Obtain the equilibrium point for the following nonlinear systems. Then linearize the differential equations about the equilibrium point. Do not solve these linearized equations.

(a) Mass-spring-damper system with a nonlinear spring: Mx¨ + bx˙ + k(1 + a 2x 2 )x = F. Obtain the equilibrium position xo for a force Fo, then linearize around this equilibrium point.

(b) Mass-spring-damper system with a nonlinear damper: Mx¨ + bxx˙ + kx = F. Obtain the equilibrium position xo for a force Fo, then linearize around this equilibrium point.