Relative Motion Analysis

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Chapter 6 Relative Motion Analysis

In this chapter, we seek to find out how the kinematics of a point can be obtained from one referential to another. More specifically, we consider a particle \(P\) whose motion is observed in two referentials \(\cE\) and \(\cF\text{,}\) which are themselves in relative motion. Assuming that the motion of \(\cF\) relative to \(\cE\) is entirely known, and that the kinematics (the velocity and acceleration) of \(P\) relative to \(\cF\) has been obtained, we ask whether the kinematics of \(P\) relative to \(\cE\) can be found. By defining the concept of coinciding point, we will obtain a relationship which relate velocity \(\vel_{P/\cE}\) to velocity \(\vel_{P /\cF}\text{.}\) As a consequence of this relationship, we will clearly identify the condition for which velocity \(\vel_{P\in \cF/\cE}\) is distinct from velocity \(\vel_{P/\cE}\text{.}\) Likewise, we will obtain a second relationship between accelerations \(\ba_{P/\cE}\) and \(\ba_{P /\cF}\text{,}\) and understand why the three expressions \(\ba_{P/\cE}\) and \(\ba_{P\in \cF/\cE}\text{,}\) and \((d \vel_{P\in \cF/\cE} /dt)_\cE\) are, in general, not identical.

These kinematical formulas will prove to be fundamental to the kinematic analysis of mechanisms and to the study of the effect of a rotating referential (such as Earth) on the dynamical behavior of a material system.