**4. The textile rotor with damping supports**

Gas bearings suffer from instability problems at high speed. A method to increase the stability threshold (the speed at which the unstable whirl occurs) is to increase the damping of the rotor-bearings system by introducing external damping supports [16]. A design guideline for the selection of the support parameters that insure stability in an aerodynamic journal bearing with damped and flexible support is given in paper [2].

The prototype described in this paragraph was designed with the priority of increasing the stability at high speeds [17]. The method adopted for this purpose was the use of rubber Orings.

The prototype consists on a rotor (1) made of hardened 32CrMo4 steel with mass 0.96 kg, diameter 37 mm and length 160 mm. The rotor is supported by a radial air bearing mounted on rubber O-rings and an axial thrust bearing (Figure 16). It was designed to rotate in stable conditions up to 150 krpm. At one end of the rotor an air turbine (2) was machined and at the other end a nose (3) was screwed to the rotor. The housing (4) is fixed to the base and has four circumferential slots in which the O-rings are inserted. The bushing (5) incorporates the rubber rings and has four sets of supply nozzles (diameter 0.2±0.01 mm) fabricated by EDM. The total length of the bearings is 57 mm. In the middle plane of the bushing a

discharge slot (6) is vented by a radial hole in the housing (see Figure 17). A central annular discharge chamber separates the radial bearings.

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 93

nozzles (8) machined on distributor (9). Annular chamber (10) connected to the nozzles is supplied through an axial hole on pre-distributor (11). Air is exhausted after actuating the turbine. Open loop speed control is maintained by setting the turbine supply pressure. The rotational speed was measured by an optical tachometer consisting of an emitter and a receiver facing the rotor at the turbine side. A retro-reflector stuck to a portion of the rotor

Radial and axial forces are applied to the nose by means of loading systems (12) similar to the ones previously described. Eight capacitive displacement transducers are inserted radially in the housing, the pre-distributor and cover (16) to sense the rotor and bushing positions. An axial transducer can be inserted near the nose to monitor the axial position of

The O-Rings have 41 mm inside diameter and 70 Shore hardness. The three materials used for testing are NBR (Butadiene Acrylonitrile), Viton® (Fluorinated Hydrocarbon) and

Accurate dimensional checks were carried out to evaluate axial and radial clearances, supply holes diameter and O-ring interference. The total diametral gap between them was found to be 35±2 μm. The difference between the thickness of central ring (14) and rotor

To measure the diameter of the nozzles supplying the bearings an optical fibre camera with 200X magnifying lens was used. The measurements were accurate and repeatable, thus proving the superiority of EDM technology over micro-drilling. Figure 18 shows a sample

The supply hole diameter, after fixing the radial clearance, is selected on the basis of numerical investigation conducted to simulate the dynamic behaviour of the system. The mathematical model used for this purpose is described in a separate paragraph at the end of this chapter.

O-ring grooves in the housing have a medium diameter of 43.5 mm, while external diameter of the bushing is 41 mm. The cross section diameter *d* of the rings was determined by a

flange was 19±2 μm, giving an axial clearance of approximately 9.5 μm.

face reflects emitted signal once per revolution.

the rotor with respect to the thrust bearing.

photographic record at 200X magnification.

**Figure 18.** Supply hole magnification (200X)

shadow comparator.

Silicone (Polysiloxane).

**Figure 16.** Test bench of the floating bushing

**Figure 17.** Enlargement of the floating bushing

The air from supply slots (7) flows to the radial clearance through the nozzles to reach the vent centrally in the discharge slot and laterally. The purpose of the O-rings, besides providing a seal between supply slots and discharge chamber, is to introduce damping in the rotor-bearing system. The turbine is driven by tangential jets discharged through 8 nozzles (8) machined on distributor (9). Annular chamber (10) connected to the nozzles is supplied through an axial hole on pre-distributor (11). Air is exhausted after actuating the turbine. Open loop speed control is maintained by setting the turbine supply pressure. The rotational speed was measured by an optical tachometer consisting of an emitter and a receiver facing the rotor at the turbine side. A retro-reflector stuck to a portion of the rotor face reflects emitted signal once per revolution.

