Abstract:
A method of processing a rotor. The rotor is formed by casting an ingot to have first and second regions formed of different alloys that intermix during casting to define a transition zone therebetween. The ingot is forged to yield a rotor forging that contains axially-aligned first and second alloy regions and a transition zone therebetween. A three-dimensional approximation of the transition zone is generated, which can be used to predict the effects of the transition zone on the dynamic performance of a rotor machined from the forging.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This is a continuation-in-part patent application of U.S. patent application Ser. No. 10/463,441, filed Jun. 18, 2003, now U.S. Pat. No. 6,807,860. 

   BACKGROUND OF INVENTION 
   1. Field of the Invention 
   The present invention generally relates to turbine rotors, such as those used in steam turbines, gas turbine engines, and jet engines. More particularly, this invention relates to a rotor and method of producing a monolithic rotor containing two or more alloys within separate regions of the rotor resulting in a transition zone between different alloy regions, and to a method of determining the shape of the transition zone to predict the dynamic performance of the rotor. 
   2. Description of the Related Art 
   Rotors used in steam turbines, gas turbines and jet engines typically experience a range of operating conditions along their lengths. The different operating conditions complicate the selection of a suitable rotor material and the manufacturing of the rotor because a material optimized to satisfy one operating condition may not be optimal for meeting another operating condition. For instance, the inlet and exhaust areas of a steam turbine rotor have different material property requirements. The high temperature inlet region typically requires a material with high creep rupture strength but only moderate toughness. The exhaust area, on the other hand, does not demand the same level of high temperature creep strength, but suitable materials typically must have very high toughness because of the high loads imposed by long turbine blades used in the exhaust area. 
   Because a monolithic rotor (i.e., a rotor that is not an assembly) of a single chemistry cannot meet the property requirements of each of the LP, IP and HP stages for the reasons discussed above, rotors constructed by assembling segments of different chemistries are widely used. For example, large steam turbines typically have a bolted construction made up of separate rotors contained in separate shells or hoods for use in different sections of the turbine. More recently, the steam turbine industry has favored CrMoV low alloy steels for use in the HP stage and NiCrMoV for use in the LP stage, though NiMoV low alloy steels have also been widely used as materials for the various stages. Smaller steam turbines may make use of a mid-span coupling to bolt high and low temperature components together within one shell. Finally, rotors for gas turbines and jet engines are often constructed by bolting a series of disks and shafts together. While rotors having a bolted construction are widely used, they suffer from several disadvantages including increased numbers of parts, increased assembly requirements, increased length of the rotor assembly, and more balance complexity. 
   Another method of combining different materials in a single rotor is to weld together rotor segments formed of dissimilar materials, forming what may be termed a multiple alloy rotor (MAR). However, a welded rotor construction also has disadvantages, such as high investment costs for the welding equipment, additional production costs for weld preparation and welding, long production times to produce, inspect and upgrade the weld, and increased cost and production time caused by the need for post weld heat treatment. The strength of rotors having a welded construction can also be limited due to a need to maintain a low carbon content in the weld, and the propensity for high numbers of small non-metallic inclusions that reduce load carrying capability. 
   The capability of producing a monolithic MAR would address the above-noted shortcomings of assembled MAR&#39;s. Furthermore, monolithic MAR&#39;s would be particularly well suited for meeting the demand for higher efficiency steam turbines whose requirements include low pressure (LP), intermediate pressure (IP) and high pressure (HP) stages (or combinations thereof) with additional stages in areas normally occupied by couplings. Consumable electrode remelting techniques such as electro-slag remelting (ESR) and vacuum arc remelting (VAR) methods offer flexibility for producing components that contain alloy combinations, and therefore has been considered for producing monolithic MAR&#39;s. As an example, U.S. Pat. No. 6,350,325 to Ewald et al. discloses an ESR method of producing a dual alloy rotor from 12Cr-type alloys that have different levels of alloying constituents, but are sufficiently close in composition so as to have substantially identical austenitizing temperatures. Ewald et al. also disclose that, because the alloys have similar compositions, problems can be avoided that are associated with mixing of alloys having significantly different material properties, which results in the formation of a transition zone (TZ) between regions of the rotor formed by the different alloys. 
   One such problem is thermal stability arising from the massive size of a rotating rotor supported by bearings at each end of the rotor. When supported in this manner, a rotor behaves as a simply supported beam structure and will deflect in reaction to the centrifugal load always present at operational conditions, with the largest deflection being near the center of the rotor. Because of the inherent asymmetry of the transition zone within a MAR rotor, deflection significantly increases when the rotor is at its elevated operating temperatures. As the rotor rotates about its bent centerline, the rotor material is subjected to high cycle fatigue as a result of being in tension and then in compression with each rotation. Consequently, reducing deflection by minimizing material asymmetry is necessary to maximize the life of a MAR rotor and the turbine in which it is installed. One solution is to limit the rotor to alloys with similar compositions. However, this restriction limits the ability to optimize the compositions of the LP, IP and HP rotor sections for their operating environments and cost. For example, such a limitation has dissuaded the manufacture of a monolithic MAR whose HP stage is formed of CrMoV and its LP stage is formed of NiCrMoV. Therefore, it would be desirable if an improved process were available for producing turbine rotors of different alloy compositions. 
   SUMMARY OF INVENTION 
   The present invention provides a method for processing a rotor. The rotor is formed by machining a single rotor forging to have at least two axially-aligned rotor regions and a transition zone therebetween. According to a particular aspect of the invention, the rotor is a monolithic multiple alloy rotor (MAR), wherein the rotor regions are formed of different alloys and the transition zone has a composition that differs from and varies between the rotor regions. 
   The process of this invention involves casting a multiple-alloy ingot having at least first and second ingot regions axially aligned within the ingot, with the first and second ingot regions being formed of different alloys so that intermixing occurs during casting to define a transition zone therebetween having a composition that differs from and varies between the first and second ingot regions. The ingot is then forged to produce a rotor forging containing first and second forging regions and a transition zone therebetween corresponding to the first and second ingot regions and the transition zone of the ingot, i.e., the first and second forging regions are formed of the different alloys and the transition zone of the rotor forging has a composition that differs from and varies between the first and second forging regions. The first and second forging regions and the transition region therebetween are axially aligned along a geometric centerline of the rotor forging. Following heat treatment, the rotor forging is machined to produce a machined rotor containing first and second rotor regions and a transition zone therebetween corresponding to the first and second forging regions and the transition zone of the rotor forging, i.e., the first and second rotor regions are also formed of the different alloys and the composition of the transition zone within the machined rotor differs from and varies between the first and second rotor regions. 
   According to one aspect of the invention, the transition zone within the rotor forging is asymmetrical about the geometric centerline of the rotor forging. This asymmetry of the transition zone may be attributable to asymmetry of the transition zone within the ingot and/or as a result of the forging operation, the latter of which always degrades the symmetry of the transition zone to some degree. Because the material properties of the rotor vary with the composition of the transition zone, asymmetry of the transition zone causes asymmetrical variations in the mechanical and physical properties of the rotor, which if not mitigated promotes bending of the rotor and thermal instability during operation. According to the invention, the effect of the transition zone on the dynamic performance of the rotor is predicted by generating a three-dimensional approximation of the transition zone. The three-dimensional approximation of the shape of the transition zone can be produced by combining measurements of the chemistry on the surface of the rotor with either knowledge of the shape of the solidified transition zone melt pool after forging or by ultrasonic inspection to identify the three-dimensional shape of the transition zone. With this knowledge, the rotor forging can be machined so that its axis of rotation is more centrally located with respect to the transition zone of the rotor, and therefore more centrally located with respect to the material properties of the rotor. As an optional additional step following rough machining of the rotor forging, a standard heat indication test can be performed on the rotor to measure its tendency to bend when heated and, if the tendency is larger than desired, the results of the heat indication test can be used to optimize the final machining of the rotor to reduce the bending tendency. 
   In view of the above, it can be seen that a significant advantage of this invention is that a multiple alloy rotor can be produced by casting and forging without the limitations previously placed on the alloys used to form such rotors. In particular, the present invention permits the use of dissimilar alloys such as NiCrMoV and CrMoV alloys, which form transition zones that, using prior art processing approaches, would result in a rotor that exhibits unacceptable thermal instability. The invention is able to overcome this restriction with a process that identifies the asymmetrical property variations attributable to the transition zone, thereby permitting different sections of the rotor to be formed of alloys that have the potential for optimizing the different rotor sections (e.g., HP, IP, and LP) for their operating environments. 
   Other objects and advantages of this invention will be better appreciated from the following detailed description. 

