Governing Equation Set of Lateral Motion

Lateral motion analysis

《Governing Equation Set of Lateral Motion》

--- Side force equation

--- Rolling moment equation

--- Yawing moment equation

--- Note that we have includedandin for completeness, but we will treatandas constants, i.e. and, in lateral motion.

《Variables of lateral motion》

: Roll rate in inertial space

: Yaw rate in inertial space

 v: Sideslip velocity (bottom figure)

--- Also define the sideslip angle as .

《Force and moments of lateral motion》

, and

《Control surfaces of lateral motion》

: Aileron deflection ( >0 for right aileron down )

: Rudder deflection ( same sign as ).

C.1 Linearization of the lateral equations

○We will linearize the inertial terms and expand the external force and external moments into their gravitational and aerodynamic components. We will also linearize the aerodynamic terms by perturbations in and from their equilibrium values. --- At equilibrium, , therefore and are perturbations as well as their absolutes values.

○For the side force equation

a) Linearization of:

--- and ; both are considered constants.

--- Then, --- We have used .

. b) Expansion of: ---

--- Aerodynamic force :

--- Gravitational force : --- at trim.

c) The final side force eq.: --- .

○For the rolling moment equation:

a) Inertial terms, , are already linear in and.

b) Expansion of: --- Only aerodynamic terms appear on .

--- We normally denote this aerodynamic rolling moment as.

--- is a function of and ; hence, we can expand into:

--- At trim condition, .

c) Normalize the equation by dividing by and define

the final rolling moment equation becomes:

○For the yawing moment equation:

a) Again, the inertial terms,, are already in linear form, and only aerodynamic terms appear on the external yawing moment.

--- We normally denote this aerodynamic yawing moment as .

b) Expand into:

c) Normalize the equation by dividing by , set=0, and define

the final yawing moment equation becomes:

《Linearized equation set of the perturbed lateral motion》

《Laplace transform of the lateral equation set》

C.2 Static Lateral Analysis

○We will analyze the equilibrium conditions of various steady lateral motions.

○By steady lateral motions we meant lateral maneuvers in which all variations in variables have vanished; hence, the static equation set becomes as follows:

, .

○Steady turn:

 In an ideal turn, v=0.

 Side force equation gives ; hence, .

 We will need

and .

Discussions:

--- Require both to maintain a turn.

--- Usually , and .

 is positive for positive   Aileron is held against turn

 is negative for positive   Rudder is held with turn

 Both will decrease with increasing .

○Straight side slip:

In a straight side slip,.

Remaining equation set:

 Side force equation gives:; hence, ; or since , we also have

--- Tilted lift to balance aerodynamic drag due to sideslip velocity v.

 Rolling equation gives:.

--- is mostly the result of Dihedral, so we need to balance Dihedral.

 And yawing equation gives: .

---to balance the yawing moment from vertical tail due to sideslip angle.

○Steady roll: --- No yaw, no sideslip, or r = v = 0.

Remaining equation set:

--- Note that p=s.

From first equation:

--- Normally, for p > 0 to balance the aerodynamic drag

From the second equation: .

--- We need to balance the roll-induced yawing moment.