Rigid body

As a rigid body, a physical model of a non-deformable body is called. The body may have a continuous mass distribution, or a system of discrete mass points (eg, atoms, molecules in quantum mechanics ). The non-deformability is an idealization depend on outside forces always have the same distance to each other, so no deflection or internal vibration occurs when any three points on the body. Certain mass distributions with a specially privileged moment of inertia can be used as gyro.

The rigid body mechanics concerned with the motion of rigid bodies under the influence of external forces. By the model assumptions it only come to movements of the whole body in one direction ( translational motion ) and rotation movements. Additional forms of movement, such as oscillations of individual mass points or deformations of the body are treated solid body by a more general mechanism.

The concept of the rigid body is inconsistent with the predictions of the theory of relativity, since according to him always, the entire body while responding to forces and torques, which implies that their effects spread within the body with infinite speed, ie, in particular faster than the vacuum speed of light c. In real bodies will spread, however, usually with the specific body for the speed of sound, which is still far below c.

Rotation of a rigid body about a stationary axis

The axis of rotation defined as the angle of rotation is the only degree of freedom of rotation. The rotation is described by the angular velocity. This quantity can be written as a vector and link with location and path velocity of a point:

This equation is valid if and only if the axis of rotation is chosen as the direction of the vector. Seen in the direction of the vector takes place while the rotation in the clockwise direction.

If a body is rotating about two axes can be defined as the angular velocity vectors for both axes. Their sum then gives the total rotation of the body. It is generally undertaken only a rotation around an axis. This ensures that the angular velocity is additive as a vector and therefore it is useful to represent this size as a vector.

General motion of rigid body

The movement of the body relative to an inertial system can be combined with a uniform translation of all points of the body (and thus the center of gravity ) and the rotation of all points of the body to pose a body axis.

The position of an arbitrary point P is obtained by addition of the vector from the origin O to the origin of the inertial system S of the body-fixed coordinate system and the vector from the origin of the body-fixed system to the point P.

The derivative with respect to time is calculated ( Euler rate equation ):

The acceleration is given by:

Here, the angular velocity and the angular acceleration of a rigid body.

The vector is constant in the body-fixed coordinate system K. The conversion to the inertial frame 0 is via the rotation matrix A.

Degrees of freedom and configuration space

The degrees of freedom of an n -particle system form a so-called configuration space. If there is a rigid bodies of three degrees of freedom with respect to the position and three other collaborating to orientation. In addition to various stationary coordinate systems which allow a description of the position, the Euler angles provide a way to describe the orientation that plays an important role especially in the aerospace industry.

To illustrate, a free body can be used as a ( aerobatic ) aircraft, which has three degrees of freedom of a linear motion, as it is free to move in three spatial dimensions. There are also three further degrees of freedom of rotation about spatial ( independent ) axes of rotation.

Obviously now reduces any restriction of movement possible, the number of degrees of freedom. For example, if a mass point of the rigid body fixed in space, so you can put in this the origin of the reference system. This eliminates the three translational degrees of freedom. This reduces the movement to a mere change in the orientation and it will have only three degrees of freedom. If a further point noted so the body can only rotate about a spatially fixed axis of rotation and thus has only one degree of freedom, namely the rotation angle. Finally Specifies a third point of the body fixed, is not located on the axis of the first two points, he loses even the last degree of freedom and is therefore motionless. Each additional spatial fixing of points now leads to a so-called static overdetermination, which plays an important role in the structural analysis.

Approaches for determining the equation of motion

According to the model assumption constant distances apply between the particles. From the center of gravity principle is now possible to draw some conclusions:

• To the action of a system of external forces acting on a rigid body, only the resultant force F and the resulting torque M are essential. All systems with the same resultant forces are thus equivalent in their effect.
• The inertia tensor of a rigid body with respect to a gravity system constant.

Often, the model will also be placed further idealizations underlying that allow so-called conservation laws for determining the equation of motion to introduce:

If a closed system is adopted, it follows from the conservation of angular momentum that the vectorial total angular momentum of the system is constant, and we have:

This call:

• The inertia tensor of the rigid body
• (), The vectorial angular velocity at the time

The change of the angular momentum corresponding to the outer moment. It is the Euler equation:

If a conservative force field basis, it follows from the energy conservation law, that the total energy is constant, and we have:

This call:

• The kinetic energy of translation and the potential energy at the time
• The kinetic energy of rotation or the rotation energy at the time