First, "ringing" and diaphragm tension are not related, unless you used the term to describe something other than a component causing unwanted resonances.
The most common form of resonance, or ringing, in microphones is called microphonics: a sound pressure wave of a certain intensity* and frequency mechanically resonates a component inside the mic. That resonance is then picked up and amplified by the mic's electronics, and superimposes itself over the music being recorded. The result is an audible non-linearity in the frequency response with unwanted peaks that stick out and ring on.
Another form of ringing would be an abnormality (non-linearity) of the mic's frequency response, due to an electrical or capsule defect. It's not strictly "ringing" defined as a resonance lasting longer than the signal it triggered.
I will address mechanical ringing, as the latter is not easily cured by the uninitiated.
It's easy to isolate microphonics in mics: all you need to do is press the mic against one ear, knock hard against the housing tube with your knuckle, and observe any ringing of a component inside. if you can hear it, the resonance frequency may contaminate the signal being recorded.
To isolate the exact location of the component that is ringing, open the mic.
As you continue knocking on different sections of the mic's frame with your knuckle, press with a finger of your other hand that is holding the mic sightly on areas containing resistors, capacitors and other non-secured components, and you will find that the ringing will get less in one area. Then zero in on the offending culprit until the ringing stops.
Sometimes there may be more than one component that rings, but it will be clear from your knocking exploration and the difference in ringing frequency which is what.
I use a small dab of E6000 (
https://eclecticproducts.com/product/e6000-craft/) to secure the ringing component(s) - applied either to bridge the adhesive to an adjacent component or to the component's mounting surface.
* the closer the source of the ringing is to the capsule/front end of the mic's amplification chain, the stronger the audible resonance
