Pitching moment

Last updated July 13, 2023

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 on the airfoil if that aerodynamic force is considered to be applied, not at the center of pressure, but at the aerodynamic center of 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]

Related Research Articles

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In aeronautics, the chord is an imaginary straight line joining the leading edge and trailing edge of an aerofoil. The chord length is the distance between the trailing edge and the point where the chord intersects the leading edge. The point on the leading edge used to define the chord may be the surface point of minimum radius. For a turbine aerofoil the chord may be defined by the line between points where the front and rear of a 2-dimensional blade section would touch a flat surface when laid convex-side up.

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In fluid mechanics, the center of pressure is the point where the total sum of a pressure field acts on a body, causing a force to act through that point. The total force vector acting at the center of pressure is the surface integral of the pressure vector field across the surface of the body. The resultant force and center of pressure location produce an equivalent force and moment on the body as the original pressure field.

<span class="mw-page-title-main">Angle of attack</span> Angle between the chord of a wing and the undisturbed airflow

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.

In aeronautics, the aspect ratio of a wing is the ratio of its span to its mean chord. It is equal to the square of the wingspan divided by the wing area. Thus, a long, narrow wing has a high aspect ratio, whereas a short, wide wing has a low aspect ratio.

An airfoil or aerofoil is a streamlined body that is capable of generating significantly more lift than drag. Wings, sails and propeller blades are examples of airfoils. Foils of similar function designed with water as the working fluid are called hydrofoils.

In fluid dynamics, the lift coefficient is a dimensionless quantity that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity and an associated reference area. A lifting body is a foil or a complete foil-bearing body such as a fixed-wing aircraft. $C L$ is a function of the angle of the body to the flow, its Reynolds number and its Mach number. The section lift coefficient $c l$ refers to the dynamic lift characteristics of a two-dimensional foil section, with the reference area replaced by the foil chord.

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<span class="mw-page-title-main">Downforce</span> Downwards lift force created by the aerodynamic characteristics of a vehicle

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<span class="mw-page-title-main">Flap (aeronautics)</span> Anti-stalling high-lift device on aircraft

A flap is a high-lift device used to reduce the stalling speed of an aircraft wing at a given weight. Flaps are usually mounted on the wing trailing edges of a fixed-wing aircraft. Flaps are used to reduce the take-off distance and the landing distance. Flaps also cause an increase in drag so they are retracted when not needed.

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In aeronautics and aeronautical engineering, camber is the asymmetry between the two acting surfaces of an airfoil, with the top surface of a wing commonly being more convex. An airfoil that is not cambered is called a symmetric airfoil. The benefits of cambering were discovered and first utilized by George Cayley in the early 19th century.

The NACA airfoils are airfoil shapes for aircraft wings developed by the National Advisory Committee for Aeronautics (NACA). The shape of the NACA airfoils is described using a series of digits following the word "NACA". The parameters in the numerical code can be entered into equations to precisely generate the cross-section of the airfoil and calculate its properties.

In aerodynamics, the torques or moments acting on an airfoil moving through a fluid can be accounted for by the net lift and net drag applied at some point on the airfoil, and a separate net pitching moment about that point whose magnitude varies with the choice of where the lift is chosen to be applied. The aerodynamic center is the point at which the pitching moment coefficient for the airfoil does not vary with lift coefficient, making analysis simpler.

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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