Extension of dual equivalent linearization to analysis of deterministic dynamic systems: Part 2—multi-parameter equivalent linearization

Authors

• Nguyen Ngoc Linh, Thuyloi University
• Nguyen Tay Anh, Robert R. McCormick School of Engineering and Applied Science
• Nguyen Cao Thang, Vietnamese Academy of Science and Technology Institute of Mechanics
• N. D. Anh, Vietnamese Academy of Science and Technology Institute of Mechanics
• Isaac Elishakoff, Florida Atlantic University

Author Type

Outside Researcher

Co-Author Type 1

Outside Researcher

Co-Author Type 2

Outside Researcher

Co-Author Type 3

Outside Researcher

College

Engineering and Computer Science

Department

Ocean and Mechanical Engineering

Document Type

Article

Publication/Event/Conference Title

Nonlinear Dynamics

Publication Status

Version of Record

Abstract

This paper developes the multi-parameter equivalent linearization (mpEL) investigating the replacement of the original nonlinear part A(x) with the linear function kdB(x) where A(x) can be expressed as an algebraic sum of nonlinear functions Ai(x),i=1,2,..,N. When many pairs (Ai,B),i=1,2,..,N are considered together there can be a case where there exist two functions Ai and Aj,i≠j such that Ai,B are on the same side while Aj,B are on different sides. Therefore, DEL is first extended to the pair (A,B) whose correlation coefficient r is a real number and the proposed weighting coefficient p depends not only on the absolute value of r² but also on the sign of r. Further, considering A(x) as the sum of Ai(x), each term Ai(x) is replaced by linear function kisB(x) using spEL. An alternative to spEL, the proposed mpEL consider the sum of equivalent linearization coefficients kis as an equivalent linearization coefficient km when replacing A(x) with kmB(x). It turns out that for an algebraic sum of nonlinear functions, DEL has two approaches, spEL or mpEL. Using spEL, the sum of nonlinear functions is considered as the entire nonlinear function while mpEL requires applying spEL to each nonlinear term. These two approaches lead to two different equivalent linearization coefficients for the original nonlinear function. The accuracy of spEL and mpEL is tested for nonlinear free vibration frequency analysis. A generalized nonlinearity measure is built based on the difference in direction and magnitude of the nonlinear function compared to the linear function. For tested nonlinear systems, it is obtained that among the solutions obtained from several approximate methods, spEL provides the lowest maximal errors within weak and weak moderate nonlinearities, and mpEL mainly provides the lowest maximal errors within moderate strong and strong nonlinearities.

First Page

18001

Last Page

18030

DOI

10.1007/s11071-024-09912-1

Publication Date

10-1-2024

Recommended Citation

Linh, Nguyen Ngoc; Anh, Nguyen Tay; Thang, Nguyen Cao; Anh, N. D.; and Elishakoff, Isaac, "Extension of dual equivalent linearization to analysis of deterministic dynamic systems: Part 2—multi-parameter equivalent linearization" (2024). Faculty Scholarship. 158.
https://digitalcommons.fau.edu/faculty_papers/158