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3.6 The resonance peak - qualitatively
We see from Fig. 10 above and Eqns (50) and (63), that higher Q's result in taller and more narrow resonance peaks. Why is this? In this section we explore an alternate way to view the process of a sinusoidal force exciting a resonator, a view that will qualitatively explain the effect that Q has on the resonance peaks.
A series of impulses
This way to view a sinusoidal driving force is that it is similar to a long string of impulses. Each cycle of the sinusoidal excitation is equivalent of a shock impulse to the resonator as discussed in an earlier posting and reviewed above. We saw there, that a single impulse will result in the resonator ringing at a frequency of ωdecay given by Equation (24) and being described by a damped sinusoidal equation, of the form of Equation (23). Because we have a linear system, the series of ringings made by a series of impulses will simply add up to give the total response of the resonator. See an earlier posting for more on the addition of sinusoidal functions.
We illustrate this in the following animations where we show each ringing separately as well as their sum (the actual response of the resonator). Read the captions and experiment with the animations to understand this concept further.
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Fig. 15a. Excitation with a series of impacts (of the men jumping onto the mass of a mass/spring resonator.) Mouse over the figure to see the action, mouse off to suspend it, or click on it to restart it. It can be speeded up by clicking on the grayed out "medium" or "fast". Combinations of Q and impact frequency can be selected by clicking on the circles on the resonance curves at the bottom. The frequencies are not the actual frequencies that the viewer will see on any of the three viewing speeds.
The response of the mass is shown in the middle of the graphing window. The series of curves at the top are the responses (or ringings) of the mass to each impact graphed separately. These are vertically offset from each other for clarity. Because this is a linear system, the motion of the mass is the sum of these separate ringings. Note that higher Q's result in longer decays of the individual ringings shown at the top. The maximum response of the mass occurs when the impacts are timed so that the peaks and valleys of the various ringing curves all line up, so there is constructive interference of all the ringings. The higher the Q and longer the decay time, the more carefully the impact frequency must adjusted to equal the resonance frequency of the resonator over the longer ringing decay times. The time dependent behavior of the resonator when the driving impulses are not optimally timed is a little complex and displays some oscillations in the amplitude, especially at the larger Q values. Note that the smooth resonance curves such as shown on the bottom part of this animation and in Fig. 10 (and other graphs above) only represent the eventual (i.e. equilibrium) behavior of the resonator after considerable time has passed and do not show the "transient" behavior of the system. Note also that one may need to examine the multiple impact ringing curves at the top very carefully to see the difference in phases between the on resonance and off resonance aligning of the peaks and valleys. Off resonance makes the timing of the peaks drift somewhat as shown in more detail below in Figs. 15b, 15c and 15d. | |
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| ↑ Fig. 15b. Alignment of the ringing response curves from the above animation for Q = 20 and the frequency of impacts set equal to the resonant frequency, i.e. at the peak of the resonance curve. We see that the red and green dots, "A" and "B" (both at time t = t1) are in the peak of their respective ringing curves, meaning that these peaks add constructively. | ↑ Fig. 15c. Alignment of the ringing curves for Q = 20 and the frequency of impacts set to less than the resonant frequency. Here we see that the peaks are not aligned. The yellow line drawn through one set of peaks drifts to earlier times as we proceed down the page. Also while point "A" is at a peak (or crest) of the top ringing curve, point "B" is near the valley of its ringing curve, even though both points occur at the same time, time t = t1. Thus the ringing curves partially subtract from each other and we get less than the full response at this frequency. |
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↓ Fig. 15e. (below) This animation is similar to Fig. 15a except that it does not show the build up with time of the response. Instead it allows the viewer to instantly see the results. The drive frequency can be entered numerically in the box, or the yellow slider can be dragged horizontally across the frequency axis of the bottom graph to adjust the drive frequency.
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| ↑ Fig. 15d. Alignment of the ringing curves for Q = 20 and the frequency of impacts set to more than the resonant frequency. Again we see that the peaks do not align. The yellow line drawn through one set drifts to later times as we proceed down the page. Also while point "A" is at a peak (or crest) of the top ringing curve, point "B" is near the valley of its ringing curve, even though both points occur at the same time, time t = t1. Thus again the ringing curves partially subtract from each other and we get less than the full response at this frequency.
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Frequency shift due to phase slippage
The above discussion brings up the question: How do the ringing curves, all oscillating at a frequency ωdecay, add up to a total response that oscillates at a different frequency ω = ωdrive ?
This question relates to a resonator being driven at a frequency different than the resonator's resonant frequency. In reference to Fig. 15e, this means that we have dragged the yellow rectangle to the left or right of the peak, perhaps to frequencies of 2.6 or 3.0Hz. It also relates to the fact that in the steady state response, i.e. after the resonance is nicely oscillating in a steady fashion, the resonator must oscillate at whatever frequency it is driven at and not at the frequency it may "wish" to oscillate at. You can see this as you drag the yellow rectangle to the extreme right or left, that the peaks and valleys in the "total resonator response" window bunch up or become drawn out in spite of the fact that ω0 and ωdecay do not change.
The change in frequency from that of the individual ringing curves (all oscillating at ωdecay) to that of the sum response (which is oscillating at ω = ωdrive ) is due to the fact that off-resonance, the ringing curves do not all line up. There is a phase slip between successive ringing curves. Since frequency is defined as the rate of change of phase, this slippage in phase changes the overall frequency. In fact, the amount of phase slippage per time exactly equals the frequency difference between ωdecay and ωdrive.
Another question that might come up is: why is the "total resonator response" in Fig. 15e rather jagged when the drive frequency is 2Hz and not a smooth sinusoid? The reason is that when pushed to this limit our series of impulses is a rather poor approximation to a sinusoidal drive force. We will do better in our approximation in the next posting (which uses impulses in two directions) and still better yet in the postings after that (which use calculus to sum an infinite number of impulses that vary in amplitude and direction to perfectly model a sinusoidal driving force).
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