Longitudinal Static Stability Calculator

Q: How to calculate the neutral point and static stability distance?

The neutral point is the position of the center of gravity where the derivative of dC_M with respect to α is zero. It is calculated using the following formula: h_np = h_ac + V_H·η_t·(a_t/a)·(1 − dε/dα). In this case, a and a_t are the slopes of the lift curves of the wing and tail respectively, and the last term accounts for downwash. The static stability is determined by h_np − h; a positive value indicates stable flight. If the option to calculate the static stability reserve is enabled, then the calculator will perform both calculations.

Q: Why is there a subtraction in the expression of the tail term?

The horizontal tail produces a moment that pitches the nose of the aircraft down when it is pitched up beyond its trimmed condition. By convention, where pitching up is defined as positive, the tail acts in opposition to the wing and so is given a negative sign. On conventional aircraft trimmed for positive lift, the tail actually experiences a small downward force. This is also why the lift coefficient of the tail is usually small or negative.

Q: What is the typical static stability and why isn't it chosen to be higher?

For conventional certified airplanes, static stability is usually about 5 to 15 percent of the mean aerodynamic chord (MAC). The larger the stability, the more stable the airplane will be but it becomes heavier in pitch, requires a higher trim drag and needs more control surface deflection for maneuvers. The designer's goal is not necessarily to have as large a value as possible, but rather an adequate positive value.

Q: Why do some fighter jets fly with reduced or negative static stability?

Because aircraft that do not return quickly to trim can change their orientation rapidly, combat aircraft position their center of gravity close to or behind the neutral point and thus trade natural stability for maneuverability. They are able to fly because a fly-by-wire flight control computer continually makes small corrections in order to guide the inherently unstable aircraft in the desired direction.

Q: Can this formula be applied to canard-configured aircraft?

The physical laws are the same but because of the canard being in front of the main wing some signs change resulting in opposite lever arms and direction of action for lift. A negative tail arm input can be handled by this calculator with a canard configuration by treating the canard as a tail surface, but a special formula for stability of canard-configured aircraft is more accurate and less prone to error.

Q: Does this formula also apply to large angles of inclination?

This is an approximation that holds within a linear region and assumes the flow is attached. It is generally applicable when the angle of attack does not change by more than about ten degrees. Outside this region, the aerodynamic characteristics of the wing become nonlinear, the moment curve bends over and may reverse near stall. Most flight conditions for which certification is issued are within the linear region and provide sufficient margin.

Horizontal tail area (S_t)

Tail moment arm (l_t)

Wing reference area (S)

Mean aerodynamic chord (c)

Tail volume coefficient (V_H)

Zero-lift pitching moment (C_M,0)

Wing lift coefficient (C_L)

Tail lift coefficient (C_L,t)

Tail efficiency factor (η_t)

CG position h (fraction of chord)

Wing AC position h_ac (fraction of chord)

Pitching moment coefficient (C_M)

Show static margin and neutral point

Enter the wing and tail lift-curve slopes and the downwash factor to locate the neutral point and read the static margin — the real measure of pitch stability.

Wing lift-curve slope (a, per radian)

Tail lift-curve slope (a_t, per radian)

Downwash factor (1 − dε/dα)

Neutral point h_np (fraction of chord)

Static margin (SM)

% MAC

Turn the coefficient into a real moment

Enter the dynamic pressure to scale the pitching-moment coefficient into an actual moment in newton-metres for your flight condition.

Size the tail for a target static margin

Enter a static margin you want to hit and read back the horizontal tail area that delivers it for the rest of your geometry.

Find the aft CG limit

Enter the minimum static margin you will accept and read the aftmost CG position that still keeps the aircraft stable — the aft limit a POH would print.

Pitching moment coefficient C_M

Statically stable in pitch. The CG sits ahead of the neutral point, so the static margin is positive and dC_M/dα is negative. If a gust raises the nose, the aircraft makes a nose-down moment that brings it back to trim on its own — exactly what a negative slope means.

The geometry gives a tail volume coefficient of 0.625 and a pitching-moment coefficient of -0.143 at this condition.

