# Experimental and Numerical Dynamic Identification of an Aerostatic Electro-Spindle
## 1 Introduction
A rotordynamic model of an air spindle with a diameter of 1/8\" was considered [14], which involves coefficients that depend on the rotational speed. Here, the critical rotational speeds were calculated as a result of the rotordynamic analysis. A test rig to measure the rotordynamic response of a spindle supported on porous gas bearings is described in [15]. In [16; 17; 18; 19], the dependence of stiffness and damping properties on both rotational speed and perturbation frequency is expressed with analytical formulations. With this technique, it is possible to calculate the critical speeds and verify the stability of rigid or flexible rotors. During the design process of a spindle supported on gas bearings, it is of great importance to estimate the conical and cylindrical natural frequencies. A clear design guideline is illustrated in [20] to avoid critical speeds within the operating speed range. The influence of temperature on the dynamic behavior of the spindle is also discussed. Conical and cylindrical modes are investigated in [21] for a non-uniform slot-clearance journal bearing, together with the onset speed of whirl instability. The orbit method is an alternative to the linearized coefficient analysis employed in the aforementioned papers. It includes the complete nonlinear equations of the rotor and the bearings, which are integrated numerically to obtain the shaft center orbits corresponding to any set of geometrical, running, and initial conditions. This technique essentially uses the computer as an accurate experimental rig; it operates exactly in accordance with the assumed governing equations [22]. This method is currently used to calculate the unbalance response, the onset speed and the non-linear rotor behavior in case of large displacements [23]. Ref. [24] makes use of this method for a spindle with herringbone-grooved journal bearings. Ref. [25] highlights the differences between linear and non-linear analyses of a rotor supported by plain journal bearing. Past research by the authors concerns an electro-spindle of 50 mm shaft dia. supported on aerostatic bearings [26], a textile spindle [27] and a mesoscopic spindle of 10 mm diameter [28]. However, the experimental campaigns to validate the models are not always time- and cost-effective. This paper proposes a cheap and simple experimental identification procedure for aerostatic spindles. Impulse and static tests are performed to identify the natural frequencies, damping ratio and static stiffness evaluated at the nose of an electro-spindle for PCB drilling. The natural frequencies and damping factors are measured by considering the impulse-response function of the system, whereas the stiffness at the nose spindle is detected through the application of static forces. These spindle features are evaluated at zero rotational speed and with different supply pressures. Moreover, the paper describes the non-linear numerical model of the spindle, which consists of a 4 DOF rigid and unsymmetrical rotor supported by two aerostatic bearings. The equations of motion of the rigid rotor are solved simultaneously with the isothermal time-dependent Reynolds equation through [PERSON]'s explicit method to obtain the lubricant pressure distribution. The main goal of the work is to validate the developed non-linear numerical model through the proposed identification procedure and the performed experimental tests.