12/13/2023 46
Table A3-8 (continued) 1791 1792 gtest/gref for track angle (Coriolis correction) for ground speeds from 500 to 700 knots GS (kts) 500 600 700 Latitude (either North or South) (degrees ) 0 30 60 90 0 30 60 90 0 30 60 90 True Track Angle (degrees) 0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 30 0.9981 0.9983 0.9990 1.0000 0.9977 0.9980 0.9988 1.0000 0.9973 0.9977 0.9987 1.0000 60 0.9967 0.9971 0.9983 1.0000 0.9960 0.9965 0.9980 1.0000 0.9953 0.9960 0.9977 1.0000 90 0.9961 0.9967 0.9981 1.0000 0.9954 0.9960 0.9977 1.0000 0.9946 0.9953 0.9973 1.0000 120 0.9967 0.9971 0.9983 1.0000 0.9960 0.9965 0.9980 1.0000 0.9953 0.9960 0.9977 1.0000 150 0.9981 0.9983 0.9990 1.0000 0.9977 0.9980 0.9988 1.0000 0.9973 0.9977 0.9987 1.0000 180 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 210 1.0019 1.0017 1.0010 1.0000 1.0023 1.0020 1.0012 1.0000 1.0027 1.0023 1.0013 1.0000 240 1.0033 1.0029 1.0017 1.0000 1.0040 1.0035 1.0020 1.0000 1.0047 1.0040 1.0023 1.0000 270 1.0039 1.0033 1.0019 1.0000 1.0046 1.0040 1.0023 1.0000 1.0054 1.0047 1.0027 1.0000 300 1.0033 1.0029 1.0017 1.0000 1.0040 1.0035 1.0020 1.0000 1.0047 1.0040 1.0023 1.0000 330 1.0019 1.0017 1.0010 1.0000 1.0023 1.0020 1.0012 1.0000 1.0027 1.0023 1.0013 1.0000 1793 1794 Step 6) Determine Massgrav, as defined below. Couple this mass with the tested SAR values, corrected for 1795 reference specifications from Step 7 above, in a regression model to determine the SAR values at each of 1796 the three reference masses of the FEM, and to determine the 90 percent confidence intervals. 1797 1798 Massgrav = Masstest x (gtest/gref)lat x (gtest/gref)alt x (gtest/gref)cent x (gtest/gref)Coriolis 1799 1800 where: 1801 1802 Massgrav is the mass at which the reference SAR value should be associated in a regression model to 1803 provide a SAR versus mass corrected for gravitational acceleration; 1804 1805 Masstest is the test mass, as defined in paragraph A3-1.1, in kg; 1806 1807 (gtest/gref)lat is gtest/gref for the test latitude, from Table A3-5; 1808 1809 (gtest/gref)alt is gtest/gref for altitude, from Table A3-6; 1810 1811 (gtest/gref)cent is gtest/gref for headwind/tailwind (centrifugal correction) from Table A3-7; and 1812 1813 (gtest/gref)Coriolis is gtest/gref for track angle (Coriolis correction) from Table A3-8. 1814 1815 A3-1.3 Acceleration/deceleration (energy). Drag determination is based on an assumption of steady, 1816 unaccelerated flight. Acceleration or deceleration relative to the ground occurring during a test condition 1817 affects the assessed drag level. The reference specification is steady, unaccelerated flight. 1818
12/13/2023 47
1819 The correction for the change in drag force resulting from acceleration during the test condition can be 1820 determined from the following equation: 1821 ∆𝐷𝐷𝑎𝑎𝑐𝑐𝑐𝑐𝑟𝑟𝑙𝑙 = −𝑀𝑀𝑣𝑣𝑒𝑒𝑒𝑒𝑡𝑡𝑟𝑟𝑡𝑡𝑡𝑡 �𝑠𝑠𝑉𝑉𝑎𝑎𝑠𝑠𝑇𝑇 � 1822 where: 1823 1824 ΔDaccel is the drag correction in N due to acceleration occurring during the test condition; 1825 1826 Masstest is the average mass of the airplane during the test condition, as defined in paragraph A3-1.1 in 1827 kg; and 1828 1829 (dVg/dT) is the change in ground speed over time during the test condition in m/s2. 