APPENDIX 3
CORRECTION OF AIR MINIMUM CONTROL SPEED TO STANDARD CONDITIONS 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 4.
Theoretical Basis Given the static lateral/directional equations of motion for straight line, unaccelerated flight:
Fy∑ = 0 Cyb b⋅ + Cy aad d⋅ + Cy rrd d⋅ = CL · sin φ (1) Mx∑ = 0 Clb b⋅ + Cl aad d⋅ + Cl rrd d⋅ = 0 (2) Mz∑ = 0 Cnb b⋅ + Cn aad d⋅ + Cn rrd d⋅ = Cna (3) where:
CL = 295 2 ⋅ ⋅ W V Se Cna = 295 2 ⋅ ⋅ ⋅ ⋅ F l V S b na e e 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 Constant Cn Method For the case where full rudder deflection is achieved, δr is a constant, and the system of equations can be resolved to an identity which shows that Cna is a linear function of CLsin φ.
Cna = A·CL ⋅ sinφ + B (4)
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 V MCA can be corrected to standard conditions by the relationship:
VMCAs = VMCAt · F F na na s t for Turbojets VMCAs = VMCAt · THP THP s s t t ⋅ ⋅ s s 1 3 for Propeller driven where:
*THPs = Maximum AFM scheduled Brake/Shaft horsepower multiplied by sta ndard day propeller efficiency.
THPt = Test day brake/shaft horsepower where V MCA was achieved multiplied by test day propeller efficiency.
s s = Atmospheric density ratio at standard conditions.
s t = Atmospheric density ratio at test conditions.
*Windmilling SHP is not considered, because the current 14 CFR 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.
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 V MCA due to variations in bank angle, weight, sideslip angle, or adverse yaw, its use is limited to corrections of 5% or less in asymmetric net thrust or power.
For corrections beyond 5%, the relationship shown in equation (4) must be used, and enough flight test data must be obtained to define the correlation between Cna and CLsin φ.
Graphical Method In theory these data could be obtained by varying any combination of asymmetric thrust, airspeed, weight, and bank angle that would provide a representative variable set. However, since V MCA 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 thrust will cause V MCA 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, V MCA tests are normally done at the lightest possible weight, at the maximum allowable asymmetric 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 V MCA 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 thrust, testing at three bank angles is required for the definition of the Cna vs CL sin φ relationship. These data should be obtained by shutting down the critical engine (normally the left), setting maximum allowable 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.
Data obtained using this method with a typical business jet and a large jet transport are shown in the following two graphs:
2-ENGINE BUSINESS JET 0
0.01
0.02
0.03
0.04
0.05
0.06 -0.04 0 0.04 0.08 0.12 0.16 CLsinφ Cna Kβ=0.20 4-ENGINE TRANSPORT 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08 0 0.05 0.1 0.15 0.2 0.25 CLsinφ Cna Kβ=0.14
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 C L, and since standard bank angle will be 5°, the only remaining variable in CLsin φ is Ve. By choosing the appropriate values of Fn available versus Ve, a plot of Cna versus CLsinφ can be made which represents the applied yawing moments. If the weight chosen represents some standard minimum weight, and the available net thrust values represent the maximum allowable 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 V MCA.
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 φ values are based on a gross weight of 9000 lbs.
Ve Fna Cna CLsin φ 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 CLsinφ Cna The intersection of the airframe and the engine curve shows values of:
Cna = 0.048 CLsin φ = 0.124 Since the engine Cn curve was based on W = 9000 lbs, the standard day value of V MCA can be determined from:
CLsin φ = 0.124 = 295 WV Se2 ⋅⋅ V .1108 9000
0.124e 2 = ⋅
VMCA = 89.7 KEAS If the airframe data is obtained from flight test, there are no assumptions or simplifications, and the value of V MCA 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 thrust that may be used.
Equation Method A single test day value of V MCA 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 φ relationship is known (from wind tunnel, or analytical estimates), or one is willing to use a default (conservative) value. Thrust extrapolation using slope values not based on flight test is limited to 10% of the test day 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 V MCA 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 φ 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
The following Cna vs CLsin φ plot shows the airframe curve, the standard day engine curve, and the test day curve:
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 CLsinφ Cna From the airframe/engine curve intersections:
Cnt = .052 Cns = .048 CLsinφt = .147 CLsinφs = .124 VMCAt = 82.4 kts VMCAs = 89.7 kts This is a thrust correction of approximately 8%. If the constant Cn method had been used, the test value of .052 would have applied, and the corresponding VMCAt would have been 85.7 kts, an error of 4 knots (5%) in the nonconservative direction.
