Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass's acceleration. This interplay of forces is mathematically described using Newton's second law, which states that the sum of all forces acting on the mass must be zero.

In translational mechanical systems, the behavior is captured by a unique differential equation derived from Newton's law. This equation accounts for all the forces acting on the mass. To solve the system analytically, the Laplace transform is applied to this differential equation under zero initial conditions. The Laplace transform, a powerful mathematical tool, converts the time-domain differential equation into an algebraic equation in the Laplace domain. Simplifying this equation yields the system's transfer function, a crucial concept that relates the output response to the input force in the frequency domain. The transfer function is essential for analyzing system stability and dynamics.

Rotational mechanical systems parallel translational systems but involve rotational motion. In these systems, torque replaces force, angular displacement substitutes for translational displacement, and rotational inertia takes the place of mass. The analogous differential equation for a rotational system, derived similarly using Newton's second law for rotation, describes the dynamics of rotational motion. By applying the Laplace transform to this second-order differential equation, and simplifying, the transfer function for the rotational system is obtained. This function provides insights into the rotational system's behavior, similar to how the transfer function in translational systems aids in understanding linear motion dynamics.