This thesis is titled "A Framework for Enhancing the Accuracy of Ultra Precision Machining." In this thesis unwanted relative tool/ workpiece vibration is identified as a major contributor to workpiece inaccuracy. The phenomenon is studied via in situ vibrational measurements during cutting and also by the analysis of the workpiece surface metrology of ultra precision diamond face turned aluminum 6061-T6.

The manifestation of vibrations in the feed and in-feed directions of the workpiece was studied over a broadband of disturbance frequencies. It is found that the waviness error measured on the cut workpiece surface was significantly larger than that caused by the feed marks during cutting. Thus it was established that unwanted relative tool / workpiece vibrations are the dominant source of surface finish error in ultra precision machining.

By deriving representative equations in the polar coordinate system, it was found that the vibrational pattern repeats itself, leading to what are referred to in this thesis as surface finish lobes. The surface finish lobes describe the waviness or form error associated with a particular frequency of unwanted relative tool / workpiece vibration, given a particular feed rate and spindle speed. With the surface finish lobes, the study of vibrations is both simplified and made more systematic. Knowing a priori the wavelength range caused by relative tool / workpiece vibration also allows one to extract considerable vibration content information from a small white light interferometry field of view. It was demonstrated analytically that the error caused by relative tool / workpiece vibration is always distinct from the surface roughness caused by the feed rate. It was also shown that the relative tool / workpiece vibration-induced wavelength in the feed direction has a limited and repeating range. Additionally, multiple disturbance frequencies can produce the same error wavelength on the workpiece surface. Since the meaningful error wavelength range is finite given the size of the part and repeating, study then focussed on this small and manageable range of wavelengths. This range of wavelengths in turn encompasses a broadband range of possible disturbance frequencies, due to the repetition described by the surface finish lobes.

Over this finite range of wavelengths, for different machining conditions, the magnitude of the waviness error resulting on the cut workpiece surface was compared with the actual relative tool / workpiece vibrational magnitude itself. It was found that several opportunities occur in ultra precision machining to mitigate the vibrational effect on the workpiece surface. The first opportunity depends only on the feed rate and spindle speed. Essentially, it is possible to force the wavelength resulting from an unwanted relative tool / workpiece vibration to a near infinite length, thus eliminating its effect in the workpiece feed direction. Further, for a given disturbance frequency, various speed and feed rate combinations are capable of producing this effect. However, this possibility exists only when a single, dominant and fixed disturbance frequency is present in the process.

By considering the tool nose geometry, depth of cut, and vibrational amplitude over the surface finish lobe finite range, it was found that the cutting parameters exhibit an attenuating or filtering effect on vibrations. Thus, cutting parameters serve to mitigate the vibrational effect on the finished workpiece over certain wavelengths. The filter curves associated with various feed rates were compared. These filter curves describe the magnitude of error on the ultra precision face turned workpiece surface compared with the original unwanted tool / workpiece vibrational magnitude. It was demonstrated with experimental data that these filter curves are physically evident on the ultra precision diamond face turned workpiece surface. It was further shown that the surface roughness on the workpiece surface caused by the feed rate was reduced with relative tool / workpiece vibrations, and in some cases the feed mark wavelength was changed altogether.

Mean arithmetic surface roughness curves were also constructed, and the filtering phenomenon was demonstrated over a broadband of disturbance frequencies. It is well established that a decrease in the feed rate reduces the surface roughness in machining. However, it was demonstrated that the improved surface finish observed with a slower feed rate in ultra precision diamond face turning was actually because it more effectively mitigated the vibrational effect on the workpiece surface over a broadband of disturbance frequencies. Experimental findings validated this observation. By only considering the effect of vibrations on the surface finish waviness error, it was shown that the workpiece diamond face turned with a feed rate of 2 μm / rev has a mean arithmetic surface roughness, R_a, that was 43 per cent smaller than when a feed rate of 10 µm / rev was used.