Equations of Motion of a Physical Pendulum I would like to formulate the equations of motion of a particular type of physical pendulum, shown in the attached figure.

The physical pendulum consists of a wheel rolling on a narrow board.  A pendulum arm grips the wheel and hangs down to a weight.  As the pendulum swings, the wheel rolls along the wooden surface.   
I am having trouble working the knowledge that the centre of rotation of the wheel is undergoing translation in the horizontal direction.   
System parameters:
- L, the  distance from the wheel axis to the pendulum's centre of mass, is known 
- the mass of the wheel, m, and the mass of the pendulum assembly, M, are known
Goal:
 - to find the coefficient of rolling friction (mu_rolling) between the wheel and the wooden board
 - to find the moment-of-inertia of the wheel (Iw) alone
Method:
- I would like to determine the equations of motion of the pendulum-wheel assembly in dynamical form using x, d(x)/dt, theta and d(theta)/dt as state variables.
Can anyone offer any help in building the equations?
 A: 
Let $x_1,\theta_1$ etc. with subscript 1 be coordinates of the upper mass, and variables with subscript 2 be coordinates of the second mass. So,the Lagrangian can be constructed as:
$$ L = 1/2M\dot x_1^2 + 1/2 \ I\dot \theta_1^2+ 1/2m\dot x_2 + 1/2m\dot y_2^2 - mgy_2$$
Now, $$ y_2 = L \ cos(\theta_2) \\ x_2 = x_1 + L \ sin(\theta_2)$$
I think that accounts for all relevant constraints. I think that leaves us with 3 generalised coordinates, and thus 3 equations, I think (sorry, my memory is a bit rusty on this topic). Substituting in and using Euler-Lagrange equation would give the right result, if there was no friction:
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot x_1} = \frac{\partial L}{\partial x_1} \ \ \ \ (1)$$
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot \theta_1} = \frac{\partial L}{\partial \theta_1} \ \ \ \ (2)$$
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot \theta_2} = \frac{\partial L}{\partial \theta_2} \ \ \ \ (3)$$
Now you just need to take the derivatives and that would give you your 3 equations of motion. So even without friction you get 3 coupled differential equations, so I would expect something very chaotic. 
 To account for friction you need to insert a dissipative function into Euler-Lagrange equation and that would depend on what kind of friction we are modelling. Or we could find some other clever way to reintroduce the friction afterwards as some sort of correction (after we have our equations of motion). Because this feels is kinda the same result that Newtonian approach would produce, but derived in a more straightforward fashion,  I was thinking to just stick friction as drag and torque terms into equation 1 and 2, but I am not sure if that is allowed (if someone, who has better understanding, can tell me whether or not I am allowed to do that I would really appreciate that). 
EDIT: Just realised that if you model this system with simple static, velocity independent friction there is no reason to actually treat the upper mass as wheel, it could as well be a point mass on a rail, rotational motion of upper mas plays no role on motion of the lower mass.
A: 
Consider the arm to be massless and of length $L$.
Two torques now act on the wheel:
$F_1L=mgL \sin \theta $
And:
$F_fR=\mu NR$
With:
$N=Mg+mg=(M+m)g$
So:
$F_f=\mu (M+m)gR$
Newton's second law (here on rotation) tells us:
$$I\alpha=-mgL\sin \theta+\mu (M+m)gR$$
Or:
$$I\frac{d^2\theta}{dt^2}+mgL\sin \theta=\mu (M+m)gR$$
If we accept the small angle approximation:
$$\sin \theta \approx \theta$$
$$I\frac{d^2\theta}{dt^2}+mgL\theta=\mu (M+m)gR$$
This the equation of motion of a simple harmonic oscillator and can of course not be the true equation of the system, as the rolling resistance definitely acts as a damper.
The problem is that $F_f$ as defined here doesn't even change direction.
To obtain something like the OP's second sketch, that is an Underdamped Oscillation, $F_f$ would have to be of the form:
$$F_f=-c\frac{d\theta}{dt}$$
