Using inductive proximity sensors to measure displacement.

The problem is that magnetic fields diminish as the square of distance.

You could try an analog Hall effect sensor - place a magnet so that as the steel moves it channels more magnetic flux though the sensor. If you sketch out the geometry of the shaft movement I can indicate where to place the sensor and magnet.

You can also use a capacitance sensing technique - set up two electrode pads (capacitor plates) near the steel shaft such that as the shaft moves it gets closer to one or both pads - this will effectively reduce the "distance" between the plates an the capacitance between them will increase. This method has the drawback that anything else nearby may affect the reading.

A first-order answer to your question is yes, but a magnetic lens doesn't perform as well as an optical lens.  This shape in ferrite will do it.

It is often called a 'flux concentrator.'

It might be helpful if you provided a little more detail about your application.

Actually I am making an apparatus to measure the deflection of shafts at their critical frequencies along with the vibrations. I want to measure how much the shaft is deflecting at different points to verify the modes being formed at that particular frequency. The sensor would measure deflection in the axis which is perpendicular to the rotational axis.

I don't know too much about this.

What is your shaft made of?  Is it smooth?  Is it shiny?  Is it magnetic?  How much and how so?  How long is it, how thick is it?  How do these modes happen in space?  Don't we need to sense in two directions, perpendicular both to each other and the shaft rotation?

Do the modes have individual acoustic signatures?

Can we use a hi-res camera and video analysis software to glean the information we need?

The shaft is smooth and shiny and is made of Stainless Steel. It is 0.8 m long and I'll be using three different shafts of different diameters, 5mm, 8mm, 10mm. The maximum deflection that each shaft will experience is around 8mm from the rotational axis. Yes we do need sense in two directions both perpendicular and to the shaft rotation as well. I highly doubt we can use a hi-res camera in this case since the deflection being measured is quite small.

These are the modal shapes that are formed at different frequencies. Although the image is 2D but the whole shaft flexes like that in 3D. I need to verify this deflection by it's measurement. My design includes a pair of sensors, one measuring deflection in the vertical axis and the other in the horizontal axis. These sensors will be attached to an arm which will have the capability to slide in a slot along the length of the shaft, enabling me to measure the deflections at different positions along the shaft. But since I was mistaken about the sensors, I want to find another alternative to it. The major problem is such a small displacement to be measured.

Engineering is a mater of taste, to a degree, I'm no authority, consider any suggestions I make, but no hurt feelings should you discard them.

SS.  This sets off alarm bells.  At the risk of being a drudge or pedant, lemme talk SS a little.  SS is Fe + Cr and sometimes Ni. The crystal structure in some SSs cause the magnetic domains to cancel.  This diminishes the permeability, potentially greatly, as regards, say, tool steels.  What alloy are you using?  They call the high-grade stuff 'non-magnetic stainless,' sometimes.

You can take, say, a small neodymium and hold it close to the rods, all the while avoiding creating a flesh sandwich.  Does the magnet wanna jump from your grasp?

Your problem is just that you have the wrong kind of sensor:

look at this link:

http://www.micro-epsilon.co.uk/displacement-position-sensors/eddy-current-sensor/index.html

and Google for "eddy current sensors"

MK

Those sensors are really cool, MK, thanks for hipping me to 'em!

A simple way to convert distance to voltage in this geometry is to use a capacitance-to-voltage circuit.

Form your variable capacitance by connecting one electrode to the metal bar and the other to a metal probe positioned 1 cm from the bar. Build a capacitance to voltage circuit that uses a fairly high frequency - well above the highest vibration frequency of the bar, and view the output voltage on an oscilloscope. As the bar vibrates the capacitance will oscillate and the scope will show both the frequency of vibration and the amplitude at the monitoring point. For best results ensure the probe wire is kept far from the bar except for the probe tip. You can look up capacitance to voltage circuits or capacitance meter circuits on the internet but be sure the one you choose has enough bandwidth to track the maximum vibration frequency.