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Hindawi Advances in Tribology Volume 2018, Article ID 9016906, 12 pages https://doi.org/10.1155/2018/9016906 Research Article Fabrizio Stefani , Ramon Francesconi, and Andrea Perrone Department of Mechanical, Energy, Management, and Transportation Engineering, University of Genova, Via all’Opera Pia 15, 16145 Genova, Italy Correspondence should be addressed to Fabrizio Stefani; stefani@unige.it Received 3 January 2018; Accepted 7 February 2018; Published 15 March 2018 Academic Editor: Huseyin C ¸ imenogl ˇ u Copyright © 2018 Fabrizio Stefani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The design of the support system (shaft, bearings, and mechanical coupling devices) of the rotor plays a key role in the development of efficient micro-gas turbines (micro-GTs) for distributed power generation. Foil air bearings are the most widespread technical solution well suited to design a reliable support system, although they cannot withstand a large number of start-stop cycles of the units. In order to overcome such limitation, we have recently proposed an innovative support system that takes advantage of spline couplings and two bearing types (e.g., air and rolling-element bearings). The devised support system employs splines as both convenient coupling systems and actuators for the load partition between the two bearing types. In the present work, the helical spline coupling is studied by means of structural FEM analyses including contact simulation in order to design the support system. Numerical results confirm previous findings in that the load transfer through the spline coupling is mainly a function of the helix angle. In addition, friction factor and structural stiffness cannot be neglected in the accurate design of the spline coupling. Such design parameters are now included in the proposed design procedure, which formerly assumed frictionless contact and rigid bodies. 1. Introduction However, in the design of a novel support system for the shaft, it is reasonable to consider magnetic bearings as Due to the high speeds of rotation (in the order of 10 rpm) a convenient alternative. Indeed, their advantages over con- and operating temperatures (up to 1000 C), bearings are ventional bearings are as follows: oil-free operation, extreme among the most stressed mechanical components in micro- temperatureaswellasactivecontrol foractivemagnetic gas turbine (micro-GT) systems, and therefore lubrication as bearings (AMBs) [1], ease of miniaturization [2], and inde- well as a support system plays a key role in the operation of the pendence of external energy input for passive magnetic bear- machine. ings (PMBs) [3]. Usually, the advantage of the resistance to eTh adoption of a single type of bearing is typical of high temperatures makes AMBs the preferred choice for the bearing arrangements that are usually employed in commer- application at hand. cial micro-GT units. Existing micro-GT installations employ Despite such features of magnetic supports, foil air foil (aerodynamic) and rolling-element bearings, where the bearings are considered the rfi st choice for oil-free micro-GT former solution is certainly more innovative and promising. applications. Indeed, in comparison with AMBs, foil bearings Conventional bearings (rolling-element type) are still used in have lower upfront cost and fewer failure modes [4] and micro-GT systems for reasons of size and cost; for example, do not consume energy to supply electromagnets. However, since1999,Capstonehaveoeff redaversionoftheirModel magnetic bearings ensure longer life, lower friction, and 330 microturbine with a ball-bearing-based compressor and potentially less severe failures. continued to offer the ball bearing compressor as an econom- eTh refore, magnetic bearings are preferred for large ical option for medium-pressure applications. Differently, oil- engines operating at high loads and relatively lower speed, free supportsystems rely onthemainpatentofCapstone while the opposite is true for foil bearings. Indeed, air technology, which has been developed by the US National bearings do not carry high loads at lower speeds, as gas Aeronautics and Space Administration (NASA). viscosity is low, and adhesion of their tribological coatings to 2 Advances in Tribology large diameter shafts under large centrifugal forces is difficult couplings provide higher load carrying capacity that oen ft [1]. turns into better durability and allows for a certain amount With regard to conventional bearings (rolling-element ofangularmisalignmentaswellasrelativesliding between type), oen ft used in low-power micro-GT units for reasons of the shaft and the hub [15]. size and cost, the fundamental problem is the duration that is In this work, the use of a helical spline as an actuator limited by the phenomenon of fatigue and is load-dependent. capable of transferring axial load between two bearings of Indeed, for these supports, the “carrying capacity” is the convenient type is studied. The operation of helical splines ability of the bearing to carry a given load for a predetermined is analyzed by means of a nonlinear structural model of the number of cycles or revolutions [5]. hub-shaft assembly and the n fi ite element method (FEM). On the other hand, foil bearings have a life dependent onthenumberofstartsand stopsofthemachine.Rubbing 2. Case Study betweenthetopfoilandjournalsurfaces,whicharepreloaded againsteachother,occursatalowspeedwhentheshaftis Table 1 reports the main data of a 100 kW micro-GT unit not airborne. er Th efore, starts and stops are the only events in designed in previous works [12, 16]. It also includes the which