Radial and axial forces are applied to the nose by means of loading systems (12) similar to the ones previously described. Eight capacitive displacement transducers are inserted radially in the housing, the pre-distributor and cover (16) to sense the rotor and bushing positions. An axial transducer can be inserted near the nose to monitor the axial position of the rotor with respect to the thrust bearing.

The O-Rings have 41 mm inside diameter and 70 Shore hardness. The three materials used for testing are NBR (Butadiene Acrylonitrile), Viton® (Fluorinated Hydrocarbon) and Silicone (Polysiloxane).

Accurate dimensional checks were carried out to evaluate axial and radial clearances, supply holes diameter and O-ring interference. The total diametral gap between them was found to be 35±2 μm. The difference between the thickness of central ring (14) and rotor flange was 19±2 μm, giving an axial clearance of approximately 9.5 μm.

To measure the diameter of the nozzles supplying the bearings an optical fibre camera with 200X magnifying lens was used. The measurements were accurate and repeatable, thus proving the superiority of EDM technology over micro-drilling. Figure 18 shows a sample photographic record at 200X magnification.

**Figure 18.** Supply hole magnification (200X)

92 Tribology in Engineering

discharge chamber separates the radial bearings.

**Figure 16.** Test bench of the floating bushing

**Figure 17.** Enlargement of the floating bushing

discharge slot (6) is vented by a radial hole in the housing (see Figure 17). A central annular

The air from supply slots (7) flows to the radial clearance through the nozzles to reach the vent centrally in the discharge slot and laterally. The purpose of the O-rings, besides providing a seal between supply slots and discharge chamber, is to introduce damping in the rotor-bearing system. The turbine is driven by tangential jets discharged through 8 The supply hole diameter, after fixing the radial clearance, is selected on the basis of numerical investigation conducted to simulate the dynamic behaviour of the system. The mathematical model used for this purpose is described in a separate paragraph at the end of this chapter.

O-ring grooves in the housing have a medium diameter of 43.5 mm, while external diameter of the bushing is 41 mm. The cross section diameter *d* of the rings was determined by a shadow comparator.

Table 3 lists the interferences on the O-rings calculated using the equation

$$int \%= \left(d - \frac{D\_l - D\_e}{2}\right) \frac{100}{d}$$

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 95

**Figure 19.** Scheme of the O-ring test rig

**Figure 20.** O-ring radial stiffness (a) and damping (b)

are considered, neglecting the dependence on the frequency.

The response to a rotor radial step jump displacement of 1 μm from coaxial position is calculated. As a first approximation, average values of O-ring stiffness *k*OR and damping *c*OR

The parameters introduced in the model are shown in Table 5. *L*1 and *L*2 are the axial lengths

*m*rot=0,97 kg *h*0=17 μm *L*2=23 mm *T*0=293 K *m*b=0,1 kg *d*s=0,2 mm *μ*=1,81e-5 Ns/m2 *k*OR=4·106 N/m *R*=18.5 mm *L*1=25 mm *R*0=287 J/kgK *c*OR=1·103 Ns/m

**4.2. Stability** 

of the two radial bearings.

**Table 5.** Input values of the model

where *D*i and *D*e are the inside diameter of the grooves machined in the housing and the external diameter of the bushing respectively. With 0.6 MPa pressure differential the sealing function of the rubber rings between chambers 7 and 6 was realized with interference about 10% or more.

In Table 4 the measures of the inner diameter *d* and the cross-section diameter *d*c are shown.


**Table 3.** Interference values

**Table 4.** Dimensions of the O-rings

#### **4.1. Measured rubber dynamic stiffness**

The dynamic stiffness of rubber O-rings is measured in order to introduce into the model the stiffness and the viscous equivalent damping. These parameters depend on the vibration frequency and also on the radial displacement imposed. Tests were made under different conditions, varying the diametral interference on the O-ring and the displacement amplitude *x*0 imposed, in the frequency range 300÷800 Hz. In Figure 19 is visible the scheme of the test rig, in which the cylinder is fixed and the casing is mounted on the shaker plate. The force amplitude *F*0 is measured by the load cell mounted between two fixed parts.

