Mathematical modeling in vibration mechanics from linear to nonlinear analysis
ZHAO Wen
LI Yinshan
DENG Qibo
Abstract：Taking the mass-spring system as the research object,the mathematical modeling methods for linear and nonlinear vibration models were systematically explored.Emphasis was placed on analyzing the stability of equilibrium points,the dynamic behaviors of micro-vibrations and large-amplitude vibrations,and an unconventional phenomenon was revealed.A nonlinear vibration mathematical model of the mass-spring system was established using Lagrange's equations.The number and positions of the system's equilibrium points were solved,and micro-vibrations were treated in three categories.For the conventional case(with only one stable equilibrium point and a quadratic term in potential energy),a linear vibration equation was obtained via an approximate method.For the case with multiple stable equilibrium points,multiple linear vibration equations were derived using the approximate method near each equilibrium point.For the stable equilibrium position with essential nonlinearity(without a quadratic term in potential energy),only the quartic term of potential energy was retained,leading to the acquisition of an approximate micro-vibration equation with pure cubic displacement.The analytical solution of the approximate micro-vibration equation with pure cubic displacement was solved using the harmonic-energy balance method.For micro-vibrations,the analytical solution of the approximate vibration equation was compared with the numerical solution of the exact nonlinear vibration equation,and the solutions exhibited high similarity.For large-amplitude vibrations,numerical simulation was adopted;comparative analyses were conducted under different parameters and initial conditions by integrating time-history curves,phase portraits,and FFT analysis,which verified the rationality of the model and the effectiveness of the method.Meanwhile,an unconventional phenomenon was discovered:small-displacement initial conditions induced large-amplitude vibrations,whereas large-displacement initial conditions induced small-amplitude vibrations.A reasonable explanation for this unconventional phenomenon was also provided.
Keywords：Micro vibrationUnconventional phenomenonMultiple equilibrium pointIntrinsic nonlinearityHarmonic energy balance method
Publication Date：2025-11-15
Online Publishing Date：2026-09-12(First online date of this platform, not the publication date of the document)
Pages：9( 1-9 )