Derivation of Pertinent Aircraft Performance Equations

This section derives the relevant performance equations used in this book. For more details, readers may consult references [2] through [6].

Using the engine and aircraft data developed thus far during the conceptual design phase, the next section verifies whether the configured aircraft satisfies the airworthiness (i. e., FAR) and customer requirements in the takeoff/landing, the ini­tial climb rates, and the maximum initial cruise speed, as well as the payload-range capability (i. e., civil aircraft). Certifying agencies have mandatory requirements to ensure safety at takeoff and landing. Airworthiness regulations differ among coun­tries. For further details, readers may refer to the respective regulation – most of which appear in the official Web sites.

13.4.1 Takeoff

During a takeoff ground run, an aircraft under maximum thrust accelerates, gaining speed until a suitable safe speed is reached. The pilot then initiates rotation of the aircraft by gently pulling back the control stick or wheel (i. e., the elevator is going up) for a liftoff.

Designers must know the sequence of the takeoff speed schedules stipulated by the certifying agencies. To ensure safety, the agencies demand mandatory require­ments for taking off with one engine inoperative to clear a 35-ft height that repre­sents an obstacle. A one-engine inoperative TOFL is computed by considering the balanced field length (BFL) when the stopping distance after an engine failure at

Figure 13.8. Takeoff, first – and second-segment climb

the decision speed, V1, is the same as the distance taken to clear the obstacle at the MTOM (Figure 13.8). Figure 13.8 gives the various speed schedules during a takeoff run, which are explained as follows:

Vi: This is the decision speed. An engine failure below this speed would

result in an aircraft not being able to satisfy takeoff within the specified field length but able to stop. If an engine fails above the V1 speed, the aircraft should continue the takeoff operation.

Vmc: This is the minimum control speed at which the rudder is effective to con­trol the asymmetry created by a one-engine failure. It should be lower than Vi; otherwise, at the loss of one engine at Vi, an aircraft cannot be controlled if it continues the takeoff operation.

VR: This is the speed at which a pilot initiates the action to rotate an aircraft

for liftoff; it should be >1.05 Vmc. Once this is accomplished, reaching VLo and V2 occurs as an outcome of the action. VR should be more than

Vstall.

Vmu: There is a minimum “unstick” speed, above which an aircraft can be

made to lift off. The speed should be slightly above VR. In fact, Vmu determines VR. If a pilot makes an early rotation, then Vmu may not be sufficient for liftoff and the aircraft tail drags until it gains sufficient speed for liftoff.

VLO: This is the speed at which the aircraft lifts off the ground; it is closely associated with VR. If one engine is inoperative, it should have a VLO > 1.05Vmu.

V2: This is the takeoff climb speed at a 35-ft height, also known as the first-

segment climb speed; it is also closely associated with VR. FAR require that V2 = 1.2Vstall (at a minimum; it can be higher).

Vr: This is the brake-application speed with a one-engine failure (VB > V1).

The first-segment speed schedules are interrelated and expressed in terms of the ratios of Vstall, as given in Table 13.4. The velocity ratios in Table 13.4 comprise a typical range and can deviate a little as long as there is compliance with the FAR stipulation (marked with an asterisk in the table). Table 13.5 provides details of the climb segments.

Table 13.4. Civil aircraft takeoff-speed schedule

Two-engine

Three-engine

Four-engine

Percent loss at an engine failure

50

33.3

25

Minimum climb gradient at 1st segment*

0%

0.3%

0.5%

Minimum climb gradient at 2nd segment*

2.4%

2.7%

3%

ViO/Vstall (approximate)

1.12 to 1.14

1.15 to 1.16

1.17 to 1.18

Vstall/VR (approximate)

1.10 to 1.18

1.14 to 1.18

1.16 to 1.18

Vmu/VR (approximate)

1.02 to 1.04

1.02 to 1.04

1.02 to 1.04

V1/VR (approximate)

0.96 to 0.98

0.93 to 0.95

0.90 to 0.92

Vmc/V1 (approximate)

0.94 to 0.98

0.94 to 0.98

0.94 to 0.98

V2 /Vstall

>1.2

>1.2

>1.2

* FAR requirements

Some engines at the takeoff rating have an APR that could generate a 5% higher thrust than the maximum takeoff thrust for a short period. These types of engines are not considered in this book.

The higher the thrust loading (T/W), the higher is the aircraft acceleration. For smaller changes, VR/Vstall and Vbo/Vstall may be linearly decreased with an increase in T/W. The decision speed V1 is established through iterations, as described in Section 13.5.1. In a family of derivative aircraft, the smaller variant can have a V1 close to the VR.

Table 13.5 lists the aircraft configurations and power settings for the climb seg­ments. The first – and second-segment climb schedule has FAR requirements; how­ever, the initial enroute climb capability is a customer requirement, not a FAR requirement.

Military aircraft requirements (i. e., MIL-C5011A) are slightly different than civil aircraft requirements; the first-segment clearing height is 50 ft rather than 35 ft. Many military aircraft have a single engine in which the concept of BFL is not applicable. Military aircraft must satisfy the critical field length (CFL) as described in Section 13.6.1. The second-segment rate of climb must meet a minimum of 500 ft/minute for a multiengine aircraft.

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