Like a jump rope, a string tends to swing in an ellipse, a fact well known to children. When holding both ends of a rope or string, it is difficult to shake it so that motion is confined to a single transverse plane. Instead of remaining confined to planar oscillations, strings appear to prefer elliptical or whirling motions like those found when playing jump rope. Borrowing terminology from optics, we would say that a string prefers circular polarization to planar polarization. In addition to whirling, other phenomena are easily observed in forced strings including bifurcations between planar and nonplanar periodic motions, transitions to chaotic motions, sudden jumps between different periodic motions, hysteresis, and periodic and aperiodic cycling between large and small vibrations.
In this chapter we will begin to explore the dynamics of an elastic string by examining a single-mode model for string vibrations. In the process, several new types of nonlinear phenomena will be discovered, including a new type of attractor, the torus, arising from quasiperiodic motions, and a new route to chaos, via torus doubling. We will also show how power spectra and Poincaré sections are used in experiments to identify different types of nonlinear attractors. In this way we will continue building the vocabulary used in studying nonlinear phenomena [1].
In addition to its intrinsic interest, understanding the dynamics of a string can also be important for musicians, instrument makers, and acoustical engineers. For instance, nonlinearity leads to the modulation and complex tonal structure of sounds from a cello or guitar. Whirling motions account for the rattling heard when a string is strongly plucked [2]. Linear theory provides the basic outline for the science of the production of musical sounds; its real richness, though, comes from nonlinear elements.
When a string vibrates, the length of the string must fluctuate. These fluctuations can be along the direction of the string, longitudinal vibrations , or up and down, vibrations transverse to the string. The longitudinal oscillations occur at about twice the frequency of the transverse vibrations. The modulation of a string's length is the essential source of a string's nonlinearity and its rich dynamical behavior. The coupling between the transverse and longitudinal motions is an example of a parametric oscillation. An oscillation is said to be parametric when some parameter of a system is modulated, in this case the string's length. Linear theory predicts that a string's free transverse oscillation frequency is independent of the string's vibration amplitude. Experimental measurements, on the other hand, show that the resonance frequency depends on the amplitude. Thus the linear theory has a restricted range of applicability.
Think of a guitar string. A string under a greater tension has a higher pitch (fundamental frequency). Whenever a string vibrates it gets stretched a little more, so its pitch increases slightly as its vibration amplitude increases.
We begin this chapter by describing the experimental apparatus we've used to study the string (section 3.2). In section 3.3 we model our experiment mathematically. Sections 3.4 to 3.6 examine a special case of string behavior, planar motion, which gives rise to the Duffing equation. Section 3.7 looks at the more general case, nonplanar motion. Finally, in section 3.8 we present experimental techniques used by nonlinear dynamicists. These experimental methods are illustrated in the string experiment.