## 1 Introduction  
Aerostatic bearings are used in precision machine tool spindles. Their use is attractive, with minimal friction losses, high durability, and environmental friendliness. Unlike hydrostatic bearings, they minimize the difficulties associated with the supply and return of air lubricant [1, 2, 3, 4].  
A typical disadvantage of aerostatic bearings is their high compliance and low bearing capacity due to limited supply pressure [5, 6]. To reduce compliance, improved designs are used, which, according to the principle of operation, can be divided into bearings with active compensation of air flow rate and active compensation of the moving element (shaft) displacement. These types of bearings allow for a decrease in compliance to zero and even negative values (in the latter case, the shaft is displaced in the direction opposite to the current load relative to the housing), which allows the structures to be used not only as supports but, also, as active deformation compensators of the machine tools' technological system (TS) in order to reduce the time and improve the accuracy of metalworking.  
In bearings with active flow rate compensation (AFC), variable resistance throttles are used, which include membrane compensators and elastic orifices that can change their resistance depending on the pressure drop [7; 8; 9; 10; 11; 12; 13; 14; 15]. These constructions have been studied in sufficient detail. Their disadvantage is a significant air flow rate, which is necessary to ensure negative compliance. In addition, they are able to compensate for the deformation of the vehicle only in a narrow range limited by the size of the working gap, which is 5-20 \(\upmu\)m. This is often not enough to exclude the negative effect of a TS deformation.  
In this respect, the relatively poorly studied aerostatic bearings with active displacement compensation (ADC) have the best prospects [16; 17; 18; 19]. Their use makes it possible to compensate for significantly larger displacements, whose magnitude can exceed the AFC's by an order or more, while, as studies show, bearings with ADC are less energy-intensive [16; 18]. In theoretical terms, the operation of such bearings was investigated only for the mode of small shaft deviation from its central position [16; 17]. This is due to the fact that the modes of arbitrary position of the moving elements of the bearing are described by nonlinear mathematical models containing one or more partial differential equations, which creates significant difficulties for the development of effective algorithms for calculating the performance of the structure. Currently, there are no studies on the operation of bearings at moderate, high and ultimate loads on bearings with ADC, as well as the influence of the configuration of the bearing gap and types of flow restrictors on their performance--in particular, on the possibility and prospects of increasing their load capacity.  
In this paper, we consider the design of a radial aerostatic bearing with a displacement compensator and various feeder options--slots and diaphragms. The design is equipped with a system of external combined throttling (SECT), which allows the bearing to provide satisfactory dynamic properties when operating at low, zero and negative compliance modes. A mathematical model of the static state of the bearing is proposed, and methods and algorithms are developed for calculating the load capacity and compliance of a bearing with an arbitrary eccentricity of the shaft and support sleeve. A model of the dynamic state of the structure is formulated, and the criteria for its dynamic quality are calculated.