Resonance

5 Key excerpts on "Resonance"

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  Introductory Physics for Biological Scientists

  • Christof M. Aegerter(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  Different couplings and/or restoring forces (shearing, compression) lead to correspondingly different sound waves in gases, liquids, and solids. In the case of electrical waves in cables, light, and other electromagnetic waves, the disturbance occurs in temporally and locally variable electric and magnetic fields. 4.2 Forced Oscillations and Resonance 4.2.1 The Forced Harmonic Oscillator We have described vibratory systems and oscillations in detail in Section 3.2 when the oscillation is left to evolve on its own. However, if we want to describe how an external excitation can drive an oscillating system, we have to understand the motion of a driven or forced oscillation. For this, we apply an oscillating, external force F ext = F 0 cos(t) onto an oscillatory system. Thus, the equation describing the motion for the oscillation (without damping) is given by the following: m d 2 x dt 2 = −kx + F ext = −kx + F 0 · cos(t) If, as in the preceding, we convert this equation into a generic equation with the eigenfrequency ω 0 = √ k/m, we get the following: 100 Resonances and Waves d 2 x(t) dt 2 + ω 2 0 x(t) = F 0 /m cos(t) Because the pendulum is excited with a certain fixed frequency, we would expect that the pendulum will also oscillate with this externally prescribed frequency (), at least after a long time. This is because we start with the system at rest and then nothing actually happens in the absence of an excitation, therefore the excitation will decide how the system behaves. However, we also know that on its own, the oscillation will have a natural frequency (or eigenfrequency) ω 0 . Therefore, we might expect that something special happens when these two frequencies are the same, i.e., when  = ω 0 . Since ω 0 is the frequency at which the oscillation runs by itself, it will be particularly easy to excite the pendulum at this frequency.

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  Physics for Scientists and Engineers with Modern Physics

  • Raymond Serway, John Jewett(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  We find that if a periodic force is applied to such a system, the amplitude of the resulting motion of the string is greatest when the frequency of the applied force is equal to one of the natural frequen- cies of the system. This phenomenon, known as Resonance, was discussed in Sec- tion 15.7 with regard to a simple harmonic oscillator. Although a block–spring system or a simple pendulum has only one natural frequency, standing-wave sys- tems have a whole set of natural frequencies, such as that given by Equation 17.7 for a string. Because an oscillating system exhibits a large amplitude when driven at any of its natural frequencies, these frequencies are often referred to as Resonance frequencies. Consider Figure 17.16, which shows a string being driven by a vibrating blade. When the frequency of the blade equals one of the natural frequencies of the string, Vibrating blade When the blade vibrates at one of the natural frequencies of the string, large-amplitude standing waves are created. Figure 17.16 Standing waves are set up in a string when one end is connected to a vibrating blade. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 466 Chapter 17 Superposition and Standing Waves standing waves are produced and the string oscillates with a large amplitude. In this Resonance case, the wave generated by the oscillating blade is in phase with the reflected wave and the string absorbs energy from the blade. If the string is driven at a frequency that is not one of its natural frequencies, the oscillations are of low amplitude and exhibit no stable pattern.

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  Nonlinear Dynamics and Stochastic Mechanics

  • Wolfgang Kliemann(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  10 ]. The presence of these Resonances influences the free as well as forced response of these systems in a very profound way, and the resulting dynamics are extremely complex and interesting. The aim of the present chapter is to introduce the reader to the concept of an “internal Resonance” and its role in shaping the response of the system, and to present some new results of studies involving a model of harmonically excited rectangular plates with a specific internal Resonance. We also summarize some of the more recent studies on the response of systems with internal Resonances. In order to describe the essential ideas behind the concept of an internal Resonance and its consequences for the system response, without resorting to lengthy algebra, we first consider an example system with only quadratic nonlinear terms. We restrict the subsequent general discussion to two-degree-of-freedom systems having cubic nonlinearities. We also consider only those systems with harmonic external excitations, and restrict the presentation to the determination of first-order approximations to the response.

