Thesis - Beaver simulation

and:

The equations contain no assumptions of either symmetric mass distributions or aerodynamic properties and are therefore applicable to asymmetric aircraft as well as to conventional symmetric aircraft. Table A-2 gives the definition of the inertial parameters P,, ,P ,...,R, [ll].The notation is analogous to the notation that is used in ref.[6f

A3 Using the angle of attack, sideslip angle, and total airspeed in stead of the velocity components in FB.

A.3.1 Why use flight-pathaxes for the state equations?

In aerodynamics, it is more convenient to use the airspeed V (TAS), the angle of attack a , and the sideslip angle P in stead of the components of the airspeed along the body axes. For solving the equations of motion, it is possible to calculate V, a, and P from u, u, and w or vice versa. In other words, both the body-axes velocity components and the flight-path axes V, a , and f3 can be used as state variables. The question is now which set of state variables is best suited for simulation purposes.

From a physical point of view it is logical to express the aerodynamic forces and moments in terms of the flight-path variables a , P, and V. A difficulty may arise because the aerodynamic forces and moments themselves can contain

symbol definition

Table A-2( i ) . Definition of inertia parameters. (See also next page).

definition

Table A-2(ii) . Definition of inertia parameters. See previous page for definition of II I,I,, ... ,16.

Substituting the definitions (A-26)for u, u, and w,and cancelling terms yields:

If we now substitute the equations for z i , v , and w in (A-25a),the terms involving the vehicle rotational rates p, q, and r are identically zero, and the resulting equation becomes:

+ (Zgr+ Zt + Za)sin a cos p ]

and using the lift and drag:

A.3.4 Derivation of the &-equation.

Using (A-27), the time derivative of a can be written as:

& = uw - ziw u 2 + w 2

(see ref.[25]). Substituting for u and w and using:

u 2 + u 2 = v2- u 2 = v2(1- sin2p) = v2cos2p

yields :

a= wcosa - u s i n a

vcos p

Substituting for w and u and rewriting terms now yields:

If we substitute equations (A-26),many terms can be cancelled, and we find:

- (Zg,+ Z t+ Z a ) s i n a 11 + p s i n a - rcosa

which yields:

Am4 Equations of motion in a non-steady atmosphere.

The equations of motion (A-25)are valid only if the components u, u, and w of the translational velocity VC.+,of. the aircraft's centre of gravity are measured relative to a non-rotating system of reference axes having a constant translational speed in inertial space. Under the assumptions 3 and 4 of section A.2.4, it is possible to select a reference frame that is fixed to the surrounding atmosphere, as long as the wind velocity vector V, is constant. In that case, the components u, v, and w of the velocity vector V,. = Va express the aircraft's velocity with respect to the surrounding atmospgere.

However, if the wind velocity vector Vw is not constant during the time interval over which the aircraft's motions are studied, it is not possible to fix the frame of reference to the surrounding atmosphere. This situation can, for instance, arise during approach and landing of a n aircraft, due to variation of wind velocity with altitude. Again using assumptions 3 and 4 of section A.2.4, the most obvious choice of the reference frame in this case turns out to be the earth-fixed reference frame FE [16].

In the subsequent part of this section, the symbols u, v, and w will be reserved for the velocity components of the vector V,, i.e., the velocity of the aircraft's c.g. with respect to the surrounding atmosphere. The vector Vc.g=.Ve is used within the equations of motion. V, denotes the velocity of the aircraft's c.g. with respect to the earth-fixed reference frame. The wind velocity V, is also measured with respect to FE.Hence:

or:

where u, ,v , ,and w , are the components of V , along the body axes of the aircraft and u, ,v, ,and w eare the components of V , along FB.The equations of motion (A-23a)now become:

Fx = m ( u , + q w , - rue)

Fy = m ( v , + rue - p w , )

F, = m (we+ p v , - qu,)

