Nonlinear and chaotic oscillations of an india-rubber band R. Chaco ´ n, P. Sua ´ rez, and F. Sa ´ nchez-Bajo Departamento de Electro ´nica e Ingenierı ´a Electromeca ´nica, Escuela de Ingenierı ´as Industriales, Universidad de Extremadura, Apartado Postal 382, 06071 Badajoz, Spain P. Mun ˜ iz Departamento de Fı ´sica Aplicada, Universidad de Castilla – La Mancha, Campus Universitario, 13071 Ciudad Real, Spain ~Received 30 September 1996; revised manuscript received 8 July 1997! The restoring force of a length of stretched elastic band is studied experimentally and the comparison with that of a spring is discussed. It is demonstrated that the simplest model of elastic band oscillations is capable of showing nonlinear phenomena including crisis, periodic, and chaotic motions, as well as spatial symmetry breaking. @S1063-651X~97!09211-8# PACS number~s!: 05.45.1b, 05.40.1j I. INTRODUCTION A great deal of attention has been paid in the past few decades to the study of nonlinear oscillations of elastic springs @1–9#, although the earliest studies go back to the past century @10#. In addition to their scientific interest, springs appear in almost all physics textbooks, at secondary and university levels @11,12#, as the simplest model for both linear and nonlinear oscillations. However, a length of ordi- nary ~cut! elastic india-rubber band ~henceforth termed ‘‘elastic band’’! is another of the simplest spatially distrib- uted nonlinear systems imaginable: In addition to its intrinsic theoretical interest, the nonlinear dynamics of an elastic band could also shed light on the dynamics of more complicated systems with spatial distribution ~e.g., hydrodynamic sys- tems!. The main purpose of this work is to reveal the char- acteristics of the restoring force of a common elastic band. The paper is organized as follows. In Sec. II we propose an analytical expression for the restoring force of a stretched elastic band that is in excellent agreement with our experi- mental results. A comparison is also made with the restoring force corresponding to a stretched nonlinear elastic spring. In Sec. III we study a very simple single-mode model of elastic- band oscillations deduced from the phenomenological restor- ing force postulated in Sec. II and demonstrate that the vi- brating elastic band can undergo spatial symmetry breaking and crisis as well as periodic and chaotic motions. We con- centrate on the structural stability of the model system under changes in an elastic-band characteristic parameter. Finally, Sec. IV gives a summary of the results. II. EXPERIMENTAL ELASTIC-BAND RESTORING FORCE We studied experimentally the elastic response of lengths of ~common! stationer’s elastic bands subjected to stretching. The samples were in the form of ribbons with different di- mensions. The experiments were carried out in a mechanical testing machine controlled by computer, in all cases at the same temperature ( ;25 °C). The specimens were gripped between a stationary lower and a movable top clamp. The movable part contained a force controller that measured up to 100 N. The experiments were carried out in two ways: ~a! controlling the displacement of the movable part with a con- stant stretching rate ~50 mm/min! and ~b! controlling the added load rate ~0.1 N/min or 0.5 N/m!. In both cases the computer records the load and longitudinal stretch every 0.05 s. We found very similar results for the two types of experi- ments: A typical example is plotted in Fig. 1 ~thick line!. Observe the initial smooth hump, forming a first nonlinear region of elasticity, which is the characteristic feature of the elastic band distinguishing it from, for example, an elastic spring @13#. A second broader region of nonlinearity ~after the inflection point! is common to elastic springs @13#. To take these characteristics into account we propose the following analytical expression for the restoring force: F ~ x ! 52k F x S 2 2tanh U x a U D 1bx 3 G , ~1! where x is the displacement from the relaxed position, k is a strength parameter, a is an elastic-band characteristic pa- FIG. 1. Experimental restoring force ~thick line! of a stretched elastic band and theoretical fit ~thin line! @Eq. ~1!# for k 50.0068 N/mm, a 5236 mm, and b 51.23310 26 mm. PHYSICAL REVIEW E NOVEMBER 1997 VOLUME 56, NUMBER 5 56 1063-651X/97/56~5!/5321~6!/$10.00 5321 © 1997 The American Physical Society