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Eulcr Rate Eqruktion
DCM
Quatcmions
Fig. 4.14 Euler angles (SZ = 30 deg/s, a = 60 deg).
Eqs. (4.122-4.124) are satisfied. Substituting, we find
Cizl + C~2 + C321 = 1.00004
C1?2 + C~2 + C~2 = 1.000006
C~3 + Cln + C~3 = 0.999996
Hence, the calculated values of missing elements are satisfactory.
4.3 Equations of Motion
The equations governing the aircraft motion are based on Newton's laws of
motion. We have the force and moment equations in the form
(4.223)
F = m'(ddV, ).
M = (ddH)
(4.224)
Eakr Rrite Eqr=tion
DCM
Qurkrmon
360 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
where V is the velocity and H 'is the angular momentum. The suffix / implies that
the acceleration dV/dt and the rate of change of angular momentum dHldt are
supposed to be measured with respect to an inertial frame of reference. We have
H -. It.o where I is the moment ofinertia of the body and co is the angular velocity
of the body with respect to an inertial system. Then Eq. (4.224) takes the form
M -.
GI),, +i (dd ),
(4.225)
If a space-fixed inertial system of reference is used to compute the angular mo-
mentum, the moment ofinertia I will continuously vary with time as the aircraft
translates and rotates in space. As a result,it will be extremely difficult to solve the
equations of motion because, at each time step, we have to evaluate the moment
o~:nertia I and its time derivative (dl/dt),. For the aircraft, I is a dyad with nine
components, three principal moments of inertia lxx, Iyy, Izz and six products of
inertia Ixy, Iyx, Iyz, Izy, Ixz, Izx.
One way of o'verconung this difficulty of computing time-varying moments
and products of inertia is to use a moving or body axes system, which is fixed
to the aircraft all the time and moves with it. Then the moments and products of
inerLia calculated with respect to this axes system will be constant except for such
variations as fuel consumption or control surface detlections that can be easily
accounted for.
Theintroduction ofa moving axes system avoids the problem ofcomputing time-
varying moments and products ofinertia but creates another problem because the
accelerations measured in the moving coordinate system are not the accelerations
with respect to an inertial frame of reference. Fortunately, this problem is a simpler
one to solve, and the following theorem of vector analysis helps us obtain the
accelerations with respect to an inertial frame of reference given the accelerations
in a moving coordinate system.
4.3.7 Moving Axes Theorem
Let Ab be a vector observed in the moving axes system Oxt,ybzb and (dAldt)b
be the time rate of change of Ab recorded in the moving axes system. Furthermore,
let tOlb be the angular velocity of the moving axes system measured with respect
to an mertial system but having components in the body axes system. As said
before, the order of the subscripts and superscript is as follows: the subscripts
/, b have the meaning: of the body axes with respect to the inertial axes system
and the superscript b means that the vector /3 iS resolved or has components in
the body_axes system. The problem is to deternune the time rate of change of a
d_
vector A in the inertial frame of reference OxiyLz, given its time rate of change
in the body axes system Oxbybzt . This can be accomplished using the following
theorem known as the moving axes theorem:
(-,,)t = ( d ) +COl,b X Ab
(4.226)
To understand the concept of moving axes theorem, consider an inertial frame
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