Lagrange equations with friction with indirect velocity-dependency

I have a simple pendulum (a point-mass attached to a rigid massless rod) and want to model friction between the rod and the axis it’s attached to.

My assumption is that it can be modeled as surface friction between the rod and it’s axis with the normal-force being equal to the tension in the rod.
$$
T = m frac{v^2}{r} – mg
$$

T: tension in the rod
v: velocity in Cartesian coordinates
r: length of the rod
m: mass of the bob
g: gravity constant

The Rayleigh Dissipation Function allows you to model velocity-dependent friction:
$$
D = frac{1}{2} c v^2
$$

c: damping constant

So as you can see, the friction is indirectly velocity-dependent(it’s actually dependent on the centrifugal force), so how can I plug that into the Rayleigh Dissipation function? The following seems incorrect to me:
$$
D = frac{1}{2} c T^2
$$