Coupled Oscillators

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  Classical Mechanics

  • 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.

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  The Physics of Vibrations and Waves

  • 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 are

  and

  Figure 4.1

  Two 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 displacement

  These 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

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  Concepts and Methods in Modern Theoretical Chemistry

  Statistical Mechanics

  • Swapan Kumar Ghosh, Pratim Kumar Chattaraj(Authors)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Some examples include the emulsion of light pulses by a population of fireflies in a synchronous manner, synchronization in populations of electrochemical oscillators, synchronous chirping by populations of crickets, syn-chronous clapping of audiences in auditoria, synchronous firing in populations of neurons, synchronized pacemaker oscillations in cardiac cells, etc. The aforemen-tioned instances of synchronization can be explained by systems of coupled oscilla-tors, which are represented typically as dotnosp X F X A N X X j N j j j k j k N = ( ) + -( ) = … = ∑ , , , , , , , ε 1 1 2 3 =       d dt , (5.1) where X j is the state vector of the j th oscillator, A is the coupling parameter that indi-cates the strength of the coupling between the oscillators in the system, ε j represents the system parameters, and F ( X j , ε j ) describes intrinsic self-sustained oscillation. Other forms of couplings such as weighted coupling, nonlocal coupling, etc., are also possible and important, although we will not discuss them here. The phenomenon of 81 Synchronization in Coupled Nonlinear Oscillators phase synchronization was first reported by the famous Dutch physicist Christiaan Huygens in the 17th century [6,7]. His report was based on the observation of two pendulum clocks (see Figure 5.1) attached to a common wooden bar that continued with precise antiphase oscillation, seemingly indefinitely. When the system of interest involves a large number of Coupled Oscillators, the dynamics of the system can be reduced to that of the mean field, that is, effectively of a single oscillator. When an individual oscillator in the population emits or absorbs energy, this will alter the physical states of the oscillators to which it is coupled. In particular, the periods of its neighbors are altered.

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  Emergence 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

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  Coupled-Oscillator Based Active-Array Antennas

  • Ronald J. Pogorzelski, Apostolos Georgiadis(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  He is involved in a number of technical program committees and serves as a reviewer for several journals, including IEEE Transactions on Antennas and Propagation and IEEE Transactions on Microwave Theory and Techniques. He was the co-recipient of the EUCAP 2010 Best Student Paper Award and the ACES 2010 2nd Best Student Paper Award. He is the Chairman of COST Action IC0803, RF/Microwave communication subsystems for emerging wireless technologies (RFCSET), and he is the Coordinator of the Marie Curie Industry-Academia Pathways and Partnerships project Symbiotic Wireless Autonomous Powered system (SWAP). Part I: Theory and Analysis Chapter 1 Introduction—Oscillators and Synchronization Oscillation is among the simplest of dynamic behaviors to describe mathematically and has thus been conveniently used in modeling a wide variety of physical phenomena ranging from mechanical vibration to quantum mechanical behavior and even neurological systems. Certainly not the least of these is the area of electronic circuits. Many years ago, van der Pol created his classical model of an oscillator including the nonlinear saturation effects that determine the amplitude of the steady-state oscillation [9]. Soon afterward, Adler provided a simple theory of what is now known as injection locking, and Coupled Oscillators became a valuable design resource for the electronics engineer and the antenna designer [10]. Moreover, circuit theorists were able to apply these principles to long chains and closed rings of Coupled Oscillators to model biological behaviors such as intestinal and colorectal myoelectrical activity in humans [11, 12]. 1.1 Early Work in Mathematical Biology and Electronic Circuits Biologists, in trying to understand how neurons coordinate the movements of animals, have defined what is known as a central pattern generator, or CPG for short. A CPG in this context is a group of neurons that produce rhythmic or periodic signals without sensory input.

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  Vibrations and Waves

  • 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

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  Biological Oscillators: Their Mathematical Analysis

  • 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].

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  Ecological Complexity and Agroecology

  • 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

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  RF Power Amplifiers

  • Marian K. Kazimierczuk(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Chapter 12 Oscillators

  12.1 Introduction

  An oscillator is a circuit that produces a periodic output signal without any ac input signal. Sinusoidal-tuned resonant oscillators [1–58] are widely used in instrumentation and test equipment as function generators and local signal oscillators in amplitude modulation (AM), frequency modulation (FM), and wireless receivers and transmitters operated in a wide range of frequencies. In radio systems, sinusoidal oscillators establish the transmitter carrier frequency and drive the mixer stages that convert signals from one frequency to another. oscillators also form an integral part of voltage-controlled oscillators (VCOs) used in communication and digital systems. Power oscillators also find a wide range of applications, such as radio transmitters, radars, and electronic ballasts. When the frequency of oscillations is required to be accurate and stable, crystal oscillators are used. In these oscillators, the oscillation frequency is precisely controlled by the vibrations of a quartz crystal. The crystal is a mechanical resonator. Electronic watches and other critical timing applications use crystal oscillators because they provide an accurate clock frequency. tuned oscillators are also used as power oscillators because power losses in reactive components are low, yielding high efficiency. An electronic oscillator is a nonlinear circuit with at least two energy storage components. A resonant circuit is a fundamental building block of oscillators because it acts a bandpass filter and determines the frequency of oscillations. In many circuits, it also forms a part of the feedback network or the amplifier or both. Operation of oscillators is based on the following principles:

  • Positive feedback.
  • Negative resistance.

  Oscillators with positive feedback consist of a forward two-port network and a positive feedback two-port network. An oscillator family may be divided into three groups:

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