Appendix F F-6 F.4.9 The intersection of the airframe and the engine curve shows values of:
𝐶𝑛𝑎 = 0.048 𝐶𝐿 𝑠𝑖𝑛 ∅ = 0.124 F.4.10 Since the engine Cn curve was based on W = 9000 lbs, the standard day value of VMCA can be determined from:
𝐶𝐿 𝑠𝑖𝑛 ∅ = 0.124 = 295 ⋅ 𝑊𝑉𝑒2 ⋅ 𝑆 𝑉𝑒2 = 0.1108 ⋅ 90000.124 𝑉𝑀𝐶𝐴 = 89.7 𝐾𝐸𝐴𝑆 F.4.11 If the airframe data is obtained from flight test, there are no assumptions or simplifications, and the value of VMCA derived from this method includes all the effects of bank angle, sideslip, adverse yaw, angle-of-attack, etc. Also, since the standard day value of Cna will always be less than the test value, no extrapolation is required, and there is no restriction on the value of standard day power or thrust that may be used.
F.5 Equation Method F.5.1 A single test day value of VMCA can also be corrected to standard conditions (using all the appropriate variables) without using this graphical method, provided either the slope of the Cna versus CL sin Ø relationship is known (from wind tunnel, or analytical estimates), or one is willing to use a default (conservative) value. Power or thrust extrapolation using slope values not based on flight test is limited to 10 percent of the test day power or thrust. The following analysis shows the derivation of this single-point correction equation:
F.5.2 If the test day engine Cna curve was added to the previous plot, it would be possible to see how far, and in what direction, the correction from test to standard day was made.
Assuming that a single value of VMCA was determined at 3000 feet at a weight of 9000 lbs, a test day engine Cna curve could be plotted using the same technique used for the standard day curve, except substituting the 3000-foot thrust values from the engine specification:
Appendix F F-7 Table F-2. Example Data for Typical Business Jet Engine at Altitude of 3,000 Feet Ve Fna Cna CLsin 70 2,634 0.073 0.204 90 2,589 0.043 0.123 110 2,554 0.029 0.082 130 2,528 0.02 0.059 150 2,510 0.015 0.044 F.5.3 The following Cna versus CL sin Ø plot shows the airframe curve, the standard day engine curve, and the test day curve:
Figure F-4. Yawing Moment—Engine and Airframe at Altitude of 3,000 Feet (Test Day) Airframe .048 .052 Standard Day Engine Test Day Engine .124 .147
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09 0 0.05 0.1 0.15 0.2 0.25 CLsin Cna
F.5.4 From the airframe/engine curve intersections:
𝐶𝑛𝑡 = 0.052 𝐶𝑛𝑠 = 0.048 𝐶𝐿 𝑠𝑖𝑛 ∅𝑡 = 0.147 𝐶𝐿 𝑠𝑖𝑛 ∅𝑠 = 0.124 𝑉𝑀𝐶𝐴𝑡 = 82.4 𝐾𝐸𝐴𝑆 𝑉𝑀𝐶𝐴𝑠 = 89.7 𝐾𝐸𝐴𝑆 F.5.5 This is a thrust correction of approximately 8 percent. If the constant Cn method had been used, the test value of .052 would have applied, and the corresponding 𝑉𝑀𝐶𝐴𝑡
Appendix F F-8 would have been 85.7 KEAS, an error of 4 knots (5 percent) in the non-conservative direction.
F.5.6 Noting that the correction from test day to standard day is along the Cn versus CL sin Ø curve, which is a straight line, and denoting the slope of the airframe curve as Kβ, the intersection of the standard day thrust line with the airframe curve as Cns and CL sin Ø, and the intersection of the test day thrust line with the airframe curve as Cnt and CL sin Ø, the following equation can be derived:
Given the point slope form of a straight line, 𝑌2 − 𝑌1 = 𝑚(𝑋2 − 𝑋1) Correspondingly, 𝐶𝑛𝑠 − 𝐶𝑛𝑡 = 𝐾𝛽(𝐶𝐿 𝑠𝑖𝑛 ∅𝑠 −𝐶𝐿𝑠𝑖𝑛 ∅𝑡) 𝐶𝑛𝑠 − 𝐾𝛽 ⋅ 𝐶𝐿 𝑠𝑖𝑛 ∅𝑠 = 𝐶𝑛𝑡 − 𝐾𝛽 ⋅ 𝐶𝐿𝑠𝑖𝑛 ∅𝑡 𝐹𝑛𝑎𝑠 ⋅ 𝑙𝑒 𝑞𝑠 ⋅ 𝑆 ⋅ 𝑏 − 𝐾𝛽 ⋅ 𝑊𝑠 𝑠𝑖𝑛 ∅𝑠𝑞𝑠 ⋅ 𝑆 = 𝐹𝑛𝑎𝑡 ⋅ 𝑙𝑒𝑞𝑡 ⋅ 𝑆 ⋅ 𝑏 − 𝐾𝛽 ⋅ 𝑊𝑡 𝑠𝑖𝑛 ∅𝑡𝑞𝑡 ⋅ 𝑆 Substituting 𝑞 = 𝑉𝑒2 295 And then multiplying through by 𝑉𝑒𝑠2 ⋅ 𝑆295 𝐹𝑛𝑎𝑠 ⋅ 𝑙𝑒𝑏 − 𝐾𝛽𝑊𝑠 𝑠𝑖𝑛 ∅𝑠 = 𝑉𝑒𝑠2𝑉𝑒𝑡2 ⋅ [𝐹𝑛𝑎𝑡 ⋅ 𝑙𝑒𝑏 − 𝐾𝛽𝑊𝑡𝑠𝑖𝑛 ∅𝑡] And finally, 𝑉𝑀𝐶𝐴𝑠 = 𝑉𝑀𝐶𝐴𝑡 [ 𝐹𝑛𝑎𝑠 ⋅ (𝑙𝑒𝑏 ) − 𝐾𝛽𝑊𝑠 𝑠𝑖𝑛 ∅𝑠 𝐹𝑛𝑎𝑡 ⋅ (𝑙𝑒𝑏 ) − 𝐾𝛽𝑊𝑡 𝑠𝑖𝑛 ∅𝑡 ] 1 2 𝑓𝑜𝑟 𝑡𝑢𝑟𝑏𝑜𝑗𝑒𝑡𝑠 And 𝑉𝑀𝐶𝐴𝑠 = 𝑉𝑀𝐶𝐴𝑡 [
