lands are assumed to be flat and parallel to the axis of the

roller as shown in Figure 3.1. The widths of the grooves and the lands are designated as 2LG and

2LL , respectively; and, these two parameters are considered to be constant along the roller surface. Tape/web is assumed to be wrapped around the roller with a sufficiently large wrap- angle, so that the transition effects from the flat-to-curved regions of the tape can be neglected. Tape-to-roller contact problem in this case can be reduced to analyzing the contact mechanics of a single groove-land pair. The simplified version for the equation of equilibrium for a thin, tensioned tape/web, wrapped around a cylindrical guide is given as the modified version of (2.1.34) [51].

TT Dw(2+ w + w ) +− Dw w =− p (3.1.1) ,xxxx ,, xxyy yyyy s b,xx Rb

where x and y are the coordinate directions in the longitudinal and axial directions, respectively, w is the out-of-plane deflection, T is the constant tensile force applied along the longitudinal boundaries, b is the width, and p is the external pressure acting on the tape/web. The shell

*2 *3 stiffness and bending rigidity are defined as Ds = Ec/ R and D= Ec /12 , respectively, where

c is the thickness, , E is the elastic modulus, ν is the Poisson’s ratio of the

tape/web and R is the outer radius of the roller. Note that in this dissertation ( ),i indicates partial differentiation with respect to i (= x, y). If the wrap-angle is large, the tape/web

54 deflections can be assumed to be independent of the circumferential direction and equation (3.1.1) becomes

T Dw+=− D w p (3.1.2) , yyyy s Rb

Width, b

Web/tape Roller radius, R radius, Roller

Symmetry plane

LL LG z

2LL 2LG 2LL

Figure 3.1 y-z cross section of tape and roller configuration

The shell stiffness Ds and belt-wrap pressure (T/Rb) exist in the wrap region, and stem from the effects of hoop strain (w/R) and initial tension in the material, respectively [84]. While the belt- wrap pressure provides a constant effect to pull the tape over the cylindrical guide, the hoop strain (w/R) provides a resistance that is proportional to deflection, w. In equation (3.1.2) the effect of hoop strain appears in a form that is similar to a linear-elastic foundation force acting on the system.

By defining the following non-dimensional variables,

1/4 3 3 y x w 4D pC 12TC y = x = w = C = p = Γ= D *  *  C , C , C , s , Ecand RbE c (3.1.3) equation (3.1.2) is expressed in the following form,

w+4 wp = −Γ , yyyy (3.1.4)

The parameter C represents a non-dimensional characteristic length for the problem, and is used as the scaling factor for all of the length parameters. In order to reduce the total number of problem parameters, the parameter Γ is chosen to represent the combination of a non- dimensional belt-wrap pressure (T/ RbE* ) and non-dimensional tape thickness (c/C). For the sake of simplicity, the parameter Γ is called the non-dimensional belt wrap pressure in this work. The non-dimensional bending moment and shear force resultants are defined as follows,

VC 2 Vw= = − , yyy and D , (3.1.5) respectively. On a smooth roller, the contact pressure balances the belt-wrap pressure [85], and we have . In the case of a grooved roller, the belt wrap pressure would cause the tape to bend into the grooves and lift away from the land surfaces. Figure 3.2 shows three possible contact states that can develop. A symmetrical contact problem can be defined as shown in this figure. The solution domain spans the half-lengths of a length and a groove defined as follows,

LL LG LL = LG = C and C (3.1.6)

LL LG z a

y Symmetry plane Symmetry plane Symmetry

z V c Vi Vc T/(bR) Vi Vc

a) b) c)

Figure 3.2 Tape-roller contact cases a) case-1 b) case-2 c) case-3

Solution to Eqn. (3.1.4) is found separately over the land and the groove,

 yy(1) (1) − (1) (1) G eA( 12sin yA++ cos y) e( A 34sin yA +−≤≤ cos y) , 0 yLL  4 wy()=  G eAyy(2) sin yA++(2) cos y e− A(2) sin yA +−≤≤+(2) cos y , L yL L  ( 12) ( 34) L LG  4 (3.1.7)

()i where, A14− are constants coefficients, and i = 1, 2 indicate the solutions over the land and the groove, respectively. Thus, it can be seen that the problem is controlled by the three parameters,

LL , LG and Γ .

3.2 Boundary Conditions

Figure 1 shows the case of zero tension, Γ = 0. This results in the trivial solution, with zero contact pressure. One of the possible contact states when Γ > 0 is depicted in Figure 3.1a, where the tape makes contact only along the edges of the lands. This type of contact is called the contact state-1 (CS-1). In this state, the tape deflection is symmetrical with respect to the centers of the land and the groove resulting in the following boundary conditions,

(1) (1) ww,,y (0)= 0, yyy (0)= 0 (a, b) wLL(2) (+= ) 0, w(2) ( LL += ) 0 (c, d) ,,y L G yyy L G (3.2.1)

At the corner of the land and the groove tape deflection is zero, and the slope and curvatures are continuous. These conditions are expressed as follows,

(1) (2) (1) (2) (1) (2) wL()0L = (a); wL()0L = (b); wL,,yL()= wL y () L (c); wL,,yy() L= wL yy () L (d)(3.2.2)

It should be noted that tape/web deflection over the land should not be less than zero,

(1) w>0, 0 ≤< yLL (3.2.3)

Consequently, even though solution of equations (3.1.7)– (3.2.2) can be found in closed form (Appendix-A) the constraint given in equation (3.2.3) should be satisfied.

Different parameters can cause a contact condition where the tape makes contact with the land along its center ( ). This contact state, depicted in Figure 3.1b, is called contact state-2 (CS- 2). In this case, equations (3.2.1) a, b are replaced by the following relationships,

(1) = (1) w (0) 0 (a); w, y (0)= 0 (b) (3.2.4)

Equations (3.2.1) c, d and (3.2.2) remain unchanged.

In contact state-3 (CS-3), the tape makes contact with the land over the contact length 2a , and also contacts along the edges of the land, as presented schematically in Figure 3.1c. The contact half-width ( a ) is not known a priori, for a given set of problem parameters. Therefore, the problem is nonlinear, and along with a there are nine unknowns. Six of the boundary conditions are common (Equations ((3.2.1)c,d) and (3.2.2)) with the previous contact states; and, the three conditions that define the contact state-3 are given as follows,

(1) = (1) (1) wa() 0 (a); wa, y ()= 0 (b); wa, yy ()= 0 (c) (3.2.5)

Solution of the problem is described in Appendix-B.

In all three contact states described above, the tape contacts the corners of the land. The distributed contact force that is associated with contact along the corner of the land is equal to the jump in the shear force resultant as follows,

V=−− ww(2) (1) c ( ,,yyy yyy ) = yLL (3.2.6)

In contact state-3, the tape contacts the center of the land and the resulting contact pressure is given by the following relationship,

p=Γ for 0 ≤≤ya c (3.2.7)

In addition, a concentrated force develops along the edge of the inner contact region for both contact states-2 and -3. These are defined as follows,

and (3.2.8)

The contact force per the unit circumferential length, for the symmetric problem described by the three contact states can therefore be summarized as follows,

V , for CS-1  c q= VVci + (0), for CS-2 V+ Va( ) +Γ a , for CS-3  ci (3.2.9)

The total contact force per unit circumferential length then becomes,

 2NV , for CS-1  pc qt= N p(2 VV ci + (0)) , for CS-2  2Npc( V+ Va i ( ) +Γ a) , for CS-3  (3.2.10)

where Np= b/ 2( LL LG + ) is the number of land-groove pairs along the roller. Note that the effects of the free lateral edges of the tape are neglected in equation (3.2.10)

Figure 3.3 Forces applied on a tape-roller system

In general, for a given set of problem parameters ( ) the contact state is unknown. In order to obtain a solution, the algorithm used in this work checks the feasibility of each contact state by evaluating the following conditions,

(3.2.11)

In CS-3 the analytical solution is obtained by using the BCs given in equations (12b) and (12c), and the value of is determined by using equation ((3.2.5)a). This results in a transcendental equation and is found numerically by using a MATLAB script. The solution quality is confirmed by checking the balance of forces acting on the tape, as follows,

LLLG+ + + − −G + <ε Vci V qa4∫ wdy () LLG L (3.2.12) 0

where ε <10−3 . Figure 3.3shows the effect of this approach for a single value of land width and

groove width. The blue, black and green lines represent the reaction, pull-down, and foundation

forces respectively. The values are added up together and presented as a check line, which is the

line with constant value of ~0.

