Calculation Example – Undamped free Vibration (Part B).

Calculate the modal frequencies for the shear frame at the previous exercise Undamped free Vibration (Part A). The story weight of the first floor is W1=60 KN end at the second floor W2=50 KN. The cross section of the columns is circular with radius r1=0.5m at the first floor and r2=0.2m at the second floor. The columns are considered to have no mass. (Modulus of elasticity E=2.1*107 KN/m2, Story height h1=5m, h2=3m.)

 Solution The shear frame has two degrees of freedom u1, u2. We calculate the masses for each floor:

The Mass Matrix is:

The moment of Inertia for the columns is:

The lateral stiffness for each column is

The Stiffness Matrix is:

 From the previous exercise Undamped free Vibration (Part A) we use the stiffness matrix.

In order to find the modal frequencies the determinant of the matrix K-?2? must be zero so the unknown variable is ?i .

We keep only the positive values so the modal frequencies are ?1 =71.0252 rad/sec, ?2 =192.1029 rad/sec.