326 ⋅ 𝑆𝐻𝑃𝑠 × 𝜂𝑠√𝜎𝑠𝑉𝑀𝐶𝐴𝑠 ⋅ (𝑙𝑒𝑏 ) − 𝐾𝛽𝑊𝑠 𝑠𝑖𝑛 ∅𝑠 326 ⋅ 𝑆𝐻𝑃𝑡 × 𝜂𝑡√𝜎𝑡𝑉𝑀𝐶𝐴𝑡 ⋅ (𝑙𝑒𝑏 ) − 𝐾𝛽𝑊𝑡 𝑠𝑖𝑛 ∅𝑡]
1 2 𝑓𝑜𝑟 𝑡𝑢𝑟𝑏𝑜𝑝𝑟𝑜𝑝𝑠
Appendix F F-9 F.5.7 In the simplified form of the lateral directional equations, K is 𝐶𝑛𝛽 𝐶𝑦𝛽⁄ , which is the directional static margin. The value of K typically varies from approximately 0.14 to
0.19, depending on the lateral/directional characteristics of the airplane being tested. As with the graphical method, if K is determined by flight test, power or thrust corrections to VMCA are based on an interpolation of flight test defined airframe capability, and there is no limit on the amount of the power or thrust correction to VMCA. A somewhat conservative default value of 0.20 for K may be used if flight test data is not available;
however, in this case, any power or thrust extrapolation is limited to 10 percent of the test day power or thrust. To assure that corrections for bank angle and weight do not result in standard day VMCA values at or below stall speed, the corrections made by either the graphical or equation method should not result in a VMCA, which is based on a CL sin Ø that is greater than CLMAX sin 5°.
Appendix G G-1 APPENDIX G. RUDDER PEDAL FORCE-LIMITED AIR MINIMUM CONTROL SPEED G.1 One Acceptable Method.
The following analysis presents one method of addressing rudder pedal force limited air minimum control speed. This method is applicable to either jet or propeller driven airplanes. The effect of banking into the operating engine is accounted for, and the method will work with either fixed pitch or constant speed propellers, including the effects of windmilling drag. For rudder deflection limited VMCA, see appendix F of this AC.
G.1.1 Given the static lateral/directional equations of motion for straight line, unaccelerated flight:
∑ 𝐹𝑦 = 0 𝐶𝑦𝛽 ⋅ 𝛽 + 𝐶𝑦𝛿𝑎 ⋅ 𝛿𝑎 + 𝐶𝑦𝛿𝑟 ⋅ 𝛿𝑟 = 𝐶𝐿 𝑠𝑖𝑛 ∅ (1) ∑ 𝑀𝑥 = 0 𝐶𝑙𝛽 ⋅ 𝛽 + 𝐶𝑙𝛿𝑎 ⋅ 𝛿𝑎 + 𝐶𝑙𝛿𝑟 ⋅ 𝛿𝑟 = 0 (2) ∑ 𝑀𝑧 = 0 𝐶𝑛𝛽 ⋅ 𝛽 + 𝐶𝑛𝛿𝑎 ⋅ 𝛿𝑎 + 𝐶𝑛𝛿𝑟 ⋅ 𝛿𝑟 = 𝐶𝑛𝑎 (3) G.1.2 For a reversible control system, rudder force versus deflection is:
𝐹𝑅 = 𝐺𝑅 ⋅ 𝑞 ⋅ 𝑆𝑅 ⋅ 𝐶𝑅 ⋅ 𝐶ℎ𝛿𝑅 ∙ 𝛿𝑅 and δ𝑅 = kR ⋅ FR Ve2 G.1.3 Substituting for FR in equations (1) through (3), and solving, results in an identity of the form:
𝐹𝑅 = 𝐴 ⋅ 𝐹𝑛𝑎 − 𝐵 ⋅ 𝑊 𝑠𝑖𝑛 ∅ G.1.4 All the airspeed terms cancel, indicating that the engine-out rudder force required for straight flight is not a function of airspeed, but only of asymmetric power or thrust, weight, and bank angle. If asymmetric power or thrust did not vary with airspeed, it would be possible to stabilize at any airspeed with the same rudder force. At higher speeds, less rudder deflection would be required (varies inversely with Ve2), but the same force would be required (varies directly with Ve2). When a force limited VMCA is determined during flight test, the variation in rudder force with airspeed results solely from the change in net power or thrust with speed, and if an airspeed (i.e., power or thrust level) is reached at which the rudder force is 150 lbs, there is no way to correct this force limited VMCA to any other power or thrust level. Therefore, if VMCA is rudder pedal force limited, takeoff power or thrust at all flight conditions should be limited to the test value of asymmetric power or thrust.
