Transfer Function of a second order system

from what I recall...

Input torque $T(t)=K_tI(t)$

Frictional torque is $T_f=b_m~dtheta(t)/dt$ for viscosity b times rotational velocity dθ(t)/dt , or sometimes Ω(t) by Mech Eng. and acceleration d²θ(t)/dt with rot. inertia, J

$J~frac{d^2θ(t)}{dt^2} = T(t) − T_f$

Spring Torque is $T_s=K_s*frac{dtheta}{dt}$

Back EMF, Vb is proportional to RPM

$V_{emf}=K_bfrac{dtheta}{dt}$

and Current is the difference between applied voltage and Vemf into the motor coil resistance, R at DC

$I(t)=frac{V(t)-V_{emf}}{R_{dc}}$

hope that gets you started to get $G(t)=frac{theta}{I(t)}$ .... oh and gear ratio Kg=N1/N2 reduces velocity and increases torque.

Hint: current is proportional to acceleration