91Ó°ÊÓ

The given values are the kinetic energy is the sum of the translational energy $\frac{1}{2}\mathrm{m}{\stackrel{˙}{\mathrm{z}}}^2$and the rotational energy $\frac{1}{2}\mathrm{I}{\stackrel{˙}{\mathrm{θ}}}^2$.

The Lagrange equations are used to construct the equations of motion of a solid mechanics issue in matrix form, including damping.

Since they are not independent. Assuming that at$\mathrm{z}=0$

the coordinate$\mathrm{θ}=0$as well,

$$\begin{gathered} \mathrm{z}=-\mathrm{²¹Î¸} \\ \stackrel{Ë™}{\mathrm{θ}}=\frac{-\stackrel{Ë™}{\mathrm{z}}}{\mathrm{a}} \end{gathered}$$

rotational energies, therefore:

The potential energy has the following form:

$\mathrm{V}=\mathrm{mgz}$

And therefore the Lagrangian is:

Observe the Euler equation for$\mathrm{z}$degree of freedom. The Euler equations reads:

$\frac{\mathrm{d}}{\mathrm{dt}}\frac{∂\mathrm{L}}{∂\mathrm{z}}-\frac{∂\mathrm{L}}{∂\mathrm{z}}=0$

Calculate the required derivatives.

Use all of the equations above,

$\left(\mathrm{m}+{\mathrm{la}}^{-2}\right)\stackrel{¨}{\mathrm{z}}+\mathrm{mg}=0$

For a cylinder with inner radius $\mathrm{a}$and outer radius $\mathrm{b}$the moment of inertia is:

$\mathrm{I}=\frac{1}{2}\mathrm{m}\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)$

This can be inserted into the Euler equation but is not necessary.

Therefore, the relation between $\stackrel{˙}{\mathrm{z}}$and $\stackrel{˙}{\mathrm{θ}}$ is$\left(\mathrm{m}+{\mathrm{la}}^{-2}\right)\stackrel{¨}{\mathrm{z}}+\mathrm{mg}=0$.