Flight Dynamics
AAE 6710
Lesson 02-7
Longitudinal Control
Factors Affecting Design
β’ Control Effectiveness: Measure of how effective the control is in producing the desired control moment
β’ Hinge Moments: The aerodynamic moments that must be overcome to rotate the control surface
β’ Aerodynamic & Mass Balancing: Techniques to keep the control forces within an acceptable range
Effect of Elevator Deflection
πͺπ³π = πͺπ³πΆππΌπ‘ = πͺπ³πΆπ πΌπ€ β ππ€ β ν + ππ‘
πͺππππ= πΌπ½π―πͺπ³πΆπ π0 + ππ€ β ππ‘ β πΌπ½π―πͺπ³πΆπ 1 β
πν
ππΌπΆπ
πΏπ = 0
For arbitrary πΏπ
πͺπ³π = πͺπ³πΆππΌπ‘ = πͺπ³πΆπ πΌπ€ β ππ€ β ν + ππ‘ + π πΏπ
πͺππππ= πΌπ½π―πͺπ³πΆπ π0 + ππ€ β ππ‘ β π πΏπ β πΌπ½π―πͺπ³πΆπ 1 β
πν
ππΌπΆπ
Increment in Lift When the elevator is deflected, it changes the lift of the airplane.
π³ = πΏπ€ + πΏπ‘
πͺπ³1
2ππ2π = πΆπΏπ€
1
2ππ2π + πΆπΏπ‘
1
2πππ‘
2ππ‘
πͺπ³ = πΆπΏπ€ + πππ‘ππΆπΏπ‘
πΆπΏπ‘ = πΆπΏπΌπ‘πΌπ‘ = πΆπΏπΌπ‘
πΌπ€ β ππ€ β ν + ππ‘ + π πΏπ
= πΆπΏπΌπ‘πΌπ€ β ππ€ β ν + ππ‘ + πΆπΏπΌπ‘
π πΏπ
βπͺπ³ = βπͺπ³π = πππ‘ππΆπΏπΌπ‘
π πΏπ π πͺπ³ππ πΉπ
= πΆπΏπΌπ‘π π =
ππΌπ‘ππΏπ
Elevator Effectiveness
π πͺπ³ππ πΉπ
= πͺπ³πΆππ
β’ π πͺπ³π/π πΉπ is the elevator effectiveness
β’ The elevator effectiveness is proportional to the size of the flap being used as an elevator and can be estimated from the equation: where π is the change in tail angle of attack per unit deflection of the elevator and is called the elevator effectiveness parameter
π =ππΌπ‘ππΏπ
Elevator Effectiveness Parameter
Elevator Effectiveness is proportional to the size of flap being used as an elevator
Increment in Pitching Moment
If βπΆπ = πΆππΏππΏπ π€βπππ πΆππΏπ
=ππΆπππΏπ
βπͺπ = βπͺππ= βπππ»πΆπΏπΌπ‘
π πΏπ = βππ»πππΆπΏπ‘ππΏπ
πΏπ
πͺππππ= πΌπ½π―πͺπ³πΆπ π0 + ππ€ β ππ‘ β π πΏπ β πΌπ½π―πͺπ³πΆπ 1 β
πν
ππΌπΆπ
πͺππΉπ= βππ»π
π πͺπ³ππ πΉπ
= βππ»ππΆπΏπΌπ‘π
The derivative πͺππΉπ is called the elevator control power
Influence of Elevator on πΆπ vs πΌ curve
πͺπ = πΆπ0+ πΆππΌ
πΌ + πΆππΏππΏπ
Elevator Angle to Trim
An airplane is said to be trimmed if the forces and moments acting on the plane are in equilibrium
πͺπππππ= π = πΆπ0
+ πΆππΌπΌπ‘πππ + πΆππΏπ
πΏππ‘πππ
πΉπππππ = βπΆπ0
+ πΆππΌπΌπ‘πππ
πΆππΏπ
πͺπ³ππππ = πΆπΏπΌπΌπ‘πππ + πΆπΏπΏππΏππ‘πππ
πΆππππ =πΆπΏπ‘πππ β πΆπΏπΏππΏππ‘πππ
πΆπΏπΌ
πΆπΏπ‘πππ =π
1
2ππ2π
Elevator Angle to Trim
If we substitute the value of πΆππππ in the πΉπππππ equation, we get:
πΉπππππ = βπΆπ0
πΆπΏπΌ + πΆππΌπΆπΏπ‘πππ
πΆππΏππΆπΏπΌ β πΆππΌ
πΆπΏπΏπ
πΆππππ = πΆπ0
πΆπΏπΏπ + πΆππΏππΆπΏπ‘πππ
πΆππΏππΆπΏπΌ β πΆππΌ
πΆπΏπΏπ
Elevator Angle to Trim
πͺπππππ= π = πΆπ0
+ πΆππΌπΌπ‘πππ + πΆππΏπ
πΏππ‘πππ
πΆπΏπ‘πππ =π
1
2ππ2π
β΄ πΉπππππβ πΏπ0 βπΆπΏπ‘ππππΆππΏπ
π πͺπ
π πͺπ³ πππ₯ where πΏπ0 = πΏπ ππ‘ π§πππ πΆπΏ = β
πΆπ0
πΆππΏπ
= πΆπ0+
ππΆπππΆπΏ πππ₯
πΆπΏπΌπΌπ‘πππ + πΆππΏππΏππ‘πππ
0 β πΆπ0+
ππΆπππΆπΏ πππ₯
πΆπΏπ‘πππ + πΆππΏππΏππ‘πππ
Elevator Angle for Trim
Remarks
β’ For a given c.g. location πΏπ‘πππ is a linear function of πΆπΏπ‘πππ
β’ When an airplane is trimmed at a chosen πΆπΏ by setting the elevator at corresponding πΏπ‘πππ, to fly at a lower speed which implies higher πΆπΏ the pilot would need to apply more negative elevator deflection or the incremental lift on the tail would be negative or ΞπΏπ would be negative
Remarks
β’ In trim (balanced) condition, the setting of the elevator determines the airplane's trim speed - a given elevator position has only one lift coefficient (and one speed at a given altitude) at which the aircraft will maintain a constant (un-accelerated) condition
Remarks
β’ Military airplanes which are highly maneuverable, sometimes have the following features:
β An all movable tail in which the entire horizontal tail is rotated to achieve higher Ξπππ
β Relaxed static stability wherein may have a small positive value πΆππΌ
Example 1
An airplane has elevator power πΆππΏπ of -0.010 per degree.
