Problem $3$ Consider a solid circular membrane with radius a and tensile force per unit length $T.$ The outer periphery

is supported on a smooth, massless, guide which provides tension but allows the membrane to freely move in the

transverse direction; hence, the edge $r=$a can be treated as being "free".

Show that the stiffness operator is positive semi$-$definite.

Derive and sketch the characteristic equation. Make a table of the lowest half$-$dozen or so natural frequencies

and indicate the number of nodal diameters $(ND)$ and circles $(NC)$ present for each mode. Sketch by hand the

nodal lines and diameters for those modes.

The membrane is excited at its center by the point force $F_{o}sin(t).$ Plot the steady state response amplitude at

the center of the membrane $($e$.$g$.,$ the collocated response$)$ as a function of excitation frequency. Describe

physically the response and which modes are active.