Appendix 6 A6-3 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 CLsin.
Graphical Method
In theory these 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 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.
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 (2000-3000 feet) and prototype gross weights, it will usually still not be possible to define VMCA with the full 5° bank, because of stall buffet.
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 vs CL sin relationship. These 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° - 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° - 3° into the inoperative engine.
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.
Appendix 6 A6-4 Data obtained using this method with a typical business jet and a large jet transport are shown in the following two graphs:
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 CLsin is Ve. By choosing the appropriate values of Fn available versus Ve, a plot of Cna versus CLsin 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
Appendix 6 A6-5 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.
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 CLsin values are based on a gross weight of 9000 lbs.
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
Plotting both the airframe and engine yawing moment curves on the same graph looks like:
Airframe .048 Engine .124 0
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
The intersection of the airframe and the engine curve shows values of:
Cna = 0.048 CLsin = 0.124
Since the engine Cn curve was based on W = 9000 lbs, the standard day value of VMCA can be determined from:
CLsin = 0.124 = 295 W V Se2
Appendix 6 A6-6
0.124 90000.1108V 2e
VMCA = 89.7 KEAS
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.
Equation Method
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 vs CLsin 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:
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 ft thrust values from the engine spec:
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
Appendix 6 The following Cna vs CLsin plot shows the airframe curve, the standard day engine curve, and the test day curve:
From the airframe/engine curve intersections:
Cnt = 0.052 C ns = 0.048 CLsint = 0.147 C Lsins = 0.124 = 82.4 KEAS = 89.7 KEAS 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.
Noting that the correction from test day to standard day is along the Cn vs CLsin curve, which is a straight line, and denoting the slope of the airframe curve as K the intersection of the standard day thrust line with the airframe curve as Cns and CLsins, and the intersection of the test day thrust line with the airframe curve as Cnt and CLsint, the following equation can be derived:
Given the point-slope form of a straight line,
Appendix 6 A6-8 Cns - Cnt = K (CLsins - CLsint ) Cns - K CLsins = Cnt - K CLsint F l q S b na e s s
- K W sin q S s s s F l q S b na e t t K W sin q S t t t Substituting q = V 295 e 2 and then multiplying through by V S 295 e 2 S Fnas l b e - K Ws sins = V V e 2 e 2 s t F lb K W sinnat e t t and finally, VMCAs = VMCAt F lb K W sin F lb K W sin na e s s na e t t s t 1 2 for Turbojets VMCAs = VMCAt 326 SHPV lb K W sin 326 SHPV lb K W sin S S S MCA e s s t t t MCAt e t t s 1 2 for Turboprops In the simplified form of the lateral directional equations, K is 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 CLsin that is greater than CLMAX sin5.
C Cn y
Appendix 7 A7-1
Appendix 7
Rudder Pedal Force-Limited Air Minimum Control Speed
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 6.
Given the static lateral/directional equations of motion for straight line, unaccelerated flight:
Fy = 0 yC + Cy aa + Cy rr = CL · sin
Mx 0 lC + a laC Cl r r = 0
Mz = 0 Cn + Cn aa Cn rr = Cna
For a reversible control system, rudder force versus deflection is:
FR = GR · q · SR · cR · C h R · R
and R = kR · F V R e 2
Substituting for R in equations (1) through (3), and solving, results in an identity of the form:
FR = A·Fna - B·Wsin
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.
Appendix 7 A7-2
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 percent of the part 25 limit (i.e., 135 lbs after amendment 25-42; 162 lbs prior to amendment 25-42).
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.
2. Plot the rudder force vs Wsin for each of the test points.
3. Calculate Wssin 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 Wsin and the curve from step (2). For example:
Fnat = 2600 lbs V = 1.13 VSR Ws = 9000 lbs FR=166 lbs Wsin = 784 lbs -50 0 50 100 150 200 250 400 800 1200 1600 Wsinlbs Rudder Pedal Force (FR) - lbs
Appendix 7 A7-3
4. Determine the maximum allowable asymmetric thrust from the relationship:
F Fna namax t 150 FRs * *( 180 FRs prior to amendment 25-42)
Assuming a 180 lb force limit:
1661802600F maxna 2820 lbs
5. Plot the maximum scheduled AFM thrust vs airspeed, and determine VMCA at the intersection of this curve and Fnamax :
80.5 KEAS 2820 lbs 2600 2650 2700 2750 2800 2850 2900 2950 3000 60 80 100 120 140 160 Ve - kts.
Fna - lbs.
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:
F 2600 150140namax 2686 lbs
Appendix 7 A7-4 Using the same maximum asymmetric thrust available vs airspeed as before:
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.
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 which provides a maximum asymmetric thrust value of F at the desired VMCA (80 knots in this example).
The amount of derate required can be determined from the following plot:
Maximum Thrust Derated Thrust (5%) 80 KEAS 2686 lbs 2400 2500 2600 2700 2800 2900 3000 50 100 150 200 Ve - kts Fna - lbs
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 which results in a pedal force no greater than 150 lbs (180 lbs prior to amendment 25-42).
na max