how do I find the modes in my signal?
You can't find it from a single signal: it's a property of how the oscillation mechanically propagates through your material. The sensor only senses at a single point!
With two sensors at different positions, assuming their sensing is strictly coherent (i.e. the time difference between the sampling instants is constant, and you can know that offset), you can at least reduce the set of possible modes, if you know the geometry of the vibrating piece of material very exactly.
You probably don't have that information!
To make a comparison:
Imagine you stand in front of an empty shed, whose exact size and shape you don't know. You hear a 8 kHz tone when you put your ear to the shed's front wall. You hear an 8 kHz tone when you put your ear on the back wall.
Can you tell, based on that information, which modes (how many standing waves) are inside the shed?
You can't; the sensor you have can't look inside the shed.
What you'd need is many sensors, mounted on and inside the vibrating material. completely lacking any knowledge of geometry, you'd actually need at least two per acoustic wavelength within your machinery to be able to exactly describe the modes that exist. In steel (which I assume your machinery is made of), acoustic wavelengths are probably too short to make that feasible.
What you can do is move one sensor on the edges of your machinery, trying to get a feeling for where there are standing wave nulls and maxima (simply from the observation of the amplitudes). However, since you're trying to do that exactly at the boundaries, and doing it on a mix of many oscillations, this will not give you an inherently clear picture without extensive mechanical simulation.
I'm afraid what you want to do – looking inside a complex block of metal – is harder than you think.