The frequency response of a system provides critical information about how the system will respond to input forcing.  The frequency response is usually characterized in terms of amplitude and phase functions of frequency.  The swept-sine testing that will be conducted in these laboratories provides some insight into the physical nature of these functions.  Basically, you force a system with a sine wave of a given frequency and measure the output.  The amplitude function is simply the ratio of the output amplitude to the input amplitude (you might take a measure as the peak values or the root-mean-square; sometimes it does not matter, but you should always assess each case independently).  As these measurements are made for different forcing frequencies you can generate a graph of amplitude (ratio) versus frequency and phase versus frequency (the phase is the difference between when the input crosses a given level and when the output crosses that level, typically we look at the zero level).  Phase is usually given in degrees.

Why measure frequency response?  The frequency response gives you direct information on what the system will do if you input a sinusoidal forcing function.  Consider a mass-spring-damper system excited by a force that can be represented by F(t) = F_osin(wt). If you have the amplitude function, A(w) (this is a function of frequency, w (rad/sec) or f (Hz)), that relates the displacement of the mass, x(t), to the input force, F(t), then you can find the amplitude of the motion for any forcing frequency.  Specifically, after any transients have died out the harmonic motion is,

Practical calculation: Remember, A(w) and f(w) provide numerical values, so if you want to know the amplitude x_o at some w where A(w) = 2.2 with u_o = 0.6, then you just find it by  x_o = u_oA(w) = (0.6)(2.2) = 1.32 (should be units of length).

Hopefully this brief explanation of how you use the amplitude and phase functions will provide at least one reasone why so much effort is given to finding transfer functions. Transfer functions for systems can be used to find the amplitude and phase functions.  Sometimes the amplitude and phase functions are referred to as Bode plots.  Bode plots refer to a specific way of plotting the A(w) and f(w) functions; amplitude is plotted in decibels (dB) and phase in degrees.  To some people, Bode plots refer strictly to the approximate plots of the amplitude and phase functions drawn "by hand".  With modern computing tools, the term Bode plots has been retained to refer to "exact" plots.  The figure below, for example gives the "Bode plots" for a transfer function: