Damped Driven Oscillator

9 Key excerpts on "Damped Driven Oscillator"

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  Oscillations and Waves

  An Introduction, Second Edition

  • Richard Fitzpatrick(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 2

  Damped and Driven Harmonic Oscillation

  2.1 INTRODUCTION

  In the previous chapter, we encountered a number of energy conserving physical systems that exhibit simple harmonic oscillation about a stable equilibrium state. One of the main features of such oscillation is that, once excited, it never dies away. However, the majority of the oscillatory systems that we encounter in everyday life suffer some sort of irreversible energy loss while they are in motion, due, for instance, to frictional or viscous heat generation. We would therefore expect oscillations excited in such systems eventually to be damped away. The aim of this chapter is to examine so-called damped harmonic oscillation, and also to introduce the differential equation that governs such motion, which is known as the damped harmonic oscillator equation. In addition, we shall examine the phenomenon of resonance in periodically driven, damped, oscillating systems. In this chapter, examples are again drawn from simple mechanical and electrical systems.

  2.2 DAMPED HARMONIC OSCILLATION

  Consider the mass–spring system discussed in Section 1.2 . Suppose that, as it slides over the horizontal surface, the mass is subject to a frictional damping force that opposes its motion, and is directly proportional to its instantaneous velocity. It follows that the net force acting on the mass when its instantaneous displacement is x(t) takes the form

  f

  (

  x ,  

  x ˙

  )

  = − k x − m v

  x ˙

  ,

  (2.1)

  where m > 0 is the mass, k > 0 the spring force constant, and v > 0 a constant (with the dimensions of angular frequency) that parameterizes the strength of the damping. The time evolution equation of the system thus becomes [cf., Equation (1.2) ]

  x ¨

  + v

  x ˙

  +

  ω 0 2

  x = 0 ,

  (2.2)

  where

  ω 0

  =

  k / m

  is the undamped oscillation frequency [cf., Equation (1.6) ]. We shall refer to the preceding equation as the damped harmonic oscillator equation

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  Simulations of Oscillatory Systems

  with Award-Winning Software, Physics of Oscillations

  • Eugene I. Butikov(Author)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  In the case of unforced (free, or natural) oscillations of an isolated system, motion is initiated by an external influence acting before a particular instant. This influence determines the mechanical state of the system, that is, the displacement and the velocity of the oscillator, at the initial instant. These in turn determine the amplitude and phase of subsequent free oscillations. Frequency and damping of such oscillations are determined by the physical properties of the system. On the other hand, the characteristics of forced oscillations generated by a periodic external influence depend not only on the initial conditions and physical properties of the oscillator but also on the nature of the external disturbance, that is, on its amplitude and (primarily) on frequency. 3.1.2 Discussion of the Physical System To study forced oscillations in a linear system excited by a sinusoidal external force, we consider here the same torsion spring pendulum described in Chapter 1 (which is devoted to free oscillations), namely, a balanced flywheel attached to one end of a spiral spring. The flywheel turns about its axis of rotation under the restoring torque of the spring, much like the devices used in mechanical watches. However, unlike the situation of free oscillations in which the other end of the spring is fixed, now this end is attached to an exciter, which is a rod that can be turned back and forth about an axis common with the axis of rotation of the flywheel. A schematic diagram of the driven torsion oscillator is shown in the left-hand panel of Figure 3.1. The right-hand panel of Figure 3.1 shows an oscillatory LCR -circuit with al-ternate input voltage. This circuit can be regarded as an electromagnetic analog of the mechanical device. Both systems are described by identical differential equa-tions and thus are dynamically isomorphic.

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  College Physics

  • Michael Tammaro(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  The damping constant b has SI units of N s m ⋅ / and depends on the object’s size and shape, as well as the nature of the resistance. When a 308 | Chapter 11 simple harmonic oscillator is subject to a force given by Equation 11.6.1, the amplitude of the oscillations will decrease in time and in some cases there will be no oscillations at all. If b is small enough (or the oscillator’s mass is large enough), then there will be multiple oscillations, and their amplitude will depend on time t in the following way: A A e b m t 0 2 = −         (11.6.2) The oscillator’s mass is m and A 0 is the amplitude at t 0 = . An oscillator described by Equa- tion 11.6.2 is said to be underdamped. As the ratio b m / increases, the amplitude decays more rapidly. There is a critical value of b m / such that the system will not oscillate at all and will simply move back to equilibrium in the shortest possible time without passing equilibrium. This situation is called critical damping. Still larger values of b m / result in overdamping. 11.7 Driven Oscillations, and Resonance If an additional force is applied to an oscillating system, at periodic intervals, it is poss- ible to counteract the effects of damping and possibly increase the amplitude to very large values. These periodically applied forces are called driving forces and the resulting oscil- lations are called driven oscillations. There is a particular frequency of the driving force, called the resonant frequency, for which the amplitude of the oscillations will be a maxi- mum. This phenomenon is called resonance. For small damping, the resonant frequency is very close to the natural frequency of the oscillating system—that is, the frequency with which the system would oscillate if there were no damping.

