Patent Publication Number: US-2023160346-A1

Title: Midshaft rating for turbomachine engines

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 17/328,795 filed on May 24, 2021, the contents of which are hereby incorporated by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     This application is generally directed to turbomachine engines, including turbomachine shafts, and a method of driving such turbomachine shafts in such turbomachine engines. 
     BACKGROUND 
     A turbofan engine, or turbomachinery engine, includes a core engine and a power turbine that drives a bypass fan. The bypass fan generates the majority of the thrust of the turbofan engine. The generated thrust can be used to move a payload (e.g., an aircraft). A turbomachine shaft coupled to the power turbine and fan (either directly or through a gearbox) may be characterized by its first-order beam bending mode, the fundamental resonance frequency of this mode, and a critical speed of rotation that corresponds to the fundamental frequency. If this bending mode occurs within the standard operating range of the engine, undetected vibration as well as an increased risk of whirl instability may result. There is a continuing need to address vibrations induced by rotating shafts in turbomachinery engines. 
     BRIEF SUMMARY 
     According to an embodiment, a turbomachine engine includes a core engine having one or more compressor sections, one or more turbine sections that include a power turbine, and a combustion chamber in flow communication with the compressor sections and turbine sections. The turbomachine engine also includes a shaft coupled to the power turbine and characterized by a midshaft rating (MSR) between two hundred (ft/sec) 1/2  and three hundred (ft/sec) 1/2 . In one aspect, the shaft has a redline speed between fifty and two hundred fifty feet per second (ft/sec). In another aspect, the shaft has a length L, an outer diameter D, and a ratio of L/D between twelve and thirty-seven. 
     Additional features, advantages, and embodiments of the present disclosure are set forth or apparent from consideration of the following detailed description, drawings, and claims. Moreover, it is to be understood that both the foregoing summary and the following detailed description are examples and intended to provide further explanation without limiting the scope of the disclosure as claimed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and other features and advantages will be apparent from the following, more particular, description of various embodiments, as illustrated in the accompanying drawings, wherein like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. 
         FIG.  1    shows a schematic, cross-sectional view of a gas turbine engine according to some embodiments of the present disclosure. 
         FIG.  2    shows an enlarged view of a portion of the cross-sectional view of  FIG.  1   . 
         FIG.  3 A  shows a cross section of a steel shaft. 
         FIG.  3 B  shows a cross section of a composite shaft. 
         FIG.  4 A  shows a cross section of a uniform shaft with a constant diameter and thickness. 
         FIG.  4 B  shows a cross section of a concave shaft with a constant diameter and a variable thickness. 
         FIG.  4 C  shows a cross section of a convex shaft with a variable diameter and a variable thickness. 
         FIG.  5 A  shows a shaft using a four-bearing straddle configuration. 
         FIG.  5 B  shows a shaft using a four-bearing outbound configuration. 
         FIG.  5 C  shows a shaft using an inbound duplex configuration. 
         FIG.  5 D  shows a shaft using an outbound duplex configuration. 
         FIG.  5 E  shows a shaft using a two-bearing configuration. 
         FIG.  6 A  shows a plot depicting an exemplary range of a midshaft rating relative to an exemplary range of outer diameter redline speeds. 
         FIG.  6 B  shows a plot depicting an exemplary range of a midshaft rating relative to an exemplary range of length-diameter ratio. 
     
    
    
