A homogeneous disk of mass m and outer radius R rolls without slipping on an inclined surface, with the center of the disk supported by two springs, as shown below.

Say we let *x* represent the displacement of the center O from where the springs are unstretched. The dynamical equation of motion (EOM) in terms of the coordinate *x* is shown below. As we can see, the resulting EOM is inhomogeneous.

From this EOM, we also see that the static equilibrium position of O is given by: *x*_{static} = *mg *sin(*theta*)/3*k*. That is, if the disk were placed at a position of *x* = *x*_{static}, the disk remains there in equilibrium.

Suppose that we choose a new coordinate of *z* = *x* - *x*_{static} , which represents the motion of O relative to the static equilibrium position. The EOM in terms of his new coordinate is shown above to be homogenous.

**Physical interpretation**

Shown below is an animation from a simulation of the free response of the system with initial conditions of *x*(0) = *x*_dot(0) = 0. The red curve shown below is the response of the system, *x*(t). Note that the response is harmonic with a non-zero mean value, as expected. From the above analysis, we know that the frequency of response is given by *omega*_{n} = sqrt(2*k/m*). Relevant to our discussion above is that the oscillations occur about the static equilibrium position of the disk (shown in grey). The motion about that position is *z*(*t*), which has a zero mean value.