Perhaps I can help a bit.
You are measuring acceleration.
Acceleration is rate of change of speed.
Inches per second (for example) is speed.
Inches per second per second would be acceleration.
Vibration is a position measurement (position vs. time).
If you integrate the accelerometer signal once, you would change IPSPS (accel) into IPS (Speed).
If you integrate the result again, you would change IPS (speed) into Inches (position).
THAT is the vibration signal you should analyze with FFTs, etc.
You will probably have to remove the DC offset from your signal before EACH integration. Otherwise the integration process generates a ramp function which is not real.