Damped harmonic oscillator

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  Vibroacoustic Simulation

  An Introduction to Statistical Energy Analysis and Hybrid Methods

  • Alexander Peiffer(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  The damping is supposed to be viscous, thus proportional to the velocity ? ?? = − ? ? ̇ ? . The equation of motion ? ̈ ? + ? ? ̇ ? + ? ? ? = 0 (1.1) Vibroacoustic Simulation: An Introduction to Statistical Energy Analysis and Hybrid Methods , First Edition. Alexander Peiffer. © 2022 John Wiley & Sons, Inc. Published 2022 by John Wiley & Sons, Inc. 2 1 Linear Systems, Random Process and Signals = Equilibrium = 0 a) b) Figure 1.1 Damped harmonic oscillator with initial conditions a) and external force excitation b). Source: Alexander Peiffer. is a homogeneous second order equation with a solution of the form ? = ?? ?? . Entering this into Equation (1.1) leads to the characteristic equation ?? 2 + ? v ? + ? ? = 0 (1.2) with the two solutions ? 1∕2 = − 1 2 ? [ − ? v ± √ ? 2 v − 4 ?? 1∕2 ? ] (1.3) Hence, ? ( ? ) = ? 1 ? ? 1 ? + ? 2 ? ? 2 ? (1.4) with ? 1 and ? 2 depending on the initial conditions. The root in Equation (1.3) is zero when ? ? equals √ 4 ?? ? . This specific value is called the critical viscous damping ? ?? = √ 4 ?? ? (1.5) We use the following definitions: 𝜔 2 0 = ? ? ? ? ? ? = 2 ?𝜔 0 ? = ? ? √ 4 ?? ? = ? ? ? ?? (1.6) 𝜔 0 is the natural angular frequency, ? is ratio of the viscous-damping to the critical viscous-damping. There are additional expressions for the period and frequency ? 0 = 𝜔 0 2 𝜋 ? 0 = 1 ? 0 (1.7) where ? 0 is the natural frequency and ? 0 the oscillation period. Equations (1.1)–(1.3) can now be written as ̈ ? + 2 ?𝜔 0 ̇ ? + 𝜔 2 0 ? = 0 (1.8) ? 2 + 2 ?𝜔 0 ? + 𝜔 2 0 = 0 (1.9) ? 1∕2 = − ?𝜔 0 ± 𝜔 0 √ ? 2 − 1 (1.10) The problem falls into three cases: ? > 1 overdamped ? < 1 underdamped ? = 1 critically damped. 1.1 The Damped harmonic oscillator 3 The first case leads to two real roots, and no oscillation is possible. The second case gives two complex roots, which means that (damped) oscillation occurs. The third case is a transition case between the two other. Subsections 1.1.2–1.1.4 deal with each case in detail.

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  An Introduction to Mechanics

  • Daniel Kleppner, Robert Kolenkow(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  THE HARMONIC OSCILLATOR 11 11.1 Introduction 412 11.2 Simple Harmonic Motion: Review 412 11.2.1 Nomenclature 413 11.2.2 Energy of the Harmonic Oscillator 414 11.3 The Damped harmonic oscillator 414 11.3.1 Energy Dissipation in the Damped Oscillator 416 11.3.2 The Q of an Oscillator 418 11.4 The Driven Harmonic Oscillator 421 11.4.1 Energy Stored in a Driven Harmonic Oscillator 423 11.4.2 Resonance 424 11.5 Transient Behavior 425 11.6 Response in Time and Response in Frequency 427 Note 11.1 Complex Numbers 430 Note 11.2 Solving the Equation of Motion for the Damped Oscillator 431 Note 11.3 Solving the Equation of Motion for the Driven Harmonic Oscillator 434 Problems 435 412 THE HARMONIC OSCILLATOR 11.1 Introduction The harmonic oscillator plays a loftier role in physics than one might guess from its humble origin: a mass bouncing at the end of a spring. The harmonic oscillator underlies the creation of sound by musical in-struments, the propagation of waves in media, the analysis and control of vibrations in machinery and airplanes, and the time-keeping crystals in digital watches. Furthermore, the harmonic oscillator arises in numer-ous atomic and optical quantum scenarios, in quantum systems such as lasers, and it is a recurrent motif in advanced quantum field theories. In short, if there were a competition for a logo for the universality of physics, the harmonic oscillator would make a pretty strong contender. We encountered simple harmonic motion—the periodic motion of a mass attached to a spring—in Chapter 3 . The treatment there was highly idealized because it neglected friction and the possibility of a time-dependent driving force. It turns out that friction is essential for the analysis to be physically meaningful and that the most interesting ap-plications of the harmonic oscillator generally involve its response to a driving force.

