achapt 5 - MOTION IN A STABILITY REGION(PART II 5 MOTION IN...

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Unformatted text preview: MOTION IN A STABILITY REGION (PART II) 5 MOTION IN A STABILITY REGION (PART II) Figure 5 – 1 MAE 461: DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION (PART II) This section begins by showing how to find the steady-state response of a one degree-of-freedom system acted on by a periodic excitation. The periodic excitation has a period T , or equivalently, a frequency T π ω 2 = . The periodic excitation is represented as a linear combination of harmonic functions. The linear combination of harmonic functions is called a Fourier series . Once a periodic excitation is expressed as a Fourier series, the steady-state response of a system acted on by a periodic excitation is found. By the principle of linear superposition, the response of the system to the periodic excitation is a linear combination of the responses of the harmonic functions that make up the Fourier series. After showing how to represent a periodic excitation by a Fourier series and how to determine the associated response, it’s shown how to represent a periodic excitation by a complex Fourier series. The complex Fourier series is used to develop a method of finding the steady-state response to a non-periodic excitation. Excitations are generally non-periodic. Earthquakes and wind produce non-periodic excitations on buildings. Wind, road surfaces and tracks and guides produce non-periodic excitations on vehicle systems. Like the periodic excitation, the non-periodic excitation is represented as a linear combination of harmonic functions. However, instead of its frequencies being multiples of the frequency of an excitation, there is a continuous range of frequencies. The non- periodic excitation is an integral of harmonic functions instead of a discrete sum of them. The integral of harmonic functions is called the Fourier integral . The coefficients in the integral are called the frequency response . The frequency MAE 461: DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION (PART II) response represents the amplitude of the response as a function of the frequency of the harmonics that make up the non-periodic excitation. Since any non-periodic excitation can be expressed in terms of its frequency response and, conversely, a frequency response can be found for any non- periodic excitation, the two expressions are also called the Fourier transform and the inverse Fourier transform ; together they’re called the Fourier transform pair . The Fourier transform pair is important in engineering. It applies not only to excitations in dynamical systems but can be used to characterize how any physical quantity changes in time. After the Fourier transform pair is developed, this section develops the procedure for finding the transient and steady-state responses of systems acted on by non-periodic excitations. The procedure uses a variation of the Fourier transform called the Laplace transform ....
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achapt 5 - MOTION IN A STABILITY REGION(PART II 5 MOTION IN...

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