I am currently designing a motorized zipline trolley that will carry a person on a horizontal cable, so I am trying to calculate the friction on the wheels to determine what kind of motors I need. I initially approached the problem using a static friction model using the coefficient of static friction between two steels. However, my professor told me to replace my model with a rolling friction model, so I took the coefficient of rolling friction between two steels. This greatly reduced the calculated value of friction; however, I feel like there should be more friction because the weight of the person will cause deflection in the cable, so a portion of the wheels' circumference will be in contact with the cable. I want to assume no slipping for the wheels, so I don't know how to account for the static friction while keeping the rolling friction model. How can I calculate the additional frictional force that I am not modeling?
There are a two items I think you are wondering about, the effect of static friction and the extended length of the contact area between the rope and the pulley.
Start up torque on a motor can be greater than the design torque of the motor, so your professor may be suggesting to ignore the rolling friction because of that. I would say it's a valid assumption to assume the motor can overcome the static friction if it's overcoming the dynamic friction. Check out motor-torque curves on engineering toolbox for example. I can't remember the complete explanation and terms for this, but when you start a motor there is not a strong magnetic field produced by the rotor, because it is not moving yet, and so the magnetic field that is being produced by the AC current in the stator gets stronger and produces more torque than when the rotor is spinning at its nominal rating. That's why you have high inrush current when starting a motor, no matter what (assuming you have no starter controls on it).
Secondly, the contact area of the rope on the pulley is irrelevant. You have assumed no slip for starters, so the total frictional force between the rope and the pulley that would be altered by contact length is not going to change. Draw a free body diagram of the pulley. You will have to have tension in the rope between the two pulleys, and you can theoretically make that tension to whatever you want. The tension in the rope (due to the tension you set as well as the mass of the trolley load) will act as the torque on the pulley of the motor. That is the torque the motor has to overcome.
There will be some deflection in the rope due to the load, which you can solve for. Then it's just the added tension in the rope from the load that the motor has to account for since the rope tension is always perpendicular to the pulley.
Break that out into components to and balance then solve for the new tension. I'm on my phone right now and can't enter the equation. But I will try once I have access.