As with the first order system, there is a general differential equation that governs the response of a second order system.  The equation is of the form:

Where:

So how is the second order differential equation for the spring mass damper generated?

Well, if an input is applied to the mass, the equation of motion for the system can be written as:

The general equation for a second order system can be manipulated to give the equation above and vice versa.

Performing the numerical integration for the second order system

The second order equation order is dealt with by rewriting the equation as pair of first order equations.  The numerical integration can be performed in a similar way to the first order system where:

Of course as all the hardware information is available or selected by the engineer, the value for (d²y/dz²)  can be calculated by manipulating the equation of motion.