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  Classical Dynamics of Particles and Systems

  • Jerry B. Marion(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  Note that the period of the simple harmonic oscillator is independent of the amplitude (or total energy) ; a system which exhibits this property is said to be isochronous. The plane pendulum is an example of an oscillator that is approximately isochronous, if the amplitude of oscillation is small. (In Section 7.4 we shall treat the general case of finite amplitudes.) If Θ represents the angle that the pendulum makes with the vertical, then for small Θ (i.e., sin θ = Θ) the equation of motion is the familiar expression θ + ωΐθ = 0 (6.14) where ω £ Ξ | (6.15) * Henceforth we shall adhere to the convention of denoting angular frequencies by ω and frequencies by v. Usually, ω will be referred to as a frequency for brevity, although angular frequency is to be understood. 132 6 · OSCILLATORY MOTION Thus, τ = 2π^ 1 -(6.16) The period is therefore independent of the mass of the pendulum bob and depends only on the length / and the gravitational acceleration g. 6.3 Damped Harmonic Motion The motion represented by the simple harmonic oscillator [see Eq. (6.5)] is termed a free oscillation ; once set into oscillation, the motion would never cease. This is, of course, an oversimplification of the actual physical case in which dissipative or frictional forces would eventually damp the motion to the point that the oscillations would cease. It is possible to incorporate into the differential equation which describes the motion a term which represents the damping force. Such forces which are encountered in Nature frequently depend upon some power of the instantaneous velocity.* One of the more important cases is that in which the damping force is directly proportional to the velocity and can therefore be described by a term — bv, where b > 0. The traditional example of such a damping force is that encountered by a particle moving with a relatively low velocity through a viscous fluid.

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  Quantum Dissipative Systems

  • Ulrich Weiss(Author)
  • 1993(Publication Date)
  • World Scientific

    (Publisher)

  79 PART II SIMPLE DISSIPATIVE QUANTUM SYSTEMS Up to this point, we have used functional integral methods to get to the exact formal solution for the dynamics of a damped quantum sys-tem. Now we move to new ground and consider the properties of simple exactly solvable linear models. 6. Damped harmonic oscillator In this chapter, I study static and dynamical properties of a damped quantum-mechanical oscillator. As I have already briefly discussed in Section 4.2.1, the Hamil-tonian of the composite system can be diagonalized directly. Here, I neither pursue that way nor use the path integral approach. I rather consider a simple stochastic modelling of the process of thermal and quantal fluctuations exerted on a heavy central oscillator. The problem of a damped harmonic quantum-mechanical oscillator has been studied intensively since the early 60's [19,20,24,25,34]. In the more recent work [31,33,96], [108] -[114] attention was put on the asymptotic low temperature prop-erties, which show anomalous behavior. In the presence of a time dependent external force F(t) coupled linearly to the particle, the potential is V{q,t)= l -MuW-qF{t). (6.1) Clearly, with the form (6.1) the equations of motion (5.81) and (5.82) are linear and can be solved. Further, the fluctuations about the classical path are strict Gaussians. Thus, in general, the functional integral expression for the propagating function can be evaluated exactly. This road was taken in [95]. 6.1 Stochastic modelling A simple direct insight into the properties of a damped quantum-mechanical oscil-lator can be gained by a simple stochastic modelling [108], which is based upon the following three principles: i. According to the Ehrenfest theorem the mean values obey the classical equa-tions of motion. 80 CHAPTER 6. Damped harmonic oscillator ii. The response function and the equilibrium correlation function are related by the fluctuation-dissipation theorem (FDT).

