Perform the following tasks for the structure shown in Figure 1.

m1 1 kip s²/in

m2 = m3 = 2m₁

m4 = 3m₁

k₁ = 800 kip/in

k₂ = 2k1

k3 = 3k₁

k4 = 4k₁

(a) Determine the natural frequencies and mass normalized modal matrix of the structure. The [V,D] = eig (A,B) command in MATLABTM pro- duces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that A*V = B*V*D. Verify that the eigenvalues and eigenvectors are sorted such that the natural frequencies are in increasing order. Confirm that the eigenvectors are mass normalized, or if necessary, mass normalize the eigenvectors to produce the modal matrix.

(b) Normalize the mode shapes such that each mode is scaled so that its maximum value is 1. Plot the mode shape vectors.

(c) Determine the modal mass, modal stiffness, and modal forces for each mode using the mass normalized modes if the mass m₁ is subjected to a harmonic excitation P₁ cos wt.

(d) Determine an expression for the undamped steady-state response in modal coordinates, y(t).

(e) Determine an expression for the top floor displacement response of x₁(t) and clearly label the contribution from each mode.

(f) Prepare a table comparing the undamped steady-state displacement amplitude x₁(t) for w=0, w = 0.5wn,1, and w = 1.3wn,3 using the first, first and second, first through third, and all four modes. Quantify the effect of truncating the higher order modes assuming the solution that uses all four modes is exact. Comment on why this error varies depending on the forcing frequency.