A single point with mass m is vibrating harmonically. The displacement x from point of balance is governed by the differential equation

A single point with mass m is vibrating harmonically. The displacement x from point of balance is governed by the differential equation

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.)

Calculate the Mass and the Stiffness matrix for the shear frame below. The mass is lumped at each level. The supports at end A,B are fixed . The columns are considered to have no mass. (Modulus of elasticity E, Storey masses m1,m2, Column moment of inertia I₁,I₂. )

Each of the following columns supports a block of identical mass, m. The columns are fixed at the bottom and free at the top. The height of the first column is twice the height of the second column. The modulus of elasticity, E, and moment of inertia, I, for both columns are the same. The systems have natural periods of vibration of T1 and T2, respectively. Neglecting the weight of the columns, what is the natural period of vibration for the first system?