Thesis - Beaver simulation

d. Gravity forces.

The forces due to gravity are calculated with the block from figure F-15:

Gravit.~Assuming. that the mass of the aircraft remains constant under the motions considered, the body axes gravity forces are calculated. If the mass is not constant, it is necessary to implement the mass as a n additional input to this block. Type help gravity for on-line help a t the MATLAB command line.

The variable m has been defined a t the Mask-definitions line:

See table F-3 for the definition of GM1.

e. Wind forces.

It is possible to use the nonlinear equations of motion in non-steady atmosphere, if additional forces due to the wind and turbulence are included. This is done with the following block (see figure F-16):

Fwind. This block includes contributions to the body axes forces due to non-steady atmosphere. Type help fwind a t the MATLABcommand line for on-line help.

The variable m has been defined a t the Mask-definitions line:

See table F-3 for the definition of GM1.

f. Total forces and moments.

The following block is used to create two vectors, with the total body axes forces and moments respectively:

FMsort. This block first separates the moments from the forces, and then adds all forces and moments to eachother. This block only functions properly if the aerodynamic forces and moments are expressed in their body axes components. If the aerodynamic lift, drag, and sideforce are used in stead, this block needs to be changed. It is also necessary to change this block if more contributions to the total forces and moments are included.

Type help fmsort a t the MATLABcommand line for on-line help. The internal structure of FMsort is shown in figure F-17.

p. Equations of motion,

The internal structure of the block called Equations of motion is shown in figure F-18. This block contains the subsystem State derivatives, which is shown in figure F-19.

First the twelve time derivatives of the state variables are computed. In order to determine the position of the aircraft relatively to the earth, the body axes velocities of the aircraft relatively to the surrounding atmosphere are calculated and added to the wind vecolity components in the body-axes, u, ,v, , and w, . Then corrections to make the implicit equation for P explicit are made, and some state derivatives are artificially set to zero if that is required by the user. Finally, the state derivatives are integrated with respect to time, which yields the state trajectories.

The integrator block is a standard SIMULINKblock, which uses the variable xinco as initial value of the state vector (xinco = x,). The Gain block multiplies the resulting vector x element-by-element with the vector xfix of length 12, which has elements that equal either one or zero. The gain vector xfix can be changed manually, or by calling the helpfunction FIXSTATE (section F.2.6), which makes it possible to fix one or more states, i.e., set the time derivatives of one or more states to zero. This may, for instance, be convenient if the user wants to neglect longitudinal or lateral motions. By default, xfix is set to 1, which is equivalent to xfiz = [ 1 1 1 1 1 1 1 1 1 1 1 1 ] T.

The equations for the time derivatives of V, a,and P are valid only if the aerodynamic forces and moments are expressed in terms of contributions along the body axes. If the aerodynamic lift, drag, and sideforce are used in stead, it is necessary to make some minor changes to these equations, see appendix A.

The subsystem Equations of motion consists of the following blocks:

1- State derivatives. This is a subsystem block, which contains a number of 'basic' blocks, needed to compute the time derivatives of the state variables, see figure F-19. The following 'basic' blocks are included:

1.1 - Hlpfcn. This block is used to compute some sines and cosines which are used more than once in the calculation of the different state derivatives. See figure F-20 for the internal structure of this block. Type help hlpfcn a t the MATLABcommand line for on-line help.

Inputvector: x

Outputvector: Y h l p

1.2 - Vabdot. This block calculates the time derivatives of V, a , and P. On-line help is available (type help vabdot a t the MATLAB com-

1.6 - x.yHdot.This block is used to calculate the time derivatives of the coordinates of the aircraft x, ,y e , and H (= --z,). Type help xyhdot a t the MATLABcommand line for on-line help. Figure F-25 shows the internal structure of xyHdot.

2 - xdotcorr. This block is used to made the implicit P-equation (section 2.2.3) explicit. This correction is valid for the 'Beaver', but if models of other aircraft which also exhibit implicit state equations are implemented in this stucture, it is relatively easy to change xdotcorr. Type help xdotcorr a t the MATLABcommand line for on-line help. The internal structure of this block is shown in figure F-26.

The block beta-dot correction contains the following expression:

where the variables b, S, m, and CYbdot have been defined in the Maskdefinitions line:

b = GMl(2); S = GMl(3); m = GMl(10); CYbdot = AM(2,19);

See tables F-1 and F-3 for the definitions of AM and GM1.

h. Other outputs.

Three other outputs blocks have been added to this list, to demonstrate the flexibility of the model. The blocks calculate flightpath variables and accelerations & specific forces, respectively:

fl~ath.This block calculates the flightpath angle y, flightpath acceleration fpa, azimuth angle X , and bank angle 0.Type help flpath a t the MATLAB command line for on-line help. The internal structure of flpath is shown in figure F-27.

Accel. This block can be used to calculate accelerations and specific forces (outputs of acceleratometers) in the aircraft's centre of gravity. Type help accel a t the MATLABcommand line for on-line help. The internal structure of accel is shown in figure F-28.