Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 32

The motion of a body is described by the motion of the particles constituting the body. A pure translation of a body is a motion in which all the particles constituting the body are moving in parallel straight lines with the same velocity. If the body is rigid, then the distance between any two of its particles is fixed; hence it is possible for the body to have a pure rotation, defined as the motion in which all the particles describe concentric circles about a fixed axis, and thus have velocities that are proportional to their respective distances from the axis of rotation. A rigid body in a combined translation and rotation has its constituent particles travelling in curved paths of different shapes relative to a stationary reference frame. A non‐rigid body can have structural deformation as it translates and rotates, wherein the relative distances of the particles varies with time. The general motion of a body therefore consists of a combination of translation, rotation, and structural deformation, whose complete description requires a determination of the spatial trajectories of the particles constituting the body.

Consider a body with the centre of mass, o, with a particle of elemental mass, d, located at relative to o (see Fig. 2.3). Also consider an inertial reference frame, OXYZ, with origin at , and unit vectors, , along , , and , respectively. The positions of o and the elemental mass relative to the origin, , are given by and , respectively, whose time derivatives in the inertial frame are the respective velocities, and . If the net force experienced by the elemental mass is d, then its equation of motion by Newton's second law is expressed as follows:

(2.55)

where the net force, d, is a sum of all internal (d) and external (d) forces applied to the elemental mass, . The velocities, and , are related by the following kinematic equation:

(2.56)

where the time derivative is taken relative to the inertial frame, (OXYZ). Integrated over all the mass particles constituting the body, Eq. (2.55) yields the following result:

(2.57)

where all the internal forces (consisting of equal and opposite pairs) cancel out by Newton's third law, and is the net external force acting on the body.

The term on the left‐hand side of Eq. (2.57) can be simplified by moving the time derivative outside the integral, because the total mass, , of the body is a constant. This fact, along with Eq. (2.56), leads to the following:

(2.58)

where we have by the virtue of being the centre of mass of the body. Thus Eqs. (2.57) and (2.58) result in the following equation of the body's translation:

(2.59)

Thus the translational motion of the body is described by the motion of its centre of mass, as if all the mass were concentrated at that point.

Figure 2.3 A body as a collection of large number of particles of elemental mass, , with centre of mass o.

The rotational kinetics of the body are described by taking moments of Eq. (2.55) about the centre of mass, , and integrating over the body as follows:

(2.60)

where all the internal torques cancel out (being equal and opposite pairs), resulting in only the net external torque, , appearing on the right‐hand side. The left‐hand side of Eq. (2.60) is derived as follows, once again by the virtue of Eq. (2.56) and the fact that is the centre of mass:

(2.61)

A substitution of Eq. (2.61) into Eq. (2.60) yields the following equation of rotational kinetics of the body:

(2.62)

where

(2.63)

is the angular momentum of the body about its centre of mass, o. Thus a net external torque about the centre of mass of a body equals the time derivative of its angular momentum about the centre of mass.

If the body is rigid, then the distance between any two of its particles is a constant. Hence, the velocity of the elemental mass relative to is given by

(2.64)

where because Here is the angular velocity of a local reference frame, oxyz, rigidly attached to the body at , with unit vectors along , , and , respectively (Fig. 2.3), and is measured relative to the inertial frame, (OXYZ). Such a reference frame, oxyz, is termed a body‐fixed frame.