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  Signal Processing, Speech and Music

  • Stan Tempelaars(Author)
  • 2014(Publication Date)
  • Taylor & Francis

    (Publisher)

  HAPTER 3

  The Harmonic Oscillator 3.1 Undamped vibrations - the time-continuous case

  A. The vibrating string; equation and solution

  We shall now study a few versions of the simple system called ‘the harmonic oscillator’. First we will analyse the behaviour of a vibrating string via a simplified model, in which we place a metal ball in the middle of the string. See fig.3.1.1 . Thanks to this ball we can ignore the mass of the string itself, which simplifies the calculation considerably. On the ball in both directions a force S is exercised by means of the tension of the string. We now assume now that during the vibration the displacement and therefore the changes in the length of the string are so small that we may consider S as a constant. We indicate the displacement of the ball with y(t). If the string at the place of the ball is pulled to the side the two forces S no longer lie in each others direction and a restoring force F occurs, which is shown in the diagram via the parallelogram construction and can be calculated as follows:

  If we combine this with

  Figure 3.1.1 Vibrating string.

  The factor 4S/L is abbreviated to b. In this expression both the magnitude and the direction of force F are represented, the latter with the minus sign that shows that the force and the displacement are in opposite directions. We combine this result with Newton’s law:

  This is a differential equation of the second order, which can be considered as the mathematical model of the system (see also Morse 1948). For the determination of the solution we start with a ‘trial’ function. This is a function that still contains several unknown coefficients, which we attempt to adjust in such a way that the function provides us with a solution for the equation.

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  An Introduction to Mathematics for Engineers

  Mechanics

  • Stephen Lee(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  The quantity 2 is called the natural frequenc y of the system: it is the same whether the system is damped or undamped. The solution of this differential equation can give several different types of motion, depending on the relative sizes of the parameters and . The auxiliary equation is 2 2 0. This has two solutions, 1 2 2 4 2 and 2 2 2 4 2 . The discriminant , 2 4 2 , determines the nature of the solution. There are three possibilities, as follows. ● Overdamping : 2 4 2 , is positive, and the system does not oscillate (figure 15.7). Figure 15.7 y t y t x O t DAMPED AND FORCED OSCILLATIONS 347 ● Underdamping : 2 4 2 is negative, and oscillations occur (figure 15.8). Figure 15.8 ● Critical damping : 2 4 2 0 (figure 15.9). Figure 15.9 Critical damping is the borderline between overdamping and underdamping. It is not obvious in a physical situation when damping is critical, since the pattern of motion for critical damping can be very similar to that in the overdamped case. 15.3 Modelling forced vibrations: the undamped case In order to understand the mathematics of forced oscillations, including resonance, we look at the simplest suitable case, that of an object hanging on a light, perfectly elastic spring, without damping. (The case in which both forcing and damping occur is considered later in this chapter, on page 354.) In figure 15.10 the top end A of the spring is forced to vibrate so that its displacement at time t is y a sin t . (This can be achieved experimentally, to a reasonable approximation, by attaching the supporting string over a pulley to a rotating cam, as shown in figure 15.14 on page 354.) If the natural length of the spring is l 0 , the stiffness of the spring is k and the object has mass m , then in equilibrium, point A coincides with O, the object is at rest and the extension of the spring is e m k g . At a general time t during the forced motion, the extension of the spring below the equilibrium position is denoted by x (figure 15.11).