  According to Sethna [38 , 40 ], “internal Resonances occur when one of the stronger harmonics in any one of the modes has a frequency close enough to the natural frequency of another mode to cause a first-order coupling effect. Internal Resonance is also said to occur when a combination tone between two or more modes matches the natural frequency of one of the other modes to cause strong coupling between the modes.” Thus, internal Resonances provide a mechanism for the different modes in a system to interact in a sufficiently strong manner, to allow for significant exchange of energy between the modes, and to strongly influence the overall response of the system. Systems that possess internal, as well as external, Resonances are found to exhibit even more interesting responses, arising because of this exchange of energy between the modes in internal Resonance. Through the external Resonance, energy can be fed to one or many of the modes in internal Resonance. Even when only one mode is directly excited, the system can exhibit the “coupled-mode response” due to the exchange of energy between the directly excited mode and the modes in internal Resonance. This modal coupling in the response is caused by the nonlinearities present in the system model. A classic example of this behavior is the response of a stretched nonlinear string, which exhibits nonplanar whirling motions even when it is subjected to a resonant harmonic excitation restricted to a plane [18 , 36 , 46

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  The Speech Chain

  The Physics And Biology Of Spoken Language

  • Dr. Peter B. Denes, Dr. Elliot N. Pinson(Authors)
  • 2016(Publication Date)
  • Hauraki Publishing

    (Publisher)

  It can be no larger than the initial displacement, and it will decay slowly because of losses in the system. In forced vibration, for a given spring-mass combination, the amplitude of the vibration depends on both the amplitude and the frequency of the motion impressed on the free end of the spring. For a given amplitude of forcing motion, the vibration of the mass is largest when the driving frequency equals the natural frequency of the system. This phenomenon, whereby a body undergoing forced vibration oscillates with greatest amplitude for applied frequencies near its own natural frequency, is called Resonance. The frequency at which the maximum response occurs is called the resonant frequency, and it is the same as the system’s natural frequency. We can show graphically the amplitude with which the mass oscillates in response to a driving motion of any frequency. Such a graph is called a frequency response curve. Two frequency response curve are shown in Fig. 3.4. The horizontal axis shows the frequency of the driving motion. The vertical axis shows the amplitude of the response (the motion of the mass) for a constant amplitude of applied motion. At 1 cps—the natural frequency of the vibrating body in our example—the response is much larger than the applied motion. This is due to the phenomenon of Resonance. The curves in the figure show the behavior of two oscillators having the same natural frequency, but different damping (different amounts of energy loss). The smaller the energy losses, the greater the increase in movement produced by Resonance. SOUND WAVES IN AIR All objects on earth are surrounded by air. Air consists of many small particles, more than 400 billion billion in every cubic inch. These particles move about rapidly in random directions. We can explain the generation and propagation of most sound waves without considering such random motions

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  Physical Foundations of Technical Acoustics

  • I. Malecki(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  400 PHYSICAL FOUNDATIONS OF TECHNICAL ACOUSTICS Perfectly rigid walls of the tank correspond in the mechanical system to fixing the spring to the immobile point of reference. If the resonator is excited into vibrations, e.g. by a single short increase of pressure at / = 0, during the observation time ( t > 0) the changes in the external pressure are equal to zero, that is, the right-hand side of Eq. (9.63) equals zero. The resonator then shows free damped vibrations, in accordance with Eq. (9.3), and further considerations concerning these vibrations can be fully applied (Section 9.2). The case is somewhat different with forced vibrations. The equations (9.34) and (9.63) are in fact formally identical; however, there is a difference in the physical conditions under which these two systems works. It is assumed for a mechanical parallel system that the power of the source is very high compared with the power taken by the system, and this is equivalent to assuming that the internal impedance of the source nearly equals zero. Such an assumption is rarely satisfied in an acoustic system if e.g. the outlet of the resonator becomes directly coupled by a short pipe to a surface of an electroacoustic transducer of sufficiently high power. The resonator usually takes energy from the acoustic field of the sur-rounding medium, and at the same time gives back part of it. The technical problem consists then in a determination of the parameters of the resonator in such a manner that it shall take maximum power from the environment, since its action is then the most effective. The resonator will take maximum power at the frequency, the value of which—on account of Eqs. (9.68) and (9.73)—will be a ^ O ^ C J -^ -^ '. (9.83) It will change however, depending on the ratio of the mechanical internal resistance R t of the resonator to the radiation resistance R s of its outlet (x = Ri/R s ).

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