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 2 a frequency The periodic excitation is T 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 nonperiodic 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 nonperiodic 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 1 Fourier Series A periodic excitation f t is represented as a Fourier series of sine functions and cosine functions as 5 1 f t 1 B0 Br cos r t C r sin r t 2 r 1 r r in which Br and Cr are constants called Fourier coefficients and T is the period of the excitation 2 The the frequency of the excitation is T 2 T periods Tr of the individual harmonics r r in Eq 4 1 are integer fractions of the period of MAE 461 DYNAMICS AND CONTROLS 2 T MOTION IN A STABILITY REGION PART II the excitation Therefore the period of the series is T See Fig 5 1 The Fourier coefficients are 1 5 2 2 T 2 T 2 f t cos r tdt r 0 1 2 T 2 C r TT 2 2 f t sin r tdt r 1 2 T Br As an illustration consider the square wave excitation shown in Fig 5 2 It is defined over the interval T 2 T 2 as A f t 0 A0 0 t T 2 T 2 t 0 Substituting f t into Eq 5 2 Br 0 r 0 1 2 Cr 1 2 A0 1 cos r r 1 2 r To find the Fourier coefficients Br r 0 1 multiply T 2 both sides of Eq 5 1 by T 2 cos r tdt Similarly Cr r 1 2 are found by multiplying both sides of Eq T 2 5 1 by T 2 sin r tdt On the right sides one then notices that that T 2 T 2 T 2 cos r t sin s tdt 0 T 2 for any r and s and T 2 cos r t cos s tdt T 2 sin r t sin s tdt 0 for any r and s that are distinct r s MAE 461 DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION PART II Figure 5 2 The Fourier series of the square wave excitation truncated to n terms is f t 2 A0 1 r 1r n 1 cos r sin 2r t T Figure 5 2 shows the Fourier series of the square wave truncated to n 2 terms and to n 10 terms The square wave itself was actually generated by truncating the Fourier series to n 500 terms 2 Steady State Response to Periodic Excitation When the steady state responses xr t to individual excitations fr t are known then the steady state response to any linear combination of the excitations fr t can be found by the principle of linear superposition Let a system be acted on by the excitation f t A1 f1 t A2 f 2 t A3 f 3 t MAE 461 DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION PART II The system s steady state response is simply x s t A1 x1 t A2 x 2 t A3 x3 t Notice that the coefficients in the linear combination of the excitation are the same as the coefficients in the linear combination of the steady state responses Consider the damped system described by Eq 5 2 Its steady state response to a constant force of F0 was found to be x F0 k Its steady state response to f F0 cos t is x X 0 cos t and its steady state response to f F0 sin t is x X 0 sin t It follows from the principle of linear superposition that the steady state response to the period excitation in Eq 5 1 is 5 3 x s t 1 1 B0 Br x0 r cos r t r C r x0 r sin r t r 2 k r 1 1 B0 x0 r Br cos r t r C r sin r t r 2 k r 1 in which from Eq 4 41 MAE 461 DYNAMICS AND CONTROLS MOTION IN A STABILITY REGION PART II 5 4 X 0r 1 1 k 2 2 1 r 2 r 2 n n r r 2 n tan 1 2 1 r n For example look again at the square wave excitation illustrated after Eq 5 2 Assume now that it acts on an undamped system that has mass m and stiffness k From Eq 5 4 its steady state response is x s t 1 1 cos r 2 …