Lateral Static Stability Calculator

Watch how a paper airplane drifts sideways after a strong gust of wind and how its wings stabilize as it wobbles. The same thing happens with real airplanes. This property that causes this phenomenon is called lateral static stability.

This calculator will determine the roll moment coefficient given the sideslip angle and the roll stability derivative (sometimes referred to as "dihedral effect"). If one of the fields is left blank, it will automatically calculate that value.

What is lateral static stability?

Lateral static stability describes the tendency of an aircraft to return its wings to a level position when it is deflected sideways by some external influence. When the airplane is banked, the lift vector also banks and the airplane begins to move sideways through the air. This sideways flow of air hits the left and right wing unevenly, and due to the shape of the wing one wing produces more lift than the other.

This difference in lift creates a rolling moment that raises the lowered wing back up. A major factor is the angle of attack of the wings. This refers to the slight upward V-shape you see when looking at an airliner from above. The wing facing into the crosswind has a greater effective angle of attack and therefore produces more lift than the other wing. As a result, the plane rolls in the opposite direction of the crosswind and approaches level again.

How to use this calculator.

If two of the three values are entered, then the third value is automatically calculated. The attenuation coefficient can be expressed in either degrees or radians and the cross drift angle can also be expressed in either degrees or radians. This calculator will ensure that both units match, thus avoiding the most common errors made when doing calculations manually.

These three values are connected by a single equation so they can be solved for from any side. If the damping ratio and the sideslip angle are inputted, the coefficient is obtained. If the desired coefficient and the sideslip angle are inputted, the required damping ratio for design is calculated. If the damping ratio and the coefficient are inputted, the corresponding sideslip angle that produces this coefficient is calculated.

Understanding of formula and its symbols.

The relationship between sideslip and roll is linear.

As a concrete example, let's say that the roll damping coefficient of a high-wing trainer is -0.0020 per degree and a gust causes a sideslip of 3 degrees to the right. The resulting roll moment coefficient would be:

The sign determines everything. Also the convention is reversed to the directional stability (yawing stability). A positive sideslip, i.e. a sideslip to the right, requires a negative roll moment coefficient, thus a roll motion to the left, in order to raise the lowered right wing again. This only applies if the damping coefficient itself is negative. Therefore, a negative damping coefficient means lateral stability. A positive damping coefficient would mean that the aircraft further amplifies the roll motion with every sideslip and deviates from level flight, which leads to loss of control.

How to calculate actual torque from the coefficient.

As the coefficient is dimensionless, its value alone cannot directly determine the moment in Newton-meters. To obtain the actual aerodynamic rolling moment, the coefficient must be multiplied by the dynamic pressure, wing reference area and wingspan, all of which vary with flight conditions.

This explains why the airplane feels stable at high speeds and mushy at low approach speeds. The coefficient does not change with speed but dynamic pressure is proportional to the square of airspeed which causes the restoring moment to be larger as well. If you open up the actual moment values and enter in the dynamic pressure (q), area (S) and wing span, you can read off the moment in Newton meters. Negative values indicate a restoring moment that will return the wing back to level position.

Where does the effect of the upper V-angle come from?

There are three design choices that affect the roll stability coefficient, of which the first two can be derived from shape. The geometric angle of attack, or V-shape with the wing bent upwards is an obvious factor. For straight wings the angle of attack can easily be deduced from the angle of incidence and the lift curve for the airfoil using strip theory.

Here 'a' is the slope of the wing's lift curve in radians and Γ is the upper v-groove angle in radians. The position where the wing attaches to the fuselage also plays an important role. In high-wing aircraft like the Cessna 172, the fuselage turns the airflow sideways so that it flows over the top of the wing going up, which gives a natural extra stability. In low-wing aircraft this is reversed, so a larger geometric upper v-groove angle is required to compensate for the shadowing of the root of the wing by the fuselage.

A third factor is wing sweep. With a swept back wing, the top of the wing experiences more relative airflow, which alone produces a very strong negative damping coefficient that increases with increasing lift coefficient. When you enter the angle of attack, lift curve and wing position in the geometry section, a calculation tool estimates the damping coefficient produced by the wing so you can compare it to your input values.

Balance: spiral stability and dutch roll

The designers balance the angle of attack, wing position and wing sweep to control the effect of the angle of attack. If the effect is too small, then the aircraft will enter into a slow spiral without pilot input. If it is too large, then Dutch Roll can occur. This is a coupled oscillation between yawing and rolling where the tail section swings from side to side and the wings move up and down.

Modern transport aircraft with swept-back wings often have a low angle of incidence due to the high negative pitching moment alone from wing sweep, i.e. a shape where the wings are slightly angled down to bring the Dutch roll into a safe region. A damper and stability augmentation system eliminates the remaining coupled motions. In contrast, combat aircraft use very small angles of incidence. For combat aircraft, agile roll control during manoeuvres is more important than stability in uncoupled flight, as the fly-by-wire flight control system can provide this damping.

Typical values

Actual values can help in fine tuning. The following ranges are rough guidelines for conventional fixed wing aircraft and do not represent design data.

Recognize the sign of small coefficients.

A negative low value means that the aircraft is stable and will automatically return its wings to a level position. A value close to zero indicates little stability, which can cause the aircraft to slowly become uncontrollable, entering into a spiral and wandering off course without control. A positive value means that the aircraft will continue to increase its roll with each lateral movement, making it basically unairworthy without controls.

The magnitude indicates the strength of the restoring reaction. Both too strong and too weak values are problematic. A very large negative damping ratio can cause the design to enter into a Dutch roll, so the goal is for a moderately negative value rather than necessarily the most negative possible value.

Why the formula no longer works with large lateral deflection angles.

If the sideslip angle exceeds about 15 degrees, there is a risk that the front wing will stall, reducing the additional lift and causing the actual curve to deviate from the straight line.

The graphs of the calculation tools show a linear model over the entire range of the cross-drift angle so that you can see where the model is reliable and when the actual aircraft performance deviates from the model.