Two Coupled Oscillators

10 Key excerpts on "Two Coupled Oscillators"

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

   11.5 Coupled Oscillators In the real physical world, oscillators are seldom isolated from their environment and an oscillating body will set up oscillations in nearby bodies. A very good example of coupled oscillators is the motion of atoms in a crystal. A vibrating atom will cause neighboring atoms to start oscillating. The analysis of coupled oscillators can be quite complicated. This section is a study of the simplest of such systems, namely two undamped oscillators connected by springs. Figure 11.11 illustrates two masses attached to rigid walls by springs of constants k 1 , k 2 , and k 3 . Let x 1 and x 2 be the displacements of masses m 1 and m 2 measured from their equilibrium positions. The equations of motion for the masses are easily determined from the Lagrangian or from Newton’s second law. We obtain the following coupled differential equations: 7 m 1 ¨ x 1 = −k 1 x 1 − k 3 (x 1 − x 2 ), (11.29) m 2 ¨ x 2 = −k 2 x 2 − k 3 (x 2 − x 1 ). For simplicity, assume that m 1 = m 2 = m and that k 1 = k 2 = k, but allow k 3 to be different. Then the equations of motion reduce to m ¨ x 1 + (k + k 3 )x 1 − k 3 x 2 = 0, m ¨ x 2 + (k + k 3 )x 2 − k 3 x 1 = 0. 7 When applied to the atoms in a solid, we often assume all the masses are equal and all the “spring constants” are equal, yielding n equations of the form m ¨ x i = k(x i − x i −1 ) + k(x i +1 − x i ) i = 1, . . . ,n. These relations reduce to Equations (11.29) for the situation pictured in Figure 11.11. 310 11 HARMONIC MOTION k 1 m 1 m 2 k 3 X 1 X 2 k 2 Figure 11.11 Coupled oscillators. Note that the positions of the two masses are measured from their equilibrium points. Denote k + k 3 by k  . Dividing by m yields: ¨ x 1 + (k  /m)x 1 − (k 3 /m)x 2 = 0, ¨ x 2 + (k  /m)x 2 − (k 3 /m)x 1 = 0. Now replace k  /m by ω 2 0 . The reason for this notation is that ω 0 is the frequency of oscillation of either one of the masses if the other mass is held at rest.

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  Asymptotic Multiple Scale Method in Time Domain

  Multi-Degree-of-Freedom Stationary and Nonstationary Dynamics

  • Jan Awrejcewicz, Roman Starosta, Grażyna Sypniewska-Kamińska(Authors)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  105 ]).

  This chapter examines the motion of coupled oscillators with two degrees of freedom under periodic excitation. The dynamics of such a system is very complicated (Awrejcewicz [17]). The method used here consists of two major steps. The first is to simplify the problem by extracting internal motion in the system and describing it with an appropriate, one effective equation of motion. The effective equation for a similar system was investigated by Kyziol and Okniński [111 ], Okniński and Kyziol [159 ]. The second step is to analyze the dynamics of the effective equation. For this purpose, the main idea contained in the work (Okniński and Kyziol [160 ]) was adapted. In the latter work the authors used an innovative approach allowing for qualitative and quantitative study of the behavior of nonlinear systems with one degree of freedom, exchanging energy with external excitation.

  It is obvious that in general, reduction of 2 DoF systems to 1 DoF systems is impossible. Besides the mentioned work, we have found the earlier investigation of Hough (1988) where a 2 DoF conservative mechanical system was reduced to 1 DoF kinematic system under the change of an independent variable. With the help of Hamiltonian integral, the problem was simplified to study a second-order ODE for function specifying the orbital curve.

  7.2 Harmonic oscillator with added nonlinear oscillator

  In many cases, the displacement of real mechanical system is relatively small during operations, so their dynamical behavior can be described using linear differential equations with good approximation. The elimination of undesirable vibrations can be achieved by attaching to such a device an oscillator with a low mass and appropriately selected nonlinear characteristics. Therefore, an important issue from the point of view of potential practical applications is the dynamic analysis of Two Coupled Oscillators, one of which the higher mass is linear.

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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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  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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  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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  Concepts and Methods in Modern Theoretical Chemistry

  Statistical Mechanics

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

    (Publisher)

  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. If we consider a system of limit cycle oscillators (where the oscillators are evolving in a globally attracting limit cycle of constant amplitude), their phase dynamical equations can be represented by a system of coupled first-order nonlinear differential equations dotnosp θ ω θ θ i i i j j N A N f j N = --( ) = … = ∑ 1 1 2 , , , , . (5.2) This model is called the Kuramoto model (named after Yoshiki Kuramoto who first proposed the model [2]), where θ i s are the phases of the individual oscillators in the system and ω i s are the natural intrinsic frequencies of the oscillators, while f is a 2 π periodic function. The model assumes weak coupling between the oscillators and that the oscillators have their own intrinsic frequencies. The collective dynamics of the system can now be represented by the dynamics of the mean field, that is, essentially that of a single oscillator. The complex mean field parameter can be given as Z X iY re N e i i j N j = + = = = ∑ ψ θ 1 1 . (5.3) FIGURE 5.1 Original drawing of Christiaan Huygens, illustrating his observation of pen-dulum clock synchronization. (Data from Horologium, The Hague , 1658 (Oeuvres XVII); Bennett, M. et al., Proc. R. Soc. A. , 458(2019), 563, 2002.) 82 Modern Theoretical Chemistry: Statistical Mechanics When r = 1, there is complete synchronization when the phases of all the oscilla-tors are the same. When B = 0.1, there is no synchronization. When r takes a value between 0 and 1, there is either partial synchronization or clustering (formation of multiple small synchronized groups) in the system.

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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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  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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  Wave Optics

  Basic Concepts and Contemporary Trends

  • Subhasish Dutta Gupta, Nirmalya Ghosh, Ayan Banerjee(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  In the presence of damping in off-resonant cases, the beating persists for some time and finally the os-cillations settle down to constant amplitude A . In the resonant case there is no beating, and the final amplitude is reached in a monotonic fashion. These features are shown in Fig. 1.14. 20 Wave Optics: Basic Concepts and Contemporary Trends FIGURE 1.14 : Transients in forced harmonic oscillation (temporal evolu-tion of the amplitudes). (a) Corresponds to very large Q ( ∼ 20000). Almost nonexistent damping leads to beating of the natural frequency ω 0 and the driving frequency ω . (b) and (c) are when Q = 20 and show the off-resonant ( ω = 0 . 85 ω 0 ) and the resonant ( ω ≈ ω 0 ) behavior, respectively. 1.5 Coupled oscillations and normal modes In most of our earlier discussions, we concentrated on the type of oscilla-tions that have mostly the same frequency. In reality the system may have components that oscillate with different frequencies. Each oscillating compo-nent has specific effects on the others and vice versa. For example, a solid body is composed of many atoms or molecules. Every atom may behave like an oscillator, vibrating about the equilibrium position. Motion of each atom affects the neighbors. Thus all the atoms are coupled together. A question results: How does the coupling affect the behavior of individual oscillators? 1.5.1 Two coupled pendulums Consider the system of two identical pendulums A and B joined by a spring of rest length equal to the distance between the bobs (see Fig. 1.15). This sys-tem serves as a prototype toward understanding more complicated phenomena involving many oscillators. Let pendulum A be pulled to a distance, keeping B held at equilibrium, and then let both be released. Oscillations of A will decrease while those of B will gain in amplitude. Finally, the motion of A will

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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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