The Guitar Bridge

As a practical lumped-modeling example, consider the simulated admittance at a guitar bridge. A highly simplified simulated example is shown in Fig. K.2.

Figure K.2: 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.

Like all lightly damped mechanical systems, the bridge must ``look like a spring'' at zero frequency and ``look like a mass'' at infinite frequency. This implies the driving point admittance must have a zero at DC and a pole at infinity; equivalently, the driving-point impedance must have a pole at DC and a zero 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''). Below the first resonance, we say the system is stiffness controlled, while above the first resonance, we say the system is mass controlled. This is a completely general characterization of any lightly damped, linear dynamic system. As we proceed up the $j\omega$ axis, we'll next encounter a zero, or ``anti-resonance,'' above which the system again appears ``stiffness controlled,'' or spring-like, and so on in alternation to infinity.

Figure K.3: Input admittance of a real classical guitar bridge measured by striking the bridge with a force hammer in the vertical direction.

A measured driving-point admittance [253] for a real guitar bridge is shown in Fig. K.3. Note that at very low frequencies, the phase information does not look like it should. This indicates a poor signal-to-noise ratio at very low frequencies. This can be verified by computing the coherence function for multiple measurements,^K.2 as shown in Figures K.4 and K.5. A coherence of 1 means that all the measurements are identical, while a coherence less than 1 indicates variation from measurement to measurement, implying a low signal-to-noise ratio. As can be seen in the figures, at frequencies for which the coherence is very close to 1, successive measurements are strongly in agreement, and the data are reliable. Where the coherence drops below 1, successive measurements disagree, and the measured admittance is not even positive real at very low frequencies.

Figure K.4: Overlay of three successively measured input admittances of a classical guitar bridge, displayed over a wider frequency range, and showing the coherence function, normalized to lie on the same dB scale as the admittance magnitude data.

Figure K.5: Overlay of the phase data for three successively measured input admittances of a classical guitar bridge, displayed over the frequency interval for which the coherence is very close to 1 (also shown).