Triggering a relay for a set period of time when voltage drops below a certain voltage
Here you go:







simulate this circuit &ndash; Schematic created using CircuitLab

Notes:


You need to supply an appropriate voltage to the 555 timer and comparator. I put 5V as a placeholder but there are other options available.
Vref can be the +5V for the 555, or if you need more precision, use a reference IC. You cannot use the supply voltage itself, so if there is nothing else, use a zener and resistor.
Vcoil depends on your relay coil.
Calculate R1, C1 like this
I put in a pot so you can adjust the trigger like you said. Just turn it so that the voltage on the - input of the comparator is the trigger voltage/4.
I selected R4, R5, C3 for a couple ms response time. Adjust as needed.
Depending on your application, you will likely want a comparator with some hysteresis.
Many comparators have open drain outputs. Add a pull up resistor if this is the case.


Theory:


Block 1: A reference voltage, adjustable by potentiometer R3 provides a comparator with a trigger point. The voltage to be monitored is scaled by R4 and R5 to fall within the range of the comparator and easy to produce reference voltages. C3 and C4 filter the voltages. When the monitor falls below the trigger, the comparator output changes from 1 to 0.
Block 2: The 555 is triggered by the falling edge on the trigger pin. The 555 is configured as a monostable multivibrator. It will provide a pulse for the desired amount of time.
Block 3: Standard relay driver. R2 determines the base current of Q1 which operates either in saturation or cutoff to drive the relay coil. Select appropriately for required coil current. D1 protects Q1 from transient voltages during coil de-energization.