Using the values from Problem TB1.1 and x # (0) = 0, plot the response x(t) for a range of values of x(0) to see how the response depends on the initial displacement.

Problem TB1.1

Fix [your choice or use the values from Example 1.3.1 with x(0) = 1 mm] the values of m, c, k, and x(0) and plot the responses x(t) for a range of values of the initial velocity x #

(0) to see how the response depends on the initial velocity. Remember to use numbers with consistent units.

Example 1.3.1

Recall the small spring of Example 1.2.1 (i.e., ωn = 132 rad>s). The damping rate of the spring is measured to be 0.11 kg>s. Calculate the damping ratio and determine if the free motion of the spring–bolt system is overdamped, underdamped, or critically damped.

Example 1.2.1

Consider a small spring about 30 mm (or 1.18 in) long, welded to a stationary table (ground) so that it is fixed at the point of contact, with a 12-mm (or 0.47-in) bolt welded to the other end, which is free to move. The mass of this system is about 49.2 * 10^−3 kg (equivalent to about 1.73 ounces). The spring stiffness can be measured using the method suggested in Figure 1.4 and yields a spring constant of k = 857.8 N>m. Calculate the natural frequency and period. Also determine the maximum amplitude of the response if the spring is initially deflected 10 mm. Assume that the spring is oriented along the direction of gravity as in Window 1.1. (Ignore the effect of gravity; see below.)