Primary resonance of an Euler-Bernoulli nano-beam modelled with second strain gradient

Document Type: Full Length Article


1 Assistant Professor, School of Mechanical Engineering, Shiraz University, Shiraz, Islamic Republic of Iran

2 Phd. student, School of Mechanical Engineering, Tehran University, Tehran, Islamic Republic of Iran



In the present manuscript, the second strain gradient (SSG) is utilized to investigate the primary resonance of a nonlinear Euler-Bernoulli nanobeam is analyzed in this paper for the first time. To that end, the second strain gradient theory, a higher-order continuum theory capable of taking the size effects into account, is utilized and the governing equation of the motion for
an Euler-Bernoulli nanobeam is derived with sixteen higher-order material constants. Then by implementing the Galerkin’s method,the Duffing equation for the vibration of a hinged-hinged nanobeam is obtained and its primary resonance is studied utilizing the method of multiple scales. The size effects and impact of various system parameters on the amplitude of the response are then investigated for three different materials and the results are compared to that
of the first strain gradient and classical theories. The results of this manuscript clearly shows that the nonlinear vibration of a second strain gradient nanobeam is size-dependent and although the difference between the results obtained by the second strain gradient theory and the first strain gradient theory is negligible for thicker beams, as the thickness decreases, the difference becomes more prominent. Also, the effects of nonlinearity on the forced vibration nonlinear response of an SSG beam are investigated and some observations are reported.


  • The second strain gradient theory is used in order to capture size effects.
  • Primary resonance behavior of a nonlinear nanobeam made of three different materials is investigated.
  • The size effects and influence of the excitation amplitude on the response of the system is studied.
  • Having multiple answers (up to three) and the jump phenomenon are important characteristics of nonlinear vibrating systems.


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