Answer Happy

A vehicle's suspension uses a combination of springs (restoring force) and shocks (energy dissipation) which can be modeled together as a single combined spring with the traditional spring-mass model. Real springs, however, have limits to their practical use. The elastic limit of a material is the furthest point it can be stretched or deformed while being able to return to its previous shape. Once a material has gone past its elastic limit, its deformation is said to be inelastic. This idea is demonstrated in the exercises in section 3.7. Let's model the force of a soft spring as k(x-sx^3) so that the model equation becomes mx"+bx'+k(x-sx^3) = f(t), but we will only be looking at unforced (f=0) motion in this option. Using your vehicle parameters and your chosen values of b in (A), for each of the six listed damping types, *Use pplane.jar (or Bluffton) to graph your solution for each case given the initial condition x(0)=0, x'(0)=1. For each case, show the phase plane with s=0 and a positive value to show the effect that "softness" has on the spring. *Use your phase plane to estimate the value of s (increasing from 0) to soften the spring to a point where it just barely restores to zero with initial conditions x(0)=0, x'(0)=1. *Discuss how the behavior changes when spring is softened (for the parts when the solution converges). Also, what does it mean when the solution diverges?