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 $x_1$ , 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.
I know that the mass spring damper system has the equation $mx_1'' + bx_1' + kx = 0$, 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 $V = Gx_1$. 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?