Coupled Oscillators

Physics Education, 2010

The topic of coupled oscillations is rich in physical content which is both interesting and complex. The study of the time evolution of coupled oscillator systems involves a mathematical formalization beyond the level of the upper secondary school student's competence. Here, we present an original approach, suitable even for secondary students, to investigate a coupled pendulum system through a series of carefully designed hands-on and minds-on modelling activities. We give a detailed description of these activities and of the strategy developed to promote both the understanding of this complex system and a sound epistemological framework. Students are actively engaged (1) in system exploration; (2) in simple model building and its implementation with an Excel spreadsheet; and (3) in comparing the measurements of the system behaviour with predictions from the model.

One of the authors (M.S.) has been teaching the physics laboratory courses (from the Sophomore Laboratory to the Advanced (Graduate) Laboratory) for many years at the Binghamton University. Undergraduate students spend a lot of time in studying the fundamental physics in the physics courses such as the introductory physics, mechanics, electromagnetism, and so on. Even if they understand the theories in a sufficient depth to be able to apply it readily, they encounter some difficulties in understanding the essence of the experiments in the physics laboratory courses, since the conditions of the experiments are more complicated than the ideal conditions of the physics models. As far as we know, so far there are very few opportunities in discussing the gap between the real physics and ideal physics for the physics laboratory courses. The mathematics which is required for the real physics is much more complicated than the mathematics for the ideal physics. In nature there are many examp...

This paper looks at the physics behind oscillatory motion and how this can be applied to many different scenarios including using different types of pendulums to explain the phenomena of SHM and DHM (simple and damped harmonic motion respectively). Written as part of my A-Level Physics course (OCR B Advancing Physics) in Autumn 2014 (Nov 2014-Jan 2015)

2009

The main task of an introductory laboratory course is to foster students' manual, conceptual and statistical ability to investigate physical phenomena. Needing very simple apparatus, pendulum experiments are an ideal starting point in our first-year laboratory course because they are rich in both physical content and data processing. These experiments allow many variations, e.g. pendulum bobs can have different shapes, threads can be tied to a hook at their edge or pass through their centre of mass, they can be hanged as simple or bifilar pendulums. In these many variations, they emphasize the difference between theory and practice in the passage from an idealized scheme to a real experimental asset, which becomes evident, for example, when the pendulum bob cannot be considered an idealized point mass. Moreover, they require careful observation of details such as the type of thread used and its rigidity or the bob initial slant, which leads to different behaviors. Their mathematical models require a wide range of fundamental topics in experimental data analysis: arithmetic and weighted mean, standard deviation, central limit theorem application, data distribution, and the significant difference between theory and practice. Setting the mass-spring experiment immediately after the pendulum highlights the question of resonance, revises the gap between theory and practice in another context, and provides another occasion to practice further techniques in data analysis.

Some oscillations are fairly simple, like the small-amplitude swinging of a pendulum, and can be modeled by a single mass on the end of a Hooke'slaw spring. Others are more complex, but can still be modeled by two or more masses and two or more springs. Examples include compound mechanical systems, oscillating electrical circuits with several branches, multi-atom molecules, and elastic solids. Here we display the techniques used to understand oscillations like these.

European Journal of Physics, 2020

Experimental analysis of the motion in a system of two coupled oscillators with arbitrary initial conditions was performed and the normal coordinates were obtained directly. The system consisted of two gliders moving on an air track, joined together by a spring and joined by two other springs to the fixed ends. From the positions of the center of mass and the relative distance, acquired by analysis of the digital video of the experiment, normal coordinates were obtained, and by a non linear fit the normal frequencies were also obtained. It is shown that although the masses of the springs are relatively small compared to that of the gliders, it is necessary to take them into consideration to improve the agreement with the experimental results. This experimental-theoretical proposal is targeted to an undergraduate laboratory.

Revista Brasileira de Ensino de Física

The Wilberforce pendulum is a mechanical oscillator often used to demonstrate the phenomenon of coupled oscillations. It is a spring-mass system whose pendulum bob contains lateral rods to vary its moment of inertia, being possible to verify the coupling of the rotational and longitudinal motions of the resonant system. In the present work, we present a experimental study of the coupled oscillations on the Wilberforce pendulum using easily accessible materials. With a Slinky spring toy, a wood rod containing masses at the extremities and getting images of the movements using a mirror and the tracker software on a smartphone, we could analyze quantitatively the properties associated with the coupled oscillations. The final result has indicated excellent agreement with the theoretical modeling, and therefore it can be used in teaching of wave physics.

Arabian Journal for Science and Engineering, 2012

We consider periodic solution for coupled systems of mass-spring. Two practical cases of these systems are explained and introduced. An analytical technique, called the Hamiltonian approach, is applied to calculate approximations to the achieved nonlinear differential oscillation equations. The concept of the Hamiltonian approach is briefly introduced, and its application for nonlinear oscillators is studied. The method introduces an alternative to overcome the difficulty of computing the periodic behavior of the oscillation problems in engineering. The results obtained employing first-order and second-order Hamiltonian approach are compared with those achieved using two other analytical techniques, named the energy balance method and the amplitude frequency formulation, and also to assess the accuracy of solutions, the results were compared with the exact ones. The results indicate that the present analysis is straightforward and provide us a unified and systematic procedure which is simple and more accurate than the other similar methods. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared with the previous approaches such as the perturbation and the classical harmonic balance methods.

Nonlinear Dynamics, 2012

We investigate the dynamics of a simple pendulum coupled to a horizontal mass-spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the massspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasiperiodic orbits in which the pendulum oscillates about an angle between zero and π/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.

Journal of Sound and Vibration, 2005

The dynamics of a linear oscillator, coupled to an essentially nonlinear attachment of substantially lower mass, is investigated. The essential (nonlinearizable) nonlinearity of the attachment enables it to resonate with the oscillator, leading to energy pumping phenomena, e.g., passive, almost irreversible transfer of energy from the substructure to the attachment. Feasibility of this process for possible applications depends on relative mass of the attachment, the obvious goal being to minimize it while preserving the efficiency of the pumping. Two different models of the attachment coupled to the main single-degree-of-freedom body are proposed and analyzed both analytically and numerically. It is demonstrated that efficient energy pumping may be obtained for a rather small value of the attachment mass. Two mechanisms of energy pumping in the system under consideration are revealed. The first one is similar to previously studied resonance capture; a novel analytic framework allowing explicit account of the damping is proposed. The second mechanism is related to nonresonant excitation of high-frequency vibrations of the attachment. Both mechanisms are demonstrated numerically for a model consisting of a linear chain with a nonlinear attachment. r