Longitudinal Static Stability

Longitudinal Static Stability

Longitudinal Static Stability: the Moment Balance, the Neutral Point, the Fuselage and Downwash Estimates and the Centre-of-Gravity Limits

The problem

An airplane in trimmed flight has zero pitching moment about its centre of gravity. If a gust raises its angle of attack, the lift of the wing and tail changes; the airplane is statically stable in pitch when the resulting moment is nose down, bringing the angle of attack back, and unstable when it is nose up. In coefficient form the requirement is dC_m/dα < 0 together with C_m0 > 0, so that the moment can be zero at a positive lift coefficient. How strongly stable the airplane is depends above all on where the centre of gravity sits relative to the point at which the moment slope vanishes, the neutral point; the distance between them in chord lengths is the static margin. The pilot does not feel C_mα directly: Gilruth and White showed in NACA Report 711 (1941), from flight tests of fifteen airplanes, that what the pilot appreciates is the rate at which the elevator must move with angle of attack, dδ_e/dα, the ratio of the restoring moment to the elevator control moment, which sets both the stick travel and the stick-force gradient. This tool computes the moment balance for a conventional airplane (wing, fuselage with or without nacelles, horizontal tail with elevator), stick fixed, propeller off: the derivatives, the neutral point, the static margin, the trim, the tail load, the Report 711 criterion and the c.g. range.

Conventions

The angle of attack α is that of the fuselage reference line. The wing chord line sits at the incidence i_w above it and has the zero-lift angle α_L0 (negative for positive camber), so the wing operates at α_w = α + i_w − α_L0 from its zero-lift direction. The tail chord line sits at i_t above the reference line (negative, leading edge down, is the usual setting). Chordwise positions are fractions h = x/c̄ of the mean aerodynamic chord measured from its leading edge: h_ac for the wing aerodynamic centre (0.25 in thin-airfoil theory), h for the centre of gravity, h_np for the neutral point. The tail arm l_t is entered from the wing aerodynamic centre to the tail aerodynamic centre because that is the fixed geometric quantity; the arm from the c.g., l_t − (h − h_ac) c̄, follows. The elevator angle δ_e is positive trailing edge down, which increases the tail lift and gives a nose-down moment; Report 711 writes its criterion with the elevator positive up, and the tool prints both. Nose-up pitching moments are positive. Angles are entered in degrees and the slopes a_w and a_t per radian, as in the NACA reports; the library works in radians.

The moment equation

With the wing lift coefficient C_Lw = a_w α_w, the downwash at the tail ε = (dε/dα) α_w (proportional to the wing lift, hence zero at the wing's zero lift: in lifting-line theory ε = 2 C_Lw/(π AR) far behind an elliptical wing), the tail angle of attack α_t = α + i_t − ε, the tail lift coefficient C_Lt = a_t (α_t + τ δ_e) with τ the elevator effectiveness, and the tail efficiency η_t = q_t/q₀, the airplane lift and the moment about the c.g. are

C_L = C_Lw + η_t (S_t/S) C_Lt,

C_m = C_m,ac + (h − h_ac) C_Lw + C_m0,f + C_mα,f α − η_t V_H C_Lt, V_H = S_t l_cg/(S c̄),

where the fuselage (and nacelles) enter as a moment slope C_mα,f about the reference angle and a camber moment C_m0,f; a slender body carries no net lift, so its moment is the same about any point. The tail drag, the vertical offsets of the thrust line, the wing and the tail from the c.g., and the lift of the fuselage are neglected, as in the classical treatment (Perkins and Hage, Etkin, Nelson). Collecting terms,

a = a_w + η_t (S_t/S) a_t (1 − dε/dα), C_Lδ = η_t (S_t/S) a_t τ,

C_mα = a_w (h − h_ac) + C_mα,f − η_t V_H a_t (1 − dε/dα), C_mδ = −η_t V_H a_t τ,

C_m0 = C_m,ac + C_m0,f + (h − h_ac) a_w (i_w − α_L0) − η_t V_H a_t [i_t − (dε/dα)(i_w − α_L0)].

