4.1: Definition of Frequency Response

We now consider system excitation that varies with time sinusoidally, as either \(\sin \omega t\) or \(\cos \omega t\), and persists for an indefinitely long duration. The frequency of excitation is \(\omega\) radians/s, or \(\omega / 2 \pi\) Hz (for hertz, which means cycles/s, named for German physicist and engineer Heinrich Rudolf Hertz, 1857-1894). After any transients due to initial conditions have decayed away, a stable linear system responds in the same sinusoidal fashion. That is, the steady-state response of a stable linear system to sinusoidal excitation also varies sinusoidally, as either \(\sin (\omega t+\phi)\) or \(\cos (\omega t+\phi)\), where the frequency \(\omega\) is the same as the excitation frequency, and \(\phi\) is a phase angle. This steady-state sinusoidal response is generally called frequency response. Although the frequency of response is the same as that of excitation, the magnitude of response can vary greatly for different excitation frequencies; therefore, in order to prevent the overloading of a system, it is important to know the frequencies of excitation to which the system is most sensitive.

Both mathematical and experimental analyses of system frequency response are common in engineering practice. We shall study the basic methods of mathematical analysis. The two primary unknowns in the analysis are the magnitude and phase of response as functions of excitation frequency. We can consider the excitation to vary as either \(\sin \omega t\) or \(\cos \omega t\); the steady-state frequency response magnitude and phase are the same in either case. For consistency, we will consider primarily \(\cos \omega t\) excitation.