Nutation

Nutation (from Latin nūtātiō, 'nodding, swaying') 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 behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. 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 is constant. In spacecraft dynamics, precession (a change in the first Euler angle) is sometimes referred to as nutation. Nutation (from Latin nūtātiō, 'nodding, swaying') 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 behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. 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 is constant. In spacecraft dynamics, precession (a change in the first Euler angle) is sometimes referred to as nutation. 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. If the top has mass M and its center of mass is at a distance l from the pivot point, its gravitational potential relative to the plane of the support is 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