the bearing sheets are prone to wear. Hence, expensive relevant loads, that is, the turbine and compressor thrusts tribological coatings on both the top foil and the shaft are 𝑇 and𝑇 , respectively, as well as the vertical load𝑊.Such 𝑡 𝑐 required [6, 7]. eTh start/stop performance is important as thrusts, whose positive direction is from the compressor to micro-GT units are expected to withstand the wear of daily the turbine, are the result of the axial forces exerted on the starts and stops. A US government/industry program has got blades andonthe backside oftheimpellers by theworking recently underway, where the machines are tested to assess uid fl . Differently, load 𝑊 is due to the weight of the rotor. Two their performance including start and stop capabilities. load cases (referred to as A and B) characterized by the same eTh proof of the research eor ff t to improve the start/stop modulus (a reference value of 500 N) of the resultant thrust, performance of air bearing yields is given by many published 𝑇 =𝑇 +𝑇 , (1) works. For example, solid lubrication has been studied in ref 𝑟 𝑐 many pertinent papers [6, 8–10]. Advances in such field are studied. Particularly, in case A, both the turbine and the are turned into new patents. For example, a powder having compressor impeller thrusts are directed toward the external lubricating properties can be interposed between the bearing side of the unit (𝑇 >0,𝑇 <0). Differently, in case B, 𝑡 𝑐 surfaces in order to provide lubrication of foil bearings at both thrusts are directed toward the inner side of the machine low speed, when the journal is not airborne [11]. It forms a and the total thrust direction reverses too (𝑇 <0,𝑇 >0). 𝑡 𝑐 film whose lubricity and adhesion properties are capable of As detailed in [12], for different impeller geometries, which reducing torque and increasing bearing life. eTh powder acts may yield different pressures on the clearances between the during mixed lubrication and it is displaced when the journal casing cover and impeller back shroud (backside pressures), is airborne. Nevertheless, it may influence aerodynamic the impeller thrusts can be either external or internal, as in lubrication to a certain extent. In any case, solid lubrication cases A and B, respectively. cannot get rid of wear completely. Compressor and turbine axial loads are the most signif- In [12],wehavepresentedtheconceptualdesignofa icant in a microturbine, since they are 10x greater than the new support system that is capable of removing axial load radial ones (shaft weight). For the application in this study, from the air thrust bearing during mixed lubrication regime, due to high axial loads, durations of rolling-element bearings thus eliminating start/stop wear. The axial load is shared by correlated to fatigue phenomenon are very low compared two axial bearings of different types. Indeed, the invention is with the life of a micro-GT unit (60.000–80.000 hours) [12]. based on the idea that since each bearing type has dieff rent strengths and weaknesses, using different types of bearings in the same support system yields benefits. Accordingly, the 3. The Innovative Layout proposed innovation overcomes the above-cited flaws of the As already explained in detail in [12], the main goal of support systems adopted in micro-GT and, at the same time, the novel assembly is to separate the axial load exerted by takes advantage of the best qualities of different bearing types. the impellers into two parts. eTh rst fi part of the thrust Indeed, it is capable of matching different types of bearings directly loads the main axial bearing, without getting through and managing the relevant loads by means of suitable sha-ft the sha.ft This type of load application will be referred to hub couplings. as “direct” in the following. Differently, the second part is A convenient solution for the coupling between the shaft managed like in conventional support systems; that is, the and hubs of compressor and turbine rotors may be the use of impellers exert the thrust on the shaft that, in turn, transfers splines. Indeed, aircraft engines adopt splines and many gas it to an additional (auxiliary) axial bearing. turbines are derivatives of aircraft engines [13]. In comparison with shrink ts, fi such design solution, without heavily limiting Particularly, for the sake of simplicity, only the axial load the transmissible torque as in the case of keys, boasts the exerted by the turbine impeller is separated, as it is higher considerable advantage of ease in assembly and disassembly than the compressor one. eTh direct application of the axial [14]. This may be particularly interesting for a small machine load is not a new idea, as reported in [17]. Differently, in like a microturbine, in order to facilitate maintenance and such reference patent, the compressor impeller in place of the inspections. In addition, in comparison with keys, spline turbine one directly exerts thrust on the axial bearing. Advances in Tribology 3 Table 1: Design data of the reference micro-GT unit as well as the spline coupling. Design variable [unit] Value Rotational speed𝑁 [rpm] 70,000 Shaft diameter 𝐷 [mm] 15 Power P [kW] 110 Pressure ratio 4 Turbine thrust𝑇 [N] 1100 (case A),−1700 (case B) Micro-GT data 𝑡 Compressor thrust𝑇 [N] −600 (case A), 1200 (case B) Total axial load𝑇 [N] 500 (case A),−500 (case B) ref Radial load (rotor weight)𝑊 [N] 40 Turbine torque𝑀 [N m] 30 Compressor torque𝑀 [N m] −15 IS0 141982 (E), shaft and hub Type 6×11×14 Length𝐿 [mm] 220 Outer diameter𝐷 [mm] 220 Spline inner (or pitch) radius𝑟 [mm] 5.5 Helical spline coupling data Spline outer radius𝑟 [mm] 7 Number of splines𝑛 6 Side distance𝐵 [mm] 3 Young’s modulus [GPa] 206 Poisson’s ratio 0.3 (3) that the main axial bearing (4) exerts on the turbine impeller (9) (1) (2) Fa (1),while𝐹 is thethrustthatthe auxiliaryaxial bearing(3) exerts on the shaft (5). Considering force equilibrium in static Tt (5) condition yields Tc 𝐹 =−𝑇 −𝑅 R 𝑎 𝑡 Mt (2) 𝐹 =𝑇 −𝑅. (7) 𝑠 𝑐 (6) 4A Fs Constraints (4A) and (4B) in Figure 1 simulate in case either A or B, respectively, the main axial bearing (4) that 4B (9) carries the load𝐹 . eTh axial forces 𝑅 are the (equal) action and reaction that the turbine impeller exerts on the shaft Figure 1: Static scheme (axial forces) of the innovative rotor through the helical spline and will be referred to as load operating in nominal conditions. transfer. Its modulus is lower than the ones of turbine and compressor thrust. eTh total thrust that acts on the shaft is 𝐹 and is carried by the auxiliary axial bearing(3).Thetorque In order to take advantage of the above-cited peculiar 𝑀 is the resisting torque of the turbine due to the pressure attitudes of each bearing type, main and auxiliary bearings exerted on the relevant blades. are different (e.g., the main thrust support is a foil air bearing, In the case of positive resultant thrust (case B,𝑇 <0), while the auxiliary one is rolling-element type). equation (2) yields that the constraint modeling the main eTh refore, such thrust load separation allows the sup- thrust bearing in Figure 1 must be (4B); that is, the axial force port system to switch between the two types of bearings exerted by the bearing on the impeller is positive (𝐹 >0). automatically when the unit ends the transient operation. In Contrarily, in case A, since𝐹 <0,the constraint must be addition, it employs helical splines as both convenient shaft- (4A). In other words, the location of the thrust main bearing impeller coupling systems and actuators suitable to adjust the (4) with reference to the machine frame should be different entity of the load transferred to each thrust bearing. Indeed, in cases A and B. a convenient partitioning of the turbine thrust is required in For example, the runner of the main thrust bearing (4) order to optimize the behavior of the support system [12]. might be mounted on the high-pressure side of the impeller Figure 1 depicts all of the axial forces acting on the rotor (1) and on its opposite side in cases B and A, respectively. components in nominal conditions, according to the modi- The location of the pads of bearing (4) might be modified fications resulting from the innovation. 𝐹 is the axial force accordingly. Nevertheless, such solution including single 𝑎 4 Advances in Tribology (2) (11) (1) (12) (10) (13) (3) (17) (14) (16) (8) (5) (7) (6) (4) (9) Figure 2: Section of the micro-GT support system assembly designed according to the invention. Table 2: Components and details numbered in the assembly figures. (1) (1) Turbine (impeller) ca1 (2) Compressor (impeller) ca2 (3) Journal bearing/auxiliary thrust bearing (4) Main thrust bearing (12) (5) Shaft (11) (6) Helical spline coupling (7) Spline cs1 cs2 (8) Shaft shoulder (13) (9) Frame (14) (16) (10) Pad spacer (17) (11) Thrust runner (8) (12) Thrust pads (10) (15) (6) (13) Turbine impeller spacer Figure 3: Clearances between the components of the support (14) Journal bearing system. (15) Inner ring shoulder of journal bearing (16) Flat washer (17) Locking ring nut is not included in the scheme of Figure 1, is added to the assembly near the turbine. In this case, at low speeds, the annular shoulder(8) does not receive the thrust load directly effect thrust bearings would not be optimal from the point from the turbine impeller(1).Indeed,itexertsthe thrust on of view of the machine layout. the inner ring side (15) (Figure 3) of the bearing(14).Anyway, A more straightforward solution consists in manufac- the behavior of the invention does not change, as the bearing turing a thrust collar/runner that is either an integral part (14) is not constrained in the axial direction and, therefore, of the turbine impeller or rigidly fastened to it, so that no axial load is transferred to the frame (9). pads of bearing (4) can be located on both sides of the The flat washer (16) is fastened to the shaft (5) with a runner regardless of the thrust direction. Accordingly, the locking ring nut (17). eTh spacer (13) is mounted between assembly drawing of the invention is reported in Figure 2. the washer (16) and the impeller(1) in order to adjust the Such support system is suited to both positive and negative axial gap of the turbine hub-shaft coupling. Similarly, two nominal thrusts (with a consistent choice of the helix angle spacers (10) are employed to adjust the axial clearance of the of the spline coupling) as well as transient loading conditions. airbearings(theactiveaswellastheinactive one). The numbered components are listed in Table 2. In this case, Particularly, as shown in Figure 3, the total axial clearance the main thrust bearing(4) is a double-effect foil air bearing 𝑐 of the turbine hub-shaft coupling is the sum of the 𝑠𝑡 insteadofasingleeeff ctone.Inother words,thepads (12) clearances𝑐 and𝑐 (𝑐 =𝑐 +𝑐 ). In the following, the gap 𝑠1 𝑠2 𝑠𝑡 𝑠1 𝑠2 are located on both sides of the thrust collar/runner(11). between pads and the runner of the air bearings is the desired The group of pads on the side that carries the thrust load in operating clearance and it is referred to as “hot” clearance. nominal working conditions is termed the “loaded” or active The gap 𝑐 is the clearance between the spacer (13) and the 𝑠1 bearing, while the other group, on the opposite side of the turbine impeller(1),while𝑐 is the gap between the turbine 𝑠2 thrust