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[[Image:Flight dynamics with text.png|thumb|right|200px|Yaw motion in an aircraft]]

A '''yaw rotation''' is a movement around the [[Aircraft principal axes|yaw axis]] of a rigid body that changes the direction it is pointing, to the left or right of its direction of motion. The '''yaw rate''' or '''yaw velocity''' of a car, aircraft, projectile or other rigid body is the [[angular velocity]] of this rotation, or rate of change of the heading angle when the aircraft is horizontal. It is commonly measured in degrees per second or radians per second.

Another important concept is the [[Moment (physics)|yaw moment]], or yawing moment, which is the component of a [[torque]] about the yaw axis. 

==Measurement==
Yaw velocity can be measured by measuring the ground velocity at two geometrically separated points on the body, or by a [[gyroscope]], or it can be synthesized from [[accelerometers]] and the like. It is the primary measure of how drivers sense a car's turning visually. 

[[Image:Rotations.png|thumb|left|Axes of a ship and rotations around them]]

It is important in [[Electronic stability control|electronic stabilized]] vehicles. The yaw rate is directly related to the lateral acceleration of the vehicle turning at constant speed around a constant radius, by the relationship

:tangential speed*yaw velocity = lateral acceleration = tangential speed^2/radius of turn, in appropriate units

The sign convention can be established by rigorous attention to coordinate systems.

In a more general [[Basic fighter manoeuvre|manoeuvre]] where the radius is varying, and/or the speed is varying, the above relationship no longer holds.

==Yaw rate control==
The yaw rate can be measured with accelerometers in the vertical axis. Any device intended to measure the yaw rate is called a [[yaw rate sensor]].

Yaw rate control is a system that maintains the yaw rate of a vehicle, for example a car, according to the values requested by the driver. It differs from simple [[attitude control|Yaw control]] because is not the heading angle the controlled variable, but its variation rate instead.

It is one of the main targets of [[electronic stability control]] systems. Arrows, darts, rockets and airships have tail surfaces to achieve stability. A road vehicle does not have elements specifically designed to maintain stability, but relies primarily on the distribution of [[mass]].

==Road vehicles==
Studying the stability of a road vehicle requires a reasonable approximation to the equations of motion.

[[Image:Car0 stabilty.png|thumb|left|Dynamics of a road vehicle]]

The diagram illustrates a four-wheel vehicle, in which the front axle is located a metres ahead of the [[centre of gravity]] and the rear axle is b metres aft of the cg. The body of the car is pointing in a direction <math>\theta</math> (theta) whilst it is travelling in a direction <math>\psi</math> (psi). In general, these are not the same. The tyre treads at the region of contact point in the direction of travel, but the hubs are aligned with the vehicle body, with the [[steering]] held central. The tyres distort as they rotate to accommodate this mis-alignment, and generate side forces as a consequence. 

From [[directional stability]] study, denoting the angular velocity <math>\omega</math>, the equations of motion are:

:<math>\frac{d\omega}{dt}=2k\frac{(a-b)}{I}\beta-2k\frac{(a^2+b^2)}{VI}\omega</math>
:<math>\frac{d\beta}{dt}=-\frac{4k}{MV}\beta+(1-2k)\frac{(b-a)}{MV^2}\omega</math>

The coefficient of <math>\frac{d\beta}{dt}</math> will be called the '[[damping]]' by analogy with a mass-spring-damper which has a similar equation of motion. By the same analogy, the coefficient of <math>\beta</math> will be called the 'stiffness', as its function is to return the system to zero deflection, in the same manner as a spring.

The form of the solution depends only on the signs of the damping and stiffness terms. The four possible solution types are presented in the figure.

[[Image:Second Order Solutions.png|600px]]

The only satisfactory solution requires both stiffness and damping to be positive. If the centre of gravity is ahead of the centre of the [[wheelbase]] (<math>(b>a)</math>, this will always be positive, and the vehicle will be stable at all speeds. However, if it lies further aft, the term has the potential of becoming negative above a speed given by:

::<math>V^2=\frac{2k(a+b)^2}{M(a-b)}</math>

Above this speed, the vehicle will be directionally (yaw) [[unstable]]. Corrections for relative effect of front and rear tyres and steering forces are available in the main article.

==Relationship with other rotation systems==
These rotations are [[moving frame|intrinsic rotations]] and the calculus behind them is similar to the [[Frenet-Serret formulas]]. Performing a rotation in an intrinsic reference frame is equivalent to right-multiply its characteristic matrix (the matrix that has the vector of the reference frame as columns) by the matrix of the rotation.

==History==
The first aircraft to demonstrate active control about all three axes was the [[Wright brothers#1902 Glider|Wright brothers' 1902 glider]].<ref>{{cite web|url=http://www.grc.nasa.gov/WWW/K-12/airplane/rotations.html|title=Aircraft rotations|accessdate=2008-08-04}}</ref>

==See also==
* [[Adverse yaw]]
* [[Aircraft principal axes]]
* [[Coriolis acceleration]]
* [[Directional stability]]
* [[Flight dynamics]]
* [[Six degrees of freedom]]
* [[Vehicle dynamics]]
* [[Yaw rate sensor]]

==References==
<references />

[[Category:Dynamics]]