Physics
Three Coupled Oscillators
Three coupled oscillators refer to a system of three oscillators that are connected to each other through springs. The motion of each oscillator is dependent on the motion of the other two, resulting in complex and interesting behavior. This system is commonly used in physics to study wave phenomena and resonance.
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7 Key excerpts on "Three Coupled Oscillators"
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
249 © 2010 Taylor & Francis Group, LLC Coupled Oscillations and Normal Coordinates We have learned in some detail how a single vibrating system will behave. However, oscillators rarely exist in complete isolation. We now consider systems of interacting oscillators. This subject is of general interest because numerous physical systems are well approximated by coupled har-monic oscillators. Coupled oscillators, in general, are able to transmit their energy to each other because two oscillators share a common component, capacitance or stiffness, inductance or mass, or resistance. Resistance coupling inevitably brings energy loss and a rapid decay in the vibration, but nonresistance coupling consumes no power, and continuous energy transfer over many oscilla-tors is possible. We shall investigate first a mechanical example of stiffness coupling between two pendulums. Two atoms set in a crystal lattice experience a mutual coupling force and would be amenable to a similar treatment. Motion of this type can be quite complex if it is described in ordinary coordinates that describe the geometrical configuration of the system. Fortunately, as we shall see, it is always possible to describe the motion of any oscillatory system in terms of normal coordinates that are constructed from the original position coordinates in such a way that there is no coupling among the oscillators. Thus, each normal coordinate oscillates with a single, well-defined frequency. Before we take up the general analytic approach, let us illustrate the concepts of normal coordinates and normal frequencies with a very simple example: the coupled pendulum. 8.1 COUPLED PENDULUM Consider a pair of identical pendulums of mass m suspended on a massless rigid rod of length b . The masses are connected by a massless spring whose spring constant is k and whose natural length equals the distance between the masses when either is displaced from equilibrium.
eBook - ePub- H. John Pain(Author)
- 2013(Publication Date)
- Wiley(Publisher)
4
Coupled Oscillations
The preceding chapters have shown in some detail how a single vibrating system will behave. Oscillators, however, rarely exist in complete isolation; wave motion owes its existence to neighbouring vibrating systems which are able to transmit their energy to each other.Such energy transfer takes place, in general, because two oscillators share a common component, capacitance or stiffness, inductance or mass, or resistance. Resistance coupling inevitably brings energy loss and a rapid decay in the vibration, but coupling by either of the other two parameters consumes no power, and continuous energy transfer over many oscillators is possible. This is the basis of wave motion.We shall investigate first a mechanical example of stiffness coupling between two pendulums. Two atoms set in a crystal lattice experience a mutual coupling force and would be amenable to a similar treatment. Then we investigate an example of mass, or inductive, coupling, and finally we consider the coupled motion of an extended array of oscillators which leads us naturally into a discussion on wave motion.Stiffness (or Capacitance) Coupled Oscillators
Figure 4.1 shows two identical pendulums, each having a mass m suspended on a light rigid rod of length l . The masses are connected by a light spring of stiffness s whose natural length equals the distance between the masses when neither is displaced from equilibrium. The small oscillations we discuss are restricted to the plane of the paper.If x and y are the respective displacements of the masses, then the equations of motion areandFigure 4.1Two identical pendulums, each a light rigid rod of length l supporting a mass m and coupled by a weightless spring of stiffness s and of natural length equal to the separation of the masses at zero displacementThese represent the normal simple harmonic motion terms of each pendulum plus a coupling term s (x − y ) from the spring. We see that if x > y the spring is extended beyond its normal length and will act against the acceleration of x but in favour of the acceleration of y
eBook - ePub- A.P. French(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
1 This whole chapter may be bypassed if it is preferred to proceed directly to the discussion of vibrations and waves in effectively continuous media. On the other hand, an acquaintance with the contents of the present chapter, even in rather general terms, may help in appreciating the sequel, for the many-particle system does provide the natural link between the single oscillator and the continuum. And it is not as mathematically formidable as it may appear at first sight.Fig. 5-1 First three normal modes of vertical chain with upper end fixed. (The tension is provided at each point by the weight of the chain below that point and so increases linearly with distance from the bottom. )How do we go about the job of accounting for these numerous modes and calculating their frequencies? The clue to this question lies in the fact that an extended object can be regarded as a large