Patent Document

CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application Ser. No. 61/054,863 filed on May 21, 2008. 
    
    
     TECHNICAL FIELD 
     The present invention generally relates to a compressor component having an airfoil and more specifically to an improved airfoil profile that has an increased stiffness in order to raise the natural frequency and reduce airfoil stress at regions around an airfoil root. 
     BACKGROUND OF THE INVENTION 
     Compressor components having an airfoil, such as a compressor blade, are held within a rotating disk or shaft and are designed to rotate at a high rate of speed in order to compress a fluid passing through, such as air. A compressor typically comprises a plurality of stages, or rotating disks of blades, of diminishing diameter that raise the pressure and temperature, of the working fluid at each stage to a pre-determined level at the compressor exit. 
     Axial compressors having multiple stages are commonly used in gas turbine engines for increasing the pressure and temperature of air to a pre-determined level at which point fuel can be mixed with the air and the mixture ignited. The hot combustion gases then pass through a turbine to provide either a propulsive output or mechanical output. 
     Compressor components, such as blades and vanes, are located near an inlet of the engine, and as such, are exposed to impact by foreign objects, dirt, and other debris, that can lead to erosion of the airfoil, especially along its leading edge. Significant erosion of this leading edge can result in that location being a life-limiting factor for the blade. Furthermore, as a blade rotates, it will have a series of vibratory modes that can cause high stress regions depending on the blade geometry. Where high stress regions coincide with other life-limiting factors, such as leading edge erosion, fatigue and failure of the blade can occur. 
     SUMMARY 
     In accordance with the present invention, there is provided a novel and improved airfoil for a compressor component having a redefined thickness and airfoil stacking that results in increased airfoil stiffness and reduced vibratory stress along the leading edge of the airfoil. The location of the airfoil material changes are found along the airfoil chord length and span so as to shift the frequency of the blade with minimal impact to blade aerodynamics. The airfoil is restacked so as to reduce the stress imparted to the airfoil leading edge. 
     In an embodiment of the present invention, a compressor component having an attachment, a first platform, and an airfoil extending radially outward from the first platform is disclosed. The airfoil has an uncoated profile substantially in accordance with Cartesian coordinate values of X, Y, and Z as set forth in Table 1, carried only to three decimal places, where Z is a distance measured radially from the first platform of the attachment to which the airfoil is mounted. 
     In an alternate embodiment, an airfoil for a compressor blade is disclosed having an uncoated profile substantially in accordance with Cartesian coordinate values of X, Y, and Z as set forth in Table 1, carried only to three decimal places, where Z is a distance measured in inches and the X and Y values are joined by smooth connecting splines to form a plurality of airfoil sections and the sections are joined to form the airfoil profile. 
     In yet another embodiment, a compressor is disclosed in which the compressor comprises a compressor disk and a plurality of compressor blades extending radially outward from the compressor disk. The compressor blades each have an airfoil with an uncoated nominal profile substantially in accordance with Cartesian coordinate values of X, Y, and Z in Table 1, with the Z coordinate values at perpendicular distances from planes normal to a radius from a platform of the compressor blade. 
     The enhancements made to the airfoil along its chord length and span are made so as to not have any impact on the throat area between adjacent blades. As such, a majority of the thickness increase is made to the pressure side, or concave side of the airfoil. Although disclosed as an airfoil that is uncoated, it is envisioned that an alternate embodiment of the present invention can include an airfoil that is at least partially coated with an erosion resistant coating, corrosion resistant coating, or a combination thereof. In this case, the coordinates of the airfoil as listed in Table 1 would be prior to a coating being applied to any portion of the airfoil. 
     Additional advantages and features of the present invention will be set forth in part in a description which follows, and in part will become apparent to those skilled in the art upon examination of the following, or may be learned from practice of the invention. The instant invention will now be described with particular reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       The present invention is described in detail below with reference to the attached drawing figures, wherein: 
         FIG. 1  is a front elevation view of a compressor blade having an airfoil in accordance with an embodiment of the present invention; 
         FIG. 2  is a side elevation view of the compressor blade of  FIG. 1 ; 
         FIG. 3  is a top elevation view of the compressor blade of  FIG. 1 ; 
         FIG. 4  is a perspective view illustrating a plurality of airfoil sections generated by the Cartesian coordinates of Table 1; 
         FIG. 5  is a perspective view of a comparison between the airfoil generated by airfoil sections of  FIG. 4  and a prior art airfoil; 
         FIG. 6  is a cross section view overlaying a plurality of airfoil sections of the present invention with airfoil sections of the prior art; 
         FIGS. 7-9  are enlarged views of various cross sections overlaying the airfoil in accordance with an embodiment of the present invention with an airfoil of the prior art; and, 
         FIG. 10  is a chart depicting a change in airfoil thickness versus percent span of the airfoil 
     
