Thesis - Beaver simulation

Appendix A. Nonlinear equations of motion for a rigid aircraft.

A1 Introduction.

The general nonlinear equations of motion for a rigid aircraft in steady atmosphere will be established in this appendix. There are many textbooks on the matter, see for instance refs.[ll], [12], [17], [22], [23], or [29]. Still, a thorough knowledge of the general equations of motion is considered to be such vital for any discussion about the implementation and use of aircraft models, that a complete derivation of the equations of motion will be presented here. One should particulary take notice of all the assumptions which shall be made during the derivation.

Am2 Derivation of the equations for velocities and rotational velocities in the body-fixed reference frame.

A.2.1 General force equation for a rigid body.

Consider a mass point 6m that moves with time varying velocity V (with respect to a right-handed orthogonal reference frame OXYZ) under the influence of a force 6F.Applying Newton's second law yields:

For a rigid body it is possible to write:

in which the contributions of all mass points are summed across the rigid body. Let the centre of gravity of the rigid body have a velocity V,., with components u,v, and w along the X, Y, and Z-axis of the right-handed reference frame. The velocity of each mass point then equals the sum of V, and the velocity of the mass point with respect to the centre of gravity. If t%e position of the mass point with respect to the c.g. is denoted by the vector r , the following vector equation is found:

so:

The equation for the resulting moment G, about the c.g. becomes:

where h denotes the resulting angular momentum of the body about the c.g.

A d 3 Angular m o m e n t u m a b o u t the centre of gravity.

If the body has a n angular velocity o, with components p , q, and r about the X, Y, and Z axes of the right-handed reference frame respectively:

(A-13)

(where i, j, and k are unity vectors about the X, Y, and Z axes), the velocity of a mass point of the rotating body becomes:

hence, the angular momentum of the rigid body about the c.g. can be written as:

The first term of the right hand side of (A-15) can be written as:

( C r a m ) x vCeg= .o

and for the second term we can write:

C r x ( o x r ) G m = C ( o ( r * r )- r ( o * r ) ) G m= C ( o r 2 - r ( o - r ) ) 6 m (A-17)

Substitution of r = i x + j y + kz, (A-16),and (A-17)in (A-15)yields:

(A-18)

Let the body now be a n airplane that is considered to be rigid. Applying (A-20) to (A-6) and (A-12), we find:

and:

The force and moment equations (A-6) and (A-12) can be written out into their components along the XB , YB, and ZB-axes (i.e. F = i F, + j F, + k F, and

+ k N), using equations (A-21) and (A-22). This yields:

F, = m ( ~ i r+ pu - q u )

and:

I t should be noted that the derivation of these equations is only valid when the following restrictive assumptions are made:

Assumptions 3 and 4 were made in the definition of the inertial reference frame in which the aircraft's motions will be considered. The description of the vehicle motion under assumptions 3 and 4 is accurate for relatively short-term guidance and control analysis purposes. I t does have practical limitations when very long term navigation or extra-atmosphere operations are of interest [23].

For aircraft flying in steady atmosphere, the forces about the XB,YB,and ZB- axes will be divided in three components:

In principle, it is possible to add other contributions, such as forces coming from the landing gear if the aircraft is taxiing, but only the contributions mentioned above will be considered here.

Gravity does not contribute to the moments about the body-axes, because the centre of gravity is located in the origin of this reference frame. So the moments can be divided in contributions from aerodynamic properties of the aircraft (subscript a)and contributions from operation of the engine (subscript t). With this division of the forces and moments in different components, and using (A-19), the equations of motion (A-23) can be written as:

(see ref.[ll]). It is possible to move the derivatives of the motion variables to the left hand side of the equations if the forces and moments don't contain any contributions from derivatives of the velocities u, v, and w, and/or the angular velocities p, q, and r. This yields a set of Ordinary Differential Equations (ODES). If the forces and moments do contain such time derivatives, a set of implicit differential equations results. For instance: for the 'Beaver' aircraft, the time derivatives of u, u , and w still appear on the right hand side of the v - equation, since a stability derivative to f3 is present in the aerodynamic model which determines the side force Ya, see ref.[25]. We get the following expressions: