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A Two-Resonance Guitar Bridge

Now let's consider a two-resonance guitar bridge, as shown in Fig. 9.6.

Figure 9.6: Synthetic input admittance of a passive, linear, dynamic system using a pair of resonating two-pole filters, a pair of zeros between the resonances, and a zero near dc.
\includegraphics[width=\twidth]{eps/lguitarsynth2simp2}

Like all mechanical systems that don't ``slide away'' in response to a constant applied input force, the bridge must ``look like a spring'' at zero frequency. Similarly, it is typical for systems to ``look like a mass'' at very high frequencies, because the driving-point typically has mass (unless the driver is spring-coupled by what seems to be massless spring). This implies the driving point admittance should have a zero at dc and a pole at infinity. If we neglect losses, as frequency increases up from zero, the first thing we encounter in the admittance is a pole (a ``resonance'' frequency at which energy is readily accepted by the bridge from the strings). As we pass the admittance peak going up in frequency, the phase switches around from being near $ \pi /2$ (``spring like'') to being closer to $ -\pi /2$ (``mass like''). (Recall the graphical method for calculating the phase response of a linear system [452].) Below the first resonance, we may say that the system is stiffness controlled (admittance phase $ \approx\pi/2$ ), while above the first resonance, we say it is mass controlled (admittance phase $ \approx-\pi/2$ ). This qualitative description is typical of any lightly damped, linear, dynamic system. As we proceed up the $ j\omega $ axis, we'll next encounter a near-zero, or ``anti-resonance,'' above which the system again appears ``stiffness controlled,'' or spring-like, and so on in alternation to infinity. The strict alternation of poles and zeros near the $ j\omega $ axis is required by the positive real property of all passive admittances (§C.11.2).


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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