Development of an algorithm for estimation of the parameters of the distribution function of the limit of endurance at fatigue tests

Mechanical Engineering


Аuthors

Agamirov L. V.1*, Agamirov V. L.2**, Vestyak V. A.3***

1. Russian State Technological University named after K.E. Tsiolkovsky, MATI, 3, Orshanskaya str., Moscow, 121552, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
3. ,

*e-mail: mmk@mati.ru
**e-mail: avl095@mail.ru
***e-mail: kaf311@mai.ru

Abstract

One of the ways of representing characteristics of fatigue resistance of materials and structural elements is the distribution function of the limit of endurance. It order to calculate structural elements, to conduct test planning and to perform statistical analysis of their results it is recommended to use the normal law of distribution , of the limit of endurance or the normal variable distribution (logarithmic normal where m — the number of levels of voltage amplitude distribution) [2, 3]. For steels Weibull-Gnedenko distribution of the limit of endurance is widely used. Statistical estimation of the parameters of the distribution function of the limit of endurance is performed by «up- and-down» and «probit» methods based on the maximum likelihood method (MLM), as well as by other methods.
According to the «up-and-down» method [19], the first sample of a series of n objects is tested at the expected level of endurance. If the first sample is not destroyed to the basic number of cycles , then the second sample is tested at a higher voltage, and if it is destroyed, then the second sample is tested at a lower voltage. Voltage level for testing the third sample is chosen depending on the test results of the second sample.
MLM-estimations of distribution parameters are defined by solving the system of (according to the number of parameters) equations:
where m — the number of levels of voltage amplitude cycles during the tests;
the number of destroyed objects at i-th level;
the number of undestroyed objects at i-th level;
the number of tests at i-th level;
the total number of tested objects;
  probability of destroying at i-th level of amplitude;
сontinuous differentiable function of the level of endurance distribution with parameters g, which are to be estimated according to the theoretical law of the level of endurance distribution.
When tested by the «up-and-down» method, are random variables.
Derivatives define the specific form of the system of equations (1).
According to the «probit» method [13], a series of n samples is divided into 4 or 5 groups. Samples fr om each group are tested to the basic value of the number of cycles at the appropriate voltage level. As a result of the tests there appear destroyed and undestroyed objects at each level. Estimations of parameters of the endurance level distribution are defined by solving the system of equations (2). Covariance matrix is defined by the equation (8). As opposed to the «up-and-down» method, the number of levels of voltage amplitudes m, as well as the number of tested samples at each level ni, are defined in advance based of the test plan.
The system of nonlinear equations of maximum likelihood (1) for estimation of parameters of normal, logarithmic normal and Weibull-Gnedenko distributions has, as a rule, several local minimums. That is why any standard method of numerical solution of the system of nonlinear equations will not be able to provide a single- valued solution, as the solution will depend significantly on the initial approximation. In these circumstances the primary task is, first, to establish exact approximations in order to justify initial approximations, and, second, to select the most acceptable numerical method of solving the system of nonlinear equations.
In this paper we recommend to use the method of deformable polyhedron (by Nalder-Mead) in order to solve this system, which, in practice, makes it possible to find the roots of the system of nonlinear equations rather fast and with the required accuracy. The authors have developed the program of implementation of this method (when writing the module, the authors used the data of the cross-platform numerical analysis library ALGLIB according to license terms GPL 2+).
Conclusions
1. It has been established that the system of maximum likelihood for estimation of parameters of normal, logarithmic normal and Weibull-Gnedenko distributions of the level of endurance at fatigue tests with the use of «up-and-down» and «probit» methods has, as a rule, several local minimums which makes the numerical solution difficult, as the solution will significantly depend on the initial approximation.
2. In these circumstances there has been developed an algorithm and computer programs for calculating initial approximation and subsequent numerical solution
3. The algorithm developed by the authors allows fast and accurate calculation of estimation values and lower confidence bounds of the limit of endurance at fatigue tests.
The approach, described in work is especially actual in problems of calculation of a limit of endurance of the various expensive designs meeting in aviation systems and systems of rocket and space equipment, wh ere a large number of mechanical tests is economically inexpedient.

Keywords:

limit of endurance, endurance tests, «upand-down method», «probit» method, statistical analysis

References

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