   
     BRIEF DESCRIPTION OF DRAWINGS 
       FIG. 1  schematically represents processing steps performed to produce a multiple alloy rotor forging having a transition zone in accordance with a preferred embodiment of this invention. 
       FIG. 2  schematically represents a partial cross-sectional view of a rotor forging formed by the process of  FIG. 1 , wherein chemical analysis is performed at points within and adjacent the transition zone in accordance with an embodiment of the invention. 
       FIG. 3  shows a rotor forging formed by the process of  FIG. 1 , wherein metallurgical characteristics within and adjacent the transition zone are determined by ultrasonic inspection in accordance with another embodiment of the invention. 
       FIG. 4  is a chart showing data collected from chemical analysis of a rotor forging in accordance with the embodiment of  FIG. 2 . 
       FIG. 5  is a two-dimensional plot generated from the data of  FIG. 4  and identifying the boundaries of regions within the transition zone having different nickel contents. 
       FIG. 6  represents a two-dimensional plot of one of the boundaries of  FIG. 5  measured at a single cross-section and rotated to various circumferential positions to illustrate transformations used to model the asymmetry of the transition zone. 
       FIG. 7  is a graph comparing the predicted and measured centerline deflection along the axis of a rotor as a result of the asymmetry of the transition zone within the rotor, in which the predicted centerline deflection was based on a chemical analysis performed in accordance with the embodiment of  FIGS. 2 and 4  through  6 . 
       FIG. 8  is a chart showing data collected from an ultrasonic scan of the transition zone at one axial position of a rotor forging in accordance with the embodiment of  FIG. 3 . 
       FIG. 9  is a three-dimensional plot generated using multiple ultrasonic scans similar to  FIG. 8  and taken at multiple axial positions of a rotor forging. 
       FIG. 10  represents an analytical technique for generating a three-dimensional mathematical model of the transition zone using the data of  FIGS. 8 and 9 . 
       FIG. 11  is a graph comparing the predicted and measured centerline deflection along the axis of a rotor as a result of the asymmetry of the transition zone within the rotor, in which the predicted centerline deflection was based on an ultrasonic inspection performed in accordance with the embodiment of  FIGS. 3 and 8  through  10 . 
       FIG. 12  represents an analytical technique for transforming the coordinate system of a rotor to a new coordinate system whose center is more centrally located within the transition zone to reduce rotor centerline deflection. 
   

