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Nutation (from Latin: nūtāre, to nod) is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behavior of a mechanism. If it is not caused by forces external to the body, it is called free nutation or Euler nutation. A pure nutation is a movement of a rotational axis such that the first Euler angle (precession) is constant. In spacecraft dynamics, precession is sometimes referred to as nutation.
Rigid body dynamics
If a top is set at a tilt on a horizontal surface and spun rapidly, its rotational axis starts precessing about the vertical. After a short interval, the top settles into a motion in which each point on its rotation axis follows a circular path. The vertical force of gravity produces a horizontal torque τ about the point of contact with the surface; the top rotates in the direction of this torque with an angular velocity Ω such that at any moment
where L is the instantaneous angular momentum of the top.
Initially, however, there is no precession, and the top falls straight downward. This gives rise to an imbalance in torques that starts the precession. In falling, the top overshoots the level at which it would precess steadily and then oscillates about this level. This oscillation is called nutation. If the motion is damped, the oscillations will die down until the motion is a steady precession.
The physics of nutation in tops and gyroscopes can be explored using the model of a heavy symmetrical top with its tip fixed. Initially, the effect of friction is ignored. The motion of the top can be described by three Euler angles: the tilt angle θ between the symmetry axis of the top and the vertical; the azimuth φ of the top about the vertical; and the rotation angle ψ of the top about its own axis. Thus, precession is the change in φ and nutation is the change in θ.
In terms of the Euler angles, this is
If the Euler-Lagrange equations are solved for this system, it is found that the motion depends on two constants a and b (each related to a constant of motion). The rate of precession is related to the tilt by
The tilt is determined by a differential equation for u = cos θ of the form
where f(u) is a cubic polynomial that depends on a and b as well as constants that are related to the energy and the gravitational torque. The roots of f(u) are the angles at which the rate of change of θ is zero. One of these is not related to a physical angle; the other two determine the upper and lower bounds on the tilt angle, between which the gyroscope oscillates.
The nutation of a planet happens because of gravitational attraction of other bodies 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 James Bradley, 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 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 currents, wind systems, and hypothesised motions in the liquid nickel-iron outer core of the Earth.
Values of nutations are usually divided into components parallel and perpendicular to the ecliptic. The component that works along the ecliptic is known as the nutation in longitude. The component perpendicular to the ecliptic 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. These items are affected both by precession 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 for calculating the apparent positions of astronomical objects.
In the case of the Earth, the principal sources of tidal force are the Sun and Moon, 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'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". In the theory, parameters are adjusted in a more or less ad hoc method to obtain the best fit to data. Simple rigid-body mechanics do not give the best theory; one has to account for deformations of the solid Earth.
The principal term of nutation is due to the regression of the moon's nodal line and has the same period of 6798 days (18.61 years). It reaches plus or minus 17″ in longitude and 9″ in obliquity. 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.
A nutating motion is similar to that seen in a swashplate 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 and pumps. 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.
A Nutating disc engine was patented in 1993 and a prototype was reported in 2006. The engine consisted of an internal disk wobbling on a Z-shaped shaft. Nutation has also been used in drive systems for gearboxes, with proposed uses including helicopter rotors, seat recliners and a European Space Agency probe to Mercury.
In upright vertebrates, the sacrum is capable of slight independent movement along the sagittal plane. When you bend backward the superior aspect of the sacrum (with sacral promontory) moves anteriorly relative to the ilium; when you bend forward the superior aspect of the sacrum moves posteriorly. The anterior motion of the sacral base is called nutation, and the posterior motion is counter-nutation.
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