1830 1831 A3-1.4 Reynolds number. The RE needs to be calculated to determine its contribution to airplane drag. 1832 For a given test condition, the RE is a function of the density and viscosity of air at the test altitude and 1833 temperature. The reference RE is derived from the density and viscosity of air from the ICAO standard 1834 atmosphere at the reference altitude. 1835 1836 The value of the drag coefficient correction for RE during the test can be expressed as: 1837 1838 ∆𝐶𝐶𝐷𝐷 𝐷𝐷𝐸𝐸 = −𝐵𝐵 𝐹𝐹𝑜𝑜𝑊𝑊 � 1 𝑀𝑀 �𝑆𝑆𝐸𝐸𝑒𝑒 �𝑡𝑡𝑟𝑟𝑡𝑡𝑡𝑡 1 𝑀𝑀 �𝑆𝑆𝐸𝐸𝑒𝑒 �𝐷𝐷𝑟𝑟𝑟𝑟 � 1839 1840 where: 1841 1842 ∆CD RE is the change in drag coefficient when correcting from test RE to the reference RE; 1843 1844 B is a value representing the variation of drag with RE for the specific airplane (see guidance below); 1845 1846 M is Mach number; and 1847 1848 RE/m is RE per meter. 1849 1850 One method to obtain B is to use a drag model to obtain the incremental drag variation in response to 1851 changing Mach and altitude from a reference cruise condition. The value for B is the value of a single 1852 representative slope of a plot of the drag variation, ∆Drag versus Log10� 1 𝑀𝑀 � 𝐷𝐷𝐸𝐸 𝑚𝑚 � 𝑥𝑥10−6�. 1853 1854 The term � 1 𝑀𝑀�𝑅𝑅𝑅𝑅𝑚𝑚 �𝑡𝑡𝑟𝑟𝑡𝑡𝑡𝑡 1 𝑀𝑀�𝑅𝑅𝑅𝑅𝑚𝑚 �𝑅𝑅𝑟𝑟𝑟𝑟�is the term 1 𝑀𝑀 � 𝐷𝐷𝐸𝐸 𝑚𝑚 � determined at the temperature and altitude for the test condition, 1855 divided by the same term determined at the standard day temperature and the reference altitude for the 1856 test mass/δ using the following equation: 1857 1858 1 𝑀𝑀 𝑆𝑆𝐸𝐸 𝑒𝑒 = 4.7899 𝑥𝑥 105 𝑃𝑃𝐷𝐷 �𝑇𝑇𝐷𝐷 + 110.4𝑇𝑇𝐷𝐷2 � 1859 1860 where: 1861 1862 RE/m is RE per m; 1863 1864 PS is static pressure in Pa; and 1865 1866
12/13/2023 48
TS is static temperature in K. 1867 1868 The effect on airplane drag can then be determined from ∆CD RE and the airplane drag equation as 1869 follows: 1870 1871 ∆𝐷𝐷𝐷𝐷𝐸𝐸 = 12 𝜌𝜌 𝑉𝑉2∆𝐶𝐶𝐷𝐷 𝐷𝐷𝐸𝐸 𝑆𝑆 1872 1873 where: 1874 1875 ∆DRE is the airplane drag correction in N due to the test RE being different than the reference RE; 1876 1877 ρ is the density of air at the test altitude and test temperature in kg/m3; 1878 1879 V is the airplane’s average true airspeed during the test condition in m/s; 1880 1881 A is the airplane’s reference wing area in m2; and 1882 1883 ∆CD RE is the change in drag coefficient due to being off the reference RE.. 1884 1885 A3-1.5 CG position. The position of the airplane center of gravity (CG) affects the drag due to longitudinal 1886 trim. 1887 1888 The drag correction for CG position during the test is the difference between the drag at the reference CG 1889 position and the drag at the test CG position. This drag correction can be determined by adjusting the 1890 longitudinal trim drag, determined from the airplane’s drag model at the reference CG, to adjust for the 1891 test CG position. For example, testing at a CG position aft of the reference CG used in the drag model 1892 would yield a positive drag correction (that is, a decrease to the test SAR). 1893 1894 A method to determine the change in longitudinal trim drag for CG position from the reference is to first 1895 determine the amount of longitudinal trim drag (CD Trim Ref CG) that exists at the reference CG position as a 1896 function of Mach and CL. This can be done using wind tunnel testing and analytical methods. The change 1897 of the longitudinal trim drag with CG, which is specific to an airplane type design, can be estimated with 1898 wind tunnel and analytical methods, and verified by flight test data. This relationship can then be used to 1899 determine the longitudinal trim drag (CD Trim Test CG) at the test CG position. 