Use EPOXY.
Epoxy provides the best strength for a Wood or Cardboard to Plastic bond.
"Rough Up" both surfaces to be bonded with 120 grit sandpaper. The Epoxy " bites" better that way. Short of Epoxy, I would use Medium CA glue. Either are far better than wood glue or plastic cement for bonding dis-similar materials.

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When in doubt, WHACK the GAS and DITCH the brake !!!

Yes, there is such a thing as NORMAL, if you have to ask what is "NORMAL" , you probably aren't !
Failure may not be an OPTION, but it is ALWAYS a POSSIBILITY.
ALL systems are GO for MAYHEM, CHAOS, TURMOIL, FIASCOS, and HAVOC !