How to use an accelerometer to correctly quantify the magnitude of vibrations Consider the following setup:
An accelerometer is placed inside a helicopter, measuring $g$ values along three axes $(x,y,z)$ every $10~\text{ms}$ (100 times a second), during a helicopter flight.
If one wants to obtain some measure of the maximum $g$ force exerted on an object inside the helicopter, what would be the best way to do this?
Make the analysis for each axis independently? Or treat $x,y,z$ as components of a vector ?
Is it correct to consider the maximum difference between two consecutive measurements as an approximation of the maximum $g$ force exerted on an object inside the helicopter
 A: It depends a little bit on how you define "g force".
An accelerometer will measure acceleration as force divided by mass, $\vec F/m$. I have no idea how the accelerometer in your smartphone works but in principle you can visualize it as some test mass $m$ mounted on springs along the three axes. For an object at rest or in uniform motion, the acceleration measured will be $-g\hat z$ where $g\approx 9.8\mathrm m/\mathrm s^2$ is the gravitational acceleration and $\hat z$ is the unit vector pointing "up".
When talking about g-factors, one usually thinks about the kinematic acceleration, i.e. $\ddot{\vec x}(t)$, the second derivative w.r.t. time of the trajectory $\vec x(t)$. For example, when saying something like "when the car accelerated, we've been pushed in our seats with 1g", you mean that the car accelerated with $g$ in a horizontal direction. However, inside the accelerating car, the passenger is pushed into the seat with a force of $m\sqrt{2}g$ at a $45^\circ$ angle. In the car, the kinematic acceleration of the car will appear as an opposite fictitious force in the rest frame of the car, i.e. to the passenger.
Thus, the g-factor which refers only to the kinematic acceleration of the accelerometer can be defined as
$\frac{\ddot{\vec x}(t)}{g} = \frac{\|\vec F/m+g\hat z\|}{g}$.
This is also a plausible definition if you are sitting in a helicopter where the kinematic acceleration may also be in the horizontal direction. Note, however that some accelerometers (at least the one on my phone) will do this calibration automatically for you.
A: The maximum acceleration experienced by your accelerometer (assuming that it does not automatically zero out the acceleration due to gravity) will be the same as the maximum acceleration experienced by any other object in the helicopter, as long as the helicopter attitude is maintained (no rotation).
You would simply take the instantaneous acceleration along the three axes, and compute the length of the composite vector:
$$g_{eff}=\sqrt{g_x^2 + g_y^2 + g_z^2}$$
No need to take differences - you are already working in acceleration space. If you had the velocities instead, you would have to differentiate to get acceleration.
Note also that if you do have a rotation, you would need to take into account that ("g") forces on objects in that case could be lower or higher than the value obtained with the above. For steady rotation, you need to add a radial component to the acceleration ("centrifugal acceleration") $\omega^2r$, and for a change in rotation, there will be a tangential component $\dot\omega r$. You would have to compute each of these and add them (again, vector sum) to whatever your accelerometer gave you.
