The corresponding Mach number for the fastest climb is around 0.62, which corresponds to a velocity of 210 m/s. Note that this is the sustained (steady) climb rate. At this Mach number, the climb angle would be around 17 deg, marginally lower than the steepest climb angle of 20 deg.
While the steepest climb angle may be of interest in clearing an obstacle after take-off, it is usually the fastest climb (maximum climb rate) that is more useful as a performance figure of merit.
In Figure 4.10, the SBA analysis reveals that the fastest climb trim is stable, whereas the steepest climb trim occurs at a point of phugoid insta- bility. This is not an unusual occurrence and some airplanes are known to have a mildly unstable phugoid dynamics during climb following take-off.
Pilots can usually handle this instability manually. (See Exercise 2 for an interesting case study.)
These maneuvers are called pull-up or push-down and are pictured in Figure 4.11.
Unfortunately, these maneuvers are necessarily accelerated ones, that is, they do not satisfy all the equilibrium (trim) conditions in Equation 4.6.
Hence, they cannot be directly analyzed per se by using the bifurcation and continuation method applied to the dynamic Equation 4.5 or the trim Equation 4.6. However, pull-up/push-down flight can still be analyzed using what is called a quasi-steady-state approximation.
Let us begin with the longitudinal dynamics equations in the form given by Equation 4.5 (reproduced below).
V
V g
V qS
W C Ma e C C c
V T
W
Dsta Dq D
= −sinγ− ( , , )α δ +
(
1+ α)
( )2α + cooscos ( , , ) ( )
α
γ γ α δ α
α
= − + +
(
+)
g
V qS
W C Ma e C C c
V
Lsta Lq1 L 2
+
=
( )
+(
+)
WT qScI C Ma e C C
yy msta mq m
sin
, , (
α
θ α δ α
α
1
V))c
2
(4.5) Consider a pull-up maneuver (a push-down is very similar). Begin with the airplane in a steady trim flight state (γ= 0 and known values of δe, η, V and all the other flight variables) at the bottom of the curved flight path. Neglect changes in velocity; hence, the first of Equation 4.5 is to be ignored.
XB
(a) (b)
V XE
ZE ZB q
ZE ZB XE
XB V
q
FIGURE 4.11 Sketch of (a) pull-up and (b) push-down maneuver in longitudinal plane (airplane is at the bottom/top of the curved flight path in each case).
Now imagine an upward (negative) deflection of the elevator. The con- sequent nose-up moment will pitch the airplane up to a higher angle of attack, hence a larger lift (in general, pre-stall). The additional lift will pro- duce a centripetal acceleration, which curves the flight path up as shown by the pull-up sketch in Figure 4.11a. From the second of Equation 4.5, still considering the flight state at the bottom of the curve where γ= 0, we have
γ = g − +
V{ 1 nz} (4.13)
where,
n qS
W C Ma e T W
z = Lsta( , , )α δ + sinα (4.14) is called the load factor and the rate derivatives with respect to CL in the second of Equation 4.5 have been dropped. Clearly, nz= 1 corresponds to level flight with γ = 0 , as suggested by Equation 4.13. With reference to Figure 4.11, γ is the angular velocity of the airplane velocity vector in the pull-up maneuver. Note that γ is not necessarily equal to θ—the angular velocity of the airplane’s body-fixed axis—at all times; the dif- ference between the two depends on the rate of change of angle of attack.
However, at a quasi-steady state, α = 0 and then θ and γ are the same, that is, the body-axis and the velocity vector follow the curved flight path in sync.
The stability of the quasi-steady state is related to perturbations in the angle of attack and must be deduced from the last of Equation 4.5. For a fixed value of velocity and throttle setting, for a point at the bottom of the curved flight path (γ= 0), the quasi-steady state and its stability can be obtained by solving
θ α δ α
= + + α
qScI C Ma e C C c
yy mstab( , , ) ( mq1 m )2V (4.15)
Effectively, the perturbations in angle of attack represent short-period dynamics about a curved flight path (in contrast to the straight-line flight path considered previously). Each value of the elevator deflection
parameter decides the load factor as given by Equation 4.14, which in turn sets the angular velocity of the curved flight path in Equation 4.13. These different quasi-steady states, distinguished by their load factor and angu- lar velocity, can have different stability properties since the aerodynamic coefficient/derivatives in Equation 4.15 are themselves functions of the trim angle of attack and elevator deflection.
