How does piezoeletric accelerometer sensors work technically? I know that piezoeletric accelerometer sensors have a piezoeletric crystal, a mass and a spring.
But where is the voltage at the ouput coming from?
What makes a crystal a piezoelectric crystal is that it produces a voltage when deformed. This is the defining characteristic of a piezo electric crystal.
Are you sure you're asking about accelerometers and not gyroscopes? Gyros need a piezo because they need a resonator but I don't know why an accelerometer would need a resonator. Almost every accelerometer from Analog Devices has this paragraph under "Theory of Operation" which does not mention any piezo crystals.
From ADXL1005 datasheet (an accelerometer)
Typical piezoelectric accelerometers do not use a resonator.
But there is research regarding accelerometers using differential resonators. This is described in the paper High Frequency FM MEMS Accelerometer Using Piezoresistive Resonators:
The accelerometer uses two differential resonators, connected to the accelerometer proofmass by an amplifying leverage mechanism. The piezoresistive double-mass resonators are electrostatically driven in anti-phase and the output signal is measured piezoresistively by applying a bias current to the connecting microbeam of the double-mass resonators.
The mass plates vibrate in-plane and in opposite directions (anti-phase) and they are electrostatically driven at resonance. When the proof mass is subjected to an external acceleration in the sensitive axis (Figure 1a), the springs sustaining the resonator are axially loaded, changing their acceleration moves the proof mass, the springs of one resonator are under compression while the other resonator springs are under tension. The tensile/compressive axial loading shifts the resonance frequency, the tensile force increasing it and the compressive force decreasing it . This shift is proportional to and a measure of the external acceleration. The differential design scheme also allows to cancel device thermal mismatches and nonlinearities. [...]
The output signal is measured piezoresistively, using the modulation of the DC bias current in the vibrating connection beam.
So in this acceleromter they do not use a piezoelectric sensor, but a piezoresisitve one. This is the prinziple from strain gauge strips. The length of the piezoresistive material is changed and by that the resistance is changes. If you apply an external voltage, this voltage is modulated with the resonance frequency, but there is no voltage generated inside the sensor.
I'm not sure if you really meant that, but that is the only accelerometer with resonators I know of.
A piezoelectric material converts between stress/strain and voltage, that's the defining feature of it.
An accelerometer can be made with a piezoelectric spring, anchored to the case, with a test mass on the end. When the case is accelerated, the test mass has to be accelerated to match, this acceleration force being provided through the spring. The strain in spring produces the output voltage through piezo action.
The output from this type of accelerometer is very low. Much more sensitive ones use capacitive sensing to measure the deflection of a silicon spring.
If you're concerned about how straining a piezo crystal produces a voltage ... Consider that a crystal is built from repeating units that can have some assymetry of charge on them. When crystals are bent, some types (piezo types) produce a net shift of charges within the structure.