METHODOLOGY

Preface METHODOLOGY

Throughout this book, we will make extensive use of mathematical objects called screws. They will provide a simple yet powerful formalism which will unify all aspects of rigid body mechanics. Specifically, we take advantage of the fact that all relevant vector fields in rigid body mechanics have the following form

\begin{equation*} \bvv : P\in \cE \mapsto \bvv_P \in E \qquad \text{ such that } \qquad\bvv_Q = \bvv_P + \bV \times \br_{PQ} \end{equation*}

Such fields define screws which will be denoted in a simple array form:

\begin{equation*} \{ \cV \} =\begin{Bmatrix}\bV \\ \bvv_A \end{Bmatrix} \end{equation*}

Various operations (addition, multiplication, scalar product, \(\ldots\)) on screws can then be defined and their essential properties will be determined. Four fundamental screws will be introduced:

\(\{ \cV_{\cB/\cE} \}\) as the kinematic screw of a rigid body \(\cB\) (Chapter 5) and \(\{ \cV_{\cB/\cE}^{q_i} \}\) as its partial kinematic screw in analytical dynamics (Chapters 13 and 14),
\(\{ \cH_{\cB/\cE} \}\) as the kinetic screw \(\{ \cH_{\cB/\cE} \}\) of a rigid body \(\cB\text{,}\) and \(\{ \cH_{\Si/\cE} \}\) as kinetic screw of a system of rigid body \(\Si\) (Chapter 9),
\(\{ \cD_{\cB/\cE} \}\) as the dynamic screw of a rigid body \(\cB\text{,}\) and \(\{ \cD_{\Si/\cE} \}\) as the dynamic screw of a system of rigid bodies \(\Si\) (Chapter 9),
\(\{\cA_{\Si_1\to \Si_2}\}\) as the action screw of material system \(\Si_1\) on material system \(\Si_2\) (Chapter 10).

The theorems underlying the methods of this book will all be derived and put to practical use by employing the formalism of screws. The starting point will be D’Alembert Principle of Virtual Power, which, in the language of screws can be stated as follows:

\begin{equation*} \underbrace{ \{ \cD_{\Si/\cE} \} }_{\text{dynamic screw}} \quad \cdot \underbrace{ \{\cV^*\} }_{\text{virtual velocity screw}} = \underbrace{ \{ \cA_{\bSi \to \Si} \}}_{\text{action screw}} \qquad\cdot \underbrace{ \{\cV^*\} }_{\text{virtual velocity screw}} \end{equation*}

in a Newtonian referential \(\cE\text{.}\) By specifying the virtual velocity screw \(\{\cV^*\}\text{,}\) various theorems are obtained:

choosing \(\{\cV^*\}\) arbitrary yields the Fundamental Theorem of Dynamics of the Newton-Euler formalism relative to a Newtonian referential (Chapter 11) or to a non-Newtonian referential (Chapter 16),
choosing \(\{\cV^*\}\) as the kinematic screw \(\{ \cV_{\cB/\cE} \}\) of rigid body \(\cB\) yields the Kinetic Energy Theorem (Chapter 12),
choosing \(\{\cV^*\}\) as the partial kinematic screws \(\{ \cV_{\cB/\cE}^{q_i} \}\) of rigid body \(\cB\) yields Lagrange equations (Chapter 13) or Gibbs-Appell equations (Chapter 14),
choosing \(\{\cV^*\}\) as the generalized partial kinematic screws \(\{ \cV_{\cB/\cE}^{u_i} \}\) of rigid body \(\cB\) yields Kane equations (Chapter 14),
choosing \(\{\cV^*\}\) as the kinematic screw \(\{ \cV_{\cB/\cE} \}\) relative to a non-Newtonian referential yields the Kinetic Energy Theorem in a non-Newtonian referential (Chapter 16).

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