## 3 An Appraisal of the Past Literature  
The archival literature on PGBs is abundant with a sound beginning in the early 1960s and continuous developments through the 1980s. Only in the 2000s did commercial applications abound, in particular in the flat panel manufacturing industry, semiconductor manufacturing industry, coordinate measuring machines, computed tomography imaging machines, and precision machine tools and spindles [3]. From 1965 to 1967, [PERSON] et al. publish three seminal papers [6; 7; 8] to introduce the theoretical framework for the analysis of gas lubricated PGBs. The authors derived analytical solutions for the flow rate and load capacity of PGBs under aerostatic conditions and produced experimental verification for both the load capacity as well as the flow coefficients. The authors noted the importance of surface roughness that modifies the PGB nominal clearance to produce agreement with the measured load and flow. The later reference [8] noted that shaft surface speed aids to increase the load capacity of PGBs and presents ultimate solutions for operation under very large speed numbers (\(\Lambda_{\Omega}>>1\)). [PERSON] et al. [6; 7; 8] did not study the effect of frequency on the bearing stiffness coefficients, and altogether ignored damping coefficients. At about the same time, in 1968, [PERSON] et al. [11; 12] developed a similar solution, albeit for an incompressible fluid, and produced predictive formulas for load capacity and gas flow that agreed well with experimental results for finite length PGBs (\(L/D\sim\
icefrac{{1}}{{2}}\)). Both [PERSON] et al. and [PERSON] et al. found that the PGB load capacity was a fraction of \(W^{*}=((p_{S}-p_{a})\times(L\;D))\), i.e., the product of the pressure difference (\(p_{S}-p_{a}\)) times the bearing projected area. Having introduced an equivalent clearance for the layer of porous material, \(c_{x}=(12\;\kappa\;t_{p})^{1/3}\), [PERSON] et al. reported a maximum aerostatic load capacity -(0.7 \(\times\;W^{*}\)) for the range (\(c_{x}/c\)) \(\sim\) 0.6-1.0; hence, the practical flow feeding number \(\Lambda_{x}\sim(R/\!\!\!\!/_{p})^{2}>>1\). A decade later (1978), [PERSON] and [PERSON] [13] presented a perturbation analysis and numerical solution to calculate the stiffness (\(K\)) and damping (\(C\)) force coefficients of aerostatic PGBs (\(\Lambda_{\Omega}=0\)). The unique predictions showed \(C<0\) for small frequency numbers (\(\Lambda_{\omega}\)). Interestingly enough, the analysis did not include the volume of the porous material, which, if large enough and under dynamic conditions, traps the gas to produce a pneumatic-hammer effect, i.e., a self-excited instability due to the absence (even negative) of effective damping [14]. Just a year later, in 1979, [PERSON] [5] presented a comprehensive analysis that included both steady-state performance parameters and the dynamic force coefficients via a perturbation analysis that produces analytical expressions as a function of \(\Lambda_{\Omega}\), \(\Lambda_{\omega}\), and \(\Lambda_{x}\), and includes the effect of the material porosity volume. For very small porous volumes, [PERSON] [1] reported a hardening bearing stiffness as frequency grows while the damping coefficient decreases steadily. The findings are similar to those for orifice-compensated aerostatic gas bearings, as published by [PERSON] [15] in 1968. Large porous volumes could lead to a negative direct stiffness but not negative damping, in opposition to [PERSON] and [PERSON]'s findings [13]. Under hybrid operations, i.e., operation with shaft speed (\(\Lambda_{\Omega}>0\)), the analysis reports cross coupled stiffness and damping coefficients, growing in proportion to the shaft speed until \(\Lambda_{\Omega}\to 10\), to then drop for operation at higher shaft speeds. [PERSON] [5] left for the future (the 21 st century) the dynamic stability of PGBs operating in a hybrid mode.