Original Paper

Nonlinear Dynamics

, Volume 70, Issue 2, pp 1147-1172

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

  • Ahmad MamandiAffiliated withDepartment of Mechanical Engineering, Parand Branch, Islamic Azad University Email author 
  • , Mohammad H. KargarnovinAffiliated withDepartment of Mechanical Engineering, Sharif University of Technology
  • , Salman FarsiAffiliated withDepartment of Mechanical Engineering, Tarbiat Modares University


In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of three-to-one (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the Euler–Bernoulli beam model, the governing nonlinear PDE of the beam’s transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximate-analytical solution techniques, namely the method of Multiple Time Scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis. The obtained results at this stage using four different methods which are all in time–space domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are obtained. Furthermore, the effect of axial compressive force in the dynamic analysis of such a beam is studied. Finally, under three-to-one-internal resonance condition the NNMs of the beam and the steady-state stability analysis are performed. Then the effect of changing the values of different parameters on the beam’s dynamic response is also considered. Moreover, 3-D plots of stability analysis in the steady-state condition and the beam’s amplitude frequency response curves are presented.


Beam’s nonlinear dynamics Nonlinear elastic foundation The 3:1 internal resonance Steady-state stability analysis Beam’s frequency response