**4. Innovative support system**

(S′ > 6) of the operating map plotted in **Figure 3** in agreement with the suitable regression proposed by the reference authors. Particularly, the nominal working point should be located in the high-speed (lightly loaded) regime, but significantly far from the shaft strength and

In order to locate in the operating map the nominal working point of a single radial foil bearing supporting the reference microturbine shaft, the data reported in **Table 1** are used together with unit (axial) length to diameter ratio (L/D = 1), load coefficient equal to 2.7 × 10−4 N/(mm3 krpm) (third-generation bearings) and isoviscous behavior assumption. In addition, the total load is approximated by the external one (40 N) as first estimate. By means of such assumptions, the assessment of S′ according to reference [32] suggests that the working conditions are not suitable (S′ = 1.6 < 6). Therefore, the journal diameter must be increased (optimal values range between 25 and 40 mm, as suggested in **Figure 3**). Indeed, transitioning over to oil-free lubrication requires suitable design solutions in that thin shafts are not required anymore to avoid rolling element bearings from operating above their DN threshold. On the contrary, large diameter hollow shafts must be used in order to increase the

**Figure 3.** Performance map of foil bearings and operating points of the reference micro-GT for different journal diameters.

As far as the remaining air film bearings are concerned, the different fluid film bearing designs (multilobe and tilting-pad geometries) achieve better stability than plain bearings at the expense of load-carrying capacity. On the contrary, gas-lubricated grooved bearings

peripheral speed and, as a consequence, the load capacity of air bearings [31].

promise stability with minor reduction of lift [33].

).

thermal limits (specific power loss lower than 155,000 W/m2

12 Bearing Technology

The conclusion of the review study is that each bearing type has different strengths and weaknesses. Therefore, by using different types of bearings in the same support system, a proper design of their arrangement should be capable of taking the maximum advantage of the peculiarity of each bearing.

To this end, the conceptual design of an innovative support system, which takes advantage of both foil and rolling element bearings, is presented.

### **4.1. Layout and components**

A simplified scheme of the assembly of the cutting-edge support system is depicted in **Figure 4**. In the simplified scheme of the invention used, hereafter load direction is assumed as in the Case B of **Table 2**.

The angular contact ball bearing (3), at the compressor side (**Figure 4**), is capable of carrying both radial and axial loads. As axial load may reverse during start-up of the unit, the bearing must have double effect, i.e., it is made up of two (or more) matched single-row bearings in back-to-back arrangement. Although four-point contact ball bearings are normally not available for precision (high speed) applications, a single bearing is depicted in **Figure 4** for the sake of conciseness of the scheme and clarity. As shown in **Figure 5**, such a bearing is mounted by inserting the external ring inside a suitable bore on the machine frame (9) and by placing the inner ring onto the relevant seat on the shaft with proper tolerances. Particularly, the outer ring must be axially constrained in order to carry thrust loads (locating bearing).

**Figure 4.** Simplified assembly of the innovative support system.

**Figure 5.** Ball bearing seat and spline coupling at compressor side.

A possible second radial support (always of rolling element type, e.g., a set of angular contact ball bearings in back-to-back arrangement) has not been included in **Figure 4** for simplicity. Nevertheless, it would be useful to avoid that the shaft is in a cantilever configuration, and it should be able to carry radial load solely. Therefore, such additional radial bearing should not be constrained axially to the housing on the frame (nonlocating bearing) so that the axial thermal expansion of the shaft would be allowed.

The plate of the foil bearing, i.e., component (4) listed in **Figure 4**, is fixed to the frame as shown in **Figure 6**, where the runner surface is located on the back of the turbine rotor (1) and the opposite sliding pairs, runner (11) and pads (12), are separated by the clearance c<sup>z</sup> . A suitable spacer (10) must allow a proper adjustment of the clearance (or preload in static conditions) to ensure optimal operation of the foil bearing.

**Figure 6.** Foil air bearing installation and adjustment.

**Figure 5.** Ball bearing seat and spline coupling at compressor side.

**Figure 4.** Simplified assembly of the innovative support system.

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When the foil bearing has no aerodynamic load-carrying capacity, the turbine thrust acts on the shaft shoulder (8). Differently, after an air film is formed and the relevant aerodynamic pressure is generated, its micrometric thickness causes the runner/impeller assembly (1) to move accordingly in the axial direction so that the contact between the turbine impeller hub and the shaft shoulder does not occur anymore. Consequently, in nominal conditions and starting from the rotation speed at which the runner is airborne, the only axial load transfer from the turbine to the shaft may occur through the helical spline coupling, as a function of the design helical angle.

