

Question 1 Suppose m-2 kg, k- 32 N/m and F() is an odd periodic force with a period of 2 s given in one period by ON if O<t<I Construct the steady periodic motion, Xsn) using the Fourier series...
The Fourier series of a periodic signal s(2) of period T can be expressed as k s(x) = cxexp ( 21 - where the coefficients Ck are given by 7/2 CR 1 T -T/2 | $(z) exp (-27 k -27=cdc T (i) Consider s(2) of period T = 6 and amplitude A= 2: 8(z) = 2 * |< T 2 Compute the Fourier coefficients ok. (ii) Use the identities exp(Trik) + exp(-rik) cos(Tk) = 2 sin(Tk) exp(Trik) – exp(-rik) 2i...
Problem 32: (20 points) Consider a periodic signal f(t), with fundamental period To, that has the exponential Fourier series representation f(t) = Σ Dnejuont . where wo 2T/To and 1. (2 points) When f(t) is a real-valued, show that DD This is known as the complex conjugate symmetry property or the Hermitian property of real signals. 2. (2 points) Show that when f(t) is an even function of time that Dn is an even function of n 3. (2 points)...
2, For the following periodic series, f(t) = t^2, ( -1<t<1), period = 2 (a) graph this periodic function, range is ( -8<t<8), and find a key point (b)Even function or odd function? (c ) find a Fourier series
The sketch of the following periodic function f (t) given in one period f(t) t2 -1, 0s t s 2 is given as follows f(t) 2 -1 We proceed as follows to find the Fourier series representation of f (t) (Note:Jt2 cos at dt = 2t as at + (a--)sina:Jt2 sin at dt = 2t sin at + sin at. Г t2 sin at dt-tsi. )cos at.) Please scroll to the bottom of page for END of question a) The...
1. Find the Fourier series for the following 1-periodic function f(t) = t, t < -- 2. Find the sum 24 3444 (Hint: Consider the Fourier series for the function f(t)-t2 on [- integer k.) 1) and f(t-k)-f(t) for all
1. Find the Fourier series for the following 1-periodic function f(t) = t, t
l, f) is a periodic signal with period f(t)-n(t)-u(t-t/2 ) for 0 2π a.) Find the exponential Fourier series of f() and sketchf). What is the fundamental radian frequency. b.) Evaluate and sketch |Dml, the magnitude of the Fourier series coefficients vs.o in the range of -4s n S4 c.) Evaluate and sketch the phase angle of D, vs. co in the same range (-4S n S4) d.) Find the signal average power e) Find the approximate average power of...
F(N) 2. A 15 kg oscillator with a stiffness of k = 960 N/m and damping coefficient c = 60 Ns/m is driven by a square- wave excitation F(t) shown in the figure. Determine and plot the steady state response for 12 s using 100 terms in the Fourier series solution. 100 -100
Consider the periodic function defined by 1<t0, 0<t<1, f(t)= f(t+2) f(), and its Fourier series F(t): Σ A, cos(nmi) +ΣB, sin (nπί), F(t)= Ao+ n1 n=1 (a) Sketch the function f(t) the function is even, odd or neither even nor odd. over the range -3<t< 3 and hence state whether (b) Calculate the constant term Ao
Consider the periodic function defined by 1
The periodic function so(t) with period 28 given by if 14 t<-0.5 1 if _ 0.5 〈 t < 0.5 so(t)- 0if 0.5 t< 14. has the Fourier series defined by So(0)-0.0357143 and for n 0 0.0357143 * sin(nTl/28) nT1/28 Use linearity and the shifting property to find the Fourier Series for s(t), defined by f -14 t <4.5 -5 if 4.5 t< 5.5 3 if 5.5 t<6.5 if 6.5 < t < 14. s(t) S(0) and for n S(n)...
Find a Fourier series expansion of the periodic function 0 -T -asts 2 - f(t) = 6 cost T <<- 2 2 0 I SISE 2 f(t) = f (t +21) Select one: a f(t)= 12 12 5 (-1)** cos nt 1 2n-1 b. f(t) = 12.12 F(-1)** cos 2nt T 4n-1 C 6 12 =+ 125 (-1) C05 211 472-1 6 12 (-1) * cosm d