    
    
     DETAILED DESCRIPTION 
     The subject matter of the present invention is described with specificity herein to meet statutory requirements. However, the description itself is not intended to limit the scope of this patent. Rather, the inventors have contemplated that the claimed subject matter might also be embodied in other ways, to include different components, combinations of components, steps, or combinations of steps similar to the ones described in this document, in conjunction with other present or future technologies. 
     Referring initially to  FIGS. 1-3 , a compressor blade  100  is shown in accordance with an embodiment of the present invention. The compressor blade  100  comprises an attachment  102 , which can also be referred to as a root. The attachment  102  utilizes one or more attachment surfaces  104  that are oriented so as to correspond with a slot in a compressor disk (not depicted) that has a matching profile. Such an engagement maintains the blade within the disk, preventing it from moving outward due to radial pulling forces associated with the rotation of the compressor disk. For the compressor blade  100 , the upper surface of the attachment  102  serves as a first platform  106 , which aligns with an adjacent surface on an outer diameter of the blade disk to provide a uniform inner wall surface for the incoming air flow to the compressor. 
     Extending radially outward from the platform  106  is an airfoil  108 . For the compressor blade  100 , the airfoil is solid, and fabricated from a material such as a hardened steel alloy. The airfoil has an uncoated profile substantially in accordance with Cartesian coordinate values of X and Y, for each distance Z, in inches, as set forth in Table 1 below, and carried to three decimal places. The distance Z is measured radially outward from the platform  106 . The X and Y coordinates are distances relative to coordinate plane origin established at each of the radial Z heights. 
     A plurality of airfoil sections  110  are established by applying smooth continuing splines between the X, Y coordinate values in each Z height. Smoothly joining each of the airfoil section  110  together form the profile of the airfoil  108 . The airfoil  108  can be fabricated by a variety of manufacturing techniques such as forging, casting, milling, and electrochemical machining (ECM). As such, the airfoil has a series of manufacturing tolerance for the position, profile, twist, and chord that can cause the airfoil  108  to vary by as much as +/−0.090 inches from a nominal state. 
     In addition to manufacturing tolerances affecting the overall size of the airfoil  108 , it is also possible to scale the airfoil  108  to a larger or smaller airfoil size. However, in order to maintain the benefits of this airfoil shape and size, in terms of stiffness and stress, as will be discussed further below, it is necessary to scale the airfoil uniformly in X and Y directions, but Z direction may be scaled separately. 
     As previously discussed, the profile generated by the X, Y, and Z coordinates of Table 1 is an uncoated profile. While an embodiment of the present invention is an uncoated compressor blade  100 , it is possible to add a coating to at least a portion of the airfoil  108  in an alternate embodiment. This coating would have a thickness of up to approximately 0.010 inches 
     Depending on the blade configuration, it is possible that a second platform can be positioned at the tip  112  of the airfoil  108 . A second platform located at the tip  112 , is commonly referred to as a shroud and interlocks with a shroud of an adjacent blade. The shrouds provide an outer airpath seal that increases efficiency by preventing air from passing over the blade tip  112  and also serves to reduce the vibration of the airfoils  108 . The use of a second platform, or a shroud, is common in airfoils having a relatively long radial length. 
     A compressor blade is typically fabricated from a relatively low temperature alloy since air temperature of the compressor only reaches upwards of 700 deg. F. One such material for the compressor blade  100  is a hardenable stainless steel alloy. Although the compressor blade has been discussed as having an attachment, at least one platform, and an airfoil, it is to be understood that all of these features of the blade are typically fabricated from the same material and are most likely integral with one another. 
     In an alternate embodiment of the invention, a compressor comprises at least one compressor disk (not depicted) having a plurality of compressor blades  100  that extend radially outward from the compressor disk. As one skilled in the art understands, a compressor typically comprises a plurality of alternating stages of rotating and stationary airfoils that raise the pressure and temperature of a fluid passing through. While the compressor blade  100  having the airfoil  108  can operate in a variety of locations within a compressor, depending on the compressor size, one such location that suits this blade, is adjacent an inlet of the compressor. 