**Figure 4** also illustrates the coupling between the impeller hubs and the shaft (5). The compressor impeller (2) is fixed by means of a conventional spline pair (7) (made up by equally spaced straight grooves), the profile of which is depicted with parallel sides (or straight teeth). Differently, on the turbine side, a helical spline pair (6) (in which each groove forms a helix around the shaft) is machined. Such particular spline fit provides the additional function of axial load distributor. Of course, involute instead of parallel-side profiles may be chosen for both compressor and turbine wheel spline pairs. The purpose of the helical spline pair is to distribute the axial load between main (4) and auxiliary (3) thrust bearings, as explained in the following paragraphs.

### **4.2. Main thrust bearing relief**

The innovative layout and, particularly, the direct matching of turbine impeller and main thrust bearing (4) allow for its relief during start/stop of the unit.

With reference to thrust loads and symbols given in **Table 2** (Case B), let F<sup>t</sup> = −T<sup>t</sup> and Fc = T<sup>c</sup> be the turbine and compressor thrusts, respectively (**Figure 7**). They are caused by the pressure of the evolving fluid on the relevant impellers.

**Figure 7.** Static scheme (axial forces) of the innovative rotor operating in nominal conditions.

At start-up and until the onset of the aerodynamic lift, the auxiliary bearing (3) carries the whole thrust so that the main thrust bearing (4) is unloaded, and therefore, it works with minimal or no wear. Indeed, in such a condition, the turbine impeller (1) exerts the thrust F<sup>t</sup> on the shaft shoulder (8) (visible in **Figures 4** and **6**) instead of on the foil bearing (4). Through the shoulder and the shaft, the thrust Ft is then transferred to the shaft support, i.e., the auxiliary bearing (3), like in a conventional bearing layout. Once the runner of the main bearing becomes airborne at sufficient speed and a consequent translation of the turbine impeller occurs, automatically it relieves the auxiliary one, as the shaft shoulder (8) does not receive thrust anymore.

#### **4.3. Load partition**

starting from the rotation speed at which the runner is airborne, the only axial load transfer from the turbine to the shaft may occur through the helical spline coupling, as a function of

**Figure 4** also illustrates the coupling between the impeller hubs and the shaft (5). The compressor impeller (2) is fixed by means of a conventional spline pair (7) (made up by equally spaced straight grooves), the profile of which is depicted with parallel sides (or straight teeth). Differently, on the turbine side, a helical spline pair (6) (in which each groove forms a helix around the shaft) is machined. Such particular spline fit provides the additional function of axial load distributor. Of course, involute instead of parallel-side profiles may be chosen for both compressor and turbine wheel spline pairs. The purpose of the helical spline pair is to distribute the axial load between main (4) and auxiliary (3) thrust bearings, as explained in

The innovative layout and, particularly, the direct matching of turbine impeller and main

the turbine and compressor thrusts, respectively (**Figure 7**). They are caused by the pressure

At start-up and until the onset of the aerodynamic lift, the auxiliary bearing (3) carries the whole thrust so that the main thrust bearing (4) is unloaded, and therefore, it works with minimal or no wear. Indeed, in such a condition, the turbine impeller (1) exerts the thrust F<sup>t</sup>

**Figure 7.** Static scheme (axial forces) of the innovative rotor operating in nominal conditions.

the shaft shoulder (8) (visible in **Figures 4** and **6**) instead of on the foil bearing (4). Through the

bearing (3), like in a conventional bearing layout. Once the runner of the main bearing becomes

is then transferred to the shaft support, i.e., the auxiliary

= −T<sup>t</sup>

and Fc

 = T<sup>c</sup> be

on

thrust bearing (4) allow for its relief during start/stop of the unit.

of the evolving fluid on the relevant impellers.

shoulder and the shaft, the thrust Ft

With reference to thrust loads and symbols given in **Table 2** (Case B), let F<sup>t</sup>

the design helical angle.

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the following paragraphs.