Noting that the correction from test day to standard day is along the Cn vs CLsin φ curve, which is a straight line, and denoting the slope of the airframe curve as K β , the intersection of the standard day thrust line with the airframe curve as Cns and CLsin φs, and the intersection of the test day thrust line with the airframe curve as Cnt and CLsinφt, the following equation can be derived:
Given the point-slope form of a straight line, Y2 - Y1 = m(X2 - X1) Correspondingly, Cns - Cnt = Kβ ⋅ (CLsinφs - CLsinφt ) Cns - Kβ ⋅ CLsinφs = Cnt - Kβ ⋅ CLsinφt F l q S b na e s s ⋅ ⋅ ⋅ ⋅ - Kβ ⋅ W sinq Ss ss f⋅ = F lq S bna et t ⋅⋅ ⋅ − Kβ ⋅ W sinq St tt f⋅
Substituting q = V 295 e 2 and then multiplying through by V S 295 e 2 S ⋅ Fnas ⋅ lbe - Kβ ⋅ Ws ⋅sinφs = VVe 2 e 2 s t ⋅ F lb K W sinnat e t t⋅ − ⋅ ⋅ b f and finally, VMCAs = VMCAt ⋅ F lb K W sin F lb K W sin na e s s na e t t s t ⋅ − ⋅ ⋅ ⋅ − ⋅ ⋅ b b f f 1 2 for Turbojets VMCAs = VMCAt ⋅ 326 SHP V l b K W sin 326 SHPV lb K W sin S S S MCA e s s t t t MCAt e t t s ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ − ⋅ h s f h s f b b 1 2 for Turboprops In the simplified form of the lateral directional equations, Kβ is C Cn yb b which is the directional static margin .
The value of Kβ 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 Kβ is determined by flight test, 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 thrust correction to VMCA. A somewhat conservative default value of 0.20 for Kβ may be used if flight test data is not available; however, in this case, any thrust extrapolation is limited to 10% of the test day 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 CLsinφ that is greater than CLMAX ⋅sin5°.
APPENDIX 4
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 V MCA , see Appendix 3.
Given the static lateral/directional equations of motion for straight line, unaccelerated flight:
Fy∑ = 0 Cy b b⋅ + C y aad d⋅ + Cy rrd d⋅ = CL · sinφ (1) M x∑ = 0 Clb b⋅ + Cl aad d⋅ + Cl r rd d⋅ = 0 (2) ΣMz = 0 Cnb b⋅ + Cn aad d⋅ + Cn rrd d⋅ = Cna (3) For a reversible control system, rudder force versus deflection is:
FR = GR · q · SR · cR · Ch Rd · δR and δR = kR · F V R e 2 Substituting for δR in equations (1) through (3), and solving, results in an identity of the form:
FR = A·Fna - B·Wsinφ 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 thrust, weight, and bank angle. If asymmetric 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 V e2), but the same force would be required (varies directly with V e2).
When a force limited V MCA is determined during flight test, the variation in rudder force with airspeed results solely from the change in net thrust with speed, and if an airspeed (thrust) is reached at which the rudder force is 150 lbs, there is no way to correct this force limited V MCA to any other thrust level. Therefore, if V MCA is rudder pedal force limited, takeoff thrust at all flight conditions must be limited to the test value of asymmetric thrust.
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 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 FAR limit (i.e., 135 lbs after Amendment 25-42; 162 lbs prior to Amendment 25-42).
1. At any convenient airspeed (typically 1.2V s 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 Wsinφ for each of the test points.
3. Calculate Wssin 5°, where W s 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 Wsinφ and the curve from step (2). For example:
Fnat = 2600 lbs V = 1.2Vs Ws = 9000 lbs FR=166 lbs Wsinφ = 784 lbs -50 0 50 100 150 200 250 400 800 1200 1600 Wsinφ − lbs Rudder Pedal Force (FR) - lbs
4. Determine the maximum allowable asymmetric thrust from the relationship:
F Fna namax t= ⋅ 150FRs * *( 180FRs prior to Amendment 25-42) Assuming a 180 lb force limit: F 2600 180 166namax = ⋅ = 2820 lbs
5. Plot 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 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 150 140namax = ⋅ = 2686 lbs 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 must be derated to a level which 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:
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 must be limited to a value which results in a pedal force no greater than 150 lbs (180 lbs prior to Amendment 25-42).
