Flight Dynamics and Control. Lecture 3: Longitudinal stability Derivatives G. Dimitriadis University of Liege
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1 Flight Dynamics and Control Lecture 3: Longitudinal stability Derivatives G. Dimitriadis University of Liege
2 Previously on AERO We developed linearized equations of motion Longitudinal direction ( ) mgcosθ e ( ) mgsinθ e " m X w 0 0% " u % " X u X w X q mw e $ $ 0 ( m Z ' $ w ) 0 0' w ' $ $ ' $ Z $ $ 0 M ' + u Z w Z q + mu e w I y 0 $ q ' $ ' $ ' $ M u M w M q 0 # $ & ' # θ & $ # " m % " v % " Y v Y p + mw e Y r mu e $ 0 I x I xz 0 0 ' $ p ' $ $ ' $ ' $ L v L p L r 0 0 $ 0 I xz I z 0 0' $ r ' + $ N $ ' $ ' v N p N r 0 0 $ $ ' $ ϕ ' $ # $ & ' # $ ψ &' $ # % " u % " ' $ ' w ' $ $ ' ' = $ $ q ' $ ' $ ' $ ' θ & # & # Lateral direction ( ) ( ) mg cosθ e mgsinθ e X η Xτ % Z ' η Zτ ' " η M η M τ ' $ % # τ & ' ' 0 0 & % " v % " Y ξ Yς % ' $ ' p ' $ $ ' L ' $ ξ Lς ' ' $ r ' = N " ξ $ ' ξ Nς ' $ % $ ' $ ' ' $ ϕ # ς & ' ' $ 0 0 ' ' & # $ ψ &' $ # 0 0 ' &
3 Longitudinal stability derivatives It has already been stated that the best way to obtain the values of the stability derivatives is to measure them. However, it is still useful to discuss simplified methods of estimating these coefficients. Such estimates can be used, for example, in the preliminary design of aircraft. This lecture will treat longitudinal stability derivatives.
4 Derivatives to be determined The aerodynamic and control derivatives to be determined are: Force derivatives in x-direction: Force derivatives in z-direction: Pitching moment derivatives
5 USAF DATCOM The most complete set of data for empirical determination of the derivatives of subsonic aircraft is the USAF Stability and Control Data Compendium (DATCOM). It is used in Roskam, Methods for Estimating Stability and Control Derivatives of Conventional Subsonic Airplanes. It dates from 1977 so the aircraft used in developing the tables and figures are not up to date. It is also available in software form here: You need a Fortran compiler.
6 Units The USAF DATCOM is a US document and therefore all calculations are carried out using imperial units. You can use SI units on a case-by-case basis if you are very careful to check that none of the empirical coefficients used in each calculation are dimensional. Angles and angular velocities are generally measured in rad/s except when stated otherwise.
7 Airplane references (1) s = b /2 c(y) c c /4 cg c /4 y l T c Aspect ratio
8 Airplane references (2) c /4 h H c /4 Tail moment arm (l T ) Tail volume ratio V T = S T l T Sc
9 Airplane references (3) c /4 cg c /4 l F lf Fin moment arm (l F ) Fin volume ratio V F = S Fl F Sc
10 Axes and forces The USAF DATCOM uses lift and drag forces instead of X and Z. It also uses angle of attack instead of u and w. Lift U e X e θ e α e γ e V 0 Drag M mg Z e W e
11 Angle of attack Consider an aircraft in equilibrium at angle of attack α e, perturbed by small airspeeds u, w and small pitch rate q. U e U e +u W e α e α t V 0 or W e +w where
12 X derivatives Using the definition of the non-dimensional derivatives: where and are the derivatives of the lift and drag coefficients with respect to a horizontal velocity perturbation.
13 X derivatives continued Using the definition of the angle of attack perturbation Differentiating X with respect to a: and, finally,
14 X derivatives continued Using similar arguments, the other two X force derivatives become and
15 Z derivatives Using the definition of the non-dimensional derivatives: where and are the derivatives of the lift and drag coefficients with respect to a horizontal velocity perturbation.
16 Z derivatives (continued) As was done for the X force derivatives: and
17 M derivatives Clearly,,. The pitching moment derivative with respect to vertical velocity perturbation is: And the derivative with respect to vertical acceleration perturbation is
18 Derivatives to be determined This means that the X, Z and M derivatives can be determined if we calculate the: Drag derivatives: Lift derivatives: Pitching moment derivatives
19 Derivative The drag coefficient is given by the drag polar Differentiating with respect to angle of attack, a, we obtain The variation of the zero-lift drag with angle of attack can be neglected so that where (1)
20 Derivative The lift curve slope of the entire aircraft is given by where (2) d is the fuselage diameter, b is the wing span, S is the wing surface, S H is the horizontal tail surface, is the wing lift curve slope, is the horizontal tail lift curve slope and.
