Purpose This project aimed to construct approximate polynomials with high approximation accuracy, reflecting the force–displacement characteristics of 1DOF and 2DOF Quasi–zero stiffness (QZS) systems. Method First, for the force–displacement characteristics of the 1DOF QZS systems, the regression method was used to construct approximate polynomials with higher approximation accuracy than approximate polynomials obtained by the traditional Taylor series expansion. Second, approximate polynomials representing the force–displacement characteristics of 2DOF QZS systems were newly constructed by using Taylor series expansion and regression method. Finally, an error analyses between approximate polynomials and exact curves representing the force–displacement characteristics of QZS systems are carried out. Result Using regression methods, the force–displacement characteristic approximation polynomials of a 1DOF QZS system with better approximation accuracy than those obtained by traditional Taylor series expansion were obtained. Also, for the 2DOF QZS system discussed in this paper, novel approximate polynomials of force–displacement characteristics were first constructed by using Taylor series expansion and regression methods. In case of 1DOF QZS system, the approximate polynomial obtained by the regression method for the dimensionless force–displacement characteristics reduces the RMSE value by 77.9% on average for any value of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> </InlineEquation> than the Taylor series expansion. In case of 2DOF QZS system, the average reduction is 88.9% on the entire area except minus values in the plane with the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha ,\beta\)</EquationSource> </InlineEquation> value. Conclusion The approximate polynomial construction method using regression method approximates the force–displacement characteristics of QZS systems with a higher accuracy than the Taylor series expansion. The high approximation accuracy of the polynomials constructed by using regression method is crucial to increase the dynamic solution' accuracy of the Duffing system.

Static Analysis and Numerical Approximation on Force–Displacement Characteristics of 1DOF and 2DOF Quasi–Zero Stiffness Systems

期刊论文
发表日期：

2026年2月24日

查看全文

Hyon–Sang Kang,
Il–Hwan Ri

摘要

Purpose

This project aimed to construct approximate polynomials with high approximation accuracy, reflecting the force–displacement characteristics of 1DOF and 2DOF Quasi–zero stiffness (QZS) systems.

Method

First, for the force–displacement characteristics of the 1DOF QZS systems, the regression method was used to construct approximate polynomials with higher approximation accuracy than approximate polynomials obtained by the traditional Taylor series expansion. Second, approximate polynomials representing the force–displacement characteristics of 2DOF QZS systems were newly constructed by using Taylor series expansion and regression method. Finally, an error analyses between approximate polynomials and exact curves representing the force–displacement characteristics of QZS systems are carried out.

Result

Using regression methods, the force–displacement characteristic approximation polynomials of a 1DOF QZS system with better approximation accuracy than those obtained by traditional Taylor series expansion were obtained. Also, for the 2DOF QZS system discussed in this paper, novel approximate polynomials of force–displacement characteristics were first constructed by using Taylor series expansion and regression methods. In case of 1DOF QZS system, the approximate polynomial obtained by the regression method for the dimensionless force–displacement characteristics reduces the RMSE value by 77.9% on average for any value of \(\alpha\) than the Taylor series expansion. In case of 2DOF QZS system, the average reduction is 88.9% on the entire area except minus values in the plane with the \(\alpha ,\beta\) value.

Conclusion

The approximate polynomial construction method using regression method approximates the force–displacement characteristics of QZS systems with a higher accuracy than the Taylor series expansion. The high approximation accuracy of the polynomials constructed by using regression method is crucial to increase the dynamic solution' accuracy of the Duffing system.