The page prints each derivative with its wing, fuselage and tail parts so that the contributions can be read off: the wing is destabilizing when the c.g. is behind the aerodynamic centre, the fuselage always, the tail stabilizing in proportion to its volume and to (1 − dε/dα).

The neutral point and the static margin

The neutral point is the c.g. position at which C_mα = 0. Because the tail arm from the c.g. is itself l_t − (h − h_ac) c̄, the tail term contains h too, and substituting it gives C_mα = a (h − h_ac) + C_mα,f − η_t V_H,ac a_t (1 − dε/dα) with V_H,ac = S_t l_t/(S c̄) the tail volume about the aerodynamic centre: the slope is exactly linear in h with the airplane lift-curve slope a as coefficient. Hence

h_np = h_ac + [η_t V_H,ac a_t (1 − dε/dα) − C_mα,f] / a, C_mα = −a (h_np − h), dC_m/dC_L = C_mα/a = −(h_np − h).

The static margin SM = h_np − h is therefore exactly the negative of the slope of C_m against C_L at fixed elevator, which is how Figure 2 of the results page should be read. Textbooks often give the approximate form h_np ≈ h_ac − C_mα,f/a_w + η_t V_H (a_t/a_w)(1 − dε/dα) with V_H evaluated at the c.g. and a_w in place of a (Nelson eq. 2.37); the tool prints it beside the exact value. The two agree when the c.g. is at the aerodynamic centre and the tail's share of the lift slope is neglected; otherwise Nelson's form places the neutral point a few percent of the chord too far aft for a c.g. behind the aerodynamic centre. The wing-body aerodynamic centre, h_ac − C_mα,f/a_w, is the neutral point of the airplane with the tail off and is also printed.

Trim and the tail load

At a flight condition the lift coefficient to trim is C_L,trim = n W/(q S) with q = ½ ρ V² from the true airspeed and the 1976 Standard Atmosphere density at the altitude (level flight, n = 1). Trim requires C_L = C_L,trim and C_m = 0, two linear equations in α and δ_e:

a α + C_Lδ δ_e = C_L,trim − C_L0, C_mα α + C_mδ δ_e = −C_m0,

solved exactly (determinant a C_mδ − C_Lδ C_mα). From the solution follow the wing and tail angles of attack, the downwash, the wing and tail lift coefficients and the tail load η_t q S_t C_Lt, negative when the tail pushes down (the usual case for a c.g. ahead of the neutral point with a cambered wing). The page checks the balance by recomputing C_L and C_m from the component build-up at the solution.

The elevator gradients and the criterion of NACA Report 711

Along the trim curve C_mα dα + C_mδ dδ_e = 0, so the elevator moves with angle of attack at dδ_e/dα = −C_mα/C_mδ, and with lift coefficient at dδ_e/dC_L = −C_mα/(a C_mδ − C_Lδ C_mα); both vanish at the neutral point, which is the flight-test definition of it (the c.g. at which the elevator angle to trim no longer changes with speed). Gilruth and White derived dδ_e/dα for the propeller-off airplane from the moments of the wing, fuselage, nacelles and tail as their equation (1),

dδ_e/dα = (1/τ) [ (1 − dε/dα) + ( S_w d dC_L/dα_w − K_f w_f² L_f − K_f N w_n² L_n ) / ( (q_t/q₀) l_t S_t dC_Nt/dα_t ) ],