We want to express the equations of motion in terms of V , , a, and f3 in the sense of equations (A-32),(A-38),and (A-44),because these three quantities are required for calculating the aerodynamic forces and moments. Expressions (A-26)and (A-27)are still valid if V is set equal to the velocity with respect to the atmosphere, V,. Rewriting (A-47)yields:

u = -Fx - q w , + rue - u, m

In a manner analogous to sections A.3.3, A.3.4, and A.3.5, expressions for the time derivatives of V, ,a, and

+ ( q w , - r v , + u , ) c o s a s i n p + ( p w , - r u , - Ij,)cosp +

+ ( p v , - q u w + w w ) s i n a s i n p + p s i n a - r c o s a

The differences between these expressions and (A-32), (A-38), and (A-44) can be modelled by adding a 'wind' component to the forces along the aircraft's body-axes. The total forces and moments along the body-axes now become:

(A-52)

where X,, Y,, and 2, represent corrections to the body-axes forces due to nonsteady atmosphere, according to the following equations:

Due to these additional force components, the responses of V, a, and f3 in nonsteady atmosphere will be different from the responses in steady atmosphere. The aerodynamic forces and moments can be expressed as functions of V, a , f3, etc., which implies that these forces and moments will also differ from the results that would have been obtained in steady atmosphere.

If the wind velocity or direction changes very fast, for instance in atmospheric turbulence, the aerodynamic model sometimes needs to be extended with terms, which bring into account aerodynamic lags, such as the gustpenetration effect, which is caused by the finite dimensions of the aircraft. This effect is described and modelled in ref.[24], but it will be neglected in this report I).

Appendix B outlines some common methods to model atmospheric turbulence.

The corrections (A-53) again contain terms, involving the vehicle rotational rates p, q, and r. Unlike the term u-q, the magnitude of uWsqwill not become large with respect to the normal acceleration F,/m, because the maximum windspeed under which it is allowed to fly is limited. Hence, expressions (A-53) can be used without any problems regarding computer precision. Often, windspeeds are expressed with respect to the earth-fixed reference frame, which means that a transformation of axes from FEto FBis necessary.

In ref.[24] the responses of an aircraft to atmospheric turbulence are modelled by adding'gust corrections' to u, a, and p, which means that a and P themselves are not measured relatively to the surrounding atmosphere as within this report. Furthermore, the most recent version of ref.[24] uses other expressions to model the gust penetration effect than earlier versions of these Lecture Notes, but it is not clear which version is the best. To model the gust penetration effect, knowledge about contributions of certain specific parts of the airframe to the stability derivatives is required, but this knowledge is not readily available. I t is possible to approximate these contributions, but this introduces errors. Also, care has to be taken if a nonlinecu*aerodynamic model is used, because the expressions are usually derived assuming linearity. For these reasons, this report will not model the responses to atmospheric turbulence in the fashion of ref.[24]. This is considered to be beyond the scope of this report. Modifications to the aerodynamic models might, however, be needed in the future.

A.5 Kinematic relations and transformation of gravity forces from earth-axes to body-axes.

A.5.1 The Euler angles.

The heading of the body reference frame with respect to the Earth-fixed reference frame is defined by the Euler angles v , 0, and cp, see section 0.5. In order to calculate the contribution of the aircraft weight to the external forces, it is necessary that these angles are known. The kinematic relations needed for this calculation are summarized in the followingformulas (see for instance refs.[12] or [17]):

0 = qcoscp - rsincp

A.5.2 Gravity forces in the body-fixedreference frame.

Using the Euler angles, the following expressions for the components of the aircraft's weight W = m-g in the body-fixed reference frame FB can be derived:

X , = - W*sinB

Yg,= W*cosesincp

2, = W~cosecoscp

A.5.3 The position of the aircraft.

The position of the aircraft with respect to the earth bounded reference frame is given by the coordinates x, ,ye ,and ze ,defined by the following equations:

i, = -u,sine + (u,sincp+w,coscp)cos~

(A-56)

which results after a transformation of axes from FBto FE:

(A-57)