APPENDIX 5
ONE-G STALL COMPLIANCE BASIS
1. APPLICABLE REGULATIONS .
§ 25.103 Stalling speed.
§ 25.201 Stall demonstration.
§ 25.203 Stall characteristics.
§ 25.207 Stall warning.
2. EXPLANATION.
a. Section 25.201 provides the regulatory defi nition of airplane stall. In particular, § 25.201(d)(1) defines the aerodynamic stall as existing "at an angle of attack measurably greater than that for maximum lift. . .." The resulting stall speed is defined in § 25.103 as the minimum speed (V S MIN) attained in the maneuver. For many high speed swept wing transport category airplanes this has resulted in the V S MIN stall speed being defined with a flight path load factor considerably less than one, which has led to inconsistent and non repeatable reference stall speeds.
Additionally, because the minimum speed in the maneuver typically occurs near the point at which the airplane spontaneously pitches nose down or where the pilot defines the level of buffet as deterrent, inconsistencies can also be introduced into the definition of the minimum speed by pilot technique and judgment. Since 14 CFR part 25 defines operating speeds as multiples of the V S MIN stall speed, the resulting operating speed margins to stall are not representative of the actual lift margin available (i.e., the margin to the speed at which wing lift alone can support the weight of the airplane in 1-g flight); the net result of this inadequate lift margin being inconsistent operating speed margins and maneuvering margins. To remedy this situation, procedures and requirements that are interpreted as being equivalent to the current part 25 stall-related regulations have been developed to define a reference stall speed that exists when the wing is generating maximum usable lift in a 1-g flight condition; this speed is referred to as the 1-g stall speed (V S -1g).
b. Service history has not indicated any safety problem from the use of V S MIN stall speeds and their related operating speed factors as currently defined in part 25. A survey of various swept wing transport category airplanes was conducted to establish what load factor existed at V S MIN; the average load factor was determined to be 0.88, which equates to the V S MIN speed being 94 percent of the speed at which the wing lift is equal to the weight of the airplane (i.e., VS -1g). Since service history indicates the existing operating speeds have provided an acceptable level of safety, the existing part 25 operating speed factors were factored by the ratio VS MIN/VS -1g = 0.94, thereby resulting in essentially the same operating speeds, regardless of the basis used for determining reference stall speeds.
c. The 1-g stall speed criteria has been applied to most type certification projects since the mid-1980s by making a finding of equivalent safety. The FAA Issue Paper process, as described in Order 8110.4A, "Type Certification Process" and Order 8100.5, "Aircraft Certification Directorate Procedures," has been used to document the equivalent safety finding and resolve any airplane-specific issues related to the application of the 1-g stall speed methodology.
3. PROCEDURES.
a. A generic Issue Paper is presented in this appendix that presents equivalent interpretations of the affected part 25 regulations to enable application of the 1-g stall speed compliance basis.
b. The following generic Issue Paper contains the necessary equivalent interpretations for making a finding of compliance for airplanes having stall defined by inherent aerodynamic characteristics (e.g., uncommanded nose down pitching moment) or by a system that provides an abrupt nose down pitch (e.g., a stick pusher). Because
airplanes with systems such as stick pushers would realize a considerable reduction in operating speeds and operating speed margins from stall with the basic 1-g stall speed criteria, the design characteristics of such airplanes and their systems should be evaluated with regard to providing a satisfactory level of safety.
c. For airplanes that have stall defined by a system that provides an abrupt nose down pitch (e.g., a stick pusher) that do not gain a significant increase in wing lift with power on, it may not be feasible to permit full reduction of the operating speed factors as contained in the 1-g stall speed criteria. Other factors such as susceptibility of the wing to stall angle of attack variability due to leading edge contamination should also be investigated before applying the full 1-g stall operating speed reductions to these airplanes. Such investigations may be accomplished by review of analytical data and in conjunction with other certification testing (e.g., windtunnel and icing flight test data could be used to investigate the effects of leading edge contamination). For assistance in determining allowable operating speed factor reductions for these airplanes, the FAA Transport Standards Staff of the Transport Airplane Directorate should be contacted.