Amplitude and Phase of the Frequency Response Function

A useful tool in the dynamic analysis of structures is the frequency response function, or frf. The frf is a plot of the amplitude and phase-lag of a particular floor of a building as a function of the forcing frequency. The first part of the frf is the plot of amplitude versus frequency. At resonant frequencies, the frf will have a sharp peak. If the damping is high, the the resonant peaks of the frf will be lower and broader. The second part of the frf is the plot of phase-lag versus frequency. At or near the natural frequencies, the phase-lag will be 90 degrees. Between the natural frequencies, the phase-lag will be close to 0 degrees or 180 degrees. An example of a frequency response function for a three-story building model is shown in Figure 9.

Figure 9: The frequency response function of a lightly-damped three-story building model, N/m, N/m/s, kg.

Quick Quiz: Figure 9 illustrates three lightly-damped resonant peaks in the frequency response function. At each resonant frequency, determine the amplitudes of the resonant peak at each floor. Use the phase plot to determine if the amplitude should be positive or negative. (Note how your answer compares to Table 1.)

Story 1 7.8 14 6.2 2 14 6.1 7.9 3 17 11 3.6

Story 1 2.1 8.7 5.7 2 5.7 2.1 -8.7 3 8.7 -5.7 2.1

Story 1 7.8 14 6.2 2 14 6.1 -7.9 3 17 -11 3.6

Story 1 90 90 90 2 90 90 270 3 -270 -90 90