1900 1901 Once the trim drag coefficients are determined from the airplane drag model, the airplane drag correction 1902 for the CG position can be determined from the drag equation: 1903 1904 ∆𝐷𝐷𝑇𝑇𝑟𝑟𝑖𝑖𝑚𝑚 𝐶𝐶𝐴𝐴 = 12 𝜌𝜌 𝑉𝑉2�𝐶𝐶𝐷𝐷 𝑇𝑇𝑟𝑟𝑖𝑖𝑚𝑚 𝐷𝐷𝑟𝑟𝑟𝑟 𝐶𝐶𝐴𝐴 − 𝐶𝐶𝐷𝐷 𝑇𝑇𝑟𝑟𝑖𝑖𝑚𝑚 𝑇𝑇𝑟𝑟𝑡𝑡𝑡𝑡 𝐶𝐶𝐴𝐴�𝑆𝑆 1905 1906 where: 1907 1908 ∆DTrim CG is the airplane drag correction in N due to the test CG being different than the reference CG; 1909 1910 ρ is the density of air at the test altitude and test temperature in kg/m3; 1911 1912 V is the airplane’s average true airspeed during the test condition in m/s; 1913 1914 A is the airplane’s reference wing area in m2; and 1915 1916 CD Trim Ref CG and CD TrimTest CG are the trim drag coefficients from the airplane’s drag model at the reference 1917 CG and test CG positions, respectively. 1918 1919 A3-1.6 Aeroelastics. Wing aeroelastics may cause a variation in drag as a function of airplane wing mass 1920 distribution. Airplane wing mass distribution will be affected by the fuel load distribution in the wings and 1921
12/13/2023 49
the presence of any external fuel stores. 1922 1923 There are no simple analytical means to correct for different wing structural loading conditions. If 1924 necessary, corrections to the reference specification should be developed by flight test or a suitable 1925 analysis process, subject to the approval of the FAA. 1926 1927 The reference specification for the wing structural loading is to be selected by the applicant based on the 1928 amount of fuel and/or removable external stores to be carried by the wing, based on the airplane’s 1929 payload capability and the manufacturer’s standard fuel management practices. The reference to the 1930 airplane’s payload capability is for establishing the zero fuel mass of the airplane, while the reference to 1931 the manufacturer’s standard fuel management practices is for establishing the distribution of that fuel and 1932 how that distribution changes as fuel is burned. 1933 1934 The reference specification for the wing structural loading reference specification should be based on an 1935 operationally representative empty weight and payload which defines the zero fuel mass of the airplane. 1936 The total amount of fuel loaded for each of the three reference masses would be the reference mass 1937 minus the zero fuel mass. Standard fuel management practices will determine the amount of fuel present 1938 in each fuel tank. An example of standard fuel management practice is to load the main (wing) fuel tanks 1939 before loading the center (body) fuel tanks and to first empty fuel from the center tanks before using the 1940 fuel in the main tanks. This helps keep the CG aft and reduces trim drag. 1941 1942 Commercial freighters may be designed from scratch, but more often are derivatives of or converted from 1943 passenger model airplanes. In determining aeroelastic effects, it is reasonable to assume that the 1944 reference loading for a freighter is the same as the passenger model it was derived from. If there is no 1945 similar passenger model, the reference zero-fuel-mass of a freighter can be based on its payload design 1946 density. The payload design density is defined by the full use of the volumetric capacity of the freighter 1947 and the highest mass it is designed to carry, expressed in kg/m3. For example, a typical payload design 1948 density for large commercial freighters is 160 kg/m3. 