   DETAILED DESCRIPTION 
   The present invention generally pertains to the production of a multiple alloy rotor (MAR) using a casting technique, preferably a consumable electrode remelting technique such as electro-slag remelting (ESR) or vacuum arc remelting (VAR), and various combinations of alloy chemistries to achieve properties suitable for different regions of the rotor, such as the high, intermediate and low pressure turbine stages of an advanced power generation steam turbine, gas turbine, or aircraft engine. In so doing, a transition zone is present within the rotor between regions that have substantially uniform compositions that differ from region to region. 
     FIG. 1  represents such a process, by which a monolithic (one-piece) rotor forging  10  is produced by ESR in accordance with the invention. The forging  10  (of which only half is depicted in  FIG. 1 ) is represented as having been rough machined to define high pressure (HP), intermediate pressure (IP), and low pressure (LP) sections  12 ,  14  and  16 , respectively, along a geometric centerline  18  of the forging  10 . The forging  10  is also represented as having a transition zone  20  between two regions  22  and  24  of the forging  10  formed of an IP/HP Alloy and an LP Alloy, respectively.  FIG. 1  shows the LP alloy region  24  and the transition zone  20  generally limited to the LP section  16  of the forging  10 , while the IP/HP alloy region  22  is generally limited to the HP and IP sections  12  and  14  of the forging  10 . Various other configurations are possible, e.g., the transition zone  20  could extend into the IP section  14  of the forging  10 . The alloys are preferably selected to have mechanical and physical properties that are optimized for their respective locations within a rotor that will be subsequently machined from the forging  10 . As such, the IP/HP and LP alloy compositions are different but substantially uniform within their respective regions  22  and  24 . In contrast, the composition within the transition zone  20  varies between each of its boundaries with the regions  22  and  24  in view of the compositional differences between the LP and IP/HP alloys and the process by which the forging  10  is cast. 
   As with conventional ESR techniques,  FIG. 1  shows an electrode  30  suspended over a melt pool  32  contained in a chilled crucible  34 . In the ESR process, resistance heating between the electrode  30  and molten slag (not shown) on the surface of the pool  32  causes melting of the electrode  30 . During melting, molten droplets of the electrode  30  drop through the slag, where they solidify to produce a directionally solidified ingot  50  having discrete regions  60 ,  62  and  64  along its axial length. The electrode  30  is depicted as comprising least three distinct regions  36 ,  38  and  40 . The lower and upper regions  36  and  40  of the electrode  30  are formed of alloys corresponding to the IP/HP and LP alloys desired for the regions  22  and  24  of the forging  10 . In accordance with an optional aspect of the invention, the intermediate region  38  of the electrode  30  may have a composition that differs from the other two electrode regions  36  and  40  for the purpose of tailoring the composition of the transition zone  20  within the forging  10 . The regions  36 ,  38  and  40  of the electrode  30  can be produced using air or vacuum melting processes, followed by the use of a welding process or an ESR or VAR process to join the regions  36 ,  38  and  40  end-to-end. Alternatively, a separate piece of each alloy could be melted individually and in sequence to form the ingot  30 . 
   As the process of this invention is represented in  FIG. 1 , the lower region  62  of the ingot is represented as being formed by the IP/HP alloy (lower region  36 ) of the electrode  30 , and the upper region  64  is indicated as being formed by the LP alloy (region  40 ) of the electrode  30 , though the order of the IP/HP and LP alloys could be reversed from that shown. The lower and upper regions  36  and  40  of the electrode  30  have lengths appropriate to create the corresponding regions  62  and  64  within the ingot  50 . The middle region  60  of the ingot  50  is a transition zone between the regions  62  and  64 . Regardless of whether the electrode  30  includes the intermediate region  38 , the transition zone  60  inherently forms as a result of intermixing of the IP/HP and LP alloys during the ESR process. If the electrode  30  is formed to include the intermediate region  38 , the intermediate region  38  can be formulated to have a special intermediate chemistry to control the chemistry gradient and/or the width of the transition zone  60  within the ingot  50 . However, the shape and size of the transition zone  60  cannot be precisely controlled in view of variables in the ESR process. Therefore, the transition zone  60  may have a shape whose axial boundaries, though corresponding to the shape of the melt pool  32 , can be asymmetrical to some degree about the axis of the ingot  50 . The regions  60 ,  62  and  64  of the ingot  50  form the transition zone  20  and the regions  22  and  24 , respectively, of the rotor forging  10  forged from the ingot  50 . Of particular significance to the present invention, any forging operation performed on the multiple-alloy ingot  50  of this invention will inherently degrade the symmetry of the transition zone  60  within the ingot  50 , such that the shape of the transition zone  20  within the rotor forging  10  can be significantly asymmetrical relative to the centerline  18  of the forging  10 . 
   Various characteristics are required for the different sections  12 ,  14  and  16  of the forging  10  in order to meet the properties required of a rotor machined from the forging  10 , such as tensile strength, fracture toughness, rupture strength, thermal stability, and high process capability (repeatability and reproducibility), as well as cost targets. In order to achieve the mechanical properties desired for the rotor, the chemistries of the multiple alloys of the forging  10  are likely to be sufficiently different to require different heat treatment temperatures and durations, such that a differential heat treatment must be performed on the forging  10  prior to machining. For this purpose, a furnace with multiple temperature zones is used to provide an appropriate heat treatment temperature for each region  20 ,  22  and  24  of the rotor forging  10 . The heat treatment may include a differential temperature for both the solution or austenitizing treatment and the aging or tempering treatment of the particular alloys. For steels, a higher temperature austenitizing treatment is preferably used if higher creep rupture strength is desired (e.g., for the high pressure rotor section  12 ), while relatively lower temperatures are used if higher toughness is needed (e.g., for the low pressure rotor section  16 ). Differential cooling from the solution or austenitizing temperature is also preferably used. Rapid cooling can be used to achieve full section hardening, to avoid harmful precipitation reactions, and/or to enhance toughness (e.g., for the low pressure rotor section  16 ). Slow cooling can be used to achieve beneficial precipitation reactions, to reduce thermal stresses, and/or to enhance creep rupture strength (e.g., for the high pressure rotor section  12 ). Particular temperatures, durations, and heating and cooling rates suitable for the forging  10  will depend on the materials used, and such heat treatment parameters will generally be within the capability of one skilled in the art. 
   Notable commercial alloys that are suitable for use as the LP alloy include the conventional NiCrMoV-type low alloy steels and chromium-containing martensitic stainless steels such as the 12Cr-3Ni—Mo—V type alloy (M 152 ). Alloys that are suitable for use as the IP/HP alloy of the IP and HP sections  12  and  14  include a variety of alloys having increasing high temperature capability, such as the conventional CrMoV low alloy steels, 9–14 Cr-type alloys with varying levels of Mo, V, W, Nb, B and N, Fe—Ni alloys (such as A286), and nickel-base alloys (such as Alloy  706  or  718 ). Particularly preferred alloys for the LP section  16  include the following: 
   
