Nutation
Encyclopedia
Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope
Gyroscope
A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

, planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

, or bullet
Bullet
A bullet is a projectile propelled by a firearm, sling, or air gun. Bullets do not normally contain explosives, but damage the intended target by impact and penetration...

in flight
External ballistics
External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bullet after it exits the barrel and before it hits the target.-Forces...

, or as an intended behavior of a mechanism. A pure nutation is a movement of a rotational axis such that the first Euler angle (precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...

) is constant.

Astronomy

The nutation of a planet happens because of tidal force
Tidal force
The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...

s that cause the precession of the equinoxes to vary over time so that the speed of precession is not constant. The nutation of the axis of the Earth was discovered in 1728 by the British astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...

James Bradley FRS was an English astronomer and served as Astronomer Royal from 1742, succeeding Edmund Halley. He is best known for two fundamental discoveries in astronomy, the aberration of light , and the nutation of the Earth's axis...

, but this nutation was not explained in detail until 20 years later.

Because the dynamic motions of the planets are so well-known, their nutations can be calculated to within arcseconds over periods of many decades. There is another disturbance of the Earth's rotation called polar motion
Polar motion
Polar motion of the earth is the movement of Earth's rotational axis across its surface. This is measured with respect to a reference frame in which the solid Earth is fixed...

that can be estimated for only a few months into the future because it is influenced by rapidly and unpredictably varying things such as ocean current
Ocean current
An ocean current is a continuous, directed movement of ocean water generated by the forces acting upon this mean flow, such as breaking waves, wind, Coriolis effect, cabbeling, temperature and salinity differences and tides caused by the gravitational pull of the Moon and the Sun...

s, wind systems, and motions in the liquid nickel-iron core of the Earth
Core of the Earth
Core of the Earth may refer to:* The inner core of the Earth* The outer core of the Earth...

.

Values of nutations are usually divided into components parallel and perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

. The component that works along the ecliptic is known as the nutation in longitude. The component perpendicular to the ecliptic
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...

is known as the nutation in obliquity. Celestial coordinate systems are based on an "equator" and "equinox," which means a great circle in the sky that is the projection of the Earth's equator outwards, and a line, the Vernal equinox intersecting that circle, which determines the starting point for measurement of right ascension
Right ascension
Right ascension is the astronomical term for one of the two coordinates of a point on the celestial sphere when using the equatorial coordinate system. The other coordinate is the declination.-Explanation:...

. These items are affected both by precession
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...

of the equinoxes and nutation, and thus depend on the theories applied to precession and nutation, and on the date used as a reference date for the coordinate system. In simpler terms, nutation (and precession) values are important in observation from Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

for calculating the apparent positions of astronomical objects.

Earth

In the case of the Earth, the principal sources of tidal force are the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

and Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

, which continuously change location relative to each other and thus cause nutation in Earth's axis. The largest component of Earth's nutation has a period of 18.6 years, the same as that of the precession of the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

's orbital nodes. However, there are other significant periodical terms that must be calculated depending on the desired accuracy of the result. A mathematical description (set of equations) that represents nutation is called a "theory of nutation" (see, e.g., http://www.iers.org/nn_10382/IERS/EN/Science/Recommendations/resolutionB3.html). In the theory, parameters are adjusted in a more or less ad hoc method to obtain the best fit to data. As can be seen from the IERS
International Earth Rotation and Reference Systems Service
The International Earth Rotation and Reference Systems Service , formerly the International Earth Rotation Service, is the body responsible for maintaining global time and reference frame standards, notably through its Earth Orientation Parameter and International Celestial Reference System ...

publication just cited, nowadays simple rigid-body mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

do not give the best theory; one has to account for deformations of the solid Earth.

Values

The principal term of nutation is due to the regression of the moon's nodal line and has the same period of 6798 days (18.6 years). It reaches plus or minus 17″ in longitude
Longitude
Longitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....

and 9″ in obliquity
Axial tilt
In astronomy, axial tilt is the angle between an object's rotational axis, and a line perpendicular to its orbital plane...

. All other terms are much smaller; the next-largest, with a period of 183 days (0.5 year), has amplitudes 1.3″ and 0.6″ respectively. The periods of all terms larger than 0.0001″ (about as accurately as one can measure) lie between 5.5 and 6798 days; for some reason they seem to avoid the range from 34.8 to 91 days, so it is customary to split the nutation into long-period and short-period terms. The long-period terms are calculated and mentioned in the almanacs, while the additional correction due to the short-period terms is usually taken from a table.

In mechanical engineering

A nutating motion can be seen in a swashplate
Swashplate
A swashplate is a device used in mechanical engineering to translate the motion of a rotating shaft into reciprocating motion, or to translate a reciprocating motion into a rotating one to replace the crankshaft in engine designs.- Construction :...

mechanism. In general, a nutating plate is carried on a skewed bearing on the main shaft and does not itself rotate, whereas a swashplate is fixed to the shaft and rotates with it. The motion is similar to the motions of coin or a tire wobbling on the ground after being dropped with the flat side down.

The nutating motion is widely employed in flowmeters
Flow measurement
Flow measurement is the quantification of bulk fluid movement. Flow can be measured in a variety of ways.Positive-displacement flow meters acumulate a fixed volume of fluid and then count the number of times the volume is filled to measure flow...

and pump
Pump
A pump is a device used to move fluids, such as liquids, gases or slurries.A pump displaces a volume by physical or mechanical action. Pumps fall into three major groups: direct lift, displacement, and gravity pumps...

s. The displacement of volume for one revolution is first determined. The speed of the device in revolutions per unit time is measured. In the case of flowmeters, the product of the rotational speed and the displacement per revolution is then taken to find the flow rate.

In physiology

In upright vertebrates, the sacrum
Sacrum
In vertebrate anatomy the sacrum is a large, triangular bone at the base of the spine and at the upper and back part of the pelvic cavity, where it is inserted like a wedge between the two hip bones. Its upper part connects with the last lumbar vertebra, and bottom part with the coccyx...

is capable of slight independent movement along the sagittal plane. When you bend backward the top (base) of the sacrum moves forward relative to the ilium
Ilium (bone)
The ilium is the uppermost and largest bone of the pelvis, and appears in most vertebrates including mammals and birds, but not bony fish. All reptiles have an ilium except snakes, although some snake species have a tiny bone which is considered to be an ilium.The name comes from the Latin ,...

; when you bend forward the top moves back. The anterior motion of the sacral base is called nutation, and the posterior motion is counter-nutation.