The semianalytic foil-bearing solution algorithm of Eshel and Elrod (1965) is extended to the solution of the linearized, free vibration problem for one-dimensional self-pressurized foil bearings. The results demonstrate that unwanted variations in the spacing between the moving foil and the stationary bearing surface can be eliminated through proper design. The penetration depth through which vibration of the free span penetrates into the foil bearing is determined by two exponential exponents, one describing inlet penetration, the other describing outlet penetration. When the inlet exponent is large and negative and the outlet exponent is large and positive, there is negligible coupling between the vibration of the free spans and the vibration of the spacing between the foil and the stationary bearing surface. This decoupling is desirable in magnetic recording and web handling applications and can be achieved by properly selecting two dimensionless parameters, one describing the ratio of the viscous forces to the tape tension, the other describing the ratio of the tape transport speed to the wave speed in the tape. The values of these two parameters in current designs of both magnetic tape recording and web-handling devices are consistent with the design goal of minimizing foil vibration over the bearing. The inlet and outlet exponents are the roots of a fourth-order polynomial, and, in most cases, good estimates for these roots can be found without explicitly solving the foil-bearing problem. The effects of the air compressibility, tape bending stiffness, and slip flow are also investigated. Tape bending stiffness is found to play a significant role in vibration coupling. These results provide new insight into the influence of vibration on foil-bearing design.

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