collar, is called the “slack” side or inactive bearing. impeller(1) and the inner ring side (15) of the bearing (14). In Figures 2 and 3, a second set of (nonlocating) angular Similarly, the total hot clearance𝑐 of the double-effect air contact bearings(14) in back-to-back arrangement, which bearing (4) is given by the sum of two contributions𝑐 and 𝑎1 𝑎𝑡 Advances in Tribology 5 components are matched and their interface is modeled by means of surface-to-surface contact elements (CONTA173 and TARGE170) with zero initial gap and isotropic friction (friction factor𝑓=0 , 0.15, 0.3, and 0.45). eTh resulting nonlinear model of the coupling includes 103,566 nodes and 104,160 elements. The default augmented Lagrangian solver is employed. eTh elapsed time required to calculate results for all the considered ranges of 𝛽 with a step of 5 deg (a single curve in the plots discussed below) is roughly 4-5 hours (6) on a conventional PC (AMD FX-8350 eight-core processor, 4 GHz clock, 16 GB RAM). Although a second-order mesh (including SOLID186, CONTA174 element types) has been also tested, due to the much higher elapsed time, the relevant resultshavebeenonlyusedinordertoconrfi m theresults obtained by means of the first-order mesh. Some second- order calculations have conrfi med the trends published in the following.Acyclicmodel hasalsobeentestedinorderto Figure 4: Helix angles and reference system of spline coupling. reduce the elapsed computational time. Nevertheless, it does not give consistent results under torque load and when the lead angle is small. Initially, case B (the case with higher thrust) is simulated 𝑐 , which are the hot clearances between runner (11) and pads 𝑎2 aer ft it is subdivided into two simpler load cases; that is, the (12) of theactiveand theinactivebearing(𝑐 =𝑐 +𝑐 ). 𝑎1 𝑎2 hub is loaded either by an axial force𝐹 or by a torque𝑀 In order to avoid wear of the double-effect air bearing (Figure 4). Particularly, as suggested by Figure 1 the nominal (4) due to dry contact between the runner and pads at low thrust𝐹=𝑇 and torque𝑀=𝑀 (Table 1) are applied to 𝑡 𝑡 speed, its total hot clearance 𝑐 must be higher than the one side of the hub. Therefore, in the former load case, the clearance of the turbine hub-shaft coupling 𝑐 (𝑐 >𝑐 ). 𝑠𝑡 𝑠𝑡 hubisloadedbyacompressivestress(𝐹<0 ), while in the Indeed, the axial clearance𝑐 of the coupling (6) must be 𝑠𝑡 latter the torque acts on the hub so that, due to the threaded very little (in the micron-length scale). Anyway,𝑐 must be 𝑠𝑡 connection, it tends to move with reference to the shaft either greater than the equivalent RMS roughness of the two contact in the positive𝑧 direction if𝛽<90 deg or in the negative one surfaces at the impeller (1)/inner ring side (15) interface if𝛽>90 deg. Since reversing the direction of the torque load (or impeller/annular shoulder (8), if the bearing (14) is not yields periodic results (with period𝛽=90 deg) and the hub employed), in order to provide the relief of the secondary stress due to the thrust load relevant to case B is compressive axial bearing (3) over the speed at which the runner (11) between the blades and the bearing, unidirectional loads are becomes airborne. considered. Finally, a combined load case is simulated, where force and torque are simultaneously applied. 4. FEM Analysis of the Spline Coupling ea Th xialloadonthehubsideisobtainedbydistributing a suitable uniform pressure on the corresponding surface, The law of load distribution followed during nominal oper- in which the axial section localized at𝑧=𝐿 is shown ation by the helical spline pair, employed as a mechanical in Figure 6(a). Differently, the torque is generated on the actuator besides a simple coupling system, must be deter- same section (𝑧=𝐿 )eitherbymeans of amultipoint mined. To this end, a campaign of FEM structural analyses constraint (MPC) contact region that is bonded to a pilot has been carried out on a model of a helical spline coupling node transmitting the torque or, equivalently, by applying with parallel-side profiles by varying the design lead angle circumferential nodal forces proportional to the radial coor- 𝛽 from 45 to 135 deg (admissible range for helical gears). dinate (Figure 6(b)). At the opposite side of the assembly Design data of the spline coupling are reported in Table 1. (section localized at𝑧=0 ), the nodes lying on the shaft are eTh reference system, the lead angle 𝛽,andthehelixangle fully constrained, while those on the hub section cannot move 𝛼 (geometric complement of𝛽) of the spline coupling model in the axial direction, as summarized by Figure 6(c). are shown in Figures 1 and 4. In agreement with the lead angle definition, the middle of the range ( 𝛽=90 deg) corresponds to a spline with rectilinear generatrices (straight teeth). 5. Results and Discussion ev Th irtualmodel,fully developedbymeans ofthe Figures 7 and 8 plot the partition of axial load between the ANSYS Parametric Design Language (APDL) in ANSYS 15.0, hubandtheshaftasafunction oftheleadangleforaxial is parametric in that the geometry is completely defined by force and torque load cases, respectively. In these gfi ures, parameters (diameters, axial length, number of keys, and if not differently specified by means of a suitable friction helix angle). The three-dimensional models of hub and shaft are meshed by means of first-order isoparametric structural factor label, FEM results are obtained by means of frictionless solid elements (SOLID185), as shown in Figures 5(a) and contact elements. The plotted FEM model axial reactions 𝑅 5(b), respectively (for𝛽 = 135 deg). Then, the two meshed and𝑅 of the hub and the shaft, respectively, are computed by 𝑎𝑡 𝑎𝑡 𝑎𝑡 6 