Rotation

Rotation, and rotational movement, rotation, rotational movement, or circular motion is the motion of a point or the body about an axis of rotation.

The concept of rotation is mainly used in physics, and here in particular in mechanics and kinematics. In astronomy, the Earth's rotation changes are an important research topic. Applications from everyday life and often used to clearly explained examples, in which the rotation plays an important role, are centrifugal, or carousel.

Orbit and rotation axis

Characteristic of a rotation is the circular path of the respective particles on a two-dimensional plane of motion. For all points of the system is excellent Especially, the axis of rotation, perpendicular to the plane of rotation.

Axis and degrees of freedom of movement

A rigid body in space has a maximum of three rotational degrees of freedom.

Rotation with non-constant axis is possible and is colloquially known as " staggering " or " eggs ". Technicians and scientists speak, however - depending on the type of axis motion - from wobbling of the axis or secondary axis errors, or from precession or nutation.

Compared with the translation movement

The following table compares the characteristic quantities and the equations of motion in a translational movement with that in a rotary movement. Due to the similarities each set of the translation by replacing the appropriate sizes can be converted into a theorem on the rotation.

Rotation of Rigid Bodies

The rotation of rigid bodies follows the Eulerian equations, for which there is no solution in the form of a simple formula. Even when no external forces act on the body, showing the axis of rotation, in most cases a complex motion, which is called nutation. There are, however, for the technical application important special cases in which the Eulerian equations are simplified to such an extent that there are simple solutions. In these cases, the trajectories of the system periodically.

Case of Euler

The case of Euler describes a gyroscope that is suspended exactly in its center of gravity. Regardless of the shape of the top of the case is integrable, since there are more conserved quantities as degrees of freedom: the energy and the angular momentum with respect to all three main axes of inertia of the body.

If the mass of the rotating body rings symmetrically distributed about the axis of rotation, so act on the axis no springs from the rotational forces, since the inertia ( centrifugal force ) of each mass-particle is canceled by an equal and opposite; such an axis is called a free axis. Since each order a free axis rotating mass particles of inertia seeks following to remain in its plane perpendicular to the axis of rotation plane and the free axis must themselves have a tendency to maintain their direction in space and is a force that will bring them from this direction, a more greater resistance, the greater the moment of inertia and the angular velocity of the rotating body. Hence it is that a sufficiently rapidly spinning top does not fall over, even if its axis is wrong, as well as wheels, coins, etc. will not fall over if you let them roll on its edge or " dance " around the vertical diameter.

The effect of the disturbing force on the gyroscope is expressed rather by the fact that the axis of which escapes in a direction perpendicular to the direction of the disturbing force direction and in slow motion, the surface of a cone describes, without the axis changes its inclination to the horizontal plane. This movement is referred to as precession.

The Euler gyroscope is technical application eg in gyroscopes and gyroscopic control systems. The wheels of bicycles and motorcycles are Euler gyroscope to a good approximation, and serve alongside the tracking of the vehicle by their desire to preserve the angular momentum to stabilize the vehicle. See also: cycling.

Case of Lagrange

In the event of the conformity of the Lagrangian with respect to two major axes of inertia, it is assumed. This is accomplished by radially symmetrical bodies. In this case there are three conserved quantities as many degrees of freedom such as: the energy, the total angular momentum and the angular momentum with respect to the z-axis ( in the direction of the force field ). This case is realized by a typical toy gyroscope, if one fixes the touchdown point at the bottom.

Case of Kovalevskaya

The Kovalevskaya gyroscope has with respect to two of its major axes equal moments of inertia and a precisely twice as large with respect to the third principal axis. The conserved quantities are the energy, the total angular momentum and a complex mathematical expression for which there is no universally understandable representation.

Case of Goryachew - Chaplygin

The case of Goryachew - Chaplygin is a modification of the Kovalevskaya - case, which is a four times as large calls instead of twice as large third moment of inertia. In this case, however, there is only a third conserved when the rotary pulse disappears around the z-axis.

Regardless of other influences each gyro is quasi- integrable, plugged in which either very little or a lot of energy (compared to the potential energy difference between upper and lower dead point ) in the rotation. The chaotic motions in the non- integrable types occur regardless of the form when the kinetic energy of the gyro is just sufficient to reach the top dead center.