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Exercise 1 Determine the amplitude, the frequency, the phase shift and the period of the motion given by u(t)3 cos(2)3 sin(2t) Hint: rewrite u into the form u(t) = Rcos(wt - 6) Exercise 2 A mass of 0.5 kg stretches a spring 49 cm. Suppose that the mass is also attached to a viscous damper with a damping constant 0.5 N -s/cm. If the mass is pulled down 5 cm below its equilibrium and then released, find its position u at any time t. Plot u versus t. Remark: for the plot, you can do a sketch by hand or use the Matlab script provided on the website of the course. Exercise 3 A mass of 1 kg stretches a spring 9.8 m. The mass is acted on by an external force of 4 cos (t) N. If the mass is set in motion from its equilibrium position with a downward velocity of 2 m/s, find the position of the mass at any time. Identify the transient (i.e., complementary) and steady state (i.e., particular) solutions. Does the motion exhibit resonance or a beat?

Answer 1

(1) u(t)= -3 cos(27) + 3 8in (2+) Multiply and divide by. 352. ult = [-3 Cos (24) +B sinet) ] 352 uit & B S2 - Cos (24) + Sin (24) .. = 3821 Sin (24-1) sin (a - b) = sina. Cosb-cosa. Sinb .... sin (24-II = Sinet. COSTI - Cos (2 t) - SinI - Sinet - Ces(24) ...uct = 362 Sin (2t - I ) = 312 Cos I - (2t - II) = 3/2 Cos (-24 + 317) = 352 Cos - ( 2t - 317 - Cos (0) = cose = cos(-(27-36)] - Cos[28-30 Ult'= 3 62 Cos 12t-3 iT Ult) = 3/2 Cos (2t - 317) It is in the form ult = R Cos (wat d ... R = 352 ; w=2 Amplitude = R=362 ; d = 31 I Phase shift = da 3 π frequency = w = 2/5 = 1 2 TT period= = frequency π bes

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