The Kinetic Energy Theorem

Section 12.6 The Kinetic Energy Theorem

In this section, we generalize Theorem 12.1.3, the Kinetic Energy Theorem, starting with the case of a single rigid body.

Subsection 12.6.1 Case 1: Single Rigid Body

Consider a rigid body \(\cB\) in motion relative to a Newtonian referential \(\cE\) under the effect of gravitational and contact mechanical actions. Let us apply D’Alembert Principle of Virtual Power by choosing the actual velocity field \(P \in \cB \mapsto \vel_{P/\cE}\) for the virtual velocity field \(P \mapsto {\bf v}^*_P\) (since \(\cB\) is a rigid body, this vector field defines a screw):

\begin{equation*} \int_\cB \vel_{P/\cE} \cdot\bF_{ \coB \to \cB}^g (P) \, dV + \int_{\partial \cB} \vel_{Q/\cE} \cdot\bof_{ \coB \to \cB}^c (Q) \, dA = \int_\cB \vel_{P/\cE} \cdot \ba_{P/\cE} \, dm \end{equation*}

In the left-hand-side of this equation, we recognize the powers \(\Pow ^g _{ \coB \rightarrow \cB / \cE }\) and \(\Pow ^c _{ \coB \rightarrow \cB / \cE }\) generated by the gravitational and contact forces during the motion of \(\cB\) relative to \(\cE\text{:}\) we can denote the sum of these two contributions simply as \(\Pow _{ \coB \rightarrow \cB / \cE }\text{.}\) In the right-hand-side, we recognize the time rate-of-change of the kinetic energy of \(\cB\)

\begin{equation*} \int_\cB \vel_{P/\cE} \cdot \ba_{P/\cE} \, dm = {d \over dt} \int_\cB {1\over 2}\vel^2_{P/\cE} \, dm= {d\over dt} \kin_{\cB/\cE} \end{equation*}

We conclude with the following theorem.

Theorem 12.6.1. Kinetic Energy Theorem (Single Body).

The power generated by all external mechanical actions exerted on rigid body \(\cB\) is equal to the time rate-of-change of the kinetic energy of \(\cB\text{:}\)

\begin{equation} {d \over dt}\kin_{\cB/\cE} = \Pow _{ {\coB} \rightarrow \cB / \cE }\tag{12.6.1} \end{equation}

relative to Newtonian referential \(\cE\text{.}\)

Remark 12.6.2.

A simple way to derive the KET consists of taking the scalar product of both sides of the FTD by the kinematic screw \(\{ \cV_{\cB/\cE} \}\text{:}\)

\begin{equation*} \{ \cD_{\cB/\cE} \} \cdot \{\cV_{\cB/\cE} \} = \{ \cA_{\coB\to \cB} \} \cdot \{\cV_{\cB/\cE} \} \end{equation*}

The term \(\{ \cA_{\coB\to \cB}\} \cdot \{\cV_{\cB/\cE} \}\) is of course the power \(\Pow _{ \coB \rightarrow \cB / \cE }\text{.}\) It is also straightforward to show that the scalar term \(\{ \cD_{\cB/\cE} \} \cdot \{\cV_{\cB/\cE} \}\) is nothing but \(\tfrac{d}{dt}\kin_{\cB/\cE}\) (see Section 9.7).

Subsection 12.6.2 Case 2: System of Rigid Bodies

Now consider a system \(\Sigma= \{\cB_1 , \cB_2, \ldots , \cB_N \}\) of \(N\) rigid bodies in motion relative to a Newtonian referential \(\cE\text{,}\) and let us apply again the Principle of Virtual Power to system \(\Sigma\text{,}\) by choosing the virtual velocity field \({\bf v}^{*}\) as follows:

\begin{equation*} {\bf v}^{*}_P = {\bf v}_{P\in \cB_{j} /\cE} \qquad {\rm if} \;\; P\in \cB_{j} \end{equation*}

