Pitching moment

Last updated December 31, 2024

A graph showing coefficient of pitching moment with respect to angle of attack for an airplane. Cm-alpha-curve-Cessna-182.svg — A graph showing coefficient of pitching moment with respect to angle of attack for an airplane.

In aerodynamics, the pitching moment on an airfoil is the moment (or torque) produced by the aerodynamic force with respect to the aerodynamic center on the airfoil . The pitching moment on the wing of an airplane is part of the total moment that must be balanced using the lift on the horizontal stabilizer.^[1]^{: Section 5.3} More generally, a pitching moment is any moment acting on the pitch axis of a moving body.

The lift on an airfoil is a distributed force that can be said to act at a point called the center of pressure. However, as angle of attack changes on a cambered airfoil, there is movement of the center of pressure forward and aft. This makes analysis difficult when attempting to use the concept of the center of pressure. One of the remarkable properties of a cambered airfoil is that, even though the center of pressure moves forward and aft, if the lift is imagined to act at a point called the aerodynamic center, the moment of the lift force changes in proportion to the square of the airspeed. If the moment is divided by the dynamic pressure, the area and chord of the airfoil, the result is known as the pitching moment coefficient. This coefficient changes only a little over the operating range of angle of attack of the airfoil.

The moment coefficient for a whole airplane is not the same as that of its wing. The figure on the right shows the variation of moment with AoA for a stable airplane. The negative slope for positive α indicates stability in pitch. The combination of the two concepts of aerodynamic center and pitching moment coefficient make it relatively simple to analyse some of the flight characteristics of an aircraft.^[1]^{: Section 5.10}

Measurement

The aerodynamic center of an airfoil is usually close to 25% of the chord behind the leading edge of the airfoil. When making tests on a model airfoil, such as in a wind-tunnel, if the force sensor is not aligned with the quarter-chord of the airfoil, but offset by a distance x, the pitching moment about the quarter-chord point, $M_{c/4}$ is given by

M_{c/4}=M_{\text{indicated}}+\mathbf {x} \times (D_{\text{indicated}},L_{\text{indicated}})

where the indicated values of D and L are the drag and lift on the model, as measured by the force sensor.

Coefficient

The pitching moment coefficient is important in the study of the longitudinal static stability of aircraft and missiles.

The pitching moment coefficient $C_{m}$ is defined as follows^[1]^{: Section 5.4}

C_{m}={\frac {M}{qSc}}

where M is the pitching moment, q is the dynamic pressure, S is the wing area, and c is the length of the chord of the airfoil. $C_{m}$ is a dimensionless coefficient so consistent units must be used for M, q, S and c.

Pitching moment coefficient is fundamental to the definition of aerodynamic center of an airfoil. The aerodynamic center is defined to be the point on the chord line of the airfoil at which the pitching moment coefficient does not vary with angle of attack,^[1]^{: Section 5.10} or at least does not vary significantly over the operating range of angle of attack of the airfoil.

In the case of a symmetric airfoil, the lift force acts through one point for all angles of attack, and the center of pressure does not move as it does in a cambered airfoil. Consequently, the pitching moment coefficient about this point for a symmetric airfoil is zero.

The pitching moment is, by convention, considered to be positive when it acts to pitch the airfoil in the nose-up direction. Conventional cambered airfoils supported at the aerodynamic center pitch nose-down so the pitching moment coefficient of these airfoils is negative.^[2]

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<span class="mw-page-title-main">Drag coefficient</span> Dimensionless parameter to quantify fluid resistance

In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.

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In fluid dynamics, angle of attack is the angle between a reference line on a body and the vector representing the relative motion between the body and the fluid through which it is moving. Angle of attack is the angle between the body's reference line and the oncoming flow. This article focuses on the most common application, the angle of attack of a wing or airfoil moving through air.

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<span class="mw-page-title-main">Decalage</span> Aeronautical engineering measurement

Decalage on a fixed-wing aircraft is a measure of the relative incidences of wing surfaces. Various sources have defined it in multiple ways, depending on context:

On a biplane, decalage can refer to the angle difference between the upper and lower wings, i.e. the acute angle contained between the chords of the wings in question.
On other fixed-wing aircraft, decalage can refer to the difference in angle of the chord line of the wing and the chord line of the horizontal stabilizer. This is different from the angle of incidence, which refers to the angle of the wing chord to the longitudinal axis of the fuselage, without reference to the horizontal stabilizer.

In flight dynamics, longitudinal stability is the stability of an aircraft in the longitudinal, or pitching, plane. This characteristic is important in determining whether an aircraft pilot will be able to control the aircraft in the pitching plane without requiring excessive attention or excessive strength.

Forces on sails result from movement of air that interacts with sails and gives them motive power for sailing craft, including sailing ships, sailboats, windsurfers, ice boats, and sail-powered land vehicles. Similar principles in a rotating frame of reference apply to windmill sails and wind turbine blades, which are also wind-driven. They are differentiated from forces on wings, and propeller blades, the actions of which are not adjusted to the wind. Kites also power certain sailing craft, but do not employ a mast to support the airfoil and are beyond the scope of this article.

The dynamic stall is one of the hazardous phenomena on helicopter rotors, which can cause the onset of large torsional airloads and vibrations on the rotor blades. Unlike fixed-wing aircraft, of which the stall occurs at relatively low flight speed, the dynamic stall on a helicopter rotor emerges at high airspeeds or/and during manoeuvres with high load factors of helicopters, when the angle of attack(AOA) of blade elements varies intensively due to time-dependent blade flapping, cyclic pitch and wake inflow. For example, during forward flight at the velocity close to V_NE, velocity, never exceed, the advancing and retreating blades almost reach their operation limits whereas flows are still attached to the blade surfaces. That is, the advancing blades operate at high Mach numbers so low values of AOA is needed but shock-induced flow separation may happen, while the retreating blade operates at much lower Mach numbers but the high values of AoA result in the stall.

References

1 2 3 4 Clancy, Laurence J. (1978). Aerodynamics. Pitman. ISBN 978-0-273-01120-0 . Retrieved 1 July 2022.
↑ Ira H. Abbott, and Albert E. Von Doenhoff (1959), Theory of Wing Sections, Dover Publications Inc., New York SBN 486-60586-8

Bibliography

L. J. Clancy (1975), Aerodynamics, Pitman Publishing Limited, London, ISBN 0-273-01120-0
Piercy, N.A.V (1943) Aerodynamics, pages 384–386, English Universities Press. London
Low-Speed Stability Retrieved on 2008-07-18

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[Clancy-1] 1 2 3 4 Clancy, Laurence J. (1978). Aerodynamics. Pitman. ISBN 978-0-273-01120-0 . Retrieved 1 July 2022.

[2] Ira H. Abbott, and Albert E. Von Doenhoff (1959), Theory of Wing Sections, Dover Publications Inc., New York SBN 486-60586-8

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