Contact conditions are evaluated for a range of parameters commonly found in magnetic tape recording and roll-to-roll web handling applications. Typical parameter values are presented in Table 3.1, in both dimensional and non-dimensional forms. Motivated by values in Table 3.1, analysis is carried out in the following ranges: non-dimensional belt wrap pressure

0.02<Γ< 0.2 range; half groove width in the 0.005<

the 0.05<

Table 3.1 An example of dimensional and non-dimensional parameter comparison for the studies on tape and the web

Dimensional Variables Tape Web Tension per width (N/m) 19.7 44 Roller radius (mm) 7.5 156 Thickness (μm) 5 300 Poisson's ratio 0.3 0.3 Elastic modulus (GPa) 9 4

Groove width (LG) (mm) 0.25 0.5

Land width (LL) (mm) 0.15 0.25 Non Dimensional Variables C (1/mm) 0.15 5.2 Γ 0.0816 0.032

Groove width ( LG ) 0.85 0.024

Land width ( LL ) 0.51 0.048

3.3 Solution of Tape Pressure for Contact States

The solution outlined in Equations (3.1.7) – (3.2.5) shows that the contact-state is independent of

the belt-wrap pressure ( Γ ), but only depends on the geometric parameters ( LLLG, ). In other words for a fixed geometry, changing the value of the tension does not cause the contact

conditions to transition between different contact states. This is numerically depicted in Figure

3.4, where for fixed values of LL and LG , different Γ values are shown to cause self-similar deflection profiles. For example, for (,LLLG ) = (0.5, 1) (Figure 3.4a) analysis shows that only contact state-1 is feasible. Similarly, for (,LLLG ) = (1, 1.5) contact state-2 (Figure 3.4b) and for (1, 0.5) contact state-3 (Figure 3.4c) are found.

The effects of using different groove width values on the contact state, are analyzed further in Figure 3.5, for Γ = 0.01 and LL = 0.5. This figure shows that increasing the groove width from 0.13 to 2.5 causes the contact to transition from state-3 to state-1. This behavior can be attributed to the moment equilibrium about the corner of the land. The moment balance due to belt-wrap pressure over the land and over the groove is a factor that determines the contact state. Having a relatively short groove width does not provide sufficient moment to lift the tape off the land as shown in Figure 3.5c. In contrast, when the groove width is large, the tape can lift off of the land surface as shown in Figure 3.5a.

Γ = 0.005

0.06

0.11

0.16

0.20

(a)

Γ = 0.005

0.06

0.11

0.16

0.20

(b)

Γ = 0.005

0.06

0.11

0.16

0.20

(c)

Figure 3.4 Deflection profiles for the same belt-wrap pressure values Γ = 0.005, 0.05, 0.11, 0.16,

and 0.2 for different land and groove width ( LLLG, ) values, a) (0.5, 1), b) (1, 1.5), c) (1, 0.5).

LG = 1.07

1.52

2.07

(a) 2.50

LG = 0.51

0.55

(b) 0.59

LG = 0.13

0.23

0.36 (c)

Figure 3.5 Dry contact cases for LL = 0.5different LG values for a) case-1, b) case-2, c) case-3

Figure 3.6 shows the effects groove width LG on i) contact-width a , ii) the shear force resultant

at the inner edge of contact Vi , iii) the shear force resultant at the corner of the land Vc and iv) the total contact force q for a fixed value of belt-wrap pressure Γ = 0.01 and four different land

width values of LL = 0.05, 0.5, 1, 1.25, and 2.5.

Figure 3.6a shows that as the groove width LG approaches zero the contact width a approaches

the land width LL . This is the expected solution for the non-grooved roller. For increasing values

of the groove width LG , contact length decreases somewhat monotonically. For land width values

of LL = 0.05, 0.5, and 1 a decreases from ~ LL to 0, as LG increases from 0 to LL . It is interesting to note that for these cases, the central contact area a is lost as the groove width

equals the land width ( LG = LL ). For longer lands, LL = 1.25, and 2.5, the length of the central contact length asymptotically approaches a ~0.03 and 1.5, respectively. Thus we see that in case

the ratio LLGL/ is less than one, contact state-3 is established in the center of the land ( a > 0)

for all land width LL values; but, in case LL is less than one, contact in the center of the land can

be lost ( a = 0) and CS-1 be established for LLGL/ greater than one.

The nature of contact on the land can be investigated in more depth by looking at the variation of

inner shear force Vi in Figure 3.6b, where different colors are used to indicate different contact states. This figure shows that in the narrow land width limit the tape transitions from contact

state-3 to contact state-1 where it loses contact along the center of the land; and, LL = 1 marks an approximate transition point in contact behavior where contact state transitions from state-3 to state-2. In the wide land-width limit contact remains in state-3, as mentioned above.

(a)

LL = 2.5

1.25

0.5 0.05

(b) LL = 0.05

0.5

1.25, 2.5

(c)

LL = 0.05

0.5

1.25, 2.5

(d) LL = 0.05 Reaction force 0.5

1.25

2.5

Figure 3.6 Effects of different groove ( LG ) and land width ( LL ) values on (a) half-contact width

Figure 3.6c shows that the magnitude of shear force Vc along the corner of the land increases with increasing groove width, but its value asymptotically approaches a constant, as the contact states transition as explained above. Two different physical effects are at play in the narrow- and

wide groove-width regimes for the reported behavior ofVc . In the narrow groove-width regime, increasing groove width causes a net increase in the total force due to belt-wrap pressure; and, the shear force increases in order to balance this increased force. In the wide groove-width regime, the tape/web deflection is dominated by the elastic foundation stiffness; therefore, even though the wider groove provides larger pull-down force, the elastic foundation prevents larger

deflections to occur, and absorbs the increasing force. As a result, the corner shear force Vc asymptotically approaches a constant value.

Variation of contact force per unit circumferential length q given in Eqn. (3.2.9) is presented as a function of groove-width in Figure 3.6d. In the narrow groove-width regime, both corner and inner reaction forces approach zero, therefore, the reaction force q is dominated by the belt wrap force Γa . As the groove width increases the corner and inner shear forces increase and contribute to balance the increasing total pull-down force; the curves of total-pull down force and

the contact force are indistinguishable for LG less than ~0.75. In the large groove width regime, the reaction force q approaches a constant value, whereas the total pull down force is increasing. This is again due to the tape behaving as a beam on elastic foundation.

(a)

(b)

Figure 3.7 Predicted deflection profiles for dimensional roller parameters

Tape PET-Web Lg CS V V Lg CS V V c i Vc Vi c i Vc Vi (µm) (N/m) (N/m) (µm) (N/m) (N/m)

LL = 50 µm LL = 150 µm 50 2 0.120 0.01 0.025 0.0028 150 2 0.079 8.8×10-3 9.4×10-4 2×10-3 100 1 0.200 0 0.042 0 250 1 0.117 0 1.4×10-3 0 200 1 0.327 0 0.069 0 500 1 0.190 0 2.3×10-3 0 500 1 0.486 0 0.102 0 1000 1 0.324 0 4.0×10-3 0

Media deflection and contact states are investigated for two practical cases in Figure 3.7 and Table 3.2. A 5 µm thick, 12.7 mm wide tape under 0.25 N tension is wrapped around a roller with 15 mm diameter. The elastic modulus and Poisson’s ratio of the tape are 9 GPa and 0.3, respectively. The land width is fixed at 50 µm; and the effects of using 50, 100, 200 and 500 µm wide grooves are investigated. It is seen that central contact takes place only for the case of 50 µm wide groove; all other groove widths cause the tape to make contact only along the corners

of the land. The contact force per unit width Vc increases with increasing groove width (Table 2), because the overall belt-wrap pressure is distributed over a fewer lands. Details of the contact of a 100 µm thick, 1 m wide PET web with a grooved roller of 15 cm diameter are also presented in Figure 7b and Table 2. Tension per unit width is 44 N/m. Elastic modulus and the Poisson’s ratio of this web are 4 GPa and 0.3, respectively. Even though for all practical purpose, the web appears flat, contact is still established along the edges of the lands. The contact force per unit width also increases with increasing contact width (Table 3.2).