Measurements were made at different radial amplitudes. Increasing *x*0 both stiffness and damping decrease. The results visible in Figure 20 were obtained with *x*0=25 μm and a diametral interference of 11%, that are similar to that occur in the air bearing test bench. The results obtained with Silicone are not reported because the FRF of transfer function *F*/*x* was very noisy.

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 95

**Figure 19.** Scheme of the O-ring test rig

94 Tribology in Engineering

10% or more.

**Table 3.** Interference values

**Table 4.** Dimensions of the O-rings

very noisy.

**4.1. Measured rubber dynamic stiffness** 

Table 3 lists the interferences on the O-rings calculated using the equation

���� � �� � �� � ��

where *D*i and *D*e are the inside diameter of the grooves machined in the housing and the external diameter of the bushing respectively. With 0.6 MPa pressure differential the sealing function of the rubber rings between chambers 7 and 6 was realized with interference about

In Table 4 the measures of the inner diameter *d* and the cross-section diameter *d*c are shown.

d

NBR 41 1.80 Viton 41 1.83 Silicone 41 1.83

The dynamic stiffness of rubber O-rings is measured in order to introduce into the model the stiffness and the viscous equivalent damping. These parameters depend on the vibration frequency and also on the radial displacement imposed. Tests were made under different conditions, varying the diametral interference on the O-ring and the displacement amplitude *x*0 imposed, in the frequency range 300÷800 Hz. In Figure 19 is visible the scheme of the test rig, in which the cylinder is fixed and the casing is mounted on the shaker plate. The force amplitude *F*0 is measured by the load cell mounted between two fixed parts.

Measurements were made at different radial amplitudes. Increasing *x*0 both stiffness and damping decrease. The results visible in Figure 20 were obtained with *x*0=25 μm and a diametral interference of 11%, that are similar to that occur in the air bearing test bench. The results obtained with Silicone are not reported because the FRF of transfer function *F*/*x* was

NBR-Silicone 1.78±0.01 30% Viton 1.73±0.01 28%

dc

<sup>2</sup> �

100 �

Cross section diameter *d* [mm] Interference

*d* (mm) *d*c (mm)

**Figure 20.** O-ring radial stiffness (a) and damping (b)

#### **4.2. Stability**

The response to a rotor radial step jump displacement of 1 μm from coaxial position is calculated. As a first approximation, average values of O-ring stiffness *k*OR and damping *c*OR are considered, neglecting the dependence on the frequency.

The parameters introduced in the model are shown in Table 5. *L*1 and *L*2 are the axial lengths of the two radial bearings.


**Table 5.** Input values of the model

In Figure 21 the theoretical supply pressure values in correspondence to the stability threshold are plotted vs. the rotational speed for the cases of fixed bearing and bearing mounted on O-rings. Each curve divides the plane into two regions: the upper one relative to a stable behaviour of the rotor-bearing system, the lower one relative to an unstable behaviour. In the first case, as a result of an initial step jump displacement of the rotor, the system evolves to the centred position (punctual stability); in the second case the rotor trajectory is an open spiral and causes the contact between the rotor and the bushing. In correspondence to the threshold curves the system evolves to a condition of orbital stability. The stabilizing effect of the rubber rings is evident because the pressure that guarantees the stability is lower.

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 97

**Figure 23.** Comparison between experimental and simulated whirling frequency ߥ

**Figure 24.** Comparison between experimental and simulated whirling ratio ߛ

**Figure 25.** Orbit amplitude versus supply pressure; *ω*=30000 rpm

sudden and considerable in both cases.

It is possible to approach this threshold by decreasing the supply pressure or by increasing the rotational speed. Both possibilities are treated: Figures 25 and 26 show the change of the orbit amplitude vs these parameters. The increase in amplitude near stability threshold is

In Figure 22 the simulated values are compared with the experimental ones, relative to three kinds of rubber: NBR, Viton and Silicone. There is good agreement between the experimental and the simulated stability threshold also if the experimental data are influenced by the rotor imbalance and in calculations the effect of imbalance is neglected (the rotor was dynamically balanced to a grade better than ISO quality grade G-2.5).