     
       
             
             
           
         
             
               TABLE 1 
             
             
                 
             
             
               ALLOY TYPE 
               COMPOSITION (approximate by weight) 
             
             
                 
             
           
           
             
               NiCrMoV low 
               2–4 Ni, 1–2 Cr, 0.2–0.5 Mo, 0.05–0.2 V, 0.18–0.35 C, 
             
             
               alloy steel 
               balance Fe and incidental impurities. 
             
             
               Martensitic 
               2–3.5 Ni, 11–14 Cr, 0.2–1.2 Mn, 1–2.5 Mo, 0.2–0.5 V, 
             
             
               stainless steel 
               0.05–0.2 C, balance Fe and incidental impurities. 
             
             
                 
             
           
        
       
     
   
   Particularly preferred alloys for the HP and IP sections  12  and  14  include the following: 
   
     
       
             
             
           
         
             
               TABLE 2 
             
             
                 
             
             
               ALLOY TYPE 
               COMPOSITION (approximate by wt. %) 
             
             
                 
             
           
           
             
               CrMoV low 
               0.25–0.75 Ni, 0.8–2.5 Cr, 0.51–1 Mn, 1–2.5 Mo, 
             
             
               alloy steel 
               0.2–0.35 V, 0.15–0.35 C, balance Fe and incidental 
             
             
                 
               impurities. 
             
             
               Martensitic 
               0–1.3 Ni, 9–14 Cr, 0.1–1 Mn, 0.2–2 Mo, 0.1–0.7 V, 
             
             
               stainless steel 
               0–3 W, 0–6 Co, 0.03–0.20 C, balance Fe and incidental 
             
             
                 
               impurities. 
             
             
               Fe—Ni alloy 
               24–27 Ni, 13–16 Cr, 1.8–2.5 Ti, 1–1.5 Mo, 0.1–0.5 V, 
             
             
                 
               less than 2 Mn, less than 1 Si, less than 0.5 Al, 
             
             
                 
               less than 0.08 C, balance Fe and incidental 
             
             
                 
               impurities. 
             
             
               Nickel-Base 
               16–20 Fe, 17–21 Cr, 2.5–3.5 Mo, 4.5–5.5 Nb, 
             
             
               Alloy 
               0.6–1.2 Ti, 0.2–0.8 Al, 0–1 Co, less than 0.35 Mn, 
             
             
                 
               less than 0.35 Si, less than 0.08 C, balance Ni 
             
             
                 
               and incidental impurities. 
             
             
                 
             
           
        
       
     
   
   On the basis of combinations of the above alloys, chemistries that are believed to be well suited for the intermediate region  38  of the electrode  30  include the following: 
   
     
       
             
             
           
         
             
               TABLE 3 
             
             
                 
             
             
               ALLOY COMBINATION 
                 
             
             
               (LP Alloy - HP/IP Alloy) 
               COMPOSITION (approximate by wt. %) 
             
             
                 
             
           
           
             
               NiCrMoV—CrMoV 
               0.25–8 Ni, 0.8–6 Cr, 0.2–1.0 Mn, 0.2–1.5 Mo, 
             
             
                 
               0.05–0.35 V, 0.1–0.4 C, balance Fe and incidental impurities. 
             
             
               NiCrMoV - Martensitic 
               4–8 Ni, 0–1 Mn, 14–30 Cr, 1.5–5 Mo, 3–7 W, 4–12 Co, 
             
             
                 
               0.2–0.35 C, balance Fe and incidental impurities. 
             
             
               NiCrMoV—FeNi alloy 
               16–32 Cr, 1–4 Mn, 1.5–4 Mo, 0.1–1.0 V, 0–0.5 Al, 
             
             
                 
               0.5–5 Ti, 0–2 Si, 0–0.08 C, balance Ni and incidental impurities. 
             
