Problem 3 Consider a solid circular membrane with radius a and tensile force per unit length 7. 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. [3 pts] 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. [6 pts] The membrane is excited at its center by the point force Fosin(at). 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. [6 pts]