OHARRAK (1868-1936) 1
0. Iberismoa(k), nazionalismoa(k)
The non-dimensional longitudinal derivatives in Eqns (4.70) and 71) may he expressed in terms of the lift, drag, moment coefficients, and thrust. Using the dimensionalizing relations of Section 4.4.7 the relevant expressions follow.
(a) X,,, longitudinal force due to forward velocity
where is the overall aircraft drag coefficient.
For the case where the drag coefficient and are independent of velocity is usually less than
Rigid airframe dynamics
(b) longitudinal force to the effect of incidence change:
where is the overall lift coefficient.
For the case when the drag coefficient is independent of the incidence, is of the order of unity.
longitudinal force due to of pitch:
where is the volume coefficient of the horizontal stabilizer, see Eqn.
and suffix 'T' refers to the horizontal stabilizer. In most instances is negligibly small.
longitudinal force due to the lag in wing at the tail:
where is the rate of change of with wing angle of attack.
As a consequence of the negligible value of and the fact that is typically less than 0.5, is also negligible.
normal force due to velocity:
=
-
This term has a typical value of or less.
force due to change in effective incidence:
where is the wing-body lift curve slope.
The term is usually negligible and the derivative has a typical value in the range of - 2 to - 6.
force due to rate of pitch:
where is the horizontal stabilizer lift curve slope. A typical value for is -2.
Aircraft loading a n d structural layout
(h) normal force due to lag:
A typical value for this term is of the order of unity (i) pitching moment due to forward velocity:
At low speed where the pitching moment coefficient, is largely independent of forward speed this is negligible. It may be significant in compressible flow.
pitching moment due to change in effective incidence:
where is the controls fixed static margin, see (4.30). If compressibility is present the expression for is more complex than that given in (4.30) see, for example, A typical value of for a statically stable aircraft is of the order of - 1 or rather less.
pitching moment due to pitch rate, see also Section
where
is the ratio of the horizontal stabilizer moment arm to the wing aerodynamic mean chord
is the pitch damping effect of the wing-body combination. This term is usually relatively small when the aircraft employs a horizontal stabilizing surface but it is significant on tailless
where the absolute value of the pitching moment due to of pitch of the wing-body combination. Note is the same as
For a conventional tailed has a typical value of the order of ten.
(1) Mw, pitching moment due to lag:
So that for a conventional design Mw is typically around -2.
Rigid airframe dynamics
X,,
longitudinal force due to pitch control deflection:where
is the ratio of the horizontal stabilizer to wing areas
is the rate of change of the drag of the horizontal stabilizer with control deflection, and is normally negligible
For a tailless design the pitch control characteristics are based on the wing area since the horizontal stabilizer is not present and effectively zero, see Section
normal force due pitch control deflection:
where is the lift curve slope due to the control deflection,
may have a typical value of less than one. In the case of a tailless design.
Z,
=M,, pitching moment due to pitch control:
M,
has a typical value of - 2 or less. As mentioned in (m) above, in the case of a tailless design is based on the wing area so that for this class of aircraft:where is the chord-wise distance between the centre of pressure due to elevator deflection and the of gravity of the aircraft, see Section
longitudinal force due to increment in Assuming that the is normally aligned along the Ox direction:
where AT is the increment in thrust.
normal force due to the effective incidence change of the thrust:
It is to be noted that this term depends upon the 8, of the disturbed motion. This introduces a complexity in the solution of the equations unless it is justified to assume that the term is negligible, see Section 4.6.4.
loading and structural layout
pitching moment due to the change in thrust:
where is the vertical offset of the thrust line from the Ox axis (positive down).
