Frequency Response and Practical Resonance In the previous note in this session we found the periodic solution to the equation mx �� + bx � + kx = B cos(ωt). (1) The solution was x p = gB cos(ωt − φ), where g is the gain g = g(ω ) = � 1 (2) (k − mω 2 )2 + b2 ω 2 and φ is the phase lag φ = φ(ω ) = Arg( p(iω )) = tan−1 (bω/(k − mω 2 )). (3) The gain or amplitude response is a function of ω. It tells us the size of the system’s response to the given input frequency. If the amplitude has a peak at ωr we call this the practical resonance frequency. If the damping b gets too large then, for the system in equation (1), there is no peak and, hence, no practical resonance. The following ﬁgure shows two graphs of g(ω ), one for small b and one for large b. g g ω0 ω ωr Fig 1a. Small b (has resonance). ω0 ω Fig 1b. Large b (no resonance) In ﬁgure (1a) the damping constant b is small and there is practical reso nance at the frequency ωr . In ﬁgure (1b) b is large and there is no practical resonant frequency. Finding the Practical Resonant Frequency. We now turn our attention to ﬁnding a formula for the practical resonant frequency -if it exists- of the system in (1). Practical resonance occurs at the frequency ωr where g(w) has a maximum. For the system (1) with gain (2) it is clear that the maximum gain occurs when the expression under the radical has a minimum. Accordingly we look for the minimum of f (ω ) = (k − mω 2 )2 + b2 ω 2 . 18.03SC Setting f � (ω ) = 0 and solving gives f � (ω ) = −4mω (k − mω 2 ) + 2b2 ω = 0 ⇒ ω = 0 or m2 ω 2 = mk − b2 /2. We see that if mk − b2 /2 > 0 then there is a practical resonant frequency � k b2 ωr = − . m 2m2 Phase Lag: In the picture below the dotted line is the input and the solid line is the response. The damping causes a lag between when the input reaches its maxi mum and when the output does. In radians, the angle φ is called the phase lag and in units of time φ/ω is the time lag. The lag is important, but in this class we will be more interested in the amplitude response. . . . ...... � • ... . . . ..... φ/ω time lag . ... . . . .. ..... ....... ...... . . .... ..... . . . . . . . .. . .... ... ..... . . . . . ........ . . . . ........... . . ... ..... t