Читать книгу The Practice of Engineering Dynamics - Ronald J. Anderson - Страница 14

Vectors have two distinct properties – magnitude and direction. Either or both of these properties may change with time and the time derivative of a vector must account for both.

The rate of change of a vector with respect to time is therefore formed from,

1 The rate of change of magnitude .

2 The rate of change of direction .

Figure 1.1 A vector changing with time.

Figure 1.1 shows the vector that changes after a time increment, , to .

The difference between and can be defined as the vector shown in Figure 1.1 and, by the rules of vector addition,

(1.1)

or,

(1.2)

Then, using the definition of the time derivative,

(1.3)

Imagine now that Figure 1.1 is compressed to show only an infinitesimally small time interval, . The components of for the interval are shown in Figure 1.1. They are,

1 A component aligned with the vector . This is a component that is strictly due to the rate of change of magnitude of . The magnitude of is where is the rate of change of length (or magnitude) of the vector . The direction of is the same as the direction of . Let be designated¹ as .

2 A component that is perpendicular to the vector . That is, a component due to the rate of change of direction of the vector. Terms of this type arise only when there is an angular velocity. The rate of change of direction term arises from the time rate of change of the angle in Figure 1.1 and is the magnitude of the angular velocity of the vector. The rate of change of direction therefore arises from the angular velocity of the vector. The magnitude of is where is the length of . By definition the rate of change of the angle (i.e. ) has the same positive sense as the angle itself. It is clear that is the “tip speed” one would expect from an object of length rotating with angular speed .

The angular velocity is itself a vector quantity since it must specify both the angular speed (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)

where the right handed set of unit vectors, , 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,

Figure 1.2 Even 2D problems are 3D.

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 a direction that, according to the right hand rule² used for cross products, is perpendicular to both and and, in fact, lies in the direction of .

Combining these two terms to get and substituting into Equation 1.3 results in,

(1.5)

The time derivative of any vector, , can therefore be written as,

(1.6)

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.