Advances in Tribology (a) (b) Figure 5: Finite element mesh of the helical spline coupling (𝛽=135 deg): (a) hub, (b) shaft. −F (a) (b) d =d =d =0 r z d =0 (c) Figure 6: Loads and boundary conditions applied to the FEM model of the helical spline coupling ( 𝛽 = 135 deg): (a) axial force loading (axial section𝑧=𝐿 ), (b) torque loading (axial section𝑧=𝐿 ), and (c) boundary conditions (axial section𝑧=0 ). Advances in Tribology 7 45 55 65 75 85 95 105 115 125 135 Lead angle (deg) −2000 −4000 40 50 60 70 80 90 100 110 120 130 140 −200 −6000 Lead angle (deg) Rs, rigid model Rs, rigid model Rh, rigid model Rh, rigid model Rs, FEM model Rs, FEM model Rh, FEM model Rh, FEM model Rs, FEM model f = 0.15 Rs, FEM model f = 0.15 Rh, FEM model f = 0.15 Rh, FEM model f = 0.15 Rs, FEM model f = 0.3 Rs, FEM model f = 0.3 Rh, FEM model f = 0.3 Rh, FEM model f = 0.3 Rs, FEM model f = 0.45 Rs, FEM model f = 0.45 Rh, FEM model f = 0.45 Rh, FEM model f = 0.45 Figure 8: Reactions of shaft and hub constraints for different lead Figure 7: Reactions of shaft and hub constraints for different lead angles under torque load calculated by means of frictionless rigid angles under axial force load calculated by means of frictionless rigid and FEM model without and with friction. andFEM modelwithoutandwithfriction. force and torque load cases reach a maximum of 15% for summing the nodal reactions of the constrained nodes that 𝛽=45 or 135 deg and 140% at𝛽=45 deg, respectively. lie on the corresponding surfaces. Figure 1 enables us to relate eTh refore, the influence of friction on the load transfer is very the FEM model reactions to the axial thrusts of the device as significant so that the manageable analytical rigid model can follows: be only used as a rfi st approximation in the machine design. 𝑅 =𝐹 ℎ 𝑎 With respect to the straight-tooth spline geometry (𝛽= (3) 90 deg) for the axial force load case, 𝑅 and 𝑅 curves ℎ 𝑠 𝑅 =𝑅. are symmetric, while for the torque load case they are antisymmetric only when friction is neglected. Particularly, Accordingly,𝑅 is the reaction that constraints exert on the theloadtransfer𝑅 due to torque is not zero for the straight- grooved part of the shaft and it represents the load transfer tooth spline when friction is taken into account. Indeed, 𝑅 from the hub to the remaining part of the shaft through sinceunder torque loadtherotations of theaxialsections thesplinesurfaces. In thesamefigures, Figures7and8,the increase with their distance from the constrained section, corresponding results for a frictionless rigid model are added the deformed shape of the straight generatrix of a tooth for the sake of comparison. becomes similar to a helix, as shown in Figure 9, so that the eTh analytical expression of 𝑅 for the frictionless rigid resultingtangentialstressesdueto friction yieldanaxialload model has been already defined in [12], where it has been component. found that deformations exert little effects on load transfer in frictionless spline couplings. Indeed, in Figures 7 and For both axial force and torque load cases, maximum 8, the rigid model trends of 𝑅 and 𝑅 (dashed lines) tfi equivalent stress and maximum contact pressure in the cou- ℎ 𝑠 the frictionless FEM model corresponding points (circle pling model are plotted as a function of lead angle in Figures markers) with low and negligible relative errors, in the order 10 and 11, respectively. In comparison with the axial force, of 1%and0.1%oftheappliedaxial forceandtransmittedload, the torque load case yields much higher stresses (2 orders of respectively. On the contrary, when a friction factor𝑓=0.3 magnitude), as in this condition the coupling behaves like an is assumed, these relative errors (difference between results actuator that without the constraints would cause a relative of frictionless rigid and FEM models with𝑓=0.3) for axial motion between the shaft and the hub. eTh refore, stress due Load (N) Load (N) 8 Advances in Tribology 9 0 40 60 80 100 120 140 Figure 9: Deformed shape of the straight-teeth spline shaft (lead angle𝛽=90 deg) under torque local case (magnification factor = Lead angle (deg) 800). Maximum VM stress-force Maximum VM stress-force load f = 0.15 Maximum VM stress-force load f = 0.3 Maximum VM stress-force load f = 0.45 to torque load can exceed yield stress of the material and its Max. VM stress-torque load effect must be considered in structural design. Max. VM stress-torque load f = 0.15 For the force load case, maximum equivalent stress and Max. VM stress-torque load f = 0.3 maximum contact pressure trends (Figures 10 and 11) are Max. VM stress-torque load f = 0.45 symmetric with respect to𝛽=90 deg. At the same angle, Figure 10: Maximum Von Mises stress for different lead angles they reach minimum values, since the load transfer𝑅=𝑅 under axial force load (left ordinate) and torque load (right ordinate) (Figure 7) is either zero for the frictionless model or very calculated by means of FEM model without and with friction. little when friction is considered. Maximum Von Mises stress variations (Figure 10) with helix angle are greater when the lead angle𝛽 is lower than 70 deg or higher than 110 deg and values of 𝛼 and 𝑓. er Th efore, for a fixed helix angle, the thecurvestendtobecomeroughly linear.Accordingly,the curve of maximum contact pressure as a function of friction maximum contact pressures (Figure 11) are higher when helix factor has a minimum, as shown in Figure 12 for𝛽=45 deg. angles 𝛼 are greater in magnitude, in agreement with the eTh rise of the contact pressure at high friction factors load transfer rise (Figure 7). Indeed, an increase of𝑅 causes consequently reduces the variation of maximum equivalent contact pressure