LECTURE 3

Coordinate systems used in flight dynamics

For the characteristic of coordinate system it is necessary to set coordinate origin po-

sition some basic direction and the basic plane. All coordinate systems used in a flight dy-

namics are right handed one.

For setting equations of motion in projections the moving coordinate systems are

used, origin of which is located in the centre of weights of an aircraft.

Normal, body-axes, wind and trajectory coordinate systems are relative to such mov-

ing coordinate systems.

1. Normal coordinate system (Fig. 3.1).

The vertical axis is directed along the prolongation of radius-vector , which

is connecting the centre of the Earth with the aircraft weights centre. The basic plane

coincides with local horizontal one, i.e. plane passing through the point and per-

pendicular to axis . The axis is directed to the north parallel to tangent to the geo-

graphical meridian, the axis is located parallel to tangent to a geographical parallel in a

direction from west to east.

Fig. 3.1

2. Body-axes coordinate system

The basic plane (Fig. 3.2) is a plane of symmetry of an aircraft. The axes of a

body-axes coordinate system coincide with the longitudinal , the normal and the

transversal axes of an aircraft. The direction of longitudinal axes can be chosen ac-

cording to the basic axes of the aircraft or according to the projection of a mean aerody-

namic chord to a plane of symmetry of a aircraft, or along the main longitudinal axis of in-

ertia of an aircraft. The position of the longitudinal axis should specially be stipulated. The

connected system is rigidly fixed with respect to the aircraft and its position with respect to

normal system determines attitude of an aircraft. It is characterized by Euler yaw, pitch

and roll angles.

The yaw angle is the angle between axis of normal coordinate system and

projection of longitudinal axis on a local horizontal plane of normal coordinate

system. Yaw angle is positive, when axis is combined with the projection of a longi-

tudinal axis to horizontal plane by rotation around the axis counter-clockwise if to

look from the end of this axis.

Pitch angle is the angle between a longitudinal axis of an aircraft and local

Fig. 3.2

horizontal plane. It is positive, when the longitudinal axis is above the horizontal plane.

Roll angle is the angle between a transversal axis and axis of normal co-

ordinate system removed in the horizontal plane in position appropriates to zero yaw angle,

or between a normal axis and local vertical plane containing a longitudinal axis .

Roll angle is positive, when the removed axis is combined with transversal axis by a

rotation round a longitudinal axis counter-clockwise if to look from the end of this axis. as

a rule, the connected system, is used at the analysis of the aircraft attitude. The aerody-

namic forces (longitudinal, normal, transversal) and moments (roll, yaw, pitch) which acts

on aircraft in flight of this system can be set in projections on these axes.

3. Wind coordinate system (Fig. 3.3).

This system is applied basically for determination of aerodynamic forces which are

acted on the aircraft. Therefore the basic direction in this system - direction of air speed of

an aircraft , i.e. the speed of an aircraft relative to air environment. If the air is motion-

less, the air speed coincides with the Earth. At presence of wind, having speed relative

to the Earth:

.The wind axis (Fig. 3.3) is directed along air speed of an aircraft, the axis

of lift lies in a plane of symmetry of an aircraft and is directed to a top of an aircraft, Fig. 3.3

the lateral axis forms the right of coordinate system with axes and .

The position of an aircraft relative to air flow determining quantity of aerodynamic

forces , is set by two angles, and , which describe relative attitude of the connected

and wind coordinate systems. The angle of attack is the angle between a longitudinal

axis of an aircraft and projection of air speed on an aircraft symmetry plane.

The yaw angle - is the angle between vector of air speed and an aircraft symmetry

plane.

Usually when determining angles of attack and sliding, the body-axes coordinate

system is used, the axis is oriented along a projection of wing mean aerodynamic

chord (MAC). In relation to normal coordinate system the wind coordinate system is ro-

tated on angles , and : the wind yaw angles, pitch angles and roll angles have the

same definitions as the Euler angles , and in the connected system (fig. 3.4).

4. Trajectory coordinate system

Analysing an aircraft motion reflective to the Earth it is convenient to build a system

of coordinates on the basis of the earth rate instead of air speed .

