Thesis - Beaver simulation
.pdfa(.)
derivative with respect to dimensionless sideslip rate,
derivative with respect to angular deflection of ailerons, -a (.I
a%
derivative with respect to 6;a , -a (.I
a @,*a)
derivative with respect to angular deflection of elevator, -a (.I
a%
derivative with respect to 6;f12, -a o
ac6..pP)
derivative with respect to angular deflection of flaps, -a (.) as,
derivative with respect to angular deflection of rudder, -a (.I
a 4
derivative with respect to |
a ( 1 |
, - |
|
|
a( 6 , 4 |
derivative with respect to dimensionless pressure increase across
the propeller, (+)a ") = -a (') a dpt
a
0.6 Superscripts.
T |
transpose of matrix or vector |
|
derivative with respect to time |
0.7 Abbreviations.
AACS |
Automatic Aircraft Control System |
CD |
Course Datum |
c.g. |
centre of gravity |
DHC |
De Havilland of Canada Ltd. |
DME |
Distance Measuring Equipment |
DUT |
Delft University of Technology |
fpa |
flight path acceleration |
FCC |
Flight Control Computer |
FCS |
Flight Control System |
ILS |
Instrument Landing System |
NLR |
Nationaal Luchten Ruimtevaartlaboratorium (Dutch Aerospace |
|
Laboratory) |
ODE |
Ordinary Differential Equation |
SA |
Standard Atmosphere |
STOL |
Short Take Off and Landing |
TAS |
True Airspeed, V |
VOR |
Very high frequency Omnidirectional Range |
0.8Reference frames and sign conventions.
0.8.1Definitions.
The definitions of the reference frames used within this report QW given below. FMand F, have been added to this list, although they will only be used in table 2-2 in section 2.2.2. The equations for translational and rotational velocities are referenced to the body axes FB.The aircraft attitude is defined by the Euler angles 9,0, and cp, for which purpose the vehiclecarried vertical reference frame Fv is introduced. The aircraft position is defined with respect to the earth-fixed reference frame FE.
Measurement reference frame F M.
The measurement reference frame OMXMYMZMis a left-handed orthogonal reference system, used for correcting the stability and control derivatives of the aircraft if the c.g. position differs from the one used during the flight tests. For the 'Beaver' aircraft, the origin OMlies in a point, resulting from the perpendicular projection of the foremost point of the wingchord parallel to the OBXBZB-plane,on this OBXBZB-plane(ref.[30]). The XMOMZM-plane coincides with the OBXBZ,-plane. The positive XM-axispoints backwards, the positive Y,-axis points to the left, and the positive 2,-axis points upwards.
The body-fixed reference frame F ,.
The body-fixed reference frame of the aircraft is a right-handed orthogonal system OBXBYBZBThe. origin 0, lies in the centre of gravity of the aircraft. The XBOBZBplane coincides with the aircraft's plane of symmetry if it is symmetric, or is located in a plane, approximating what would be the plane of symmetry (ref.[ll]). The XB-axis is directed towards the nose of the aircraft, the YB-axis points to the right wing (starboard), and the 2,-axis towards the bottom of the aircraft. The positive directions forp, q, r, u, v, w, F,, F,, F z , L, M, and N are shown in figure 0-1.
The special body-fixed reference frame F for the 'Beaver'.
The body-fixed reference frame F,, which is defined specially for the 'Beaver' aircraft, is identical to FB, with one exception: the origin OR is placed in a body-fixed reference point, which has been selected to coincide with a c.g. position which was actually used during one flight. It has the following coordinates in FM:x = 0.5996 m, y = 0 m, z = -0.8815 m (ref.[30]). FRis used only to define the moments and products of inertia for the aircraft condition on which the aerodynamic model is based (see table 2-2 in chapter 2).
The stabilitv reference frame F s.
The stability reference frame Os&YsZs is a special body-fixed frame, used in the study of small deviations from a nominal flight condition. The reference frames FBand Fs differ in the orientation of the X-axis. The &-axis is chosen parallel to the projection of V on the OBXBZB-plane(if the aircraft is
Symbols and definitions. |
9 |
Figure 0-1. The body-axesreference frame
directions of the body-axesforces F,, F, ,F, ,moments L, M, N, velocities u, v, w, and rotational velocities p, q, r.
symmetric, this is the plane of symmetry), or parallel to V itself in case of a symmetrical nominal flight condition. The Ys-axis coincides with the Y,-axis.
Flight path reference frame F.,
The flight path reference frame OwXwYwZw,also called the wind reference frame, has its origin a t the c.g. of the aircraft. The Xw axis is aligned with the velocity vector of the aircraft. -&-c&ncide6-&.t*
The 2, and Zs-axes are parallel.
