Section I: Measured variability data from manufactured parts
2. Uncertainty and variability in components and built-up structures
2.2 Variability in a single complex component: An alloy wheel rim
The statistical distribution of the response of a single complex manufactured component, an alloy wheel, is investigated. The original test work was conducted and reported by Brown and Gear [2.1]. The first four natural frequencies
(
f1,f2,f3,f4)
and the mass were recorded for a set of 79 nominally identical alloy wheel rims. Figure 2-1 shows the distribution of the results. For comparison equivalent sample numbers from various probability distributions with the same mean and standard deviation as the normalised data set are also shown.A chi-squared
( )χ
2 test [2.8] was conducted to test the goodness-of-fit of a selection of standard probability distributions to the data sets. Of particular interest is the goodness-of-fit toGaussian or close to Gaussian distributions. All of the non-Gaussian distributions selected to be trialled can under certain combinations of their parameters, become close to Gaussian in form. Further information on each of the distributions can be found in [2.9]. The maximum likelihood method was used to estimate the parameters for each of the distributions [2.10]. A summary of the subsequent χ2 results can be found in Table 2-1. The χ2 test is a null hypothesis test and a probability of below 95% represents a 95% confidence that the sample set cannot be rejected as having come from the distribution being tested against, see [2.10] for further information. The
2
χ test is conducted on classified (binned) data and outlying bins are summed to ensure at least five counts in each; this reduces the skewing effect of out-lying results.
860 865 870 875
0 5 10 15
Distribution of Second Mode, Hz
C o u n ts 11900 1195 1200 1205 1210 1215 1220 1225 1230 5 10 15
Distribution of Third Mode, Hz
C o u n ts 13450 1350 1355 1360 1365 5 10 15
Distribution of Fourth Mode, Hz
C o u n ts 8 8.1 8.2 8.3 8.4 8.5 8.6 0 5 10 15 Distribution of Mass, kg C o u n ts 414 416 418 420 422 424 426 428 0 5 10 15
Distribution of First Mode, Hz
C o u n ts Measured Data Gaussian Lognormal Gamma Weibull
Figure 2-1 Alloy wheel rims, frequency distribution of the first four natural frequencies and the total mass with equivalent sample numbers from various standard probability distributions.
Table 2-1 Summary of χ2 probability results for goodness-of-fit tests of alloy wheel data to various
distributions.
A
χ
2 probability of less than 95% represents a 95% confidence that the alloy wheel response cannot be rejected as having come from the distribution being tested against, in Table 2-1 thesef1 f2 f3 f4 Mass
Gaussian 0.751 0.985 0.650 0.937 0.999
Lognormal 0.740 0.984 0.645 0.938 0.999
Gamma 0.741 0.983 0.635 0.938 0.999
values are highlighted. From examination of Table 2-1, one cannot reject the hypothesis that the first, third and fourth modal frequencies fit to a Gaussian distribution. Conversely it can be seen from Figure 2-1 that the distribution for the second natural frequency and the mass have a high number of samples close to the mean with a low uneven spread, and a Gaussian distribution can be rejected as a likely fit.In general, a lognormal distribution closely approximates a Gaussian distribution for data sets where the σ µ<<1. This condition is approached for all of the alloy wheel data sets and the first, third and fourth modal frequencies fit either distribution. Both distributions rejected as a fit for the second modal frequency and the mass. A Gamma distribution also approaches a Gaussian distribution as the ratio σ µ becomes increasingly small. This condition is also satisfied for the alloy wheel results and the Gamma distribution can also be rejected for the second mode and the mass. As can be seen in Figure 2-1 there is no difference between a gamma, lognormal and Gaussian distribution. It can be seen from the results in Table 2-1 that a Weibull distribution can be rejected as a good fit to all of the data sets except the fourth mode. The data sets were also tested against a Rayleigh probability distribution; the results are not included in Table 2-1 as they were all rejected above the 95% probability confidence level as not having come from this distribution.
In conclusion, a Gaussian distribution was found to be a good fit to the distribution for three of the first four natural frequencies of an alloy wheel. Due to the relative sizes of the mean and variance in the distribution of the data, a Gamma distribution approaches a Gaussian distribution and hence also fits well. A lognormal distribution fits two of the natural frequencies.
The levels of normalised standard deviation, as discussed in section 2.1, ranged from 0.002 to 0.005 for the first four modes, and 0.007 for the mass.
Brown and Gear [2.1] investigated the cross correlation between the natural frequencies and the wheel rim mass, but no correlation was found. They did not examine the possible correlation between inverse square root of the mass and the natural frequencies, but investigation of this did not yield any correlation. They also examined the cross correlations between the natural frequencies and found some correlation between the first and second mode. It was surmised that this could be due to the mode shapes for both modes being similar, but this did not appear to be valid for the third and fourth mode which, although similar in shape, did not display any significant cross correlation.