1. Verify Eqs. 1 through 5. Figure 1: mass spring-damper In class, we have studied mechanical systems of this type. Here, the main results of our in-class analysis are reviewed. The dynamic behavior of this system is deter- mined from the linear second-order ordinary differential equation: d'a + + kr = 0 dt² dt (1) where r(t) is the displacement of the mass, m is the mass, b is the damping coefficient, and k is the spring stiffness. Equations like Eq. 1 are often written in the "standard form" C M m where d'a (2) dt2 The variable wn is the natural frequency of the system and is the damping ratio. tan If the system is underdamped, i.e. < 1, and it has initial conditions (0) = o-o=0, then the solution to Eq. 2 is given by: x(t) = Lele dr +26wn+wn²x = 0 Ta xo (-) 2π Wd -Cunt sin (wat +) is the damped natural frequency. In Figure 2, the normalized plot of the response of this system reveals some useful information. Note that the amount of time Ta between peaks is constant. The elapsed time, Ta, is the period of oscillation. From Eq. 3 it can be shown that: (3) , and wa=w₁√√1-C² (4) 2T wn√√√1-(² (5)