and, consequently, equivalent stress to grow. stress (Figure 12). As far as the effect of friction in the force load case In the torque load case, maximum equivalent stress and is concerned, Figure 7 shows that a growth of the friction maximum contact pressure trends are not symmetric with factor𝑓 tendstoleveloffthehub andshaftreactions ( 𝑅 respect to the straight-tooth spline configuration (Figures 10 and𝑅 )byincreasingtheload transfer𝑅 in agreement with and 11). Indeed, stress and contact pressure intensity depend equations (2) and (3). Nevertheless, such increase of𝑅 due on the handedness of the spline helix with reference to the to theriseof 𝑓 does not yield an increase of maximum torque direction. Similarly, in [15], the numerical simulations stress. On the contrary, it decreases the maximum equivalent of helical gears mounted on shafts by means of helical spline stress (Figure 10), since part of the load transfer is carried couplings have proved that the selection of a helical spline out by the friction forces, whose magnitude and direction with the same helix direction as that of the helical gear yields a are still unable to cause deformation of the pair members. reduction of the load concentration. Due to such asymmetry, Such phenomenon can also be observed in Figure 11, where, the minima of peak equivalent stress for the frictionless in comparison with the frictionless model results, limited model are reached for𝛽=60 deg or lower angles when the increases of𝑓 cause the maximum contact pressure to drop. friction factor is high (Figure 10). Nevertheless, for high values of the friction factor (𝑓>0.3), especially for high magnitude of the helix angle𝛼,maximum The friction in the torque load case reduces hub and shaft contact pressure rises with𝑓 duetothe action ofthehigh thrusts as well as the load transfer𝑅 (Figure 8) except for friction forces, which globally increase the deformations. the lead angles in the middle of the considered range where Such trend is also visible in the maximum equivalent stress 𝑅 rises with 𝑓. Accordingly, maximum contact pressure plot (Figure 10, same values at𝛽=45 or 135 deg for𝑓= (Figure 11) decreases by increasing𝑓 except for the above- 0.3 and 0.45), although it is less evident and requires higher cited 𝛽 middle range. Maximum Von Mises stress curves Stress-thrust load (MPa) Stress-force load (MPa) Advances in Tribology 9 30 2500 18 20 12 0 0 0 0 0.1 0.2 0.3 0.4 0.5 40 60 80 100 120 140 Friction factor f Lead angle (deg) Maximum VM stress-force load Maximum contact pressure-force load Maximum contact pressure-force load Maximum contact pressure-force load f = 0.15 Figure 12: Maximum Von Mises stress and maximum contact Maximum contact pressure-force load pressure for different friction factors at a constant lead angle ( 𝛽= f = 0.3 45 deg) under axial force load calculated by means of FEM model. Maximum contact pressure-force load f = 0.45 Max. contact pressure-torque load Max. contact pressure-torque load f = 0.15 maximum Von Mises stress exceeds 1000 MPa for lead angles Max. contact pressure-torque load f = 0.3 greater than 110 deg. Max. contact pressure-torque load f = 0.45 Figure 11: Maximum contact pressure for different lead angles 6. Design of the Support System under axial force load (left ordinate) and torque load (right ordinate) calculated by means of FEM model without and with friction. A simple method to choose the helix angle has been previ- ously published in [12], where it is only applied to load case B, and the eeff cts of friction, axial force, and deformations of (Figure 10) follow a similar trend to maximum contact the helical spline coupling (6) are neglected. In the present paper, all of these effects are taken into account in order to pressure plots, although they are less sensitive to friction. eTh improve the design method. middle region of lead angles where stress increases with𝑓 is eTh relevant graphicalconstruction, showninFigures 15 wider and more shifted to low angles. and 16, is aimed at n fi ding the possible design range for the Figure 13 depicts the reactions𝑅 and𝑅 calculated at the ℎ 𝑠 lead angle𝛽. To this end, the load transfer 𝑅 through the different lead angles under combined load case (simultaneous coupling together with the corresponding axial loads𝐹 and application of force and torque). Such gfi ure compares the 𝐹 of the bearings(4) and(3),respectively, is plottedasa trends obtained by taking advantage of the frictionless rigid function of the lead angle𝛽 in nominal working conditions. model (dashed curves), FEM model with friction (𝑓 = 0.3, In Figures 15 and 16, the constant trends (thick solid lines), data marked with triangles), and the superimposition of whichplotthemagnitudeofthe resultantload𝑇 ,canbe ref effects (square labels), that is, the sum of the values obtained compared with𝐹 in order to assess for which lead angles for𝑓 = 0.3 by meansoftheFEMmodelintwoseparate the main thrust bearing is either penalized or favored in analyses, under force and torque loads. The plot confirms that comparison with a conventional support system [12]. eTh friction and deformations must be taken into account and trends𝑅 are computed by means of either the FEM model the superimposition of effects is not valid due to the non- for a friction factor𝑓 = 0.3 and combined load case or the linear behavior of the model. eTh corresponding maximum frictionless rigid model. eTh corresponding curves of 𝐹 and equivalent stress and maximum contact pressure trends are 𝐹 are found by means of equation (2). In case B (Figure 15), plotted in Figure 14. As far as such parameters are concerned, the curves of the load transfer𝑅 for frictionless rigid and