BEST NOTES

1. EQUATIONS OF MOTION
Body - fixed Coordinate system
Thus coordinate system is having origin at the center of gravity of the aircraft in which x- axis has been
taken along the length of the aircraft i.e. along the length of fuselage in the airplane’s plane of symmetry;
y-axis has been taken as perpendicular to the aircraft’s plane of symmetry. It has been denoted as (xb, yb,
zb) as shown below.
Earth - fixed Coordinate system
This coordinate system is fixed to Earth and with respect to this coordinate system aircraft is moving. It is a
fixed frame and coordinates on Earth with x-axis pointing north, y-axis pointing east and z-axis pointing
downwards. It is denoted as (xf, yf, zf) as shown above.
Atmosphere - fixed Coordinate system
This coordinate system is always parallel to Earth – Fixed Coordinate system which moves at a constant
velocity relative to Earth – Fixed Coordinate system. When t = 0, then Atmosphere – fixed Coordinate
system coincides with Earth – Fixed Coordinate system.
Here, Velocity of the body – fixed frame relative to the Earth – fixed coordinate frame which is also
known as ground speed i.e. speed of aircraft with respect to the ground.
V Velocity of the body – fixed frame relative to the atmosphere – fixed coordinate frame which is also
known as air speed.
Vw velocity of the atmosphere relative to the Earth which is also known as wind speed.

2. Airspeed and Ground speed are related as –
Total airspeed is expressed in terms of body - fixed coordinate system as –
Similarly, angle of attack( and side slip angle(β) can be expressed as –
Similarly,
EULER ANGLE FORMULATION
The position of the aircraft is specified by the location of the origin of the body - fixed coordinate frame
relative to the Earth – fixed coordinate frame.
 The orientation of an aircraft relative to the Earth can be described in terms of what are called
EULER ANGLES.
 The orientation of the body fixed coordinate frame (xb, yb, zb) relative to the Earth fixed coordinate
frame (xf, yf, zf) is described in terms of three consecutive rotations through three Euler angles, in
the specific order as follows –
 Rotate the (xf, yf, zf) coordinate system about Zf axis through an angle Ѱ to the
coordinate system (x1, y1, z1)
 Rotate the (x1, y1, z1) coordinate system about y1 axis through an angle ϴ to the
coordinate system (x2, y2, z2)
 Rotate the (x2, y2, z2) coordinate system about x2 axis through an angle ɸ to the
coordinate system (xb, yb, zb)
 The three Euler angles ɸ, ϴ and Ѱ are respectively, called the bank angle, the elevation angle and
the azimuth angle or heading.
 There is a subtle but very important difference between the Euler angles and roll, pitch and yaw.
 Roll, Pitch and Yaw are orthogonal whereas, the Euler’s angles are not.