What is a fsecond order system/Process? and Second system transfer function

A second-order system is one whose output, y(t), is described by the solution of a second-order differential equation. For example, the following equation describes a second-order system:

If a0 # 0, then eqn. yields

where

Equation  is in the standard form of a second-order system where

T = the natural period of oscillation of the system
ζ = the damping factor, and
KP = the steady state, or static, or simply gain of the system.

The physical meaning of the parameters r and 5 will become clear in the next two sections, while KP as the same significance as for the first-order system.

If eqn. is in terms of deviation variables, the initial conditions are zero and the Laplace transformation of eqn.  yields the following standard transfer function for a second-order system;