A Statistical Approach for Study of Roundness in Commercially Produced Round Metal Bars

: Commercially produced ferrous and non-ferrous metal round bars are available at various sizes. These bars have tremendous engineering use in multiple areas like automobile, aerospace, power, manufacturing, paper, printing, packaging and construction industries. Due to its wide range of utility of round bars, it is naturally pertinent to understand and ascertain quality aspect of these bars. Quite often it has been observed that round bars as available are not completely geometrically round at various sections along the length, also straightness is also somewhat compromised. Therefore, it is necessary to check the uniformity of diameters at various sections. In this paper statistical process control has been used to ascertain whether such round bars as commercially available can be utilized for industrial application or not. Control limits in terms of diameter measurement along the length of bars help to evaluate the characteristics of data variation which is needed to take decision about confirmation of utility of commercially produced bars.


Introduction
The measurement of roundness is of critical importance for many applications.One of the most important fundamental forms for engineering components is the circular crosssection.Circular forms arise in many applications, particularly in bearing surface such as rotating shafts and ball bearings.A rotational bearing and shaft whose components are not accurately round will tend to be noisy and is likely to fail prematurely.Accurate roundness measurement is therefore vital to ensure proper functioning of such parts.The measurement of roundness is an extremely important assessment.
In the present investigation an attempt has been made by choosing various different types of materials to ascertain the actual availability of bar sizes at commercial market and also to understand the level of roundness of these bars along the axis of the bar.Statistical process control technique is also used here to find out whether such bars as commercially available can be utilized for industrial applications or not.The results obtained indicate that the commercial bars are actually with statistical process control limits (in terms of diameter) or not.295 may be considered almost fundamental form tolerance for cylindrical components and has direct influence on the product performance (Whitehouse,1997).The stability of a measurement system for precision working of ball screw had been investigated by using a Monte Carlo simulation to establish bias, linearity accuracy (Pai et. al., 2015).A study (Reddy & Rao, 2016) on optimization of surface roughness and circularity deviation during drilling on CNC machine of different Aluminium alloys was conducted.Waviness measurement of cylindrical surfaces had been conducted (Adamctak et. al. 2016) by the use of V-block method and the accuracy of the measurements had also been analyzed.Statistical tool ANOVA was used in the design of experiment and presented to predict optimum value of surface roughness and circularity deviation and experimental value.An interesting study by (Raghu et. al. 2020) on circularity or roundness had been investigated by V-Block method during turning of a bar where effect of roundness had been analysed by surface plot.(Cuesta et.al., 2019) investigated dimensional accuracy in precision additive manufacturing for direct metal printing machines.An important investigation had been carried out (Görög, 2022) by measuring the roundness of cylindrical and conical surface.It had been indicated that measurement accuracy is defined as the closeness of agreement between a measured quantity value and a real quantity value of a measure i.e., the quantity intended to be measured which is often limited by calibration errors.
In the present investigation it has been attempted to study a number of samples of commercially produced metal round bars for ascertaining of production processes.Moreover, the effects of segmental length and angular rotation of round bars on the measurement of roundness have also been analysed for the control of quality improvement to display the performance metrics of various objects.

Measurement of Roundness of Bars
To study and analyze roundness of bars, an experimental study has been conducted by taking sample of such bars.Sample bars are taken one by one and have been placed over V-Blocks.Each bar is then considered for measurement of deflections, if any through dial gauges over small length segments along the bar length.The experimental arrangement for measurement of straightness and roundness of bars is shown in Figure 1.
Small length segments have been chosen to understand the variation of deviations along the length of rod.The process is repeated throughout each bar at an interval of about 6mm to 10mm along the bar length.Readings through dial gauges are taken and data recorded.These data have been plotted along the bar axis for further analysis.Four types of round bars are chosen of different diameters.Bar diameters are measured by micrometre screw gauge at different positions of a section and mean diameters at each section are evaluated.Materials of bars chosen are Mild steel (MS), Stainless steel (SS), Aluminium (Al) and Copper (Cu).

