A mass-spring system is shown in the figure on the right. Using Newton's laws, d'y the equations of motion are given by M +F +F = r(t). Common dt friction spring Inertial force models of the spring force include Fring = ky (linear spring), Fapring -k(1ay)y (softening spring, common in micromachined mechanical resonators), and F Fpring - K(1 + ay)y (hardening spring). Assuming a hardening spring, and that the (wall) friction is of viscous type, i.e., dy F =b- the equations of motion then are 3 dt friction Force (6+9 = 15 pts) (a) For the numerical values M = 1,b= 0.2, k = 0.5, a-0.1, r(t)=5, verify that the system has an 6.8. (rough calculations are sufficient) equilibrium point with y - y (b) Linearize the system around this equilibrium point, defining all variables of interest clearly. dy dt M Mass M dy + b +k(1 + ay)y=r(t)| dt