             
               NiCrMoV—Ni alloy 
               0–20 Fe, 15–40 Cr, 0–0.35 Mn, 2–10 Mo, 3–12 Nb, 
             
             
                 
               0–2 Al, 0–3 Ti, 0–1 Co, 0.01–0.08 C, balance 
             
             
                 
               (30–60) Ni and incidental impurities. 
             
             
               Martensitic—Martensitic 
               0.5–8 Ni, 9–18 Cr, 0–4 Mn, 0.8–4 Mo, 0.1–0.5 V, 
             
             
                 
               0–0.5 Al, 0.02–0.05 N, 0.15–0.35 C, balance Fe 
             
             
                 
               and incidental impurities. 
             
             
               Martensitic 0 FeNi alloy 
               0–60 Ni, 9–24 Cr, 0.5–2.0 Mn, 0.5–3.0 Mo, 0–0.5 V, 
             
             
                 
               0.01–0.35 C, balance Fe and incidental impurities. 
             
             
               Martensitic - Alloy 718 
               8–12 Cr, 0.5–1.2 Mn, 0–2 Mo, 0.2–0.5 V, 0–1 Co, 
             
             
                 
               0.01–0.20 C, balance Fe and incidental impurities. 
             
             
               Martensitic - Alloy 718 
               9–50 Cr, 0–2.0 Mn, 0–8 Mo, 0–12 Nb, 0–2 Al, 0–3 Ti, 
             
             
                 
               0–1 Co, 0.01–0.08 C, balance Fe and incidental impurities. 
             
             
                 
             
           
        
       
     
   
