Page 38 - Summer 2010

THE BANJO: THE “MODEL” INSTRUMENT
Joseph W. Dickey
3960 Birdsville Rd. Davidsonville, Maryland 21035
If you stare at a banjo hard enough, you can see two interacting, wave- bearing systems; specifically, strings and a circular membrane. What’s more, they are systems for which we can solve the equations of motion and thereby describe the propagation of structural vibrations; i.e., waves. Why this has not been done before is a mystery to me.
At its lowest level of abstraction, the
“simplified” banjo shown in Fig. 1 is basically two interacting vibrating systems: a plucked string and a circular membrane. By “interacting” I mean that waves in the string generate waves in the membrane, and visa versa, so that the whole instrument vibrates together. The five-string banjo is a little more complicated as it has six interacting systems—five strings and one membrane. The equations which describe the way waves travel in these systems, how they radiate sound, and how they interact are fairly straightforward. For example, a string has only two coordinates, position and time, and the equation which describes the propagation of a disturbance along the string, the wave equation, links these two variables in such a way that if you know the wave’s position at any one time, you know it at any other time. It is an analogous situa- tion in the membrane but complicated somewhat by it being a two dimensional surface and by having the bridge being placed away from the membrane’s geometric center.
Another complication, common to both the strings and the membrane, is that every time a wave hits something, it changes as it reflects and/or passes through. With this, the math gets pretty messy.1 Part of the “art” of modeling is to decide where you can simpli-
fy things without throwing
out the essence of what you
are looking for. For example,
there are undoubtedly waves
traveling in the neck as
though it were a sixth string:
are they important? And
there are also reverberations
within the bridge, tone ring,
and other parts: are they
important? The model which
I developed and describe here
does not consider reverbera-
tions of waves in these parts,
but does account for their
influence on the reverbera-
tion of waves in the head and
strings. As banjo modeling
gets more sophisticated, there
are many such considerations
“Developing a model is a little like collecting garbage; you really should know what you are going to do with it before you start.”
which will become important, and with this will come more pressure to keep the model as simple a possible while retain- ing what you set out to discover. In this sense, developing a model is a little like collecting garbage; you really should know what you are going to do with it before you start.
Why do we go to so much trouble to model the banjo? Partly to build better banjos or help people who fix or play banjos get a better sound by understanding how the various components of the instrument work together to produce the sound. More gen- erally, we do this sort of thing (modeling) to build a better anything. This story describes the why and wherefore of ana-
lytical modeling.
The strings
Given the tension in the string (provided by the tuning peg) and its mass we know how an initial displacement (a pluck) travels, and given the properties of the nut, bridge and clamp we know how this traveling disturbance reflects off the ends. We call this a structural vibration, or a wave, and this wave will travel back and forth in the string reflecting alter- nately from one end and the other. And just as it takes some force to contain the end of a jump-rope, it takes force to keep the string connected (through the bridge) to the membrane. We can calculate all of this. Figure 2 describes a pluck in the upper part of the figure as, initially, a triangular shaped dis- placement shown as a dotted line. The vertical scale in the fig- ure is greatly exaggerated to make it easier to see what is going
on. Normally, the displace- ment of the string is only about one-thousandth of its length. The figure also shows what the string shape would be after about 12 reverbera- tions. Note that the response of the string has diminished in time because there has been energy shared with the other systems and because all the systems have losses. Note, also, that the triangular shape of the pluck has smoothed to the point of looking like our jump-rope; specifically, it resembles the sinusoidal shape we generally expect from a vibrating string. There are several factors which con- tribute to this smoothing
Fig. 1. The banjo simplified. Breaking the banjo down into interacting parts that are individually modeled and solvable.
The Banjo 37