Lecture #13 Topics
• Define Trim
• Stability (Static & Dynamic)
• Develop Static Stability Criterion – for Passive
Control
FAR Part 25.161 Trim
General. Each airplane must meet the trim requirements of this section after being
trimmed, and without further pressure upon, or movement of, either the primary controls or
their corresponding trim controls by the pilot or the automatic pilot.
I.
Longitudinal trim.
A. A climb with maximum continuous power at a speed not more than 1.3 VSR1, with the
landing gear retracted, and the flaps
i.
retracted and
ii.
in the takeoff position;
B. Either
i.
a glide with power off at a speed not more than 1.3 VSR1 ,
ii.
or an approach within the normal range of approach speeds appropriate to the weight
and configuration with power settings corresponding to a 3 degree glidepath,
whichever is the most severe, with the landing gear extended, the wing flaps
i.
retracted and
ii.
extended, and with the most unfavorable combination of center of gravity position and
weight approved for landing;
C. Level flight at
i.
any speed from 1.3 VSR1 to VMO / MMO, with the landing gear and flaps retracted,
ii. And from 1.3 VSR1 to VLE with the landing gear extended.
Trim – Special Case of
Reference Condition
How many equations do we need to
characterize the reference set?
m (U 0  Q 0W 0  V 0 R0 )  mg sin  0  FAX 0  FPX 0
m (V0  R0U 0  P0W 0 )  mg cos 0 sin  0  FAY 0  FPY 0
m (W0  P0V 0  Q 0U 0 )  mg cos 0 cos  0  FAZ 0  FPZ 0


I xx P0  I xz  R0  P0Q 0   I zz  I yy Q 0R0  LA 0  LP 0
I yy Q 0   I xx  I zz  P0 R0  I xz  P02  R02   M A 0  M P 0

Assuming 𝐼𝑥𝑦 = 𝐼𝑦𝑧 = 0

I zz R0  I xz  P0  Q 0 R0   I yy  I xx P0Q 0  N A 0  N P 0
 0  P0  Q 0 sin  0 tan  0  R0 cos  0 tan  0
 0  Q 0 cos  0  R0 sin  0
 0  Q 0 sin  0  R0 cos  0  sec 0
X 0  U 0 cos 0 cos 0  V 0  sin  0 sin  0 cos  0  cos  0 sin  0 
 W 0  cos  0 sin  0 cos  0  sin  0 sin  0 
Y0  U 0 cos 0 sin 0  V 0  sin  0 sin  0 sin  0  cos  0 cos  0 
 W 0  cos  0 sin  0 sin  0  sin  0 cos  0 
h0  U 0 sin  0  V 0  sin  0 cos 0 
 W 0  cos  0 cos 0 
How many equations do we need to
characterize the reference set? – 8
m (U 0  Q 0W 0  V 0 R0 )  mg sin  0  FAX 0  FPX 0
m (V0  R0U 0  P0W 0 )  mg cos 0 sin  0  FAY 0  FPY 0
m (W0  P0V 0  Q 0U 0 )  mg cos 0 cos  0  FAZ 0  FPZ 0


I xx P0  I xz  R0  P0Q 0   I zz  I yy Q 0R0  LA 0  LP 0
I yy Q 0   I xx  I zz  P0 R0  I xz  P02  R02   M A 0  M P 0

Assuming 𝐼𝑥𝑦 = 𝐼𝑦𝑧 = 0

I zz R0  I xz  P0  Q 0 R0   I yy  I xx P0Q 0  N A 0  N P 0
 0  P0  Q 0 sin  0 tan  0  R0 cos  0 tan  0
 0  Q 0 cos  0  R0 sin  0
 0  Q 0 sin  0  R0 cos  0  sec 0
X 0  U 0 cos 0 cos 0  V 0  sin  0 sin  0 cos  0  cos  0 sin  0 
 W 0  cos  0 sin  0 cos  0  sin  0 sin  0 
Y0  U 0 cos 0 sin 0  V 0  sin  0 sin  0 sin  0  cos  0 cos  0 
 W 0  cos  0 sin  0 sin  0  sin  0 cos  0 
h0  U 0 sin  0  V 0  sin  0 cos 0 
 W 0  cos  0 cos 0 
Total number of unknowns: 8 states 4 controls
x 0  [U 0 V 0 W 0 P0 Q 0 R0  0  0 ]
u 0  T0  E 0  A 0  R 0 
Special case: Straight equilibrium
condition
Equilibrium Condition
All state derivatives =0
U 0  V0  W0  0
P0  Q 0  R0  0
 0  0  0
Straight line flight ->
0  0
Reference Set
mg sin  0  FAX 0  FPX 0
(written in
stability axes frame)
mg cos 0 cos  0  FAZ 0  FPZ 0
mg cos 0 sin  0  FAY 0  FPY 0
Unknowns – 6 [ 0  0 T0  E 0  A 0  R 0 ]
0  LA 0  LP 0
0  M A0  MP0
0  N A0  NP0
Trim solution trends
Stability of a Trim
Concept of Aerodynamic Static
Stability
A system is statically stable with respect to a given equilibrium condition
if the force and/or moment acting on the system due to a small static displacement
from the equilibrium condition is such a direction that would tend to return the
system to a given condition
Aerodynamic Static Stability
Perturbation causes the nose to go –up
Static stability in pitch requires there be a nose-down moment
 Criterion for static stability in pitch  M A  M P |  0

 

 
0
Other longitudinal criterion
Lateral-directional static stability
criterion