3.4 Summary and Conclusion

Contact of a tensioned, flexible tape with a circumferentially grooved cylindrical roller is investigated. It is shown that the contact conditions are governed by three non-dimensional variables related to land-width, groove-width and belt-wrap pressure. The solution is self-similar with respect to the non-dimensional belt wrap pressure, and the three distinct contact states depend on the land-width and the groove-width. The tape makes contact along the edge of the

69 land in all contact states; and it could also make contact along the center of the land depending on the relative sizes of the grooves and the lands. In case the groove-width-to-land-width ratio,

LLGL/ , is less than one, the tape/web is expected to contact the land surface ( a > 0), but this characteristic could change for larger values of this ratio ( LLGL/ > 1) if the non-dimensional land width, LL , is less than one. The belt-wrap pressure, Γ , is balanced by contact forces and the internal forces due to stretching of the tape. The later component, which appears as an elastic foundation in the equation of equilibrium, becomes significant for groove width LG values greater than one. Investigations of the contact conditions of a thin tape and a relatively thick web show that for practical land and groove geometries the media tends to deflect into the first contact state. While the deflections of the relatively thick web are very small, contact is still established along the corners of the land. This could have influence on creating streaks and or scratches in the web. The goal of this chapter is to analyze the fundamental characteristics of contact. We show that this is controlled by three non-dimensional parameters. Additional, effects such as surface roughness, land shape and the effects of air lubrication in the tape/web-roller interface will be reported separately.

4 Mechanics of Tape Grooved Roller Interface with Air Lubrication Effects: Part-1

4.1 Introduction

In most applications where the tape is transported over a roller, the surrounding air is dragged into the tape-roller interface. As a result of this entrainment, air acts like a lubricant and develops a super-ambient pressure distribution in the interface. Increased air pressure can partially or fully support the tape. Thus, air entrainment can eventually cause reduction in the contact pressure, and hence in the tractive capacity of the interface [51]. Grooved rollers are used to prevent excessive built-up of air pressure [43, 44, 72, 86, 87]. In Chapter 3, mechanics of a thin tape wrapped around a grooved roller was presented without considering the effects of air lubrication. In the present chapter, a mathematical model for the tribology and mechanics of a tape-grooved- roller interface is presented by including the effects of the self-acting air lubrication. The tape roller-interface is first investigated for the full width of the tape. It is shown that, as in Chapter 3, it is possible to simplify the problem to analysis of a single land.

4.1.1 Traction

In the tape-roller interface, the belt-wrap pressure, T/(RgLy), is balanced partially by rigid body contact of the asperities and partially by air pressure, where T/Ly is the tape tension per unit

width and Rg is the roller radius. With increasing tape speed, Vt, the interfacial air pressure p increases, decreasing the proportion of the belt-wrap pressure shared by the asperity contact pressure pc. Thus, by considering Coulomb's friction law, FFfn= µ , it can be seen that the frictional force required to sustain high traction can be effectively reduced, without altering the

value of the coefficient of static friction, µs. An equivalent coefficient of friction µe can be established by using the proportion of the computed contact force to belt-wrap force [44],

RL µµ= gy p dA (4.2.1) esTA ∫ c c Ac

where Ac is the apparent contact area. In this work we compute the µµes/ ratio, where the values of 1 and 0 indicate no- loss and full-loss of traction, respectively.

Figure 4.1 A schematic of the tape and guide.

4.2 Mathematical Model of the Tape-Roller Interactions with Air Lubrication

Tape is modeled as a tensioned, translating shell (Figure 4.1) and effects of air lubrication are modeled by using the modified, version of the incompressible Reynolds equation. In addition, tape-to-guide contact is modeled by using a non-linear contact pressure function. Both governing equations are expressed in their transient forms, and coupling is achieved by a fully transient coupling algorithm [38]. The governing equations are summarized below.

Equation of motion of a tensioned, translating shell given by equation (2.1.34) is used model the out-of-plane tape dynamics [51],

∂2w ∂∂ 22 w w Tw∂2 T ρ 24+ + +∇ + − =+ − AtVV222 t D w Dws 2 p pc (4.2.2) ∂x ∂∂ xt ∂ t LYY∂ x LR

where w is the out-of-plane tape deflection, x and y are the longitudinal and lateral coordinate directions, respectively (Figure 4.1) and t is time. Note that the coordinate axes are fixed on the

tape. Lx and Ly are the tape’s length and the width, respectively. The inertial effects are

represented by the mass density per unit area ρA of the tape and acceleration computed by using

4 material time derivative, where Vt is the tape transport velocity along the x-axis. ∇ represents the bi-harmonic derivative operator. The bending rigidity of the tape is D= Ec3212(1 −ν ) and

22 the shell stiffness of the tape is Ds = Ec R (1 −ν ) with elastic modulus E, Poisson’s ratio ν and thickness c of the tape. Tape radius is defined as follows,

Rx()= Rg ( Hx ( −− L12 ) Hx ( − L )) (4.2.3)

where Rg is the roller radius, H is the Heaviside step function, and L1 and L2 are the tangency

points of the tape (Figure 4.1). The contact pressure pc is given as follows [43, 88],

2 h (4.2.4) pco=− P1 (1( −−Hh σ )) σ

where Po is the is the asperity stiffness and σ is the asperity engagement height. The tape-to- roller clearance, h, is found as follows,

hw= +δ (4.2.5)

The guide shape, δ, is defined in the tape based (x, y) coordinate system. The following separation is possible, because the grooves are assumed to be oriented circumferentially,

δ(,xy )= δδxy () x + () y (4.2.6)

Figure 4.2 Guide shape δ y ( y) .

The functions δx(x) and δy(y) describe the guide shape variations in the running and lateral

directions, respectively. In particular, the groove shapes are defined by δy(y) as shown in Figure

4.2, and characterized by the variables; groove width, LG and land width, LL. The groove depth dg is assumed to be sufficiently deep, O(mm’s), therefore it is not a factor in this work.

In general, the lateral edges of the tape are free and if the analysis includes the entire width of the

tape we need to impose the free edge boundary conditions where the bending moment My and the

shear force resultant Vy vanish,

Myy= V =0 at 0 ≤≤ xLx and y = 0, Ly (4.2.7)

On the other hand if we only consider a section of the tape around a land, with a repeating pattern in the lateral direction, then the lateral edges of the tape should be modeled with symmetry boundary conditions. In this case the slope and the shear force resultant should vanish,

∂w =V =0 at 0 ≤≤ xL and y = 0, L (4.2.8) ∂y y xy

The boundary conditions along the longitudinal edges of the tape are the same in both of the cases considered by the above two scenarios. The simply supported edge conditions are given as follows,

wM=xxy =0 at x = 0, L and 0 ≤≤ yL (4.2.9)

where Mx is the bending moment resultant acting normal to the simply supported edges.

Air pressure, p, is modeled by using the compressible Reynolds equation, with slip flow corrections (equation (2.4.17)) given as follows,

∂∂33p λλaa ∂∂p ∂ph ∂ ph hp 16+ +hp 16 + = 6µ  2 ++(VVtr) (4.2.10) ∂∂x x hy ∂∂ y  h∂ t ∂ x

where λa is the molecular mean free path, μ is the viscosity of air and Vr is the roller velocity. As the grooves are deep in the application of interest, the air pressure is assumed to be ambient inside the grooves. Therefore, ambient pressure boundary condition is assigned around the periphery of the lands and along the outer edges of the tape.

Equations (4.2.2) and the associated boundary conditions are discretized in space by finite difference approximations and in time by using an explicit version of the alpha-method of time integration [51]. Equation (4.2.10) is solved by the alternating direction implicit (ADI) algorithm as described in reference [51]. The dependent variables w, p and pc are found by the fully transient, coupled solution of equations (4.2.2)–(4.2.9) [51]. The same mesh was used for the solution of the tape deflections and air pressure.