     DETAILED DESCRIPTION 
     Various embodiments are discussed in detail below. While specific embodiments are discussed, this is done for illustration purposes only. A person skilled in the relevant art will recognize that other components and configurations may be used without departing from the spirit and scope of the present disclosure. 
     As used herein, the terms “first”, “second”, and “third” may be used interchangeably to distinguish one component from another and are not intended to signify location or importance of the individual components. 
     The terms “forward” and “aft” refer to relative positions within a gas turbine engine or vehicle, and refer to the normal operational attitude of the gas turbine engine or vehicle. For example, with regard to a gas turbine engine, forward refers to a position closer to an engine inlet and aft refers to a position closer to an engine nozzle or exhaust. 
     The terms “upstream” and “downstream” refer to the relative direction with respect to fluid flow in a fluid pathway. For example, “upstream” refers to the direction from which the fluid flows, and “downstream” refers to the direction to which the fluid flows. 
     The terms “coupled,” “fixed,” “attached to,” and the like, refer to both direct coupling, fixing, or attaching, as well as indirect coupling, fixing, or attaching through one or more intermediate components or features, unless otherwise specified herein. 
     The term “propulsive system” refers generally to a thrust-producing system, which thrust is produced by a propulsor, and the propulsor provides the thrust using an electrically-powered motor(s), a heat engine such as a turbomachine, or a combination of electrical motor(s) and a turbomachine. 
     The singular forms “a”, “an”, and “the” include plural references unless the context clearly dictates otherwise. 
     Approximating language, as used herein throughout the specification and claims, is applied to modify any quantitative representation that could permissibly vary without resulting in a change in the basic function to which it is related. Accordingly, a value modified by a term or terms, such as “about”, “approximately”, and “substantially”, are not to be limited to the precise value specified. In at least some instances, the approximating language may correspond to the precision of an instrument for measuring the value, or the precision of the methods or machines for constructing or manufacturing the components and/or systems. For example, the approximating language may refer to being within a one, two, four, ten, fifteen, or twenty percent margin in either individual values, range(s) of values and/or endpoints defining range(s) of values. 
     As used herein, “redline speed” means the maximum expected rotational speed of a shaft during normal operation of an engine. The redline speed may be expressed in terms of rotations per second in Hertz (Hz), rotations per minute (RPM), or as a linear velocity of the outer diameter of the shaft in terms of feet per second. 
     As used herein, “critical speed” means a rotational speed of the shaft that is about the same as the fundamental, or natural frequency of a first-order bending mode of the shaft (e.g., the shaft rotates at eighty Hz and the first-order modal frequency is eighty Hertz). When the shaft rotates at the critical speed, the shaft is expected to have a maximum amount of deflection, hence instability, due to excitation of the first-order bending mode of the shaft. The critical speed may be expressed in terms of rotations per second in Hertz (Hz), rotations per minute (RPM), or as a linear velocity of the outer diameter of the shaft in terms of feet per second. 
     As used herein, “critical frequency” is a synonym for the fundamental, or natural frequency, of the first-order bending mode of the shaft. 
     The term “subcritical speed” refers to a shaft redline speed that is less than the fundamental, or natural frequency of the first-order bending mode of the shaft (e.g., the shaft rotates at a redline speed of 70 Hz while the first-order modal frequency is about 80 Hertz). When the rotational speed is subcritical the shaft is more stable than when rotating at a critical speed. A “subcritical shaft” is a shaft that has a redline speed below the critical speed of the shaft. 
     The term “supercritical speed” refers to a shaft rotational speed that is above the fundamental, or natural frequency of the first-order bending mode of the shaft (e.g., the shaft rotates at eighty Hz while the first-order modal frequency is about seventy Hertz). A supercritical shaft is less stable than a subcritical shaft because the shaft speed can pass through the critical speed since its fundamental mode is below the redline speed. A “supercritical shaft” is a shaft that has a redline speed above the critical speed of the shaft. 
     Here and throughout the specification and claims, range limitations are combined, and interchanged, such ranges are identified and include all the sub-ranges contained therein unless context or language indicates otherwise. For example, all ranges disclosed herein are inclusive of the endpoints, and the endpoints are independently combinable with each other. 
     One or more components of the turbomachine engine described herein below may be manufactured or formed using any suitable process, such as an additive manufacturing process, such as a three-dimensional (3D) printing process. The use of such a process may allow such a component to be formed integrally, as a single monolithic component, or as any suitable number of sub-components. In particular, the additive manufacturing process may allow such a component to be integrally formed and include a variety of features not possible when using prior manufacturing methods. For example, the additive manufacturing methods described herein enable the manufacture of shafts having unique features, configurations, thicknesses, materials, densities, passageways, headers, and mounting structures that may not have been possible or practical using prior manufacturing methods. Some of these features are described herein. 
     This disclosure and various embodiments relate to a turbomachinery engine, also referred to as a gas turbine engine, a turboprop engine, or a turbomachine. These turbomachinery engines can be applied across various technologies and industries. Various embodiments may be described herein in the context of aeronautical engines and aircraft machinery. 
     In some instances, a turbomachinery engine is configured as a direct drive engine. In other instances, a turbomachinery engine can be configured as a geared engine with a gearbox. In some instances, a propulsor of a turbomachinery engine can be a fan encased within a fan case and/or nacelle. This type of turbomachinery engine can be referred to as “a ducted engine.” In other instances, a propulsor of a turbomachinery engine can be exposed (e.g., not within a fan case or a nacelle). This type of turbomachinery engine can be referred to as “an open rotor engine” or an “unducted engine.” 
     Newer engine architectures may be characterized by faster shaft speeds for the low-pressure turbine (LPT), can have longer shafts to accommodate a longer core, and be required to operate within a more limited radial space. These requirements can result in reductions in stiffness-to-weight ratio, however, that have the effect of lowering the critical speed and/or limiting the available options for increasing the critical speed for the LPT shaft. Accordingly, different approaches for engine types, midshaft geometry, bearing support, and material compositions are required for next-generation turbomachine engines, to permit high-speed operation without resulting in an unstable bending mode during regular operation. The inventors, tasked with finding a suitable design to meet these requirements, conceived and tested a wide variety of shafts having different combinations of stiffness, material, bearing type and location, shaft length, and diameter in order to determine which embodiment(s) were most promising for a variety of contemplated engine designs. The various embodiments, as described herein and as shown in the figures, include turbomachine shafts that employ one or more of these techniques to increase the critical speed of the first-order bending mode. 
       FIG.  1    shows a schematic, cross-sectional view of an embodiment of a gas turbine engine  100  for an aircraft, and may incorporate one or more embodiments of the present disclosure. The gas turbine engine  100  can include a fan assembly  102  and a turbomachine  103 . In some embodiments, the turbomachine  103  is a two-spool turbomachine that includes a high-speed system and a low-speed system. 
     The high-speed system of the turbomachine  103 , which is not shown in  FIG.  1   , can include a high-pressure compressor, a high-pressure turbine, a combustor, and a high-speed shaft supported by bearings and connecting the high-pressure compressor and the high-pressure turbine. The high-speed shaft, components of the high-pressure compressor, and components of the high-pressure turbine all rotate around a centerline axis  112  of the gas turbine engine  100 . The high-pressure compressor (or at least the rotating components thereof), the high-pressure turbine (or at least the rotating components thereof), and the high-speed shaft may be collectively referred to as a high-pressure spool of the gas turbine engine  100 . The combustor is located between the high-pressure compressor and the high-pressure turbine. The combustor can include one or more configurations for receiving a mixture of fuel and air, and for providing a flow of combustion gases through the high-pressure turbine for driving the high-pressure spool. 
     The low-speed system of the turbomachine  103  can include a low-pressure turbine  120 , a low-pressure compressor or booster  121 , and a low-speed shaft  122  extending between and connecting the booster  121  and the low-pressure turbine  120 . In some embodiments, the low-speed shaft  122  may extend further along the centerline axis  112  than is shown in  FIG.  1   . 
     The low-speed shaft  122  is supported on bearings  123   a,    123   b,    123   c,    123   d,  which are mounted to support structures (not shown) of the gas turbine engine  100 . At each position, only two bearings are shown in  FIG.  1    for clarity, though it is contemplated that more than two bearings, e.g. 3 or 4 bearings forward and/or aft of the respective illustrated locations, may be arranged to support the low-speed shaft  122  at the respective positions, and may be evenly spaced or irregularly spaced depending on the geometry of the bearing supporting structure, and available space and clearances. 
     The low-speed shaft  122 , components of the booster  121 , and components of the low-pressure turbine  120  all rotate around the centerline axis  112  of the gas turbine engine  100 , in either the same direction or a counter-rotating direction as that of the high-pressure spool. The booster  121  (or at least the rotating components thereof), the low-pressure turbine  120  (or at least the rotating components thereof), and the low-speed shaft  122  may collectively be referred to as a low-pressure spool  200  of the gas turbine engine  100 , and is further described in  FIG.  2   . 
     The fan assembly  102  includes an array of fan blades  124  extending radially outward from a rotor disc  126 . The gas turbine engine  100  has an intake side  128  and an exhaust side  130 . In some embodiments, the gas turbine engine  100  is a multi-spool turbofan engine, in which a low-pressure spool connects the rotor disc  126  to a rotor (not shown in  FIG.  1   ) of the low-pressure turbine  120 , and a high-pressure spool connects the high-pressure compressor to a rotor (not shown in  FIG.  1   ) of the high-pressure turbine. 
     The turbomachine  103  is generally encased in a cowl  131 . Moreover, it will be appreciated that the cowl  131  defines at least in part an inlet  132  of the turbomachine  103  and an exhaust  135  of the turbomachine  103 , and includes a turbomachinery flow path extending between the inlet  132  and the exhaust  135 . In operation, air flows through fan assembly  102 , and compressed air is supplied from the inlet  132  to the booster  121  and the high-pressure compressor. The highly compressed air proceeds along the turbomachinery flow path and is delivered to the combustor. The airflow from the combustor drives the high-pressure turbine and the low-pressure turbine  120 , and the low-pressure turbine  120  drives the fan assembly  102  via the low-speed shaft  122 . 
     For the embodiment shown in  FIG.  1   , the inlet  132  has an annular or an axisymmetric three hundred sixty-degree configuration, and provides a path for incoming atmospheric air to enter the turbomachinery flow path, as described above. Such a location may be advantageous for a variety of reasons, including management of icing performance as well as protecting the inlet  132  from various objects and materials as may be encountered in operation. It will be appreciated, however, that in other embodiments, the inlet  132  may be positioned at any other suitable location, e.g., arranged in a non-axisymmetric configuration, etc. 
