Damped free vibrations can bemodeled by a block of mass m thatis attached to a spring

Chapter 11, Problem 31

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Damped free vibrations can bemodeled by a block of mass m thatis attached to a spring and a dashpotas shown. From Newton's secondlaw of motion, thedisplacement x of the mass as a=.!LJ--tfunction of time can be determined by solving the differential equationm-d2x +cd-x +kx dt2 dt = 0where k is the spring constant and c is the damping coefficient of the dashpot.If the mass is displaced from its equilibrium position and then released, it willstart oscillating back and forth. The nature of the oscillations depends on thesize of the mass and the values of k and c.For the system shown in the figure, m = 10 kg and k = 28 N/m. At timet = 0 the mass is displaced to x = 0.18 m and then released from rest. Deriveexpressions for the displacement x and the velocity v of the mass, as a functionof time. Consider the following two cases:(a) c = 3 (N s)/m.(b) c = 50(N s)/m.For each case, plot the position x and the velocity v versus time (two plots onone page). For case (a) take 0 t 20 s, and for case (b) take 0 t 10 s.

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