4.3 Contact Analysis for a Full Roller with Air-lubrication Effects

The traction state of a roller-tape interface is evaluated by using the parameters of a typical data tape recording application. The parameters reported in Table 4.1 were used in this work.

Figure 4.3 shows the air and contact pressure distributions in the tape roller interface for the different groove designs, 0.7 N tape tension and 6 m/s tape speed. The corresponding lateral (y-) direction cross-sections of the tape deflection, air and contact pressure variations are presented in

Figure 4.4. These cross-sections are taken at the center of wrap. In these graphs the air and

contact pressure values are normalized with respect to the belt wrap pressure T/Rg Ly. This normalization allows us to evaluate the interface contact stress and air pressure conditions with respect to a common reference. In fact, in the classical foil bearing, where the interface is without any grooves, the air pressure should be equal to the belt wrap pressure.

As expected, it is observed that air pressure develops to super-ambient values over the lands, and drops to ambient value along the edges of the lands. The air pressure profiles in the first and the last lands in the axial (y-) direction are slightly different than those on the inner lands, because the land widths are different on the two lateral edges of the roller. But, the solution is symmetrical otherwise. Along the edges of the lands the air pressure is insufficient to prevent contact, and substantial contact pressures are predicted.

In these three cases the air pressure in the center of the wrap is not necessarily equal to the belt- wrap pressure. For the case of (LL, LG) = (0.3, 0.5) mm the air pressure is ~80% of the belt-wrap- pressure, as a result the highest contact pressure values, that are more than 10 times the belt-wrap pressure develop along the edges of the grooves. For the case of (LL, LG) = (0.5, 0.3) mm the air pressure is approximately equal to the belt-wrap pressure at the center of wrap, and for the case

of (LL, LG) = (0.25, 0.25) the air pressure is ~20% higher than the belt-wrap pressure. The edge contact pressures scale accordingly as shown in

Figure 4.4. An in depth analysis of the effects of groove and land widths on the mechanics of the problem is presented in the next Chapter.

According to the theory of air lubrication, as air enters the shell, since the space is limited due to the belt wrap pressure, air pressure starts to develop. At the end of contact region (exit region), due to the pressure difference between contact region and atmospheric pressure, air pressure subsequently first increases and decreases at the exit region due to the pressure difference between the atmospheric and interfacial pressures.

Table 4.1 Parameters used in the study presented in Section 4.3.

Tape parameters E (GPa) 5 ν 0.3 ρ (kg/m3) 1,400 c (µm) 7

Ly (mm) 12.7

Lx (mm) 7.1 T (N) 0.5, 0.7, 0.9 (or as specified) Air lubrication parameters µ (Pa*s) 1.81x10-5 -8 λa (m) 6.35x10 Pa (kPa) 101.3 Contact parameters σ (nm) 90

Po (Pa) 100 Roller geometry

Rg (mm) 7.5

2LL (µm) 250, 300, 600 (or as specified)

2LG (µm) 250, 300, 500 Numerical parameters Δx (µm) 31.8 Δy (µm) 31.8 Δt (ns) 5

a) b)

c) d)

e) f)

Figure 4.3 Air pressure and contact pressure distributions for three different groove types: (LL,

LG) a,b) (0.3, 0.5) mm, c,d) (0.5, 0.3) mm, e,f) (0.25, 0.25) mm. Tape running direction is along the + x-axis. Tape velocity Vt = 6 m/s and tape tension Tx = 0.7 N. Other variables are reported in the text.

c) Figure 4.4 Lateral (y-)direction cross-sections of the tape deflection, air pressure and contact pressure distributions obtained at the center of wrap, for three different groove types: (LL, LG) a,b) (0.3, 0.5) mm, c,d) (0.5, 0.3) mm, e,f) (0.25, 0.25) mm. Tape velocity Vt = 6 m/s and tape

tension Tx = 0.7 N. Other variables are reported in the text.

4.4 Contact analysis for a single land with air-lubrication effects

In the previous section we simulated the entire width of the grooved roller interface and included all of the grooves in the analysis. While this practice gives valuable information, the computations are time consuming. A close look at the results presented in the previous section would reveal that, except for the two outermost lands, the pressure profiles in the central section of the roller are very similar, if not identical. Therefore, most of the analysis presented in this thesis is carried out based on analyzing a single land. The boundary conditions that enable this analysis are given in Section 4.2. The “zoom-in” provided by this approach reveals important information related to the coupling between the tape deflection, air pressure and contact pressure at different design and operation parameters. The effects of land and groove widths are investigated in Figure 4.5 - Figure 4.7, for

Vt = 0.5, 2 and 6 m/s and T = 0.5 N. The remaining parameters of these cases are given in Table 4.2. These parameters will also be used in the next chapter. Results are shown by looking at tape-to-land clearance h, contact and air pressure distributions over a single land, at steady state. In Figure 4.5, where (2LL, 2LG) = (50, 50) µm, air lubrication has very small effect in the interface and full contact is predicted even at 6 m/s tape velocity (Figure 4.5c). In this case, we conclude that the tape mechanics in the static conditions, where the bending moment over the grooves controls the deflection, remains dominant even at fast transport velocities. No appreciable air pressure develops, and this leaves the tape sitting on the lands.

In Figure 4.6, the groove width 2LG is 450 µm, while the land width 2LL is still 50 µm. The air pressure rises near the entrance of the wrap but collapses due to side flow of air. Thus, it is clear that in this case too, the clearance is dominated by the static bending conditions as in the previous case. Nevertheless, it is worth noting that the contact is made only along the edges of the land, whereas in the previous case contact was made over the entire land surface.

However, in the case of Figure 4.7 where (2LL, 2LG) = (600, 50) m, air pressure is evenly distributed in most of the interface, and its amplitude increases gradually휇 with tape speed from Vt = 0.5 to 6 m/s. For this case, the analytical analyses show that tape is expected to contact the middle of the land (CS-1). This supports the assertion that if tape makes contact with the land at zero tape speed, the coupling between the tape and the air lubrication is stronger, resulting in

lifting of the tape and widening of the tape-land clearance as the speed of the tape increases. As a result, this trend has a strong adverse influence on the traction capacity of tape on the roller.

Figure 4.6 and Figure 4.7 indicate that regardless of the of coupling level between the tape deflection and air pressure, increasing tape speed causes the contact pressure to decrease. In case

of Figure 4.6, where (2LL, 2LG) = (50, 450) µm non-negligible contact pressure along the edges of the lands remain even at 6 m/s tape speed. On the other hand, in Figure 4.7, contact pressure

magnitude is reduced very drastically with speed such that all contact is lost at Vt = 6 m/s. This trend has a strong influence on the traction capacity.

Table 4.2 Parameters used in Section 4.4.

Tape parameters E (GPa) 9 ν 0.3 ρ (kg/m3) 3000 c (µm) 6.4

Ly (mm) 12.7

Lx (mm) 7.23 T (N) 0.3, 0.5, 0.7 (or as specified) Air lubrication parameters µ (Pa*s) 1.81x10-5 -8 λa (m) 6.35x10 Pa (kPa) 101.3 Contact parameters σ (nm) 90

Po (Pa) 100 Roller geometry

Rg (mm) 7.615

2LL (µm) 50, 300, 600 (or as specified)

2LG (µm) 50, 300, 600 (or as specified) Numerical parameters Δx (µm) 9.028 Δy (µm) 9.028 Δt (ns) 5

a) b)

Figure 4.5 Tape-land clearance h, contact pressure pc and air pressure p variations for tension T =

0.5 N, land width 2LL =50 m and groove width 2LG = 50 m, tape speeds a) 0.5 m/s, b) 2 m/s,

휇 c) 6 m/s. 휇

a) b)

Figure 4.6 Tape-land clearance h, contact pressure pc and air pressure p variations for tension T =

0.5 N, land width 2LL =50 m and groove width 2LG = 450 m, tape speeds a) 0.5 m/s, b) 2 m/s,

휇 c) 6 m/s. 휇

a) b)

Figure 4.7 Tape-land clearance h, contact pressure pc and air pressure p variations for tension T =

0.5 N, land width 2LL =600 m and groove width 2LG = 50 m, tape speeds a) 0.5 m/s, b) 2 m/s,

휇 c) 6 m/s. 휇

4.5 Summary and Conclusions

In this chapter, contact of a tensioned flexible tape with a circumferentially grooved cylindrical roller is investigated with two different approaches. In the first approach, we look at the tape on the whole width scale. This approach helps us to visualize the shape on the grooved roller during reading and recording process. However, on the second approach, we model only a portion of the width, i.e. two half grooves and one land. Since the tape takes a symmetrical shape on every two half groove-one full land pair, tape is assumed to be symmetrical at the edges of both half lands. This approach is very advantageous since it takes much shorter time to analyze a smaller piece of tape instead of the whole width. Nevertheless, first approach is also useful to investigate the effects at the outermost locations of the lands. In the next chapter, air lubrication between a tape and a grooved roller will be investigated on various land and groove width combinations using symmetrical analysis approach.