Response of the aircraft to pitch control input 4.6.3.1 Aircraft having inherent static stability
As discussed in Section 4.6.1 it is acceptable for the purpose of loading calculations to neglect the effect of any change in forward speed when analysing the response to pitch control of an inherently statically stable aircraft. The motion of the aircraft is now described by the latter two of Eqns (4.69) namely the normal force and the pitching moment equations. Eqn. (4.71) reduces to:
The terms are defined at Eqn. (4.71) and some simplification is possible. The longitudinal relative density factor, has a magnitude of the order of and when this is considered the typical values of the relevant derivatives as given in Section 4.6.2 it is reasonable to assume that:
<<
1.0 and 1.0Further it is acceptable to assume that is negligibly small. Thus, approximately:
Applying Cramer's to Eqn. see Eqn. (4.73). gives the solution for as:
where
Rigid airframe dynamics
Substitution from Eqn. with simplifications of Eqn. (4.76) gives the expression for the first of the above determinants as:
Likewise the second of the determinants is:
Substituting these expressions into the determinants of Eqn. gives
where
Further development of Eqn. (4.79) is achieved by substituting the expressions for the aerodynamic derivatives given in Section Noting that, from is
to and neglecting in comparison with in the the of w becomes:
The coefficient of becomes:
Aircrafl loading and structural layout
where use has been made of Eqn.
The coefficient of is The coefficient of becomes:
Therefore:
Finally = = a, where here a is the increment in the body angle due to the departure from the trimmed condition. Hence:
where
is the damping coefficient in the short-period motion
is the natural damped frequency in the short-period motion.
=
is the forcing function due to the of change of the pitch control and may often be taken as zero.
Rigid airframe dynamics
An alternative form of Eqn. is:
where is the forcing function due to pitch control deflection. The in is negligible except, possibly, for a tailless aircraft. The first term in is comparable in value to the relative density factor, and is likely to be an order of magnitude greater than the middle term so that this latter also may often be neglected. Thus approximately:
The natural undamped frequency of the short-period motion, is given by:
and the damping ratio is:
Thus Eqn. may be written in the form:
where is the right-hand side of Eqn. and is given either by Eqns or or, approximately, by Eqn.
The effect of the value of the damping ratio, is considered in Appendix A4.
Suffice it to note that if is negative, system is unstable in the short-period mode.
The solution in terms of the non-dimensional pitching velocity, is derived from Eqn. (4.75) as:
loading and structural layout
The expansion of the determinate on the left-hand side of this equation is given by Eqn. The expansion of the determinate on the right-hand side is:
In this case the left-hand side of Eqn. the equivalent of (4.79).
becomes:
where the expressions for the coefficients in terms of the aerodynamic derivatives are
given in and
The right-hand side of Eqn. becomes:
It will be seen that the coefficient of is identical to that of the coefficient of in Eqn.
When the expressions for the derivatives are substituted, the coefficient of is:
This is similar to the middle term in the expression for given in Eqn.
except for the From the comments made there it may he concluded that this term is negligible i n comparison with the term except, possibly, when the pitch control is applied rapidly.
Rigid airframe dynamics
The coefficient of is:
where use has been made of the approximate expression for from Eqn.
Therefore, using the same development as for the left-hand side of Eqn. (4.79). the solution of the equations in terms of the non-dimensional pitching velocity is:
It should be noted that when the deflection of the pitch control is constant and the aircraft has reached a steady state condition such that the increment in the angle, a.
and the pitching velocity, are both constant:
and
where the suffix refers to the steady state condition
4.6.3.2
Statically unstable aircraftWhile Eqns (4.81). (4.82). and (4.84) apply to a statically or dynamically unstable system they are of little value for loading calculations due to the fact that the response of the aircraft to a pitch control input will cause it to depart from naturally restrained flight.
In practice, an inherently unstable aircraft has to have the stability augmented through the control system as briefly mentioned in Section The characteristics of the airframe and the control system become parts of a closed loop system utilizing feedback of the motion of the aircraft unlike the open loop system described by Eqn. (4.84). For a further discussion of this matter refer, for example, to
However, for the purpose of loading actions, it is not usually necessary to consider the closed loop dynamic characteristics of the aircraft since the system is used to place limits on the displacements, velocities, and accelerations achieved by the aircraft.
This enables limits to he placed on the parameters which determine the loads as discussed in Chapter 3, Section
Aircraft loading and structural layout