Lecture 8

Lecture 8

Ballistic and dynamic ceilings of an airplane. The summary diagram of range of possible speeds and airplane flight altitudes. Climb and descent of an airplane.At the ballistic ceiling altitude the airplane has only potential energy.

Fig. 8.1In order to find graphically a ballistic ceiling value it is necessary to construct the boundary of possible speeds and altitudes range for horizontal steady-state flight (Fig. 8.1), and then plot a set of curves on the obtained diagram (Fig. 8.1), calculated on expression (7.10) or (7.11). That curve, which is tangent to curve

will determine

.

In the point of contact the airplane will have optimal values of

and

, which determine the greatest reserve of full energy, i.e. provide obtaining maximum power amount. The curve

is the boundary of inaccessibility area, i.e. the airplane will be not capable to get in this area located to the right of curve

, without using additional means.

In a cross point of curve

with an ordinate axis we obtain altitude of a ballistic ceiling at absence of power losses.

In practice it is impossible to realize a lost-free dynamic maneuver, and that is why above mentioned altitude is called as a conditional ballistic ceiling

. In actual conditions while dynamic maneuvering

drops with the change of geometric altitude

due to resistance forces action. If we calculate

with taking into account losses at climb and plot an obtained curve

on Fig. 8.1, then we shall receive the value of real ballistic ceiling in a cross point of curve with the ordinate axis

. Frequently for simplification of ballistic and dynamic ceilings calculation an amount of power losses is set

on a site

or

. For example, let's assume that

. Then, for

the curve

is calculated by formula (8.11) and built. Its cross point with an ordinate axis will define the ballistic ceiling altitude. Of course, such method of

definition is approximate.

Obviously reaching the ballistic ceiling altitude, that requires a drop of speed up to zero, is not desirable for airplanes, because the airplane loses the controllability. For keeping the airplane controllability its speed should be not less than minimum control speed

or

. While flying with minimum control speed, aerodynamic forces arose on airplane controls are sufficient for flight mode change. Minimum control speed increases with altitude (refer to Fig. 8.1), at that it is remaining less than minimum theoretical speed.

Maximum altitude, which the airplane can reach during unsteady controlled flight, is called the dynamic ceiling

. At the dynamic ceiling altitude the flight speed is equal to minimal control speed. In other words, an altitude the airplane can reach using a kinetic energy up to a moment, when the flight speed becomes equal to minimum control speed is called an airplane dynamic ceiling.

If one neglects losses, then the conditional dynamic ceiling

will be determined by a cross point

of curve

with the curve

. The real dynamic ceiling

will be determined by the point

which is located below the point

on the amount of power losses (refer to Fig. 8.1).

Thus, flight altitude increasing over the static ceiling is reached by transformation of a portion of the kinetic or all energy inherent to the airplane, into potential.

The maneuver, which provides reaching the ballistic and dynamic ceilings, is called "zoom".

The summary diagram of range of possible speeds and altitudes of an airplane flight If one generalizes all previous reasoning, then it is possible to construct the diagram of possible altitudes and flight speeds of the airplane, which is shown in Fig. 8.2.

It is possible to allocate the following areas on the diagram:

Area located between curve

, curve of ceilings, limitations of maximum speed on limit dynamic pressure

and

with an abscissas axis is the area of steady-state flight modes and modes of flight with acceleration and deceleration.

Fig. 8.2The following factors are taken into account while selecting

:

a) Aerodynamic heating;

b) Stability and controllability;

c) Vibration;

d) Flutter critical speed;

e) Reliability of engine operation etc.

The most exactly value of

is determined by airplanes flight tests. At absence of flight tests data it is possible to accept

for supersonic airplanes.

The following recommendations can be used for selection of maximum dynamic pressure:

- For maneuverable airplanes

;

- For heavy airplanes

.

The shaded zones correspond to the second modes, on which the long duration flight is not recommended.

2. The area

- an area of forbidden flights, since the given area is limited at the left by the curves limiting possible flight modes on maximum dynamic pressure

and

.

3. There is an area of inaccessibility on the right and above the line

. In this area the airplane can not get because of the limited capability of the power plant (engine) and presence of limitations put on maximum speed of steady-state horizontal flight

.

4. Unsteady controlled flight with

and

is possible in the area between curve

, curve of ceilings and curve

, and in the area between curve

and

- unsteady controlled flight with

is probable.

5. Area between curves

,

, ordinate axis and axis of abscissas is the area of uncontrolled ballistic flights.

The airplane has optimal values of speed and altitude corresponded to maximum power altitude in point of contact of a line of equal energy and curve

. For reaching maximum ballistic and dynamic altitudes it would be necessary to begin executing "zoom" (climb) with speed

, being on altitude

. But as the point

, in this case, is in the field of forbidden flights, climb will begin with parameters corresponded to point

.

