Is this a problem... just a very short chirp... I removed panel cover and could not find any charring or loose connections

Have you removed the breaker and looked at the buss? If all is clean and OK, try a new breaker. Are you using the right size breaker?

I've found it's normal for a breaker to make a sound when a high current appliance engages. If it's not normal, then all the breakers at my church, my house, and my parents house need to be replaced. And some of them are new.

It's funny, we got a new central A/C yesterday and the same problem appeared today: a chirping sound that seemed to come from the electric panel (that's how I came to this site). It turned it out to be the smoke detector... the battery had to be changed.