The modeling equation gathered from the free body diagram is in the time domain. Some analysis are easier to perform in the frequency domain. In order to convert to the frequency domain, apply the Laplace Transform to determine the transfer function of the system.

The Laplace Transform converts linear differential equations into algebraic expressions which are easier to manipulate. The Laplace Transform converts functions with a real dependent variable (such as time) into functions with a complex dependent variable (such as frequency, often represented by s).

The transfer function is the ratio of the output Laplace Transform to the input Laplace Transform assuming zero initial conditions. Many important characteristics of dynamic or control systems can be determined from the transfer function.

The general procedure to find the transfer function of a linear differential equation from input to output is to take the Laplace Transforms of both sides assuming zero conditions, and to solve for the ratio of the output Laplace over the input Laplace.