9.8: Summary

In this chapter we have analysed the kinematics of rotating rigid bodies. The motion of a rigid body can be considered as the combination of a translation of a certain reference point on the rigid body, and the rotation of the rigid body around an axis through that point. Velocity and acceleration vectors of the reference point can change in time, and so can the angular velocity and acceleration vectors of the rotation around the axis. In this textbook we mainly focus on the planar kinematics where the point masses move in the two-dimensional \(x y\)-plane and angular velocity and acceleration vectors point in the \(z\)-axis direction. We present the expressions for the velocities \(\overrightarrow{\boldsymbol{v}}_{B}\) and \(\overrightarrow{\boldsymbol{a}}_{B}\) for points in the rigid body, or point masses moving inside a rotating reference frame.

Now we are ready to analyse the dynamics of rigid bodies under the influence of forces in the next chapter, dealing with the kinetics of rigid bodies.

• Orientation and motion of a rigid body

\(-\overrightarrow{\boldsymbol{r}}_{B, 2 D}(t)=\overrightarrow{\boldsymbol{r}}_{A}(t)+\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \cos \phi_{B / A}(t) \hat{\boldsymbol{\imath}}+\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \sin \phi_{B / A}(t) \hat{\boldsymbol{\jmath}}\)

• Angular velocity and acceleration
• Angular velocity: \(\vec{\omega}_{2 D}=\omega \hat{\boldsymbol{k}}=\dot{\phi} \hat{\boldsymbol{k}}\)
• Angular acceleration: \(\overrightarrow{\boldsymbol{\alpha}}_{2 D}=\alpha \hat{\boldsymbol{k}}=\ddot{\phi} \hat{\boldsymbol{k}}\)
• Angular velocity is a property of a rigid body and independent of the points used to determine the angle \(\phi\)
• Differentiation and integration: \(\phi(t) \leftrightarrow \omega(t) \leftrightarrow \alpha(t)\)
• Kinematics rigid body

\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B} & =\overrightarrow{\boldsymbol{v}}_{A}+\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.118} \label{9.118}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B} & =\overrightarrow{\boldsymbol{a}}_{A}+\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / A}+\overrightarrow{\boldsymbol{\omega}} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A}\right) \tag{9.119} \label{9.119}\\[4pt] \overrightarrow{\boldsymbol{v}}_{B, 2 D} & =\overrightarrow{\boldsymbol{v}}_{A}+\omega\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A} \tag{9.120} \label{9.120}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B, 2 D} & =\overrightarrow{\boldsymbol{a}}_{A}+\alpha\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A}-\omega^{2} \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.121} \label{9.121}\end{align}\]

- Special types of motion and methods

• Pure translation: \(\overrightarrow{\boldsymbol{\omega}}=\overrightarrow{\mathbf{0}}\) and \(\overrightarrow{\boldsymbol{v}}_{A} \neq \overrightarrow{\mathbf{0}}\).
• Pure rotation: \(\overrightarrow{\boldsymbol{v}}_{A}=\overrightarrow{\mathbf{0}}, \overrightarrow{\boldsymbol{a}}_{A}=\overrightarrow{\mathbf{0}}\) and \(\overrightarrow{\boldsymbol{\omega}} \neq \overrightarrow{\mathbf{0}}\).
• Instantaneous centre of rotation: \(\overrightarrow{\boldsymbol{v}}_{I C}=\overrightarrow{\mathbf{0}}\).

-3 methods to find \(I C\).

- Combining kinematics with constraint equations

• Analyse mechanisms using multiple kinematic equations.
• Combine these kinematic equations with constraint equations to determine the motion of mechanisms and rigid bodies. See e.g. Example 9.3.
• Motion in a rotating reference frame \(x^{\prime} y^{\prime} z\) ’

\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B}= & \overrightarrow{\boldsymbol{v}}_{B / O^{\prime}}^{\prime}+\overrightarrow{\boldsymbol{v}}_{O^{\prime}}+\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / O^{\prime}} \tag{9.122} \label{9.122}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B}= & \overrightarrow{\boldsymbol{a}}_{B / O^{\prime}}^{\prime}+\overrightarrow{\boldsymbol{a}}_{O^{\prime}}+\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / O^{\prime}} \tag{9.123} \label{9.123}\\[4pt] & +2 \overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{v}}_{B / O^{\prime}}^{\prime}+\overrightarrow{\boldsymbol{\omega}} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / O^{\prime}}\right) \\[4pt] \overrightarrow{\boldsymbol{v}}_{B, 2 D}= & \overrightarrow{\boldsymbol{v}}_{B / O^{\prime}}^{\prime}+\overrightarrow{\boldsymbol{v}}_{O^{\prime}}+\omega\left|\overrightarrow{\boldsymbol{r}}_{B / O^{\prime}}\right| \hat{\boldsymbol{\phi}}_{O^{\prime}} \tag{9.124} \label{9.124}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B, 2 D}= & \overrightarrow{\boldsymbol{a}}_{B / O^{\prime}}^{\prime}+\overrightarrow{\boldsymbol{a}}_{O^{\prime}}+\alpha\left|\overrightarrow{\boldsymbol{r}}_{B / O^{\prime}}\right| \hat{\boldsymbol{\phi}}_{O^{\prime}} \tag{9.125} \label{9.125}\\[4pt] & +2 \omega\left(v_{B / O^{\prime}, \rho}^{\prime} \hat{\boldsymbol{\phi}}_{O^{\prime}}-v_{B / O^{\prime}, \phi}^{\prime} \hat{\boldsymbol{\rho}}_{O^{\prime}}\right)-\omega^{2} \overrightarrow{\boldsymbol{r}}_{B / O^{\prime}}\end{align}\]