# Transfer functions when the input is not a factor of every component of the equation

1. Jul 14, 2009

### majin_andrew

(edit: sorry I made a mistake in the thread title. It's the output that isnt a factor of every component in the equation, that is giving me trouble)

Hi! I'm new here, and any help would be greatly appreciated.

1. The problem statement, all variables and given/known data
I am modelling a mass-spring-damper system in which the damping is coulomb damping, as in the damping force is constant (once the system is in motion) and independent of the velocity of the mass. There is an external force (f(t)) acting on the mass.

I think the differential equation describing the motion of the mass is: f(t) = mx'' + kx + c
where x is the displacement of the mass in the direction of f(t), m is the mass, k is the spring constant and c is the constant damping force.

I have found a Laplace transform of this function, which is F(s) = (ms^2+k)X(s) + c/s

I am required to write a transform function for this, in the form of T(s) = output / input.

I am having difficulty with this as my output (X(s)) is not a factor of all components of the right hand side of the Laplace transform, so I cannot eliminate the input (F(s)) or the output (X(s)) from this ratio.

Can someone please offer me some guidance?

2. Relevant equations

Differential equation describing motion: f(t) = mx'' + kx + c

Laplace transform: F(s) = (ms^2+k)X(s) + c/s

Transform function = T(s) = output / input = X(s) / F(s)

3. The attempt at a solution
My attempt has so far been the creation of the differential equation and the Laplace transform, I am stuck with the transfer function.

here goes:
T(s) = X(s) / F(s)
= 1 / [ (ms^2+k) + c/(sX(s)) ]

As you can see, I still have transfer function written in terms of the output.

Thanks a lot for your time!
Andrew

2. Jul 14, 2009

### turin

Re: Transfer functions when the input is not a factor of every component of the equat

What you've got there is a nonlinear system. The transfer function that you describe applies to linear systems.

Also, I don't think that you have the coulomb damping part quite correct, assuming that I get your meaning. I think that you should replace c -> - c sgn(x'). Unfortunately, that only confounds the nonlinearity.

Look into Green Functions (but I think that those are restricted to linear differential operators as well).

OK, I thought about this a bit, and I think that you have two options, but neither one of them is going to exactly achieve your original goal. In both options, the motion, x', must have a constant direction.

1) You can simply subtract c from both sides, and find the "adjusted" transfer function: Fadj(s)=X(s)/(F(s)-c/s). This is basically a nonlinear version of the transfer function in that different input frequencies must be adjusted by -c/s. Of course, this seems rather obvious.

2) You can discretize the system, and then extend the discretized differential operator to an affine transformation. I will explain further if this approach is acceptable.

The main problem with these options is that I don't know how you would garuntee that the direction of x' is constant.

Last edited: Jul 14, 2009
3. Jul 14, 2009

### majin_andrew

Re: Transfer functions when the input is not a factor of every component of the equat

Thanks a lot Turin!
I believe your first option will suffice for what I am trying to achieve. And yes, I thought about the direction of c depending on the direction of x' but I wasnt sure of the notation for it.

Thanks again!!!