     For reference purposes,  FIG.  1    also depicts a forward or thrust direction with arrow F, which in turn defines the forward and aft portions of the system. As in the embodiment shown in  FIG.  1   , the fan assembly  102  is located forward of the turbomachine  103  in a “puller” configuration, and the exhaust  135  is located aft. As is depicted, the fan assembly  102  is driven by the turbomachine  103 , and, more specifically, is driven by the low-pressure spool. More specifically, the gas turbine engine  100  in the embodiment shown in  FIG.  1    includes a power gearbox (not shown in  FIG.  1   ), and the fan assembly  102  is driven by the low-pressure spool of the turbomachine  103  across the power gearbox. The power gearbox may include a gearset for decreasing a rotational speed of the low-pressure spool relative to the low-pressure turbine  120 , such that the fan assembly  102  may rotate at a slower rotational speed than does the low-pressure spool. In such a manner, the rotating fan blades  124  of the fan assembly  102  may rotate around the centerline axis  112  and generate thrust to propel the gas turbine engine  100 , and, hence, an aircraft to which it is mounted, in the forward direction F. Other configurations are possible and contemplated within the scope of the present disclosure, such as what may be termed a “pusher” configuration embodiment in which the turbomachine  103  is located forward of the fan assembly  102 . 
     It will be appreciated, however, that the gas turbine engine  100  depicted in  FIG.  1    is by way of example only. In other embodiments, the gas turbine engine  100  may have any other suitable configuration, including, for example, any other suitable number of shafts or spools, fan blades, turbines, compressors, etc., and the power gearbox may have any suitable configuration, including, for example, a star gear configuration, a planet gear configuration, a single-stage, a multi-stage, epicyclic, non-epicyclic, etc. The fan assembly  102  may be any suitable fixed-pitched assembly or variable-pitched assembly. The gas turbine engine  100  may include additional components not shown in  FIG.  1   , such as vane assemblies and/or guide vanes, etc. 
     It should be appreciated that the terms “low” and “high,” or their respective comparative degrees (e.g., “lower” and “higher”, where applicable), when used with the compressor, turbine, shaft, or spool components, each refers to relative pressures and/or relative speeds within an engine unless otherwise specified. For example, a “low-speed shaft” defines a component configured to operate at a rotational speed, such as a maximum allowable rotational speed, which is lower than that of a “high-speed shaft” of the engine. Alternatively, unless otherwise specified, the aforementioned terms may be understood in their superlative degree. For example, a “low-pressure turbine” may refer to the lowest maximum pressure within a turbine section, and a “high-pressure turbine” may refer to the highest maximum pressure within the turbine section. It should further be appreciated that the terms “low” or “high” in such aforementioned regards may additionally, or alternatively, be understood as relative to minimum allowable speeds and/or pressures, or minimum or maximum allowable speeds and/or pressures relative to normal, desired, steady state, etc., operation of the engine. 
       FIG.  2    shows an enlarged view of a portion of the cross-sectional view of  FIG.  1   , that includes the low-pressure spool  200  according to some embodiments of the present disclosure. For example, a portion of the booster  121  and a portion of the low-pressure turbine  120  are shown mounted to the low-speed shaft  122  of the turbomachine  103 , which in this example is a two-spool turbomachine. Alternatively, the low-speed shaft  122  may be an intermediate shaft in a three-spool turbomachine (not shown). The low-speed shaft  122  is supported by at least bearings  123   a  to  123   d,  which are located at mounting points  205   a,    205   b  associated with a booster  121  location and a low-pressure turbine  120  location, respectively, for providing shaft rotational support at these locations. In the example of  FIG.  2   , bearings  123   a ,  123   b,    123   c,  and  123   d  are all positioned inside of the mounting points  205   a  and  205   b,  which is referred to as an inbound bearing layout, or alternatively referred to as an overhung configuration for the booster  121  and low-pressure turbine  120 . If the bearings were positioned outside of the mounting point  205   b,  then that would be referred to as an outbound layout. The bearings  123   a  to  123   d  can, however, be positioned at any point along the low-speed shaft  122 , and may both be inbound, both be outbound, or one inbound and the other outbound. 
     The low-speed shaft  122  has a length “L” (indicated by arrow  208 ) and an outer diameter “D” (indicated by arrow  210 ). The low-speed shaft  122  can be hollow, with an inner diameter “d” indicated by arrow  211 ). In cases when the diameter of the low-speed shaft  122  varies along the length L, the outer diameter “D” and the inner diameter “d” may be defined at a midpoint of the low-speed shaft  122  (also referred to as the midshaft  215 ). The thickness may be defined as the thickness of the walls of the low-speed shaft  122  in embodiments in which the low-speed shaft  122  is hollow. A difference between a stated outer diameter D and inner diameter d of the low-speed shaft  122  may be understood as the shaft&#39;s wall thickness. In cases when the wall thickness varies along the length of the low-speed shaft  122 , the thickness may be defined as the difference between the inner diameter and the outer diameter at the midshaft  215 . 
     The length L can be understood as the portion of the low-speed shaft  122  between the bearings  123   a  to  123   d  and/or the mounting points  205   a,    205   b  of engine components such as the booster  121  and the low-pressure turbine  120 . For example, in the two-bearing arrangement of  FIG.  2   , the length L may be measured as the distance between midpoints of the bearings  123   a  to  123   b,  as indicated by the dashed vertical lines and arrow  208 . For a four-bearing arrangement, there may be additional bearings along the shaft, in which case the length L may be measured as the distance between the midpoints of an innermost pair of bearings, or the distance between pairs or other groupings of bearings. In some embodiments, the length may be measured relative to specific bearings associated with specific engine components such as the booster  121  and the low-pressure turbine  120 . 
     During operation, the low-speed shaft  122  rotates with a rotational speed that can be expressed in either rotations per minute (RPM), or as an outer diameter (OD) speed expressed in units of linear velocity, such as feet per second (ft/sec). The rotational stability of the low-speed shaft  122  relative to its operational range may be characterized by the resonance frequency of the fundamental or first order bending mode. When an operational speed is the same as this resonance frequency, the shaft is operating at its critical speed. The low-speed shaft  122 , when supported by bearings  123   a  to  123   d,  has a mode shape for this first-order bending mode that may be generally described as a half-sinusoid, with a midshaft  215  location undergoing maximum displacement (indicated by arrow  220 , which is exaggerated for clarity and is not to scale) and, therefore, having a maximum kinetic energy of displacement relative to other portions of the low-speed shaft  122 . The fundamental mode shape is illustrated by dashed line  225  extending from bearing  123   c  to bearing  123   d  in  FIG.  2   , though this is only half of the amplitude of oscillation. This unstable mode is a standing wave across the length L of the low-speed shaft  122 . The maximum deflection occurs when the excitation source has a periodicity or cyclic component near to the fundamental frequency. Since the bending mode is not active at the location of innermost bearings  123   a  to  123   d  for the low-speed shaft  122 , the instability cannot be mitigated with the use of bearing dampers. When an engine is designed, the shaft speed expected to produce the highest deflection or instability at the midshaft is the shaft speed that equals the critical speed. 
     If the critical speed of the shaft critical speed falls within the standard operational range, i.e., if the critical speed is below the redline speed or the low-speed shaft  122  is a supercritical shaft, then during routine operation, the low-speed shaft  122  may at times operate at or pass through the critical speed, which induces an unstable condition. Even if the engine is operated at the critical speed temporarily, there is a possibility of undetected vibration, whirl instability, and some likelihood of damage. For low vibration and stability, it is preferable to have an operating range free of any intervening critical speeds. 
     There is a desire to pursue engines capable of operating at higher redline speeds. This pursuit of higher operating speeds requires that the low-speed shaft  122  have a higher strength to weight characteristic if it is also desired that the shaft remain subcritical. The inventors sought this end result—higher speed shafts while remaining subcritical. To this end, a large number of engine designs were evaluated. Depending on the architecture, the positions and numbers of bearings relative to mounting points  205   a,    205   b  were varied, and the resulting impact, not only on the critical speed but also the feasibility of such configurations given competing requirements (clearance, spacing, sump locations, oil supply lines), were taken into consideration, as will be readily apparent in view of the disclosure. A discussion of these embodiments follows. In the following discussion, strength to weight ratio is represented as E/rho, calculated as the ratio of Young&#39;s modulus E for the material (expressed, for example, in pounds per square foot) divided by the density rho (expressed, for example, in pounds per cubic inch). The shaft bending mode is represented as the critical rotational speed expressed in rotations per minute (RPM), though it could alternatively be expressed as the fundamental frequency of the bending mode in Hertz. 
     In some embodiments, high strength steel alloys, advanced and/or composite materials were contemplated. For example, high strength-to-weight ratio materials such as titanium boride (TiB), a titanium metal matrix composite (TiMMC), provided 30% to 50% increased strength-to-weight ratio relative to steel or titanium alloys. In addition, coatings with materials such as graphene were found to improve strength by a factor of two in lab tests, without impacting weight. These types of changes in material composition may be characterized in some embodiments by the ratio of E/rho. 
       FIG.  3 A  shows a cross section of a steel shaft  305 , with a standard hollow interior  306  surrounded by a steel layer  307 , and geometry defined by a length L, outer diameter D, inner diameter d, etc. 
       FIG.  3 B  shows an example of a composite shaft  310 , with identical geometry to the steel shaft  305 . Rather than being composed entirely of steel, the composite shaft  310  has an inner layer  315  surrounding a hollow interior  317 , a middle layer  320 , and an outer coating  325 , all of different materials. The middle layer  320  in this example is also steel, though in other embodiments the composite shaft could use no steel at all, or have a different layer be steel. 
     For example, both the steel shaft  305  and the composite shaft  310  have length L of seventy-six inches and outer diameter of three inches, along with a standard inbound two-bearings configuration as depicted in  FIG.  2   . The fundamental frequency of the unstable mode for the steel shaft  305  is eighty Hertz (Hz), whereas the fundamental frequency for the composite shaft  310  is ninety Hz. 
     In other embodiments, more layers or fewer layers may be used. Some or all of these layers and coatings may be of numerous alternative materials to steel, including but not limited to TiB, TiMMC, other metals and metal matrix composites, silicon carbide alloys, aluminum alloys, and/or graphene. The concepts of the present disclosure are not limited by the particular materials used for the layers and/or coatings. For the composite shaft  310 , the critical speed corresponding to the unstable mode is increased relative to the (otherwise identical) steel shaft  305 , which means that relative to the steel shaft  305 , the composite shaft  310  can attain a higher rotational speed, before reaching the critical speed. 
     Depending on the type of composite materials chosen and the relative thickness and arrangement of the layers, the ratio of stiffness to weight can be modified, and, therefore the critical speed can be increased. The inventors conceived of a variety of embodiments resulting from the selection of different composite materials, thicknesses, and bearings configurations to allow for operation at higher speed. Two such embodiments are listed in TABLE 1. These embodiments were considered as possible designs that could increase the shaft stiffness to weight ratio in such a way to be compatible with engine architecture and without requiring modifications or limitations on the targeted operating range for a subcritical shaft. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Embodi- 
                 L 
                 D 
                   