with the elevator angle positive up, d the distance from the wing aerodynamic centre to the c.g. (positive when the c.g. is ahead; their Table I lists d = −0.45 ft for airplane 1, whose c.g. was behind the aerodynamic centre), l_t from the c.g. to the elevator hinge line and q_t/q₀ = 0.9. Dividing the numerator and denominator of the second term by q S c̄ shows that it is exactly −C_mα/C_mδ of the equations above, with K_f w_f² L_f/(S c̄) the fuselage slope; the tool prints the equation in the report's symbols with the substituted values and confirms the identity. The report found the computed values to agree with wind-tunnel measurements on seven models (its Figure 10) and, after adding an empirical term K_p N_p D² l_p with K_p = 0.65 for the idling propeller, with the flight values of fifteen airplanes (Figure 12). It suggests dδ_e/dα of about 0.5 as the lower limit for design in the gliding condition, giving about 4 inches of stick travel in power-on manoeuvres, and finds that about 0.2 must be exceeded for stick-free stability with conventional tails. Low-wing monoplanes lost stability with power on as the angle of attack rose, high-wing monoplanes little; the propeller term is not included in this tool, which treats the propeller-off airplane.

Estimating the fuselage moment

Multhopp's strip method. Munk showed (NACA Report 184, 1924) that a slender body in a stream inclined at α experiences a pure couple, dM/dα = 2 (k₂ − k₁) q × volume, where k₁ and k₂ are the apparent-mass coefficients for longitudinal and transverse motion; for an ellipsoid of revolution Lamb gave them exactly (Report 184, Table I: k₂ − k₁ = 0.49 at a fineness ratio of 2, 0.78 at 4, 0.92 at 8, tending to 1), and Munk applies the ellipsoid's value to any body of the same length and volume. Multhopp (TM 1036, 1942) noticed that the moment of the fuselage of an airplane is dominated not by the free-stream angle but by the flow the wing imposes on it: ahead of the wing the body sits in the upwash, which raises its local angle of attack, over the wing the lift is carried by the wing itself, and behind the trailing edge the body sits in the downwash, which reduces the local angle from zero at the trailing edge to (1 − dε/dα) α at the tail. Writing the volume integral as a sum over segments of width w_i and length Δx_i (the cross-section area π w²/4 for a round body),

C_mα,f = (π/2)(k₂ − k₁) Σ w_i² (dα_local/dα)_i Δx_i / (S c̄) per radian,

which is the 1/36.5 form of Perkins and Hage and Nelson with the angle in degrees. The textbooks read the upwash gradient ahead of the wing from a chart derived from Multhopp's work; this tool uses instead the thin-airfoil closed form for the upwash on the chord line extended ahead of a lifting flat plate, w/(V α) = √((ξ + 1)/(ξ − 1)) − 1 with ξ = 1 + 2x/c_r, x the distance ahead of the leading edge and c_r the root chord, scaled by a_w/(2π) for the wing's actual bound vorticity and averaged analytically over each segment (its integral is √(ξ² − 1) + acosh ξ, finite at the leading edge). The closed form has the square-root rise of the chart at the leading edge and the bound-vortex far field c/(2r); the segment immediately ahead of the wing, where the upwash is largest, dominates the sum, so the fuselage slope depends on how that segment is divided: finer segments near the leading edge give a slightly larger value. Segments lying over the wing chord contribute nothing; segments behind the trailing edge take the downwash factor (x/l_h)(1 − dε/dα) with l_h = l_t − 0.75 c_r from the trailing edge to the tail aerodynamic centre. The result is a destabilizing slope of the order of 0.1 to 0.5 per radian for light airplanes, larger for fat fuselages and rear-mounted wings.

Report 711's factor K_f. Gilruth and White take the fuselage slope as dM/dα = K_f w_f² L_f q with K_f read from their Figure 8 against the position of the wing-root quarter chord in percent of the body length, a curve fitted to the 209 wing-fuselage combinations of Jacobs and Ward (NACA Report 540); the same factor applies to each nacelle with its own width and length. K_f rises from about 0.1 for a wing at 10 % of the length to 1.0 at 40 % and 3.0 at 62 %, because a rear wing leaves a long forebody in the upwash. The tool's copy of the curve was digitized from the NTRS scan at 2 % intervals (readings in the References folder); its implied width-squared-times-length form is Multhopp's sum with the upwash weighting folded into one empirical factor, which is why the two methods give values of the same order but not the same number for the same fuselage.