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Show the pitching-moment-vs-angle-of-attack graph

A line of the pitching-moment coefficient across an angle-of-attack disturbance. A downward slope is the signature of a statically stable aircraft.

A downward slope is the signature of a statically stable aircraft: a nose-up gust makes a nose-down moment that returns it to trim. The steeper the line, the larger the static margin and the more sluggish the aircraft feels in pitch; a flat line is neutral stability and an upward slope is divergence.

An airplane maintains a stable attitude for good reason. When a strong wind pushes the nose up, it takes some force to bring it back down into its original position and that force results from the balance of three aerodynamic moments.

This calculator computes the total angle of attack for pitch moment based on geometric parameters of wings and tail surfaces, flight condition coefficients, and center of gravity position. It also determines values for neutral point and static stability, which are the numbers that actually determine aircraft stability with respect to the pitching axis. This helps in determining size of tail surfaces and establishing limits on how far aft the center of gravity can be moved.

Longitudinal static stability

Longitudinal static stability refers to the tendency of an aircraft to return to its trimmed pitch attitude after a disturbance. If strong winds raise the nose and aerodynamic forces cause it to come back down on its own, then the airplane is longitudinally stable.

If instead the same aerodynamic forces push the nose further up, then the aircraft is unstable. Without active control it would diverge. The mathematical analysis depends on a single sign.

\frac{dC_M}{d\alpha} < 0

The pitch angle gradient for the pitching moment must be negative. This is consistent with the stability derivative concept in roll and yaw, and there are similar pitfalls. As with roll a negative value is required to ensure stability. A positive gradient indicates an unstable condition.

Instructions for using the calculator:

When you input the geometric parameters of the wings and tailplanes, namely the tailplane area, the lever arm of the tailplane moment, the wing area and the mean wing length, the calculation tool will return the tailplane volume coefficient. If you also input the coefficients for flight conditions and the center of gravity position, the total moment coefficient will be calculated.

When the static stability surplus determination layer is activated, the calculation tool identifies the neutral point, reads out the static stability surplus and determines whether or not the aircraft is stable. In the two option design layers you can reverse equations. One option calculates the size of the horizontal tail based on a chosen static stability surplus, while the other one determines the most rearward center of gravity position at which the aircraft remains stable despite a fixed surplus.

Understanding of formulas.

The equation for longitudinal static stability breaks down the total moment into three physical components.

C_M = C_{M,0} + C_L\,(h - h_{ac}) - V_H\,\eta_t\,C_{L,t}

The first part, C_M,0, represents the pitching moment of the wing at zero lift. For a cambered wing this value is almost always negative. This means that by itself the wing has a tendency to pitch the nose down.

The second term, C_L·(h − h_ac), is the moment produced by the lift of the main wing about the center of gravity. If the center of gravity is ahead of the aerodynamic center, an increase in lift will cause the nose to point down, which contributes to stability. If the center of gravity is behind the aerodynamic center, an increase in lift will cause the nose to point up, which reduces stability.

The third part represents the recovery moment of the side wings and is scaled by the side wing volume coefficient. The side wing volume coefficient indicates how large the lever arm of the side wing is.

V_H = \frac{S_t\,l_t}{S\,c}

The larger the side wing or the longer the lever arm, or the smaller the main wing is, the greater V_H and the stronger the control effect of the side wing. Below you will find a calculation example for a light aircraft training plane with S = 16 m², S_t = 3 m², l_t = 5 m and c = 1.5 m.

V_H = \frac{3 \times 5}{16 \times 1.5} = 0.625

If C_M,0 = −0.05, C_L = 0.4, h = 0.30, h_ac = 0.25 and η_t = 0.9, C_L,t = 0.2 then the total pitching moment coefficient is:

C_M = -0.05 + 0.4\,(0.05) - 0.625 \times 0.9 \times 0.2 = -0.1425

A negative value indicates that there is a moment in this state that pushes the nose down.

Neutral point and static stability.

