Edge states and frequency response in nonlinear forced-damped model of valve spring

  • Majdi Gzal
  • O. V. GendelmanEmail author
Original paper


This study explores the nonlinear dynamics of helical compression valve springs. To this end, the spring is mathematically modeled as a finite nonhomogenous one-dimensional mass–spring–damper discrete chain. Periodic displacement, which mimics the actual camshaft profile, is assumed at the upper end of the chain, while the other end is fixed. For the linear dynamics, the amplitudes of the periodic response are determined directly; they decrease toward the fixed end of the spring. Then, in order to meet more realistic conditions, the displacement of the upper mass is assumed to be nonnegative. This condition is realized by introducing an appropriate impact constraint. We assume that the impact is described by Newton impact law with restitution coefficient less than unity. For the case of one impact per period (1IPP) of excitation, exact periodic solutions are derived. The interplay between the nonhomogenous structure, multi-frequency excitation and nonlinearity leads to two qualitatively different states of the periodic responses; we refer to them as propagating states and edge states. The propagating states are characterized by weak localization, and the edge states—by strong localization at the forced edge. The stability of the system is analyzed using Floquet theory. Generic pitchfork and Neimark–Sacker bifurcations are observed. Analytical solutions conform to numerical simulations and experimental tests conducted on real valve springs.


Valve spring Frequency response Impacts Edge states Discrete breathers Stability 



We acknowledge Nordia Springs Ltd for the help with the experimental setting used in the paper.


The authors are grateful to Israel Science Foundation (Grant No. 1696/17) and to the Neubauer Family foundation for financial support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Mechanical EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael

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