Electrical Circuit Equations (13)
Since must be real we see that the coefficients must be
complex. If we let
where are real-valued, then we have
and all terms in Equation 17 are real. Therefore we see that the
two real-valued fundamental solutions associated with complex
x
21 ,cc
2/)(
2/)(
432
431
iccc
iccc
−=
+=
43 ,cc
Eq’n 17
Electrical Circuit Equations (14)
Using Equation 17, we determine the coefficients from initial
conditions:
Using Equations 15 :
43 ,cc
0
Electrical Circuit Equations (15)
L
R
C
i
i
)(tV
0
)0( VV =
In summary, dynamics of the circuit to the
right are defined by the equations
where voltage over the capacitor
and current through the inductor,
4=R
1=L
16/1=C
=
1
x
=
2
x
If , and then
Electrical Circuit Equations (16)
L
R
C
)(tV
and
where
Electrical Circuit Equations (17)
A 2-state system with complex eigenvalues
and eigenvectors has just one dynamic
R
1
i
Systems of First Order ODEs
Linear Time Invariant Example:
Aircraft Longitudinal Dynamics
Aircraft Dynamics Example
Aircraft Dynamics: Equilibrium
Lift
Drag
Thrust = Drag
T
0
D
Aircraft Dynamics: Perturbations
Longitudinal perturbations from straight and level flight:
u
V
Aircraft
centerline
Aircraft Longitudinal Dynamics:
Linearized Equations of Motion
The Supplement to Class 18 Notes defines terms, derives the linear
equations and estimates numerical values. The linear equations are
L
m
CSV
L
2
0
=
D
V
g
Du
V
g
L
D
dt
du
−−+−=
0
)(2
Linearized Equations of Motion
For Example Aircraft
−−−
u
DVgDVgLD
u
0)/()/)(/(2
0000
−−−
u
u
00235.0310.00101.0
xAx =
Aircraft Dynamic Modes
Aircraft longitudinal dynamics captured in these equations exhibit two
modes:
These modes relate to eigenvalues and eigenvectors of the system of
ODEs.
Phugoid Mode
The natural frequency of the phugoid oscillation is approximately
and the “damping ratio”
The natural frequency for an aircraft traveling 350 nautical miles per
Phugoid Mode (2)
The phugoid motion (vertical scale exaggerated):
Note the zero angle of attack.
Short-Period Mode
The short-period mode has natural frequency and damping factor
approximately given by
In the short-period mode the aircraft oscillates in the pitch plane
without changing flight path (approximately).
Short-Period Mode (2)
Short-period motion (approximate):
Homework Assignment 18
Read:
Chapter 8, Sections 8.5.1.2 and 8.5.1.3