Lecture 5 : Static Lateral Stability and Control or how not to move like a crab
1.0 Lateral static stability Lateral static stability refers to the ability of the aircraft to generate a yawing moment to cancel disturbances in sideslip V Question : Which direction should the yawing moment act to align the aircraft with the velocity vector? positive sideslip
1.1 Yawing moment variation with sideslip Cn 1. Cn should be an anti symmetric function of 2. Cn / ( denoted as Cn ) must be > 0 for lateral static stability
Figure 1.1 : F4 Wind tunnel data - yawing moment ( 0 o ) Observe 1. Positive slope 2. Quasi linear variation 3. Effect of rudder Source : NASA TN D6425
Figure 1.2 : F4 Wind tunnel data - Cn Question : What happens at high AOA? Source : NASA TN D6425
1.3 Quantifying static lateral stability X The aircraft yawing moment is: V n ½ V2 S b Cn cg Y lv Lv
The dominant contribution to the aircraft yawing moment comes from the vertical tail ntail ½ V2 Sv CLv lv ½ V2 Sv lv av [(1- ) + r ] Where av : vertical tail lift curve slope : vertical tail sidewash factor
Hence the aircraft yawing moment coefficient is approximately : Cn ntail / (1/2 V2 S b) [(Sv lv)/ (Sb) ] av [(1- ) + r ] The yawing moment curve slope is Cn Vv av (1- )
1.4 : Interpreting the vertical tail volume ratio Similarly the effectiveness of the vertical tail is related to the vertical tail volume ratio Vv Sv lv ------Sw b lv : distance from aerodynamic center of vertical tail to cg
Question : What are typical values for Vv? 0.16 0.20 0.17 0.18 0.17 0.13 0.30 0.14 0.26 Comments? 0.23 0.19
2.0 FAR Part 23.147 Directional and lateral control (a) For each multiengine airplane, it must be possible, while holding the wings level within five degrees, to make sudden changes in heading safely in both directions. with the (1) Critical engine inoperative and its propeller in the minimum drag position; (2). (b) For each multiengine airplane, it must be possible to regain full control of the airplane without exceeding a bank angle of 45 degrees, reaching a dangerous attitude or encountering dangerous characteristics, in the event of a sudden and complete failure of the critical engine,
2.1 Directional stability - One Engine Inoperative (OEI) flight For OEI flight, the rudder has to deflect to cancel the torque from the asymmetric thrust from the operative engine. Is the aircraft now trimmed?
The aircraft is rolled towards the operative engine Y Z
2.2 Coupled rudder-aileron controls for OEI X Torque from operative engine -T le Rudder Torque T ½ V2 Sv av ( r) lv Assume steady flight T D Y Hence or lv rtrim le D Fr
How much should you roll to maintain directional stability? Force balance in the Y direction Fr W sin Force balance in the Z direction L W cos The trim rudder force is Fr Equating ½ V2 Sv av ( rtrim ) Fr tan
2.3 FAR Part 25.147 Directional and Lateral Control (d) Lateral control; airplanes with four or more engines. Airplanes with four or more engines must be able to make 20 banked turns, with and against the inoperative engines, from steady flight at a speed equal to 1.3 VSR1, with maximum continuous power, and with the airplane in the configuration prescribed by paragraph (b)* of this section. Notes * Two critical engines inoperative VSR1 : reference stall speed for the specific configuration in paragraph (b)
3.0 FAR Part 23. 177 - Static directional and lateral stability (b) The static lateral stability, as shown by the tendency to raise the low wing in a sideslip, must be positive for all landing gear and flap positions. Question? What s raise the low wing in a sideslip?
3.1 Roll static stability the dihedral effect In the OEI example, a sideslip disturbance was countered by rolling the aircraft. Now we are concerned with a roll disturbance and how to counter it by using the resulting sideslip. (!? ) L L is vectored off Y +ve roll disturbance Z Z Y
Define Cl (Cl )/ as the rate of change of rolling moment with respect to sideslip Question : For the aircraft to be statically stable in roll, i.e. have a self correcting tendency, what should the sign of Cl be? Wings return to level poisiton Need a ve rolling moment to counter disturbance Y Z Y Z
3.2 Estimating the dihedral effect Consider a +ve sideslip Y Vs Vb Z Resolve the sideward velocity Vs into components parallel Vb and normal Vn to the wing.
Hence the angle of attack for is effectively Vn Right wing section V Left wing section
X For small sideslip, the sideward velocity is: V Vs Y For small wing dihedral angle, the normal component of Vs is : Vn The change in angle of attack is:
Now consider the wing section at location y of chord c(y) and with dy. X The incremental rolling moment dl due to y dy this section and the corresponding one on the left wing is : Y dl -(½ V2 ds CL) x (2y)
Hence the total rolling moment due to the dihedral effect is: l dl - V2 a b/2 0 The dihedral effect is measured by : Cl -a 2 b/2 0 y c(y)dy Sb For a linearly tapered wing, this works out to Cl - (a/6) (1+2 )/(1+ ) y c(y)dy
Example : Assessing roll static stability So dihedral is a good thing, right? What about this