The whirling frequency� ߥ increases with the rotational speed, see Figure 23. It is interesting to observe that the whirling ratio ߛ ൌ ߥȀ߱ at the stability threshold (Figure 24) decreases with the rotational speed.

**Figure 21.** Theoretical results with fixed bearing and bearing mounted on OR

**Figure 22.** Comparison between experimental and simulated threshold stability

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 97

**Figure 23.** Comparison between experimental and simulated whirling frequency ߥ

96 Tribology in Engineering

stability is lower.

with the rotational speed.

In Figure 21 the theoretical supply pressure values in correspondence to the stability threshold are plotted vs. the rotational speed for the cases of fixed bearing and bearing mounted on O-rings. Each curve divides the plane into two regions: the upper one relative to a stable behaviour of the rotor-bearing system, the lower one relative to an unstable behaviour. In the first case, as a result of an initial step jump displacement of the rotor, the system evolves to the centred position (punctual stability); in the second case the rotor trajectory is an open spiral and causes the contact between the rotor and the bushing. In correspondence to the threshold curves the system evolves to a condition of orbital stability. The stabilizing effect of the rubber rings is evident because the pressure that guarantees the

In Figure 22 the simulated values are compared with the experimental ones, relative to three kinds of rubber: NBR, Viton and Silicone. There is good agreement between the experimental and the simulated stability threshold also if the experimental data are influenced by the rotor imbalance and in calculations the effect of imbalance is neglected

The whirling frequency� ߥ increases with the rotational speed, see Figure 23. It is interesting to observe that the whirling ratio ߛ ൌ ߥȀ߱ at the stability threshold (Figure 24) decreases

(the rotor was dynamically balanced to a grade better than ISO quality grade G-2.5).

**Figure 21.** Theoretical results with fixed bearing and bearing mounted on OR

**Figure 22.** Comparison between experimental and simulated threshold stability

**Figure 24.** Comparison between experimental and simulated whirling ratio ߛ

It is possible to approach this threshold by decreasing the supply pressure or by increasing the rotational speed. Both possibilities are treated: Figures 25 and 26 show the change of the orbit amplitude vs these parameters. The increase in amplitude near stability threshold is sudden and considerable in both cases.

**Figure 25.** Orbit amplitude versus supply pressure; *ω*=30000 rpm

**Figure 26.** Orbit amplitude versus rotational speed; *p*s=0.22 Mpa

Figure 27 shows the change in orbit shape with decreasing the supply pressure. Whirl motion is conical for any supply pressure at the stability threshold. Frequency spectra for turbine displacement in the two conditions are visible in Figure 28.

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 99

Another method to increase the bearing stability is to modify the film geometry from the circular journal bearing profile. Non-circular journal bearings can assume various geometries: elliptical [20-23] offset halves [24] and three-lobe configuration [25,26] are the most common geometries. Paper [27] shows a comparative analysis of three types of hydrodynamic journal bearing configurations namely, circular, axial groove, and offset-

There is an extensive literature about the study of the dynamic stability of hydrodynamic journal bearings with non-circular profile, but very few papers consider gas journal bearings of this type. The wave bearing with compressible lubricants was introduced in the early

In the present paragraph the design of the elliptical and multi-lobes gas bearings for a ultra-

The bearings were designed to have a stable regime of rotation up to 500 krpm with acceptable stiffness and load characteristics. A computerized design was used for optimization of the rotor-bearing characteristics. The bearing clearance was represented by

where *c*form is the profile form factor, *h*0 is the maximum clearance and *n* is the number of

