You have an interesting dilemna. 
 
First, let's clear some things up. It is not up to the property management company. They do not make the decisions. Your board of directors makes those decisions. It isn't the property management that doesn't want to do anything, but often that is the impression. Second, as a portfolio property manager for a firm with multiple communities, any legal questions would go to the association's legal counsel. The legal counsel is for the association - they do not work for the board or the management company - they work for the organization which by wnership you are a member of. However, if you have a dispute with the organization, you would need your own counsel to represent you. 
 
If you believe the sound is coming from a heating pump, and you had yours checked, it would leave me to believe without consulting your documents that the heating is a homeowner responsibility. The first step is to speak to your neighbor. Tell him your concerns (particularly about fire) and urge him/her to have the heating pump checked out. 
 
Second, get legal counsel and review your documents with them to see what the association's obligatinos in this matter are. Third, discuss with your board your concerns. Fourth, don't assume that it could make your health problems worse. See a doctor and confirm if it will or not. My best guess is that it is affecting you mentally and emotionally, and that will definitely affect you. You may also consider, during this time, getting a white noise machine for your bedroom, which may help when you sleep. I would also look at altermatives for your living room while working with your board and neighbor on any long range solutions.  
 
You can also request, as a member, the written opinion of the attorney. I don't know any attorney that would say what you have said. If it was an association responsibility, the association is required to take care of it in a reasonable time frame if they have the money or not. If the association does not have funds, they must assess for it if it needs to be done and they know about it - not "they don't have to do anything if they don't have the budget for it." By that logic, you might as well not budget for anything so you don't have to do anything.

In most situations I've dealt with the condominium lawyer responds only to request/direction from the board of directors. The lawyer should not respond to inquiries from unjit owners because that could result in exorbitant bills for services from the lawyer. 
 
Your problem is cleasrly a "maintenance" issue and such costs are included as a line item in your annual budget. If the cause/solution cannot be determined the board should direct the management company to provide/procure the services of an appropriate licensed engineering firm to provide a professional anaklysis of this difficulty.

I had sounds coming from the bedroom unit below me that drove me crazy, and I had trouble sleeping at night. It turned out to be their ceiling fan that they turned on at night. It was unbalanced and therefore thumped, thumped, thumped. My sister, while visiting overnight, recognized the sound because she once had a ceiling fan that did the same thing.  

Please better define your property. Hown many stories are in your building. You refer to 2 but are there more above you. Does each unit have their own jheat pump or does the multi-storied building have a large common unit which is connected to the individual units via ducts??

I just finished serving as an engineering expert witness for acoustic litigation regarding a multi-family building. Feel free to call me @ 208-699-0080 I may be able to offer you some assistance.

I am on the second floor of three floors and the units to the right of me have their bathroom fan vents running to the outside via my bedroom since my unit is at the end, so when they are running their bathroom fans, I get a vibration and noise.

I have had problems in apts where if the person on the third floor never changed their air filter, there was vibration all the way to the first floor. These are heat pumps also. What about a valve in the hot water heater also. I would check all of those, of neighbors above your unit. Tell your neighbors and ask them if they have had regular summer and winter check ups on the heating, cooling and hot water heater systems.Many owners think if its not broke don't fix it. But being a homeowner, condo owners requires checking many things in your home on a regular basis.

From the information you posted there is no doubt in my mind that the elimination of this noise is wholly the responsibility of your Condo association. Your Board should direct the management company to engage the professional engineering services who can then determi8ne the cause and recommend a fiox. How capable is you condo chief enginerr?? Or do you even have such???

I am having a similar low frequence rumbling noise in a new condo which I purchased just a little less than a year ago. The noise is getting more frequent - almost once every 2 minutes now. It sounds like a cruise ship leaving port. We called the builder and they sent someone out who wasn't able to figure out what the issue is.  
 
 
 
Thank you. 
 
 
 
I was wondering if the originator of this post was able to figure out what the issue was.

If you are still experiencing this problem please contact me 
thanks