in comparison with reaction curves, superimposition of eeff ct FEM model (𝑓 = 0.3) are the same as plotted in Figure 13, yields better agreement with the FEM model results obtained where both torque (𝑀=𝑀 ) and compressive axial force under combined load.Forthesakeofstructuralstrength, the (𝐹=𝑇 <0) are considered in the numerical analysis. design lead angle should be as low as possible. Particularly, Differently, in case A (Figure 16), a new plot of 𝑅 is calculated Stress-thrust load (MPa) Stress-force load (MPa) Stress (MPa) 10 Advances in Tribology Table3:Designrangesofleadanglesinthetwoloadcasescalculated by means of the different models. Load case Model 𝛽 [deg] 𝛽 [deg] min max B Frictionless rigid 72.0 77.4 1400 BFEM𝑓=0.3 81.0 88.2 A Frictionless rigid 95.4 100.8 AFEM𝑓=0.3 108.9 115.2 45 55 65 75 85 95 105 115 125 135 Lead angle (deg) 45 55 65 75 85 95 105 115 125 135 Max. VM stress-combined load f = 0.3 Lead angle (deg) −2000 Max. contact pressure-combined load f = 0.3 Max. VM stress-combined load superimposition f = 0.3 −4000 Max. contact pressure-combined load superimposition f = 0.3 −6000 Figure 14: Maximum equivalent stress and contact pressure for different lead angles under combined load calculated by means of −8000 the FEM model (friction factor = 0.3) as well as by summing results from analyses under force and torque loads. Rs, rigid model Rh, rigid model Rs, FEM model f = 0.3 Rh, FEM model f = 0.3 Rs, superimposition f = 0.3 thrust can vary noticeably in intensity and reverse. er Th efore, Rh, superimposition f = 0.3 the range of design helix angles may considerably change Figure 13: Reactions of shaft and hub constraints for dieff rent lead with the impeller design. As a rule, the design limits of𝛽 angles under combined load calculated by means of the frictionless must be included within the range of lead angles feasible rigid and the FEM model (friction factor = 0.3) as well as by for helical spline couplings and gears; that is,𝛽 must be in summing results from analyses under force and torque loads. the range of 45–90 deg when both compressor and turbine thrusts are directed toward the inner side of the machine and the resultant thrust is negative, as in case B. bysimultaneouslyapplyingtorqueandtensileaxialforce(𝐹= Differently, when the resultant thrust is positive ( 𝑇 >0, ref 𝑇 >0)tothe FEMmodel(𝑓 = 0.3). Such calculation is as in case A), according to the proposed graphical construc- compulsory, as the coupling model is nonlinear. tion, the trend of the load transfer𝑅 must reach negative For the reasons explained in [12], the method of locating values in order to make the thrust functions (𝐹 and 𝐹 ) 𝑎 𝑠 the limits for the possible design values of 𝛽 consists in zero, as shown in Figure 16 and suggested by equation (2). finding the zeros of the functions 𝐹 =𝐹 (𝛽) and𝐹 =𝐹 (𝛽). 𝑎 𝑎 𝑠 𝑠 According to load transfer results, this is possible if the lead Such zeros enclose the design range of the lead angles and angle𝛽 is greater than 90 deg, that is, by means of a reversal they are represented by star symbols in Figures 15 and 16, of the helix handedness. Again, the lead angle cannot exceed wherewhite andgraystars arerelatedto FEMandrigid 135 deg, which is a safe operation limit for the actuator, like in model, respectively. eTh corresponding design limits of the helical gears. lead angle 𝛽 forloadcases AandBaresummarizedin In a nutshell, with reference to Figure 2, if the total thrust Table 3. Such table evidences that the helix angles admissible load in nominal conditions is negative (𝑇 <0,asincaseB), for design purpose roughly span either 5 deg (rigid model) ref theactivebearingislocatedontherightsideoftherunner or 6 deg (FEM model), and friction and deformation tend to and the lead angle𝛽 of the spline coupling (6) is lower than increase the angles that mark the boundaries of the design 90 deg. Conversely, if the resultant nominal thrust is positive range. (𝑇 >0,asincaseA),theactivebearing isontheleft sideof As explained in the case study paragraph, for different ref impeller geometries, nominal compressor, turbine, and total the runner and𝛽>90 deg. Load (N) Stress (MPa) Advances in Tribology 11 8000 8000 6000 6000 4000 4000 2000 2000 0 0 −2000 −2000 −4000 −4000 −6000 −6000 −8000 −8000 45 55 65 75 85 95 105 115 125 135 45 55 65 75 85 95 105 115 125 135 (deg) (deg) R FEM model f = 0.3 R rigid model R FEM model f = 0.3 R rigid model Fa FEM model f = 0.3 Fa rigid model Fa FEM model f = 0.3 Fa rigid model Fs FEM model f = 0.3 Fs rigid model Fs FEM model f = 0.3 Fs rigid model −|Tref| |Tref| Figure 16: Graphical design method: trends of axial bearing thrusts Figure 15: Graphical design method: trends of axial bearing thrusts and load transfer as a function of lead angle in nominal operating and load transfer as a function of lead angle in nominal operating conditions in case A. conditions in case B. In wider terms,thehelix angleofthespline coupling (iii) deformation and friction tend to level off the thrusts (6) can be chosen with reference to the nominal working transmitted by the shaft and the hub; conditions onthebasisof thetargetlifeofthebearings within a suitable range, whose limits are assessed by means (iv) the handedness of the helical spline coupling with of the proposed graphical construction. eTh resulting choice respect to the applied torque direction is influential of lead angle𝛽 must finally fall within the admissible range of on stress and contact pressure magnitude. 