The static and dynamic performances were numerically analyzed for two pairs of radial externally pressurized gas bearings. Conical and cylindrical whirl modes were considered. From numerical simulations for a 10 mm diameter rotor, bearing clearances non less than 5

 the maximum rotor speed obtained with circular bearing (clearance 5 μm) with 4 supply orifices of 0.1 mm diameter in circumferential direction was 150 krpm, while

 with elliptical bearing profile the maximum rotor speed obtained with stable operation was 500 krpm for bearings with 4 supply orifices of 0.2 mm diameter in circumferential

the rotor with the multi-lobe bearings were less stable in comparison with the rotor

the positioning of supply orifices at 45° with respect to the principal axes of the elliptic

The final bearing geometry is defined by the parameters summarized in Table 6. Each elliptical journal bearing presents two rows of 4 supply orifices positioned at 45° with

<sup>2</sup> (���(��) − 1)�

ℎ=ℎ� �1 � �����

μm and supply pressure 0.6 MPa the following results were obtained:

with 32 supply orifices of 0.2 mm diameter was 250 krpm;

**5. The mesoscopic spindle** 

high speed spindle is described [30].

halves.

1990's [28,29].

expression

lobes of the profile.

direction;

with elliptical bearings;

respect to the principal axes.

profile improved bearing characteristics.

**Figure 27.** Rotor orbits in stable condition (a) and at stability threshold (b); *ω*=50000 rpm

**Figure 28.** Frequency spectra of rotor displacement; *ω*=50 krpm

#### **5. The mesoscopic spindle**

98 Tribology in Engineering

**Figure 26.** Orbit amplitude versus rotational speed; *p*s=0.22 Mpa

**Figure 28.** Frequency spectra of rotor displacement; *ω*=50 krpm

0

0,4

0,8

amplitude [micron]

1,2

1,6

2

turbine displacement in the two conditions are visible in Figure 28.

**Figure 27.** Rotor orbits in stable condition (a) and at stability threshold (b); *ω*=50000 rpm

0 200 400 600 800 1000 frequency [Hz]

Figure 27 shows the change in orbit shape with decreasing the supply pressure. Whirl motion is conical for any supply pressure at the stability threshold. Frequency spectra for

0.3MPa 0.5 MPa Another method to increase the bearing stability is to modify the film geometry from the circular journal bearing profile. Non-circular journal bearings can assume various geometries: elliptical [20-23] offset halves [24] and three-lobe configuration [25,26] are the most common geometries. Paper [27] shows a comparative analysis of three types of hydrodynamic journal bearing configurations namely, circular, axial groove, and offsethalves.

There is an extensive literature about the study of the dynamic stability of hydrodynamic journal bearings with non-circular profile, but very few papers consider gas journal bearings of this type. The wave bearing with compressible lubricants was introduced in the early 1990's [28,29].

In the present paragraph the design of the elliptical and multi-lobes gas bearings for a ultrahigh speed spindle is described [30].

The bearings were designed to have a stable regime of rotation up to 500 krpm with acceptable stiffness and load characteristics. A computerized design was used for optimization of the rotor-bearing characteristics. The bearing clearance was represented by expression

$$h = h\_0 \left( 1 + \frac{c\_{form}}{2} \left( \cos(n\theta) - 1 \right) \right)$$

where *c*form is the profile form factor, *h*0 is the maximum clearance and *n* is the number of lobes of the profile.

The static and dynamic performances were numerically analyzed for two pairs of radial externally pressurized gas bearings. Conical and cylindrical whirl modes were considered. From numerical simulations for a 10 mm diameter rotor, bearing clearances non less than 5 μm and supply pressure 0.6 MPa the following results were obtained:


The final bearing geometry is defined by the parameters summarized in Table 6. Each elliptical journal bearing presents two rows of 4 supply orifices positioned at 45° with respect to the principal axes.


High Speed Rotors on Gas Bearings: Design and Experimental Characterization 101

**Figure 30.** Test benches realized to measure the radial and axial bearing stiffness

**Figure 31.** Test benches realized to measure the radial and axial bearing stiffness; supply gauge

pressure 0.6 MPa

**Table 6.** Final bearing parameters

Figure 29 shows the prototype of ultra-high speed spindle. The rotor, of mass 0.07 kg, is supported by two pairs radial elliptical bearings and a double thrust bearing. The calculated radial stiffness on the rotor end is 3 N/μm and the air consumption is 3.65·10-4 kg/s.