Tail slope, elevator effectiveness and downwash

The tail lift-curve slope may be entered or taken from the lifting-line form Silverstein and Katzoff fitted to their seventeen tail surfaces (NACA Report 688, eq. 2), a_t = a₀/(1 + r a₀/(π A_t)) with the section slope a₀ per radian and the end-plate factor r (1 without end plates, 0.63 for twin fins of height 0.32 of the span, their Figure 25); Report 711 writes it as 5.7/(1 + 1.8 r/A_t) with a₀ = 5.7 per radian. The elevator effectiveness τ, the ratio of the lift produced per unit elevator angle to that per unit tail angle, depends mainly on the ratio of elevator area behind the hinge line to tail area; Report 711 Figure 3 (from the Report 688 tests, open gap) gives 0.15 at S_e/S_t = 0.1, 0.34 at 0.2, 0.45 at 0.3, 0.54 at 0.4 and 0.62 at 0.5, well below the thin-airfoil flap theory, and the tool offers the curve as digitized. The downwash gradient at the tail may be entered (from a wind tunnel or from the charts of NACA Report 648, which Report 711 reduces to its Figures 4 to 6 and applies in a worked example to its airplane 1: 0.56 at the centre line corrected to 0.53 over the tail span), taken as the lifting-line far-field value of an elliptical wing dε/dα = 2 a_w/(π AR), or from the DATCOM fit (section 4.4.1, as given by Raymer) dε/dα = 4.44 [K_A K_λ K_H √cos Λ]^1.19 with K_A = 1/AR − 1/(1 + AR^1.7), K_λ = (10 − 3λ)/7 and K_H = (1 − |h_H/b|)/(2 l_H/b)^1/3, which accounts for the taper, the tail height above the wake and the sweep. The lifting-line value is a far-field, centre-line figure and is usually a little high for a tail at a finite distance and above the wake; Report 711's tests gave 0.3 to 0.6 for conventional layouts.

Centre-of-gravity limits

The aft limit follows from the required minimum static margin, h_aft = h_np − SM_min; 5 % of the chord is a common design minimum, Report 711 would rather specify dδ_e/dα ≈ 0.5. The forward limit comes from the elevator: as the c.g. moves forward the nose-down arm of the wing lift grows and the tail arm shortens, so more up elevator is needed to trim a given lift coefficient, most at C_Lmax in the landing approach. The tool solves the trim equations for δ_e at C_Lmax as a function of h and finds by bisection the c.g. at which it reaches the up-elevator limit entered; when the limit is not reached down to 1.5 chord lengths ahead of the aerodynamic centre, or is already exceeded at the neutral point, the page says so. Ground effect (which reduces the downwash and increases the up elevator needed in the flare), the landing-flap pitching moment and the stick-free limits move the real forward limit further forward and are not included.

Reading the results

Figure 1 marks the aerodynamic centre, the c.g., the neutral point and the limits on the chord scale of a side view. The derivative section gives a, C_L0, C_Lδ, C_mα (with its wing, fuselage and tail parts), C_m0 and C_mδ; the neutral-point section the exact h_np, Nelson's approximation, the wing-body aerodynamic centre and the static margin; the trim section the angle of attack, elevator angle, downwash, wing and tail lift coefficients and loads with a check of the balance; the Report 711 section the equation (1) in the report's symbols with its value, the same from the derivatives, and the gradients in this page's sign convention; the c.g. section the aft and forward limits and the range. Figure 2 draws C_m against C_L for the c.g. entered (elevator at trim and neutral), the neutral point and the limits, each line's slope being minus its static margin; Figure 3 the elevator angle to trim against speed for the same c.g. positions with the up-elevator limit, which the forward-limit curve meets at the stall speed. Not covered: stick-free stability and stick forces, power effects, ground effect, the manoeuvre point, flaps, compressibility, and unconventional (canard, tailless, T-tail with a different downwash) layouts beyond what the entered dε/dα and η_t can represent.