53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 1 - Lecture 12: Discrete Fourier Transform (Optional) Reading materials: Section 4.5 1. Introduction All experimental vibration time signals are available as numerical data at discrete time steps. For actual dynamic loading, including wind or earthquakes, the results can only be obtained through numerical integration of equations of motion. It is important to create a discrete form of the Fourier series that can be used on discrete time signals. The complex Fourier series53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 2 - Period and frequency Numerical integration (Trapezoidal rule) Here, tS = sΔt,53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 3 - In summary, 2. Example53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 4 -53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 5 -53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 6 - 3. Data smoothing Discrete Fourier transform can be used to remove noise from measured time-history data. An Example List of data over one period53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 7 - Discrete Fourier frequency coefficients Smoothed solution53/58:153 Lecture 12 Fundamental of Vibration ______________________________________________________________________________ - 8