The axial and the radial stiffness of the bearings were measured with test benches realized at the purpose (Figure 30). The axial and radial displacement of the rotor due to an imposed load was measured by laser beams. The axial stiffness of the thrust supplied at 0.6 MPa is 2.8 N/μm, while the radial stiffness is 1 N/μm. This value can be increased with a better dimensional control of the bearings internal profile.

**Figure 29.** Ultra-high speed spindle prototype

By means of start-up (acceleration) and coast down (deceleration) tests on the spindle the bearing friction torque was estimated as a function of the speed up to 150000 rpm. The deceleration tests from different rotational speeds are depicted in Figure 31. The friction torque was found to be proportional to the rotational speed with the rate of 10-4 Nm every 10000 rpm. The dynamic runout of the shaft was measured by means of laser beams at different rotational speeds in correspondence of the nose.

The unbalance response was synchronous and unstable whirl was not encountered. In Figure 32 the waterfall diagram, obtained with the FFT of the shaft radial vibration, is shown. There is a critical speed at 34000 rpm, to which corresponds a maximum spindle runout of ±9 μm.

**Figure 30.** Test benches realized to measure the radial and axial bearing stiffness

**Table 6.** Final bearing parameters

dimensional control of the bearings internal profile.

**Figure 29.** Ultra-high speed spindle prototype

runout of ±9 μm.

different rotational speeds in correspondence of the nose.

Maximum clearance *h*0, μm 15 Rotor diameter, mm 10 Supply orifice diameter, mm 0.2 Number of supply orifices for each bearing 8 Number of bearings 4 Profile form factor *c*form 0.7 Number of profile lobes *n* 2

Figure 29 shows the prototype of ultra-high speed spindle. The rotor, of mass 0.07 kg, is supported by two pairs radial elliptical bearings and a double thrust bearing. The calculated

The axial and the radial stiffness of the bearings were measured with test benches realized at the purpose (Figure 30). The axial and radial displacement of the rotor due to an imposed load was measured by laser beams. The axial stiffness of the thrust supplied at 0.6 MPa is 2.8 N/μm, while the radial stiffness is 1 N/μm. This value can be increased with a better

By means of start-up (acceleration) and coast down (deceleration) tests on the spindle the bearing friction torque was estimated as a function of the speed up to 150000 rpm. The deceleration tests from different rotational speeds are depicted in Figure 31. The friction torque was found to be proportional to the rotational speed with the rate of 10-4 Nm every 10000 rpm. The dynamic runout of the shaft was measured by means of laser beams at

The unbalance response was synchronous and unstable whirl was not encountered. In Figure 32 the waterfall diagram, obtained with the FFT of the shaft radial vibration, is shown. There is a critical speed at 34000 rpm, to which corresponds a maximum spindle

radial stiffness on the rotor end is 3 N/μm and the air consumption is 3.65·10-4 kg/s.

**Figure 31.** Test benches realized to measure the radial and axial bearing stiffness; supply gauge pressure 0.6 MPa

$$\frac{p}{R^{\bullet}T^{\bullet}} \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{R \partial \vartheta} \right) + \frac{\partial p}{\partial z} = -\mu \frac{12u}{h^2} \tag{1}$$

$$\frac{p}{R^{\bullet T^{\bullet}}} \left( \frac{\partial v}{\partial t} + \mu \frac{\partial v}{\partial z} + \nu \frac{\partial v}{R \partial \vartheta} \right) + \frac{\partial p}{R \partial \vartheta} = \mu \frac{6(R\omega - 2\nu)}{\hbar^2} \tag{2}$$

$$\frac{\partial(phu)}{\partial x} + \frac{\partial(phv)}{R\partial\theta} + \frac{\partial(ph)}{\partial t} - R^0 T^0 q = 0\tag{3}$$

