fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...

, the center of pressure is the point on a body where a single

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

acting at that point can represent the total effect of the

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...

field acting on the body. The total force

vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

acting at the center of pressure is the

surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, o ...

of the pressure

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

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. Pressure fields occur in both static and dynamic fluid mechanics. Specification of the center of pressure, the reference point from which the center of pressure is referenced, and the associated force vector allows the moment generated about any point to be computed by a translation from the reference point to the desired new point. It is common for the center of pressure to be located on the body, but in fluid flows it is possible for the pressure field to exert a moment on the body of such magnitude that the center of pressure is located outside the body.

Hydrostatic example (dam)

Since the forces of water on a dam are hydrostatic forces, they vary linearly with depth. The total force on the dam is then the integral of the pressure multiplied by the width of the dam as a function of the depth. The center of pressure is located at the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

of the triangular shaped pressure field

\tfrac

from the top of the water line. The hydrostatic force and tipping moment on the dam about some point can be computed from the total force and center of pressure location relative to the point of interest.

Historical usage for sailboats

Center of pressure is used in

sailboat A sailboat or sailing boat is a boat propelled partly or entirely by sails and is smaller than a sailing ship. Distinctions in what constitutes a sailing boat and ship vary by region and maritime culture. Types Although sailboat terminology ...

design to represent the position on a

sail A sail is a tensile structure, which is made from fabric or other membrane materials, that uses wind power to propel sailing craft, including sailing ships, sailboats, windsurfers, ice boats, and even sail-powered land vehicles. Sails may b ...

where the

aerodynamic force In fluid mechanics, an aerodynamic force is a force exerted on a body by the air (or other gas) in which the body is immersed, and is due to the relative motion between the body and the gas. Force There are two causes of aerodynamic force: * ...

is concentrated. The relationship of the aerodynamic center of pressure on the sails to the hydrodynamic center of pressure (referred to as the center of lateral resistance) on the hull determines the behavior of the boat in the wind. This behavior is known as the "helm" and is either a weather helm or lee helm. A slight amount of weather helm is thought by some sailors to be a desirable situation, both from the standpoint of the "feel" of the helm, and the tendency of the boat to head slightly to windward in stronger gusts, to some extent self-feathering the sails. Other sailors disagree and prefer a neutral helm. The fundamental cause of "helm", be it weather or lee, is the relationship of the center of pressure of the

sail plan A sail plan is a drawing of a sailing craft, viewed from the side, depicting its sails, the spars that carry them and some of the rigging that supports the rig. By extension, "sail plan" describes the arrangement of sails on a craft. A sailing c ...

to the center of lateral resistance of the hull. If the center of pressure is astern of the center of lateral resistance, a weather helm, the tendency of the vessel is to want to turn into the wind. If the situation is reversed, with the center of pressure forward of the center of lateral resistance of the hull, a "lee" helm will result, which is generally considered undesirable, if not dangerous. Too much of either helm is not good, since it forces the

helmsman A helmsman or helm (sometimes driver or steersman) is a person who steering, steers a ship, sailboat, submarine, other type of maritime vessel, airship, or spacecraft. The rank and seniority of the helmsman may vary: on small vessels such as fis ...

to hold the rudder deflected to counter it, thus inducing extra drag beyond what a vessel with neutral or minimal helm would experience.

Aircraft aerodynamics

A stable configuration is desirable not only in sailing, but in

aircraft An aircraft ( aircraft) is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or the Lift (force), dynamic lift of an airfoil, or, i ...

design as well. Aircraft design therefore borrowed the term center of pressure. And like a sail, a rigid non-symmetrical

airfoil An airfoil (American English) or aerofoil (British English) is a streamlined body that is capable of generating significantly more Lift (force), lift than Drag (physics), drag. Wings, sails and propeller blades are examples of airfoils. Foil (fl ...

not only produces lift, but a moment. The center of pressure of an aircraft is the point where all of the aerodynamic pressure field may be represented by a single force vector with no moment. A similar idea is the aerodynamic center which is the point on an

where the

pitching moment In aerodynamics, the pitching moment on an airfoil is the Moment (physics), 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 ...

produced by the aerodynamic forces is constant with

angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a Airfoil#Airfoil terminology, reference line on a body (often the chord (aircraft), chord line of an airfoil) and the vector (geometry), vector representing the relat ...

. The aerodynamic center plays an important role in analysis of the

longitudinal static stability 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 pl ...

of all flying machines. It is desirable that when the pitch angle and angle of attack of an aircraft are disturbed (by, for example

wind shear Wind shear (; also written windshear), sometimes referred to as wind gradient, is a difference in wind speed and/or direction over a relatively short distance in the atmosphere. Atmospheric wind shear is normally described as either vertical ...

