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Extra resources for Applied structural and mechanical vibrations: theory, methods, and measuring instrumentation

Example text

6a) converges to x(t) at all the points where x(t) is continuous. At jumps (discontinuities) the Fourier series converges to the midpoint of the jump. Moreover, if x(t) is complex (a case of little interest for our purposes), the conditions apply to its real and imaginary parts separately. Two things are worthy of note at this point. First, the usefulness of this theorem lies mainly in the fact that we do not need to test the convergence of the Fourier series. We just need to check the function to be expanded, and if the Dirichlet conditions are satisfied, then the series (when we get it) will converge to the function x(t) as stated.

69) The formal ‘proofs’ of all the above properties are not difficult and are left to the reader. Other properties will be considered if and whenever needed in the course of future chapters. Copyright © 2003 Taylor & Francis Group LLC Nonetheless, it is worth noting that the above properties do not follow from the definition of δ(t), but they are mere assumptions consistent with the formal properties of the integral. 70) is not justified in the ordinary sense. Let us ignore this for the moment and proceed in our discussion by giving some examples of how we can see δ(t) as a limit of this kind.

2. 41), the Laplace transform provides automatically for the initial conditions at t=0. This latter characteristic will be considered (and exploited) in the following examples and in future chapters. 3. 9) by standard methods. 19) with the appropriate modifications). Partial differential equations can also be solved with the aid of Laplace transforms, the effect of Laplace transformation being the reduction of independent variables by one. Also, Laplace and Fourier transforms can be used together, as in the following example.

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