$$q = \frac{G}{\Delta x R \Delta \theta} \tag{4}$$

$$\frac{\partial}{\partial x}\left(ph^3\frac{\partial p}{\partial x}\right) + \frac{\partial}{R\partial\theta}\left(ph^3\frac{\partial p}{R\partial\theta}\right) + 12\mu R^0 T^0 q = 6\mu\omega\frac{\partial(ph)}{\partial\theta} + 12\mu\frac{\partial(ph)}{\partial t} \tag{5}$$

$$h(\theta, \mathbf{z}) = h\_0 - e\_\mathbf{x}(\mathbf{z})\cos\theta - e\_\mathbf{y}(\mathbf{z})\sin\theta \tag{6}$$

$$\mathbf{e}\_{\mathbf{x}}(\mathbf{z}) = \mathbf{x}\_{G} + (\mathbf{z} - \mathbf{z}\_{G})\boldsymbol{\theta}\_{\mathbf{y}};\ \mathbf{e}\_{\mathbf{y}}(\mathbf{z}) = \mathbf{y}\_{G} - (\mathbf{z} - \mathbf{z}\_{G})\boldsymbol{\theta}\_{\mathbf{x}} \tag{7}$$

$$r^2 \frac{\partial}{\partial r} \left( p h^3 \frac{\partial p}{\partial r} \right) + \frac{\partial}{\partial \theta} \left( p h^3 \frac{\partial p}{\partial \theta} \right) + 12 \mu R^0 T^0 r^2 q = 6 \mu \omega r^2 \frac{\partial (p h)}{\partial \theta} + 12 \mu r^2 \frac{\partial (p h)}{\partial t} \tag{8}$$

$$G = \begin{cases} c\_s k\_T \rho\_N p\_s & \text{if } 0 < \frac{p}{p\_s} < b \\ c\_s k\_T \rho\_N p\_s \sqrt{1 - \left(\frac{p}{p\_s} - b\right)^2} & \text{if } b < \frac{p}{p\_s} < 1 \\ -c\_s k\_T \rho\_N p \sqrt{1 - \left(\frac{p}{1-b}\right)^2} & \text{if } 1 < \frac{p}{p\_s} < \frac{1}{b} \\ & -c\_s k\_T \rho\_N p & \text{if } \frac{p}{p\_s} > \frac{1}{b} \end{cases} \tag{9}$$

$$c\_s = 0.686 \frac{c\_d S}{\rho\_N \sqrt{R^0 T^0}} \tag{10}$$

$$c\_d = 0.85 \left( 1 - e^{-8.2 \frac{h}{d\_s}} \right) \left( 1 - e^{-0.005 Re} \right) \tag{11}$$

$$Re = \frac{4G}{\pi d\_s \mu} \tag{12}$$

$$3h\_{l,l}^2 \left(\frac{\partial h}{\partial x}\right)\_{l,j} \frac{p\_{l+1,j}^2 - p\_{l-1,j}^2}{2\Delta x} + h\_{l,j}^2 \frac{p\_{l+1,j}^2 - 2p\_{l,j}^2 + p\_{l-1,j}^2}{(\Delta x)^2} + 3h\_{l,j}^2 \left(\frac{\partial h}{R\partial \theta}\right)\_{l,j} \frac{p\_{l,j+1}^2 - p\_{l,j-1}^2}{2R\Delta \theta} + h\_{l,j}^3 \frac{p\_{l,j+1}^2 - p\_{l,j}^2 + p\_{l,j-1}^2}{(R\Delta \theta)^2} + \frac{p\_{l,j}^2 - p\_{l,j}^2}{(\Delta x)^2} + \frac{p\_{l,j}^2 - p\_{l,j}^2}{(\Delta x)^2}$$

$$24R^0 T^0 \mu q\_{l,j} = 12\mu\omega\theta h\_{l,j} \frac{p\_{l,j+1} - p\_{l,j-1}}{2\Delta \theta} + 12\mu\omega\mu p\_{l,j} \left(\frac{\partial h}{\partial \theta}\right)\_{l,j} + 24\mu h\_{l,j}^n \frac{p\_{l,j}^{n+1} - p\_{l,j}^n}{\Delta t} + 24\mu p\_{l,j}^n \frac{h\_{l,j}^{n+1} - h\_{l,j}^n}{\Delta t} \tag{13}$$