     For compressor blades in this location, a common durability issue exhibited by prior art blades is erosion of the blade leading edge. The leading edge of the blade (see  114  in  FIGS. 1 and 2 ) is the generally radially extending edge at the forward or upstream end of the blade where the concave and convex surfaces of the airfoil come together. This edge first receives the oncoming air flow, and therefore, is also first impacted by anything entering the compressor. Over time, this leading edge can erode away and weaken the airfoil. 
     As one skilled in the art understands, as a compressor blade is rotated by a compressor disk, the weight of the blade pulls radially outward on the disk. However, because of blade design issues such as desired compression of the airflow, blade materials, and compressor size, rarely is the only load a truly radial pulling load. For large unshrouded blades there is usually a substantial amount of blade twist from airfoil root to airfoil tip. Due to the blade&#39;s pulling load, the airfoil will tend to untwist or try to straighten itself out. The compression of the airflow also creates load on the airfoil that tries to bend the blade where the airfoil attaches to the platform. Blade pull, untwist, and aero loading result in concentrated steady stress that can occur near the blade&#39;s airfoil root leading edge and the blade attachment, as seen with blades of prior art. Airfoil unsteady stress can occur due to the vibratory nature of the blade. Specific vibratory shapes for the blade result in stress concentrations on the airfoil. Blade failure can occur when the blade steady and unsteady stress concentrations occur together. If erosion forms at a location of high steady and unsteady stress then the chance of blade failure is increased. 
     For a compressor blade that is subject to foreign object damage and erosion, the present invention provides an airfoil that increases the airfoil stiffness so as to alter the natural frequency of the blade away from resonant frequency. The airfoil stiffness has been accomplished by selectively increasing and decreasing the airfoil thickness while maintaining baseline aerodynamic performance. Material has been added to the airfoil from 0% span (adjacent the platform  106 ) up to approximately 75% span, with material having been removed from this location and radially outward to the tip  112  of the airfoil  108 . Adding mass to the lower region of the airfoil  108  has increased the stiffness of the airfoil  108  by making the airfoil  108  less subject to bending. The increase in thickness is generally uniform up to approximately 63% span. The thickness of the leading edge  114  and trailing edge  116  has not been changed. The majority of the thickness has been added along the chord length on the concave, or pressure, side of the airfoil  108 . 
     In order to reduce the stress concentrations found in the attachment of the airfoil and redistribute this stress, the airfoil  108  has been restacked relative to a prior art airfoil. Through restacking the airfoil sections (i.e. altering the tilt or lean of the radial stack of the airfoil sections), a compressive bending stress is imparted at the leading edge and a tensile bending stress on the convex side of the trailing edge at approximately 70% chord (with 0% chord=the airfoil leading edge  114  and 100% chord=the airfoil trailing edge  116 ). A parabolic bow/lean is applied to the airfoil sections  110  so as to minimize distortion of the airfoil shape, which will minimize bending stresses at the root/attachment. By reducing the stress in the leading edge  114  and applying that stress on the convex side at the trailing edge  116 , the stress concentrations have been reduced. Therefore, even if erosion damage occurs to the leading edge  114 , the blade  100  has adequate fatigue capability. 
     The modifications to the prior art airfoil, in terms of thickness and restacking, can be seen in more detail in  FIGS. 5-10 .  FIG. 5  is a perspective view depicting the present invention airfoil  108  with solid lines compared to the prior art airfoil  200 , shown in dashed lines. From  FIG. 5  it can be seen where the airfoil  108  has an increased thickness.  FIG. 6  depicts a plurality of sections taken through the airfoils  108  and  200  of  FIG. 5 . Through this overlay of sections, it can also be seen how the airfoil thickness has increased and how the airfoil sections have been restacked to alter the bending stresses.  FIGS. 7-9  are enlargements of specific sections depicted in  FIG. 6 , with  FIG. 7  taken at a radial height of Z=1.000,  FIG. 8  taken at a radial height of Z=8.000, and  FIG. 9  taken at a radial height of Z=16.9000. 
     Referring now to  FIG. 10 , another way to look at how the thickness of the airfoil  108  is changed for stiffening purposes is depicted. This chart shows a view of relative airfoil thickness (in inches) versus percent span. The solid line depicts the thickness increase or decrease over percent span for the airfoil  108  of the present invention. For this embodiment, the thickness of the airfoil is increased until approximately 60% span and then the amount of thickness increase diminishes and starts to decrease in thickness between 70%-80% span through to the blade tip. 
     