   The ability to produce a monolithic multiple alloy rotor as described above provides various advantages. For example, fewer parts are required to produce a rotor as compared to prior art rotors produced by bolting or welding rotor sections together. Additional machining that would otherwise be required to prepare the parts for assembly is not required, and the steps of assembling and welding the parts together is eliminated. Each of these advantages reduces the cost and time required to produce a rotor. By eliminating the assembly requirement, the overall length of a rotor does not need to be increased in order to accommodate a mid-span coupling, thereby minimizing related expenses such as turbine shell and site preparation costs. The ability to avoid a welding operation eliminates the requirement for a post-weld heat treatment, which is otherwise required in addition to the heat treatment performed after forging. By eliminating welding, the strength level of the rotor can be higher than that of a rotor with a welded-type construction because there are no limitations on chemical composition other than the limitations normally imposed on ingot making. 
   Facility costs are also reduced by eliminating the requirement for specialized welding equipment to weld a rotor, which is particularly significant if a massive steam rotor is being produced. Notably, existing consumable electrode remelting (ESR and VAR) furnaces are capable of producing rotors. Consumable electrode remelting is also advantageous in that it provides a rotor that is essentially free from small nonmetallic inclusions commonly found in welded construction and which if present can reduce load-carrying capability. Grinding and repair welding to upgrade defective welded rotor joints will not be required thus saving cost and production time. Little or no alloy development is needed because alloys that are currently joined by assembly, welding, etc., can be employed by this invention. 
   As discussed above, the different chemistries of the alloys within the sections  12 ,  14  and  16  of the rotor forging  10  result in the formation of the transition zone  20 , whose properties and shape affect the dynamics of the rotor machined from the forging  10 . In particular, the transition zone  20  affects the thermal stability of the rotor at high temperatures, characterized by rotor centerline deflection that is detrimental to rotor balance, turbine clearance, and high cycle fatigue life. While consumable electrode remelting techniques are able to minimize the amount of molten metal that exists at any time in the ingot crucible  34 , and thereby limit the axial extent over which alloy mixing will occur, a transition zone of some size and shape will be present within the ingot  50  and therefore within the forging  10 , particularly in view of the significant differences in the preferred alloys identified in Tables 1 and 2. Optimum chemistries for the electrode&#39;s intermediate region  38  (Table 3) can have the effect of minimizing the extent of the transition zones  20  and  60 , thus reducing thermal stability attributable to the transition zone  20  in the forging  10 . The chemistry of the electrodes intermediate region  38  can also be optimized so as to limit the extent of the transition zone  20  to assist in locating the boundaries of the chemistries of the forging regions  22  and  24 . Nonetheless, forging of the multiple-alloy ingot  50  to produce the rotor forging  10  will result in the transition zone  20  having an asymmetrical shape and therefore asymmetrical material properties that negatively affect the dynamics of a rotor machined from the forging  10 . 
   Accordingly, a preferred aspect of the invention is to mitigate the detrimental effect of the transition zone  20  by minimizing the material asymmetry within the rotor near and within the transition zone  20 , thus reducing centerline deflection of the rotor when it is heated to its operating temperatures. For this purpose, the present invention preferably includes the step of altering the geometry of the machined rotor relative to that of the rotor forging  10  by off-center machining, so that the centerline of the final machined geometry of the rotor is relocated from that of the forging  10  to minimize the adverse effects of the inhomogeneity of the rotor around the transition region  20 . To determine the extent of off-center machining required, the three-dimensional shape of the transition zone  20  is determined using a suitable approximation technique. For example, boundary points of the transition zone  20  within the rough-machined forging  10  can be ascertained to define a plurality of axially-spaced, two-dimensional cross-sectional shapes of the transition zone  20 . These two-dimensional cross-sectional shapes can then be used to generate the three-dimensional shape of the transition zone  20 , and finite element modeling (FEM) or another suitable analytical technique can be performed on the three-dimensional shape to predict the deflection of the geometric centerline (axis of rotation) of a rotor machined from the forging  10 . With this information, the centerline of the machined rotor can be shifted by off-center machining to reduce deflection. 
   Various techniques could be used to develop both two and three-dimensional shapes of the transition zone  20  within the rotor forging  10 . According to one embodiment of the invention, the three dimensional shape of the transition zone  20  is approximated by measuring the variation of chemistry at the outer surface of the rough-machined rotor forging  10 , combined with information about the likely shape of the transition zone  20  obtained by sectioning another rotor that was cast and forged under similar or identical conditions. With this approach, an axial-spaced series of two-dimensional shapes is generated by longitudinally sectioning the similarly-processed rotor forging specimen along its geometric centerline, and then detecting the level of one or more alloying constituents present in the forging specimen to identify the boundaries separating the transition zone from the adjacent regions of the specimen. Such a technique is represented in  FIG. 2 , in which a rough-machined forging specimen  70  is depicted as containing a transition zone between two axially-aligned regions of dissimilar alloys (identified with the same reference numbers as used in  FIG. 1  for convenience).  FIG. 2  represents a portion of the forging specimen  70  as having been cross-sectioned to expose a diametrical sectioned surface  72  of the specimen  70 . The chemistries of the transition zone  20  and adjacent portions of the regions  22  and  24  are then determined at any number of locations  74  on the sectioned surface  72 . The locations  74  are aligned along radials of the forging specimen  70 , with the locations  74  being sufficiently close within each radial and sufficiently close along the axis  76  of the specimen  70  to be able to detect changes in one or more of the alloying constituents, e.g., nickel, within each region  20 ,  22  and  24  of the specimen  70 . 
   In an investigation in which CrMoV and NiCrMoV alloys were used to form the HP and LP regions  22  and  24  of a rough-machined forging specimen (e.g., similar to the forging specimen  70  of  FIG. 2 ), a series of five diametrical measurements, each series axially spaced about 20 to 100 millimeters from the adjacent series, were made consisting of two measurements at the outer surfaces of the forging specimen, at the axis of the specimen, and at radial midpoints between the outer surfaces and the axis.  FIG. 4  is a graph plotting the nickel content (wt. %) versus axial location for each measured radial location of the specimen. To generate two-dimensional shapes from the data, an optimum curve was found that would fit the data points. While a continuous curve could be used, a linear three-step function is plotted in  FIG. 4  by grouping the data points into three groups according to nickel content, one group being in the range of about 0.5% to about 2.25%, the second in the range of 2.25% to 3.2%, and the third in the range of 3.2% to 3.5%. Each group of data was fitted with a straight line using least square estimation on the condition that the two lines must share a common point at 2.25% and 3.2%. 
   After the function was determined, the axial location of 0.5%, 2.25%, 3.2%, and 3.5% nickel levels were calculated at the outer surface locations, the mid-radial locations, and the axis location. The axial location at the step transition was calculated because a step function to cover a range of 0.5% to 3.5% nickel content was used. The 0.5% and 3.5% nickel levels were chosen as generally corresponding to the levels of nickel for the CrMoV and NiCrMoV alloys, respectively, forming the regions of the specimen outside the transition zone, and therefore indicative of the boundaries of the transition zone. In the present example where each radial series consisted of five measurements, each boundary (nickel levels of 0.5% or 3.5%) of the transition zone and each subdivided zone was located with at least five measured points. Several methods are available for fitting each of the five measurements to a two-dimensional curve. For example, different orders of polynomial or cubic spline curve fitting could be used.  FIG. 5  illustrates the use of a cubic spline curve fitting, which is capable of generating a continuous, smooth curve that passes through each set of measurements corresponding to the same nickel level. 
   As evident from  FIG. 5 , each two-dimensional curve has an asymmetric shape relative to the axis of the specimen. As previously noted, this asymmetry is largely attributable to the forging operation, and would be detrimental to the thermal stability of a rotor machined from the specimen. The asymmetry of each two-dimensional is such that any attempt to establish three-dimensional axial boundaries (corresponding to nickel levels from 0.5% to 3.5%) by simply rotating the two halves of the curve on each side of the centerline at each nickel level would generate two volcano-shaped three-dimensional images of each boundary. To avoid this result, a suitable approach is to rotate each half of the curve through 180 degrees, gradually changing the shape of the curve during rotation so that the curve acquires the shape of the other half of the curve at the completion of the 180-degree rotation, so that the rotated curves coincide with each other when they meet. This can be achieved by assuming a two-dimensional curve function z=f(r), where r is the value for the radial direction and z is the value for the axial direction of a given point. Rotating a two-dimensional curve to generate a three-dimensional surface adds the variable θ, which is the variable for the circumferential direction. Using this technique, the equation to use is:
 
 z ( r,θ )= z   1  cos 2 (θ/2)+ z   2  sin 2 (θ/2)= f   1 ( r,θ )cos 2 (θ/2)+ f   1 (− r,θ )sin 2 (θ/2)
 