**4.2. Main thrust bearing relief**

According to the explanation of the previous paragraph, in nominal working conditions with no helical spline fit (e.g., by adopting a straight grooved spline on turbine-side too), the whole thrust of the turbine Ft would be carried by the main thrust bearing and the compressor thrust Fc would be supported by the auxiliary bearing. In a conventional shaft-bearing assembly, the reference thrust Tref that loads the single axial bearing comes from the opposite thrusts exerted by turbine and compressor, i.e., Tref = F<sup>t</sup> −F<sup>c</sup> . Therefore, during nominal operation in comparison with a conventional support system, the new assembly design would disadvantage the main thrust bearings, whereas Ft > Tref. Differently, by taking advantage of the (turbine-side) helical spline pair as an actuator, a part of the turbine thrust can be transferred from the hub to the shaft. In such a way, any wanted division of the thrust load between the two (main (4) and auxiliary (3)) bearings can be obtained as a function of a single design parameter, i.e., the spline helix angle β shown in **Figure 8**.

**Figure 8.** Helix angle and reference system of spline coupling.

Particularly, by means of a suitable choice of the helix angle in the design phase, in nominal conditions, it is also possible to subject the main axial bearing to a thrust Tref as in a conventional support system, while the auxiliary bearing remains axially unloaded. In such a case, as this support is an angular contact ball bearing, it carries solely the radial load, the intensity of which is much lower than the axial loads. Obviously, this is only an example of load division. The new layout allows us the setting of the optimal load division in the design phase as a function of the load-carrying characteristics of the bearings, as well as expected duration and reliability of rolling element supports.

### **4.4. Law of load distribution**

First, the law of load distribution followed during nominal operation by the helical spline pair, employed as a mechanical actuator besides a simple coupling system, is determined.

**Figure 7** depicts the forces acting in nominal conditions on the rotor components according to the modifications resulting from the innovation. The constraint simulates the main axial bearing (4), which carries the load F<sup>t</sup> −R. The axial forces R are the (equal) action and reaction that the turbine impeller exerts on the shaft through the helical spline. The total thrust that acts on the shaft is Fc −R and is carried by the auxiliary axial bearing (3). The torque M<sup>t</sup> is the resisting torque of the turbine due to the pressure exerted on the relevant blades.

A campaign of FEM structural analyses has been carried out on a model of helical spline coupling (**Figure 9**) with parallel-side profiles by varying the design helix angle β from 45 to 135°. Reference system and helix angle β of the spline coupling model are shown in **Figure 8**. In agreement with the helix angle definition, the middle of the range (β = 90°) corresponds to a spline with rectilinear generatrices (straight teeth).

As shown in **Figure 9**, the spline and hub submodels are merged into the coupling model by means of contact elements. Two load cases are analyzed, where the hub section of one model end is submitted to either the axial load F<sup>t</sup> or the torque Mt according to the values in **Tables 1** and **2**. Suitable constraints are added to the other end of the model, i.e, zero displacement components dr , dθ, and dz in cylindrical coordinates.

**Figure 9.** FEM model of helical spline coupling (β = 70°): (a) mesh of shaft and hub, (b) axial force loading, (c) torque loading, (d) boundary conditions.

Numerical results evidence that statics of spline couplings obeys two important rules. First, the load transfer through the spline due to the axial thrust Ft is small (6.7% of the thrust for β = 135° deg, i.e. the helical angle of maximum load transfer), and it is caused by deformations of the kinematic pair. In other words, if the hub and the shaft were perfectly stiff, the turbine thrust would not be transmitted at all to the shaft through the spline surfaces, but it would be carried by the constraint (4), i.e., the main axial bearing. Second, the load transfer R from the hub to the shaft through the spline due to the torque Mt is actually ruled by the following relationship, valid for a spline pair with perfectly stiff members

$$R = \frac{M\_i}{r\_p \tan \beta} \tag{1}$$

where rp is either the pitch radius in case of involute splines or the inner radius for parallel key splines.

**Figure 10** compares the shaft and hub reactions computed by means of Eq. (1) and the FEM model with frictionless contact elements (torque load case). The shaft thrust is the reaction that the constraints exert on the grooved part of the shaft, and it represents the load transfer R from the hub to the remaining part of the shaft through the spline surfaces. The hub thrust is equal and opposite according to Newton's third law. Therefore, in case of compliant members, the rule defined by Eq. (1) is still valid with negligible error (0.6% of the transmitted load for β = 135°). Equivalently, the compliance of the kinematic pair members does not yield perceivable effects for the nominal value of torque Mt . Details of the FEM analyses will be published in the near future.

**Figure 10.** Reactions of shaft and hub constraints for different helix angles under torque load.

#### **4.5. Design of the support system**

**4.4. Law of load distribution**

ing (4), which carries the load F<sup>t</sup>

the shaft is Fc

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components dr

loading, (d) boundary conditions.