At absence of power losses

the ballistic ceiling will be defined by point

, dynamic ceiling - by point

. However, flight with

can not be realized, because there will always be power losses due to action of external forces on a flight path. That is why the curve

is created with taking into account power losses and both the ballistic and dynamic ceilings will be determined correspondingly by points

and

.

So:

Point

corresponds to the conditional ballistic ceiling;

Point

corresponds to the conditional dynamic ceiling;

Point

corresponds to the ballistic ceiling;

Point

corresponds to the dynamic ceiling.

In one word, having constructed the given diagram, it is possible to define the range of speeds and altitudes, where uniform steady-state horizontal flight, flight with acceleration or deceleration, unsteady flight with

and

, area of undesirable and prohibited flights, area of uncontrolled ballistic flights and area of inaccessibility are probable.

The flight and power airplanes characteristics obtained as the results of calculations are compared to the qualifying characteristics for the given class of airplanes. The conclusions about suitability of the given configuration are made and the ways of the airplane refinement are planned on the grounds of this comparison.

Climb and descent of an airplane. Equations of motion on sloped path. Required speeds and thrusts for flight on sloped path.Airplane center of mass movement can be performed on straight-line paths having various flight path angles. The flight path angle

is considered positive, if speed vertical component is directed upwards and corresponds to altitude increasing during flight. The flight mode with positive flight path angle is called climb.

Climb is performed for the different purposes:

a) Climb up to optimal altitude for flying over distance. In this case climb has to satisfy the requirement of minimum fuel consumption per kilometer. It is characteristic for heavy airplanes - passenger, transport and bombers;

b) Climb up to fighting altitude for the shortest time, it is characteristic, on the whole, for interceptors;

c) Climb up to static ceiling, it is characteristic, in general, for maneuverable airplanes;

d) Climb up to dynamic ceiling for expansion of fighting capabilities for maneuverable airplanes.

The mode of the airplane descent is characterized by a negative flight path angle. The descent at a thrust close to zero or at a zero thrust of engines is called as planning, and on very steep path - dive.

Let's record equations of motion in a vertical plane (4.1, 4.2):

,

.

If we take into account, that the angle

is usually very small (

), we shall receive:

,

.

(8.1)

Having divided the right and the left-hand parts of equations of a system (8.1) by

, we shall receive equations of motion of an airplane in a vertical plane in g-loads:

,

.

(8.2)

The diagram of forces which are acting aboard the plane in sloped flight is shown on fig. 8.3.

Fig. 8.3A linearity condition of flight on sloped path is the equality

.

Having supposed, that in sloped straight flight the angles of attack @ are small, we shall accept equations (8.1) and (and rplane, which is shown in __________________________________________________________.2) in the following form:

,

.

(8.3)

,

.

(8.4)

From the equation being a condition of a flight path linearity (the second equation of system 8.3 or 8.4) while climbing, it follows, that the lift should be less than airplane, i.e.

.

It follows from the first equation of system (8.3) that:

- climb with acceleration;

- steady-state climb;

- climb with deceleration.

The decelerating, accelerating or steady-state climb can be realized by the pilot by a engine thrust variation or change of the angle of attack of the airplane by deflection of an elevator.

For the airplane steady-state climb with the speed constant on size and direction the equation of motion will have the following form:

,

.

(8.5)

It is possible to define speed, required for airplane flight on sloped path from the second equation of system (8.5)

:

(8.6)

or

.

(8.7)

If we compare this speed with speed, required for horizontal flight at identical values of nominal load

, air density

and lift coefficient

, then it is easy to mention, that

.

(8.8)

Thus, the speed, required for sloped path flight with other identical parameters is less than horizontal required speed.

At

, what, in the main, takes place at sloped flights, it is possible to accept the amount of

, close to 1 and to assume, that the speed, required while sloping path flight, differs a little from required speed at horizontal flight, if other conditions remain identical, and to consider the following ratio as acceptable:

.

Let's express an airplane drag force

through its lift

:

.

Taking into account, that

, we shall receive

.

At small values of the angle

it is possible to consider, that the airplane drag force in straight sloped flight differs a little from the drag force at horizontal flight, and to assume

.

The adopted assumptions allow to identify the drag force relation to the speed in horizontal and sloped flights.

The amount of the required thrust for steady-state flight on sloped path is determined from the first equation of a system (8.5):

.

(8.9)

It follows from a ratio (8.9), that at flight with climb the engines thrust should be more than drag (thrust in horizontal flight) on size of weight force component

. At decent, the required thrust becomes less than drag force, as in this case

. At

, the required thrust will be equal to the required thrust of horizontal flight, i.e. to the drag force.

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