Find transfer function of a drive pulley where the output is a spring force

I have the following system

and have deduced the equations of motion to be

and with the substitutions for the spring forces we get

I understand how to find the transfer function of any single displacement/velocity/acceleration of either θ1 or θ2 with torque as the input, by taking the Laplace then setting up a Cramer's ruler matrix system. But in this instance I must find a transfer function of the spring force fs1 with respect to torque. I am very confused on how to have both θ1 and θ2 as part of the output. I have tried to rearrange the equations to eliminate theta terms, but end up just getting an endless loop where I repeatedly substituted things in and out.

Any help would be greatly appreciated. I even understand how to create a block diagram with this spring force as the output, but cannot figure out how to straight up create a transfer function.

Thank you!

• My current thought is that since I would ultimately need k(R1theta1(s)-R2theta2(s))/T(s). I would be able to first find theta1(s)/T(s), multiply that by K and R1, then find theta2(s)/T(s) multiply that by K and R2, then subtract the two terms to get k(R1theta1(s)-R2theta2(s))/T(s)? Commented Mar 13, 2023 at 6:09
• One way is to find the transfer functions $\frac{\theta_1}{\tau}$, $\frac{\theta_2}{\tau}$. Combine them in the ratio required to find the transfer function $\frac{(R_1\theta_1 - R_2\theta_2)}{\tau}$. This should then help in finding $\frac{F_1}{\tau}$; right ?
– AJN
Commented Mar 13, 2023 at 12:05
• Are you familiar with state space representation ? Please include your attempts at taking Laplace transform and setting up Cramer's matrices.
– AJN
Commented Mar 13, 2023 at 12:08