diesel heating system
Since you mentioned central air, I‚Äôm assuming forced-air heat? If so, there is a squirrel-cage fan inside the unit. Sounds like the fan or the motor (or both) are worn, so when the unit first starts up, it shakes, and the shaking causes the ducts to vibrate, and the floor to shudder. It goes away once the fan gains momentum and reached operating speed. Of course, this happens every time the heater kicks in. I would advise having the motor, capacitor, and fan inspected. Also, they should look for a broken mounting bolt or anything like that. Could also be a very dirty air filter, or malfunctioned whole-house humidifier. Also, a loose or dirty oil injector nozzle could cause the flame to be unbalanced. Since your tank is buried, when was the last time it was checked for water in it? Water in the oil causes the flame to sputter, which may cause the vibration.
All these things could cause the problem, or none. Now I don‚Äôt know where you buy your oil from (you call it diesel, we call it heating oil), but most suppliers also have service contracts, and repair people who know these units, and can help you out.
And now, a word about heat pumps. Our last house had one, and we lived there for over 7 years. It was only a little over a year old when we moved in. We live just north of Philadelphia, so we are in a colder region than you. But all we did in winter was freeze, and fix, that heat pump. From January to April it ran almost continuously. Our electric bill often was over $400, and we kept the home at 67 degrees!!! As it turns out, heat pumps have a shorter service live than a regular heater, by less than half. Of course, you are in a warmer clement in Alabama,, and the technology has improved, but I can only say what I have experienced.
And now, we have oil, in a 45-year-old home, like yours; 550 gallon drum in the yard, etc., except our heater is only a hear old, put in just before we bought the house. This is only the second heater the home has had. We love the heat, absolutely. I really wanted to change to gas originally. Heat pumps were being pushed to us by various experts, mostly due to subsidies offered by the electric company, but you can guess our opinion on that. Anyway, all that was before we found out the heater was new, so we decided to leave it and see how everything worked out. So far the service of the unit has been spectacular, as is our supplier, whom we have a maintenance contract with. We are putting in a pool as we speak, and I‚Äôm opting to get an oil heater for it, instead of the propane one supplied by the pool installer.
This is just one family‚Äôs opinion, but I thought you would find it interesting to know. I would highly suggest that you have a professional oil heat contractor go through your unit. Maybe it‚Äôs only a simple repair. But if replacement is warranted, then I would still strongly consider another oil heater.
One last question: Are you actually using diesel fuel?
Good luck -