10.3: Rotation analysis in kinetics
Since the derivation of the key equations governing rotations in rigid body dynamics is quite elaborate, we start by introducing the equations in this section without deriving them. Then we explain how they should be used in kinetic analysis of rigid bodies and finally present the derivations in sections \(10.8-10.10\).
10.3.1 Moments
The use of moments is a technique to facilitate the analysis of the effect of forces on the rotations of rigid bodies and other systems of point masses. Moments are defined as follows (see Figure 10.2).
Definition. Moment
The moment vector \(\overrightarrow{\boldsymbol{M}}_{B / P}\) of a force \(\overrightarrow{\boldsymbol{F}}_{B}\) (with point of action \(\overrightarrow{\boldsymbol{r}}_{B}\) ) with respect to a reference point \(\overrightarrow{\boldsymbol{r}}_{P}\) is defined as the cross product \(\overrightarrow{\boldsymbol{r}}_{B / P} \times \overrightarrow{\boldsymbol{F}}_{B}\).
\[\overrightarrow{\boldsymbol{M}}_{B / P} \equiv \overrightarrow{\boldsymbol{r}}_{B / P} \times \overrightarrow{\boldsymbol{F}}_{B} \tag{10.15} \label{10.15}\]
When using moments to analyse rigid body dynamics, for example with an FBD it is essential that all moments are determined with respect to the same reference point \(P\). Furthermore, it can be shown that the sum of all internal moments on the rigid body is always zero (see Figure 10.3), such that the resultant moment on the system is equal to the sum of all external moments, i.e. moments generated by external forces, as will be derived in Equation 10.63.
The resultant moment vector \(\overrightarrow{\boldsymbol{M}}_{C / P, \text { ext }}\) acting on a system of point masses \(C\) with respect to a reference point \(P\) is the sum over the moments of the external forces with respect to \(P\).
\[\overrightarrow{\boldsymbol{M}}_{C / P, \mathrm{ext}}=\sum_{i} \overrightarrow{\boldsymbol{M}}_{i / P, \mathrm{ext}}=\sum_{i} \overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{F}}_{i, \mathrm{ext}} \tag{10.16} \label{10.16}\]
Because the sums of internal moments and forces on a system or rigid body are zero, they do not affect the kinetics. For that reason only external moments and external forces \(\overrightarrow{\boldsymbol{F}}_{\text {ext }}\) should be drawn in an FBD. In fact, a key purpose of an FBD is to define the difference between internal and external forces, since all objects drawn in the FBD can be considered to be internal and their forces should not be drawn, whereas all drawn forces are generated by external objects that are not drawn.
10.3.2 Angular momentum
We now define the angular momentum vector \(\overrightarrow{\boldsymbol{L}}\) of a point mass (see Figure 10.17), which is used to analyse rotating motion, similar to the way momentum \(\overrightarrow{\boldsymbol{p}}\) is used for translational motion as described in Ch. 8.
Definition. Angular momentum of a point mass
The angular momentum vector \(\overrightarrow{\boldsymbol{L}}_{i / P}\) of a point mass at position \(\overrightarrow{\boldsymbol{r}}_{i}\) with momentum vector \(\overrightarrow{\boldsymbol{p}}_{i}\), is defined as the cross product \(\overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{p}}_{i}\).
\[\overrightarrow{\boldsymbol{L}}_{i / P} \equiv \overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{p}}_{i}=\overrightarrow{\boldsymbol{r}}_{i / P} \times\left(m_{i} \overrightarrow{\boldsymbol{v}}_{i}\right) \tag{10.17} \label{10.17}\]
The total angular momentum \(\overrightarrow{\boldsymbol{L}}_{C / P}\) of a system of point masses \(C\), with respect to reference point \(P\) is equal to the sum of the angular momentum vectors of its individual point masses.
\[\overrightarrow{\boldsymbol{L}}_{C / P}=\sum_{i} \overrightarrow{\boldsymbol{L}}_{i / P}=\sum_{i} \overrightarrow{\boldsymbol{r}}_{i / P} \times\left(m_{i} \overrightarrow{\boldsymbol{v}}_{i}\right) \tag{10.18} \label{10.18}\]
To determine the angular momentum of a rigid body one can use this equation to sum or integrate over all point masses \(i\) in the rigid body, but a
simpler way is to use the moment of inertia tensor \(\mathbf{I}_{C / P}\) of the rigid body or its \(z z\) component \(I_{C / G, z z}\) which will be discussed in Sec. 10.5. This results in:
\[\begin{align} \overrightarrow{\boldsymbol{L}}_{C / P} & =\mathbf{I}_{C / P} \overrightarrow{\boldsymbol{\omega}}_{C} \tag{10.19} \label{10.19}\\[4pt] \overrightarrow{\boldsymbol{L}}_{C / P, 2 D} & =\left(m_{\mathrm{tot}} \rho_{G / P}^{2}+I_{C / G, z z}\right) \omega_{C} \hat{\boldsymbol{k}} \tag{10.20} \label{10.20}\end{align}\]
10.3.3 Euler’s second law
We now introduce Euler’s second law that will be derived in Sec. 10.10.
Euler’s second law for a system or rigid body \(C\) states that the time derivative of the total angular momentum of \(C\) is equal to the total moment from external forces acting on \(C\). The total angular momentum should be determined with respect to the same reference point \(P\) that is fixed in an IRF or is the CoM of \(C\).
\[\overrightarrow{\boldsymbol{M}}_{C / P, \mathrm{ext}}=\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{L}}_{C / P} \tag{10.21} \label{10.21}\]
Euler’s first law, and Euler’s second law together provide the full equations of motion for a rigid body and are therefore very useful and important for analysing their dynamics. For planar kinetics of rigid bodies Euler’s second law can be simplified:
\[M_{C / P, \mathrm{ext}, 2 D} \hat{\boldsymbol{k}}=\overrightarrow{\boldsymbol{r}}_{G / P} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{a}}_{G}\right)+I_{C / G, z z} \alpha_{C} \hat{\boldsymbol{k}} \tag{10.22} \label{10.22}\]