Variations of Diameters of Round Bars
It is always intended for any producer that the process used for production should be capable of meeting customer requirement or exceeding the same thus satisfying the customer by improving the product or service quality.One standard aspect of quality is to ascertain the dimensional aspects.Present study is primarily focussed on the dimensional accuracy which is indeed a prime engineering requirement.Dimensions of any product form an important aspect in many design calculations.A quality cross check on dimensional aspect is actually always necessary to ensure that the variability of processes are within control limits.Although, it is assumed that round bars of a given size will have uniform diameter but in reality, diameter along the length varies to some extent.
Moreover, it is also assumed that naturally sagging over small length segments are insignificant and can be ignored for present study.

Roundness of Mild Steel Bar
Four round bars of mild steel are taken of diameters of 6mm, 8mm, 10mm and 12mm respectively.These bars are put on two V-Blocks supported in a fixture and deflections are measured by using Dial Gauge as shown in Figure 1.Ball of the dial gauge has been so placed that it touches the topmost point on the surface of the round bar.Data have been measured for angles from 0 o to 360 o at an interval angle of 15 o .Dial gauge readings taken at the periphery of the round bar have been noted.As the dataset is big, measurement of roundness of 12 mm diameter MS Rod only is tabulated in Table 1 (appendix 1).Round MS bars are produced in rolling mills.Elastic modulus of mild steel is about 210 GPa.It is seen that variations of deflections not only exist but also not uniform at various sections or length segments of MS round bar.This indicates that there is a variation in the circularity of the round bar even if the strength of bar is quite high.

Roundness of Stainless-Steel Bar
Similar approach has been followed in case of stainless steel round bars.Round bars of SS material were taken of diameters of 6mm, 8mm, 10mm and 12mm respectively as done for MS Bar.These bars are put on two V-Blocks supported in a fixture and deflections are measured by using Dial Gauge as shown in Figure 1.Data have been noted for angles from 0 o to 360 o at an interval angle of 15 o .Due to large dataset of all four bars, only measurement of roundness of 12 mm diameter SS Rod have been tabulated in Table 2 (appendix 1).Elastic modulus of stainless steel varies quite a lot, maximum value of which is considered as 310 GPa (MatWeb).Although stainless steel bears 297 higher value of elastic modulus, but it is seen that variations of deflections exist in SS rods also.This indicates that there is a variation in the roundness in case of strong bars too.

Roundness of Aluminium Bars
Round bars of Aluminium were taken of four diameters of 6mm, 8mm, 10mm and 12mm respectively as done previously for MS & SS Bars.
Similar experiments have been conducted as for MS & SS bars by putting the bars on V-Blocks as shown in Figure 1.Due to large dataset, measurement of 12 mm diameter Aluminium Rod has been considered at present and data tabulated in Table 3 (appendix 1).Elastic modulus of Aluminium is about 69 GPa.It is seen that variations of deflections exist in Aluminium rods also.This indicates that variation in circularity not only exist in bars of high strength materials but also in case of relatively less stronger materials or of lesser value of modulus of elasticity.

Roundness of Copper Bar
Based on same procedure a set of four round bars of 6mm, 8mm, 10mm and 12mm of copper were taken and Dial Gauge measurements were recorded.For convenience purpose, dataset of 12 mm diameter Copper rod has been taken into account and tabulated in Table 4 (appendix 1).The elastic modulus of copper is about 120 GPa which is much more that Aluminium but much lesser than steels.However, it is again seen that variations of deflections exist in copper rods also which indicates variation in the roundness.
If we consider standard deviation (σ) of the data set given in Table-1 to 4 for the dial gauge measurements of each length segment then we find the values of σ as shown in Table 5.

Statistical Process Control Analysis
Statistical process control primarily uses control charts.A control chart is a graphical tool for monitoring the activity of an ongoing process.
The values of the quality characteristics are plotted along the vertical axis, and the horizontal axis represents the samples, or subgroups, from which the quality characteristics are found (Mitra, 2016).Since variability is a part of any process, no matter how sophisticated, therefore management and employees understand that some factors must be there which contributes to the variability.The causes of variation are subdivided into common or chance causes and special or assignable causes.Once the special causes are eliminated through remedial actions, the process is again brought to a state of statistical control.Variability due to chance causes is inherent to a process which changes only when a process is changed.
A control chart is having a centerline and lower and upper control limits.The center line is usually found in accordance with data in the samples.The centerline is usually found in accordance with the data in the samples.It is an indication of the mean of a process and is usually found by taking the average of the values in the sample (Mitra, 2016).
The sampling method should maximize differences between samples and minimize differences within the samples.Control charts have been used to understand the variability of mean values of subgroups and also the variability of range of values.For this purpose, X-Bar chart and R chart is quite a reasonable useful tool to understand whether the process is actually 299 within statistical process control limits.Therefore, in the present study, X-Bar and R Chart have been used to understand the variability of diameter values of round rod.