G.1.5 In some cases, it is possible to achieve full rudder deflection at the test altitude without reaching a pedal force limit, but with the higher power or thrust at standard conditions, a force limit would exist. To preclude missing this crossover effect, the following analysis should be performed whenever test day rudder pedal forces are greater than 90
Appendix G G-2 percent of the part 25 limit (i.e., 135 lbs after amendment 25-42; 162 lbs prior to amendment 25-42).
G.1.5.1 At any convenient airspeed (typically 1.13 VSR with minimum takeoff flaps), shut down the critical engine, and leave it windmilling (propeller feathered if autofeather is required), apply maximum available power/thrust to the operating engine, and while maintaining constant heading, vary the bank angle from 10° to less than 5° in approximately 2°-3° increments, noting the rudder force at each stabilized bank angle.
G.1.5.2 Plot the rudder force versus Wsin Ø for each of the test points. See figure G-1.)
Figure G-1. Rudder Force Versus WsinΦ FR=166 lbs Wsin = 784 lbs -50 0 50 100 150 200 250 400 800 1200 1600 Wsinlbs Rudder Pedal Force (FR) - lbs
G.1.5.3 Calculate WS sin 5° where WS is either the average test weight, or the lightest weight scheduled in the AFM, and define standard day rudder force (FRs) as the intersection of this value of WS sin 5° and the curve from step (2). For example:
𝐹𝑛𝑎𝑡 = 2600 𝑙𝑏𝑠 𝑉 = 1.13 𝑉𝑆𝑅 𝑊𝑠 = 9000 𝑙𝑏𝑠 G.1.5.4 Determine the maximum allowable asymmetric thrust from the relationship:
Appendix G G-3 𝐹𝑛𝑎𝑚𝑎𝑥 = 𝐹𝑛𝑎𝑡 ⋅ ( 150 𝐹𝑅𝑠 )1 Assuming a 180 lb force limit:
𝐹𝑛𝑎𝑚𝑎𝑥 = 2600 ⋅ (180166) = 2820 𝑙𝑏𝑠 G.1.5.5 Plot the maximum scheduled AFM thrust versus airspeed (see figure G-2), and determine VMCA at the intersection of this curve and Fnamax:
Figure G-2. Plot to Determine VMCA
80.5 KEAS 2820 lbs 2600 2650 2700 2750 2800 2850 2900 2950 3000 60 80 100 120 140 160 Ve - kts.
Fna - lbs.
G.1.6 If the force limited VMCA value is high enough to adversely impact the takeoff speed schedule, it can be reduced to an acceptable value by derating takeoff power or thrust.
For example, if the standard day rudder force (FRs) was 140 lbs on an amendment 25-42 airplane, the maximum allowable asymmetric takeoff thrust would be:
𝐹𝑛𝑎𝑚𝑎𝑥 = 2600 ⋅ 150140 = 2686 𝑙𝑏𝑠
1 (180𝐹𝑅𝑠) prior to amendment 25-42.
Appendix G G-4 G.1.7 Using the same maximum asymmetric thrust available versus airspeed as before:
Figure G-3. Plot to Determine VMCA for an Amendment 25-42 Airplane 172 KEAS 2686 lbs 2600 2650 2700 2750 2800 2850 2900 2950 3000 60 80 100 120 140 160 180 200 Ve - kts.
Fna - lbs.
G.1.8 Because of the shallow thrust lapse rate with airspeed, the force limited VMCA for these conditions would be 172 knots, which is obviously unacceptable. To reduce this value back to 80 knots (or any other speed), takeoff thrust should be derated to a level that provides a maximum asymmetric thrust value of Fnamax at the desired VMCA (80 knots in this example). The amount of derate required can be determined from the following plot:
Appendix G G-5 Figure G-4. Plot to Determine VMCA for Derate Thrust Maximum Thrust Derated Thrust (5%) 80 KEAS 2686 lbs 2400 2500 2600 2700 2800 2900 3000 50 100 150 200 Ve - kts Fna - lbs
G.1.9 For a given weight and bank angle, rudder pedal force is determined solely by asymmetric thrust; consequently, takeoff thrust should be limited to a value that results in a pedal force no greater than 150 lbs (180 lbs prior to amendment 25-42).
Appendix H
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