The c.g. is placed such that the static margin is 10% of
m.a.c. Further, the tail setting , ππ‘ ,is such that the airplane is
in trim, with zero elevator deflection, at πΆπΏ = 0.5. Plot the
curves of πΆπππ vs πΆπΏ for constant elevator angles of
πΏπ = β200, β100, 00 , 100, 200. Cross plot these curves to
obtain the curve corresponding to πΏπ‘πππ vs. πΆπΏπ‘πππ. Note
πΆπΏπππ₯= 1.5.
πΆπππ= πΆπ0
+ πΆππΌπΌ + πΆππΏπ
πΏπ = πΆπ0+ ππΆπππ
ππΆπΏ πΆπΏ + πΆππΏππΏπ
Since the airplane is in equilibrium with zero elevator deflection at πΆπΏ = 0.5
0 = πΆπ0β 0.1 β 0.5 + 0.0 πΆπ0
= 0.05
πΆπππ= 0.05 β 0.1πΆπΏ β 0.01πΏπ
πΏπ = β200, πΆπππ= 0.05 β 0.1πΆπΏ β 0.01 β200
= +0.25 β 0.1πΆπΏ πΏπ = β100, πΆπππ
= +0.15 β 0.1πΆπΏ
πΏπ = 00, πΆπππ= +0.05 β 0.1πΆπΏ
πΏπ = +100, πΆπππ= β0.05 β 0.1πΆπΏ
πΏπ = +100, πΆπππ= β0.15 β 0.1πΆπΏ
Static Margin= 0.1 ππΆπππππΆπΏ = β0.1
Solution
πΆππΏπ= β0.01
πͺπππ ππ πͺπ³
πΏπ‘πππ vs πΆπΏ
πΏπ‘πππ = πΏπ0 β ππΆπππππΆπΏ πΆπΏπ‘πππ πΆππΏπ
πΏπ‘πππ = 50 β β0.1 πΆπΏπ‘πππ β0.01 = 50 β 10 β πΆπΏ
Note that πΏπ‘πππ = 0 ππ‘ πΆπΏπ‘πππ = 0.5
β΄ 0 = πΏπ0 ββ0.1
β0.01β 0.5 πΏπ0 = 50
Flight Measurement of XNP
Procedure
π€βππ πΆππΌ= 0,
π πΉππππππ πͺπ³ππππ
πππ’ππ π‘π π§πππ
π πΉππππππ πͺπ³ππππ
= βπΆππΌ
πΆππΏππΆπΏπΌ β πΆππΌ
πΆπΏπΏπ
πΉπππππ = βπΆπ0
πΆπΏπΌ + πΆππΌπΆπΏπ‘πππ
πΆππΏππΆπΏπΌ + πΆππΌ
πΆπΏπΏπ
Neutral Point
Ground Effect
β The slope of lift curve of the wing, i.e. πΆπΏπΌπ€ increases slightly
β The downwash due to wing decreases considerably
β Result: The airplane becomes more stable and requires more negative elevator deflection
Limitation on Fwd Movement of CG
β’ In Free Flight: Forward c.g. location at which the maximum negative elevator deflection would be just able to permit trim at πΆπΏπππ₯
β’ Near Ground: Requirement of more negative elevator deflection due to increased stability in ground effect
Limitation on Fwd Movement of CG
β’ Flare at Landing: Additional elevator deflection to compensate for gusts and to flare the aircraft. This requirement dictates a ππΆπ ππΆπΏπππ₯
less than the value required for
πΆπΏπππ₯only
Restrictions on C.G. Movement from Stick Fixed Stability Considerations
Example 2.3 (Nelson)
Solution
H.W. Assignment # 3
Solve problems 2.4, 2.6, 2.7, 2.9 & 2.10 from Nelsonβs book Submission date: 14 Apr. 2015 Submit at the start of class on due date (even if you plan to be absent). No credit afterwards. Do not copy any assignment.
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