Then, we obtain

\begin{equation*} \frac{d}{dt} \kin_{\Sigma /\cE} = \sum_{j=1}^{N} \{ \cA_{\coB_{j} \rightarrow \cB_{j}} \} \cdot \{ {\cal V}_{\cB_{j}/\cE} \} \end{equation*}

which amounts to summing the \(N\) equations obtained by applying the KET to each rigid body of the system. Next, we decompose each action screw \(\{\cA_{ \coB_{j} \rightarrow \cB_{j}} \}\) as the sum \(\{ \cA_{\bSi \rightarrow \cB_{j}} \} + \sum_{i\neq j}\{ \cA_{\cB_i \rightarrow \cB_{j}} \}\) of external and internal contributions to system \(\Sigma\text{.}\) This leads to two terms:

a term associated to the external actions on \(\Sigma\text{:}\)
\begin{equation*} \sum_{j=1}^{N}\{ \cA_{\bSi \rightarrow \cB_{j}} \}\cdot \{ {\cal V}_{\cB_{j}/\cE} \} , \end{equation*}
which is recognized as the power \(\Pow _{ \bSi \rightarrow \Sigma /\cE}\) generated by the external actions exerted on \(\Sigma\text{,}\)
a term associated to all possible internal actions, that is, the actions (and reactions) between all possible parts of \(\Sigma\text{:}\)
\begin{equation*} \sum_{i,j =1 \atop i \lt j}^{N} \{ \cA_{\cB_i \rightarrow \cB_j} \} \cdot \underbrace{ \left( \{ {\cal V}_{\cB_{j}/\cE} \} - \{ {\cal V}_{\cB_{i}/\cE} \} \right) }_{ \{ {\cal V}_{\cB_{j}/\cB_{i}} \} } \end{equation*}
which is recognized as the sum of all powers \(\Pow _{\cB_{i} \leftrightarrow \cB_{j}}\) of interactions within \(\Sigma\text{.}\)

In conclusion, Theorem 12.6.1 can be stated as follows:

Theorem 12.6.3. Kinetic Energy Theorem (System of Rigid Bodies).

The Kinetic Energy Theorem applied to a system \(\Sigma\) of \(N\) rigid bodies \(\{\cB_1 , \cB_2, \ldots , \cB_N \}\) takes the following form:

\begin{equation} {d\over dt} \kin_{\Sigma /\cE} = \Pow _{\bSi \rightarrow \Sigma /\cE} + \sum_{i,j =1 \atop i \lt j}^N \Pow _{\cB_{i} \leftrightarrow \cB_{j}} \tag{12.6.2} \end{equation}

relative to Newtonian referential \(\cE\text{.}\)

\(\danger\) A fundamental difference with the FTD is that the internal interactions between the rigid bodies of a system \(\Sigma\) must be accounted for.

Remark 12.6.4.

The integrated forms of the KET is known as the Work-Energy Theorem: integration of (12.6.2) over time gives

\begin{equation} \Delta \kin_{\Sigma /\cE} = \int_0^T \Pow _{\bSi \rightarrow \Sigma /\cE} dt + \sum_{i,j =1 \atop i \lt j}^N \int_0^T \Pow _{\cB_{i} \leftrightarrow \cB_{j}} dt\tag{12.6.3} \end{equation}

Hence the variation of the system’s kinetic energy is equal to the work done by all external and internal actions in the time interval \(0\leq t \leq T\text{.}\)

Remark 12.6.5.

A first integral governing the motion of system \(\Si\) can be found if the following conditions are satisfied:

if the external actions either generate no power or derive from a total potential \(\pot_{\bSi\to \Si/\cE}\text{,}\)
if the internal interactions either generate no power or derive from a total potential \(\sum_{i\lt j}\pot_{i\leftrightarrow j}\text{,}\)

then the system’s mechanical energy \(\energ_{\Sigma /\cE} = \kin_{\Sigma /\cE} + \pot_{\bSi\to \Si/\cE} + \sum_{i \lt j} \pot_{i\leftrightarrow j}\) is conserved.

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