\text { The second order dynamics } m \ddot{y}=-k \dot{y}+m g-\frac{L_{0} i^{2}}{2 a(1+y / a)^{2}} \text { relate the position } y \text { of a steel ball of } mass m to a controlling current i in an electromagnet, with g, k, a and Lo being constants. It is desired to stabilize the position at an equilibrium position ỹ = r. Derive the linearized transfer function \frac{y_{s}}{i_{s}}=\frac{A}{s^{2}+B_{s}+C}, \text { clearly defining the variables } i_{s,} y_{s} . \text { Tell me what the coefinients } A, B \text { and } C (10 pts)in the transfer function are in terms of the constants in the original differential equation.

Solved: \text { The second order dynamics } m \ddot{y}=-k \dot{y}+m g-\frac{L

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\text { The second order dynamics } m \ddot{y}=-k \dot{y}+m g-\frac{L_{0} i^{2}}{2 a(1+y / a)^{2}} \text { relate the position } y \text { of a steel ball of } mass m to a controlling current i in an electromagnet, with g, k, a and Lo being constants. It is desired to stabilize the position at an equilibrium position