Taper ratio for partial space elevator

The calculations for the taper ratio of a space elevator don't seem to be all that difficult and it is easy to find some equations to use. You can even find an expression written out for the total mass (which involves more complicated mathematical functions).

But if we have an orbiting, tidally locked, partial space elevator what would the equations for those be? Searching around, I can't find any material on this. Presumably it would be slightly more complicated than a full geosynchronous space elevator because you have an additional independent variable. Does anyone have either an approach to this problem with some of the basic math hashed out, or just a straightforward equation for taper ratio?

$$U(r) = - \frac{ GM}{r } - \frac{1}{2} \omega^2 r^2 \\ \frac{ GM}{r_0^2} = \omega^2 r_0 \\ U(r) = - \frac{ GM}{r } - \frac{1}{2} \frac{ GM}{r_0^3 } r^2 \\ \frac{ \lambda(r_{0}) }{ \lambda(r) } = \exp{ \left( \frac{ U(r_{0}) - U(r) } { \left( \frac{ \sigma}{ \rho} \right) } \right) }$$