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AC 25-7D Appendix F
APPENDIX F. CORRECTION OF AIR MINIMUM CONTROL SPEED TO
STANDARD CONDITIONS F.1 Overview.
The following analysis presents three methods of correcting a flight test derived value of air minimum control speed to standard conditions. These methods are applicable only to rudder deflection limited VMCA, for 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 pedal force limited VMCA, see appendix G of this AC.
F.2 Theoretical Basis.
Given the static lateral/directional equations of motion for straight line, unaccelerated flight:
(1)
(2)
(3) Where:
W = Weight (lbs) Ve = Equivalent airspeed (kts) S = Wing area (ft2) Fna = Asymmetric net thrust (lbs) Fna = (Fn + Dw) for engine inoperative Fna = (Fn + Fi) for engine at idle Fn = Net thrust of the operating engine (lbs) Fi = Idle engine net thrust (lbs) Dw = Windmill drag (lbs) le = Distance from aircraft center line to engine thrust line (ft) b = Wingspan (ft) F-1
Appendix F F.3 Constant Cn Method.
F.3.1 For the case where full rudder deflection is achieved, r is a constant, and the system of equations can be resolved to an identity that shows that Cna is a linear function of CL sin Ø.
(4) F.3.2 If it is assumed that test and standard day VMCA occur at the same angle-of-attack and bank angle, the asymmetric yawing moment coefficient will be constant, and VMCA can be corrected to standard conditions by the relationship:
Where:
THPs = Maximum AFM scheduled brake/shaft horsepower multiplied by standard day propeller efficiency.
THPt = Test day brake/shaft horsepower where VMCA was achieved multiplied by test day propeller efficiency.
Atmospheric density ratio at standard conditions.
Atmospheric density ratio at test conditions.
F.3.3 Windmilling shaft horsepower is not considered because the current part 25 takeoff requirements for propeller driven airplanes result in such large performance penalties with a windmilling propeller that all part 25 turboprops to date have had autofeather installed.
F.3.4 Since both net thrust and shaft horsepower vary with speed, use of these equations will require an iterative solution. Because this constant Cn method does not consider the effect on VMCA due to variations in bank angle, weight, sideslip angle, or adverse yaw, its use is limited to corrections of 5 percent or less in asymmetric net thrust or power.
F.3.5 For corrections beyond 5 percent, the relationship shown in equation (4) should be used, and enough flight test data should be obtained to define the correlation between Cna and CL sin Ø.
F.4 Graphical Method.
F.4.1 In theory this data could be obtained by varying any combination of asymmetric power or thrust, airspeed, weight, and bank angle that would provide a representative variable set. However, since VMCA and stall speed are nearly coincident for most airplanes, there are some severe constraints on most of the variables. Typically, any reduction in F-2
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AC 25-7D Appendix F maximum asymmetric power or thrust will cause VMCA to decrease below stall speed, and any increase in weight will cause stall speed to increase above VMCA. Therefore, the only parameter that can reasonably be varied is bank angle.
F.4.2 To maximize the spread between stall and minimum control speed, VMCA tests are normally done at the lightest possible weight, at the maximum allowable asymmetric power or thrust (even with a short duration overboost, if the engine manufacturer will agree). At typical test altitudes (2000 to 3000 feet) and prototype gross weights, it will usually still not be possible to define VMCA with the full 5° bank, because of stall buffet.
F.4.3 To obtain the data necessary for extrapolation to the 5° bank limit, and to maximum asymmetric power or thrust, testing at three bank angles is required for the definition of the Cna versus CL sin Ø relationship. This data should be obtained by shutting down the critical engine (normally the left), setting maximum allowable power or thrust on the operative engine, and slowing down while maintaining constant heading until full rudder deflection is achieved. The first point, a wings level condition, is easy to set up, and results in a speed well above stall buffet. A second point, at zero sideslip, will be achieved at approximately 2° to 3° bank (flown with a yaw string, or instrumented sideslip vane) and will provide an intermediate speed, still above buffet. The third data point is flown with as much bank angle that can be used without excessive buffeting (no more than would be accepted as the minimum level of stall warning). If necessary, an additional point can be obtained by banking 2° to 3° into the inoperative engine.
F.4.4 To use this method, instrumentation is necessary for the determination of net thrust/shaft horsepower, and an accurate calibrated airspeed system is required, as well as engine/propeller charts for windmill drag, charts for propeller efficiency, and the ability to measure bank angle to at least a tenth of a degree.
