The Practice of Engineering Dynamics. Ronald J. Anderson
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Название: The Practice of Engineering Dynamics

Автор: Ronald J. Anderson

Издательство: John Wiley & Sons Limited

Жанр: Физика

Серия:

isbn: 9781119053699

isbn:

СКАЧАТЬ (i.e. magnitude) and the axis of rotation (i.e. direction). In Figure 1.1, the speed of rotation is
and the axis of rotation is perpendicular to the page. This results in an angular velocity vector,

      (1.4)

, is defined in Figure 1.2. Note that it is essential that right handed coordinate systems be used for dynamic analysis because of the extensive use of the cross product and the directions of vectors arising from it. If there is a right handed coordinate system
, with respective unit vectors
, then the cross products are such that,

      Using this definition of the angular velocity, the motion of the tip of vector

, resulting from the angular change in time
, can be determined from the cross product

      which, by the rules of the vector cross product, has magnitude,

and
and, in fact, lies in the direction of
.

and substituting into Equation 1.3 results in,

      (1.5)

      The time derivative of any vector,

, can therefore be written as,

      It is important to understand that the angular velocity vector,

, is the angular velocity of the coordinate system in which the vector,
, is expressed. There is a danger that the rate of change of direction terms will be included twice if the angular velocity of the vector with respect to the coordinate system in which it measured is used instead. The example presented in Section 1.3 shows a number of different ways to arrive at the derivative of a vector which rotates in a plane.

      For this general description of kinematic analysis, we assume that we are analyzing a system that has multiple bodies connected to each other by joints and that we are attempting to derive an expression for the acceleration of the center of mass of a body that is not the first in the assembly.

      The procedure is as follows.

      1 Find a fixed point (i.e. one having no velocity or acceleration) in the system from which you can begin to write relative position vectors that will lead to the centers of mass of bodies in the system.

      2 Define a position vector that goes from the fixed point, through the first body, to the next joint in the system. This is the position of the joint relative to the fixed point.

      3 Determine how many degrees of freedom, both translational and rotational, are required to define the motion of the relative position vector just defined. The degrees of freedom must be chosen to satisfy the constraints imposed by the joint that connects this body to ground.

      4 Define a coordinate system in which the relative position vector will be written and determine the angular velocity of the coordinate system.

      5 Repeat the previous three steps as you go from joint to joint in the system, always being careful to satisfy the joint constraints by defining appropriate degrees of freedom.

      6 When the desired body is reached, define a final relative position vector from the joint to the center of mass.

      7 The sum of all the relative position vectors will be the absolute position of the center of mass and the derivatives of the sum of vectors will yield the absolute velocity and acceleration of the center of mass.

, rotating about a fixed point,
, in a plane. An expression for the velocity of the free end of the rod,
, relative to point images СКАЧАТЬ