1949 1950 Using a reference payload significantly lower than the passenger interior limits or structural limited 1951 payload could potentially provide a more beneficial aeroelastic effect. An applicant would need to justify 1952 the reference payload assumptions in the context of the capability of the airplane and what could be 1953 considered typical for the type design. 1954 1955 A3-1.7 Fuel lower heating value. The fuel lower heating value defines the energy content of the fuel. The 1956 lower heating value directly affects the fuel flow at a given test condition. 1957 1958 The fuel flow measured during the flight test is corrected to the fuel flow for the reference lower heating 1959 value as follows: 1960 1961 ∆𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹𝑜𝑜𝐹𝐹𝐶𝐶𝑐𝑐𝑟𝑟𝑟𝑟 𝐿𝐿𝐿𝐿𝐴𝐴 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹𝑜𝑜𝐹𝐹𝑡𝑡𝑟𝑟𝑡𝑡𝑡𝑡 𝐿𝐿𝐿𝐿𝐴𝐴 �𝐿𝐿𝐿𝐿𝑉𝑉𝑡𝑡𝑟𝑟𝑡𝑡𝑡𝑡𝐿𝐿𝐿𝐿𝑉𝑉𝐷𝐷𝑟𝑟𝑟𝑟 � − 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐹𝐹𝐹𝐹𝑜𝑜𝐹𝐹𝑡𝑡𝑟𝑟𝑡𝑡𝑡𝑡 𝐿𝐿𝐿𝐿𝐴𝐴 1962 1963 where: 1964 1965 ∆Fuel FlowCorr LHV is the fuel flow correction in kilograms/hour due to the fuel lower heating value being 1966 different than the reference fuel lower heating value; 1967 1968 Fuel Flowtest LHV is the measured fuel flow in kilograms/hour during the test (at the test fuel lower heating 1969 value); 1970 1971 LHVtest is the fuel lower heating value of the fuel used for the test in MJ/kg; and 1972 1973 LHVRef is the reference fuel lower heating value = 43.217 MJ/kg. 1974 1975
12/13/2023 50
A3-1.8 Altitude. The altitude at which the airplane is flown affects the fuel flow. As noted in Table A3-1, 1976 since testing at any altitude other than the reference optimum altitude increases the FEM, this correction 1977 is considered optional. For applicants choosing to make this correction, the selection of the reference 1978 altitude depends on the method used for determining SAR for the three reference airplane masses 1979 specified in § 38.13 (b). For SAR data clustered around each of the three reference masses (see § 1980 A38.5.3.3), a reference altitude is selected by the applicant for each of the specified reference masses. If 1981 SAR data is obtained over a range of masses (see § A38.5.3.4), the applicant should select a reference 1982 altitude for each test point (since none of the test points are directly associated with any of the reference 1983 masses). For applicants using a first principle model (see paragraph 6.4.4 of this AC), the applicant 1984 should select a reference altitude for each target Weight/δ value (see Figure 6-6 of this AC). 1985 1986 The engine model should be used to determine the fuel flow at the test altitude (Fuel Flowtest alt) and the 1987 fuel flow at the reference altitude (Fuel Flowref alt). The fuel flow correction for altitude is determined as 1988 follows: 1989 1990 ∆Fuel Flowalt = Fuel Flowref alt – Fuel Flowtest alt. 1991 1992 A3.9 Temperature. The ambient temperature affects the fuel flow. The reference temperature is the 1993 standard day temperature from the ICAO standard atmosphere at the reference altitude. 