Research on Diameter Tolerance of Transmission Shaft Based on Interval Analysis

Abstract

This paper describes how to discuss the uncertainty of diameter tolerance by using interval analysis. Firstly, the interval relationship among reliability, diameter tolerance and process capability index (PCI) is obtained. Considering the reasonable PCI range, then reliability range is calculated and compared by using universal gray method and combinational method, respectively. Finally, the ranges of improved tolerance and other uncertain variables are obtained. The results show that the reliability and tolerance ranges obtained by universal gray method are more reasonable. This paper provides a research thought for the uncertainty of diameter tolerance.

Appendices

Appendix A: Notation

$C_{{\rm p}}$ :

Process capability index

d :

Diameter of transmission shaft

$d_{0}$ :

Target diameter

$d_{i}$ :

Diameter of the ith sample shaft

M _T :

Torque

$\mu_{{M_{{\rm T}} }}$ :

Mean value of the torque

$n\hat{p}_{i}$ :

Excepted frequency

$\hat{p}_{i}$ :

Probability estimation

R :

Reliability

$S_{{\rm d}}$ :

Sample standard deviation

$S_{{\rm T}}$ :

Shear stress

$\mu_{{S_{{\rm T}} }}$ :

Mean value of the shear stress

T :

Dimensional tolerance

$T_{{\rm l}}$ :

Lower bound of the dimensional tolerance

$T_{{\rm u}}$ :

Upper bound of the dimensional tolerance

$W_{{\rm T}}$ :

Torsional section modulus

$Z_{{\rm R}}$ :

Coupling coefficient

$\alpha$ :

Positive constant in the range [0, 1]

$\beta$ :

Significance level

$\delta_{{\rm S}}$ :

Yield strength

$\mu_{{\delta_{{\rm S}} }}$ :

Mean value of the yield strength

$\delta_{{\rm T}}$ :

Shear static strength

$\mu_{{\delta_{{\rm T}} }}$ :

Mean value of the shear static strength

$\Delta d$ :

Diameter tolerance

$\mu_{{\rm d}}$ :

Mean value of the sample shafts diameters

$\nu_{i}$ :

Actual frequency

$\sigma$ :

Overall standard deviation

$\sigma_{{\rm d}}$ :

Standard deviation of the diameter

$\sigma_{{M_{{\rm T}} }}$ :

Standard deviation of the torque

$\sigma_{{S_{{\rm T}} }}$ :

Standard deviation of the shear stress

$\sigma_{{\delta_{{\rm S}} }}$ :

Standard deviation of the yield strength

$\sigma_{{\delta_{{\rm T}} }}$ :

Standard deviation of the shear static strength

$\phi ( \cdot )$ :

Density function of standard normal distribution

$\chi^{2}$ :

Test statistic

Appendix B: The Definition of Interval

Many engineering design problems involve imprecision or approximation or uncertainty to varying degrees. Depending on the nature of imprecision, the analysis and design of the system can be conducted using interval analysis [26], defined as follows:

$$ [x] = [\underline{x} ,\;\overline{x} ] = \{ x \in R:\;\underline{x} \le x \le \;\overline{x} \} $$

(B.1)

where $[x]$ is an non-empty bounded set of real numbers, $\bar{x}$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}$ are the upper and lower bounds of the interval, respectively.

Appendix C: Interval Arithmetic

For any intervals $[a] = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}]$ and $[b] = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{b}]$, the interval arithmetic operations are defined as follows [26]:

$$ [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}] + [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{b}] = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\bar{a} + \bar{b}] $$

(C.1)

$$ [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}] - [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{b}] = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} - \bar{b},\bar{a} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ] $$

(C.2)

$$ [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}] \cdot [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{b}] = [\hbox{min} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \bar{b},\;\bar{a}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{a}\bar{b}),\;\hbox{max} (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} \bar{b},\;\bar{a}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{a}\bar{b})] $$

(C.3)

$$ [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}]/[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{b}] = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}] \cdot [1/\bar{b},\;1/\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ]\; {\text{if}}\;0 \notin [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b} ,\;\bar{b}] $$

(C.4)

$$ k[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;\bar{a}] = \left\{ {\begin{array}{*{20}l} {[k\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ,\;k\bar{a}]} \hfill & {k > 0} \hfill \\ {[k\bar{a},\;k\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{a} ]} \hfill & {k < 0} \hfill \\ \end{array} } \right. \quad {\text{if}}\;k \in R $$

(C.5)

Appendix D: Universal Gray Mathematics

Based on gray system theory, universal gray mathematics is introduced to solve engineering problems with uncertainty and defined as follows [13]:

$$ [g] = (x,\;[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } ,\bar{\mu }]) \cdot (x,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } ,\bar{\mu } \in R) $$

(D.1)

where g is a universal number, x is an observed value and $[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } ,\bar{\mu }]$ is the gray information section of x.

For any universal numbers $[g_{i} ] = (x_{i} ,\;[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{i} ,\bar{\mu }_{i} ])\;(i = 1,2)$, the universal number arithmetic operations are defined as follows [13]:

$$ [g_{1} ] + [g{}_{2}] = \left( {x_{1} + x_{2} ,\;\left[ {\frac{{x_{1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{1} + x_{2} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{2} }}{{x_{1} + x_{2} }},\frac{{x_{1} \bar{\mu }_{1} + x_{2} \bar{\mu }{}_{2}}}{{x_{1} + x_{2} }}} \right]} \right) $$

(D.2)

$$ [g_{1} ] - [g{}_{2}] = \left( {x_{1} - x_{2} ,\;\left[ {\frac{{x_{1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{1} - x_{2} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{2} }}{{x_{1} - x_{2} }},\frac{{x_{1} \bar{\mu }_{1} - x_{2} \bar{\mu }{}_{2}}}{{x_{1} - x_{2} }}} \right]} \right) $$

(D.3)

$$ [g_{1} ] \times [g_{2} ] = (x_{1} x_{2} ,\;[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{1} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{2} ,\bar{\mu }_{1} \bar{\mu }_{2} ]) $$

(D.4)

$$ [{{g_{1} ]} \mathord{\left/ {\vphantom {{g_{1} ]} {[g_{2} }}} \right. \kern-0pt} {[g_{2} }}] = (x_{1} /x_{2} ,\;[\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{1} /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu }_{2} /,\bar{\mu }_{1} \bar{\mu }_{2} ])\quad {\text{if}}\;g_{2} \ne g^{\prime}_{(0)} $$

(D.5)