How can I find transfer function of the given servomechanism system with input $V$(voltage) and output $θ_L$(angle of the load). Schematic of the system is given below.
Schematic of the servomechanism system
There are two differential equations of the given system. $$\dot{\omega}_L=-\dfrac{k_T}{J_L}\left[θ_L-\dfrac{1}{\rho}θ_M\right]-\dfrac{β_L}{J_L}\omega_L$$ $$\dot{\omega}_M=\dfrac{k_M}{RJ_M}\left[V-k_M\omega_M\right]-\dfrac{β_M}{J_M}\omega_M+\dfrac{k_T}{ρJ_M}\left[θ_L-\dfrac{1}{\rho}θ_M\right] $$
$k_T, J_L, ρ, k_M, J_M, β_L, β_M $ and $R$ are positive constants. In this case, $$F(s)=\dfrac{θ_L(s)}{V(s)}$$ has to be found.
First note that $\dot{\theta}_L=\omega_L$ and $\dot{\theta}_M=\omega_M$, then we obtain:
$$\ddot{\theta}_L=-\dfrac{k_T}{J_L}\left[θ_L-\dfrac{1}{\rho}θ_M\right]-\dfrac{β_L}{J_L}\dot{\theta}_L$$ $$\ddot{\theta}_M=\dfrac{k_M}{RJ_M}\left[V-k_M\dot{\theta}_M\right]-\dfrac{β_M}{J_M}\dot{\theta}_M+\dfrac{k_T}{ρJ_M}\left[θ_L-\dfrac{1}{\rho}θ_M\right] $$
The Laplace transform of the first equation results in
$$\theta_L=\dfrac{\dfrac{k_T}{\rho J_L}}{s^2+\dfrac{\beta_L}{J_L}s+\dfrac{k_T}{J_L}}\theta_M=\dfrac{k_T}{\rho J_Ls^3+\rho\beta_Ls+\rho k_T}\theta_M$$
or
$$\theta_M=\dfrac{\rho J_Ls^3+\rho\beta_Ls+\rho k_T}{k_T}\theta_L.\qquad \qquad (1)$$
The Laplace transform of the second equation is given by
$$\left[s^2+\left(\dfrac{k^2_M}{RJ_M}+\dfrac{\beta_M}{J_M} \right)s+\dfrac{k_T}{\rho^2J_M}\right]\theta_M = \dfrac{k_T}{\rho J_M}\theta_L+\dfrac{k_M}{RJ_M}V \qquad (2)$$
Now, we $(1)$ into $(2)$:
$$\left[s^2+\left(\dfrac{k^2_M}{RJ_M}+\dfrac{\beta_M}{J_M} \right)s+\dfrac{k_T}{\rho^2J_M}\right]\dfrac{\rho J_Ls^3+\rho\beta_Ls+\rho k_T}{k_T}\theta_L = \dfrac{k_T}{\rho J_M}\theta_L+\dfrac{k_M}{RJ_M}V.$$
Can you complete it from here?