The quantum of elevator deflection required to change from one quasi- steady state to another (or from a level flight steady state to a curved flight quasi-steady state) can be estimated from Equations 4.14 and 4.15. In terms of perturbations in elevator deflection, angle of attack, and load fac- tor between two quasi-steady states,
(∆n )W ∆ ∆
qS C C e
z−1 = Lα α+ L eδ δ (4.16)
where the thrust term has been ignored in addition to the (CLq1+CLα) term that had been dropped previously. Similarly,
∆ ∆ ∆ ∆
θ α δ α
α δ α
= + + +
qScI C C e C C c
yy ( m m e ) ( mq1 m ) 2V (4.17)
However, since ∆θ and ∆α are individually zero at each (quasi-) steady state, Equations 4.16 and 4.17 form a pair of equations in terms of Δα, Δδe and (∆nz−1). Eliminating Δα yields a result for the elevator deflec- tion required for a certain change in load factor, popularly called “elevator deflection per g” as below:
∆
∆ δ α
α δ α δ
n e C W qS C C C C
z
m
m L e L m e
( ) ( )
− =1 − / (4.18)
Equation 4.18 offers an alternative interpretation of stability—onset of instability corresponds to the point where a vanishingly small elevator deflection can induce a unit change in load factor (“g”). This, according to Equation 4.18, matches the condition Cmα= 0, which leads to the standard definition of neutral point and static margin (see Reference 1). This would suggest that there is no fundamental difference in pitch stability between a straight-line flight path and a curved one, a conclusion that is reaffirmed
by the numerical example that follows shortly. For a traditional view of
“elevator angle per g” and the concepts of the so-called maneuver point and the maneuver margin, see Box 4.1.
The standard phugoid dynamics is no longer possible in this quasi- steady-state analysis since the velocity is being held fixed. However, note that perturbations in the angle of attack will also affect the lift coeffi- cient in Equation 4.15, thus perturbing the load factor nz and thereby the curvature of the flight path (Equation 4.13). As long as the α dynam- ics in Equation 4.15 is stable, the perturbations in Equation 4.13 should also die down and the quasi-steady state with its angular velocity γ is regained.
An initial level flight trim state at Ma = 0.4, η = 0.38 is used as the starting point for a numerical study of the pull-up maneuver. Velocity and throttle setting are held unchanged at these values. The second and third of Equation 4.5 are solved in the form below with elevator deflection vary- ing as the continuation parameter. The point at the bottom of the vertical loop is considered.
BOX 4.1 STABILITY IN PULL-UP MANEUVERS—THE TRADITIONAL VIEWPOINT
Traditionally, the pitch damping term in Equation 4.15 has been mod- eled as Cmq1(q1c/2V), where q1 is the body-axis angular velocity. In that case, expressing q1 in terms of α and γ, there appears an additional term involving Cmq1 and γ in the equation for the short-period dynamics about a curved flight path. The effect of this term on the short-period dynamics is to provide an additional source of stability in case of a curved flight path over and above the short-period stability in level flight. In terms of
“elevator angle per g,” Equation. 4.18 in the traditional version would appear as
∆
∆
δ α α
α δ α δ
e n
C W qS C c V C g V C C C C
z
m mq L
m L e L m e
( )
( ( / ( / ) ( / ))
( )
− = +
− 1
2 ) 1
with the additional term Cmq1(c/2V)CLα(g/v) in the numerator. As a result, it has been traditionally believed that an airplane in a curved flight path with load factor different from 1 somehow acquires additional stability in pitch, leading to the definition of maneuver point and maneuver margin for curved flight paths, which replace the neutral point and static margin in case of straight-line 1−g trim flight.