Strong geometric softening–hardening nonlinearities in an oscillator composed of linear stiffness and damping elements

Alireza Mojahed, Keegan Moore, Lawrence A. Bergman, Alexander F. Vakakis

Research output: Contribution to journalArticlepeer-review


A single-degree-of-freedom (SDOF) oscillator grounded through a linear spring in parallel with a linear viscous damper, and two inclined pairs of linear spring–damper elements forming an initial angle of inclination, ϕ0, with the horizontal at equilibrium, is considered. It is assumed that there is no pre-compression in any element. An impulsive excitation is applied to this system, and it is shown that, depending on the system parameters, the intensity of the applied impulse and the initial angle of inclination, there are strong stiffness and damping nonlinearities in the transient response induced solely due to geometric effects; these strong nonlinearities occur even though all elastic and dissipative elements of the system are governed by linear constitutive laws. Preliminary numerical simulations indicate that in different regimes of the dynamics the geometric nonlinearities are of hardening, hardening–softening or softening type. An analytical study is then performed to reveal two bifurcations in the dynamics with respect to the initial angle of inclination and detect the critical energy beyond which the nonlinearity changes from hardening to softening. Another effect of the initial angle of inclination is that it “slows” the decay rate of the transient response. To investigate this effect analytically, the complexification-averaging method is applied to an approximate (truncated) equation of motion, to show that, for non-zero initial angle of inclination, the time-scale of the slow dynamics of the system is directly related to the initial angle of inclination. An experimental study is then performed to verify the analytical and numerical predictions. The experimental system consists of a beam clamped at one of its ends and grounded by the inclined linear spring element at its other end. System identification is performed to identify the (linear) modal properties of the beam and detect the linear stiffness and viscous damping characteristics of the inclined spring. The experiments are performed for several different initial angles and initial conditions in order to obtain sufficient measured time series to be able to verify the theoretical predictions. The experimental results confirm the theoretical findings. This study highlights the strong hardening–softening stiffness and damping nonlinearities that may be induced by geometric (and/or kinematic) effects in oscillating systems composed of otherwise linear stiffness and damping elements.

Original languageEnglish (US)
Pages (from-to)94-111
Number of pages18
JournalInternational Journal of Non-Linear Mechanics
StatePublished - Dec 2018


  • Complexification-averaging method
  • Geometric nonlinearity
  • Hardening/softening nonlinearity
  • Nonlinear damping

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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