1994 1995 The engine model should be used to determine the difference between the fuel flow at the test 1996 temperature (Fuel Flowtest temp) and the fuel flow at the reference temperature (Fuel Flowref temp). The fuel 1997 flow correction for temperature is determined as follows: 1998 1999 ∆Fuel Flowtemp = Fuel Flowref temp – Fuel Flowtest temp. 2000 2001 A3-1.10 Engine deterioration level. When first used, engines undergo a rapid, initial deterioration in fuel 2002 efficiency. Thereafter, the rate of deterioration significantly decreases. Engines with less than the 2003 reference deterioration level may be used, subject to the approval of the FAA, as stated in 14 CFR part 2004 38, Appendix A, § 5.2.2.1.7.1. In such a case, the fuel flow is to be corrected to the reference engine 2005 deterioration level using an approved method. Engines with more deterioration than the reference engine 2006 deterioration level may be used. In this case, a correction to the reference specification shall not be 2007 permitted in accordance with 14 CFR part 38, Appendix A, 5.2.2.1.7.2. 2008 2009 As stated above, a correction should generally not be made for engine deterioration level. If an applicant 2010 proposes to use an engine or engines with less than the reference deterioration level for testing, it may be 2011 possible to establish a conservative correction level to apply to the test fuel flow to represent engines at 2012 the reference deterioration level. Such a correction should be substantiated by engine fuel flow 2013 deterioration data from the same engine type or family and approved by the FAA. 2014 2015 A3-1.11 Electrical and mechanical power extraction and bleed flow. Electrical and mechanical power 2016 extraction and bleed flow both affect the fuel flow. 2017 2018 The engine model should be used to determine the difference between the fuel flow at the test power 2019 extraction (Fuel Flowtest bleed) and bleed flow and the fuel flow at the reference power extraction and bleed 2020 flow (Fuel Flowref bleed). The fuel flow correction for electrical and mechanical power extraction and bleed 2021 flow is determined as follows: 2022 2023 ∆Fuel FlowCorr bleed = Fuel Flowref bleed – Fuel Flowtest bleed. 2024 2025 A3-1.12 Modifications and non-standard equipment that will not be included on the production airplane. 2026 The test airplane may contain modifications and non-standard equipment, such as flight test 2027 instrumentation, that will not be included on the production airplane. Examples of such flight test 2028 instrumentation includes trailing cones, precision total air temperature probes, air data booms, telemetry 2029 antennas, and engine pressure rakes. Where these modifications or non-standard equipment increase 2030 drag or fuel flow, this effect may be corrected out of the tested drag and fuel flow levels used to determine 2031
12/13/2023 51
the SAR used for calculating the FEM. 2032 2033
12/13/2023 52
