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TORSION PENDULUM : THE MECHANICAL NONLINEAR OSCILLATOR

The main purpose of talking about nonlinear phenomena in school is to gain the students ability for distinguishing the nonlinear system from the linear one by qualitative observations of the system dynamics. The complex and qualitatively different behavior of nonlinear systems should be pointed out as an remarkable consequence of non-linearity that happens to occur very often in natural systems. Formal thinking plays an essential roll when basic features of a nonlinear system and its dynamics are compared to the linear system. A qualitative approach is particularly fruitful for developing the basic idea of the phenomena and gaining a deeper understanding of both linear and nonlinear systems. The simplest example of nonlinear oscillator is the physical pendulum. The torsion pendulum [1, 2, 3] instead is considered as a didactical example of a nonlinear system that performs all the basic features of a dynamical system. According to the equation of motion the torsion pendulum is characterized as a Duffing oscillator. The Duffing equation is often numerically solved to analyze its dynamics [4, 5]. There are also known electric circuits that behave according to the Duffing equation [6]. Such systems are very precise and easy to analyze but they are abstract and hard to use for developing formal thinking. The basic idea has been initiated and first experiments performed by prof. dr. Giacomo Torzo at University of Padua. The theoretical analysis as well as design and construction of improved apparatus has been done at Physics Department, University of Ljubljana as BSc thesis [7]. The apparatus can be used as a didactical tool for demonstrating: the amplitude dependence of the period of oscillation, the hysteresis phenomena in the response of sinusoidaly driven pendulum and bifurcation of stable orbits and the transition to chaotic motion. Our construction is based on theoretical predictions and numerical simulations. The goal was to build a mechanical oscillator of large dimensions that would help the observer to recognize and understand the phenomena. The main problem was to find the design that will optimize the requirements mentioned above. For this reason we have chosen the construction that leaves all the free parameters to be varied to some extend.