An improved least square similitude method based on Lagrange energy for estimating scaling laws and eliminating coupling effects

• Published in: Thin-walled structures 2022-05, Vol.174, p.109018, Article 109018
• Main Authors: Zhang, Wen Di, Luo, Zhong, Guo, Si Wei, Li, Hong Guang
• Format: Article
• Language: English
• Subjects: Coupling effects; Lagrange energy; Least squares; Scaling laws; Support stiffness; Thin rectangular plate
• ISSN: 0263-8231; 1879-3223
• DOI: 10.1016/j.tws.2022.109018

Scaling laws refer to the relationships of the input and output parameters between prototype and scaled models. They are usually used to predict the vibration characteristics of a large-scale structure with scaled-down models. In general, scaling laws are estimated from the dimensional analysis of concerned variables or the dimensionless governing equation of vibrating structures. However, there are still some limitations that existed in the partial similitude of complex systems. The present work is concerned with determining scaling laws to predict the vibration characteristics of a thin rectangular plate containing elastic boundary conditions and even complicated geometrical features. The Lagrange energy-least squares similitude method (LE-LSSM) is proposed in this paper, the Lagrange energy approach eliminates the coupling effect of support stiffness on powers, and the LSSM can then be used to deduce output scaling laws. In numerical verifications, a perfect rectangular thin plate is simulated by finite element analysis (FEA) to verify the scaling laws, and a perforated thin rectangular plate is further applied for standing out the highlight of the LE-LSSM. Moreover, the coupling effects on scaling laws are analyzed among geometric dimensions, support stiffness, and material parameters. The coupling effects exist if powers change over the geometric dimensions of scaled models, but the inverse is not. Powers are stable and do not change over the geometric dimensions of scaled models under the free boundary condition, illustrating no coupling effect of geometric dimensions on powers. However, the support stiffness exhibits a significant coupling effect on powers of geometric scaling factors, which leads to inaccurate scaling laws. It can be effectively eliminated no matter how large the support stiffness values when LSSM is combined with the Lagrange energy approach. It is found that the results under the uncoupled powers are more accurate than those under the coupled powers. Moreover, material parameters produce a coupling effect on powers in non-zero support stiffness. Unlike support stiffness, only the power values change, but their stability not. Finally, the simulated results and proposed method are experimentally validated using the Brüel & Kjær test system, which derives consistent results. [Display omitted] •LE-LSSM does not depend on closed-form solution or profound theoretical foundation.•LE-LSSM can be applied in plates with elastic boundaries