[Paper Review] Nonlinear normal modes of a two degree of freedom oscillator with a bilateral elastic stop

El Hadi Moussi, Sergio Bellizzi+2 more|arXiv (Cornell University)|Feb 4, 2013

Bladed Disk Vibration Dynamics11 references3 citations

TL;DR

This paper investigates nonlinear normal modes (NNMs) in a two-degree-of-freedom mechanical oscillator with a bilateral elastic stop, using a regularization technique to handle non-smooth impact forces. Employing the harmonic balance method with high harmonic content and the asymptotic numerical method via MANlab software, it computes two distinct NNMs—one exhibiting complex multi-impact dynamics and the other showing localized motion on the first mass—validated through time-domain integration and energy-frequency analysis.

ABSTRACT

A study of the non linear modes of a two degree of freedom mechanical system with bilateral elastic stop is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. In order to obtain the Nonlinear Normal Mode (NNM), the harmonic balance method with a large number of harmonics, combined with the asymptotic numerical method, is used to solve the regularized problem. These methods are present in the software "package" MANLAB. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non regular problem. The two NNMs starting respectively from the two linear normal modes of the associated underlying linear system are discussed. The energy-frequency plot is used to present a global vision of the behavior of the modes. The dynamics of the modes are also analyzed comparing each periodic orbits and modal lines. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period. On the other hand, the second NNM presents a more simple dynamics with a localization of the displacement on the first mass.

Motivation & Objective

• To analyze nonlinear normal modes (NNMs) in a two-degree-of-freedom mechanical system with a bilateral elastic stop.
• To address the non-smoothness of impact forces through a regularization technique.
• To compute NNMs using high-harmonic harmonic balance and asymptotic numerical methods.
• To validate results via direct time-domain integration of the non-regularized system.
• To compare modal dynamics and energy-frequency behavior across the two identified NNMs.

Proposed method

• Regularization of the non-smooth impact force to enable smooth numerical treatment.
• Application of the harmonic balance method with a large number of harmonics to approximate periodic solutions.
• Integration of the asymptotic numerical method to solve the regularized nonlinear system efficiently.
• Use of the MANlab software package to implement and solve the regularized problem.
• Validation of computed NNMs by comparing with periodic orbits obtained from direct time-domain integration of the original non-regularized system.
• Construction of energy-frequency plots to provide a global overview of NNM behavior.

Experimental results

Research questions

• RQ1 How do nonlinear normal modes emerge in a two-degree-of-freedom oscillator with bilateral elastic stops?
• RQ2 What is the influence of impact non-smoothness on the structure and dynamics of NNMs?
• RQ3 How do the two primary NNMs—derived from the two linear normal modes—differ in their dynamic behavior and energy distribution?
• RQ4 To what extent do high-harmonic approximations accurately represent the true periodic solutions in the presence of impacts?
• RQ5 How does the energy-frequency relationship characterize the transition and complexity of the two NNMs?

Key findings

• The first NNM exhibits complex dynamics characterized by multiple impacts per oscillation period, indicating strong nonlinear coupling.
• The second NNM displays simpler dynamics with significant localization of displacement on the first mass, suggesting reduced coupling.
• The energy-frequency plot reveals distinct branches for the two NNMs, highlighting differences in stiffness and energy dependence.
• The harmonic balance method with high harmonic content successfully captures the periodic behavior of both NNMs, even under strong nonlinearity.
• Validation via direct time-domain integration confirms the accuracy and convergence of the regularized NNM solutions.
• The asymptotic numerical method enables efficient computation of NNMs across a wide range of energy levels, supporting robust modal analysis.