First, the law of load distribution followed during nominal operation by the helical spline pair, employed as a mechanical actuator besides a simple coupling system, is determined.

**Figure 7** depicts the forces acting in nominal conditions on the rotor components according to the modifications resulting from the innovation. The constraint simulates the main axial bear-

the turbine impeller exerts on the shaft through the helical spline. The total thrust that acts on

A campaign of FEM structural analyses has been carried out on a model of helical spline coupling (**Figure 9**) with parallel-side profiles by varying the design helix angle β from 45 to 135°. Reference system and helix angle β of the spline coupling model are shown in **Figure 8**. In agreement with the helix angle definition, the middle of the range (β = 90°) corresponds to

As shown in **Figure 9**, the spline and hub submodels are merged into the coupling model by means of contact elements. Two load cases are analyzed, where the hub section of one model

and **2**. Suitable constraints are added to the other end of the model, i.e, zero displacement

Numerical results evidence that statics of spline couplings obeys two important rules. First,

**Figure 9.** FEM model of helical spline coupling (β = 70°): (a) mesh of shaft and hub, (b) axial force loading, (c) torque

β = 135° deg, i.e. the helical angle of maximum load transfer), and it is caused by deformations

the load transfer through the spline due to the axial thrust Ft

in cylindrical coordinates.

or the torque Mt

−R and is carried by the auxiliary axial bearing (3). The torque M<sup>t</sup>

torque of the turbine due to the pressure exerted on the relevant blades.

a spline with rectilinear generatrices (straight teeth).

end is submitted to either the axial load F<sup>t</sup>

, dθ, and dz

−R. The axial forces R are the (equal) action and reaction that

is the resisting

according to the values in **Tables 1**

is small (6.7% of the thrust for

In the following, the above-explained laws of load distribution are used to choose the design parameters in the assumption, adopted for the sake of simplicity, that the members of the spline pair are stiff. Such an assumption does not lead to significant errors, as proved above. By means of the resulting design procedure, the load distribution in nominal operating conditions between the two axial bearings can be set by means of a proper choice of the helical angle. To this purpose, **Figure 11** shows for the data reported in **Tables 1** and **2** (Case B) the load transfer through the coupling together with the corresponding axial loads of the bearings as a function of the helix angle β in nominal working conditions.

**Figure 11.** Trends of axial bearing thrusts and load transfer as a function of helix angle in nominal operating conditions.

By assuming that positive torque acts on a turbine impeller (M = M<sup>t</sup> > 0), handedness of the helix must be chosen so that the load transfer R is directed as in **Figure 7**. In other words, the spline coupling must exert (equal) axial forces R opposite to F<sup>t</sup> and Fc on the hub and the shaft, respectively. According to Eq. (1), such a condition yields that the range of helix angle in the abscissa of the plot in **Figure 11** cannot exceed 90°.

The load transfer R (thick solid curve) is plotted in **Figure 11** according to Eq. (1). On the basis of its trend, the axial thrusts Fa and Fs that turbine and shaft, respectively, exert on the main bearing (4) and on the auxiliary bearing (3) are plotted as dashed and dash-dotted curves. They are evaluated by means of the relations Fa = F<sup>t</sup> −R and F<sup>s</sup> = F<sup>c</sup> −R, which can be deduced by the analysis of **Figure 7**, where turbine and compressor thrusts Ft and Fc are obviously constant in nominal operating conditions. The gray solid horizontal line represents the value of the reference load Tref = F<sup>t</sup> −F<sup>c</sup> , which has to be carried by the single thrust bearing of a conventional machine. As reported in the paragraph dealing with layout and clearly visible in **Figure 11**, when a straight grooved spline (β = 90°) is picked, the load acting on the main axial bearing (4) is greater than in a conventional plant. Indeed, it is equal to the total turbine thrust Ft , while the auxiliary axial bearing (3) supports the compressor thrust F<sup>c</sup> . By reducing the helix angle, the loads acting on both bearings begin to decrease, since the helix is oriented in such a way as to exert on the shaft and the turbine impeller a thrust R opposite to Fc and Ft , respectively (**Figure 7**). Particularly, by choosing β roughly equal to 74°, the main axial bearing (4) must carry the reference load (F<sup>a</sup> = Tref), while the auxiliary one is axially unloaded and, therefore, since it is only subjected to the (light) radial load, it will have an average life exceeding 6 million hours, as specified in the first row of **Table 5**. Differently, for design values of β ranging between 68 and 74°, the load of the main bearing (4) becomes lower than the reference one, at the expense of the duration of the auxiliary bearing (3), on which the shaft exerts a negative thrust Fs (directed from the turbine to the compressor). For β = 68° the load on the main axial bearing (4) is null and, consequently, the thrust exerted by the shaft on the auxiliary bearing (3) assumes its maximum value, i.e., F<sup>s</sup> = −Tref. In such a condition, the rolling element bearing (3) exhibits the same duration as in a conventional layout (e.g., see the basic rating life reported in the last two rows of **Table 5**). Obviously, since in the simplified layout of **Figure 4**, the main foil bearing is not of double effect, helix angles lower than 68°, which moreover would lead to an even more unsuitable life of the auxiliary bearing, are forbidden. A good design may require a helix angle slightly reduced in comparison with a value of 74°, where the amount of such reduction can be evaluated by taking into account the life and the reliability required for the bearing (3). In the design range, the highest life/reliability of auxiliary bearing (3) is obtained for β = 74°, while the lowest one, typical of a conventional layout, for β = 68°; the most severe loading case for the main axial bearing (4), equivalent to that of a conventional layout, occurs at β = 74°, the most favorable one (zero thrust) at β = 68°.