                 E/rho 
                 Teff 
                 Mode 
               
               
                 ment 
                 (in) 
                 (in) 
                 Bearing type 
                 (in −1 ) 
                 (in) 
                 (RPM) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 82.2 
                 2.74 
                 2-bearing 
                 1.00E+08 
                 0.35 
                 4181 
               
               
                   
                   
                   
                 outbound 
               
               
                 2 
                 60.6 
                 2.75 
                 inbound OTM 
                 1.27E+08 
                 0.35 
                 10263 
               
               
                 3 
                 82.2 
                 2.74 
                 outbound OTM 
                 1.27E+08 
                 0.35 
                 6915 
               
               
                   
               
            
           
         
       
     
     Embodiment 1 was evaluated using a high strength steel alloy and an outbound bearing layout. Embodiments 2 and 3 were evaluated with a composite material instead of steel alloy. Embodiment 2 uses overturning moment (OTM) bearings with an inbound bearing layout that is different from the layout used by Embodiment 1. Embodiment 3 uses OTM bearings with an outbound bearing layout that is similar to that used by Embodiment 1. These bearing types and layouts are described in further detail below with reference to  FIG.  5 A,  5 B , and TABLE 3. The values shown in TABLE 1 illustrate that Embodiments 2 and 3 achieve a higher strength-to-weight ratio (E/rho) when using a composite material, instead of the steel alloy used in Embodiment 1. As a result of these differences, the shaft mode critical speed occurs at 4181 RPM for Embodiment 1, at 10263 RPM for Embodiment 2 and at 6915 for Embodiment 3. 
     The inventors also modified the shaft thickness along its length, to evaluate the effect on critical speed for a strength to weight ratio of E/rho that is not constant along the length L, and for different suitable materials. An example of a shaft with a uniform E/rho along its length L is shown in  FIG.  4 A , and examples of shafts having variable E/rho are shown in  FIGS.  4 B and  4 C . 
       FIG.  4 A  conceptually shows a cross section of a uniform shaft  405  with a constant diameter and thickness. In this example, the uniform shaft  405  has a length L of seventy-six inches. The outer diameter D of the uniform shaft  405  is 3.0 inches. The uniform shaft  405  is hollow, with a constant wall thickness of 0.2 inches and corresponding constant inner radius of 1.3 inches along its length. For this example of a uniform shaft  405 , and a two-bearing outbound configuration such as in  FIG.  2   , the fundamental frequency of the unstable mode is eighty Hz. 
       FIG.  4 B  conceptually shows a cross section of a concave shaft  410  with a constant outer diameter D and a variable thickness. For comparison, the uniform shaft  405  and the concave shaft  410  have the same material (e.g., hollow steel), bearings (outbound), and length (seventy-six inches), with a constant outer radius of 1.5 inches along its length. The outer diameter D of the concave shaft  410  is, therefore, 3.0 inches. Unlike the uniform shaft  405 , however, the concave shaft  410  has a wall thickness of 0.3 inches at the ends  412 ,  414  (e.g., at the bearings, which are not shown in  FIG.  4   ), and a thinner wall thickness of 0.15 inches in the midshaft region  415 . This results in an inner radius of 1.35 inches in the midshaft region  415  and a smaller inner radius of 1.2 inches at the ends  412 ,  414 . The concave shaft  410  therefore has a reduced mass density in the midshaft region  415 . To achieve the resulting concave profile, various methods may be used to manufacture the concave shaft  410 , such as a bottle boring technique. 
       FIG.  4 C  conceptually shows a cross section of a convex shaft  420  with a variable outer diameter D and a variable thickness. For comparison, the uniform shaft  405  and the convex shaft  420  have the same material (e.g., hollow steel), bearings (outbound), and length (seventy-six inches), with a constant inner radius of 1.2 inches along its length. Unlike the uniform shaft  405 , the convex shaft  420  has a wall thickness of 0.3 inches at the ends  422 ,  424 , and a thinner wall thickness of 0.15 inches in the midshaft region  425 , just like the concave shaft  410 . Unlike the concave shaft  410 , the convex shaft  420  has an outer radius of 1.5 inches at the ends  422 ,  424 , and a smaller outer radius of 1.35 inches in the midshaft region  425 . The convex shaft  420  also has a reduced mass density in the midshaft region  425 . 
     Since the radius (and, therefore, the diameter) are variable over the length of the convex shaft  420 , the diameter D is defined in some embodiments as the diameter at the midshaft region  425 , since this has the most relevance to the bending mode and undergoes maximum deflection. In the example of the convex shaft  420 , the shaft outer diameter D is 2.7 inches in the midshaft region  425 . In other embodiments, for example, embodiments when the radius has multiple minima and/or maxima, the diameter D may be defined at any of those minima or maxima. To achieve the resulting convex profile, various methods may be used to manufacture the convex shaft  420 , such as external machining. 
     For both convex and concave thickness profiles, as well as types of variable thickness profiles, the thickness may be described using an effective thickness value, T eff . For a uniform shaft the thickness would simply be the difference between the outer diameter and the inner diameter. When these values are variable over the length of the shaft, the effective thickness is calculated as the difference between the effective outer diameter and effective inner diameter. For example, the effective thickness may be defined at the midshaft. 
     With variable thickness, in some embodiments the concave shaft  410  and the convex shaft  420  can have twenty-five to thirty percent less weight than the uniform shaft  405  in the midshaft region  415  and  425 , respectively. Note that the variation in thickness need not be continuous, for example a stepped change in geometry could also be used. As a result, the fundamental frequency of the unstable mode for both the concave shaft  410  and the convex shaft  420  is increased to ninety Hz, which is higher than the eighty Hz fundamental frequency for the uniform shaft  405 . In other words, the concave shaft  410  and the convex shaft  420  can both attain a higher rotational speed than that of uniform shaft  405 , before reaching subcritical speeds. 
     The concave shaft  410  and the convex shaft  420  are examples of different thickness profiles that may be used in some embodiments. Other thickness profiles are also contemplated, which reduce or increase the mass density of the shaft in the midshaft region. The concepts of the current disclosure are not limited by the particular thickness profile used. 
     Depending on the thickness profile, the ratio of stiffness to weight can be modified to produce significant changes in the critical speed. Exemplary embodiments are listed in TABLE 2. These embodiments were considered as possible designs that could modify the effective thickness in such a way to be compatible with engine architecture and without requiring modifications or limitations on the targeted operating range for a subcritical shaft. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Embodi- 
                 L 
                 D 
                   
                 E/rho 
                 Teff 
                 Mode 
               
               
                 ment 
                 (in) 
                 (in) 
                 Bearing type 
                 (in −1 ) 
                 (in) 
                 (RPM) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 4 
                 60.6 
                 2.75 
                 inbound OTM 
                 1.00E+08 
                 0.35 
                 9001 
               
               
                 5 
                 82.2 
                 2.74 
                 outbound OTM 
                 1.00E+08 
                 0.35 
                 6065 
               
               
                 6 
                 60.6 
                 2.75 
                 inbound OTM 
                 1.00E+08 
                 0.32 
                 10039 
               
               
                 7 
                 82.2 
                 2.74 
                 outbound OTM 
                 1.00E+08 
                 0.32 
                 6942 
               
               
                   
               
            
           
         
       
     
