Pitching Moment Calculator

Pitching moment is a torque that rotates an aircraft about its lateral axis. Both lift and drag are forces acting in straight lines while pitching moment is a moment of force which determines whether or not the wing can maintain stable flight or will tip out of trim. Because incorrect calculations could result in unsafe flying, designers balance this moment over the entire speed range.

If you enter the moment coefficient, dynamic pressure, reference area and characteristic length, then you can calculate the moment. Enter the moment and three of the above inputs leaving one field blank and the tool will calculate the missing value.

What is the pitching moment?

The moment is usually specified relative to a selected reference point (usually the center of gravity or the aerodynamic center). The moment can be directly calculated using the following formula:

In this case M is the pitching moment, C_m is the dimensionless pitching moment coefficient, q is the dynamic pressure, S is the reference area (usually wing area) and the Greek letter l is the characteristic length (usually mean aerodynamic chord).

The sign of the moment has a real meaning. A positive moment will cause the nose to point upwards while a negative moment will push the nose downwards. In most stable aircrafts, the moment produced by the wing is slightly negative which is counteracted by the tail.

Please note that this formula looks like an additional term has been added to the lift formula. Lift is a force while moment is the product of a force and a distance, where the distance is the lever arm. The lever arm allows aerodynamics to be converted into torque.

How to use this calculator:

Enter four of the five fields and the fifth field will be automatically calculated. To calculate moment enter the moment coefficient, dynamic pressure, reference area, and characteristic length. If you want to achieve a certain target moment then enter that moment along with three inputs to find either the required moment coefficient or wing length.

Each field has a separate units menu so you can use pascals, kilopascals or psi for pressure, square meters or square feet for area and meters or feet for length as well as newton meters or pound foot for moment. Additional function layers can be enabled if desired. The dynamic pressure utility calculates q from air density and true headwind speed. The stability function layer reads the sign of the slope of the moment curve and determines trim angle. The force function layer converts the moment based on lever arm to an equivalent force.

Understanding of formula:

Let's take the example of a wing on a small drone model. The moment coefficient is C_m = -0.05, which indicates natural stability as the nose points slightly downwards. The dynamic pressure is q = 50 Pa, the wing area is S = 0.2 square meters and the average chord length is 0.15 meters. If you plug these values in,

A negative value means that the wing is generating a torque of 0.075 Newton meters which rotates it downwards. This torque is very small and approximately acceptable for a drone. If scaled up to the size of a real aircraft this value would be significantly higher.

For a light general aviation aircraft with C_m = -0.08, q = 3,000 Pa (roughly 70 m/s), S = 16 square meters, and wing length of 1.5 meters the resulting values are as follows:

The tailplane must counteract all this torque to maintain level flight.

The table below shows each symbol and its example values.

Dynamic pressure and relative air speed:

If the dynamic pressure is unknown, it can be calculated using air density and velocity.

The air density near the ground is about 1.225 kilograms per cubic meter and decreases with increasing altitude. As q is proportional to the square of the speed, the moment increases faster the faster you fly. If the speed doubles, the moment quadruples. That's why the pilot has to trim the aircraft every time the flight speed changes.

Moment and stability

Stability depends on how the moment coefficient changes with angle of attack. Since a stable aircraft has a negative slope, an increase in angle of attack will create a nose-down moment that returns the aircraft to trim.

When the moment coefficient is plotted as a linear function of angle of attack, the aircraft will be in trim at the point where this line intersects the zero axis.

Engineers scan various angles of attack to create stability curves and measure this tilt in a wind tunnel. There is one particular point on the longitudinal axis of the wing, called the aerodynamic center, where the moment coefficient hardly changes with angle of attack. For thin airfoils it is located about one quarter chord from the leading edge.

Practical applications

These values affect many design decisions. The control surfaces are designed to counteract the moment of the wing and bring the aircraft into a trimmed condition for level flight. The elevator is designed such that the pilot can compensate for this moment during flight.

The same mathematics applies to model aircraft, drones and unmanned aerial vehicles used by hobbyists as well as all objects moving through a fluid (including rudders, hydrofoils and underwater vehicles). As long as the distributed pressure load can be reduced to a single force and a single moment then this moment can be quantified with this formula.

Tips for accurate calculation:

For S use the wing area, not the wetted surface i.e. the contour area when viewed from above. For the characteristic length choose the mean aerodynamic chord and not the wingspan. Watch out for signs. Positive values are nose up, negative values are nose down. Also note that the coefficients depend on the angle of attack so pick values appropriate to actual flight conditions.