By means of the resulting design procedure, the load distribution in nominal operating conditions between the two axial bearings can be set by means of a proper choice of the helical angle. To this purpose, **Figure 11** shows for the data reported in **Tables 1** and **2** (Case B) the load transfer through the coupling together with the corresponding axial loads of the bear-

ings as a function of the helix angle β in nominal working conditions.

By assuming that positive torque acts on a turbine impeller (M = M<sup>t</sup>

and Fs

by the analysis of **Figure 7**, where turbine and compressor thrusts Ft

−F<sup>c</sup>

spline coupling must exert (equal) axial forces R opposite to F<sup>t</sup>

abscissa of the plot in **Figure 11** cannot exceed 90°.

They are evaluated by means of the relations Fa

bearing (4) must carry the reference load (F<sup>a</sup>

of its trend, the axial thrusts Fa

of the reference load Tref = F<sup>t</sup>

thrust Ft

20 Bearing Technology

Ft

helix must be chosen so that the load transfer R is directed as in **Figure 7**. In other words, the

**Figure 11.** Trends of axial bearing thrusts and load transfer as a function of helix angle in nominal operating conditions.

respectively. According to Eq. (1), such a condition yields that the range of helix angle in the

The load transfer R (thick solid curve) is plotted in **Figure 11** according to Eq. (1). On the basis

bearing (4) and on the auxiliary bearing (3) are plotted as dashed and dash-dotted curves.

constant in nominal operating conditions. The gray solid horizontal line represents the value

conventional machine. As reported in the paragraph dealing with layout and clearly visible in **Figure 11**, when a straight grooved spline (β = 90°) is picked, the load acting on the main axial bearing (4) is greater than in a conventional plant. Indeed, it is equal to the total turbine

the helix angle, the loads acting on both bearings begin to decrease, since the helix is oriented in such a way as to exert on the shaft and the turbine impeller a thrust R opposite to Fc

, respectively (**Figure 7**). Particularly, by choosing β roughly equal to 74°, the main axial

and, therefore, since it is only subjected to the (light) radial load, it will have an average life exceeding 6 million hours, as specified in the first row of **Table 5**. Differently, for design values of β ranging between 68 and 74°, the load of the main bearing (4) becomes lower than the

, while the auxiliary axial bearing (3) supports the compressor thrust F<sup>c</sup>

= F<sup>t</sup>

−R and F<sup>s</sup>

> 0), handedness of the

on the hub and the shaft,

−R, which can be deduced

are obviously

. By reducing

and

and Fc

and Fc

that turbine and shaft, respectively, exert on the main

= F<sup>c</sup>

, which has to be carried by the single thrust bearing of a

= Tref), while the auxiliary one is axially unloaded

Finally, the actual assembly drawing of the invention, suited to both Cases A and B of **Table 2** as well as transient loading conditions, is reported in **Figure 12**. In this case, the total hot clearance between runner (11) and pads (12) of the double-effect air bearing must be higher than that between turbine impeller (1) and the spacer (13) used to adjust the impeller axial clearance. The second set of (nonlocating) angular contact bearings (14) cited in paragraph 4.1 is added.

**Figure 12.** Section of the micro-GT support system assembly designed according to the invention.
