Directional Static Stability Calculator

Stability derivative (C_n,β)

Sideslip angle (β)

Yawing moment coefficient (C_n)

Turn the coefficient into a real moment

Enter the flight condition (dynamic pressure, wing area, wingspan) to get the yawing moment in newton-meters.

Estimate C_n,beta from the vertical tail

Enter the fin and wing geometry to estimate the stability derivative the tail produces, then compare it with the value you entered.

Yawing moment coefficient C_n

Directionally stable. The derivative is positive, so a sideslip makes a moment that swings the nose back into the wind, the way a weather vane points itself. The vertical tail is winning.

A stability derivative of 0.0025 1/° at a sideslip of 4 deg (°) gives a yawing-moment coefficient of 0.01.

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Show the yawing-moment-vs-sideslip graph

A line of the yawing-moment coefficient across the sideslip range. Its slope is the directional stiffness.

The steeper the upward slope, the stiffer the aircraft holds its heading. A line sloping downward would mean a divergent, unstable aircraft, and a flat line would mean no weathercock tendency at all.

An aircraft automatically corrects its orientation much like a weather vane on the roof of a barn and points its nose into the wind. This self-correcting behavior is called directional stability.

This calculator will calculate the yaw moment coefficient from slip angle and directional sensitivity coefficient. If one of the three fields is left blank, then this value will be automatically calculated by the tool.

What is directional stability?

Directional stability is often referred to as weathercock stability. It describes an aircraft's tendency to align its nose into the relative airflow when it is yawed by some factor. When a strong wind yaws the airplane, the airflow impinges on the sides of the fuselage and vertical stabilizer creating forces that generate a yawing moment and return the nose back into the oncoming air stream.

The vertical stabilizer plays an important role in restoring the heading. A larger or more aft vertical stabilizer produces a greater restoring moment for any yawing motion. The forward fuselage acts like a short, wide sail and exerts a force in the opposite direction, trying to turn the nose away from the wind. In a good design, the influence of the vertical stabilizer dominates over that of the fuselage throughout the flight envelope.

Here's how to use this calculator.

If two of the three values are entered, the third value will automatically be calculated. The damping angle can be expressed in degrees or radians and the traverse drift angle can be expressed in degrees or radians. This is a common mistake when doing manual calculations as the tool adapts both values units.

The three values are connected by a single equation so that each value can be calculated. If you input the damping angle and the lateral drift angle, then you get the coefficient; if you input the target coefficient and the lateral drift angle, then the design required damping angle is calculated; and if you input the damping angle and the target coefficient, then the corresponding lateral drift angle is calculated.

Understanding of formulas and symbols.

The relationship between yaw and roll is linear.

C_n = C_{n,\beta}\,\cdot\,\beta

Assume a small aircraft has an angle of attack of 0.0025 degrees and is subjected to a crosswind gust that causes it to yaw at an angle of 4 degrees. The yaw moment coefficient would be the product of 0.0025 and 4, which is 0.010.

C_n = 0.0025 \times 4 = 0.010

A positive sideslip angle produces a positive yawing moment that should turn the nose of the aircraft back into the wind. However this only happens if the dihedral is positive. So a positive dihedral means directional stability. A negative dihedral means the aircraft will diverge in any disturbance.

From coefficients to actual moments.

Since the coefficient is dimensionless, it alone does not reveal what moment in Newton-meters is present. To obtain the actual aerodynamic torque, the coefficients must be scaled by the dynamic pressure, wing area and wingspan depending on flight condition.

N = C_n \cdot q \cdot S \cdot b, \qquad q = \tfrac{1}{2}\rho V^2

This explains why a vertical stabilizer has more effect at high speed than it does at slow approach. The coefficient doesn't change with speed but since the dynamic pressure increases as the square of relative airspeed, the restoring moment also increases. If you open up the actual moment curve and enter values for q (dynamic pressure), S (wing area) and wing length, then you can read off the moment in Newton-meters.

The source of the small coefficient is the side rudder.

Directional stability is produced mainly by the vertical stabilizer so its contribution can be estimated from its geometric dimensions. An important value is the ratio of the volume of the vertical stabilizer to wing area, which indicates the area and moment arm of the vertical stabilizer compared with that of the wing.