5 Mechanics of Tape Grooved Roller Interface with Air Lubrication Effects: Part-2

5.1 Introduction

Cylindrical rollers are often used as guiding and supporting elements in tape recording and web- handling applications. It is crucial for the tape1 to remain in contact with its supporting rollers in order for it to follow a designated path. Traction capability of a roller is compromised when air creates a lubrication layer in the tape-roller interface [44]. The no-slip condition between air and a translating tape causes air to enter and lubricate tape-to-guide interface. Surface of a spinning roller exacerbates this problem. It has been shown experimentally and theoretically that the air pressure that develops under the tape is closely coupled to tape deflections. Mechanics of the foil bearing problem has been investigated analytically [25, 27] and numerically [42, 45, 49, 51, 60, 73, 89].

Under negligible air lubrication, tape-to-roller contact pressure, pc, is equal to the belt-wrap

pressure, T/RgLy, where T is tape tension, Ly is tape width and Rg is roller radius. It is generally understood that due to rough nature of the surfaces, contact takes place on the summits of the asperities, leaving some open volume in which air can flow. Air pressure p increases with increasing tape speed and eventually provides another mechanism to support the belt-wrap pressure. Combined effects of air and contact pressures at sufficiently high tape velocities balance the belt-wrap pressure. Rice et al. investigated web-to-roller contact in view of air entrainment, experimentally and numerically for non-grooved rollers [44]. Müftü and Jagodnik presented a non-dimensional model of traction between a web and a non-grooved roller [60].

Assuming that transfer of shear stress between the tape and the roller, or traction, is governed by Coulomb's friction law, it can easily be seen that increasing air pressure would reduce the amount of contact pressure required to provide balance the belt wrap pressure. Consequently, the frictional force required to sustain high traction can be effectively reduced, without altering the

value of the coefficient of static friction, µs. An effective static friction coefficient µe is defined by using the proportion of the computed contact force to belt-wrap force [44], RL µµ= gy p dA (5.1.1) esTA ∫ c c Ac

where Ac is the apparent contact area. Effective friction coefficient is an indication of how much shear stress can be transferred between the two surfaces without inducing slip or traction-loss. In this work we compute the μe/μs ratio, where the values of 1 and 0 indicate no-loss and full-loss of traction, respectively.

Often grooves are manufactured on the roller surface in order to reduce the air lubrication effects [43, 86, 87, 90]. Ducotey and Good numerically studied optimal groove geometry for grooved rollers [43]. They focused on rollers with circumferentially oriented grooves with rectangular cross-sections. They showed that contact becomes more reliable, and traction loss is reduced with increasing groove pitch and groove depth. Tran et al. presented a numerical model to analyze the fluid-structure interactions between a web and a spirally-grooved roller [73]. Their study included the dynamics effects that occur on spirally grooved rollers, and experiments confirmed their model. Rice and Gans presented a model where they considered the loss of wrap due to the air entrainment [44, 87, 90]. They assumed that the web is rigid in the lateral direction, and has zero bending stiffness. They developed a closed form relationship for traction loss as a function of roller radius, land and groove widths, static friction coefficient, upstream tape tension, tape speed and surface asperity height. They carried out traction experiments with different grooved roller shapes. Their study is applicable to relatively thick webs, and does not consider the edge contact pressure that develops along the edges of the lands.

Traction characteristics of a thin tape, used in modern tape drives, over a grooved roller are investigated in this paper. An experimental technique to measure the tape to land spacing in a commercially available tape-drive is reported. This spacing is measured for different tape tension and velocity values. The results are compared to numerical computations. Different grooved roller designs for magnetic tape drives are investigated systematically. The analysis for the static contact conditions, presented in Chapter 3, is briefly reviewed for the physical parameters of the application. The effects of air lubrication on contact conditions are then added to analysis. Mechanics of the translating tape wrapped around a grooved roller is then investigated/explained

in detail. Finally, traction curves for different design parameters and design recommendations are presented.

5.2 Theory

Tape is modeled as a tensioned, translating shell while effects of air lubrication are modeled by using the Reynolds equation. Tape-to-roller contact is modeled by using a non-linear contact pressure function. Both governing equations are expressed in their transient forms, and solution is obtained by using a fully transient coupling algorithm [87]. Equation of motion of a tensioned, translating shell is given as follows [24, 32, 43],

∂2w ∂∂ 22 ww TT∂2w ρ 24+ + +∇ + −xx =+ − AtVV222 t D w Dws 2 p pc (5.1.2) ∂x ∂∂ x t ∂ t Lyy∂ x RL

where w is the out-of-plane deflection, ρA is mass density per unit area, Tx is tension, Vt is the tape transport velocity, and x and y are the longitudinal and lateral coordinate directions,

respectively (Figure 5.1), and t is time. Lx is tape’s length between the supports, and Ly is its width.∇4 represents the bi-harmonic derivative operator. The bending stiffness

32 22 D= Ec /12(1 −ν ) and the shell stiffness Ds = Ec/1 R ( −ν ) depend on the elastic modulus E,

Poisson’s ratio ν and thickness c of the tape. Tape radius is a discontinuous function that is non- zero only in the wrap region. This is defined as follows,

Rx()= Rg [ Hx ( −− L12 ) Hx ( − L )] (5.1.3)

x = L1 x = L2 θw Rg Vt

z x

x = 0 x = Lx

(a) (b)

(c) (d) Figure 5.1 Schematic depiction of a grooved roller and the details of the area considered in analysis. where Rg is the roller radius, H is the Heaviside step function, and L1 and L2 are the tangency points of the tape (Figure 5.1). The contact pressure pc is given as follows [49, 51],

2 h pco= P1 − (1( −−Hh σ )) σ (5.1.4) where Po is the is asperity compliance and σ is asperity engagement height. The tape-to-roller clearance, h, is found as follows,

hw= +δ (5.1.5) where guide shape, δ, is defined by superposition of two functions as follows,

δ( xy, ) = δδxy( x) + () y (5.1.6)

The functions δx(x) and δy(y) describe the guide shape variations in the running and lateral directions, respectively. The groove shapes are introduced by δy(y) as shown in Figure 5.3, and characterized by the variables; groove width, 2LG, and land width, 2LL. The grooves are assumed to be sufficiently deep to validate the assumption of ambient pressure over them.

As shown in Chapter 4, it is sufficient to consider the mechanics of the tape over a single land in order to investigate the salient characteristics of tape land interactions. Symmetry conditions around a single land are used along the lateral edges of the tape, dw =V =0 along 0 ≤≤xL , and y =±( L + L) (5.1.7) dy y x GL where Vy is the shear force resultant acting along the symmetry line. Longitudinal edges of the tape are simply supported,

wM=x =0 at x = 0, Lx and along −( LLGL +) ≤≤ yLL( GL + ) (5.1.8) where Mx is the bending moment resultant along the far end “supports”.

