The mass matrix of a 2node beam element undergoing only bending is:

a. Assume that the beam is cantilevered. State the generalized eigenvalue problem for computing the natural frequencies and mode shapes of vibration of this beam.

b. Write the characteristic equation for the eigenvalue problem, and solve for the natural frequencies.

c. Determine the mode shapes of vibration.

d. If the beam is subjected to a transverse load at the tip that is varying harmonically as: \(F=\sin t\), use the modal superposition approach to write the displacement as a weighted

sum of the first two modes of vibration, and obtain a decoupled set of equations of each mode.

e. Determine the forced response of this structure due to the applied forcing function assuming that at time \(t=0\) the beam was at rest with no deflection.

f. Redo the modal superposition using only the first mode, and compare the solutions from part (e).

156 22L 54 -13L7 22L 4L2 13L -3L Me] = PAL 420 54 13L 156 -22L L-13L -3L2-22L 4L p=4.210+, E=2 1010, 1=108, A= 10-4, L=10