       
         
               
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 X 
                 Y 
                 Z 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 −2.946 
                 1.884 
                 1.000 
               
               
                 −2.971 
                 1.899 
                 1.000 
               
               
                 −2.994 
                 1.911 
                 1.000 
               
               
                 −3.016 
                 1.920 
                 1.000 
               
               
                 −3.036 
                 1.924 
                 1.000 
               
               
                 −3.055 
                 1.925 
                 1.000 
               
               
                 −3.071 
                 1.922 
                 1.000 
               
               
                 −3.086 
                 1.916 
                 1.000 
               
               
                 −3.100 
                 1.907 
                 1.000 
               
               
                 −3.112 
                 1.896 
                 1.000 
               
               
                 −3.121 
                 1.882 
                 1.000 
               
               
                 −3.128 
                 1.867 
                 1.000 
               
               
                 −3.131 
                 1.851 
                 1.000 
               
               
                 −3.131 
                 1.835 
                 1.000 
               
               
                 −3.127 
                 1.818 
                 1.000 
               
               
                 −3.120 
                 1.801 
                 1.000 
               
               
                 −3.109 
                 1.782 
                 1.000 
               
               
                 −3.096 
                 1.763 
                 1.000 
               
               
                 −3.080 
                 1.743 
                 1.000 
               
               
                 −3.062 
                 1.720 
                 1.000 
               
               
                 −3.062 
                 1.720 
                 1.000 
               
               
                 −2.935 
                 1.568 
                 1.000 
               
               
                 −2.806 
                 1.416 
                 1.000 
               
               
                 −2.675 
                 1.267 
                 1.000 
               
               
                 −2.542 
                 1.121 
                 1.000 
               
               
                 −2.407 
                 0.977 
                 1.000 
               
               
                 −2.269 
                 0.836 
                 1.000 
               
               
                 −2.128 
                 0.698 
                 1.000 
               
               
                 −1.986 
                 0.565 
                 1.000 
               
               
                 −1.837 
                 0.431 
                 1.000 
               
               
                 −1.671 
                 0.292 
                 1.000 
               
               
                 −1.480 
                 0.141 
                 1.000 
               
               
                 −1.253 
                 −0.022 
                 1.000 
               
               
                 −0.987 
                 −0.190 
                 1.000 
               
               
                 −0.683 
                 −0.355 
                 1.000 
               
               
                 −0.352 
                 −0.505 
                 1.000 
               
               
                 −0.013 
                 −0.630 
                 1.000 
               
               
                 0.315 
                 −0.728 
                 1.000 
               
               
                 0.616 
                 −0.800 
                 1.000 
               
               
                 0.887 
                 −0.852 
                 1.000 
               
               
                 1.125 
                 −0.890 
                 1.000 
               
               
                 1.338 
                 −0.919 
                 1.000 
               
               
                 1.537 
                 −0.942 
                 1.000 
               
               
                 1.731 
                 −0.960 
                 1.000 
               
               
                 1.927 
                 −0.976 
                 1.000 
               
               
                 2.124 
                 −0.989 
                 1.000 
               
               
                 2.320 
                 −1.000 
                 1.000 
               
               
                 2.516 
                 −1.008 
                 1.000 
               
               
                 2.712 
                 −1.014 
                 1.000 
               
               
                 2.907 
                 −1.018 
                 1.000 
               
               
                 3.104 
                 −1.020 
                 1.000 
               
               
                 3.104 
                 −1.021 
                 1.000 
               
               
                 3.133 
                 −1.021 
                 1.000 
               
               
                 3.159 
                 −1.020 
                 1.000 
               
               
                 3.183 
                 −1.018 
                 1.000 
               
               
                 3.204 
                 −1.015 
                 1.000 
               
               
                 3.223 
                 −1.009 
                 1.000 
               
               
                 3.239 
                 −1.002 
                 1.000 
               
               
                 3.251 
                 −0.993 
                 1.000 
               
               
                 3.261 
                 −0.984 
                 1.000 
               
               
                 3.267 
                 −0.973 
                 1.000 
               
               
                 3.271 
                 −0.961 
                 1.000 
               
               
                 3.273 
                 −0.949 
                 1.000 
               
               
                 3.271 
                 −0.935 
