Deriving relationship between LVDT and mass spring damper

[wu_weidong]

Homework Statement

[/B]
I have this system below.

It is the schematic of a linear accelerometer moving horizontally, where m is the total mass of the slide, b denotes the viscous damping, and k represents the spring constant. The relative position between the moving mass and the case is measured by a linear variable differential transformer (LVDT).

How do I derive the dynamic equation describing the relationship between V, the voltage output of the LVDT, and x₁ , the external position, and show that V indeed can be used to measure the external acceleration? I'm supposed to also state any assumptions made and any possible problems and remedies.

The Attempt at a Solution

I know that the mass spring damper system has the equation mx₁'' + bx₁' + kx = 0, and the moving core that is pulled by the mass spring damper system will induce a voltage in the LVDT. If I was given a transfer function for the LVDT, G, then I have V = Gx₁. But I don't know how to link the 2 concepts together to derive the relationship.

As for the assumptions, I would say the damping force is constant, i.e. the case is smooth. I'm wondering how the mass hitting the stopper will affect the system. Should it be made of rubber to absorb the impact and to minimize the shock wave?

 

[rude man]

If I was given a transfer function for the LVDT, G, then I have V = Gx₁. But I don't know how to link the 2 concepts together to derive the relationship.

?
You just stated the relationship: V = Gx₁.

As for the assumptions, I would say the damping force is constant, i.e. the case is smooth.

I hope so, otherwise you're in for it!

I'm wondering how the mass hitting the stopper will affect the system. Should it be made of rubber to absorb the impact and to minimize the shock wave?

If the mass hits the stop you no longer have an accelerometer. Not until it leaves the stop again.
With any such accelerometer you have to deal with the fact that x₁'' is not available; you get something like x₁''exp(-at)sin(ωt + φ). As you know if you solved for x₁ then took the 2nd time derivative.

More fun: is the LVDT transfer function really dynamics-free? I.e. is G really free from frequency dependence? Your LVDT is excited by an ac voltage, typically 400 Hz in aircraft or 60 Hz at home, so this alone limits the bandwidth. Then there's the question of what happens even if the excitation frequency is high enough to ignore it: if you apply a step inductance change by step-moving the core, will you get an immediate step response in V?