Fisher-Tippet Law Truncated Form for Loading Modeling of Machinery Structures

Introduction. Statistical data are used as a basis for assessing the reliability of engineering structures. However, incomplete data or inaccurate modeling of random variables may lead to an overestimation of reliability indicators. In practice, laws with infinitely decreasing or increasing distribution functions of an exponential family are usually used to model random variables characterizing the bearing capacity, load, and resource of engineering structures. To improve the accuracy of modeling of random variables, truncated forms of distribution laws are often used. These forms allow us to consider the random variable within a specified interval, excluding impossible values. Several studies have suggested using the Fisher-Tippett law with three parameters for modeling random variables related to the loading of engineering structures. The advantage of this law is that it limits the range of the random variable on the right side, but the left side of the distribution function decreases indefinitely, which is not ideal for load characteristics. To improve the accuracy of predicting random variables that characterize the load, it would be helpful to have a left-sided restriction using the Fisher-Tippett law. Currently, there are no descriptions of truncated forms of the distribution law in scientific literature. This article will explore the justification and development of a three-parameter truncated form of the Fisher-Tippett law and its use in calculation methods. The goal is to create a left-sided truncated version of the Fisher-Tippett distribution with three parameters to model random variables within a specific range.
Materials and Methods. The article provides a detailed description of the history of the Fisher-Tippet law, including its three-parameter form, and justifies the need for obtaining its truncated form.
Results. As a result of the research, a truncated form of the Fisher-Tippet three-parameter law in differential and integral forms was obtained and substantiated. The findings included graphs and calculations that demonstrated the normalization of a random variable within a given range.
Discussion and Conclusion. The conclusion was drawn about the advantages and disadvantages of the truncated form of the Fisher-Tippet law. The possibility of its practical application in the schematization of random loading processes under operating conditions and testing of machine elements and structures to assess fatigue life and determine fatigue resistance characteristics was established. The direction of further research is related to the practical use of the truncated form, particularly with the need to develop a method for evaluating the parameters of the truncated distribution and verifying the consistency of the proposed model

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