Fig. 3.4

The appropriate coordinate system is called the trajectory coordinate system.

The trajectory coordinate system is shown in (Fig. 3.5).

Axis coincides with flight vehicle earth speed direction . Axis lies in a

local vertical plane which goes through axis and is directed upwards from the Earth

surface. The axis forms the right system of coordinates. Compared to normal system

the trajectory coordinate system is moved at angles and .

Angle of the path is the angle between projection on local horizontal plane (travel-

ling speed ) and axis ..

.Flight pass angle is formed by earth speed direction

(axis ) and local horizontal

plane .

If the wind is absent axis also coincides with axis and the angle of the path

coincides with the speed angle , the flight pass angle is equal pitch angle .In the

case of the straight-line flight, without rolling and sliding and at the absence of wind

between angles and the following simple ratio (fig. 3.6) is obtained.

Fig. 4.1

. (3.1)

As we have already marked, for derivation of equations of motion of an aircraft it is

often necessary to recalculate the projections of vectors (speeds, forces etc.), given in one

coordinate system, to the projections of the same vectors in other coordinate system. The

most convenient way to do this transition is to use tables of direction cosines.

Let's look at the direction cosines for transition from normal, connected and wing coordin-

ate systems to trajectory coordinate system.

Normal coordinate system

Body - axes coordinate system

Fig. 3.6

Wind - axes coordinate system

Motion equations of an aircraft as a material point generally.

As we have already mentioned the following external forces act on an aircraft during its

flight, in general case: They are: mass forces directed vertically along axis , aerody-

namic force, which can be presented by three terms wind - axes coordinate system axis, lift

force , drag force and lateral force .

And at last it is the engine thrust force , which is assumed to be directed along engine

axis generally inclined to wing chord at angle .

Let's make up aircraft motion equation in the projection on trajectory coordinate sys-

tem axis. For this purpose it is necessary to find projections of all external forces to the ap-

propriate axes. It is known that force is the product of weight on acceleration. Now it is ne-

cessary to find formula for acceleration on tangent along axis trajectory , on normal to

trajectory (along axis ) and on axis .

The acceleration of motion on tangent to flight trajectory (tangential acceleration) is

equal to derivative .

The force which is directed along axis (flight speed) will be represented by pro-

jections sum of all external forces on tangent to flight trajectory (on axis ). Taking into

account, that at wind absence axis and , coincide, , and using

tables of direction cosines, we shall obtain:

. (3.2)

The motion acceleration on normal to trajectory (on axis ) is centripetal acceleration.

We know from mechanics that it is determined by the expression:

, (3.3)

Where is the speed of flight, is the radius of trajectory curvature .

Analysing an aircraft motion it is convenient to express centripetal acceleration through

angular velocity of trajectory rotation . It is easy to make sure that at any moment of

time the equality takes place:

. (3.4)

That is why the centripetal acceleration may be represented as:

.

Quantity and sign of centripetal acceleration will be determined by quantity and sign

of all forces projections operating the flight vehicle on flight trajectory normal axis .

Projecting forces on axis we shall obtain the second equation of the longitudinal mo-

tion.

. (3.5)

The third equation is obtained in the similar way.

. (3.6)

These motion equations are set up without taking into account diurnal rotation of

the Earth and its surface curvature, because the items which take into account these factors

at actual values of flight speeds appear negligible ( especially in planes).

To the above-stated equations it is necessary to add kinematic relations. The basic kin-

ematic relations dealing with high, speed and flight pass angle is obvious and looks like

this:

. (3.7)

These equations are set up for the most common cases of the aircraft flight along

spatial trajectory when the angle of roll and sliding angle are available. The research of

such spatial motion by means of these equations presents a difficult problem, because these

equations are non-linear differential equations and the integration generally can be pro-

duced only by means of numerical integration methods.

The research of simple kinds of motion is of significant interest. It occurs when the

airplane goes along the trajectory lying in one plane, for example, in a horizontal or ver-

tical plane. For such flat motions the equation of motion becomes simple.