Can I use a chain to suspend the motor and not be in violation of the patent [reciting a strap]
It's impossible to tell without reading the patent specification and prosecution history. The content of the filings is generally more controlling than a dictionary definition.
Ideally, the examiner has cited prior art against the applicant, and the applicant has argued how the invention distinguishes over the prior art in the prosecution history. The statements made in the prosecution history might be sufficient to settle the question of whether a "strap" does or does not include a chain.
Or would I be in violation of Doctrine of equivalents?
Again, a question that could only reasonably be answered by looking into the prosecution history, and getting an idea of what the applicant has argued involving the strap. If the strap doesn't relate to the crux of the invention, I'd say it's likely that the doctrine of equivalents applies. If, on the other hand, crucial properties of a strap were relied on to argue un-obviousness, then probably not.
All the above assumes we're talking about U.S. patent law, and is purely academic discussion. You should consult a patent attorney for real legal advice.