number of simple oscillators coupled together. A solid body, for example, is composed of many atoms or molecules. Every atom may behave as an oscillator, vibrating about an equilibrium position. But the motion of each atom affects its neighbors so that, in effect, all the atoms of the solid are coupled together. The question then becomes: How does the coupling affect the behavior of the individual oscillators?We shall begin by discussing in some detail the properties of a system of just two coupled oscillators. The change from one oscillator to two may seem rather trivial, but this new system has some novel and surprising features. Moreover, in analyzing its behavior we shall develop essentially all the theoretical tools we need to handle the problem of an arbitrarily large number of coupled oscillators—which will be our ultimate concern. And this means that, from quite simple beginnings, we can end up with a significant insight into the dynamical properties of something as complicated as a crystal lattice. That is no small achievement, and it is worth the little extra amount of mathematical effort that our discussion will entail.Two Coupled Pendulums
Let us begin with a very simple example. Take two identical pendulums, A and B, and connect them with a spring whose relaxed length is exactly equal to the distance between the pendulum bobs, as shown in Fig. 5-2 . Draw pendulum A aside while holding Β
eBook - PDFEmergence Of Dynamical Order: Synchronization Phenomena In Complex Systems
Synchronization Phenomena in Complex Systems
- Susanna C Manrubia, Alexander S Mikhailov, Damian H Zanette(Authors)
- 2004(Publication Date)
- World Scientific(Publisher)
PART 1 Synchronization and Clustering of Periodic Oscillators This page intentionally left blank Chapter 2 Ensembles of Identical Phase Oscillators Dynamical systems with oscillatory motion are a basic ingredient in the mathematical modeling of a broad class of physical, physicochemical, and biological phenomena. Ensembles of interacting elements with periodic dynamics are used to represent natural systems with collective rhythmic behavior. A simplified model for the periodic evolution of each individual element is given by a single variable with cyclic uniform motion, like ari elementary clock. This simple dynamical system is called phase oscilla- tor. Coupled phase oscillators provide a phenomenological description of complex systems whose collective evolution is driven by synchronization processes. They reproduce the main features of the emergence of coherent behavior found in more elaborate models of interacting oscillators. We begin this chapter by introducing the equations of motion of cou- pled periodic oscillators, and the phase oscillator model. After discussing the synchronization properties of a system of two phase oscillators, we fo- cus the attention on large ensembles of identical oscillators subject to global coupling, where interactions are uniform for all oscillator pairs. We char- acterize the state of full synchronization induced by attractive interactions. Then, the regimes of clustering and incoherent behavior for more complex interaction models are analyzed. 2.1 Coupled Periodic Oscillators Macroscopic oscillations may emerge from the mutual synchronization of a large number of more elementary, individual oscillatory processes [Wiener (1948)]. The mechanisms governing the spontaneous organization of such cyclic elements are intricate, and may be considerably dissimilar for dif- ferent systems. However, all these systems can be phenomenologically 13
- Theodosios Pavlidis(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
(iv) The frequency of oscillations may depend significantly on the coupling strength, and therefore the frequency of synchronization in a population may be quite different than the individual frequencies of the oscillators. The biological significance of these and other features will be discussed in the next chapter. 7.8 Bibliographical Notes The concept of structural stability was investigated by the Russian mathematicians in the early 1930s [And-I, Bog-I]. Closely related to it is 158 POPULATIONS OF INTERACTING OSCILLATORS the notion of bifurcation, of the qualitative change in the behavior of a system when one of its parameters passes through a critical value. The asymptotic techniques are also due to Russian mathematicians, and the notes on Section 1.5 are pertinent here too. Technical applications of systems consisting of a large number of coupled oscillators have been scant. The best known relates to the problem of synchronization in a communications network. This has been studied by Pierce, Sandberg, and others [Pie-1, San-1, San-2]. However, the mathematical formulation they use reduces the problem to one of finding the equilibria of a high-order system rather than the periodic solutions. This is achieved by using the frequencies as state variables. (Each station measures frequency differences.) Such an approach is difficult to justify for the case of biological systems. Kemer and later Goodwin studied populations of oscillators as conservative systems which could be directly integrated [Goo-A]. Winfree studied the case of weakly coupled oscillators where the limit cycles of the individual units in the presence of interaction are close to the ones in the absence of interactions [Win-0]. Walker has studied interacting oscillators through Liapunov's method and computer simulation. The basic simplification in his study was the assumption that only one essential nonlinearity was present [Wal-1].
