$2.$ A climbing robot has been invented for many applications. The robot can be modeled by a particle mass $\backslash ($ m $\backslash )$ climbing up a uniform massless aluminium beam of length $\backslash (\backslash$ell $\backslash ),$ Young's modulus $\backslash ($ E $\backslash ),$ and circular cross$-$section with radius $\backslash ($ r $\backslash ),$ as shown in Fig. $2.$ When necessary, you may use the following numerical values. $\backslash ($ m$=20\backslash$mathrm$\{$~kg$\},\backslash$ell$=6\backslash$mathrm$\{$~m$\},$ r$=6\backslash$mathrm$\{$~cm$\}\backslash ),\backslash ($ E$=200\backslash$mathrm$\{$GPa$\}\backslash ).$ Ignore gravity. The beam will vibrate transversely. $[$Hint: it may help you to visualize it as a beam if you turn the problem around so it is horizontal.$]$

Figure $2$: Climbing robot

$($a$)$ Determine the natural frequency of the vibratory system $($for transverse vibration$)$ at the height shown, namely, $\backslash (($h$=\backslash$ell $/2)\backslash ).$

$($b$)$ Consider the fact that the robot could be at any height $\backslash ($ h $\backslash ).$ Now, determine the natural frequency as a function of $\backslash ($ h $\backslash ),$ and plot it$.$

$($c$)$ Strictly speaking, if the robot is climbing fairly quickly, the effective stiffness of the beam is changing quickly as well. In that case, $\backslash ($ k $\backslash )$ would be a function of time. Let us say the robot is climbing up at a constant rate $\backslash ($ v $\backslash ).$ Use the numerical values and carry out a numerical investigation $($using MATLAB$)$ to determine the vibrational response of the beam. For the climbing rate, investigate the response for varying values; some suggested numbers are $\backslash (0\mathrm{.}01,0\mathrm{.}1\backslash ),$ and $\backslash (0\mathrm{.}5\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ Do you see a difference between what you get here, and what you obtained in $($b$)?$ Explain. The effect of a moving mass on an elastic structure is an important practical problem that leads to a dangerous instability; did you get any of that in your numerical simulation? Explain.