45–135 deg. Finally, the previously presented method for the helical spline design [12] has been refined in order to take into 7. Conclusions account of the above-cited findings. It has been also adapted to the double-eect ff thrust air bearings, showing that the The conceptual design of an innovative support system handedness of the helical spline must be chosen on the basis capable of drastically improving start/stop performance of of which bearing is active in nominal working conditions. modern oil-free micro-GT has been concluded. The rolling- element bearings used in the paper in order to explain the Such graphical method has been applied by taking advan- device may be substituted with oil-free magnetic bearings tage of the new numerical results. eTh results show that so that a support system with lower current consumption in friction and deformation increase the design helix angles. comparison with systems totally based on AMBs is obtained. In the future, a detailed design of the invention will The invention can also be used to retrotfi the older machines, be further studied by considering involute spline profiles as whoseshaftismounted onrolling-elementbearings. Such well as by calculating the optimal hot clearances of both retrofit cannot be done by means of conventional support thrust bearings and turbine hub-shaft coupling by means systems based on foil air bearings [8]. of elastoaerodynamic analysis of the thrust foil bearing In order to perform the detailed design of the invention, lubrication. structural FEM analyses of the helical spline employed as load partition as well as coupling device have been carried out. Numerical calculations have shown that Nomenclature (i) load partition of the spline coupling strongly depends 𝐵: Spline side distance, mm on friction, which cannot be neglected in the detailed 𝑐: Clearance, 𝜇m design of the device; 𝑑: Displacement, mm (ii) axial load and deformation of the coupling are also 𝐷:Diameter, mm inu fl ent on load partition; they can only be neglected 𝑓: Friction factor as rfi st approximation; 𝐹: Axial load, N Force (N) Force (N) 12 Advances in Tribology [6] C. Dellacorte, V. Lukaszewicz, M. J. Valco, K. C. Radil, and 𝐿: Spline coupling length, m H. Heshmat, “Performance and durability of high temperature 𝑀:Torque,Nm foil air bearings for oil-free turbomachinery,” Tribology Trans- 𝑛:Numberofsplines actions,vol.43,no.4,pp. 774–780, 2000. 𝑁: Rotational speed, rpm [7] G. L. Agrawal, “Foil air/gas bearing technology - An overview,” 𝑃:Power,kW in Proceedings of the ASME 1997 International Gas Turbine and 𝑟:Radius,mm Aeroengine Congress and Exhibition (GT ’97),USA,June1997. 𝑅:Reaction,N [8] K. C. Radil and C. Dellacorte, Foil Bearing Starting Considera- 𝑇:Thrust,N tions and Requirements for Rotorcraft Engine Applications ,2009. 𝑥,𝑦,𝑧: Spline coupling Cartesian coordinate system [9] C. E. Fanning and T. A. Blanchet, “High-temperature evaluation 𝑊:Bearingradialload,N of solid lubricant coatings in a foil thrust bearing,” Wear,vol. 𝛼: Helix angle, deg 265, no. 7-8, pp. 1076–1086, 2008. 𝛽: Lead angle, deg. [10] H. Heshmat, P. Hryniewicz, J. F. Walton II, J. P. Willis, S. Jahan- mir, and C. DellaCorte, “Low-friction wear-resistant coatings Subscripts for high-temperature foil bearings,” Tribology International,vol. 38,no.11-12, pp.1059–1075,2005. 𝑐:Compressor ℎ:Hub [11] Y. Masato, Foil Bearing,2015. 𝑜:Outer [12] F. Stefani, A. Perrone, L. Ratto, and R. Francesconi, “Com- 𝑝:Pitch parative analysis of bearings for micro-GT: an innovative 𝑟:Radialdirection arrangement,” in Bearing Technology,P.H.Darji,Ed.,pp. 1–26, 𝑠:Shaft InTech, Rijeka, Croatia, 2017. 𝑡:Turbine [13] J.R.MancusoandR.Jones,“Coupling interface connection,” in Proceedings of the 30th Turbomachinery Symposium,pp.121–138, ref: Reference Texas A & M University, 2001. 𝑧: Axial direction [14] M. Balasubramaniam, E. Golaski, S.-K. Son, K. Sriram, and 𝜗: Circumferential direction. A. Slocum, “An anti-backlash two-part shaft coupling with interlocking elastically averaged teeth,” Precision Engineering, Data Availability vol.26,no. 3,pp.314–330,2002. [15] J.Hong,D.Talbot,andA.Kahraman,“Load distribution All of the data obtained from this study are contained within analysis of clearance-fit spline joints using finite elements,” the present manuscript and [12]. Mechanism and Machine eTh ory ,vol.74,pp.42–57,2014. [16] D. Barsi, T. Garbarino, A. Perrone et al., “Micro gas turbine inte- Conflicts of Interest grated design: thermodynamic cycle, combustor, recuperator and bearings (Part 1),” in Proceedings of theThe4th International eTh authors declare that there are no conflicts of interest Conference on Water Resource and Environment (WRE ’15),2015. regarding the publication of this article. [17] A. M. Davis, Thrust Balancing ,1956. Acknowledgments eTh authors would like to thank their patent consultant Dr. Andrea Grimaldo for the useful discussion of the results published in the present paper during the preparation of the documentation of the national patent and PCT. References [1] D. J. Clark, M. J. Jansen, and G. T. Montague, An Overview of MagneticBearingTechnologyfor GasTurbine Engines,NASA- TM—2004-213177, 2004. [2] R. Moser, J. Sandtner, and H. Bleuler, “Optimization of repulsive passive magnetic bearings,” IEEE Transactions on Magnetics, vol. 42, no. 8, pp. 2038–2042, 2006. [3] J. Hillyard, “Magnetic bearings,” in Proceedings of the Joint Advanced Student School Conference (JASS ’06),pp.1–15,St. Petersburg, Russia, April 2006. [4] A. Irving and J. Ibets, “High-Speed Bearing Technologies for Wastewater Treatment Applications,” in Proceedings of the Water Environment Federation (WEFTEC ’13), vol. 2013, pp. 98– [5] B. J. Hamrock and W. J. Anderson, Rolling-Element Bearings, NASA RP-1105 1–58, 1983. 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Published: Mar 15, 2018
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