$$\begin{cases} & m\_r \ddot{\mathbf{x}}\_G = F\_{cx} + F\_x + m\_r \varepsilon \omega^2 \cos(\omega t) \\ & m\_r \ddot{\mathbf{y}}\_G = F\_{cy} + F\_y + m\_r \varepsilon \omega^2 \sin(\omega t) \\ & f\_G \ddot{\mathbf{y}}\_\mathbf{x} = M\_{cx} + F\_\mathbf{y} (\mathbf{z}\_G - \mathbf{z}\_F) - f\_P \omega \dot{\vartheta}\_\mathbf{y} + \chi (f\_P - f\_G) \omega^2 \cos(\omega t - \varphi) \\ & f\_G \ddot{\boldsymbol{\varphi}}\_\mathbf{y} = M\_{cy} - F\_\mathbf{x} (\mathbf{z}\_G - \mathbf{z}\_F) + f\_P \omega \dot{\vartheta}\_\mathbf{x} + \chi (f\_P - f\_G) \omega^2 \sin(\omega t - \varphi) \end{cases} \tag{14}$$

$$\begin{cases} \mathbf{F}\_{\rm cx} = \int\_{0}^{L} \int\_{0}^{2\pi} (p\cos\theta - \tau\_{\theta}\sin\theta)R d\theta dz \\ \mathbf{F}\_{\rm cy} = \int\_{0}^{L} \int\_{0}^{2\pi} (p\sin\theta + \tau\_{\theta}\cos\theta)R d\theta dz \\ \mathbf{M}\_{\rm cx} = \int\_{0}^{L} \int\_{0}^{2\pi} (-p\sin\theta + \tau\_{\theta}\cos\theta)(\mathbf{z}\_{G} - \mathbf{z})R d\theta dz \\ \mathbf{M}\_{\rm cy} = \int\_{0}^{L} \int\_{0}^{2\pi} (p\cos\theta + \tau\_{\theta}\sin\theta)(\mathbf{z}\_{G} - \mathbf{z})R d\theta dz \end{cases} \tag{15}$$

$$
\pi\_{\vartheta} = -\frac{\hbar}{2} \frac{\partial p}{\partial \vartheta} - \frac{\mu \omega R}{\hbar} \tag{16}
$$

$$p\_{l,j}^{n+1} = p\_{l,j}^n + \Delta t \left( p\_{l,j}^n, p\_{l+1,j}^n, p\_{l-1,j}^n, p\_{l,j+1}^n, p\_{l,j-1}^n, h\_{l,j}^n, h\_{l,j}^{n-1}, \left(\frac{\partial h}{\partial \vartheta}\right)\_{l,j}^n, \left(\frac{\partial h}{\partial x}\right)\_{l,j}^n \right) \tag{17}$$

$$\begin{aligned} \mathfrak{x}(0) &= h\_0 \varepsilon\_{\mathfrak{x}}(0); \mathfrak{y}(0) = h\_0 \varepsilon\_{\mathfrak{y}}(0) \\\\ \dot{\mathfrak{x}}(0) &= h\_0 \dot{\varepsilon}\_{\mathfrak{x}}(0); \dot{\mathfrak{y}}(0) = h\_0 \dot{\varepsilon}\_{\mathfrak{y}}(0) \end{aligned}$$

$$m\_r \ddot{x} + F\_t = F \tag{18}$$

The prototypes developed operated in stable conditions in the speed range expected. Future investigations will verify the stability at higher speeds.

High Speed Rotors on Gas Bearings: Design and Experimental Characterization 107

*ε* static rotor unbalance *εx, ε<sup>y</sup>* rotor eccentricity ratios *γ* whirl ratio, *γ=υ/ω*

*υ* whirl frequency

**Author details** 

**9. References** 

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*ω* rotor angular speed

*ϕ* angle between static and dynamic unbalance *μ* dynamic viscosity, in calculations *μ*=17.89·10-6 Pa·s

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*ρ<sup>N</sup>* air density in normal conditions