The earth-fixed reference frame F ,.
The earth-fixed reference frame, also called the topodetic reference frame [ll], is a right-handed orthogonal system OEXEYEZE,which is considered to be fixed in space. Its origin can be placed a t a n arbitrary position, but will be chosen to coincide with the aircraft's centre of gravity a t the start of a flight test manoeuvre. The &-axis points downwards, parallel to the local direction of the gravitation. The XE-axis is directed north, the YE-axis east. For the simulation of ILS approaches a runway-fixed reference frame will be introduced, see section C.2 of appendix C. A beacon-fixed reference frame is used for simulating VOR navigation, see section C.5.
The vehicle-carried vertical axis reference system F ,.
The vehicle-carried vertical axis system OvXvYVZvhas its origin a t the c.g. of the aircraft. The Xvaxis is directed to the north, the Yv-axis to the east, and the &-axis downwards (parallel to the local direction of gravity).
The runway-fixed reference frame F ,.
In appendix C, a runway-fixed reference frame O&FYFZF will be introduced. The origin 0, is located a t the point of intersection of the runway treshold and runway centerline. The XF-axis is directed along the runway centerline, in take-off and landing direction, YF points to the left (as seen from a n approaching aircraft), ZF points downwards.
Figure 0-2. Relationship between the vehicle-carriedvertical reference frame Fv and the earth-fixedreference frame F,
0.8.2 Relationships between the reference frames.
In figure 0-2 the relationship between the earth-fixed and vehicle-carried vertical axis sytems is shown. FE and Fv differ only in the position of their origins. The relationship between the vehicle-carried vertical and body axes is shown in figure 0-3. The Euler angles v, 0, and cp define the orientation of F, with respect to Fv, hence they define the attitude of the aircraft with
Symbols and definitions. |
11 |
respect to the earth. The transformation matrices expressing each of the Euler rotations separately are:
r
1 0 0
0 coscp sincp
0 -sincp coscp
The total transformation matrix from Fv to FBthen becomes:
,T = @+*l= $J |
|
|
cosq cos 8 |
s i n q cos 8 |
- sin0 |
cosq s i n e sincp -s i n q cosrp |
s i n q s i n e sincp + cosq coscp |
cose sincp |
c o s ~ s i n 8 c o s r p + s i n ~ s i n c psinqsin8coscp-cos~psinrp |
cosecoscp |
|
A
(0-4)
so the relation between a vector yBin FBand yv in Fv is:
The orientation of the flight path axes with respect to the vehicle-carried vertical axes can also be expressed in terms of Euler angles (x,y and p). This is shown in figure 0-4.
The relationships between the body, flight path, and stability axes are shown in figure 0-5. All three axis sytems have their origin a t the aircraft's centre of gravity. The Xwaxis is aligned with the velocity vector of the aircraft. The orientation of the flight path axis with respect to the body axes is defined by the angle of attack a and the angle of sideslip f3. The stability axis reference system is displaced from the flight path axis system by a rotation and from the body axis sytem by a rotation -a.
0.8.3 Sign conventions for deflections of control surfaces.
Figure 0-6 shows how the positive deflections of the control surfaces are measured. The positive elevator deflection is measured downwards: a positive 6, gives a pitch-down movement of the aircraft. The rudder and ailerons deflections are positive if they force the aircraft to move leftwards. If one aileron deflection has a positive sign, the other is negative. The 'total'
aileron deflection is therefore defined as: 6, = 6,,efl.The positive flap deflection is measured downwards, similar to the elevator deflection. A positive 6/ corresponds to a n increase in lift.
- A rotation by about the &axis to the intermediate position X'Y'Zv.
- A rotation by 8 about the Y-axis to the intermediate position XB'rZ'.
- A rotation by cp about the &-axis to the fmal position XBYBZB.
Figure 0-3. Relationship between the vehicle-carried vertical reference frame F, and the body-fixed reference frame FB.
Symbols and definitions. |
13 |
- A rotation by x about the +axis to the intermediate position X'Y'G.
- A rotation by y about the Y-axis to the intermediate position XwYZ'.
- A rotation by p about the Xw-axis to the final position XwYwZw.
Figure 0-4. Relationship between the flight path reference frame Fw and the body-fixed reference frame FB.
Figure 0-5. Relationships between body-kued reference &ame FB, flight path reference frame Fw,and stability reference frame FS.
Figure 0-6. Sign conventions for control surface deflections.
Symbols and definitions. |
15 |