     Embodiments 4, 5, 6, and 7 all use a steel alloy material composition. Embodiments 4 and 6 use an inbound bearing layout with OTM bearings, and Embodiments 5 and 7 use an outbound bearing layout with OTM bearings. Embodiments 4 and 5 are uniform shafts similar to the Example of  FIG.  4 A . Embodiments 6 and 7, however, have a convex thickness profile similar to the example of  FIG.  4 C , having been manufactured with a bottle boring manufacturing technique. The values shown in TABLE 2 illustrate that Embodiments 6 and 7 achieve a lower effective thickness T eff  due to their convex profile, instead of the uniform profile for Embodiments 4 and 5. As a result of these differences, the shaft mode critical speed occurs at 9001 RPM for Embodiment 4, and at 10039 RPM for Embodiment 6. The shaft mode critical speed occurs at 6065 RPM for Embodiment 5, and at 6942 RPM for Embodiment 7. 
     The inventors also conceived of a variety of shafts with modified bearing configurations. Bearings are used to provide transverse support to the shaft along its length. Bearings may be ball-type bearings, which have a very small contact area with the shaft to provide less friction, or roller-type bearings, which have a large contact area with the shaft to provide increased rigidity and load bearing. Different types of bearings may be mixed in various bearing layouts. According to additional embodiments, different bearing layouts were considered, for different combinations of uniform, convex, and concave shafts, or varying shaft thickness profiles and material composition in order to determine which combination would work best for a given architecture and need, as well as taking into account competing engineering requirements. 
     A variety of combinations of bearing configurations were contemplated, such as embodiments when the number of bearings in duplex and/or straddling position relative to engine components (e.g., a booster  521  or a low-pressure turbine  520 ) were changed. Either or both of the engine components mounted to the shaft  522  may be straddled or overhung. It was found that these variations can improve the critical speed and/or be more suitable to accommodate space limitations, lubrication resources or other architecture-imposed limitations. The embodiments included locating bearings at different inbound or outbound positions relative to the mounting points  505   a,    505   b.    
     Specific bearing layouts were preferentially used in various embodiments. These are now described in more detail, though the concepts of the present disclosure are not limited by the particular number or arrangement of bearings described herein. 
     For example,  FIG.  5 A  conceptually shows a low-pressure turbine  520  and a booster  521  mounted on a shaft  522  supported by a four-bearing straddle configuration. Additional bearings located around the circumference of the shaft  522  are omitted from  FIG.  5 A  for clarity. In this system, one pair of bearings  523   a,    524   a  straddle (i.e., placed forward and aft of) a mounting point  505   a  of the booster  521 , and a second pair of bearings  523   b,    524   b  straddle a mounting point  505   b  of the low-pressure turbine  520 . In this example, bearings  524   a,    523   b,  and  524   b  are roller bearings, and bearing  523   a  is a ball bearing, though these bearing types may vary. The length L for shaft  522  is represented in some embodiments as the distance between the midpoints or centers of the innermost bearings  524   a,    523   b.  The four-bearing straddle layout is used in several embodiments described with reference to TABLE 3. 
     As another example,  FIG.  5 B  conceptually shows a low-pressure turbine  520  and a booster  521  mounted on a shaft  522  supported by a four-bearing outbound configuration. Additional bearings located around the circumference of the shaft  522  are omitted from  FIG.  5 B  for clarity. This system is similar to that of the straddle system shown in  FIG.  5 A , but differs in that bearings  523   a,    524   a  are both placed forward of mounting point  505   a  of the booster  521 , and bearings  523   b,    524   b  are placed aft of mounting point  505   b  of the low-pressure turbine  520 . The shaft  522  may extend beyond bearings  523   b,    524   b.  As in the example of  FIG.  5 A , bearings  524   a ,  523   b,  and  524   b  are roller bearings, and bearing  523   a  is a ball bearing, though these bearing types may vary. The length L for shaft  522  is represented in some embodiments as the distance between the midpoints or centers of the innermost bearings  524   a,    523   b.    
     As yet another example,  FIG.  5 C  conceptually shows a shaft  522  with an inbound duplex bearing configuration. Additional bearings located around the circumference of the shaft  522  are omitted from  FIG.  5 C  for clarity. According to some embodiments, a first pair of ball bearings  525   a,    526   a  is arranged in a duplex configuration aft of the mounting point  505   a  for the booster  521 . A second pair of ball bearings  525   b,    526   b  is arranged in a duplex configuration forward of the mounting point  505   b  for the low-pressure turbine  520 . Duplex bearing arrangements may also be referred to as double-row bearings, or overturning moment (OTM) bearings, since they provide moment stiffness to the shaft, i.e., provide resistance to rotation across the bearing locations. In some embodiments duplex bearing types may include tandem bearings, back-to-back bearings, face-to-face bearings, and/or tapered roller bearings. 
     In the example shown in  FIG.  5 C , both the first pair of ball bearings  525   a,    526   a  and the second pair of ball bearings  525   b,    526   b  are in an inbound position, i.e., located closer towards the midshaft  527  than the respective mounting points  505   a,    505   b.  In this position, the booster  521  and the low-pressure turbine  520  are referred to as overhung. This inbound OTM layout is used in Embodiments 2, 4, and 6, for example, described above with reference to TABLES 1 and 2. 
     Alternatively, the first pair of ball bearings  525   a,    526   a  and/or the second pair of ball bearings  525   b,    526   b  may be in an outbound position, as shown in  FIG.  5 D , i.e., located farther from the midshaft  527  than the respective mounting points  505   a,    505   b  of the booster  521  and the low-pressure turbine  520 . The length L for the duplex bearing configurations shown in  FIGS.  5 C and  5 D  may be represented in some embodiments as the distance between the midpoints or centers of the innermost ball bearings  526   a  and  525   b,  or alternatively, as the distance between the center of the first pair of ball bearings  525   a,    526   a  and the center of the second pair of ball bearings  525   b,    526   b.  The outbound OTM layout is used in Embodiments 3, 5, and 7, for example, described above with reference to TABLES 1 and 2. 
     As a further example,  FIG.  5 E  conceptually shows a shaft  522  with a two-bearing configuration. This configuration employs a first roller bearing  524   a  positioned aft of the mounting point  505   a  for the booster  521 , and a second roller bearing  524   b  positioned aft of the mounting point  505   b  for the low-pressure turbine  520 . The length L for this two-bearing configuration is represented in some embodiments as the distance between the midpoints or centers of the roller bearings  524   a,    524   b.  Alternative two-bearing configurations may position the two bearings in either an outbound configuration or an inbound configuration. An example of a two-bearing layout in an inbound configuration is shown in  FIG.  1   , that shows bearings  123   a ,  123   b,    123   c,  and  123   d  are all located inbound of the mounting points for the booster  121  and the low-pressure turbine  120 . Note that in this context, two is the number of bearings along the shaft  522 , and does not include additional bearings along the circumference of the shaft  522 . Embodiment 1, described above with reference to TABLE 1, uses a two-bearing layout in an outbound configuration (not shown). 
     In  FIGS.  5 A to  5 E , the lines connecting the booster  521  to mounting point  505   a  and low-pressure turbine  520  to mounting point  505   b  are intended only to indicate schematically the general location of a net force of the engine core components (e.g., booster  521  or low-pressure turbine  520 ) acting on the shaft  522  relative to the bearings, and is illustrated in this fashion only for purposes of illustrating a relationship between the nearest engine component relative to the bearing(s). It will be understood that the actual loading on a shaft is distributed and comes from not only the engine components represented by booster  521  and low-pressure turbine  520 , but other nearby structures as well. The primary loading for purposes of this disclosure may, however, be thought of simply in terms of engine components attached to the shaft  522  (e.g., low-pressure turbine  520  and booster  521 ). It will be understood that the representation shown in  FIGS.  5 A to  5 E  is sufficient in defining the parts of the turbomachine that mostly influence the shaft  522  behavior. 
     As discussed, at least one bearing may have an overturning moment (OTM) capability, which can resist relative rotation across the bearing in at least a lateral plane or a vertical plane. These relative rotations may occur during bending of the shaft. The position along the shaft of such bearings with OTM capabilities may directly affect the critical speed, by providing constraints to the relative rotations of the shaft, in addition to the transverse support function of the bearings. 
     Examples of embodiments with different bearing arrangements are summarized in TABLE 3. Generally, the inventors found that the number of bearings, the position of the bearings and the OTM capability of the bearings can be selected to make a full range of operations subcritical for an engine. In other words, the selection of bearing layout can affect (either increase or decrease) the shaft&#39;s critical speed. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 Embodi- 
                 L 
                 D 
                   
                 E/rho 
                 Teff 
                 Mode 
               
               
                 ment 
                 (in) 
                 (in) 
                 Bearing type 
                 (in −1 ) 
                 (in) 
                 (RPM) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 8 
                 60.6 
                 2.75 
                 4-bearing straddle 
                 1.00E+08 
                 0.35 
                 7746 
               
               
                 9 
                 60.6 
                 2.75 
                 4-bearing straddle 
                 1.00E+08 
                 0.32 
                 8555 
               
               
                 10 
                 60.6 
                 2.75 
                 4-bearing straddle 
                 1.27E+08 
                 0.35 
                 8832 
               
               
                 11 
                 82.2 
                 2.74 
                 4-bearing straddle 
                 1.27E+08 
                 0.32 
                 9703 
               
               
                 12 
                 60.6 
                 2.75 
                 inbound OTM 
                 1.27E+08 
                 0.32 
                 11386 
               
               
                 13 
                 82.2 
                 2.74 
                 outbound OTM 
                 1.27E+08 
                 0.32 
                 7873 
               
               
                   
               
            
           
         
       
     
     Embodiments 8, 9, 10, and 11 use a four-bearing straddle layout. Embodiments 8 and 9 use steel alloy, while Embodiments 10 and 11 use composite materials. Embodiments 8 and 10 have a uniform thickness profile, while Embodiments 9 and 11 have a concave thickness profile, manufactured using a bottle boring method. As a result of these differences, the shaft mode critical speed occurs at 7746 RPM for Embodiment 8, 8555 RPM for Embodiment 9, 8832 RPM for Embodiment 10, and 9703 RPM for Embodiment 11. 
     Embodiments 11, 12, and 13 all use composite material and concave thickness profile via bottle boring. However, Embodiment 11 uses a four-bearing straddle layout, Embodiment 12 uses an inbound OTM bearing layout, and Embodiment 13 uses an outbound OTM bearing layout. As a result of these differences, the shaft mode critical speed occurs at 9703 RPM for Embodiment 11, 11386 RPM for Embodiment 12, and 7873 RPM for Embodiment 13. 
     Embodiment 11 can also be compared to Embodiments 8, 9, and 10 as described with reference to TABLE 3. This allows a comparison of the impact on critical speed of using composite material, variable thickness profile, and both, on a shaft with a four-bearing straddle layout. 
     Embodiment 12 can be compared to Embodiments 2, 4, and 6 described with reference to TABLE 2. This allows a comparison of the impact on critical speed of using composite material, variable thickness profile, and both, on a shaft with an inbound OTM layout. 
     Embodiment 13 can be compared to Embodiments 3, 5, and 7 described with reference to TABLE 2. This allows a comparison of the impact on critical speed of using composite material, variable thickness profile, and both, on a shaft with an outbound OTM layout. 
     Additionally, Embodiments 2 and 3 (in TABLE 1), and 10 (in TABLE 3) can be compared, to evaluate the impact on critical speed of using different bearing layouts on shafts using composite material. Embodiments 6 and 7 (in TABLE 2) and 9 (in TABLE 3) can be compared, to evaluate the impact on critical speed of using different bearing layouts on shafts using concave thickness profiles. 
     The embodiments of turbomachine engines, and in particular the shafts associated with a power turbine described with reference to  FIGS.  3 A,  3 B,  4 A to  4 C,  5 A, and  5 B , were found to provide an improvement in the performance of a shaft vis-à-vis its operating range. In addition to the mentioned embodiments and those provided in TABLES 1 to 3, the types of improvements to the critical speed of the shaft when these features were combined, taking into consideration the various benefits, as well as down-sides, to selecting a particular configuration for a turbomachine architecture. 
     Examples of a subcritical shaft with a high redline speed include a shaft with a redline speed of, e.g., 70 ft/sec and adapted for a shaft mode of 5293 RPM, a shaft with a redline speed of, e.g., 75 ft/sec and adapted for a shaft mode of 6380 RPM, and a shaft with a redline speed of, e.g., 181 ft/sec and adapted for a shaft mode of 11410 RPM. 
     As indicated earlier, a selection of power turbine shaft and bearing arrangement, and location of those bearings for a turbomachine takes into consideration other factors, some of which can limit the selection of a shaft. The inventors however realized during the course of making the several embodiments referred to in the foregoing that there is a particular range of designs, constraints on feasible designs that provided an unexpected benefit. The interplay between components can make it particularly difficult to select or develop one component during engine design and prototype testing, especially when some components are at different stages of completion. For example, one or more components may be nearly complete, yet one or more other components may be in an initial or preliminary phase where only one (or a few) design parameters are known. It is desired to arrive at what is possible at an early stage of design, so that the down selection of candidate optimal designs, given the tradeoffs, become more possible. Heretofore the process has sometimes been more ad hoc, selecting one design or another without knowing the impact when a concept is first taken into consideration. 
     Even taken separately from the integration of a shaft design with the rest of an engine, modifying an existing shaft to increase its critical speed is challenging, and the impact of the different types of improvements and configurations on critical speed is not easily predictable without empirical experimentation and simulation, which can be enormously expensive and time-consuming. In some cases, a modification may even result in lowering the critical speed. 
     It is desirable to narrow the range of configurations or combination of features that can yield favorable results given the constraints of the design, feasibility, manufacturing, certification requirements, etc. early in the design selection process to avoid wasted time and effort. During the course of the evaluation of different embodiments as set forth above, the inventors discovered, unexpectedly, that there exists a relationship between the critical speed of the shaft and the ratio L/D, which uniquely identifies a finite and readily ascertainable (in view of this disclosure) number of embodiments suitable for a particular architecture that can avoid a supercritical or critical shaft situation during normal operation of an engine. This relationship is referred to by the inventors as the midshaft rating (MSR), and is calculated according to the following relationship (1) between length, diameter and a redline speed (ft/sec) measured at the outer diameter of the shaft: 
       Midshaft Rating MSR=( L/D ) (Shaft OD Speed at redline) 1/2   (1)
 
