A review of methods for developing similarity criteria in mechanics

Similarity theory is the theoretical basis for modeling and drafting experiments. In addition, it can be used to conduct a comparative analysis of changes in the desired parameters of the problem without solving equations and without conducting experiments. All arguments in similarity theory are based on dimensionless power complexes, which are called similarity criteria (numbers), or invariants. In the literature of different years of release, various methods of obtaining similarity criteria are described, but the author was not able to find a unified classification of these methods and their comparison.
The article provides a review of various methods for obtaining similarity criteria, their classification, which includes five methods from differential equations and seven methods from dimension analysis. All methods are compared on a single problem of mechanics about forced vibrations of the load, which leads to four similarity numbers. This approach helps you compare the labor required to output similarity numbers in different ways. For each method, a list of references is given where it is mentioned, and a brief description of the tasks that are solved there. At the end is a summary table showing which methods are considered in the mentioned works. The table shows the relative popularity of methods.