2034 Appendix 4: Confidence Interval Evaluation 2035 2036 2037 A4-1 Direct flight testing 2038 2039 If n measurements of SAR (𝐸𝐸1, 𝐸𝐸2, …., 𝐸𝐸n) are obtained under approximately the same conditions, and it 2040 can be assumed that they constitute a random sample from a normal population with true population 2041 mean, µ, and true standard deviation, σ, then the following statistics can be derived: 2042 2043 𝐸𝐸 = 𝐹𝐹𝑒𝑒𝑒𝑒𝐸𝐸𝑒𝑒𝑣𝑣𝑒𝑒𝐹𝐹 𝑜𝑜𝐸𝐸 𝑒𝑒ℎ𝐹𝐹 𝑒𝑒𝐹𝐹𝑣𝑣𝐸𝐸 = 1𝐸𝐸 �� 𝐸𝐸(𝑖𝑖) 𝑖𝑖=𝑛𝑛 𝑖𝑖=1 � 2044 𝑒𝑒 = 𝐹𝐹𝑒𝑒𝑒𝑒𝐸𝐸𝑒𝑒𝑣𝑣𝑒𝑒𝐹𝐹 𝑜𝑜𝐸𝐸 𝑒𝑒ℎ𝐹𝐹 𝑒𝑒𝑒𝑒𝑣𝑣𝐸𝐸𝑠𝑠𝑣𝑣𝑒𝑒𝑠𝑠 𝑠𝑠𝐹𝐹𝑣𝑣𝐸𝐸𝑣𝑣𝑒𝑒𝐸𝐸𝑜𝑜𝐸𝐸 𝑜𝑜𝐸𝐸 𝑒𝑒ℎ𝐹𝐹 𝑒𝑒𝐹𝐹𝑣𝑣𝐸𝐸 = �∑ (𝑦𝑦𝑖𝑖−𝑦𝑦�)2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−1 . 2045 2046 From these and the Student t-distribution, the confidence interval, CI, for the estimate of the mean, 𝐸𝐸, can 2047 be determined as: 2048 2049 CI = ȳ ± 𝑒𝑒�1−𝛼𝛼2,𝜁𝜁� 𝑒𝑒 √𝐸𝐸 2050 2051 where 𝑒𝑒�1−𝛼𝛼2,𝜁𝜁� denotes the �1 − 𝛼𝛼 2� percentile of the single-sided Student t-test with 𝜁𝜁 degrees freedom 2052 (for a clustered data set 𝜁𝜁 = 𝐸𝐸 − 1) and where α is defined such that 100(1 − 𝛼𝛼) percent is the desired 2053 confidence level for the confidence interval. In other words, it denotes the probability with which the 2054 interval will contain the unknown mean, µ. For fuel efficiency metric certification purposes, 90 percent 2055 confidence intervals are required and thus 𝑒𝑒.95,𝜁𝜁 is used. See Table A4-1 for a listing of values of 𝑒𝑒.95,𝜁𝜁 for 2056 different values of 𝜁𝜁. 2057 2058 2059 A4-2 Regression model 2060 2061 If n measurements of SAR (𝐸𝐸1, 𝐸𝐸2, …., 𝐸𝐸n) are obtained under significantly varying values of mass (𝑥𝑥1, 𝑥𝑥 2, 2062 …., 𝑥𝑥 n), respectively, then a polynomial can be fitted to the data by the method of least squares. For 2063 determining the mean SAR, µ , the following polynomial regression model is assumed to apply: 2064 2065 μ= B0 + B1𝑥𝑥 + B2𝑥𝑥2 +…..+ Bk𝑥𝑥k. 2066 2067 The estimate of the mean line through the data of the SAR is given by: 2068 2069 𝐸𝐸= b0 + b1𝑥𝑥 + b2𝑥𝑥2 +…..+ bk𝑥𝑥k. 2070 2071 Each regression coefficient (Bi) is estimated by bi from the sample data using the method of least squares 2072 in a process summarized as follows: 2073 2074 Each observation (xi,yi) satisfies the equations: 2075 2076 𝐸𝐸i= B0 + B1𝑥𝑥i + B2𝑥𝑥i2 +…..+ Bk𝑥𝑥ik + ɛi 2077 2078 = b0 + b1𝑥𝑥i + b2𝑥𝑥i2 +…..+ bk𝑥𝑥ik + ei 2079 2080 where ɛi and ei are, respectively, the random error and residual associated with the SAR. The random 2081 error is assumed to be a random sample from a normal population with mean zero and standard deviation 2082 σ . The residual (ei) is the difference between the measured value and the estimate of the value using the 2083
12/13/2023 53
estimates of the regression coefficients and 𝑥𝑥i. Its root mean square value (s) is the sample estimate for 2084 σ. These equations are often referred to as the normal equations. 2085 2086 2087 Table A4-1. 2088 2089 Student t-distribution (for 90 percent confidence) for various degrees of freedom 2090 2091 Degrees of freedom (ζ) t.95, ζ 1 6.314 2 2.920 3 2.353 4 2.132 5 2.015 6 1.943 7 1.895 8 1.860 9 1.833 10 1.812 12 1.782 14 1.761 16 1.746 18 1.734 20 1.725 24 1.711 30 1.697 60 1.671 >60 1.645 2092 2093 The n data points of measurements (𝑥𝑥i,yi) are processed as follows: 2094 2095 Each elemental vector (𝑥𝑥i) and its transpose (𝑥𝑥’i) are formed such that: 2096 2097 𝑥𝑥i = (1 𝑥𝑥i 𝑥𝑥𝑖𝑖2 . . 𝑥𝑥𝑖𝑖𝑘𝑘), a row vector; and 2098 2099 𝑥𝑥𝑖𝑖′ = ⎝ ⎜ ⎛ 1 x𝑖𝑖 𝑥𝑥𝑖𝑖2.
.