Air pressure, p, is generated by air lubrication which is governed by the compressible Reynolds lubrication equation, with slip flow corrections given as follows [51],

∂∂33p λλaa ∂∂p ∂ph ∂ ph hp 16+ +hp 16 + = 6µ  2 ++(VVtr) (5.1.9) ∂∂ ∂∂  ∂ ∂ x x hy y  h t x where λ is the molecular mean free path, μ is the viscosity of air. In this work the air pressure is a computed only over the lands. Therefore, the air pressure is ambient around the periphery of the lands and along the outer edges of the tape.

The dependent variables w, p and pc are found by the coupled solution of equations (2) – (8) by a transient coupling algorithm as described in [51].

Figure 5.2 Description of the three possible contact states when the tape is “pushed” against the grooved roller by the belt-wrap pressure.

As demonstrated later in this chapter, it is instructive to separate the effects of contact and air lubrication. An analytical solution to predict the contact states of a tape wrapped around a grooved roller when the tape is tensioned but stationary was presented in Chapter 3. This is briefly reviewed here for the sake of completeness. In that work, the following assumptions were made: i) the contact-surface of the roller consists of circumferentially oriented lands separated by grooves; ii) lands are flat and parallel to the roller axis; iii) tape is wrapped around the roller with a sufficiently large wrap-angle. Tape-to-roller interaction in this case reduces to analyzing the contact mechanics of a single groove-land pair as described in Figure 5.1. With the assumptions listed above, tape deflections can be assumed to be independent of the circumferential direction and equation (5.1.2) becomes, dw4 T +=−x D4 Dwsc p . (5.1.10) dx RLgy

Shell stiffness Ds and belt-wrap pressure (Tx/RgLy) originate from the hoop strain (w/Rg) and initial tension in the material, respectively [84]. While the belt-wrap pressure provides a constant pull over the cylindrical guide, the hoop strain provides a resistance that is proportional to deflection, w. Thus in equation (5.1.10) the effect of hoop strain appears similar to a linear- elastic foundation force.

In Chapter 3, we showed that three distinct contact states, depicted in Figure 2, are possible depending on the parameters of the problem [91]. In all of these states the belt wrap pressure pushes the tape to bend into the grooves, and the tape deflects away from the land. In contact state-1 (CS) tape makes contact only along the edge (y = LL) of the land. In contact state-2 (CS- 2) tape makes contact along the edge and the center (y = 0) of the land. In contact state-3 (CS) the central contact on the lands spreads to a finite contact area of width a, while edge contact is maintained. Tape deflection profile over the land and the groove are obtained from equation (5.1.10), with the appropriate boundary conditions, presented in Chapter 3 and reference [91].

5.3 Experiments

In order to bring some level of verification for the mathematical model, experiments were conducted to measure the tape-to-land spacing h. These experiments were carried out on a commercially available high capacity tape-drive (Quantum LTO). The drive was modified, as shown in Figure 5.3, in order to accommodate an extra grooved roller on which the measurements were taken. A grooved, flangeless roller with diameter (2Rg) of 15.23 mm was

used in the experiments. The groove (2LG) and land widths (2LL) of the rollers were 245 and 137 µm, respectively. This roller was placed between a smooth flanged and a smooth unflanged roller on the tape path (Figure 5.3). Thus, the rough side of the tape was wrapped around the experimental roller by about 105 degrees. A 6.4 µm thick, 12.7 mm wide LTO-6 tape was used during the tests.

In these experiments two types of data were collected as shown in the schematic provided in Figure 5.5. A FotonicTM sensor (MTI Instruments – 2032RX – 1544-11) was aimed at the front coat of the tape to measure the tape-to-land spacing h at the center of the wrap. In addition, a laser sensor was used to measure the spin rate of the roller. This sensor was mounted parallel to the axis of the roller, and was aimed on the edge of the rim of the roller. By monitoring the once around frequency of a marker placed on the edge of the rim of the roller, this “spin sensor” measured the rolling velocity of roller (Figure 5.5). These measurements were used to monitor presence or lack of slip between the tape and the roller.

The signals from the two sensors were collected in synchronized manner. The flying height daa is collected using a The FotonicTM sensor at the rate of 10 kHz per second. The high data collection rate caused some noise in the data, thus a 2nd order ‘Tustin’ approximation is used to design a low pass filter. This filtering method mainly ensures that there is a good domain match between continuous-time and discretized system. The optimized cutout frequency for this filter is set at 150 Hz.

The grooves on the roller had a small helical orientation (0.9 degree) with respect to the circumferential direction. This orientation turned out to be very helpful in measuring the tape-to- land spacing. As the roller spins during tape transport, the FotonicTM sensor which is fixed in space, monitors both the lands and the grooves passing by underneath it. This would not have

been possible, if the lands were oriented purely circumferentially. Thus the FotonicTM sensor, which has a relatively large measurement area that is O(0.5 mm) collects the tape deflection in and out of the grooves as a continuous function of time. Figure 5.6a shows the original and filtered FotonicTM sensor signal for five full rotations. The lowest point of the filtered data is indicative of the tape deflection into the grooves, and the highest point is indicative of the tape deflection away from the lands. At the same time the spin sensor measures the once around of the roller (Figure 5.6b). The maximum clearance detected by the FotonicTM sensor, between two consecutive spin sensor spikes, is taken as the tape-to-land clearance. The reference flying height was set to value measured at the first full spin of the roller, where the roller speed (0.6 – 0.9 m/s) is considerably slower than the full speed (2 – 8 m/s) of the roller. In order to improve the statistics, each reported data point was obtained by averaging 200 roller spins at the full speed. Experiments were conducted with the tape tension values in the range of 0.3 – 0.9 N with 0.1 N increments, and tape speed values in the range of 2 – 8 m/s with 2 m/s increments. Tape-to-roller clearance is reported as functions of these parameters in Figure 5.7.

Displacement sensor Experimental roller

Figure 5.3 Test setup showing a flanged roller and the position of the displacement sensor. Note that the spin-sensor is not shown in this figure.

Figure 5.4 Close up SEM picture of grooved roller surface. Note that measurements with laser and Fotonic sensor are taken on this roller

Figure 5.5 Tape roller and sensors configuration on the tape path

a) Photonic Sensor Before and After Filtering T=600nM

N) 휇 Flying ( Height

Figure 5.6 Fotonic Sensor(top) and laser sensor (bottom) data. On top graph, gray lines show the raw data where blue line shows the cleaned data

Table 5.1 Parameters used in this chapter.

Tape parameters E (GPa) 9 ν 0.3 ρ (kg/m3) 3000 c (µm) 6.4

Ly (mm) 12.7

Lx (mm) 7.23 T (N) 0.3, 0.5, 0.7 (or as specified) Air lubrication parameters µ (Pa*s) 1.81x10-5 -8 λa (m) 6.35x10 Pa (kPa) 101.3 Contact parameters σ (nm) 90

Po (Pa) 100 Roller geometry

Rg (mm) 7.615

2LL (µm) 150, 300, 600 (or as specified)

2LG (µm) 150, 300, 600 (or as specified) Numerical parameters Δx (µm) 9.028 Δy (µm) 9.028 Δt (ns) 5

5.4 Results and Discussion

5.4.1 Experimental results and comparison with theory Figure 5.7 shows the comparison of experimental and numerical tape-to-land clearance variation as a function of tape tension and transport velocity. In order to present the results clearly, the data is separated into two groups. Figure 5.7a shows the results for tension values of 0.3, 0.5, 0.7 and 0.9 N, and Figure 5.7b for 0.4, 0.6 and 0.8 N. Experimental data shows that the flying height increases with increasing velocity. This is expected since the air entrainment between tape and roller increases with increasing speed. Similar trend is shown with the simulation results. Simulation and experimental results fall within a reasonable range of one another. The experimental results also show that tape-to-roller clearance increases with tension. Based on previous work on the foil bearing problem it is well known that the clearance should be 2/3 proportional to tape velocity and inversely proportional to tension as h = 0.643R(6µV Ly/T) [24, 32]. Therefore, while the dependence of tape-to-roller spacing on the tape velocity, as described above, is expected, the reverse trend in tape tension is at first surprising. On the other hand the simulation results also show the same trend as the experiments. This unusual and unexpected dependence on tape tension is attributed to the presence of grooves on the rollers and is explained in the rest of this chapter.