eBook - ePub- John Vandermeer, Ivette Perfecto(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
6 Coupled oscillators An odd kind of sympathy In the mid-seventeenth century, the world’s second most famous physicist reported on a strange behavior of pendulum clocks. Having taken on the challenge of constructing clocks of sufficient accuracy to be used in determining longitude on long ocean voyages, Christian Huygens invented the pendulum clock, which he thought would be useful for this challenging seventeenth-century problem. Lying sick in bed in 1665, he noticed that two of his pendulum clocks, mounted on the wall, were oscillating approximately 180 degrees out of phase with one another. Curiosity drove him to change the pendula to be swinging in phase, and within a half hour, once again they had taken up an antiphase coherence with one another. Repeating this experiment, probably many times, he convinced himself that this phenomenon was a real element of pendula. He wrote to a confidant that his two clocks displayed “an odd kind of sympathy … [when] suspended by the side of each other.” 1 A reminder of this episode can be encountered on Leidsestraat in Amsterdam, where a mosaic of him with his tell-tale clocks can be seen (Figure 6.1). What Huygens discovered has become the foundation of a variety of applications in many sciences and a variety of engineering applications – when coupled, sometimes very subtly, two oscillators will come to “inform” one another about their oscillations and attune to each other. Many illustrations of this phenomenon can be seen in abundant video clips, and applications from signaling fireflies to electronic oscillators are well-known. 2 For ecology, as we noted in Chapter 2, one of the most evident of ecology’s rules is the consumption of one organism by another, whether bacteria in the soil or parasitoids attacking pests. This trophic connection is, at its most foundational level, oscillatory
- Joel Franklin(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
3 Coupled Oscillators Suppose we now include additional masses in our mass-on-a-spring problem. If we have two masses, m 1 and m 2 at respective locations x 1 (t) and x 2 (t) (in one dimension) attached by a spring with spring constant k and equilibrium spacing a as in Figure 3.1, the equations of motion are m 1 ¨ x 1 (t) = k(x 2 (t) − x 1 (t) − a) m 2 ¨ x 2 (t) = −k(x 2 (t) − x 1 (t) − a). (3.1) It is useful, in this situation, to change variables, since m 1 ¨ x 1 + m 2 ¨ x 2 = 0 (by Newton’s third law), the coordinate z ∝ m 1 x 1 + m 2 x 2 has ¨ z = 0. To give z(t) the appropriate dimension of length, we must divide by a mass, and to be democratic, we’ll treat m 1 and m 2 symmetrically, let z(t) ≡ m 1 x 1 (t) + m 2 x 2 (t) m 1 + m 2 . (3.2) If we define the difference coordinate d(t) ≡ x 2 (t) − x 1 (t) − a, then the equations of motion can be combined to give ¨ z(t) = 0 ¨ d(t) = −k 1 m 1 + 1 m 2 d(t). (3.3) The angular frequency ω can be read off from the equation for d(t): ω 2 ≡ k 1 m 1 + 1 m 2 , (3.4) and we know the general solution to the problem is z(t) = At + B d(t) = C cos(ωt) + D sin(ωt) (3.5) m 1 m 2 a x 1 (t) x 2 (t) k Fig. 3.1 Two masses connected by a spring. 65 66 Coupled Oscillators with four constants of integration, {A, B, C, D}, just right for a pair of second-order ODEs. We could set the constants given the initial position and velocity for each mass. The “center of mass” motion, z(t), and relative motion d(t) have been decoupled, each has its own ODE in (3.3) that makes no reference to the other. Our goal is to perform that same decoupling for a system of n masses connected by n − 1 springs, but in order to do this, we need to review some linear algebra. 3.1 Vectors As we add masses and springs, we add equations to (3.1), and the language of linear algebra allows us to think about those equations of motion as a whole. A “vector” is a collection of n numbers that could be real or complex.
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