Question

Problem 2 (25 points): Consider an undamped single-degree-of-freedom system with k = 10 N/m, 41 = 10 N 92 = 8N, and m = 10 kg subjected to the harmonic force f(t) = qı sin(vt) + 92 cos(vt), v = 1 rad/ sec. Assume zero initial conditions (0) = 0 and c(0) = 0. Derive and plot the analytical solution of the displacement of the system. mm m = f(t) WWWWWWWW No friction Problem 2 Problem 3 (30 points): Using Duhamel's intergral, determine the response of the system in Problem 2 subjected to force: f(t) = s 91 for 0 <t<ti 92 for ti <t where qı = 10 N, 92 = 15 N, t1 = 10 sec.

Answer 1

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2) FBD: Exct) krem to fet) Using Newton's second law of motion, ma= EF :. mä = fct) – KX :. m +kz= f(t) :-mž +Kx= qisin (vt)+92.Coscut) -: lož+ 10x= 10 sint + 8 cost ci) This is the equation of motion. Take Laplace transform with zero-initial conditions 108?xc9) +10%C5) = 10:21 XCS) = (64(100640) + () (1066+0) xc9= Couget e These are the standard convolution properties of Laplace transform XC+) = { cost + i sint]+0.8[Esint] act) = 0.4tsint - 0.5tcost +0.5 sint S + 8. S+1