𝑥𝑥𝑖𝑖𝑘𝑘⎠ ⎟ ⎞, a column vector. 2100 2101 A matrix X is formed from all the elemental vectors xi for i = 1, ….., n. X’ is the transpose of X. A matrix A 2102 is defined such that A = X’X and a matrix A-1 is the inverse of A. In addition, y = (𝐸𝐸1 𝐸𝐸2 … 𝐸𝐸n) and b = (b0 b1 2103
12/13/2023 54
… b2), with b determined as the solution of the normal equations: 2104 2105 y = Xb and X’y = X’Xb = Ab 2106 2107 to give 2108 2109 b = A-1X’y. 2110 2111 The 90 percent confidence interval CI90 for the mean value of the SAR estimated with the associated 2112 value of the mass x0 is then defined as: 2113 2114 CI90 = ȳ (x0) ± t.95,ζ s v(x0) 2115 2116 where 𝑣𝑣(𝑥𝑥0) = �𝑥𝑥0 𝑆𝑆−1 𝑥𝑥0′ . 2117 2118 Thus, CI90 = ȳ (𝑥𝑥0) ± 𝑒𝑒.95,𝜁𝜁 𝑒𝑒 �𝑥𝑥0 𝑆𝑆−1 𝑥𝑥0′ , 2119 2120 where: 2121 2122 𝑥𝑥0 = (1 𝑥𝑥0 𝑥𝑥02 ... 𝑥𝑥0𝑘𝑘); 2123 2124 x’ is the transpose of 𝑥𝑥0; 2125 2126 ȳ (𝑥𝑥0) is the estimate of the mean value of the SAR at the associated value of the mass xo; 2127 2128
t.95,ζ is obtained for ζ degrees of freedom. For the general case of a multiple regression analysis involving 2129 K independent variables (i.e. K + 1 coefficients), ζ is defined as ζ = n – K – 1 ; and 2130 2131 𝑒𝑒 = �∑ (𝑦𝑦𝑖𝑖− ȳ(𝑥𝑥𝑖𝑖 ))2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−𝐾𝐾−1 is the estimate of σ, the true standard deviation. 2132 2133 2134 A4-3 Worked examples of the determination of 90 percent confidence intervals 2135 2136 Direct flight testing 2137 2138 Example 1: In which the confidence interval is less than the confidence interval limit. 2139 2140 Consider the following set of 6 independent measurements of SAR obtained by flight test around one of 2141 the three reference masses of the CO2 emissions evaluation metric. After correction to reference 2142 specifications, the following clustered data set of SAR values is obtained: 2143 2144 2145 Table A4-2. 2146 2147 Measurements of SAR — Example 1 2148 2149 Measurement number Corrected SAR (km/kg) 1 0.38152 2 0.38656 3 0.37988
12/13/2023 55
Measurement number Corrected SAR (km/kg) 4 0.38011 5 0.38567 6 0.37820 2150 2151 The number of data points (n) = 6 2152 2153 The degrees of freedom (n-1) = 5 2154 2155 The Student t-distribution for 90 percent confidence and 5 degrees of freedom (𝑒𝑒(.95 ,5) ) = 2.015. (See 2156 Table A4-1.) 2157 2158 Note.— 6 is the minimum number of test points required, as stated in § A38.5.3.3. 2159 Estimate of the mean SAR ( 𝑆𝑆𝑆𝑆𝑆𝑆 ) for the clustered data set: 2160 2161 𝑆𝑆𝑆𝑆𝑆𝑆 = 1 𝑛𝑛 �∑ 𝑆𝑆𝑆𝑆𝑆𝑆(𝑖𝑖)𝑖𝑖=𝑛𝑛𝑖𝑖=1 � = 0.38282 km/kg. 2162 2163 Estimate of the standard deviation(s): 2164 2165 𝑒𝑒 = �∑ �𝐷𝐷𝐴𝐴𝐷𝐷(𝑖𝑖)−𝐷𝐷𝐴𝐴𝐷𝐷�������2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−1 = 0.00344 km/kg . 2166 2167 2168 Confidence interval determination 2169 2170 The 90 percent confidence interval (CI90) is calculated as follows (see 3.3.2): 2171 2172 𝐶𝐶𝐶𝐶90 = 𝑆𝑆𝑆𝑆𝑆𝑆 ± 𝑒𝑒(.95,𝑛𝑛−1) s √𝑛𝑛 = 0.38282 ± 2.015 x
0.00344 √6 = 0.38282 ± 0.00283 km/kg. 2173 2174 2175 Check of confidence interval limits 2176 2177 The confidence interval extends to ±0.00283 km/kg around the mean SAR value of the clustered data set 2178
