Dynamic Response of Self-Acting Foil Bearings

A new approach to the analysis of wide foil bearings is investigated. The equation of motion for a finite length of tape is coupled to the transient lubrication equation for the air film between the tape and the recording head. Compressibility and slip flow are retained in the fluid mechanics equation; flexural rigidity and high-speed dynamic effects are retained in the tape equation. The steadystate solution to the coupled equations is obtained as the limiting case of the transient initial value problem. Describing the system equations relative to the undeflected tape (as opposed to conventional foil-bearing theory, which uses the head as the reference surface) permits investigation of noncircular head geometries. In addition, wave propagation effects in the tape and the interaction of waves in the tape with the air-bearing region may be studied. Introduction Development of digital magnetic recording devices using flexible media as the recording surface is dependent on accurate positioning of the recording element relative to the medium. The demands of high-density recording with a high data transfer rate require that the separation between head and medium be very small (< 1.25 pm) while the head-to-medium velocity must be large (B2.5 m/s, in some cases). To achieve these goals with reasonable longevity of the recording medium, a hydrodynamic air bearing can be used to provide a controllable separation of the head and medium with, in the ideal case, no contact between the two. Although the configurations of flexible-media recording devices may be quite complicated, it is usually possible to formulate a system of equations that can be expected to adequately describe the elastohydrodynamic interaction of the device. Often the elasticity equations describing the motion of the medium, or the fluid mechanics equations describing the behavior of the air bearing, can be solved independently to obtain information about the system behavior. To predict the spacing between a head and a flexible medium requires the simultaneous solution of the coupled system of equations, however, and it is in this area that significant difficulties occur, because of the strong interaction of the system equations. In recent years, a relatively large amount of literature [ 1-12] has been devoted to the solution of foil-bearing problems in which the flexible medium is wrapped around a circular cylinder. The elasticity equations describing the motion of the medium and the lubrication I equation for the air film are written in a coordinate system attached to the surface of the cylinder, following the derivation given by Barlow [ 11. Thus, a tractable mathematical problem is obtained if the cross section of the cylinder be either circular or composed of circular arcs [SI. Otherwise, the mathematical difficulties inherent in the problem formulation prohibit convenient analysis. Figure 1 illustrates the geometry of an interface between head and tape typical of )-inch ( 1.27 cm) tape drives in computer systems. It is generally assumed that the width of the head and tape is sufficiently large to inhibit side flow in the air-bearing region; thus, the unknowns are defined as functions of a single spatial coordinate. The quantity that is generally of most interest in such problems is the steady-state separation between the head and the tape, dependent on the head shape and on a host of parameters describing the medium and the air film. To obtain the solution of this problem, previous investigators have derived the steady-state equations for the tape shape and ;or the air film in a coordinate system relative to the circular head, and have then solved the steady-state equations numerically to obtain the separation between head and tape. The problem formulation and solution procedure presented in this paper differ from conventional foil-bearing theory in four respects. First, the equations of motion for the tape are not written relative to the head; instead, the undeflected tape (in the absence of the head) is used as a reference surface. Second, the inertial terms in the tape equation arising from the axial motion of the tape are retained in the analysis. Third, a finite segment of the 51 3 NOVEMBER 1974 FOIL-BEARING ANALYSIS x = o x = L 1 x = L z x = L Figure 1 Geometrical aspects of the interface between head and tape in typical +inch ( 1.27 cm) tape drive. I I tape with appropriate boundary conditions is considered, rather than the usual asymptotic foil boundary conditions. Finally, all time-dependent terms are retained in both the tape and lubrication equations. The solution then consists in prescribing some arbitrary initial conditions and numerically tracking the complete dynamic transient problem to the final steady-state solution. For large penetrations of the head into the tape, the tape model, linearized on the assumption of small deflections, might be expected to predict the tape shape less accurately than does the usual foil-bearing theory. However, for tape deflections typical of tape-drive configurations, the model is expected to be adequate, and, moreover, offers the ability to analyze noncircular head geometries. In addition, we gain the ability to analyze wave-propagation effects in the axially moving tape and to analyze the interaction of waves in the tape with the air-bearing region. Problem formulation We consider a finite-length foil moving at constant velocity between two supports, as shown in Fig. l . The deflection of the foil away from its equilibrium position is denoted by u ( x , t ) . Swope and Ames [ 131 have derived the equations of motion for such a foil (disregarding flexural rigidity) and have presented numerous analytical results depicting wave-propagation effects, which will be referred to subsequently. An alternate derivation, including the effects of flexural rigidity, has been given by Mote [ 141. We wish to retain flexural rigidity in the analysis and, hence, use that equation to describe the dynamics of the moving foil: m ( u , + VU,, + V'u,J + du, + EZU,~,, Tu,, p p a , ( 1 )