Figure 5.7 Comparison of numerical and experimental flying height values for tension values of a) T= 0.3, 0.5, 0.7 and 0.9 N analyses and b) T= 0.4, 0.6 and 0.8 N.

5.4.2 Contact mechanics of non-translating tape It was shown in Chapter 3 that the contact states described above, depend mainly on the land and groove widths; and, tension only adjusts the displacement amplitude within a contact state [91]. This behavior is attributed to the competition between the moment resultants due to belt wrap pressure acting over the land and groove regions. Figure 5.8 shows the closed form solution with experimental roller parameters (2LL = 137 m and 2LG = 245 m) where the air lubrication effects are neglected. Note that, with increasing tension휇 , the clearance휇 on the land increase which supports our explanation for experimental results presented in Chapter 4. Figure 5.9 shows the contact states for different land and groove width combinations under tension T = 0.5 N where air lubrication effects are neglected. The parameters used in these calculations are reported in Table 5.1. Each row and column correspond to a different land width and groove width values (50, 300 and 600 µm), respectively. The rows of this figure correspond to constant land widths. Increasing the groove width from 50 µm to 600 µm (left to right) increases the resultant moment from the groove region, while the resultant moment from the land region remains constant. As a result of this, the tendency of the tape to deflect away from the land surface increases.

Increasing T from 0.3 – 0.8 N

Figure 5.8 Variation of tape deflection predicted by closed form analysis for 2LL = 137 m and

2LG = 245 m for T = 0.3 – 0.8 N. 휇 휇

Figure 5.9 Variation of tape deflection and contact force predicted by closed form analysis for T = 0.5 N and different land and groove width combinations.

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5.4.3 Contact mechanics of translating tape Next, we investigate the effects of the air lubrication on the contact state of the roller geometries analyzed in the previous section. In particular, we study the effects of tape speed (0.5 – 6 m/s) and tape tension (03, 0.5, 0.7 N). The other parameters of these analyses are given in Figure 5.1. The model including the lubrication effects presented in the theory section is used in this analysis.

In Chapters 3 and 4, we demonstrated that substantial information can be gained related to the mechanics of the tape-to-land interface by simulating the tape behavior around a land. This is the

approach we take in this chapter. A sample result for the case of (2LL, 2LG) = 50, 300 µm, T =

0.5 N and Vt = 6 m/s is presented in Figure 5.10. Majority of our analysis will be based on the two cross-sectional planes shown in this figure. In order present the effects of air entrainment on the contact state, the tape deflection, contact pressure and air pressure variations are presented in the lateral (y-) and longitudinal (x-) direction cross-sections of the solution domain. Figure 5.11 shows the cross-section of the tape deflection profiles along the lateral direction, at the center of

the wrap. The speed and tension values are Vt = 0.5 – 6 m/s and T = 0.5 N, respectively. Figure 5.12 and Figure 5.13 show the corresponding air and contact pressure profiles.

The parameters in Figure 5.11 are the same as those used in the analytical results presented in

(Figure 5.9). It is observed that tape-to-land clearance increases strongly for the wide lands, 2LL

= 300 and 600 µm, but this effect is negligibly small for the narrowest land, 2LL = 50 µm. The tape deflection is insensitive to transport velocity for the combinations of land and groove width

values (2LL, 2LG) = (50, 50-600) µm. An explanation for this behavior can be seen when we note that the analysis presented in Figure 5.9 shows that this combination of land and groove widths remain in contact state-1, and the tape’s deflection is dominated by the belt wrap pressure acting over the groove. Thus even for the stationary tape (Vt = 0) a “cupped” shape appears in the tape- land interface. This built-in clearance reduces the effects of air lubrication, and the air pressure does not couple to tape deflection as in the classical foil bearing. This claim is supported in Figure 5.12 which presents the lateral cross-sections of the air pressure distributions. In particular, on the first row, where (2LL, 2LG) = (50, 50-600) µm, relatively low and non-characteristic air pressure distributions are predicted in the interface. For all other cases, the air pressure, which in turn pushes the away tape away from the land, is observed to develop.

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Figure 5.13 shows the corresponding contact pressure distributions. For narrowest land, 2LL = 50 µm, the tape contacts the land through its entire width, and this contact condition is not affected by the tape velocity.

The tape-to-land clearance, h, and air pressure, p, variations that correspond to the cases presented above, are plotted along the central cross-section in the longitudinal (x-) direction in Figure 5.14. This provides another view of the results presented above. As expected, we see that on the top row of the figure (2LL = 50 µm) the deflection does not change with increasing speed, and air presure does not fully develop, even at high velocities. The interface mechanics is dominated by the tape bending moment that forces the tape into the “cupped shape over the land”

and prevents a proper coupling with the air lubrication. In the other rows, where 2LL = 300 and 600 µm, we see that air pressure is fully developed and both deflection and the air pressure are increasing with the increasing speed. A correlation can be drawn with the propensity to allow air pressure to develop, and the contact state that the unlubricated state predicts. Comparison with Figure 5.9 shows that if the contact state is predicted one where central contact to occur (CS-3) in the static case, the interface tends to be sensitive to air entrainment, and vice versa.

(a)

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(b)

(c) Figure 5.10 Tape deflection and corresponding air and contact pressure around the land at steady state for (2LL, 2LG) = (50, 300) µm, T = 0.5N at Vt = 6 m/s.

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Figure 5.11 Lateral (y-) direction cross section of tape deflection w for different land and groove width combinations.

Figure 5.12 Lateral (y-) direction cross section of air pressure p for different land and groove width combinations.

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Figure 5.13 Lateral (y-) direction cross section of contact pressure pc for different land and groove width combinations.

Figure 5.14 Axial (x-) direction cross section of tape deflection w and air pressure p variations for tension T = 0.5 N, for different land and groove width land width.

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Traction coefficients for all cases presented above are computed as described by Equation (5.1.1) as functions of tape speed and tension. Figure 5.15 shows that generally, increasing tape speed and decreasing tape tension results in the reduction of traction coefficient as expected. Especially, tape loses traction very drastically even at lower speeds for larger land width values

(2LL = 300-600 2LG = 50). Distinct differences are noticeable in the two cases where the

deflection is governed by the tape bending (2LL, 2LG) = (50, 50-600), where relatively less

traction loss occurs with increasing speed. For the other combinations of (2LL, 2LG) traction loss with tape speed is fairly large. In particular, more traction loss is predicted for the same speed

and tension values with increasing LL and increasing LL/LG ratios. Moreover, note that for a low

land width value (LL=50), traction is maintained for all groove width LG=50-600 at all speed values. This result produces a safe zone for roller designers to maintain the traction of the tape with rollers at all values of speed and tension.

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5.5 Summary and Conclusions

Traction between a thin, tensioned tape and a grooved roller is studied numerically and experimentally. In the zero tape speed limit an analytical approach can be described to model tape contact with a grooved roller. In this range, air lubrication effects are negligible and tape-to- roller contact is dominated by tape deflection in the lateral direction. This is in contrast to previous work where tape bending in the lateral direction is neglected for much thicker webs. At operational tape transport speeds a wide range of groove width, land width; tape velocity and tension values were used to characterize the traction of a thin tape over a grooved roller. It was shown that air lubrication effects reduce the contact force; however, the underlying effects of tape mechanics are not entirely eliminated.

An experimental technique is described and used to measure the tape-to-roller spacing. The technique requires synchronized measurements of the tape flying height and roller spin. The experimental and numerical results agree fairly well. The experiments show that increasing tape tension causes the tape-to-land spacing to increase. Theory shows that increasing the tape tension causes it to bend into the grooves and away from the lands, while contact is actually being maintained along the edges of the grooves.

Maps of traction coefficient for two design (2LL, 2LG) and two operation (Vt, T) parameters are obtained. It is shown that traction loss is worse for rollers with wide lands and narrow grooves. By using the principles described in this chapter, it should be possible to design land and groove widths that are relatively insensitive to tape velocity.

