Chapter 11, Problem 31P

Damped free vibrations can be modeled by a block of mass in that is attached to a spring and a dashpot as shown. From Newton’s second law of motion, the displacement x of the mass as a function of time can be determined by solving the differential equation:

$m\frac{d^{2}x}{d{t}^2}+c\frac{dx}{dt}+kx=0$

where 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 will start oscillating back and forth. The nature of the oscillations depends on the size of the mass and the values of k and c.

For the system shown in the figure, m = 10kg and k = 25 N/m. At time t = 0 the mass is displaced to x =0.18 m and then released from rest. Derive expressions for the displacement x and the velocity v of the mass, as a function of time. Consider the following two cases:

(a) c= 3(Ns)/m. (b) c = 50(Ns)/m.

For each case, plot the position x and the velocity v versus time (two plots on one page). For case (a) take $0\le t\le 20$ s, and for case (b) take $0\le t\le 10$ s.