     L/D is shaft length divided by effective shaft outer diameter, L/D. The ratio L/D is multiplied with the square root of the outer diameter (OD) rotation speed (OD Speed) at the redline speed for the engine architecture. Generally, the length L and diameter D are expressed in inches, and the shaft OD redline speed is the linear speed of the shaft surface. The OD redline speed in feet per second is calculated as the shaft mode speed (in RPM) multiplied by the outer circumference of the shaft (the outer diameter of the shaft multiplied by pi), and with additional corrections to convert from inches to feet and from minutes to seconds. Accordingly, the midshaft rating has units of (velocity) 1/2 . 
     The midshaft rating identifies embodiments for a turbomachine&#39;s power turbine that allow subcritical operation of the engine for a rated redline speed. TABLE 4 lists exemplary embodiments of the turbine shaft along with its associated MSR value. The embodiments can inform one of the dimensions or qualities of the shaft that are believed reasonable and practical for a shaft according to its basic features and the intended, rated critical speed. In other words, the midshaft rating, and, optionally, the L/D ratio and/or the OD speed at redline, indicates the operating ranges of interest, taking into account the constraints within which a turbomachine operates, e.g., size, dimensions, cost, mission requirements, airframe type, etc. 
     In other embodiments, the midshaft rating may also, or alternatively, be used to define the propulsive system operating at a relatively high redline speed. Such things as the requirements of a propulsive system, the requirements of its subsystem(s), airframe integration needs and limitations, and performance capabilities may, therefore, be summarized or defined by the midshaft rating. 
     In still other embodiments, the midshaft rating may additionally provide a particularly useful indication of the efficiency and effectiveness of the engine during initial development, e.g. as a tool to accept or reject a particular configuration. Thus, the midshaft rating can be used, for example, to guide low-speed shaft development. Therefore, the midshaft rating can also improve the process of developing a turbomachine engine. 
     Table 4 lists the bearing layout, the strength-to-weight ratio E/rho in inches&#39;, the effective thickness T eff  in inches, the critical speed corresponding to the shaft&#39;s fundamental mode in RPM, the OD linear speed at redline in ft/sec, the length-to-diameter ratio L/D (dimensionless), and MSR in (ft/sec) 1/2  for all the embodiments (1 to 13) of Tables 1 to 13, as well as a number of additional embodiments (14 to 34). As noted above, L/D represents the ratio of the length over the outer diameter. When the shaft has a variable diameter over its length, the outer diameter may be the diameter at the midshaft. E/rho represents the material composition of the shaft, and T eff  represents an effective wall thickness of the shaft. For shafts with variable thickness over their length, the wall thickness may be the thickness at the midshaft. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                   
                 E/rho 
                 Teff 
                 Mode 
                   
                 OD Speed 
                 MSR 
               
               
                 Embodiment 
                 Bearing Layout 
                 in −1   
                 in 
                 RPM 
                 L/D 
                 ft/sec 
                 (ft/sec) 1/2   
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 1 
                 2-bearing outbound 
                 1.00E+8 
                 0.35 
                 4181 
                 30 
                 50 
                 214 
               
               
                 2 
                 inbound OTM 
                 1.27E+8 
                 0.35 
                 10263 
                 22 
                 123 
                 247 
               
               
                 3 
                 outbound OTM 
                 1.27E+8 
                 0.35 
                 6915 
                 30 
                 83 
                 275 
               
               
                 4 
                 inbound OTM 
                 1.00E+8 
                 0.35 
                 9001 
                 22 
                 108 
                 231 
               
               
                 5 
                 outbound OTM 
                 1.00E+8 
                 0.35 
                 6065 
                 30 
                 73 
                 257 
               
               
                 6 
                 inbound OTM 
                 1.00E+8 
                 0.32 
                 10039 
                 22 
                 121 
                 242 
               
               
                 7 
                 outbound OTM 
                 1.00E+8 
                 0.32 
                 6942 
                 30 
                 83 
                 272 
               
               
                 8 
                 4-bearing straddle 
                 1.00E+8 
                 0.35 
                 7746 
                 22 
                 93 
                 214 
               
               
                 9 
                 4-bearing straddle 
                 1.00E+8 
                 0.32 
                 8555 
                 22 
                 103 
                 223 
               
               
                 10 
                 4-bearing straddle 
                 1.27E+8 
                 0.35 
                 8832 
                 22 
                 106 
                 229 
               
               
                 11 
                 4-bearing straddle 
                 1.27E+8 
                 0.32 
                 9703 
                 30 
                 116 
                 322 
               
               
                 12 
                 inbound OTM 
                 1.27E+8 
                 0.32 
                 11386 
                 22 
                 137 
                 257 
               
               
                 13 
                 outbound OTM 
                 1.27E+8 
                 0.32 
                 7873 
                 30 
                 94 
                 290 
               
               
                 14 
                 4-bearing outbound 
                 1.00E+8 
                 0.35 
                 6262 
                 26 
                 72 
                 219 
               
               
                 15 
                 2-bearing aft 
                 1.27E+8 
                 0.29 
                 8255 
                 21 
                 109 
                 215 
               
               
                 16 
                 2-bearing aft 
                 1.27E+8 
                 0.31 
                 13323 
                 14 
                 233 
                 216 
               
               
                 17 
                 2-bearing aft 
                 1.27E+8 
                 0.47 
                 5667 
                 23 
                 83 
                 210 
               
               
                 18 
                 2-bearing aft 
                 1.27E+8 
                 0.29 
                 6380 
                 24 
                 83 
                 215 
               
               
                 19 
                 2-bearing aft 
                 1.27E+8 
                 0.31 
                 9821 
                 17 
                 154 
                 216 
               
               
                 20 
                 2-bearing aft 
                 1.27E+8 
                 0.47 
                 4586 
                 26 
                 67 
                 211 
               
               
                 21 
                 2-bearing aft 
                 1.00E+8 
                 0.23 
                 6380 
                 24 
                 84 
                 217 
               
               
                 22 
                 2-bearing aft 
                 1.00E+8 
                 0.25 
                 13493 
                 14 
                 235 
                 218 
               
               
                 23 
                 2-bearing aft 
                 1.00E+8 
                 0.38 
                 4586 
                 27 
                 62 
                 210 
               
               
                 24 
                 2-bearing aft 
                 1.27E+8 
                 0.29 
                 6619 
                 25 
                 87 
                 231 
               
               
                 25 
                 2-bearing aft 
                 1.27E+8 
                 0.31 
                 11065 
                 17 
                 176 
                 232 
               
               
                 26 
                 2-bearing aft 
                 1.27E+8 
                 0.47 
                 4852 
                 28 
                 64 
                 224 
               