     FIG. 6  shows the two-dimensional curve defining the 0.5% nickel boundary of  FIG. 5  as its shape is modified when rotated through angles of 45, 90, 135 and 180 degrees in accordance with the above equation. From the three-dimensional contour defined by the curves of  FIG. 6 , a finite element model can be generated and used to predict the centerline deflection of a rotor machined from a similar rotor forging and then heated to some elevated temperature, e.g., a temperature to which the rotor would be subjected during its operation. More particularly, a three-dimensional approximation of the shape of the transition zone within a similarly-processed rotor forging having the same multiple alloy composition can be predicted by measuring the variation of chemistry at the outer surface of the rotor forging, and then utilizing the shape of the transition zone obtained from the analysis of the sectioned forging specimen.  FIG. 7  shows the results of such analysis using a temperature of about 1150° F. (about 620° C.), and evidences that the predicted deflection profile is very close to the actual deflection profile of a rotor machined from a rotor forging produced by the same process as the forging specimen. 
     FIG. 3  represents an alternative to the chemical analysis approach described above, in which the boundary points of the transition zone  20  are ascertained by ultrasonically examining the rough-machined forging  10 . Such an approach is disclosed in copending and commonly-assigned U.S. Pat. No. 6,807,860, incorporated herein by reference. In an investigation in which this approach was implemented in the present invention, an ultrasonic transducer  78  was placed against the outer surface of the forging  10 , and ultrasonic energy was transmitted through the forging  10  along the transition zone  20  to detect changes in the response of the forging material to ultrasonic energy. As evidenced by  FIG. 8 , ultrasonic inspection of the forging  10  produced a noise pattern corresponding to variations in the metallurgical characteristics within the forging  10 , such as differences in grain size attributable to the changes in chemistry between the transition zone  20  and the adjacent regions  22  and  24 . After filtering the peak noise at each scan line, the data is fitted to a circular or elliptical contour  26  representative of the expected cross-sectional (two-dimensional) shape of each transition zone boundary.  FIG. 9  represents a three-dimensional image  27  generated by combining multiple contours  26  taken along the length of the forging  10 . 
   Using the image of  FIG. 9 , a three-dimensional mathematical model was generated of the transition zone  20  by axially sectioning each transition zone contour  26  of the image with planes intersecting at the geometric centerline  18  of the forging  10 .  FIG. 10  represents one of the planes  28  intersecting one of the contours  26  at two points. Assuming the transition zone contour  26  has a radius r o , the center of the contour  26  is offset from the forging centerline  18  a distance A at an angle θ o , and the plane  28  at an angle θintersects the contour  26  to define radial distances r 1  and r 2  from the centerline  18 , the following equations can be used to find the radial distances r 1  and r 2 .
 
 r   1   =A  cos(θ o −θ)+( r   o   2   −A   2  sin 2  (θ o −θ)) 1/2 
 
 r   2 =−(− A  cos(θ o −θ)+( r   o   2   −A   2  sin 2 (θ o −θ)) 1/2 )
 
   If the total number of transition zone contours  26  is N c , then a maximum of 2N c  number of intercept points are obtained by cutting the transition zone  20  with a plane  28  of at an angle θ, and a maximum of 2N c N p  intercept points can be obtained if the transition zone  20  is cut with N p  number of planes  28 . These intercept points can then be curve fit using polynomial or cubic spline curve fitting techniques. If a polynomial curve fitting technique is used, different orders are preferably tried until a good curve fit of the data (e.g., R 2  above 95%) is obtained. The peak position of each polynomial at different angles should also be consistent, i.e., all curves end up with the same peak. If a cubic spline fitting technique is used, an additional data point at the peak location is preferred. A set of curve fit coefficients from the previous step can then be used to interpolate points between intercepting points on adjacent scans within two adjacent angles, so that a three-dimensional model for the transition zone  20  can be mathematically constructed. 
   In the investigation described above, the rotor forging  10  was produced from an ESR ingot in which the LP and IP/HP alloys were NiCrMoV and CrMoV, respectively. Assumptions for interpreting the ultrasonic data included a transition in the chemistry (and thus property) distribution occurred at a nickel content of about 3.0 weight percent, and that the ultrasonic inspection produced a noise pattern corresponding to variations in the metallurgical characteristics that occurred at the 3.0 wt. % nickel location. From the two-dimensional contours  26  and three-dimensional contour  27  defined in the manner described above, a three-dimensional finite element model was generated and used to predict the centerline deflection of a rotor machined from the rotor forging  10  and then heated to some elevated temperature, e.g., a temperature to which the rotor would be subjected during its operation.  FIG. 11  shows the results of such analysis using a furnace with three different temperature zones (about 400, 800 and 1000° F. (about 200, 425, and 540° C.)) to simulate the variation of temperature along the length of a rotor during operation, and evidences that the predicted deflection profile was very close to the actual deflection profile of the machined rotor. This model was also able to predict the rotor centerline deflection before and after removing (gashing) the material between wheels of the rotor. 
   Finally, to reduce the amount of centerline deflection from that predicted through the above-described modeling techniques, a new rotor axis is identified that is more centrally located within the transition zone  20  than the forging centerline  18 , while still within machining tolerances. The forging (or, more typically, a rough-machined geometry formed by rough machining the forging) can then be off-center machined to establish the new axis as the geometric centerline (and therefore the axis of rotation) of the final machined rotor geometry. The methodology involves translating the coordinates of the rough-machined forging from the geometric centerline of the forging to the new axis, which is at or near the center of a three-dimensional approximation of the transition zone  20 . 
   To explain the methodology, reference is made to  FIG. 12  which shows the machined outer profile of a rotor  90  at one axial position. For illustrative purposes, the rotor  90  is indicated as having an as-forged geometric centerline  18 , corresponding to the rough-machined forging  10  of  FIG. 3 . The location for the new rotor axis, more centrally located within the rotor  90  (and approximately centrally located within the transition zone  20  of the forging  10 ), is identified with reference number  98 . The offset of the new rotor axis  98  relative to the forging centerline  18  is exaggerated to assist in clarifying the methodology. In  FIG. 12 , the new rotor axis  98  is offset a distance A and on an angle θ o  from the forging centerline  18 . A point P is located a distance r o  and at an angle αrelative to the forging centerline  18 , and a distance r and at an angle θrelative to the new rotor axis  98 . The coordinates (r,θ) in the coordinate system with its center at the axis  98  can be obtained with the following equations:
 
 r   o =[( r   2   +A   2 −2 Ar  cos(θ−θ o )] 1/2 
 
tan α=( r  sin θ− A  sin θ o )/( r  cos θ− A  cos θ o )
 
