Buildings designed to withstand earthquakes must have natural frequencies of vibration that are not close to the ,oscillation frequency of the ground motion. A building’s natural frequencies are determined primarily by the masses of its floors and by the lateral stiffness of its supporting columns (which act like horizontal springs). We can find these frequencies by solving for the roots of a polynomial called the structure’s characteristic
polynomial (characteristic polynomials are discussed further in Chapter 8). Figure 2.5-1 shows the exaggerated motion of the floors of a three-story building. For such a building, if each floor has a mass m and the columns have stiffness k, the

Simple vibration model of a building subjected to ground motion.

polynomial is

where ex = kj4 mn? (models such as these are discussed in more detail in [Palm, 2005]). The building’s natural frequencies in cycles per second are the positive roots of this equation. Find the building’s natural frequencies in cycles per second for the case where m = 1000 kg and k = 5 X 106 N/m.

Solution

• Solution
The characteristic polynomial consists of sums and products of lower-degree polynomials. We can use this fact to have MATLAB do the algebra for us. The characteristic polynomial has the form