               
                 27 
                 4-bearing straddle 
                 1.00E+8 
                 0.29 
                 6380 
                 28 
                 75 
                 245 
               
               
                 28 
                 inbound OTM 
                 1.00E+8 
                 0.31 
                 10666 
                 19 
                 165 
                 247 
               
               
                 29 
                 outbound OTM 
                 1.00E+8 
                 0.47 
                 4586 
                 31 
                 59 
                 239 
               
               
                 30 
                 4-bearing straddle 
                 1.27E+8 
                 0.23 
                 6380 
                 35 
                 70 
                 289 
               
               
                 31 
                 inbound OTM 
                 1.27E+8 
                 0.25 
                 11410 
                 22 
                 181 
                 294 
               
               
                 32 
                 outbound OTM 
                 1.27E+8 
                 0.38 
                 5293 
                 33 
                 70 
                 276 
               
               
                   
               
            
           
         
       
     
     Embodiments 15 to 26 use a two-bearing aft layout. These embodiments differ in using composite materials, different shaft geometries, and variable thickness profiles. 
     Embodiments 15 to 17 use a composite material instead of steel alloy. These embodiments differ in shaft geometry, with different L/D ratios ranging from 14 to 23. 
     Embodiments 18 to 20 use a material composite instead of a steel alloy. These embodiments also differ from each other in shaft geometry (e.g., L/D ratio). These also differ from Embodiments 15 to 17, in being longer and thinner, resulting in a higher range of L/D ratio, from 17 to 26. 
     Embodiments 21 to 23 use a steel alloy, vary the shaft geometry (length and/or diameter), and have a concave thickness profile. These differ from each other in terms of their effective thickness. These embodiments may be compared to Embodiments 24 to 26, which use composite materials, vary the shaft geometry (length and/or diameter), and have a concave thickness profile. 
     Embodiments 27 to 32 use different bearing layouts. Embodiments 27 to 29 use steel alloy and have varying geometry. Embodiments 30 to 32 use material composites and a concave thickness profile, in addition to varying geometry. 
     Based on the experimentation described above, the inventors identified embodiments with MSR between two hundred and three hundred (ft/sec) −1  and OD redline speeds ranging from fifty to two hundred fifty ft/sec and with L/D ratio ranging from twelve to thirty-seven were possible and indicated noticeable improvements in subcritical range when the power turbine shaft incorporates the various aspects of the disclosure. 
       FIG.  6 A  illustrates exemplary ranges and/or values for a midshaft rating, with respect to OD speed at redline. The plot indicates exemplary values for the midshaft rating (MSR). Specifically,  FIG.  6 A  shows a range  615  defined by MSR between 200 (ft/sec) −1  (curve  620 ) and 300 (ft/sec) −1  (curve  625 ), for redline speeds from fifty to two hundred and fifty feet per second. 
       FIG.  6 B  illustrates exemplary ranges and/or values for a midshaft rating, with respect to L/D ratio. The plot indicates exemplary values for the midshaft rating (MSR). Specifically,  FIG.  6 B  shows a range  615  defined by MSR between 200 (ft/sec) −1  (curve  620 ) and 300 (ft/sec) −1  (curve  625 ), for L/D ratios from twelve to thirty-seven. 
     Table 5 summarizes examples of different operating ranges for embodiments, such as the embodiments listed in Table 4. For example, an embodiment can be configured with a L/D ranging between twelve and twenty may have an OD speed between one hundred and fifty and two hundred and fifty ft/sec, and a corresponding range of MSR between one hundred ninety and two hundred forty-five (ft/sec) 1/2 . As another example, an embodiment can be configured with a L/D ranging between sixteen and thirty may have an OD speed between seventy-five and one hundred seventy-five ft/sec, and a corresponding range of MSR between two hundred twelve and two hundred sixty (ft/sec) 1/2 . As still another example, an embodiment can be configured with a L/D ranging between twenty-six and thirty-seven may have an OD speed between sixty and ninety ft/sec, and a corresponding range of MSR between two hundred forty-seven and two hundred eighty-seven ft/sec) 1/2 . These low, nominal, and high ranges as summarized in Table 5 are general examples, and individual embodiments may exceed these values. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 5 
               
               
                   
                   
               
               
                   
                 Example Limits 
                 L/D 
                 OD Speed 
                 MSR 
               
               
                   
                 and Ranges 
                 (in/in) 
                 (ft/sec) 
                 (ft/sec) 1/2   
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 Low limit 
                 12 
                 250 
                 190 
               
               
                   
                   
                 20 
                 150 
                 245 
               
               
                   
                 Nominal limit 
                 16 
                 175 
                 212 
               
               
                   
                   
                 30 
                 75 
                 260 
               
               
                   
                 High Limit 
                 26 
                 90 
                 247 
               
               
                   
                   
                 37 
                 60 
                 287 
               
               
                   
                   
               
            
           
         
       