   As atan(x) is in the range of −90 to +90 degrees, and the angle in a cylindrical coordinate system is in the range of −180 to +180 degrees, the following table can be used to obtain angle α. 
                                   α =   (rcosθ − acosθ 0 ) &gt; 0   (rcosθ − acosθ 0 ) &lt; 0                   (rsinθ −asinθ 0 ) &gt; 0           a   ⁢           ⁢   tan   ⁢         r   ⁢           ⁢   sin   ⁢           ⁢   θ     -     a   ⁢           ⁢   sin   ⁢           ⁢     θ   0             r   ⁢           ⁢   cos   ⁢           ⁢   θ     -     a   ⁢           ⁢   cos   ⁢           ⁢     θ   0                             a   ⁢           ⁢   tan   ⁢         r   ⁢           ⁢   sin   ⁢           ⁢   θ     -     a   ⁢           ⁢   sin   ⁢           ⁢     θ   0             r   ⁢           ⁢   cos   ⁢           ⁢   θ     -     a   ⁢           ⁢   cos   ⁢           ⁢     θ   0             +     180   ⁢   °                         (rsinθ −asinθ 0 ) &lt; 0           a   ⁢           ⁢   tan   ⁢         r   ⁢           ⁢   sin   ⁢           ⁢   θ     -     a   ⁢           ⁢   sin   ⁢           ⁢     θ   0             r   ⁢           ⁢   cos   ⁢           ⁢   θ     -     a   ⁢           ⁢   cos   ⁢           ⁢     θ   0                             a   ⁢           ⁢   tan   ⁢         r   ⁢           ⁢   sin   ⁢           ⁢   θ     -     a   ⁢           ⁢   sin   ⁢           ⁢     θ   0             r   ⁢           ⁢   cos   ⁢           ⁢   θ     -     a   ⁢           ⁢   cos   ⁢           ⁢     θ   0             -     180   ⁢   °                              
This table can be summarized by one common equation as:
 
   
     
       
         
           
             
               
                 α 
                 = 
                   
                 ⁢ 
                 
                   
                     A 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     tan 
                     ⁢ 
                     
                       
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           A 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             0 
                           
                         
                       
                       
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           A 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             0 
                           
                         
                       
                     
                   
                   + 
                   
                     90 
                     ⁢ 
                     ° 
                     × 
                   
                 
               
             
           
           
             
               
                   
                 ⁢ 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           
                             r 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             θ 
                           
                           - 
                           
                             A 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               θ 
                               0 
                             
                           
                         
                         
                            
                           
                             
                               r 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                             - 
                             
                               A 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 0 
                               
                             
                           
                            
                         
                       
                     
                     ) 
                   
                   × 
                   
                     
                       
                         r 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         θ 
                       
                       - 
                       
                         A 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sin 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           0 
                         
                       
                     
                     
                        
                       
                         
                           r 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           A 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             θ 
                             0 
                           
                         
                       
                        
                     
                   
                 
               
             
           
         
       
     
   
   By translating the rotor geometry to the new coordinate system with the off-center magnitude A and the offset angle α from the forging centerline  18 , all points based on a three-dimensional model generated as described above are translated to the new coordinate system with its center at centerline  98 . In this manner, any axial position in the transition zone of the forging  10  can be machined using the new coordinate system so the axis of rotation of the machined rotor  90  coincides with the axis  98 , and therefore is more centrally located with respect to the three-dimensional approximation of the shape of the transition zone  20  of the rough-machined forging  10 . In so doing, the axis of rotation of the machined rotor  90  is also more centrally located with respect to the material properties of the forging  10  than was the geometric centerline  18  of the forging  10 . As a result, the machined rotor  90  is able to exhibit less deflection when heated and rotated about its axis of rotation  98  than would the machined rotor if it had been machined so that its axis of rotation coincided with the geometric centerline  18  of the forging  10 . 
   The procedure just described can be repeated at several positions along the length of the transition zone. The results of the several calculations can be analyzed using finite element techniques in conjunction with optimization techniques to select the best choice of off-center machining. Based on the new rotor axis  98  identified by the above process, thermal analysis can be performed to obtain the temperature gradient of the entire rotor  90  to permit recalculating of the predicted rotor centerline deflection. To find the optimum location of the new rotor axis  98 , the largest possible off-center machine tolerances can be calculated based on the difference between the diameter of the rotor  90  at its rough-machined geometry and the diameter of the rotor  90  at its final machined geometry. Using the off-center machine distance A and the off-center angle α as variables, and the maximum rotor centerline deflection as response, a series of runs and analysis-evaluation-modification cycles can be performed with random variable inputs for A and α. A response curve can then be generated using least square curve fitting between data points to identify the minimum centerline deflection within the allowed range for off-center shift. In one investigation implementing the above-described methodology, relocating the rotor axis  98  a distance of about 12 mm from the original rough-machined centerline  18  reduced centerline deflection by about 65%. 
   While the invention has been described in terms of one or more particular embodiments, it is apparent that other forms could be adopted by one skilled in the art. Therefore, the scope of the invention is to be limited only by the following claims.