                 1.000 
               
               
                 3.267 
                 −0.920 
                 1.000 
               
               
                 3.259 
                 −0.906 
                 1.000 
               
               
                 3.247 
                 −0.891 
                 1.000 
               
               
                 3.231 
                 −0.877 
                 1.000 
               
               
                 3.211 
                 −0.864 
                 1.000 
               
               
                 3.188 
                 −0.851 
                 1.000 
               
               
                 3.161 
                 −0.839 
                 1.000 
               
               
                 3.161 
                 −0.839 
                 1.000 
               
               
                 2.834 
                 −0.691 
                 1.000 
               
               
                 2.505 
                 −0.548 
                 1.000 
               
               
                 2.175 
                 −0.408 
                 1.000 
               
               
                 1.844 
                 −0.272 
                 1.000 
               
               
                 1.512 
                 −0.140 
                 1.000 
               
               
                 1.176 
                 −0.010 
                 1.000 
               
               
                 0.838 
                 0.118 
                 1.000 
               
               
                 0.516 
                 0.236 
                 1.000 
               
               
                 0.216 
                 0.346 
                 1.000 
               
               
                 −0.064 
                 0.449 
                 1.000 
               
               
                 −0.323 
                 0.549 
                 1.000 
               
               
                 −0.554 
                 0.642 
                 1.000 
               
               
                 −0.755 
                 0.726 
                 1.000 
               
               
                 −0.930 
                 0.803 
                 1.000 
               
               
                 −1.092 
                 0.876 
                 1.000 
               
               
                 −1.248 
                 0.949 
                 1.000 
               
               
                 −1.404 
                 1.025 
                 1.000 
               
               
                 −1.560 
                 1.102 
                 1.000 
               
               
                 −1.715 
                 1.182 
                 1.000 
               
               
                 −1.868 
                 1.263 
                 1.000 
               
               
                 −2.021 
                 1.346 
                 1.000 
               
               
                 −2.176 
                 1.431 
                 1.000 
               
               
                 −2.330 
                 1.519 
                 1.000 
               
               
                 −2.485 
                 1.608 
                 1.000 
               
               
                 −2.639 
                 1.699 
                 1.000 
               
               
                 −2.793 
                 1.791 
                 1.000 
               
               
                 −2.946 
                 1.884 
                 1.000 
               
               
                 −2.900 
                 2.035 
                 2.000 
               
               
                 −2.925 
                 2.052 
                 2.000 
               
               
                 −2.948 
                 2.067 
                 2.000 
               
               
                 −2.970 
                 2.079 
                 2.000 
               
               
                 −2.991 
                 2.088 
                 2.000 
               
               
                 −3.010 
                 2.093 
                 2.000 
               
               
                 −3.027 
                 2.095 
                 2.000 
               
               
                 −3.043 
                 2.094 
                 2.000 
               
               
                 −3.056 
                 2.090 
                 2.000 
               
               
                 −3.067 
                 2.083 
                 2.000 
               
               
                 −3.076 
                 2.075 
                 2.000 
               
               
                 −3.083 
                 2.065 
                 2.000 
               
               
                 −3.087 
                 2.053 
                 2.000 
               
               
                 −3.089 
                 2.041 
                 2.000 
               
               
                 −3.088 
                 2.027 
                 2.000 
               
               
                 −3.084 
                 2.012 
                 2.000 
               
               
                 −3.077 
                 1.996 
                 2.000 
               
               
                 −3.067 
                 1.980 
                 2.000 
               
               
                 −3.055 
                 1.963 
                 2.000 
               
               
                 −3.041 
                 1.944 
                 2.000 
               
               
                 −3.025 
                 1.924 
                 2.000 
               
               
                 −3.006 
                 1.902 
                 2.000 
               
               
                 −3.006 
                 1.902 
                 2.000 
               
               
                 −2.945 
                 1.827 
                 2.000 
               
               
                 −2.883 
                 1.753 
                 2.000 
               
               
                 −2.821 
                 1.680 
                 2.000 
               
               
                 −2.759 
                 1.606 
                 2.000 
               
               
                 −2.696 
                 1.532 
                 2.000 
               
               
                 −2.632 
                 1.459 
                 2.000 
               
               
                 −2.569 
                 1.386 
