Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: Analytical solutions

Dynamic model of large amplitude vibration of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

“…So, many new techniques have appeared in the open literature such as: variational iteration method (He, 2007a;Barari et al, 2011), homotopy perturbation method Saravi et al, 2013), homotopy analysis method (Liao, 2003;Khan et al, 2012)and some other methods Ghasempoor et al, 2102;Akbarzade and Khan, 2012;Alinia et al, 2011;Ganji, 2012;Torabi et al, 2012; …”

“…So, many new techniques have appeared in the open literature such as: variational iteration method (He, 2007a;Barari et al, 2011), homotopy perturbation method Saravi et al, 2013), homotopy analysis method (Liao, 2003;Khan et al, 2012)and some other methods Ghasempoor et al, 2102;Akbarzade and Khan, 2012;Alinia et al, 2011;Ganji, 2012;Torabi et al, 2012;Sheikholeslami et al, 2012a;2012b;Bayat et al, 2011;Rafieipour et al, 2012;Marinca and Herisanu, 2010;Salehi et al, 2012;Hamidi et al, 2012) The variational iteration method (VIM) and variational approach (VA) which were introduced by 2007a;2007b) are very powerful methods in solving non-linear differential equations and studying nonlinear vibration of beams. The Variational Approach which is also called He's variational approach, as well as the variational iteration method, was utilized here to obtain the analytical expression for the following model of nonlinear oscillations in the structural engineering problems:…”

Study of strongly nonlinear oscillators using the Aboodh transform and the homotopy perturbation method

“…Considering the arbitrary parameters as λ = α 2 , a 1 = ±k, a 2 = a 3 = a 4 = 0, b 1 = ∓k and b 2 = 0 one can arrive at eq. (35). The frequency of the nonlinear oscillator which is obtained by ATHPM from eq.…”

Extension of the Hamiltonian approach with general initial conditions

“…Xu and He (2010) used the Hamiltonian approach to determine the limit cycle motion for strongly nonlinear oscillators. The method was applied for solving some practical problems, such as nonlinear oscillations of a punctual charge in the electric field of a charged ring (Yildirim et al, 2011a), nonlinear vibrations of micro electro mechanical systems (Sadeghzadeh and Kabiri, 2016), nonlinear oscillations in an engineering structure (Akbarzade and Khan, 2012), among others. The Hamiltonian approach has also been applied to improve the accuracy of the solution for nonlinear oscillators, such as higher order approximations (Durmaz et al, 2010;Yildirim et al, 2011c), multiple coupled nonlinear oscillators (Durmaz et al, 2012) and multiple-parameter ones , among others.…”