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  Mechanical Science

  • W. C. Bolton(Author)
  • 2013(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  Chapter 17 Damped and forced oscillations 17.1 Free oscillations Mechanical systems can often be considered to be capable of being re-presented by a spring suspended from some rigid support with a weight attached to the lower end of the spring. When the mass is pulled down and then released, oscillations occur. In the ideal situation these oscillations would continue indefinitely, the motion then being referred to as free or natural oscillations . In practice the oscillations die away with time and so we have to include with the model a means of providing damping. Figure 17.1 shows a free oscillation system with no damping. The restoring force acting on the mass m when it is pulled down some distance x and then released is the force exerted by the stretched spring, i.e. − kx where k is the stiffness and a minus sign is included because the restoring force is in the opposite direction to that in which x increases. This force causes an acceleration a of the mass. Thus − kx = ma . But acceleration is the rate of change of velocity with time d v / d t and velocity v is the rate of change of displacement with time x , i.e. d x/ d t . Thus a = d ( d x/ d t)/ d t = d 2 x/ d t 2 . Hence − kx = m d 2 x d t 2 or, when rearranged, Fig. 17.1 Free oscillations. 412 Damped and forced oscillations 413 m d 2 x d t 2 + kx = 0 [1] From the previous chapter we know that this equation must describe an object moving with simple harmonic motion and thus the solution to this differential equation is x = A cos ω n t , where A is the amplitude and ω n the natural angular frequency. This gives acceleration = d 2 x d t 2 = − A ω 2 n cos ω n t = − ω 2 n x Hence equation [1] gives − m ω 2 n x + kx = 0 and so k = m ω 2 n . Thus equation [1] can be written as d 2 x d t 2 + ω 2 n x = 0 [2] Equations [1] and [2] are called second order differential equations.

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  Applied Structural and Mechanical Vibrations

  Theory and Methods, Second Edition

  • Paolo L. Gatti(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  This particular condi-tion is called undamped free vibrations and a key point of this phenomenon consists in the fact that the frequency characteristics of the motion depend on the system’s parameters – that is, its mass and elasticity – while the amplitude characteristics depend on the initial conditions. x ( t ) k c m f ( t ) Figure 4.1 Harmonic oscillator. Table 4.1 Analogies between translational and rotational systems Translation Rotation Linear displacement x Angular displacement α Force f Torque M Spring constant k Spring constant k r Damping constant c Damping constant c r Mass m Moment of inertia J Spring law F = k ( x 1 − x 2 ) Spring law M = k r ( α 1 − α 2 ) Damping law F = -c x x ( ) dotnosp dotnosp 1 2 Damping law M c = -r 1 2 ( ) dotnosp dotnosp α α Inertia law F mx = dotnospdotnosp Inertia law M = J dotnospdotnosp α Single degree of freedom systems 121 When some kind of damping is present, on the other hand, energy is lost during the motion and the amplitude of the oscillation decreases with time until it stops completely; this is the case of damped free vibrations . Once again, however, the frequency characteristics of the motion depend on the system’s parameters, and not on the initial conditions that started the motion (no musical instrument could be played in tune if this gen-eral rule did not apply). When, however, damping is ‘sufficiently high’, the system does not vibrate at all but quickly loses its initial energy and simply returns to its equilibrium position without oscillating. We will quantitatively determine the meaning of the term ‘sufficiently high’ in the following sections. 4.2.1 Undamped free vibrations Let us now consider the simple ideal system of Figure 4.2 consisting of a mass m and a massless spring k .

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  Mathematical Methods for Oscillations and Waves

