Coupled Diff Eq → Single Diff Eq

Methods for transforming from coupled differential equations to a single n^th order differential equation were discussed on the page "System Representation by Differential Equations," example 3 and example 4. Another example is included below. It shows how to start with a set of coupled differential equations and transform them into a single n^th order differential equation.

Example: Deriving a single n^th order differential equation from coupled equations

Consider the system shown with f_a(t) as input and x(t) as output. Find the differential equation relating x(t) to f_a(t).

We can write free body equations for the system at x and at y.

Freebody Diagram	Equation

Take the Laplace Transform of both equations with zero initial conditions (so derivatives in time are replaced by multiplications by "s" in the Laplace domain).

Now solve the second equation for Y(s) and substitute into the first equation and clear the fractions (so there are only positive powers of s).

Convert back to differential equation (replacing "s" in Laplace by a derivative in time).

Matlab Solution

Your browser does not support inline frames or is currently configured not to display inline frames.
Matlab code, html

Transformation: Coupled Diff Eq → Single Diff Eq

Example: Deriving a single nth order differential equation from coupled equations

Matlab Solution

Example: Deriving a single n^th order differential equation from coupled equations