Aerodynamic foil bearings represent a viable solution for supporting the rotors of small, high speed rotating machines. The lift force is created by entraining a gas film between the journal and the bearing. Minute thickness of the gas film (<20...50 µm) and high rotation speeds (>2 krpm) are needed for creating an acceptable lift force. The bearing operates with very small power loss due to the low viscosity of the gas. However, the gas film is unfavorably known for its very low damping characteristics. Therefore foil bearings have a compliant structure made of thin foils (<0.1 mm) of super-alloys (most often Inconel). In its most basic design, the bearing uses two foils: a smooth, circumferential, top foil that collects the pressures engendered in the thin gas film and a circumferential, corrugated foil carrying a pattern of bumps (a foil bearing may have 20...25 bumps). The bumps act like springs: submitted to a vertical load, they respond with both, a vertical and a lateral displacement. Both foils are welded at one end and free at the opposite. The corrugated foil is placed between the smooth foil and the bearing casing. The vibrations of the rotor are transmitted to the foils via the gas pressure field. During deformation, the corrugated foil slips between the top (smooth) foil and the bearing casing. Friction forces accompany this slip. Thus the foil bearing has good damping characteristics that allow it to operate a very high rotation speeds. However, the experimental results published since more than two decades showed that the dynamic characteristics of these bearings (for example critical speeds) could not be accurately predicted.

Although known since years, this technology became largely used only during the last decade, after the details of the manufacturing technology of the foils were disclosed. Indeed, manufacturing the corrugated and the smooth foil from a super-alloy with minute accuracy is a challenging task. As any mechanical component, these foils are affected by manufacturing errors. Designers are now becoming aware that the manufacturing errors of the foils may be responsible for the large dispersion of the dynamic characteristics of foil bearings. For example, if the bumps have different heights, then not only the stiffness of each bump will be different but the contacts between the bumps and the top foil may become loose. This means that the stiffness characteristics of the foil structure will be different from the nominal ones.

The present paper introduces an analysis aimed to quantify the impact of manufacturing errors on the structural stiffness of foil journal bearings. The stiffness of each bump is analytically calculated by using periodicity boundary conditions and Bresse relation. Due to manufacturing errors, the bump height follows a normal distribution: its average is the design value and its standard deviation is a user-defined parameter (for example a percentage of the design height).

The stiffness of the top (smooth) foil is estimated from a simple influence matrix approach. The bump foil and the top foil are assembled in a single structure with loose contacts by using a Lagrange multiplier or a penalty method.

The rotor is then pushed against this compliant structure. The resulting reaction rotor force is estimated from the structural model and the derivative of the force/displacement curve represents the stiffness of the foil structure. The results show that the bump height manufacturing errors lead to a decrease of the foil structure stiffness. The theoretical model explains why the dispersion of the dynamic characteristics of foil journal bearings found in the literature is so large. It also provides the design engineer a method for specifying the limits of acceptable manufacturing errors.