  • Joel Franklin(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Damped harmonic oscillator Let’s complicate matters – suppose we take our mass-on-a-spring, and introduce some friction. We’ll start with the familiar kinetic friction which opposes motion with constant magnitude. Calling that magnitude α, we can include the friction force in Newton’s second law, m ¨ x(t) = −kx(t) − αsign(˙ x), (2.1) and we have set the equilibrium position of the spring to be at a = 0, as shown in Figure 2.1. We’ll start the mass off from rest with initial extension p 0 > 0. The mass moves to the left, so that the sign of ˙ x is −1, and we start by solving ¨ x 1 (t) = − k m x 1 (t) + α m = − k m  x 1 (t) − α k  . (2.2) Again letting ω 2 ≡ k/m, and defining the length z ≡ α/k, we have solution x 1 (t) = A cos(ωt) + B sin(ωt) + z, (2.3) and the initial conditions, x 1 (0) = p 0 , ˙ x 1 (0) = 0, give A = (p 0 − z) and B = 0. The solution is x 1 (t) = (p 0 − z) cos(ωt) + z. (2.4) This solution only holds until the mass comes to rest on the other side of its equilibrium location. That happens when t 1 = π/ ω, at which point the mass is at x 1 (π/ ω) = −p 0 + 2z. Now we have a mass that starts from rest at p 1 ≡ −p 0 + 2z and travels to the right, so that the sign of ˙ x(t) is +1. We must solve ¨ x 2 (t) = −ω 2 (x 2 (t) + z) with x 2 (π/ ω) = −p 0 + 2z and ˙ x 2 (π/ ω) = 0. (2.5) Since this is the same problem as in (2.2), with z → −z, and p 0 → p 1 , we have solution m k x(t) α a = 0 Fig. 2.1 A mass moves under the influence of both a spring and kinetic friction. 31 32 Damped harmonic oscillator t x(t) π ω p 0 −p 0 + 2z x 1 (t) x 2 (t) Fig. 2.2 The first full “cycle”of the frictionally Damped harmonic oscillator. x 2 (t) = (p 1 + z) cos(ω(t − π/ ω)) − z = (−p 0 + 3z) cos(ω(t − π/ ω)) − z. (2.6) The process continues, each time the mass stops to turn around, we pick up a new value for “p” and the sign of z changes.

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  Vibration Damping, Control, and Design

  • Clarence W. de Silva(Author)
  • 2007(Publication Date)
  • CRC Press

    (Publisher)

  Likewise, the spring constant has been set to unity. It is generally useful to distill a given problem to its most basic form when attempting to understand the physics. Constants that provide no useful information for trend analysis purposes are conveniently “normalized.” Such is common practice, for example, in modeling chaotic systems. The second-order set can always be reduced mathematically to a first-order pair; however, the pair results naturally from the use of the Hamiltonian as opposed to the Newtonian formulation of mechanics. 2.17.7 Damping and Harmonic Content Equation 2.52 to Equation 2.54 are the nonlinear, modified Coulomb damping model forms that correspond, respectively, to (i) hysteretic, (ii) amplitude-dependent, and (iii) Coulomb damping. The damping term for each of the three cases can be expressed as follows: f m ¼ p 4 v Q ½ v ( ð 2 : 64 Þ where f is the friction force, and ½ v ( is the square wave whose fundamental in a Fourier series expansion is equal to the velocity of the oscillator times 4/ p ; i.e., for a square wave ^ h ; the amplitude of the fundamental is ^ ð 4 = p Þ h : We see that all the damping types that have been considered in this chapter, when expressed in canonical form, correspond to a fundamental friction force f ¼ m v v = Q : The simplicity of this result is probably why viscous damping has been viewed by so many physicists as “inviolate.” One must be careful, however, because (as noted in the previous section) only for the case of hysteretic damping is Q constant. For amplitude-dependent damping Q f ¼ Q f 0 ð y 0 = y Þ and for Coulomb damping 1.50 1.00 0.50 0.00 − 0.50 − 1.00 − 1.50 0 200 400 Sensor output (V) Period = 1.0 s 600 800 1000 1200 1400 Q i = 3000, Q f = 8000 b = 0.0004 s − 1 , a = 0.00120 s/v, c = 0 a = 0.00141 s/v, b = 0 time (s) .0011s − 1 , a = 0 .0017s − 1 , a = 0 FIGURE 2.22 Example of a mix of two damping types, hysteretic and amplitude-dependent. Vibration Damping, Control, and Design 2 -56

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  Classical And Quantum Dissipative Systems (Second Edition)

