Transfer Functions (General Question)

[Burner]

Homework Statement

Just a generic question. If you have Laplace-transformed differential equation like s³Y + sY = g/s + X, where g is a constant, how do you handle manipulating the equation to get the transfer function Y/X? I feel like I've done this before, but I'm having a severe mental block.

2. The attempt at a solution

I've had some thoughts, but can't get anywhere with them. The constant g has nothing to do with X, so any attempts at factoring have been unsuccessful. I thought about some kind of superposition, but am not sure how that would work. Any guidance or hints would be greatly appreciated. Thanks!

 

[kentigens]

try move the g/s over to the left side of the equation. s3Y + sY - g*s-1 = X. Then do the transformation(use the table).

 

[Burner]

Thank you, but let me try to clarify a bit.

I don't require getting it back into the time domain. I'm just trying to find the transfer function with an output Y for an input X (in the s domain). To illustrate the problem I am encountering, simplifying you'll get Y(s⁴ + s²) = g + sX. The problem is that I need the quantity Y/X in terms of only g and s, and dividing both sides by X doesn't quite get the job done.

My thought is that there might be a partial fraction expansion or something I am missing.

 

[Burner]

Correct me if I'm wrong, but after a bit more reading and thinking: For the purposes of a useful transfer function, the constant term can be dropped because it is representative of an initial condition, right?

 

[DeeNos]

As a scientist, it is important to have a clear understanding of transfer functions and how to manipulate them in order to solve differential equations. In this case, the first step would be to take the Laplace transform of the given equation, which would result in an algebraic equation in terms of the Laplace transform variables. Then, by rearranging the equation, one can isolate the transfer function Y/X and solve for it.

One approach could be to divide both sides of the equation by X and then rearrange the terms to have Y/X on one side. Another approach could be to use partial fractions to break up the equation into simpler forms and then solve for Y/X.

It is also important to keep in mind that the constant g may not have a direct influence on X, but it could still affect the overall transfer function. Therefore, it is important to consider all variables and constants when manipulating the equation.

In terms of superposition, it could be useful to break up the equation into smaller parts and solve for the transfer function for each part separately. Then, the overall transfer function can be found by combining the individual transfer functions using superposition.

Overall, the key is to understand the properties and techniques of manipulating transfer functions and to approach the problem systematically. It may also be helpful to review previous examples or seek guidance from a peer or instructor.