Spline Standards And Spline Calculation.pdf
Nurit Dardon
shaft splines and serrations are ridges, or teeth-type keys, that are an integral part of the shaft that mesh with grooves in a mating hub to transfer torque and rotational motion. For example, a bevel gear mounted on a shaft might use a male shaft spline that matches the female spline on the gear, as shown below.
Although a splined shaft looks like having a series of shaft keyways with keys pushed in, splines are considerably stronger than the keyed joint as the keyways weaken the shaft and reduce its torque-carrying capacity.
Depending on their relative axial movement, splines and serrations can also be grouped as fixed splines or flexible splines. As the name suggests, a fixed spline is a joint that does not move axially, such as gears, pullers, turbine wheels, etc.
These have straight and parallel tooth flanks, as shown in the figure below, and as per various standards, number of teeth can vary from 4 to 12. They can transmit higher torque than involute splines and serrations because of their large tooth thickness from the minor to the major diameter of the profile. But might fail due to fatigue due to stress concentration in the root of the flanks.
Naturally, it lacks centring ability because of the straight flanks, forcing it to rely on the major and minor diameter fits to manage the centring. Because of the straight-sided face, there will be a line of contact and surface contact will only exist after some wear.
Involute splines are very common and similar to internal and external involute gear teeth. They are comparably stronger than the parallel spline because of the lower stress concentration factor and have better surface quality. Involute splines can be produced by gear manufacturing techniques and have the ability to self-centre under load.
Involute splines are made with pressure angles 30o, 37.5 and 45o and can include between 60 and 100 splines per the American National Standard. Involute splines can be either Side fit or Diameter fit.
Serrations also have straight flanks but are angled, as shown in the figure below. The biggest advantage of the serrations is that the angles flanks centre the shafts and the hub resulting in self-centring splines. Flank angles are generally between 50o and 90o.
The main disadvantages of serrations are due to comparably small teeth, it can only be used for low torque applications. These are only used for non-axial moving applications. Like straight-sided splines, there will be line contact and wear.
The load is equally distributed if the transferring load is purely radial torsion and the torsional radial load is in the middle of the spline length. But if, for example, a bevel gear is used, this will put some unwanted axial loads into the spline.
If there are any axial or radial shock loading on the element connected, then care should be taken to support the external axial and radial shock loads to increase the joint life. This should also be considered during the calculations using the spline application factor.
Life factors for splines under wear conditions are based on the number of revolutions of the spline joint, not reversible cycles. Wear life factor only applies to flexible or sliding spline compressive stress calculations, as each time the spline slide back and forth, it wears the teeth.
The information theoretic concept of mutual information provides a general framework to evaluate dependencies between variables. In the context of the clustering of genes with similar patterns of expression it has been suggested as a general quantity of similarity to extend commonly used linear measures. Since mutual information is defined in terms of discrete variables, its application to continuous data requires the use of binning procedures, which can lead to significant numerical errors for datasets of small or moderate size.
In this work, we propose a method for the numerical estimation of mutual information from continuous data. We investigate the characteristic properties arising from the application of our algorithm and show that our approach outperforms commonly used algorithms: The significance, as a measure of the power of distinction from random correlation, is significantly increased. This concept is subsequently illustrated on two large-scale gene expression datasets and the results are compared to those obtained using other similarity measures.
The utilisation of mutual information as similarity measure enables the detection of non-linear correlations in gene expression datasets. Frequently applied linear correlation measures, which are often used on an ad-hoc basis without further justification, are thereby extended.
The evaluation of the complex regulatory networks underlying molecular processes poses a major challenge to current research. With modern experimental methods in the field of gene expression, it is possible to monitor mRNA abundance for whole genomes [1, 2]. To elucidate the functional relationships inherent in this data, a commonly used approach is the clustering of co-expressed genes [3]. In this context, the choice of the similarity measure used for clustering, as well as the clustering method itself, is crucial for the results obtained. Often, linear similarity measures such as the Euclidean distance or Pearson correlation are used in an ad-hoc manner. By doing so, it is possible that subsets of non-linear correlations contained in a given dataset are missed.
In this work, we discuss mutual information as a measure of similarity between variables. In the first section, we give a short introduction into the basic concepts including a brief description of the commonly used approaches for numerical estimation from continuous data. In the following section, we then present an algorithm for estimating mutual information from finite data.
The properties arising from this approach are compared to previously existing algorithms. In subsequent sections, we then apply our concept to large-scale cDNA abundance datasets and determine if these datasets can be sufficiently described using linear measurements or if a significant amount of non-linear correlations are missed.
If mutual information is indeed to be used for the analysis of gene-expression data, the continuous experimental data need to be partitioned into discrete intervals, or bins. In the following section, we briefly review the established procedures; a description of how we have extended the basic approach will be provided in the subsequent section.
In the next sections, we discuss some of the properties arising from the utilisation of B-spline functions for the estimation of mutual information and compare our approach to other commonly used estimators. We support this discussion using examples for which the underlying distributions and thereby the true mutual information is known.
As can be seen for an example of artificially generated equidistributed random numbers (Figure 3, left), mutual information calculated from the simple binning scales linearly with 1/N, with the slope depending on the number of bins M in accordance with Eq. (12). Figure 3 shows that this scaling is preserved for the extension to B-spline functions, while the slope is significantly decreased for k = 3, compared to the estimation with the simple binning (k = 1). Mutual information calculated from KDE does not show a linear behaviour but rather an asymptotic one with a linear tail for large datasets. The values are slightly increased compared to the ones from the B-spline approach. The entropy estimator gives values comparable to the ones obtained from the B-spline approach.
More importantly, a similar result also holds for the standard deviation of mutual information. As shown in Figure 3 (right), the standard deviation of the mutual information estimated with the simple binning (k = 1) scales with 1/N for statistically independent events [26, 29]. For the B-spline approach (k = 3), this scaling still holds, but the average values are decreased significantly. For the KDE approach, an asymptotic run above the values from the B-spline approach is observed, again with linear tail for large datasets. shows a linear scaling slightly below the simple binning.
The interpretation of any results obtained from the application of mutual information to experimental data is based on testing to see if the calculated results are consistent with a previously chosen null hypothesis. By following the intuitive approach that the null hypothesis assumes the statistical independence of variables, mutual information is tested against a surrogate dataset, which is consistent with this null hypothesis. As discussed previously in more detail [20], one way of generating such a surrogate dataset is by random permutations of the original data. From the mutual information of the original dataset MI(X,Y)data, the average value obtained from surrogate data , and its standard deviation σsurr, the significance S can be formulated as
For each S the null hypothesis can be rejected to a certain level α depending on the underlying distribution. With increasing significance the probability of false positive associations drops.
In the following, we address the influence of the spline order and the number of bins on the estimation of mutual information. Based on 300 data points of an artificially-generated dataset drawn from the distribution shown in Figure 1, we calculate the mutual information for M = 6 bins and different spline orders k = 1... 5 (Figure 4, left).
Mutual information calculated for a dataset of 300 data points drawn from the distribution shown in Figure 1 (crosses). The number of bins was fixed to M = 6. The average mutual information for 300 shuffled realisations of the dataset is shown (circles) together with the standard deviation as error-bars. The largest value found within the ensemble of shuffled data is drawn as a dotted line (left). The significance was calculated from Eq. (13) (right).
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