  • Mohsen Razavy(Author)
  • 2017(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 15 Path Integral Formulation of a Damped harmonic oscillator An elegant way of quantizing the dissipative systems which is specially suited for the Damped harmonic oscillator is by means of the Feynman formulation where one can obtain the propagator and from the propagator determine the time development of the wave function [1], [2]. This same method of path in-tegration can be applied to a central particle under the action of the potential V ( Q ) but coupled linearly to a bath of harmonic oscillators. We will discuss the latter problem in detail later. For the construction of the propagator we can use either the Lagrangian or the Hamiltonian, but here like other methods of quantization (Dirac or Schr¨ odinger) the Feynman approach will not lead to a unique result. The non-uniqueness arises from the fact that we can use an infinite set of classically equivalent Lagrangians as the starting point. But even if we decide on a sin-gle Lagrangian L , the ordering of x ( t ) and ˙ x ( t ) operators will not be a trivial task in the case of dissipative systems where, as we have noted, there are terms like x n ( t ) ˙ x j ( t ) in the Lagrangian. Once it was thought that the path integral method can be used to determine a unique way of ordering the operators x ( t ) and ˙ x ( t ) [3]. But later it was discovered that this is not the case and in the path integral formulation like other methods there is no unique way of ordering x and ˙ x operators [4],[5]. In addition to the ambiguity due to the ordering of x and ˙ x factors, there is also the problem of choosing the proper classical Lagrangian for the path integral quantization. By “proper Lagrangian” we mean those Lagrangians that are quadratic functions of velocity, since for equivalent Lagrangians like 1 6 p 0 m 2 ˙ x 3 e 2 λt , Eq. (2.166), the Feynman method of construction of propagator does not work [1]. 351

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  Physics with MAPLE

  The Computer Algebra Resource for Mathematical Methods in Physics

  • Frank Y. Wang(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  58 2 Oscillatory Motion > F0 := 1 . 0 m := 1 . 0 ω 0 := 1 . 0 > Amp := b → 1 . 0 √ b 2 ω 2 + 1 . 000000 − 2 . 0000 ω 2 + 1 . 00 ω 4 > pha := b → arctan ω b √ b 2 ω 2 + 1 . 000000 − 2 . 0000 ω 2 + 1 . 00 ω 4 , 1 . 0 (1 . 00 − ω 2 ) √ b 2 ω 2 + 1 . 000000 − 2 . 0000 ω 2 + 1 . 00 ω 4 > 0 2 4 6 8 Amp 0.5 1 1.5 2 2.5 3 omega 2.3 Sinusoidally Driven Oscillation 59 > 0 0.5 1 1.5 2 2.5 3 delta 0.5 1 1.5 2 2.5 3 omega > > b=0 b=0.25 b=0.5 0 1 2 3 4 Amp 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 omega 60 2 Oscillatory Motion > > b=0 b=0.25 b=0.5 0 0.5 1 1.5 2 2.5 3 delta 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 omega For one specific value b = 0 . 125 , we plot the amplitude and phase versus the driving fre-quency ω . These plots indicate that when ω = ω 0 , the amplitude is large; this phenomenon is called resonance. At this frequency, the force response lags behind the driving force by π/ 2 , as we have seen. The plot also indicates that, at small frequency, the force response and driving force are in phase; at large frequency, they are out of phase. We plot curves of the force response with various damping coefficients b in a series. For small b , the response at resonance is much larger than the amplitude of the driving force. We also plot curves of phase angle in a sequence with various damping coefficients b ; at resonance, the phase angle is π/ 2 . 2.4 Phase Space A second-order differential equation with constant coefficients is exactly solvable; it serves as a satisfactory model for many oscillatory systems, but for some situations this model is inadequate. For instance, to describe the motion of a pendulum in a gravitational field with a damping force, the angle x relative to the vertical is governed by an equation ml 2 ¨ x + bl ˙ x + mgl sin( x ) = 0 . (2.19) 2.4 Phase Space 61 When x is small, we can make the small-angle approximation such that sin( x ) ∼ = x , and reduce it to the same equation of damped oscillation. For a large angle, it is difficult to write the solution.

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