Statistical Process Control of MS Bars
Mild steel round bars of various small diameters are considered for the study of actual mean diameters and range as per X-Bar ( � )and R Charts.Nearly 15 segments of about 10 mm lengths were measured through micrometer screw gauge and diameters are noted down at each section at three different angles.It has been found that diameters are actually not uniform at every angle due to compromise in roundness of bars.Average data is calculated for each section and mean values ( � )are plotted along the length segments of the round bars. � was calculated based on average values of mean values ( � ) and a centre line  � is plotted along the bar length.
Range of mean values, R at each section is noted and average of the same is considered as centre line,  � .The process has been repeated for round bars of all samples of diameter 12 mm.Upper and Lower Control Limits (UCL & LCL) have been evaluated along with mean values.The evaluations process of control limits is as below: (1) where σ is standard deviation of the set of data at diameter measured at each section.
Graphs in Figure 6

Statistical Process Control of SS Bar
Stainless steel round bars of 12mm diameters were considered for the study of actual mean diameters and range as per X-Bar ( � )and R Charts.A similar process was carried out with nearly fifteen segments of about ten mm lengths.Diameters were measured through micrometer at each section at three different angles.It has again been found that diameters are actually not uniform at every angular direction.From the graphical representations of X-Bar and R Charts shown in the Figure 10 & 11, it can be seen that although mean diameters are varying at various sections along the bar length, but all the values are within control limits.

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Mean values  � were evaluated for each section and the same were plotted with  � along the length segments of the round bar.Range R was also evaluated and plotted along with  � .The study on mean diameters based on X-Bar and R chart as shown in Figures.5,6,7&8reveals that in all cases diameters measured at various sections are varying to some extent but variations are within control limits.In all the X-Bar charts mean diameters are well within control limits.R charts also show that variations are within control limits.Although a small length of bar was deployed for experimental data, but qualitatively small length of round bars can be considered as good enough to assess the overall average diameter of bars.
It is observed that dial gauge readings are varying at specified angles along the bar length in all the tables.This is a clear indication that bar itself is not straight.This is probably due to residual curvature which got built in during rolling operation.From the database generated for four types of round bar of different sizes it can be postulated that residual curvatures exist even in small length segments of the bar.As such our aim was to understand the level of roundness and the variation in circularity, but as residual curvatures playing a role the obvious effect finally is on the dial gauge readings.Residual curvatures can be reduced through bar straightening process by cold working, more specifically by passing the bar in cross roll arrangement (Roy & Pal, 2022).The applied load may be given through reverse kinematic bending or through bending moment at intended sections.

Conclusion
Overall, this study proposes a compact and reasonably accurate method for a straightforward and reliable means of extracting the linear deflections.It has been validated by comparing the experimental results of linear displacements and surface roughness with those obtained using Radar charts of the commercially produced bars of different diameters at various angular positions.
It can be concluded easily that round bars as produced by industries and available in market Roundness of bars is a matter of concern which is evident from various research work.Cylindrical components are indispensable part in engineering products.Therefore, roundness Suggested Citation Roy, S. & Pal, A.K. (2023).A Statistical Approach for Study of Roundness in Commercially Produced Round Metal Bars.European Journal of Theoretical and Applied Sciences, 1(5), 294-306.DOI: 10.59324/ejtas.2023.1(5).20

Figure
Figure 2. Radar Chart of 12 mm MS round bar Figure 3. Radar Chart of 12 mm SS round bar & Figure 7 have been plotted for Mild Steel round bar of diameter 12mm based on observations and readings noted.Upper Control Limits (UCL) and Lower Control Limits (LCL) have been drawn on each graph along with Mean value data i.e.  � and  � .

Figure
Figure 6.X-Bar chart for 12mm diameter MS round bar

Figure
Figure 8. X-Bar chart for 12mm diameter SS round bar

Figure
Figure 10.X-Bar chart for 12mm diameter Aluminium round bar

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Figure 12.X-Bar chart for 12mm diameterCopper round bar