/vertical gust) that the aircraft returns to its original trimmed pitch angle and

without a pilot or

autopilot An autopilot is a system used to control the path of a vehicle without requiring constant manual control by a human operator. Autopilots do not replace human operators. Instead, the autopilot assists the operator's control of the vehicle, allow ...

changing the control surface deflection. For an aircraft to return towards its trimmed attitude, without input from a pilot or autopilot, it must have positive

Missile aerodynamics

Missiles typically do not have a preferred plane or direction of maneuver and thus have symmetric airfoils. Since the center of pressure for symmetric airfoils is relatively constant for small angle of attack, missile engineers typically speak of the complete center of pressure of the entire vehicle for stability and control analysis. In missile analysis, the center of pressure is typically defined as the center of the additional pressure field due to a change in the angle of attack off of the trim angle of attack. For unguided rockets the trim position is typically zero angle of attack and the center of pressure is defined to be the center of pressure of the resultant flow field on the entire vehicle resulting from a very small angle of attack (that is, the center of pressure is the limit as angle of attack goes to zero). For positive stability in missiles, the total vehicle center of pressure defined as given above must be further from the nose of the vehicle than the

center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weighted relative position of the distributed mass sums to zero. For ...

. In missiles at lower angles of attack, the contributions to the center of pressure are dominated by the nose, wings, and fins. The normalized

normal force In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts. In this instance '' normal'' is used in the geometric sense and means perpendicular, as opposed to the meanin ...

coefficient derivative with respect to the angle of attack of each component multiplied by the location of the center of pressure can be used to compute a centroid representing the total center of pressure. The center of pressure of the added flow field is behind the center of gravity and the additional force "points" in the direction of the added angle of attack; this produces a moment that pushes the vehicle back to the trim position. In guided missiles where the fins can be moved to trim the vehicles in different angles of attack, the center of pressure is the center of pressure of the flow field at that angle of attack for the undeflected fin position. This is the center of pressure of any small change in the angle of attack (as defined above). Once again for positive static stability, this definition of center of pressure requires that the center of pressure be further from the nose than the center of gravity. This ensures that any increased forces resulting from increased angle of attack results in increased restoring moment to drive the missile back to the trimmed position. In missile analysis, positive static margin implies that the complete vehicle makes a restoring moment for any angle of attack from the trim position.

Movement of center of pressure for aerodynamic fields

The center of pressure on a symmetric

typically lies close to 25% of the chord length behind the leading edge of the airfoil. (This is called the "quarter-chord point".) For a symmetric airfoil, as

and

lift coefficient 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 co ...

change, the center of pressure does not move. It remains around the quarter-chord point for angles of attack below the stalling angle of attack. The role of center of pressure in the control characterization of aircraft takes a different form than in missiles. On a cambered airfoil the center of pressure does not occupy a fixed location. For a conventionally cambered airfoil, the center of pressure lies a little behind the quarter-chord point at maximum lift coefficient (large angle of attack), but as lift coefficient reduces (angle of attack reduces) the center of pressure moves toward the rear.Clancy, L.J., ''Aerodynamics'', Section 5.11 When the lift coefficient is zero an airfoil is generating no lift but a conventionally cambered airfoil generates a nose-down pitching moment, so the location of the center of pressure is an infinite distance behind the airfoil. For a reflex-cambered airfoil, the center of pressure lies a little ahead of the quarter-chord point at maximum lift coefficient (large angle of attack), but as lift coefficient reduces (angle of attack reduces) the center of pressure moves forward. When the lift coefficient is zero an airfoil is generating no lift but a reflex-cambered airfoil generates a nose-up pitching moment, so the location of the center of pressure is an infinite distance ahead of the airfoil. This direction of movement of the center of pressure on a reflex-cambered airfoil has a stabilising effect. The way the center of pressure moves as lift coefficient changes makes it difficult to use the center of pressure in the mathematical analysis of

of an aircraft. For this reason, it is much simpler to use the aerodynamic center when carrying out a mathematical analysis. The aerodynamic center occupies a fixed location on an airfoil, typically close to the quarter-chord point. The aerodynamic center is the conceptual starting point for longitudinal stability. The horizontal stabilizer contributes extra stability and this allows the center of gravity to be a small distance aft of the aerodynamic center without the aircraft reaching neutral stability. The position of the center of gravity at which the aircraft has neutral stability is called the neutral point.

Notes

References

* * * Anderson, John D. (1999), ''Aircraft Performance and Design'', McGraw-Hill. * Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London. {{ISBN, 0-273-01120-0 Aircraft aerodynamics Fluid dynamics

Pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...

Vehicle dynamics