Nonlinear superharmonic resonance analysis of a nonlocal beam on a fractional visco-Pasternak foundation

Abstract

<jats:p> This paper investigates the dynamic behavior of a geometrically nonlinear nanobeam resting on the fractional visco-Pasternak foundation and subjected to dynamic axial and transverse loads. The fractional-order governing equation of the system is derived and then discretized by using the single-mode Galerkin discretization. Corresponding forced Mathieu-Duffing equation is solved by using the perturbation multiple time scales method for the weak nonlinearity and by the semi-numerical incremental harmonic balance method for the strongly nonlinear case. A comparison of the results from two methods is performed in the validation study for the weakly nonlinear case and a fine agreement is achieved. A parametric study is performed and the advantages and deficiencies of each method are discussed for order two and three superharmonic resonance conditions. The results demonstrate a significant influence of the fractional-order damping of the visco-Pasternak foundation as well as the nonlocal parameter and external excitation load on the frequency response of the system. The proposed methodology can be used in pre-design procedures of novel energy harvesting and sensor devices at small scales exhibiting nonlinear dynamic behavior. </jats:p>

Nonlinear vibration of a nonlocal functionally graded beam on fractional visco-Pasternak foundation

“…( 58) has to be solved to obtain the data for the next stage. By applying the procedure established at [66,46] elements of the matrix M α 2 , and vectors R α 2 and V α 2 can be expressed as…”

“…( 42)) is introduced in nondimenzionalization process with the purpose to reduce nonlinear parameter θ in Eq. (46). Extreme high values of θ comparing to linear stiffness parameter ω 2 0 would later induce problems with solving fractional-order differential equation of motion (Eq.…”

“…Analytical perturbation methods such as the multiple scales method are usually used to solve the nonlinear fractional differential equations in the case of weak nonlinearity [55]. For strong nonlinearities, it is more common to use numerical methods such as the differential quadrature method (DQM) [40] or incremental harmonic balance (IHB) method [46]. A brief review of available numerical methods for solving the aforementioned nonlinear fractional differential equations is given by Zhou et al [74].…”

“Nonlinear Dynamics of Mechanical Systems” dedicated to 75th birthday and 52 years scientific contribution of Prof. Katica R. (Stevanovic) Hedrih

“…Mohajed et al 1 with nonlinear dynamic analysis showed the existence of a particular type of modal interactions with a self-adeptness of an attachment to narrow or broadband targeted energy transfers in low or high frequencies. Super-harmonic resonance of a nanobeam with geometric nonlinearities rested on a fractional visco-Paternak foundation in 7 with multiple scale analysis is shown.…”