Is My DC Motor Differential Equation Setup Correct?

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Scott77
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Block diagram of a DC motor driving a gear train to a load with inertia JL and damping DL, with the motor side labelled Ja, Da and the gear ratio N1/N2, followed by the printed problem statement and equations for Jm, Dm, and the motor relations.


This question involves finding the transfer function for the system, but I first need to get the differential equations correct. Have I set up the gearbox correctly?

Three sets of equations for a DC motor with gearbox: electrical part e_a(t) = R_a i_a(t) + K_b dθ_m/dt; mechanical load T_m(t) - D_a + (N1/N2)^2 D_L dθ_m/dt = J_a + (N1/N2)^2 J_L d²θ_m/dt²; and T_m = K_a i_a(t).
 

Discussion

Scott77 said:
This question involves finding the transfer function for the system, but I first need to get the differential equations correct.

Well, the transfer function is a differential equation. Using Laplace transform it may be written:

H(s) = output(s)/input(s)

When expressing the transfer function by its Laplace transform, it becomes much easier to calculate at controller.

I have some questions:

- Is it a DC-motor ?
- What is the input: Current or voltage?
- What is the preferred output: Speed or position?
- Is the armature damped ( Da )?
- What about the self induction in the armature, La?

If it's a DC-motor, you should come up with something like this, as for the transfer function:

Block diagram of a DC motor feedback loop: Va(s) enters a summing junction with plus and minus signs, then 1/(Ra + sLa) giving Ia(s), then Km producing Me, minus ML load torque summed, then 1/(sJ + B) giving Ω(s), with Km feedback forming Eb(s). Fig. 5

Use Masons rule to determine the transfer function as for the motor: ω(s) / Va(s)
 
This is a system modeling question. That diagram and the hints is all the info we got, but we were told to ignore the armature inductance. I don't think it matters what the input type and output type are for this type of question...
 
Of course the input and output type matters. If you change the type of input or output, the motors behaviour will change.

In the model, choosing ω(s) as output, the transfer function will be H(s). Choosing θ(s) as output, the transfer function will be H(s)/s:
θ(s) = ω(s)/s.
 
The input is ea(t) and the output is θL(t) like the question states, that is all I know.
 
Ok, then use these types as input/output.

Setting La = 0, you will get a 2. order transfer function.
 
Ka and Kb have the same value, but the units are [Nm/A] and [Vs] respectively.

There are people that can prove it.