Here are my observations on this circuit:
1) the sensor manufacturer, in their data sheet, shows the use of a variable threshold envelope detector after the op amp. Your circuit does not have this feature. I think you should add this in. It would be easy to add by placing a potentiometer between the opamp output (junction with R2) and the following diode.
2) if this sensor amp is recommended by the sensor maker, then pay no attention to this comment, but I'm uncomfortable with the lack of anything at the input to establish a DC level, to drain off charge and insure the input is at 0 V DC bias. It might be harmless to place a 10Mohm resistor from op amp pin 3 to ground to allay my fears.
3) The value of R4 is quite large and may be too large to provide sufficient base current for Q2. This may be why Q2 isn't turning on. You can fix this easily by raising the value of R6 which may be too low.
The way I see this working is...when the pan shakes, you get AC out of the op amp, which is turned into DC at C1 which turns on Q1. Q2 should then be held off since its base voltage is below turn-on threshold. The 555 monostable needs a falling voltage for the trigger input so it hasn't triggered yet. Q3 remains off also.
I presume this circuit is supposed to sound the alert when the machine stops moving, or in other words, when you get no output from the sensor. I would think that the sensor will continue to give some low level output even after you think the machine isn't vibrating any more, simply because it is sensitive to very minute vibrations. That's why you should have a potentiometer before the diode detector, so you can adjust for this.
Anyway, so the 555 output should remain high until the vibration decreases, right? Should this happen, then Q1 turns off, and Q2 turns on. When Q2 turns on, we get the edge to trigger the 555 and it begins trying to charge C3. What puzzles me is that you have Q3 there to short out C3 and Q3 will turn on when Q2 turns on, so the monostable output pulse is held on until 10 seconds after the motion is re-established. Is that the way its supposed to work?