10.11: Summary
In this chapter we have introduced the concepts needed to analyse the planar kinetics of rigid bodies. The moments of forces were introduced, the angular momentum of rigid bodies was calculated. The moment of inertia was defined and methods to determine it for solid continuous bodies were introduced. Finally, this allowed us to determine Euler’s second law, and the equation of motion for rotations of rigid bodies. In essence our toolbox to analyse the planar dynamics of rigid bodies and point masses is quite complete now. However, in some cases the principles of work and energy, and that of impulse and momentum, can facilitate the analysis of the kinetics of rigid bodies. We will extend these concepts and apply them to rotations in the next two chapters.
- Euler’s first law
\[\begin{align} \sum \overrightarrow{\boldsymbol{F}}_{\mathrm{ext}} & =m_{\mathrm{tot}} \overrightarrow{\boldsymbol{a}}_{G} \tag{10.94} \label{10.94}\\[4pt] \sum F_{\mathrm{ext}, x} & =m_{\mathrm{tot}} \ddot{x}_{G} \tag{10.95} \label{10.95}\\[4pt] \sum F_{\mathrm{ext}, y} & =m_{\mathrm{tot}} \ddot{y}_{G} \tag{10.96} \label{10.96}\end{align}\]
- The sum of internal forces on a rigid body or system is always zero.
- Resultant moment on a rigid body
\[\overrightarrow{\boldsymbol{M}}_{C / P, \mathrm{ext}}=\sum_{i} \overrightarrow{\boldsymbol{M}}_{i / P, \mathrm{ext}}=\sum_{i} \overrightarrow{\boldsymbol{r}}_{i / P} \times \overrightarrow{\boldsymbol{F}}_{i, \mathrm{ext}} \tag{10.97} \label{10.97}\]
- The sum of internal moments on a rigid body or system is always zero.
- Total angular momentum of a rigid body
\[\begin{align} \overrightarrow{\boldsymbol{L}}_{C / P} & =\mathbf{I}_{C / P} \overrightarrow{\boldsymbol{\omega}}_{C} \tag{10.98} \label{10.98}\\[4pt] \overrightarrow{\boldsymbol{L}}_{C / P, 2 D} & =\left(m_{\mathrm{tot}} \rho_{G / P}^{2}+I_{C / G, z z}\right) \omega_{C} \hat{\boldsymbol{k}} \tag{10.99} \label{10.99}\end{align}\]
- Moment of inertia \(I_{z z}\) of a rigid body
\[I_{G}=\sum_{i} m_{i} \rho_{i / G}^{2}=\int_{V} \rho_{m} \rho_{i / G}^{2} \mathrm{~d} V \tag{10.100} \label{10.100}\]
- Techniques for solving these kind of moment of inertia sums and integrals.
- Parallel axis theorem:
\[I_{P}=m_{\mathrm{tot}} \rho_{G / P}^{2}+I_{G} \tag{10.101} \label{10.101}\]
- Adding and subtracting moments of inertia:
\[I_{C, P}=I_{A, P} \pm I_{B, P} \tag{10.102} \label{10.102}\]
- Euler’s second law
\[\begin{align} \overrightarrow{\boldsymbol{M}}_{C / P} & =\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{L}}_{C / P} \tag{10.103} \label{10.103}\\[4pt] M_{C / P, 2 D} \hat{\boldsymbol{k}} & =\overrightarrow{\boldsymbol{r}}_{G / P} \times\left(m_{\mathrm{tot}} \overrightarrow{\boldsymbol{a}}_{G}\right)+I_{C / G, z z} \alpha_{C} \hat{\boldsymbol{k}} \tag{10.104} \label{10.104}\end{align}\]
- Only valid if reference point \(P\) is a fixed point in an IRF, or if \(P=G\).
- Only valid if the same reference point \(P\) is used for determining \(\overrightarrow{\boldsymbol{M}}_{C / P}\) and \(\overrightarrow{\boldsymbol{L}}_{C / P}\).
- Determine and solve the EoM for a rigid body
- Methodology to determine the EoM, using FBD, moments, constraints and Euler’s laws.
- Choose fixed axis as reference point when dealing with pure rotation around fixed point.
- Simplify EoMs
- Solve EoMs using kinematic techniques.
- Couples
- A couple is a number of force vectors that generate a moment on a system for which the sum of forces is zero such that \(\overrightarrow{\boldsymbol{a}}_{G}=\overrightarrow{\boldsymbol{0}}\). For a simple couple the number of forces is two.
- The resultant moment vector of a couple is independent of the choice of reference point \(P\). Such a vector is called a pure moment or couple moment.