V_v = \frac{l_v\,S_v}{b\,S_w}, \qquad C_{n,\beta,\text{tail}} = \eta_v\,a_v\,V_v

Here S_v is the area of the vertical stabilizer, l_v is the distance from the center of gravity to the vertical stabilizer, b is the wing length, S_w is the wing area, a_v is the slope of the lift curve of the vertical stabilizer and eta_v is the tailplane efficiency. The typical volume ratio of vertical stabilizers on conventional aircraft usually lies in the range 0.02 to 0.09.

Open the Vertical Tail tab and enter the dimensions of your vertical stabilizer and wing to find out the volume ratio of the vertical tail and how much coefficient it produces. By comparing this estimated value with the coefficient you entered above, you can see how much contribution the vertical stabilizer makes to stability.

Typical values:

Actual numbers help calibrate the results. The following are rough estimates for conventional aircraft and not design data.

Quantity	Symbol	Typical range
Stability derivative	C_n,beta	0.001 to 0.005 per degree (stable)
Sideslip angle	beta	within plus or minus 15 degrees (linear range)
Yawing-moment coefficient	C_n	a few hundredths in the linear range
Vertical tail volume ratio	V_v	about 0.02 to 0.09

Interpretation of the sign of the small coefficient:

A positive small coefficient means that the aircraft is stable and will return to its original heading like a wind vane. A value close to zero indicates very little stability, causing the aircraft to wander, be difficult to control, and require constant rudder inputs to maintain heading. A negative value means that the aircraft diverges from any sideslip and would not be controllable without a yaw damper or fly-by-wire flight controls system.

Some modern combat aircraft intentionally retain a small amount of instability and use computers to control the flight to restore stability, which allows for improved maneuverability. In other aircraft designers ensure that the coefficient is comfortably positive.

Reasons why this formula no longer applies at large rudder angles:

The linear relationship only holds when the airflow is attached to the vertical stabilizing surface. When the rudder angle exceeds about 15 degrees, the side fin begins to stall, reducing its contribution and causing actual turn rate to deviate from linearity. A dorsal fin, a long fairing in front of many fins, delays flow separation and extends the usable range.

The tool plots a linear model over the whole rudder angle range so you can see where that model is reliable and where the actual aircraft deviates from this model.

Frequently asked questions

What unit should be used for the stability derivative?

It doesn't matter whether you use degrees or radians but be consistent with the units for the rudder angle. Most engineering tables give values per degree because it is easier to read numbers close to 0.001 than 0.005 and smaller than the corresponding values in radians. This calculator tool will convert both values into a common base so you get the correct answer even if you mix degrees and radians on your screen.

Can the derivative be negative?

Mathematically this is possible and the tool will accept negative values if you want to explore such cases. Physically negative values represent an aircraft with directional stability problems. Instead of gliding out of a crosswind, it diverges. Most aircraft designs maintain positive values. A few fighter jets have slightly negative values relying on fly-by-wire flight control computers to compensate for this difference.

Why does the formula take small angles of attack?

This is a linear approximation that only holds true when the sideslip angle does not exceed about 15 degrees. Outside of this range there can be flow separation on the vertical stabilizer which makes the relationship nonlinear. In this case wind tunnel data or higher order models are required. A dorsal fin extends the linear region by delaying flow separation from the stabilizer.

How is the derivative actually measured?

There are three methods to do this, and the choice depends on cost and when tests can be done. These are wind tunnel tests of scale models, computational fluid dynamics (CFD) analysis, and flight tests with constant rate turns. Engineers compare results from these three methods and verify the design before flight testing the aircraft.

What is the difference between static and dynamic stability?

Static stability refers to the initial tendency to recover when a disturbance occurs, i.e. the sign and magnitude of the small coefficients that this computational tool works with. Dynamic stability describes how this response evolves over time, i.e. whether the resulting motion decays gradually or develops into a "dutch roll". Aircraft can be statically stable but still have problems with dynamic stability, which is why engineers study both aspects simultaneously.

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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: Flight dynamics (fixed-wing aircraft)
Yawing moment, directional stability, and the lateral-directional equations.
Wikipedia: Directional stability
Weathercock stability and the role of the vertical tail.
NASA Glenn: Yaw motion and the rudder
Plain-language explanation of yaw and the vertical tail.