APPLICATION OF SIMILARITY THEORY TO CONSTRUCT A DIAGNOSTIC MODEL OF THE OBJECT OF DIAGNOSTICATION

Abstract

The paper substantiates the application of the Theory of Similarity as an effective method for constructing a Diagnostic Model (DM) with a disturbance (or fault model), aimed at the operational and accurate determination of the technical condition of complex objects. The key idea lies in utilizing the dimensional information inherent in the system and process indicators as fundamental data for formalizing the model and deriving the target function.

Traditional diagnostic models often face challenges such as unwieldiness, excessive sensitivity to noise, or the impossibility of achieving precise analytical description, particularly for objects characterized by a large number of interrelated parameters. The application of the Theory of Similarity offers an elegant approach to resolving these issues through the reduction of initial data and the transition to dimensionless characteristics, which significantly simplifies the subsequent analysis. The process of constructing the diagnostic model is based on the sequential application of the First and Second Theorems of Similarity. The Second Theorem of Similarity, or the πTheorem, serves as the foundation for describing the process in its criterial form. It postulates that any complete equation describing a physical process, when written in a specific system of units, can be expressed as an equation relating only dimensionless quantities (similarity criteria), which are derived from the indicators participating in the physical process. This theorem provides a potent mathematical advantage: it allows for the reduction of the number of variables from $m$ dimensional quantities down to (m – k) dimensionless criteria, where k represents the number of fundamental units of measurement. This variable reduction significantly streamlines both experimental investigations and mathematical analysis, making complex systems more manageable for diagnostic purposes.

Once the criterial relationships are established, a necessary transition to a mathematical form is required to directly obtain the target function of the diagnostic model. For this purpose, the First Theorem of Similarity is employed. This theorem formulates the condition for similarity between processes: if two physical processes are similar, their corresponding similarity criteria are equal to each other. This principle allows for the establishment of the parametric form of the model, where the object’s fault or the change in its technical state is reflected through the variation in the values of the dimensionless criteria. Using the rule for transforming criteria allows for an efficient shift from the criterial form to the parametric form, which is the immediate objective of the diagnostic process. This shift is crucial for translating theoretical dimensionless relationships into a practical, measurable diagnostic tool.

Consequently, the Theory of Similarity provides a formalized, structured, and economical (with respect to the number of variables) approach to fault modeling. The developed model allows for the effective utilization of internal information embedded within the dimensions of physical indicators to construct a target function that accurately reflects the object’s technical condition. This methodology offers a robust alternative to purely analytical or purely data-driven models by leveraging fundamental physical relationships in a simplified, dimensionless framework, enhancing both the rigor and practicality of the diagnostic system. The integration of dimensional analysis ensures that the resulting diagnostic function is physically meaningful and independent of the chosen system of units, a major advantage in cross-system applications. The $\pi$-theorem, by consolidating many variables into a few dimensionless groups, inherently provides a necessary level of abstraction, focusing the diagnostic effort on the most influential parameters and their ratios, rather than absolute values. This is essential for building resilient and universally applicable diagnostic tools for industrial and engineering systems. The parametric form derived via the First Theorem of Similarity serves as the final, quantitative output for real-time monitoring and fault detection.