Mechanics of Continua and Structures
I. Rodrigues’ formula
\[\begin{align} \u{R} = \hat{\u{e}}\otimes\hat{\u{e}} + (\u{I}-\hat{\u{e}}\otimes\hat{\u{e}})\cos\theta + (*\hat{\u{e}})\sin\theta \end{align}\]It can be shown that the angular vector \(\omega\) is in the same direction as the axis of rotation vector \(\u{e}\). This is because
\[\begin{align} \u{\Omega}\u{a} = \u{\omega}\times\u{a} \end{align}\]II. Angular velocity tensor
In the Mathematica file, we show that
\[\begin{align} \Omega_{ij} = -\dot{\phi}\epsilon_{ijk}\hat{e}_{k} - (1 - \cos(\phi))\epsilon_{ijk}\epsilon_{kpq}\hat{e}_{p}\dot{\hat{e}}_{q} - \sin(\phi)\epsilon_{ijk}\dot{\hat{e}}_{k} \end{align}\]III. Angular velocity vector
\[\begin{align} \u{\omega} = \dot{\phi}\hat{\u{e}} + \sin(\phi)\dot{\hat{\u{e}}} + (1-\cos(\phi))(\hat{\u{e}} \times \dot{\hat{\u{e}}}) \end{align}\]In component notation we have,
\[\begin{align} \omega_{i} = \dot{\phi}\hat{e}_{i} + \sin(\phi)\dot{\hat{e}}_{i} + (1-\cos(\phi))\epsilon_{ipq}\hat{e}_{p}\dot{\hat{e}}_{q} \end{align}\]IV. Governing equation
In our last class we derived the following equations:
\[\begin{align} \lambda_{1}\dot{\omega}_{1} - (\lambda_{2} - \lambda_{3})\omega_{2}\omega_{3} &= \u{M}\cdot\hat{e}_{1}, \\ \lambda_{2}\dot{\omega}_{2} - (\lambda_{3} - \lambda_{1})\omega_{1}\omega_{3} &= \u{M}\cdot\hat{e}_{2}, \\ \lambda_{3}\dot{\omega}_{3} - (\lambda_{1} - \lambda_{2})\omega_{1}\omega_{2} &= \u{M}\cdot\hat{e}_{3} \end{align}\]where \(\omega_{i} = \u{\omega}\cdot\hat{e}_{i}\), the vector \(\u{\omega}\) is the angular velocity vector, and \(\u{M}\) is total moment acting on the solid.
V. Force free motion
Consider the case where there are no forces acting on the solid. In this case, the net moment acting on the solid is zero. Thus, in the case of no force, the governing equations simplify to
\[\begin{align} \lambda_{1}\dot{\omega}_{1} - (\lambda_{2} - \lambda_{3})\omega_{2}\omega_{3} &= 0, \\ \lambda_{2}\dot{\omega}_{2} - (\lambda_{3} - \lambda_{1})\omega_{1}\omega_{3} &= 0, \\ \lambda_{3}\dot{\omega}_{3} - (\lambda_{1} - \lambda_{2})\omega_{1}\omega_{2} &= 0 \end{align}\]1) Symmetric Solid
This is the case when \(\lambda_{i} = \lambda\). In this case, the equations of motion simplify to
\[\begin{align} \dot{\omega}_{i} =0 \end{align}\]The solution of the above set of three equations is
\[\begin{align} \omega_{i} = c_{i}, \end{align}\]where \(c_{i}\) are some constants that are determine from the initial conditions.
2) Axi-symmetric Solid
The top corresponds to the case \(\lambda_{1} = \lambda_{2} = \lambda\) and \(\lambda_{3}\neq\lambda\). When \(\lambda_{1} = \lambda_{2} < \lambda_{3}\) then the solid is called oblate and when \(\lambda_{1} = \lambda_{2} > \lambda_{3}\) then the solid is called prolate. In this case, the equations simplify to
\[\begin{align} \lambda_{1}\dot{\omega}_{1} - (\lambda_{2} - \lambda_{3})\omega_{2}\omega_{3} &= 0, \\ \lambda_{2}\dot{\omega}_{2} - (\lambda_{3} - \lambda_{1})\omega_{1}\omega_{3} &= 0, \\ \lambda_{3}\dot{\omega}_{3} &= 0 \end{align}\]from which it follows that \(\omega_{3} =c\), a constant. When \(\omega_{3} \rightarrow c\) are put in the above set of equations and simplified, we get
\[\begin{align} \dot{\omega}_{1} - \Lambda\omega_{2} &=0, \\ \dot{\omega}_{2} + \Lambda\omega_{1} &=0 \end{align}\]where
\[\begin{align} \Lambda :=c(1-\frac{\lambda_{3}}{\lambda}) \end{align}\]Differentiating the first equation w.r.t time and then substituting in it \(\dot{\omega}_{2} \rightarrow -\Lambda\omega_{1}\) from the second equation, we get
\[\begin{align} \ddot{\omega}_{1} + \Lambda^{2}\omega_{1} &=0, \\ \ddot{\omega}_{2} + \Lambda^{2}\omega_{2} &=0. \end{align}\]The solution of the above equations is
\[\begin{align} \omega_{1} &= A_{1}\cos(\Lambda t + \phi_{1}), \\ \omega_{2} &= A_{2}\cos(\Lambda t + \phi_{2}) \end{align}\]where \(A_{i}\) and \(\phi_{i}\), \(i=1, 2\) are constants.