On similitude law of sub-systems

Based on Buckingham's π-Theorem, dimensional analysis has achieved considerable success over the past near-century. Model testing has long been a powerful tool in both scientific studies and engineering applications. However, the prototype objects are becoming more and more complicated nowadays, and many of the prototype systems can contain several sub-systems. The conventional theories on model-prototype similarity and dimensional analysis have only limited application since the π -Theorem itself does not distinguish between the original system and subsystems. This is particularly true in the field of structural dynamics, where the structure is often modeled as a multi-degree-of-freedom system. In this paper, we attempt to show that, if a system can be decoupled into several nontrivial subsystems, then, in each subsystem, the number of π -terms will be reduced and therefore simplify the model testing. On the other hand, if a system cannot be decoupled into subsystems, then using model testing with reduced π -term analysis, both experimentally and theoretically, may introduce severe errors.