Calculation Example: Natural Periods of Vibration for Systems

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?

a. 0.2T₂

b. 0.2T₂

c. 2T₂

d. 3T₂

Answer D.

SI solution

The stiffness of a cantilever beam is described by

It is known that E₁=E₂ and I₁=I₂. Since h₁=2h₂, then:

Since K₂=8K₁ , then:

System 1:

System 2:

 Given that m₁=m₂, it is assumed: