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Quantifying nonlinear dynamics of a spring pendulum with two springs in series: an analytical approach


[ 1 ] Instytut Mechaniki Stosowanej, Wydział Inżynierii Mechanicznej, Politechnika Poznańska | [ P ] employee

Scientific discipline (Law 2.0)

[2.8] Mechanical engineering

Year of publication


Published in

Nonlinear Dynamics

Journal year: 2022 | Journal volume: vol. 110 | Journal number: iss. 1

Article type

scientific article

Publication language


  • Asymptotic analysis
  • Multiple scales method
  • Differential–algebraic system
  • Springs in series

EN In this paper, planar forced oscillations of a particle connected to the support via two nonlinear springs linked in series and two viscous dampers are investigated. The constitutive relationships for elastic forces of both springs are postulated in the form of the third-order power law. The geometric nonlinearity caused by the transverse motion of the pendulum is approximated by three terms of the Taylor series, which limits the range of applicability of the obtained results to swings with maximum amplitudes of about 0.6 rad. The system has two degrees of freedom, but its motion is described by two differential equations and one algebraic equation which have been derived using the Lagrange equations of the second kind. The classical multiple scales method (MSM) in the time domain was employed. However, the MSM variant with three scales of the time variable has been modified by developing new and dedicated algorithms to adapt the technique to solving problems described by the differential and algebraic equations (DAEs). The paper investigates the cases of forced and damped oscillation in non-resonant conditions, three cases of external resonances, and the internal 1: 2 resonance in the system. Moreover, the analysis of the stationary periodic states with external resonances was carried out, and investigations into the system’s stability were concluded in each case. Two methods of assessing asymptotic solutions have been proposed. The first is based on the determination of the error satisfying the equations of the mathematical model. The second one is a relative measure in the sense of the L2-norm, which compares the asymptotic solution with the numerical one determined using the NDSolve procedure of Mathematica software. These measures show that the applied MSM solves the system to a high degree of accuracy and exposes the key dynamical features of the system. It was observed that the system exhibits jump phenomena at some points in the resonance cases, with stable and unstable periodic orbits. This feature predicts chaotic vibration in the system and defines the regions for its applications.

Date of online publication


Pages (from - to)

1 - 36




License type

CC BY (attribution alone)

Open Access Mode

czasopismo hybrydowe

Open Access Text Version

final published version

Date of Open Access to the publication

in press

Ministry points / journal


Impact Factor

5.741 [List 2021]

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