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Unformatted text preview: 11.03.1 Chapter 11.03 Fourier Transform Pair: Frequency and Time Domain Introduction In Chapter 11.02, Fourier approximations were expressed in the time domain. The amplitude (vertical axis) of a given periodic function can be plotted versus time (horizontal axis), but it can also be plotted in the frequency domain [1-6] as shown in Figure 1. Figure 1 Periodic Function (see Example 1 in Chapter 11.02) In Frequency Domain. The advantages of plotting the amplitude of a given periodic function in frequency domain (instead of time domain) are due to the following reasons: For a specific value “ k ” (say 2 = k ) of the Fourier series in the time domain, one has to plot the entire curve to observe the amplitude of a given periodic function (recall ) 2 sin( ) 2 cos( ) sin( ) cos( ) ( 2 2 1 1 2 t b t a t b t a a t f + + + + = , see Example 1 in Chapter 11.02). However, in the frequency domain, the amplitude can be plotted as a single point. (see Figure 1a). In the frequency domain, one can easily identify which frequency (or corresponding to which value of “ k ”) contributes the most to the amplitude [see Figure 1(a)], where such information is not readily available if time domain is used. 11.03.2 Chapter 11.03 From the amplitude plot in frequency domain [see Figure 1(a)], one can easily identify that contributions to the amplitude beyond the 8th frequency ( 8 > k ) are not significant any more. In real-life structural dynamics problems, such as the dynamical (time-dependent) response of a (building) structure subjected to oscillated loads (for example, the operational machines attached to the structures), the displacement superposition method is often used to predict the (time dependent) displacement response of the structure. This method basically transforms the original (large, coupled) equation of motion into a reduced (much smaller size, un- coupled) equation of motion by making use of the few free vibration mode shapes and its associated frequencies. Knowledge of which frequencies (and the corresponding mode shapes) that have the most contribution to the predicted dynamical response (such as nodal displacement response) plays crucial roles for the algorithms’ efficiencies. Detailed explanations on how to obtain Figures 1(a), and 1(b) are now presented in the following sections. Explanation of Figure 1(a) and 1(b) One starts with Equation (18) and (20) of Chapter 11.02 ∑ ∞ −∞ = = k t ikw k e C t f ~ ) ( where × = ∫ − T t ikw k dt e t f T C ) ( 1 ~ For the periodic function shown in Example 1 of Chapter 11.02 (or Figure 1 of Chapter 11.02), one has f w π 2 = T π 2 = π π 2 2 = 1 = × + × = ∫ ∫ − − π π π π 2 1 ~ dt e dt e t T C ikt ikt k Define, and using “integration by parts” formula ∫ ∫ − − − + − × = × ≡...
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- Spring '08
- Fourier Series, Time domain, fourier transform pair