# Experimental and Numerical Dynamic Identification of an Aerostatic Electro-Spindle
## 1 Introduction  
Gas bearings, owing to their properties, play an essential role in turbochargers, gyroscopes, turbo blowers, gas turbines and micromachining spindles. In particular, micromachining spindles are widely used in surface finishing, printed circuit board (PCB) drilling, micro-processing, micro-milling, micro-grinding and, in general, for micromachining materials with low shear resistance. This is because compared to oil and rolling bearings, gas bearings can reach higher rotation speeds while simultaneously reducing the amount of machine maintenance required. The aim of research on spindles supported by gas bearings is to develop systems with excellent dynamic stability and stiffness, along with very low runout of the tool, even at very high rotation speeds. Poor damping is one of the main disadvantages of journal gas bearings, which can compromise the accuracy of high-precision machining. Moreover, friction power losses in motors and bearings can be a non-negligible source of heat at high speeds. The contribution of thermally induced deformations of a machine tool can exceed 50% of the total machining error [1], which is very significant in precision milling. To reduce these problems, an efficient and precise refrigeration system is required. Recent developments in manufacturing technologies have made it possible to realize innovative high-precision devices that can even be miniaturized. Research has therefore been enriched by numerous experimental contributions, some of which are discussed below. Machine tools of 4 mm and 1/8\" diameters used for micro-milling can now be supported on aerostatic bearings up to 400 krpm and 450 krpm [2, 3]. Experiments ondynamic characterization of air-bearing spindles are described in [4; 5], where the frequency-response function (FRF) of the spindle is measured. Impact tests have been carried out with an impact hammer [4] or with a custom-designed impact excitation system able to provide excitations up to 20 kHz [5]. Ref. [6] describes a test rig used to measure the dynamic stiffness and damping coefficients of a hemisphere spiral groove hybrid gas bearing. In this rig, the excitation is provided by means of two electromagnetic exciters. Mathematical models have long been used as a valuable tool to investigate the dynamic performance and stability of high-speed rotors. Early attempts at modelling this kind of system were aimed at gaining a better insight into the half-whirl instability, known today as asynchronous whirl instability [7]. At the time, this was one of the most troublesome problems in journal bearings. The first attempts to actually solve the equations of motion and the gas-lubricated equation simultaneously were reported by [PERSON] [8], [PERSON] and [PERSON] [9]. In their solution, they linearized the equations of motion of a 2 DOF rotor, along with the Reynolds equation, to obtain lubricant pressure distribution. However, in the Reynolds equation, they omitted the time derivative of pressure, and therefore, the resulting quasi-static solutions do not adequately represent the squeeze-film effects. In the same period, the literature presented many other models based on the \"_p-h-_\" linearization of the time-dependent Reynolds equation and the equations of motion of a 4 DOF rigid rotor [10; 11]. However, in most cases, these kinds of simplified models cannot replace more costly trial-and-error experimentation since they do not represent a reliable theory. The results obtained from these models have become ever more accurate with the use of finite-difference and finite-element models, where the perturbation method can be used to compute stiffness and damping coefficients for rotordynamic models. Ref. [12] investigates the impulse response of an ultra-precision raster milling spindle under the simplified hypothesis of constant coefficients. In [13], critical speeds and modal shapes of a micro-spindle machine tools were evaluated, but this analysis is limited because it only considers bearings with constant stiffnesses and neglects their damping properties.