     
     In view of the foregoing objectives, in at least certain embodiments, a propulsion system is configured to define an MSR greater than one hundred ninety (ft/sec) 1/2 , such as greater than two hundred (ft/sec) 1/2 , such as at least two hundred thirty-five (ft/sec) 1/2 , up to at least three hundred (ft/sec) 1/2 . 
     In view of the foregoing objectives, in at least certain embodiments, a propulsion system is configured to define an L/D ratio greater than twelve, such as greater than sixteen, such as at least twenty-six, up to at least thirty-seven. 
     In view of the foregoing objectives, in at least certain embodiments, a propulsion system is configured to define an OD redline speed greater than sixty ft/sec, such as greater than seventy five ft/sec, such as at least one hundred and fifty ft/sec, up to at least two hundred and fifty ft/sec. 
     Based on the teachings in this disclosure, and without limiting the disclosure to only those embodiments explicitly shown, it will be understood how both the manner and the degree to which a modification of shaft length, diameter, material composition, bearings configuration, and thickness profile affects the MSR, and, additionally, the competing requirements, or requirements for a turbomachine architecture (e.g., available spacing/packaging, clearance, sump location, lubrication, etc.) for a given MSR. 
     Further aspects of the present disclosure are provided by the subject matter of the following clauses. 
     A turbomachine engine includes a core engine having one or more compressor sections, one or more turbine sections that includes a power turbine, and a combustion chamber in flow communication with the compressor sections and turbine sections. The turbomachine engine also includes a shaft that is coupled to the power turbine and that is characterized by a midshaft rating (MSR) between two hundred (ft/sec) 1/2  and three hundred (ft/sec) 1/2 , and a redline speed between fifty and two hundred fifty feet per second (ft/sec). 
     A turbomachine engine includes a core engine having one or more compressor sections, one or more turbine sections that includes a power turbine, and a combustion chamber in flow communication with the compressor sections and turbine sections. The turbomachine engine also includes a shaft that is coupled to the power turbine and that is characterized by a midshaft rating (MSR) between two hundred (ft/sec) 1/2  and three hundred (ft/sec) 1/2 , a length L, an outer diameter D, and a ratio of L/D between twelve and thirty-seven. 
     The turbomachine engine of any preceding clause, wherein the turbomachine engine is configured to operate up to a redline speed without passing through a critical speed associated with a first-order bending mode of the shaft. 
     The turbomachine engine of any preceding clause, wherein the turbomachine engine is configured to operate up to the redline speed without passing through a critical speed associated with a first-order bending mode of the shaft. 
     The turbomachine engine of any preceding clause, wherein the MSR is between one hundred ninety (ft/sec) 1/2  and two hundred forty-five (ft/sec) 1/2 . 
     The turbomachine engine of any preceding clause, wherein the MSR is between two hundred twelve (ft/sec) 1/2  and two hundred sixty (ft/sec) 1/2 . 
     The turbomachine engine of any preceding clause, wherein the MSR is between two hundred forty-seven (ft/sec) 1/2  and two hundred ninety (ft/sec) 1/2 . 
     The turbomachine engine of any preceding clause, wherein the redline speed is between sixty and ninety ft/sec. 
     The turbomachine engine of any preceding clause, wherein the redline speed is between seventy-five and one hundred seventy-five ft/sec. 
     The turbomachine engine of any preceding clause, wherein the redline speed is between one hundred fifty and two hundred fifty ft/sec. 
     The turbomachine engine of any preceding clause, wherein the ratio of L/D is between twelve and twenty. 
     The turbomachine engine of any preceding clause, wherein the ratio of LID is between sixteen and thirty. 
     The turbomachine engine of any preceding clause, wherein the ratio of L/D is between twenty-six and thirty-seven. 
     The turbomachine engine of any preceding clause, wherein the shaft is a composite shaft made of at least two different materials. 
     The turbomachine engine of any preceding clause, wherein the shaft has a length L and a reduced mass density at a midpoint along the length L. 
     The turbomachine engine of any preceding clause, wherein the shaft has a reduced mass density at a midpoint along the length L. 
     The turbomachine engine of any preceding clause, wherein the shaft is a hollow convex shaft with a reduced wall thickness at the midpoint, a variable inner diameter, and a constant outer diameter. 
     The turbomachine engine of any preceding clause, wherein the shaft is a hollow convex shaft with a reduced wall thickness at the midpoint, a constant inner diameter, and a variable outer diameter. 
     The turbomachine engine of any preceding clause, wherein the shaft is coupled to the power turbine at a first mounting point, and wherein the shaft is also coupled to one of the compressor sections at a second mounting point. 
     The turbomachine engine of any preceding clause, wherein the shaft is supported by at least a first bearing and a second bearing. 
     The turbomachine engine of any preceding clause, wherein the shaft has a length L that is measured as the distance between the first bearing and the second bearing. 
     The turbomachine engine of any preceding clause, wherein the length L is measured as the distance between the first bearing and the second bearing. 
     The turbomachine engine of any preceding clause, wherein at least one bearing is a duplex bearing that has an overturning moment capability. 
     The turbomachine engine of any preceding clause, wherein each bearing is one of a ball bearing and a roller bearing. 
     The turbomachine engine of any preceding clause, wherein the first bearing is positioned between the first mounting point and the second mounting point, and wherein the second mounting point is positioned between the first bearing and the second bearing. 
     The turbomachine engine of any preceding clause, wherein the first bearing and the second bearing support the shaft in an inbound configuration in which the first bearing and the second bearing are positioned between the first mounting point and the second mounting point. 
     The turbomachine engine of any preceding clause, wherein the first bearing and the second bearing support the shaft in an outbound configuration in which the first mounting point and the second mounting point are positioned between the first bearing and the second bearing. 
     The turbomachine engine of any preceding clause, wherein the shaft is further supported by a third bearing and a fourth bearing. 
     The turbomachine engine of any preceding clause, wherein the first bearing and the second bearing are a first pair of duplex bearings, and the third bearing and the fourth bearing are a second pair of duplex bearings, wherein the first pair of duplex bearings and the second pair of duplex bearings support the shaft in an inbound overturning moment configuration in which the first pair of duplex bearings and the second pair of duplex bearings are positioned between the first mounting point and the second mounting point. 
     The turbomachine engine of any preceding clause, wherein the first bearing and the second bearing are a first pair of duplex bearings, and the third bearing and the fourth bearing are a second pair of duplex bearings, wherein the first pair of duplex bearings and the second pair of duplex bearings support the shaft in an outbound overturning moment configuration in which the first mounting point and the second mounting point are positioned between the first pair of duplex bearings and the second pair of duplex bearings. 
     The turbomachine engine of any preceding clause, wherein the first bearing, the second bearing, the third bearing, and the fourth bearing support the shaft in a four-bearing straddle configuration in which the first bearing and the second bearing are positioned between the first mounting point and the second mounting point, and the first mounting point and the second mounting point are positioned between the third bearing and the fourth bearing. 
     The turbomachine engine of any preceding clause, wherein the first bearing, the second bearing, the third bearing, and the fourth bearing support the shaft in a four-bearing outbound configuration in which the first mounting point and the second mounting point are positioned between a first group of bearings comprising the first bearing and the second bearing, and a second group of bearings comprising the third bearing and the fourth bearing. 
     In another aspect, a method includes using a turbomachine engine with a core having one or more compressor sections, one or more turbine sections that includes a power turbine, and a combustion chamber in flow communication with the compressor sections and turbine sections. The method also includes driving a shaft that is coupled to the power turbine and that is characterized by a midshaft rating (MSR) between two hundred (ft/sec) 1/2  and three hundred (ft/sec) 1/2 , and a redline speed between fifty and two hundred fifty feet per second (ft/sec). 
     In another aspect, a method includes using a turbomachine engine with a core having one or more compressor sections, one or more turbine sections that includes a power turbine, and a combustion chamber in flow communication with the compressor sections and turbine sections. The method also includes driving a shaft that is coupled to the power turbine and that is characterized by a midshaft rating (MSR) between two hundred (ft/sec) 1/2  and three hundred (ft/sec) 1/2 , a length L, an outer diameter D, and a ratio of L/D between twelve and thirty-seven. 
     The method of any preceding clause, wherein the turbomachine engine is configured to operate up to a redline speed without passing through a critical speed associated with a first-order bending mode of the shaft. 
     The method of any preceding clause, wherein the turbomachine engine is configured to operate up to the redline speed without passing through a critical speed associated with a first-order bending mode of the shaft. 
     The method of any preceding clause, wherein the MSR is between one hundred ninety (ft/sec) 1/2  and two hundred forty-five (ft/sec) 1/2 . 
     The method of any preceding clause, wherein the MSR is between two hundred twelve (ft/sec) 1/2  and two hundred sixty (ft/sec) 1/2 . 
     The method of any preceding clause, wherein the MSR is between two hundred forty-seven (ft/sec) 1/2  and two hundred ninety (ft/sec) 1/2 . 
     The method of any preceding clause, wherein the redline speed is between sixty and ninety ft/sec. 
     The method of any preceding clause, wherein the redline speed is between seventy-five and one hundred seventy-five ft/sec. 
     The method of any preceding clause, wherein the redline speed is between one hundred fifty and two hundred fifty ft/sec. 
     The method of any preceding clause, wherein the ratio of L/D is between twelve and twenty. 
     The method of any preceding clause, wherein the ratio of L/D is between sixteen and thirty. 
     The method of any preceding clause, wherein the ratio of L/D is between twenty-six and thirty-seven. 
     The method of any preceding clause, wherein the shaft is a composite shaft made of at least two different materials. 
     The method of any preceding clause, wherein the shaft has a length L and a reduced mass density at a midpoint along the length L. 
     The method of any preceding clause, wherein the shaft has a reduced mass density at a midpoint along the length L. 
     The method of any preceding clause, wherein the shaft is a hollow convex shaft with a reduced wall thickness at the midpoint, a variable inner diameter, and a constant outer diameter. 
     The method of any preceding clause, wherein the shaft is a hollow convex shaft with a reduced wall thickness at the midpoint, a constant inner diameter, and a variable outer diameter. 
     The method of any preceding clause, wherein the shaft is coupled to the power turbine at a first mounting point, and wherein the shaft is also coupled to one of the compressor sections at a second mounting point. 
     The method of any preceding clause, wherein the shaft is supported by at least a first bearing and a second bearing. 
     The method of any preceding clause, wherein the shaft has a length L that is measured as the distance between the first bearing and the second bearing. 
     The method of any preceding clause, wherein the length L is measured as the distance between the first bearing and the second bearing. 
     The method of any preceding clause, wherein at least one bearing is a duplex bearing that has an overturning moment capability. 
     The method of any preceding clause, wherein each bearing is one of a ball bearing and a roller bearing. 
     The method of any preceding clause, wherein the first bearing is positioned between the first mounting point and the second mounting point, and wherein the second mounting point is positioned between the first bearing and the second bearing. 
     The method of any preceding clause, wherein the first bearing and the second bearing support the shaft in an inbound configuration in which the first bearing and the second bearing are positioned between the first mounting point and the second mounting point. 
     The method of any preceding clause, wherein the first bearing and the second bearing support the shaft in an outbound configuration in which the first mounting point and the second mounting point are positioned between the first bearing and the second bearing. 
     The method of any preceding clause, wherein the shaft is further supported by a third bearing and a fourth bearing. 
     The method of any preceding clause, wherein the first bearing and the second bearing are a first pair of duplex bearings, and the third bearing and the fourth bearing are a second pair of duplex bearings, wherein the first pair of duplex bearings and the second pair of duplex bearings support the shaft in an inbound overturning moment configuration in which the first pair of duplex bearings and the second pair of duplex bearings are positioned between the first mounting point and the second mounting point. 
     The method of any preceding clause, wherein the first bearing and the second bearing are a first pair of duplex bearings, and the third bearing and the fourth bearing are a second pair of duplex bearings, wherein the first pair of duplex bearings and the second pair of duplex bearings support the shaft in an outbound overturning moment configuration in which the first mounting point and the second mounting point are positioned between the first pair of duplex bearings and the second pair of duplex bearings. 
     The method of any preceding clause, wherein the first bearing, the second bearing, the third bearing, and the fourth bearing support the shaft in a four-bearing straddle configuration in which the first bearing and the second bearing are positioned between the first mounting point and the second mounting point, and the first mounting point and the second mounting point are positioned between the third bearing and the fourth bearing. 
     The method of any preceding clause, wherein the first bearing, the second bearing, the third bearing, and the fourth bearing support the shaft in a four-bearing outbound configuration in which the first mounting point and the second mounting point are positioned between a first group of bearings comprising the first bearing and the second bearing, and a second group of bearings comprising the third bearing and the fourth bearing. 
     Although the foregoing description is directed to certain embodiments, it is noted that other variations and modifications will be apparent to those skilled in the art, and may be made without departing from the spirit or scope of the disclosure. Moreover, features described in connection with one embodiment may be used in conjunction with other embodiments, even if not explicitly stated above.