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  Vibrations and Waves

  • A.P. French(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  4 Forced vibrations and resonance THE PRECEDING CHAPTER was concerned entirely with the free vibrations of various types of physical systems. We shall now turn to the remarkable phenomena, of profound importance throughout physics, that occur when such a system—a physical oscillator—is subjected to a periodic driving force by an external agency. The key word is “resonance.” Everybody has at least a qualitative familiarity with this phenomenon, and probably the most striking feature of a driven oscillator is the way in which a periodic force of a fixed size produces very different results depending on its frequency. In particular, if the driving frequency is made close to the natural frequency, then (as anyone who has pushed a swing knows) the amplitude of oscillation can be made very large by repeated applications of a quite small force. This is the phenomenon of resonance. A force of about the same size at frequencies well above or well below the resonant frequency is much less effective; the amplitude produced by it remains quite small. To judge by the quotation at the beginning of this chapter, the phenomenon has been recognized for a very long time. 1 It is typical of this type of motion that the driven system is compelled to accept whatever repetition frequency the driving force has ; its tendency to vibrate at its own natural frequency may be in evidence at first, but ultimately gives way to the external influence. 1 As Alexander Wood remarks in his book Acoustics (Blackie & Son, London, 1940): “It seems difficult to believe that legislation should be designed to cover a situation that had never arisen.” The example does seem rather bizarre, however, and H

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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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  Vibroacoustic Simulation

  An Introduction to Statistical Energy Analysis and Hybrid Methods

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

    (Publisher)

  1 1 Linear Systems, Random Process and Signals Simple systems with properties constructed by lumped elements as masses, springs and dampers are a good playground to understand and investigate the physics of dynamic systems. Many phenomena of vibration as resonance, forced vibration and even first means of vibration control can be explained and visualized by these lumped systems. In addition, a basic knowledge of signal and system analysis is required to put the principle of cause and effect in the right context. Every vibroacoustic system response depends on excitation by random, harmonic or specific signals in the time domain and we need a mathematical tool set to describe this. An excellent test case to demonstrate and define the principle effects of vibration is the harmonic oscillator. It consists of a point mass, a spring and a damper. The combi-nation of many point masses connected via simple springs and dampers provides some further insight into dynamic systems. As those systems are described by components that have no dynamics in themselves they are called lumped systems. In principle all vibroacoustic systems can by modelled and approximated by this simplified approach. 1.1 The Damped Harmonic Oscillator A realization of the harmonic oscillator is given by a concentrated point mass ? fixed at massless spring with stiffness ? ? as in Figure 1.1. The static equilibrium is assumed at ? = 0 being the displacement in ? -direction. A damper connecting mass and fixation creates dissipation. 1.1.1 Homogeneous Solutions Without external excitation as shown in Figure 1.1 a) the motion depends on the initial conditions at time ? = 0 with the displacement ? (0) = ? 0 and velocity ? ? (0) = ? ? 0 . 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.

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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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  Fundamentals of Musical Acoustics

  Second, Revised Edition

  • Arthur H. Benade(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  10

  Sinusoidally Driven Oscillations

  When one plucks a guitar string or strikes a piano string with its hammer, the string is given a complicated motion made up of a collection of characteristic sinusoidal oscillations belonging to the string. We have already investigated this sort of composite motion in considerable detail. We have also seen that soundboards and other two-dimensional objects respond to excitation in a similar fashion. We should now inquire about the way forces exerted by any one of the characteristic sinusoidal oscillations of one object (for instance, a piano string) excite the vibrations of another object (for example, the soundboard on which it is mounted). In simplest terms this question can be reduced to one about the behavior of a single spring-and-mass system when it is driven by a sinusoidally varying force. Once we understand what goes on here, it will prove easy to generalize in familiar ways to learn what takes place when a more complicated system (having several characteristic modes) is sinusoidally driven. Each of these modes will respond to the driving force in the same basic way.

  Many coffee drinkers have observed that the liquid in a cup has a side-to-side sloshing mode which oscillates with a frequency of about two repetitions /second. This mode is, as a matter of fact, a very close cousin to the single mode of oscillation that was found to take place in a U-tube filled with water (see sec. 6.1 and fig. 6.3 ). The coffee drinker may also know from experience that if he waves his filled cup gently to and fro at a frequency that is even approximately equal to the natural frequency of the fluid, the coffee oscillates ever more wildly, and soon slops out of the cup. That is, repetitive excitation of an object can build up a very large amplitude of oscillation if the excitation frequency is roughly equal to the natural vibration frequency of the object.

  10.1. Excitation of a Pendulum by a Repetitive Force

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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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