                 2.000 
               
               
                 −2.505 
                 1.314 
                 2.000 
               
               
                 −2.441 
                 1.242 
                 2.000 
               
               
                 −2.372 
                 1.167 
                 2.000 
               
               
                 −2.297 
                 1.085 
                 2.000 
               
               
                 −2.211 
                 0.994 
                 2.000 
               
               
                 −2.114 
                 0.892 
                 2.000 
               
               
                 −2.005 
                 0.781 
                 2.000 
               
               
                 −1.883 
                 0.660 
                 2.000 
               
               
                 −1.744 
                 0.528 
                 2.000 
               
               
                 −1.587 
                 0.384 
                 2.000 
               
               
                 −1.407 
                 0.230 
                 2.000 
               
               
                 −1.199 
                 0.064 
                 2.000 
               
               
                 −0.957 
                 −0.109 
                 2.000 
               
               
                 −0.680 
                 −0.283 
                 2.000 
               
               
                 −0.389 
                 −0.440 
                 2.000 
               
               
                 −0.109 
                 −0.570 
                 2.000 
               
               
                 0.150 
                 −0.674 
                 2.000 
               
               
                 0.379 
                 −0.755 
                 2.000 
               
               
                 0.582 
                 −0.819 
                 2.000 
               
               
                 0.768 
                 −0.872 
                 2.000 
               
               
                 0.948 
                 −0.920 
                 2.000 
               
               
                 1.130 
                 −0.964 
                 2.000 
               
               
                 1.317 
                 −1.006 
                 2.000 
               
               
                 1.504 
                 −1.046 
                 2.000 
               
               
                 1.676 
                 −1.080 
                 2.000 
               
               
                 1.826 
                 −1.108 
                 2.000 
               
               
                 1.957 
                 −1.131 
                 2.000 
               
               
                 2.072 
                 −1.151 
                 2.000 
               
               
                 2.176 
                 −1.168 
                 2.000 
               
               
                 2.274 
                 −1.184 
                 2.000 
               
               
                 2.369 
                 −1.198 
                 2.000 
               
               
                 2.464 
                 −1.213 
                 2.000 
               
               
                 2.559 
                 −1.227 
                 2.000 
               
               
                 2.654 
                 −1.240 
                 2.000 
               
               
                 2.750 
                 −1.253 
                 2.000 
               
               
                 2.845 
                 −1.266 
                 2.000 
               
               
                 2.942 
                 −1.279 
                 2.000 
               
               
                 3.037 
                 −1.292 
                 2.000 
               
               
                 3.037 
                 −1.292 
                 2.000 
               
               
                 3.063 
                 −1.295 
                 2.000 
               
               
                 3.087 
                 −1.299 
                 2.000 
               
               
                 3.109 
                 −1.302 
                 2.000 
               
               
                 3.128 
                 −1.305 
                 2.000 
               
               
                 3.147 
                 −1.306 
                 2.000 
               
               
                 3.163 
                 −1.306 
                 2.000 
               
               
                 3.177 
                 −1.303 
                 2.000 
               
               
                 3.189 
                 −1.298 
                 2.000 
               
               
                 3.199 
                 −1.291 
                 2.000 
               
               
                 3.205 
                 −1.284 
                 2.000 
               
               
                 3.210 
                 −1.275 
                 2.000 
               
               
                 3.212 
                 −1.266 
                 2.000 
               
               
                 3.212 
                 −1.256 
                 2.000 
               
               
                 3.209 
                 −1.245 
                 2.000 
               
               
                 3.204 
                 −1.235 
                 2.000 
               
               
                 3.195 
                 −1.224 
                 2.000 
               
               
                 3.184 
                 −1.214 
                 2.000 
               
               
                 3.171 
                 −1.204 
                 2.000 
               
               
                 3.156 
                 −1.193 
                 2.000 
               
               
                 3.140 
                 −1.182 
                 2.000 
               
               
                 3.121 
                 −1.171 
                 2.000 
               
               
                 3.100 
                 −1.159 
                 2.000 
               
               
                 3.100 
                 −1.159 
                 2.000 
               
               
                 2.940 
                 −1.069 
                 2.000 
               
               
                 2.779 
                 −0.980 
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                 −2.089 
                 3.784 
                 16.900 
               
