Tape Mechanics over a Flat Recording Head under Uniform Pull-down Pressure

The mechanics of the "tape" over a flat-head is investigated. It has been recently shown that a thin, flexible tape, traveling over a flat recording head under tension contacts the head over the central area of the head. This phenomenon is due to the selfacting, subambient foil bearing effect. In this paper, as a first order of approximation, a simplified system is analyzed; where a uniform subambient pressure * p is assumed to be acting on the tape, over the head region. Increasing subambient pressure values represent faster tape speeds. This approach enables an independent investigation of the tape mechanics alone, and provides a closed-form solution. The tape is modeled as a tensioned infinitely wide plate, travelling at steady state. This non-dimensional solution relates the tape displacements and the reaction forces to the problem parameters, i.e., the wrap angle, tape tension, bending rigidity, head-length and external pressure. Tape and head-wear at the corners of the head, and wear of the magnetically active regions located at the central part of the flat-head are critical issues to be considered in designing a flat-head/tape interface. The closed-form solution is particularly useful in obtaining estimates of the magnitudes of the reaction forces at the corners. Tape mechanics over a flat recording head under uniform pull-down pressure 2 Nomenclature w Tape displacement w * / w c = x Coordinate axis x * / H x L = p Magnitude of uniform pressure p ( ) ( ) ( )4 2 12 1 H p E L c υ = − * LH Head length θ * H L c θ = D Bending rigidity D = Ec/12(1-ν) ∆ H L b = T Tension per unit width b eff D T = Teff Effective tension, T-ρV 0 F 3 0 * H F L Dc = V Tape speed L F 3 * L H F L Dc = E Elastic modulus U = ULH/Teff c c Thickness ν Poisson's ratio ρ Tape mass per area H Heaviside function θ Wrap angle 0 * F Edge reaction force/width * L F Inner contact reaction force/width U Strain energy density Tape mechanics over a flat recording head under uniform pull-down pressure 3 Introduction The foil bearing problem describes the self-acting lubrication phenomenon between a thin, flexible tape moving over a cylindrical surface under tension. Such applications are found over magnetic tape recording heads, and rollers in various web handling applications. The deflections of the tape and super-ambient air pressure that forms in the head/tape interface, due to the self-acting air bearing, are strongly coupled. The mechanics and solution of the foil bearing problem have been reported by many investigators, including the following references [1-7]. In order to minimize the signal loss, modern tape recording applications require the tape to be in contact with the recording head, during the read/write operations. However, keeping the tape in contact with the head is a challenging task with cylindrically contoured heads, in which the mostly superambient air pressure in the head-tape interface, caused by the self-acting air bearing, tends to separate the tape from the head. It has been shown that when a flat recording head is used instead of a cylindrical one, reliable contact is obtained over the central region of the flat-head [8-13]. This recent finding has sparked interest in the mechanics and tribology of such systems. Flatheads have been recently implemented in commercial tape recorders. There are several technical and commercial advantages to using flat-heads. In particular, faster tape speeds over flat-heads have been shown to provide more reliable contact, in direct contrast to tape behavior over cylindrical heads. Moreover, the flat contour is more forgiving for tapes with different thickness, enabling backward/forward compatibility of tapes. Finally, flat-heads are considerably easier to manufacture as compared to contoured heads. Tape and head-wear at the corners of the head, and wear of the magnetically active regions located at the central part of the flat-head are critical issues to be considered in designing a flat-head/tape interface. We address this issue among others in this paper. The mechanics of a tensioned tape moving over a flat-head is interesting. One explanation of this operation involves the assumption that the sharp leading edge of the recording head removes air from the surface of the magnetic medium, creating a vacuum under the tape; Then the pressure differential between the top and the bottom surfaces of the tape push it down toward the recording head surface [8,9]. A considerably more complete model of the head-tape interface, including the effects of a) the mechanics of a translating tape, b) air lubrication using Reynolds equation, and c) surface roughness, showed that the contact is provided due to the self-acting, subambient foil bearing effect [10-12]. The analysis showed that air entrained in the flat-head/tape interface forms a subambient pressure layer, because the tape wrapped over a flat surface creates a diverging channel at the leading edge. The subambient air pressure over the flat-head region eventually pulls the tape down to contact the head. The mechanics of the tape and air lubrication are strongly coupled in the subambient foil bearing. In the case of cylindrical heads, contact can be obtained by increasing the tape tension. In contrast, in the case of flat-heads, the subambient air bearing causes the desired contact. Thus the mechanics of the flat-head/tape interface is fundamentally different from that of the cylindrically contoured one. Müftü and Kaiser used a commercially available two-wavelength interferometer, to measure the head/tape spacing for a wide range of parameters [13]. Comparison of the measurements to the model results was very good. This work confirmed that the tape forms two displacement bumps near the leading and trailing edges of the flat-head, and Tape mechanics over a flat recording head under uniform pull-down pressure 4 these bumps are, in general, connected with a flat region where the tape contacts the head. It was shown that the length of the displacement bumps is the critical dependent variable, which increases at higher wrap angle and tape tension values; and decreases with increasing tape speed [13]. In this paper the mechanics of the tape deformation underlying the operation of a flat-head/tape interface is investigated analytically. As a first order of approximation, a simplified system is analyzed, where a uniform subambient pressure * p is assumed to be acting on the tape, only over the head region. Note that increasing subambient pressure * p , in general, can be thought to represent faster tape speed. Figure 1 shows the generic geometry that is considered. The problem is modeled with a constant coefficient, ordinary differential equation, and a closed-form solution is given in normalized coordinates. It is found that the tape mechanics for this problem needs to be analyzed in three cases for nocontact, point-contact and area-contact on the central part of the head, named later in the paper case-1, -2 and -3, respectively. Therefore, three different solutions are found for each case. The critical pressure values causing the transition between these cases are identified. Among others, it should be underlined that the solution provided here is particularly useful in obtaining the magnitudes of the reaction forces at the corners of the flat-head. In this work, the corners of the head are assumed to be unworn, hence the concentrated reaction forces develop on the corners. In practice, the corners will be beveled and the corner reaction will be distributed over an area. The Governing Equation Tape is modeled as an infinitely wide tensioned plate, travelling at steady state, ( ) ( ) 4 * 2 * 2 * * * 4 2 * * ( ) ( ) H d w d w D T V p H x H x L dx dx ρ − − = − − − . (1) The variables of this equation are defined in the nomenclature. The Heaviside function is defined such that ( ) 1 H x = * when 1 x ≥ * and zero, otherwise. Note that the first term on the left-hand side of this equation represents the bending stiffness, and the second term represents the in-plane tension effects. The 2 2 2 * * / V d w dx ρ term represents the centrifugal acceleration, which is the only remaining term of total acceleration at steady state [5]. A schematic of the modeled system is given in Figure 1; a uniform downward pressure * p acts on the tape over the head region, 0 * H x L ≤ ≤ ; the tape is wrapped symmetrically around the head with wrap angles θ. The following non-dimensional variables are introduced, ( ) 1 2 4 2 , , 12 1 , / * * * *