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Chapter 1. Vibrations of Continuous Elastic Solid Media. Chapter 2. Chapter 3.

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Equation of Motion for Beams. Chapter 4. Equation of Vibration for Plates. Chapter 5. Vibratory Phenomena Described by the Wave Equation. Chapter 6. Free Bending Vibration of Beams.

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Chapter 7. Bending Vibration of Plates.

Wearing of MR Wire Vibration Insulation Material Under Random Load

Chapter 8. Introduction to Damping: Example of the Wave Equation. Chapter 9. Calculation of forced vibrations by modal expansion. Chapter 10 Calculation of forced vibrations by forced wave decomposition. Chapter The Rayleigh-Ritz method based on Reissner's Functional. The Rayleigh-Ritz method based on Hamilton's Functional. Bibliography and Further Reading. A particular effort has been made to provide a clear understanding of the limits associated with each modeling approach.

Examples of applications are used throughout the book to provide a better understanding of the material presented. Subject Vibration. These topics will deal with frequency and the frequency content of individual sounds under more or less ideal conditions. The interaction with the environment, with other sounds, with the auditory system in the sense or hearing ability , and the manipulation by electroacoustic devices will be left to other sections of this document.

A Oscillation, the Sound Wave and its Basic Parameters: as has been stated, the basic nature of sound derives from the physical behaviour called "oscillation", a phenomenon common to many physical systems.

Oscillation in a medium gives rise to the propagation of a "wave", the series of oscillations that carry energy and, from the listener's perspective, potential information. We introduce here the basic parameters which commonly describe oscillation and waves, the most commonly used of which is the concept of "frequency".

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B Spectrum and Timbre: the most common way of describing and measuring the oscillation of a sound wave is by means of the concept of "spectrum", or what might be called the "frequency content" of the sound. Of course, the magnitude of each frequency component is a necessary part of such a description, and therefore spectrum is not independent of magnitude considerations. The spectrum of a sound is one of the main determinants of the subjective perception of the "quality" or "colour" of the sound, more properly called its "timbre".

This subjective impression of the sound is probably the most difficult to explain as it depends on many other variables as well. For convenience, we will introduce the topic in stages, with several subdivisions:. C Resonance Phenomena: the nature and behaviour of the sound producer, including the nature of its contact with the surrounding medium, determines a great deal of the nature of the resulting sound heard.

Although this topic is more fully treated under Sound-Medium Interface , the implications of the so-called "natural modes of vibration" or "resonances" of objects and enclosures is so important for understanding frequency, spectrum and timbre, that we also include the relevant information here. Aperiodic vibration is the most basic physical description of "noise".

ME 7442: Vibration of Continuous Systems

In terms of frequency content, aperiodic vibrations can only be described as having their energy spread out more or less continuously over some range or "band" of frequencies, perhaps even over the entire audible spectrum. E Pitch and Musical Pitch Systems: the subjective response of the auditory system to periodic vibration is the sense of pitch, the parameter which the auditory system and the brain seem the most developed to detect. The study of pitch perception is an extremely large and important part of psychoacoustics, and the organization of pitches is common to all musical cultures.

The adjustment of pitch called "tuning" , the limitation of musical material to specific sets of pitches, called "scales", and the concept of the distance between pitches, called the "interval", are central concerns of most musical systems.


See also Appendix C. F Electroacoustic Considerations: the basis of the electrical representation of sound is that a voltage which oscillates can be transformed into sound when it is amplified to drive a loudspeaker. Thus the audio signal is an oscillating voltage pattern that is analogous to an oscillating sound pressure pattern.