# Nutation

Rotation (green), precession (blue) and nutation in obliquity (red) of a planet

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.[1] A pure nutation is a movement of a rotational axis such that the first Euler angle (precession) is constant.[citation needed] In spacecraft dynamics, precession is sometimes referred to as nutation.[2]

## 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

$\mathbf{\tau} = \mathbf{\Omega} \times \mathbf{L},$

where L is the instantaneous angular momentum of the top.[3]

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.[3][4]

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.[5]

If the top has mass M and its center of mass is at a distance l from the pivot point, its gravitational potential is

$V = Mgl\cos\theta.$

In a coordinate system where the z axis is the axis of symmetry, the top has angular velocities ω1,ω2,ω3 and moments of inertia I1,I2,I3 about the x,y, and z axes. The kinetic energy is

$T = \frac{1}{2}I_1\left(\omega_1^2+\omega_2^2\right) + \frac{1}{2}I_3\omega_3^2.$

In terms of the Euler angles, this is

$T = \frac{1}{2}I_1\left(\dot{\theta}^2+\dot{\phi}^2\sin^2\theta\right) + \frac{1}{2}I_3\left(\dot{\psi}+\dot{\phi}\cos\theta\right)^2.$

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

$\dot{\phi} = \frac{b - a\cos\theta}{\sin^2\theta}.$

The tilt is determined by a differential equation for u = cos θ of the form

$\dot{u}^2 = f(u)$

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.[6]

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## Astronomy

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.[7]

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.

### Earth

Evolution of the position where the Tropic of Cancer crosses Carretera 83 (Vía Corta) Zaragoza-Victoria, Km 27+800 (Mexico). Nutation makes a small change to the angle at which the Earth tilts with respect to the Sun, thereby moving the location of the Tropic of Cancer, the most northerly latitude which the Sun can reach directly overhead. All four major circles of latitude that are defined by the Earth's tilt (both Tropical Circles and both Polar Circles) will shift correspondingly.

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.[1] 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.[8]

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.

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## Mechanical engineering

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.

Nutating disc engines are proposed from time to time, but there is little evidence of success.

More recently, a number of designs of drives and gearboxes have emerged that use nutating elements to provide very large reduction ratios in a compact unit.[9][10]

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## Physiology

In upright vertebrates, the sacrum 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; when you bend forward the top moves back.[11] The anterior motion of the sacral base is called nutation, and the posterior motion is counter-nutation.[12]

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## References

1. ^ a b Lowrie, William (2007). Fundamentals of geophysics (2nd ed.). Cambridge [u.a.]: Cambridge Univ. Press. pp. 58–59. ISBN 9780521675963.
2. ^ Kasdin, N. Jeremy; Paley, Derek A. (2010). Engineering dynamics : a comprehensive introduction. Princeton, N.J.: Princeton University Press. pp. 526–527. ISBN 9780691135373.
3. ^ a b Feynman, Leighton & Sands 2011, pp. 20–7
4. ^ Goldstein 1980, p. 220
5. ^ Goldstein 1980, p. 217
6. ^ Goldstein 1980, pp. 213–217
7. ^ Robert E. Bradley. "The Nodding Sphere and the Bird's Beak: D'Alembert's Dispute with Euler". The MAA Mathematical Sciences Digital Library. Mathematical Association of America. Retrieved 17 April 2013.
8. ^ "Resolution 83 on non-rigid Earth nutation theory". International Earth Rotation and Reference Systems Service. Federal Agency for Cartography and Geodesy. 2 April 2009. Retrieved 6 August 2012.
9. ^ Tom Shelley (06 November 2003). "Wobbling gears achieve high ratios". Eureka (Findlay Media). Retrieved 17 April 2013.
10. ^ Loewenthal, Stuart H; Townsend, Dennis P (Jan 1, 1972). "Design analysis for a nutating plate drive". NASA-TM-X-68117, E-7050.
11. ^ Kirkpatrick, Jeffrey Maitland ; photographs by Kelley (2001). Spinal manipulation made simple : a manual of soft tissue techniques. Berkeley: North Atlantic Books. p. 72. ISBN 978-1556433528.
12. ^ Joseph D. Kurnik, DC (December 16, 1996). "The AS Ilium Fixation, Nutation, and Respect". Dynamic Chiropractic 14 (26). Retrieved 17 April 2013.
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