The Practice of Engineering Dynamics. Ronald J. Anderson
href="#ulink_85f0f223-9943-5eb1-ae86-20c5a97fc1d5">Equation 1.20) has an interesting and, perhaps, unexpected form. In particular, the origin of the term that has twice the product of an angular velocity and a translational velocity (i.e.
In general, the derivation of an expression for the acceleration of a point (say
Let the position vector be
(1.44)
Then, applying Equation 1.6, the velocity is,
where the directions of the two components are defined. The rate of change of magnitude term is aligned with the position vector and is thus termed radial and the rate of change of direction component is perpendicular to the position vector and is therefore tangential.
Differentiating the velocity expression of Equation 1.45 yields,
and, applying Equation 1.6 to Equation 1.46, we can write,
After collecting terms and substituting
Each of the terms in Equation 1.48 has a name and a physical meaning, as follows.
1 is the radial acceleration. This is nothing more than the second derivative of the distance between and and it is aligned with .
2 is the tangential acceleration. It is called the tangential acceleration because it is aligned with the direction in which the point would move if it were a fixed distance from and were rotating about (i.e. in a direction perpendicular to a line passing through and ). Notice that is the total derivative of the angular velocity including its rate of change of magnitude and its rate of change of direction. As a result, may not be aligned with .
3 is the Coriolis acceleration5. The vectorial approach to finding the Coriolis acceleration is in many ways preferable to the scalar approach put forward in many books on dynamics. The magnitude of the radial velocity is often referred to in reference books as and the Coriolis acceleration is seen written as where the reader is left to determine its direction from a complicated set of rules. Consideration of Equations 1.45,1.46,1.47 shows that there are two very different types of terms that combine to form the Coriolis acceleration with its remarkable 2. The two terms are equal in magnitude and direction (i.e. each is ). One of these arises from part of the rate of change of magnitude of the tangential velocity of . The second arises from the rate of change of direction of the radial velocity of .
4 is the centripetal acceleration. In 2D circular motion. this is commonly written as and points toward the center of the circle. For the general points and used here, the centripetal acceleration points from to .
It is possible to visualize the acceleration components using a simple graphical construction. As an example, we can use the slider in a slot system shown in Figure 1.4 for which we have already derived both the velocity (Equation 1.19) and the acceleration (Equation 1.20) in body fixed coordinates.
Remember that rates of change of magnitude are aligned with the vector that is changing and rates of change of direction are perpendicular to the original vector and are pointed in the direction that the tip of the vector would move if it had the prescribed angular velocity and were simply rotating about its tail.
Figure 1.7 The velocity and acceleration components of the slider.
Figure 1.7 shows the two components of the velocity of the slider in the inner circle. A component is labeled with a
Between the inner and outer circles on Figure 1.7 are the components of the acceleration.