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Calculation Example Systems Vibration Natural periods # Calculation Example: Natural Periods of Vibration for Systems

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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. 0.2T_{2}

b. 0.2T_{2}

c. 2T_{2}

d. 3T_{2}

Answer **D**.

SI solution

The stiffness of a cantilever beam is described by

It is known that E_{1}=E_{2} and I_{1}=I_{2}. Since h_{1}=2h_{2}, then:

Since K_{2}=8K_{1} , then:

System 1:

System 2:

Given that m_{1}=m_{2}, it is assumed:

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