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I have the following system

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and have deduced the equations of motion to be

enter image description here

and with the substitutions for the spring forces we get

enter image description here

enter image description here

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!

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  • $\begingroup$ 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)? $\endgroup$
    – Sonite
    Commented Mar 13, 2023 at 6:09
  • $\begingroup$ 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 ? $\endgroup$
    – AJN
    Commented Mar 13, 2023 at 12:05
  • $\begingroup$ Are you familiar with state space representation ? Please include your attempts at taking Laplace transform and setting up Cramer's matrices. $\endgroup$
    – AJN
    Commented Mar 13, 2023 at 12:08

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