Dall [1946], in the first analytical approach to predicting roundness, used two basic parameters, the tangent angle в and the top workrest angle у shown in Figure 19.4. He calculated the magnitude of the error reproduced on the workpiece as a result of one already occurring at its point of contact either with the workrest or with the control wheel. The calculations assumed a perfectly rigid machine.
Yonetsu [1959] derived relationships between pregrinding and postgrinding amplitudes of harmonics of the workpiece profile. Theoretical relationships were obtained for the case of a sudden infeed and these were compared with experimental results for infeed made over three revolutions of the workpiece. The theory showed qualitative agreement with the results. Three conclusions reached by Yonetsu were
• Lobed shapes related to odd harmonics below the 11th are better removed with a large tangent angle в.
• Lobed shapes related to even harmonics below the 10th are better removed with a small angle в.
• Other errors are generated, which include even and odd order harmonics that vary with в and it was suggested that these are related to the infeed motion.
Unfortunately the technique is unsuited to a conventional method of stock removal.
Becker [1965] suggested an optimum geometrical configuration for rounding 2, 3, 5, and 7 lobe shapes. This suggestion was reached by investigating a parameter s/R. The term Ss is the difference in the apparent depths of cut when the particular shape is contained in its two extreme positions in the grinding configuration, and SR = A is the roundness error. It was proposed that the larger this parameter is, the greater will be the tendency to remove the shape. On the assumption that the optimum value of the tangent angle is b = 6°, it was suggested that the optimum workrest angle g = 23°. Decreasing the workrest angle to -10° improves the situation slightly for 3 and 5 lobes but worsens the situation for the ellipse and 7 lobes. Increasing the workrest angle is detrimental for 2, 3, and 5 lobes and if over 35° for 7 lobes too. This method enables definite results to be obtained and is simpler than the method of Yonetsu. However, it does not allow an assessment of the final shape to be made nor does it take the elasticity of the machine into account.
The stability of the process was investigated by Gurney [1964]. It was concluded that errors will be generated during the spark-out period and that the stiffer the machine the greater will be the errors. It was also suggested that the grinding wheel may not always remain in contact with the workpiece during the spark-out period.
It was realized in the early days of computers that it would be possible using a mainframe computer to carry out a realistic simulation of the plunge-grinding process taking a number of important factors into account that would otherwise be ignored such as spark-out, machine deflections, vibrations, loss of contact, grinding wheel curvature, and control wheel curvature [Rowe and Barash 1964]. Importantly, by means of simulation, it was possible to show the rate of buildup of regenerated roundness errors. In some cases, a geometric configuration might be theoretically unstable but relatively stable for practical purposes due to the high frequency of the instability and limitations of buildup due to the large arc of contact with the grinding wheel.
The technique revealed several new aspects of the process including some contradictory indications for the effect of machine stiffness. It was, for example, found that work-regenerative vibrations due to geometric instability grew more rapidly with a very stiff machine. However, the elimination of roundness errors in a stable rounding process was usually improved with a stiff machine. It was also possible to explore the introduction of roundness errors due to the plunge — feed process. All these results were verified by experiments and dynamic analysis. The simulation technique and some experimental results are reported here.
Geometric work-regenerative instability was shown to be a special case of more general work — regenerative vibration instability [Rowe 1964]. Other workers such as Richards and Rowe [1972] have since presented stability charts for use in selection of rounding geometry for use in grinding.
Miyashita [1965] analyzed the role of machine vibrations in the generation of workpiece errors. Reeka [1967], following on from Becker, demonstrated the importance of the number of workpiece revolutions in reducing roundness errors.
The rounding process has been analyzed by various workers since, as indicated by the extensive list of references given at the end of the chapter. Extensive results for centerless grinding, either below center or above center, were presented by Johnson [1989].