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.