21 Wing and tail The lift curve slopes of the wing and tail can be obtained from where and. If the sectional lift curve slope changes along the span (i.e. wing profile changes) use the mean value. (3)
22 Downwash derivative The derivative of the wing s downwash can be found from where (4)
23 Derivative The pitching moment derivative with respect to angle of attack is where and the aerodynamic centre of the aircraft is
24 Effect of body on aerodynamic centre The combined wing and body aerodynamic centre is where the shift in the aerodynamic centre due to the body is
25 Body effects on the downwash include the effects of the fuselage and/or nacelles and/or tailbooms. Body effects
26 Body downwash The individual contributions for bodies or sections of bodies behind the wing are obtained from In front of the wing, they are obtained from where is in deg-1 and comes from Figure 1.
27 Figure 1
28 Derivative The derivative of the lift coefficient with respect to a horizontal velocity perturbation is simple: Calculate the value of C L for the aircraft at the flight Mach number and angle of attack from (4) where comes from equation (2)
29 Derivative The derivative of the drag coefficient with respect to a horizontal velocity perturbation is given by: Calculate the value of C L for the aircraft at the flight Mach number and a slightly higher Mach number M+DM from equation (4). Then calculate the corresponding drag coefficients from the drag polar of equation (1). Finally,
30 Derivative The derivative of the pitching moment coefficient with respect to a horizontal velocity perturbation is given by: Calculate the wing s aerodynamic centre at M and M+DM. Calculate the derivative from
31 Aerodynamic centre and Mach The aerodynamic centre lies on the quarter of the mean aerodynamic chord for low Mach numbers. As the Mach approaches 1, the aerodynamic centre moves backwards. The position of the aerodymic centre of a wing or tail can be obtained from (5) where, and are given by figures 2-4.
32 l is the taper ratio Figure 2 Subsonic: Supersonic:
33 Figure 2 continued
34 Figure 2 continued
35 Figure 3
36 Figure 4
37 Figure 4 continued
38 Figure 4 continued
39 Derivative The derivative of the drag coefficient with respect to a pitch rate perturbation is usually neglected.
40 Derivative The derivative of the drag coefficient with respect to a pitch rate perturbation is given by: The derivative for the wing is where is the distance between the centre of gravity and the wing s aerodynamics centre. is obtained from equation (3). The derivative for the tail is Where VT is the tail volume ratio and is also obtained from equation (3). Note that.
41 Derivative The derivative of the pitching moment coefficient with respect to a pitch rate perturbation is given by: The derivative for the wing is
42 Derivative continued and where K is obtained from figure 5. The derivative for the tail is where is obtained from equation (3).
43 Figure 5
44 Derivative The derivative of the drag coefficient with respect to a perturbation in the rate of change of angle of attack is negligible.
45 Derivative The derivative of the lift coefficient with respect to a perturbation in the rate of change of angle of attack is The derivative for the wing is where is obtained from equation (5), C R is the root chord, is taken from equation (3) and from figure 6.
46 Figure 6
47 Derivative continued The derivative for the tail is where is obtained from equation (3) and from equation (4).
48 Derivative The derivative of the pitching moment coefficient with respect to a perturbation in the rate of change of angle of attack is but the wing derivative is usually negligible, i.e. The tail derivative is where is obtained from equation (3) and from equation (4).
49 Control derivatives Three different type of longitudinal control surface deflections are considered: Flap deflection,. Only plain flaps are treated. Horizontal stabilizer incidence, Elevator deflection, Only lift and moment control derivatives are calculated. Drag derivatives can often be considered negligible for flight dynamics.
50 Derivative The derivative of the lift coefficient with flap deflection is (6) where is obtained from equation (3), is the incompressible sectional lift curve slope, is obtained from figure 7. is the flap span factor, obtained from figure 8 using the instructions from figure 9.
51 Derivative continued is the section lift curve slope variation with flap deflection, calculated from K where and are obtained from figures 10 and 11. if. If, is obtained from figure 12.
52 Figure 7
53 Figure 8
54 Figure 9
55 Figure 10
56 Figure 11
57 Figure 12
58 Derivative The derivative of the pitching moment coefficient with flap deflection is neglected here.
59 Derivative The derivative of the lift coefficient with horizontal stabilizer incidence is where is obtained from equation 3.
60 Derivative The derivative of the pitching moment coefficient with horizontal stabilizer incidence is where is obtained from equation 3.
61 Derivative The derivative of the lift coefficient with elevator deflection is where is obtained from equation 6.
62 Derivative The derivative of the pitching moment coefficient with elevator deflection is where is obtained from equation 6.
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