(0.38282 km/kg). This represents ±0.74 percent of the mean SAR value, which is below the confidence 2179 interval limit of 1.5 percent, as defined in § A38.5.3.2. 2180 2181 As a result, the SAR value of 0.38282 km/kg associated to one of the reference masses of the FEM can 2182 be used for the metric determination. 2183 2184 Example 2: In which the confidence interval exceeds the confidence interval limit. 2185 2186 Consider the following set of 6 independent measurements of SAR obtained by flight test around one of 2187 the three reference masses of the FEM. After correction to reference specifications, the following 2188 clustered data set of SAR values is obtained: 2189 2190 2191 2192 2193 2194 2195
12/13/2023 56
Table A4-3. 2196 2197 Measurements of SAR — Example 2 2198 2199 Measurement number Corrected SAR (km/kg) 1 0.15208 2 0.15795 3 0.15114 4 0.15225 5 0.15697 6 0.15834 2200 The number of data points (n) = 6 2201 2202 The degrees of freedom (n-1) = 5 2203 The Student t-distribution for 90 percent confidence and 5 degrees of freedom (𝑒𝑒(.95 ,5) ) = 2.015. (See 2204 Table A4-1) 2205 2206 Estimate of the mean SAR ( 𝑆𝑆𝑆𝑆𝑆𝑆 ) for the clustered data set: 2207 2208 𝑆𝑆𝑆𝑆𝑆𝑆 = 1 𝑛𝑛 �∑ 𝑆𝑆𝑆𝑆𝑆𝑆(𝑖𝑖)𝑖𝑖=𝑛𝑛𝑖𝑖=1 � = 0.15479 km/kg. 2209 2210 Estimate of the standard deviation(s): 2211 2212 𝑒𝑒 = �∑ �𝐷𝐷𝐴𝐴𝐷𝐷(𝑖𝑖)−𝐷𝐷𝐴𝐴𝐷𝐷�������2𝑖𝑖=𝑛𝑛𝑖𝑖=1 𝑛𝑛−1 = 0.0033 km/kg . 2213 2214 2215 Confidence interval determination 2216 2217 The 90 percent confidence interval (CI90) is calculated as follows (see 3.3.2): 2218 2219 𝐶𝐶𝐶𝐶90 = 𝑆𝑆𝑆𝑆𝑆𝑆 ± 𝑒𝑒(.95,𝑛𝑛−1) s √𝑛𝑛 = 0.15479 ± 2.015 x
0.0033 √6 = 0.15479 ± 0.00271 km/kg. 2220 2221 2222 Check of confidence interval limits 2223 2224 The confidence interval extends to ±0.00271 km/kg around the mean SAR value of the clustered data set 2225
(0.15479 km/kg). This represents ±1.75 percent of the mean SAR value, which is above the confidence 2226 interval limit of 1.5 percent, as defined in § A38.5.3.2. 2227 2228 In such case, a penalty equal to the amount that the 90 percent confidence interval exceeds ±1.5 percent 2229 is applied to the mean SAR value, i.e. (1.75-1.50) = 0.25 percent. The mean SAR value is therefore be 2230 penalized by an amount of 0.25 percent as follows: 2231 2232 𝑆𝑆𝑆𝑆𝑆𝑆 = (1 −
0.25
100) 𝑥𝑥 0.15479 = 0.15440 km/kg. 2233 2234 As a result, the SAR value of 0.15440 km/kg associated to one of the reference masses of the FEM can 2235 be used for the metric determination. 2236
12/13/2023 57
2237 Regression model 2238 2239 Example 3: In which the confidence interval at each of the three reference masses of FEM is less than 2240 the confidence interval limit. 2241 2242 Consider the following set of 12 measurements of SAR obtained by flight test at optimum speed and 2243 optimum altitude as a function of the airplane gross mass. After SAR correction to reference 2244 specifications, the following data set is obtained: 2245 2246 Table A4-4. 2247 2248 Measurements of SAR — Example 3 2249 2250 Measurement number and reference mass Gross mass (mi) (kg) Corrected SAR (SARi) (km/kg) 1 17,800 0.928 Low mass (*) 17,825 2 17,970 0.905 3 18,400 0.908 4 18,850 0.884 5 19,500 0.850 6 19,950 0.845 Mid mass (*) 19,953 7 20,180 0.833 8 20,350 0.818 9 21,000 0.792 10 21,500 0.781 11 21,870 0.779 High mass (*) 22,080 12 22,150 0.771 2251 (*) Low, mid and high mass represent the reference masses of the FEM, as defined in §38.13(b). 2252 2253 The number of data points (n) = 12 2254 2255 Note.— 12 is the minimum number of test points requested, as stated in § A38.5.3.4. 2256 2257 A representation of the above measurement points is proposed in Figure A4-5. 2258 2259