               
                 −2.092 
                 3.788 
                 16.900 
               
               
                 −2.095 
                 3.791 
                 16.900 
               
               
                 −2.097 
                 3.793 
                 16.900 
               
               
                 −2.100 
                 3.796 
                 16.900 
               
               
                 −2.102 
                 3.798 
                 16.900 
               
               
                 −2.104 
                 3.799 
                 16.900 
               
               
                 −2.107 
                 3.801 
                 16.900 
               
               
                 −2.109 
                 3.801 
                 16.900 
               
               
                 −2.111 
                 3.801 
                 16.900 
               
               
                 −2.112 
                 3.801 
                 16.900 
               
               
                 −2.114 
                 3.800 
                 16.900 
               
               
                 −2.115 
                 3.798 
                 16.900 
               
               
                 −2.115 
                 3.796 
                 16.900 
               
               
                 −2.115 
                 3.793 
                 16.900 
               
               
                 −2.115 
                 3.791 
                 16.900 
               
               
                 −2.115 
                 3.788 
                 16.900 
               
               
                 −2.115 
                 3.785 
                 16.900 
               
               
                 −2.114 
                 3.781 
                 16.900 
               
               
                 −2.113 
                 3.778 
                 16.900 
               
               
                 −2.112 
                 3.774 
                 16.900 
               
               
                 −2.111 
                 3.769 
                 16.900 
               
               
                 −2.110 
                 3.765 
                 16.900 
               
               
                 −2.108 
                 3.760 
                 16.900 
               
               
                 −2.106 
                 3.754 
                 16.900 
               
               
                 −2.104 
                 3.748 
                 16.900 
               
               
                 −2.102 
                 3.742 
                 16.900 
               
               
                 −2.099 
                 3.735 
                 16.900 
               
               
                 −2.096 
                 3.728 
                 16.900 
               
               
                 −2.093 
                 3.720 
                 16.900 
               
               
                 −2.090 
                 3.711 
                 16.900 
               
               
                 −2.086 
                 3.702 
                 16.900 
               
               
                 −2.081 
                 3.692 
                 16.900 
               
               
                 −2.077 
                 3.682 
                 16.900 
               
               
                 −2.071 
                 3.670 
                 16.900 
               
               
                 −2.066 
                 3.657 
                 16.900 
               
               
                 −2.059 
                 3.644 
                 16.900 
               
               
                 −2.053 
                 3.629 
                 16.900 
               
               
                 −2.045 
                 3.613 
                 16.900 
               
               
                 −2.037 
                 3.595 
                 16.900 
               
               
                 −2.037 
                 3.595 
                 16.900 
               
               
                 −1.841 
                 3.168 
                 16.900 
               
               
                 −1.642 
                 2.744 
                 16.900 
               
               
                 −1.441 
                 2.325 
                 16.900 
               
               
                 −1.240 
                 1.913 
                 16.900 
               
               
                 −1.038 
                 1.505 
                 16.900 
               
               
                 −0.831 
                 1.094 
                 16.900 
               
               
                 −0.628 
                 0.692 
                 16.900 
               
               
                 −0.448 
                 0.334 
                 16.900 
               
               
                 −0.294 
                 0.029 
                 16.900 
               
               
                 −0.162 
                 −0.236 
                 16.900 
               
               
                 −0.047 
                 −0.470 
                 16.900 
               
               
                 0.058 
                 −0.684 
                 16.900 
               
               
                 0.157 
                 −0.888 
                 16.900 
               
               
                 0.255 
                 −1.092 
                 16.900 
               
               
                 0.351 
                 −1.297 
                 16.900 
               
               
                 0.447 
                 −1.502 
                 16.900 
               
               
                 0.543 
                 −1.707 
                 16.900 
               
               
                 0.639 
                 −1.911 
                 16.900 
               
               
                 0.739 
                 −2.115 
                 16.900 
               
               
                 0.841 
                 −2.317 
                 16.900 
               
               
                 0.946 
                 −2.518 
                 16.900 
               
               
                 1.055 
                 −2.717 
                 16.900 
               
               
                 1.166 
                 −2.917 
                 16.900 
               
               
                 1.282 
                 −3.118 
                 16.900 
               
               
                 1.401 
                 −3.320 
                 16.900 
               
               
                 1.524 
                 −3.524 
                 16.900 
               
               
                 1.648 
                 −3.727 
                 16.900 
               
               
                 1.773 
                 −3.927 
                 16.900 
               
               
                 1.773 
                 −3.927 
                 16.900 
               
               
                 1.784 
                 −3.944 
                 16.900 
               
               
                 1.793 
                 −3.959 
                 16.900 
               
               
                 1.802 
                 −3.973 
                 16.900 
               
               
                 1.810 
                 −3.986 
                 16.900 
               
               
                 1.817 
                 −3.998 
                 16.900 
               
               
                 1.823 
                 −4.009 
                 16.900 
               
               
                 1.829 
                 −4.019 
                 16.900 
               
               
                 1.835 
                 −4.029 
                 16.900 
               
               
                 1.840 
                 −4.037 
                 16.900 
               
               
                 1.844 
                 −4.045 
                 16.900 
               
               
                 1.849 
                 −4.053 
                 16.900 
               
               
