The Practice of Engineering Dynamics. Ronald J. Anderson

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.

1.2 Performing Kinematic Analysis

Before proceeding with examples of kinematic analyses we state here the steps that are necessary in achieving a successful result. This first step in any dynamic analysis is vitally important. The goal is to derive expressions for the absolute velocities and accelerations of the centers of mass of the bodies making up the system being analyzed. In addition, expressions for the absolute angular velocities and angular accelerations of the bodies will be required. It is at this first step of the analysis that degrees of freedom are defined and constraints on relative motion between bodies are satisfied.

      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