By Gatti P. L., Ferrari V.
This ebook offers with the basic recommendations, rules and techniques underlying all vibration phenomena. specific cognizance is given to the dynamics of constitution, and the analyses awarded follow effectively to many different fields.
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Extra info for Applied Structural and Mechanical Vibrations: Theory Methods and Measuring Instrumentation
Fig. 1 Periodic function. Copyright © 2003 Taylor & Francis Group LLC Fig. 2 Nonperiodic function. 4) where the quantities An, and Bn are, respectively, the cosine and sine amplitudes at the frequency ωn. When a fundamental frequency ω1 exists, then and we call ωn the nth harmonic frequency. 1 Periodic functions: Fourier series As stated before, harmonic analysis is, in essence, the reverse process of harmonic synthesis. 5) is that the value of x(t) at any instant can be obtained by finding the partial sum SN of the first N harmonics and defining x(t) as the limiting value of SN when N tends to infinity.
The addition of several sinusoidal oscillations to form a resultant (harmonic synthesis); 2. e. given a resultant, the problem of finding the sinusoidal oscillations from which it was formed (harmonic analysis). 1) leads to a harmonic oscillation of the same frequency with a sine amplitude and a cosine amplitude If the two oscillations have different frequencies, the resultant will not be harmonic. However, if the frequencies being summed are all multiples of some fundamental frequency and ω1, then the resultant oscillation will repeat itself after a time we say that it is periodic with a period of T seconds.
5) with the Copyright © 2003 Taylor & Francis Group LLC coefficients An=0, and Bn=1/n is the well-known periodic ‘sawtooth’ oscillation (see Fig. 3 representing the partial sum S6 with ). Incidentally, two short comments can be made at this point. First, the example of Fig. 3 gives us the opportunity to note that a function which executes very rapid changes must require that high-frequency components have appreciable amplitudes: more generally it can be shown that a function with discontinuous jumps (the ‘sawtooth’, for example) will have A and B coefficients whose general trend is proportional to n–1.