How to work with transfer functions that are challenging to reduce to standard form?

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Mark Rzewnicki
Mark Rzewnicki on 23 Oct 2019
Edited: Mark Rzewnicki on 29 Oct 2019
I am trying to model a DC motor driving a load in Simulink by implementing a transfer function block.
I model the system as follows:
From the electrical perspective, I consider a simple equivalent series circuit consisting of input voltage , resistance , inductance , and motor back-emf .
Applying KVL around this loop, I obtain the equation .
From the mechanical perspective, I consider a motor torque , motor moment of inertia , damping term (due to bearings), and angular displacement . The motor is connected to a load with moment of inertia by a shaft of stiffness , resulting in displacement of the load by .
Applying Newton's second law to , I obtain the equation
Applying Newton's second law to , I obtain the equation
To couple the electrical and mechanical elements, I use 2 relations:
1. for some back-emf constant
2. for some motor torque constant
Eliminating and and solving for the transfer function , I obtain
To manipulate this transfer function into a form usable in Simulink (i.e., with unity leading coefficient) will be analytically tedious. Is there an easy workaround for cases like this that I am unaware of?
I know that I can create a state-space model by substituting the coupling relations directly into the differential equations. This is usally my preferred method, but I am hoping to expand my skillset by solving the problem a different way.
Edit: Figured it out! I'll leave the (really obvious) answer here in case someone searches for a similar issue.
s = tf('s')
can be used to define a transfer function as a rational expression.
Jm = 3E-6; %% kg.m^2
Bm = 3.5E-6; %% N.m.s
J1 = 5; %% kg.m^2
R = 4; %% ohms
L = 2.75E-6; %% Henries
K1 = 1; %% N.m/rad
K = 0.0275; %% V/rad/sec OR N.m/A
s = tf('s');
sys = K/((J1*s^2+K1)/K1*(Jm*s^2 + Bm*s + K1 + (K^2/(s*L+R)))-K1)

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