# Kinematics and Relative Motion

• Lester W. Schmerr
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 254)

## Abstract

This chapter describes the kinematics of point masses and rigid bodies when non-inertial coordinate systems (frames) are used to describe their motion. We obtain relative velocity and acceleration expressions for moving frames and then apply those expressions to find the velocities and accelerations of constrained systems of rigid bodies at specific instances of time, similar to what is done in many elementary dynamics texts. However, we also show that it is possible to determine the kinematics of a constrained system of rigid bodies more completely as a function of time. In some cases this can be done analytically but in general the solution must be done numerically since the positional constraints are normally nonlinear. To solve the positional constraints we use the Newton-Raphson method. Kinematics is treated in this chapter both by the traditional vector approach and by an equivalent matrix-vector method that is more readily suited to dealing with complex systems. The matrix-vector approach for planar problems is covered in Sects. 4.4 and 4.5 while more general three-dimensional problems are treated in Sect. 4.6 and those that follow. Three-dimensional rotations are described in terms of both Euler angles and Euler parameters as these are the most commonly used generalized rotational coordinates. Some classical examples of the dynamics effects seen in rotating coordinate systems, such as the Foucault pendulum, are also given.

## References

1. 1.
A.A. Shabana, Computational Dynamics, 3rd edn. (John Wiley, Chichester, 2010)
2. 2.
A.A. Shabana, Dynamics of Multibody Systems, 4th edn. (Cambridge University Press, New York, 2013)
3. 3.
H. Baruh, Analytical Dynamics (McGraw-Hill, New York, 1999)Google Scholar
4. 4.
J.H. Ginsberg, Advanced Engineering Dynamics, 2nd edn. (Cambridge University Press, New York, 1995)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Lester W. Schmerr
• 1
1. 1.Aerospace EngineeringIowa State UniversityAmesUSA

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