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Sir Brig . . . . .
You just neglected to use the specified flashing LED(s) which is constructed with its mere fly speck size flashing electronics aspect being included inside of the same plastic domed LED case...
While you have that conventional LED installed, you might give the unit a shake to start the LED and then lay aside to see how long the FET needs to time out.
Which is what would happen, if the kiddos have tired out . . . .or ADD'd out . . . . of playing with the unit, and then it just shuts itself off after the charge on C1 has been bled down by R1 until the voltage hold on threshold of Q1 gate has occurred.
Might as well check the specs and get the very BRIGHTEST flashing LED that you can find . . . for an optimal enhanced effect !
Some LED's these days are downright blinding !
The rest of the story . . . . .
Way back in '68 . . . . . . some 45 years ago . . . .I made a bipolar Sziklai / Darlington combo *** version of this auto time out circuit. . . . . . . . . .*** ( Since Power FET's weren't exactly " in " back in those times. )
It was what I used on my Hexacon soldering station to stop the oxidizing and fouling up of its quite pricey soldering tips.
A Push Button start up switch was being equivalent to your "shock / jar / motion" detector and a power relays coil was your " LED".
Another spring lever Microswitch, in parallel with the PB switch, was mechanically linked into a two part , side by side, metal clam shell contrivance, that surrounded the soldering iron when you placed it into its small holder channel.
Sooooooo . . . . . . . if I was using the iron, and kept it in movement between work and the holder , it continually re timed the circuit.
Then, when I left it dormant, it just timed out and shut down my station.
No more tip problems.
Thasssit . . . .
73's de Edd
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