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I am trying to construct an equation that describes the output torque as a function of lambda and omega I have obtained the following bond graph but I am unsure of how to proceed. enter image description here

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It might help to organize the equations a little in steps. I think you have all this above:

Output torque of system:

$\tau_F$

Output torque at motor:

$\tau_0 = \tau_F/N$

Motor output speed:

$\omega_a = N\omega_M$

Motor torque speed relationship$^{*}$:

$\displaystyle{\frac{K_m}{R}}V_{in} = \frac{L}{R}\displaystyle{\frac{d\tau_0}{dt}} + \displaystyle{\frac{K_m^2}{R}}\omega_a$

I think it is also correct to state that:

$\lambda = LI_a$

This doesn't quite answer your question, but maybe gives you some new ideas.


$^{*}$ See, e.g., Harry Asada, Introduction to Robotics, Chapter 2

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