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6 Summary and Conclusions

In this dissertation, we analyzed the contact mechanics of a thin flexible tape wrapped around a grooved roller. The main focus of this work was to identify the factors that influence traction in the tape-roller interface. In particular, we investigated the case of dry and lubricated contacts as functions of operational and design parameters that are relevant to current data tape recording technology. This work had experimental and theoretical components. A survey of the literature that covers the topic of web-traction and the factors that influence it, such as air lubrication, is provided in Chapter 1. The governing equations of the model are presented in Chapter 2. A detailed analysis of the dry contact problem, provided in Chapter 3, sheds light onto the effects of groove/land geometry on the tape mechanics. The problem becomes considerably more complex when the effects of air lubrication are considered. In Chapter 4, a detailed account of the interface conditions for the lubricated contacts is provided. The analysis and experimental results presented in Chapter 5 show that the roller traction is a strong function of operation parameters such as tape transport velocity and roller geometry.

6.1 Wrap Pressure

A mathematical model for contact of a stationary tape on a grooved roller is presented in Chapter 3. Dry contact of the tape with a grooved roller, where air lubrication effects are neglected, is modeled in order to investigate the effects of groove and land widths on contact conditions. This work generally shows that the belt wrap pressure causes the tape to bend into the grooves and over the lands. The moment balance about the corner of the land determines what type of contact will be established. It was shown that that there are three main contact states when tape is stationary. We find that tape only contacts the edges of the land, if the non-dimensional land

= 1/4 1/2 width LLL( 3/ L( cR) ) is less than one and the groove width to land width ration LLGL/ is

greater than one; this condition is not significantly affected with the increasing LLGL/ ratio. However, as the land width increases, the tape starts to contact the roller at the mid-land and the belt wrap pressure is balanced by both the edge and mid-contact forces.

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6.2 Traction Modelling

The main concern between the tape and guiding rollers is the traction. When the tape moves slowly over the guiding components, the movement can be analyzed using analytical solution. However, when the tape speed increases, air entrainment effects are observed and they should be included in the form of numerical analyses. It’s well known that air effects reduce the contact force and consequently traction.

6.2.1 Conclusion of Traction Modelling

In our study, we did the full width analyses for some groove and land width combinations to investigate the problem. We compared the results in terms of traction coefficient and confirmed that traction is directly proportional to tension and inversely proportional to velocity. Also, traction decreased for an increasing land width value.

6.3 Air Lubrication Effects on Contact of Tape with Grooved Roller

In last part of our work, we analyzed the air lubrication effects on various combinations of groove and land width numerically. From our full width analyses, we were fully aware that the tape was bent into the grooves. Furthermore, to validate our results experimentally, we presented an experimental technique to measure the clearance between the tape and a grooved roller. The measurements were carried on grooved flangeless rollers, which showed no sign of slipping. Our experimental results show that clearance also increases with increasing tape tension and velocity. This might seem controversial to the theory where clearance is expected to decrease with increasing tension.

As a conclusion, we assessed the traction between a tape and grooved rollers with various parameters.

6.3.1 Conclusion of Air Lubrication Effects on Contact of Tape with Grooved Roller

In Chapter 5, we investigated the air lubrication effects in the tape-grooved roller interface. In particular, we focused on the effects of various combinations of groove and land widths in the range of common operational parameters. We confirmed the assumption made in Chapter3, that at very slow speed, tape bending dominates over air lubrication, which has a small effect. We

109 also showed that, as expected, air lubrication effects increase with speed. The effects of air entrainment depend on the land and groove width values. We find that for narrow lands, while some air entrainment effects can be seen, the amount of tape-lift-off may not be significant. For wider lands, air entrainment has significant effects on tape mechanics. As a general rule of thumb, we conclude that if the tape contacts the middle of the land when it is stationary it is likely to lose traction at high speeds.

The effect of air entrainment on traction was investigated numerically by testing a wide range of land and groove width values. In particular, LLLG/ ratios were investigated in the range of 1/12 to 12, systematically. This showed that traction is best maintained for all speeds over rollers that have narrow land width and wide grooves, as this with combination the tape mechanics is dominated by the belt-wrap pressure rather than air lubrication. By presenting the traction graphs for all cases, we presented a full design methodology for the industry. With these analyses, first, we are able to comment on how the land and groove parameters of the rollers should be designed and secondly, we are able to comment on at what tension and speed tape should be operated to achieve the best traction between tape and the guiding elements. This work should help design of more robust grooved rollers in the next generation tape drives.

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7 Future Work

In the future of data tape technology will involve thinner and smoother tape moving with faster transport speeds. The tape guiding elements, such as rollers, should also be modified to enable successful operation under these conditions.

In Chapter 5, the effects of different grooved roller designs on traction were investigated. This chapter involved ideal land and groove properties. However, due to the nature of the manufacturing of the grooves, some slight differences on the elevation of the lands exist. Future research should also include the waviness of the land profiles across the roller. This effect is usually neglected as the total tape to roller contact pressure remains the same. However, the local contact pressure on each land is different, creating the possibility of local slip.

In Chapter 3 a linear model for the tape displacement was presented. However, preliminary experimental evidence [92] suggests that tape deflection into the grooves is on the order of tape thickness or more. A large deflection small strain shell model should be implemented in order to improve the simulation outcome.

Another effect that was neglected in our research is the tape’s natural curl. Tape has a natural curl in the lateral (y-)direction possibly due to its composite layered manufacturing. This causes tape to have a natural cupped shape away from the roller, and this effect is visible under low tension. In the future studies, this effect should be included as an extra moment.

While in this work we investigated the factors that influence traction on grooved rollers, it would be interesting to investigate how the use of grooved rollers actually prevents the LTM, beyond improved traction. Factors to consider to this end are the effects of tape’s bending into the grooves, the effects of the small helix angle of the grooves, the effects of rounded edges of the lands.

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APPENDIX In order to demonstrate that the tape/web deflection is only linearly proportional to the parameter

Γ the coefficients of equation (7) for contact state-1 are presented below. The coefficients are found by using the boundary conditions in equations (8, 9). The contact state-2 admits a similar linear solution whose coefficients are omitted from this paper.

(1) (1) ΓΓLL(1) (1) AA=−=−+f feLL, AA= = f+ fe 1 3 44ff( 34) 24 ( 56) 00 (a-d) (A1)

G + 2(LL+ ) − (2) = 24LLGL ++GL LL A1 (e f7 e f89 fe) (a) 4 f0

G + 2(LL+ ) − (2) = 24LLGL +−GL LL A2 (e f10 e f11 fe 12 ) (b) 4 f0

G 2(L +L ) + (2) = −−2LL − L G 2LLGL A3 ( f10 ef11 e fe12 ) (c) 4 f0

G 2(LL+ ) + (2) = ++2LL GL 2LLG L A4 ( f7 efe8 fe12 ) (d) 4 f0 (A2) where,

42(LL++) (LL) = GL− + GL + ++ f0 ( e1) 2 e sin( 2( LLGL)) ff12

=2LG  +44LLLL−− fe1 ( 1sin21cos2 e) ( LGG) ( e) ( L)

=2LL  +44LLGG−− fe2 (1 e)sin( 2 LLL) ( 1 e) cos( 2 L)

42LLGL++42LLGL f3 =−+( e1)sin ( LeLL) +−( 1) cos( L)

42LLGL++42LLGL f5 = ( e+ 1) sin ( LeLL) +−( 1) cos( L)

= − 2LG + 22LLLL+− − + fe4 (1 e)sin( 2LLGL) ( e1) cos( 2 LLGL)

2LG 22LLLL f6 = e(1 + e)sin( 2 LLGL ++) ( e −1c) os( 2 LLGL +)  (A3)

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f7 =sin( 2 LLGL +−) cos( 2 LL GL +) f8 =sin23cos23( LLGL +−) ( LL GL +)

22LLLL f9 =−−(1 eLe) sin( LL) ++( 1) cos( L) f10 =sin( 2 LLGL ++) cos( 2 LL GL +) f11 =sin23cos23( LLGL ++) ( LL GL +) feLe=−122LLLL sin( ) ++ 1 cos( L) 12 ( ) LL( ) (A4)

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