The angle of attack moment coefficient gives the moment at a particular state but what is important for stability is how this moment changes with the angle of attack. An important position is the neutral point, which is the location of the center of gravity when the aircraft is in a neutrally stable condition. In that case dC_M/dα is exactly zero.

h_{np} = h_{ac} + V_H\,\eta_t\,\frac{a_t}{a}\,\left(1 - \frac{d\varepsilon}{d\alpha}\right)

Here, 'a' and 'a_t' are the slopes of the lift curves of the wing and tail respectively. The last term takes into account the effect of the wing's wake on the angle of attack at the tail. The static stability reserve is given as the ratio of the distance from the center of gravity to the neutral point to the mean wing length.

SM = h_{np} - h

If the static stability margin is positive, then the center of gravity is ahead of the neutral point and the aircraft is stable. For certificated airplanes, the static stability margin is usually held to 5 to 15 percent of the wing length over the full range of weights. A larger margin increases stability but slows down the pitch response and requires more trim drag compensation. So designers do not strive for a maximum margin, but rather an adequate positive margin.

From coefficients to actual moment

Since the pitching-moment coefficient is dimensionless, it cannot be used to calculate the actual moment in newton-metres. To obtain the actual aerodynamic moment, the coefficient must be multiplied by the dynamic pressure, wing area and mean chord length. Note that for pitch, the reference length is the chord length of the airfoil, while for roll and yaw, the reference length is the wingspan.

M = C_M \cdot q \cdot S \cdot c, \qquad q = \tfrac{1}{2}\rho V^2

This explains why the feel of flight is different when gliding than it is during cruise flight. The coefficient does not change with speed, but since dynamic pressure increases as the square of the incoming velocity, the actual moment also increases. By turning on the Actual Moment layer and typing 'q' you can see the actual moment in newton meters.

Determination of the size of the control surface and establishment of the center of gravity limits

When used inversely it serves as a design tool. Aeronautical engineers use it to determine the size of the horizontal stabilizers in conceptual design while light aircraft manufacturers use it to establish the forward and aft center of gravity limits before they are specified in the operations manuals.

If you activate the level for the tail size and enter a desired static stability reserve, the calculation tool will return the required tail area to achieve this reserve. If you activate the level for the center of gravity limits and enter an acceptable minimum reserve, the rear center of gravity position is shown where the aircraft remains stable. Model airplanes or drones with too small or short tails cannot control their longitudinal axis no matter how powerful the servos are. With this formula you can determine what actual tail volume is required.

Reduced static stability, fly-by-wire and canard arrangement

Not all aircraft require a positive static stability margin. For example, fighter jets such as the F-16 have their center of gravity intentionally placed near or behind the neutral point. This is referred to as reduced static stability. Aircraft that do not return quickly to trim can change direction more quickly which is important for fighter pilots.

However these aircraft are still flyable as a flight control computer makes tiny corrections many times per second that would be impossible for humans to do. Aircraft with canards follow the same physics but since the horizontal surface is in front of the wing rather than behind it, the sign changes. This calculation allows you to account for canard configurations by entering a negative value for the tail lever arm although a special formula for canard layouts would be simpler.

Typical values

Actual values are helpful in calibrating results. The following ranges are rough guidelines for conventional fixed wing aircraft and do not represent design data.

Quantity	Symbol	Typical range
Tail volume coefficient	V_H	0.5 to 0.7 (GA), 0.9 to 1.2 (transport)
Static margin	SM	5 to 15 percent of chord (stable)
Tail efficiency	eta_t	0.85 to 0.95
Zero-lift moment	C_M,0	small and negative for cambered wings
Wing lift-curve slope	a	about 4 to 6 per radian

Why models no longer work at large angles of attack

The linear relationship between the pitching moment and angle of attack only holds when the airflow remains attached. When the angle of attack changes by about ten degrees, the aerodynamic properties of the wing become nonlinear, and the moment curve becomes curved. Near stall it can even reverse completely.

The plots in the calculators show a linear model within the disturbance range. This allows you to see directly a negative slope that has a stabilizing effect and also where the actual aircraft deviates from this model.

Frequently asked questions

At what values of C_M is the aircraft stable?