F.4.5 Data obtained using this method with a typical business jet and a large jet transport are shown in figures F-1 and F-2, respectively:
F-3
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AC 25-7D Appendix F Figure F-1. Two-Engine Business Jet
2-ENGINE BUSINESS JET -0.04 0 0.04 0.08 0.12 0.16 CLsin 0
0.01
0.02
0.03
0.04
0.05
0.06 C K=0.20 Figure F-2. Four-Engine Transport
4-ENGINE TRANSPORT
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01 0 K=0.14 0 0.05 0.1 0.15 0.2 CLsin C F.4.6 These plots represent the capability of the airframe to produce yawing moment by a combination of rudder deflection (full, in this case), and the sideslip which results from the bank angle. In order to determine the limiting condition for VMCA, it is necessary to know what the applied yawing moment is (due to the engine-out moments), and to plot the applied moments on the same plot, in a similar form. It is possible to do this by choosing a gross weight to be used to calculate CL, and since standard bank angle will be 5°, the only remaining variable in CL sin Ø is Ve. By choosing the appropriate values F-4
Appendix F of Fn available versus Ve, a plot of Cna versus CL sin Ø can be made which represents the applied yawing moments. If the weight chosen represents some standard minimum weight, and the available net power or thrust values represent the maximum allowable power or thrust scheduled in the AFM, the intersection point of the airframe curve and the engine curve will be the desired standard day values, which can be used to calculate VMCA.
F.4.7 As an example, the following values of net thrust plus windmill drag have been extracted from a typical corporate jet engine spec. The data represent a maximum thrust engine, and have been corrected for ram drag and minimum accessory bleed and electrical load. The CL sin Ø values are based on a gross weight of 9000 lbs.
Table F-1. Example Data for Typical Business Jet Engine Ve Fna Cna CLsin˜ 70 2,846 0.079 0.204 90 2,798 0.047 0.123 110 2,764 0.031 0.082 130 2,737 0.022 0.059 150 2,710 0.016 0.044 F.4.8 Plotting both the airframe and engine yawing moment curves on the same graph looks like:
Figure F-3. Yawing Moment—Engine and Airframe
0 0.05 0.1 0.15 0.2 0.25 CLsin Airframe .048 Engine .124 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09 Cna F-5
Appendix F F.4.9 The intersection of the airframe and the engine curve shows values of:
F.4.10 Since the engine Cn curve was based on W = 9000 lbs, the standard day value of VMCA can be determined from:
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:
F-6
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AC 25-7D Appendix F 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)
0 0.05 0.1 0.15 0.2 0.25 CLsin 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 Cna F.5.4 From the airframe/engine curve intersections:
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
would have been 85.7 KEAS, an error of 4 knots (5 percent) in the non-conservative direction.
F-7
Appendix F F.5.6 Noting that the correction from test day to standard day is along the Cn versus CL sin O curve, which is a straight line, and denoting the slope of the airframe curve as Kp, the intersection of the standard day thrust line with the airframe curve as Cns and CL sinØ, and the intersection of the test day thrust line with the airframe curve as Cnt and Cr sinØ, the following equation can be derived:
Given the point slope form of a straight line, ·2 - 41 = m (X2 - 81) Correspondingly, Cns - Cnt = Kp (CL sin Ds -Cisin Øt) Ins - Ap. CisinDs = Cnt - hp Cisin D;
Fras le -Wt sin Dt 95 ' 5 • - Kg • 9: • S Substituting q = 295 And then multiplying through by 295 And finally, Fnas (F) - KgW sin Os VMCA; = VMCAL for turbojets Fna; (%) - KeW: sin Dt] And 326 - SHPs x Novos. (g) - KoWe sin Ds VACA; = MCAL 326- SHP: X NeVer. (f) - KeWesin De VuCAt VMCAS for turboprops F.5.7 In the simplified form of the lateral directional equations, Kb is Cng/ Cys, which is the directional static margin. The value of Ko typically varies from approximately 0.14 to
0.19, depending on the lateral/directional characteristics of the airplane being tested. As F-8
Appendix F with the graphical method, if Kb 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 Kb 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°.
F-9
Appendix G 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:
(1)
(2)
(3) G.1.2 For a reversible control system, rudder force versus deflection is:
G.1.3 Substituting for FR in equations (1) through (3), and solving, results in an identity of the form:
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 ), but the same force would be required (varies directly with
). When a force limited VMCA is determined during flight test, the variation in ru dder 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 G-1
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AC 25-7D Appendix G 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
400 800 1200 1600 Wsinlbs FR=166 lbs Wsin = 784 lbs -50 0 50 100 150 200 250 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:
G.1.5.4 Determine the maximum allowable asymmetric thrust from the relationship:
G-2
Appendix G Framax = Frat' 150 Assuming a 180 1b force limit:
180 Framex = 2600 • = 2820 lbs 166 G.1.5.5 Plot the maximum scheduled AFM thrust versus airspeed (see ШИ OSTALE High pitch rate Stall In Ground Effect * with low climb rate a can result in stall → Examples of
- → with positive takeoff tests Stall Margin
- > Decreasing VIOF 0 On Ground 1 (Free Air) h/b (Wing reference height above ground / Wingspan) figure ), and determine VMCA at the intersection of this curve and F namax* '(Fx.,) prior to amendment 25-42.
G-3