                 1.853 
                 −4.059 
                 16.900 
               
               
                 1.857 
                 −4.065 
                 16.900 
               
               
                 1.861 
                 −4.071 
                 16.900 
               
               
                 1.864 
                 −4.076 
                 16.900 
               
               
                 1.868 
                 −4.080 
                 16.900 
               
               
                 1.871 
                 −4.084 
                 16.900 
               
               
                 1.874 
                 −4.088 
                 16.900 
               
               
                 1.877 
                 −4.091 
                 16.900 
               
               
                 1.880 
                 −4.094 
                 16.900 
               
               
                 1.883 
                 −4.096 
                 16.900 
               
               
                 1.886 
                 −4.098 
                 16.900 
               
               
                 1.889 
                 −4.100 
                 16.900 
               
               
                 1.892 
                 −4.101 
                 16.900 
               
               
                 1.895 
                 −4.101 
                 16.900 
               
               
                 1.897 
                 −4.101 
                 16.900 
               
               
                 1.899 
                 −4.099 
                 16.900 
               
               
                 1.900 
                 −4.096 
                 16.900 
               
               
                 1.900 
                 −4.094 
                 16.900 
               
               
                 1.900 
                 −4.091 
                 16.900 
               
               
                 1.900 
                 −4.087 
                 16.900 
               
               
                 1.899 
                 −4.084 
                 16.900 
               
               
                 1.899 
                 −4.080 
                 16.900 
               
               
                 1.898 
                 −4.076 
                 16.900 
               
               
                 1.897 
                 −4.072 
                 16.900 
               
               
                 1.895 
                 −4.067 
                 16.900 
               
               
                 1.893 
                 −4.062 
                 16.900 
               
               
                 1.891 
                 −4.057 
                 16.900 
               
               
                 1.889 
                 −4.051 
                 16.900 
               
               
                 1.886 
                 −4.045 
                 16.900 
               
               
                 1.883 
                 −4.038 
                 16.900 
               
               
                 1.880 
                 −4.031 
                 16.900 
               
               
                 1.876 
                 −4.023 
                 16.900 
               
               
                 1.872 
                 −4.015 
                 16.900 
               
               
                 1.868 
                 −4.006 
                 16.900 
               
               
                 1.863 
                 −3.996 
                 16.900 
               
               
                 1.857 
                 −3.986 
                 16.900 
               
               
                 1.851 
                 −3.974 
                 16.900 
               
               
                 1.845 
                 −3.962 
                 16.900 
               
               
                 1.838 
                 −3.949 
                 16.900 
               
               
                 1.830 
                 −3.935 
                 16.900 
               
               
                 1.822 
                 −3.919 
                 16.900 
               
               
                 1.813 
                 −3.902 
                 16.900 
               
               
                 1.813 
                 −3.902 
                 16.900 
               
               
                 1.707 
                 −3.704 
                 16.900 
               
               
                 1.603 
                 −3.505 
                 16.900 
               
               
                 1.500 
                 −3.306 
                 16.900 
               
               
                 1.398 
                 −3.107 
                 16.900 
               
               
                 1.298 
                 −2.907 
                 16.900 
               
               
                 1.199 
                 −2.707 
                 16.900 
               
               
                 1.102 
                 −2.506 
                 16.900 
               
               
                 1.006 
                 −2.303 
                 16.900 
               
               
                 0.912 
                 −2.099 
                 16.900 
               
               
                 0.819 
                 −1.893 
                 16.900 
               
               
                 0.727 
                 −1.688 
                 16.900 
               
               
                 0.637 
                 −1.483 
                 16.900 
               
               
                 0.547 
                 −1.278 
                 16.900 
               
               
                 0.455 
                 −1.073 
                 16.900 
               
               
                 0.363 
                 −0.870 
                 16.900 
               
               
                 0.270 
                 −0.667 
                 16.900 
               
               
                 0.173 
                 −0.460 
                 16.900 
               
               
                 0.068 
                 −0.240 
                 16.900 
               
               
                 −0.051 
                 0.002 
                 16.900 
               
               
                 −0.186 
                 0.273 
                 16.900 
               
               
                 −0.344 
                 0.582 
                 16.900 
               
               
                 −0.526 
                 0.933 
                 16.900 
               
               
                 −0.726 
                 1.311 
                 16.900 
               
               
                 −0.930 
                 1.691 
                 16.900 
               
               
                 −1.137 
                 2.071 
                 16.900 
               
               
                 −1.348 
                 2.456 
                 16.900 
               
               
                 −1.563 
                 2.844 
                 16.900 
               
               
                 −1.779 
                 3.233 
                 16.900 
               
               
                 −1.995 
                 3.619 
                 16.900 
               
               
                   
               
             
          
         
       
     
     The present invention has been described in relation to particular embodiments, which are intended in all respects to be illustrative rather than restrictive. Alternative embodiments will become apparent to those of ordinary skill in the art to which the present invention pertains without departing from its scope. 
     From the foregoing, it will be seen that this invention is one well adapted to attain all the ends and objects set forth above, together with other advantages which are obvious and inherent to the system and method. It will be understood that certain features and sub-combinations are of utility and may be employed without reference to other features and sub-combinations. This is contemplated by and within the scope of the claims.

Technology Category: f