The stability is not determined by the value of C_M itself but rather by the slope of C_M with respect to angle of attack. A negative slope, i.e. dC_M/dα < 0, means that there is stability while a positive slope causes instability. Also the static margin must be positive and the center of gravity must be located ahead of the neutral point. This slope is directly shown in the plots from calculation tools.

How to calculate the neutral point and static stability distance?

The neutral point is the position of the center of gravity where the derivative of dC_M with respect to α is zero. It is calculated using the following formula: h_np = h_ac + V_H·η_t·(a_t/a)·(1 − dε/dα). In this case, a and a_t are the slopes of the lift curves of the wing and tail respectively, and the last term accounts for downwash. The static stability is determined by h_np − h; a positive value indicates stable flight. If the option to calculate the static stability reserve is enabled, then the calculator will perform both calculations.

Why is there a subtraction in the expression of the tail term?

The horizontal tail produces a moment that pitches the nose of the aircraft down when it is pitched up beyond its trimmed condition. By convention, where pitching up is defined as positive, the tail acts in opposition to the wing and so is given a negative sign. On conventional aircraft trimmed for positive lift, the tail actually experiences a small downward force. This is also why the lift coefficient of the tail is usually small or negative.

What is the typical static stability and why isn't it chosen to be higher?

For conventional certified airplanes, static stability is usually about 5 to 15 percent of the mean aerodynamic chord (MAC). The larger the stability, the more stable the airplane will be but it becomes heavier in pitch, requires a higher trim drag and needs more control surface deflection for maneuvers. The designer's goal is not necessarily to have as large a value as possible, but rather an adequate positive value.

Why do some fighter jets fly with reduced or negative static stability?

Because aircraft that do not return quickly to trim can change their orientation rapidly, combat aircraft position their center of gravity close to or behind the neutral point and thus trade natural stability for maneuverability. They are able to fly because a fly-by-wire flight control computer continually makes small corrections in order to guide the inherently unstable aircraft in the desired direction.

Can this formula be applied to canard-configured aircraft?

The physical laws are the same but because of the canard being in front of the main wing some signs change resulting in opposite lever arms and direction of action for lift. A negative tail arm input can be handled by this calculator with a canard configuration by treating the canard as a tail surface, but a special formula for stability of canard-configured aircraft is more accurate and less prone to error.

Does this formula also apply to large angles of inclination?

This is an approximation that holds within a linear region and assumes the flow is attached. It is generally applicable when the angle of attack does not change by more than about ten degrees. Outside this region, the aerodynamic characteristics of the wing become nonlinear, the moment curve bends over and may reverse near stall. Most flight conditions for which certification is issued are within the linear region and provide sufficient margin.

Related calculators

Directional Static Stability CalculatorFind the yawing-moment coefficient from the sideslip angle and the directional stability derivative, solve for any variable, estimate the derivative from the vertical tail, and scale it to a real yawing moment.

Lateral Static Stability CalculatorFind the rolling-moment coefficient from the sideslip angle and the roll stability derivative (the dihedral effect), solve for any variable, estimate the derivative from wing dihedral and position, and scale it to a real rolling moment.

Neutral Point CalculatorFind an aircraft's neutral point, static margin, and recommended centre of gravity from wing and tail geometry. Also gives the mean aerodynamic chord, tail volume coefficient, wing loading, and stall speed.

Static Thrust Load CalculatorFind the static thrust load capacity of pins, bearings, and fasteners from yield strength and contact area with Ft = Sy x As, and solve for any of the three. Add a safety factor for the safe working load, check an applied load, and compare materials.

Pitching Moment CalculatorCalculate the aircraft pitching moment from the moment coefficient, dynamic pressure, reference area, and characteristic length, or solve for any one of them. Includes a dynamic-pressure helper, a pitch-stability check, and charts.

Drag Coefficient CalculatorCalculate the drag coefficient from drag force, density, velocity, and area. Solve for any variable, plus dynamic pressure, drag area, power, the Reynolds number, and induced and total drag.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Longitudinal static stability
The pitching-moment equation, the neutral point, and the static margin.
Wikipedia: Static margin
Definition of static margin and its relation to the neutral point.
NASA Glenn: Aircraft pitch motion
Plain-language explanation of pitch and the pitching moment.